E X P L O R I N G D I G I TA L L O G I C
With Logisim-Evolution
george self
September 2019 – Edition 7.0
B R I E F C O N T E N T S
List of Figures xi
List of Tables xiv
i theory
1 introduction 3
2 foundations of binary arithmetic 11
3 binary arithmetic operations 35
4 boolean functions 65
5 boolean expressions 93
6 karnaugh maps 117
7 advanced simplifying methods 145
ii practice
8 arithmetic circuits 173
9 encoder circuits 189
10 register circuits 211
11 counters 223
12 finite state machines 235
13 central processing units 241
iii appendix
a boolean properties and functions 251
glossary 253
bibliography 255
iii
C O N T E N T S
List of Figures xi
List of Tables xiv
i theory
1 introduction 3
1.1 Preface 3
1.1.1
Introduction to the Study of Digital Logic
3
1.1.2 Introduction to the Author 3
1.1.3 Introduction to This Book 4
1.1.4 About the Creative Commons License 5
1.2 About Digital Logic 5
1.2.1 Introduction 5
1.2.2 A Brief Electronics Primer 6
1.3 Boolean Algebra 9
1.3.1 History 9
1.3.2 Boolean Equations 9
1.4 About This Book 10
2 foundations of binary arithmetic 11
2.1 Introduction to Number Systems 11
2.1.1 Background 11
2.1.2 Binary Mathematics 12
2.1.3 Systems Without Place Value 12
2.1.4 Systems With Place Value 13
2.1.5 Summary of Numeration Systems 16
2.1.6 Conventions 18
2.2 Converting Between Radices 19
2.2.1 Introduction 19
2.2.2 Expanded Positional Notation 19
2.2.3 Binary to Decimal 20
2.2.4 Binary to Octal 22
2.2.5 Binary to Hexadecimal 23
2.2.6 Octal to Decimal 24
2.2.7 Hexadecimal to Decimal 25
2.2.8 Decimal to Binary 26
2.2.9 Calculators 30
2.2.10 Practice Problems 30
2.3 Floating Point Numbers 31
3 binary arithmetic operations 35
3.1 Binary Addition 35
3.1.1 Overflow Error 37
3.1.2 Sample Binary Addition Problems 37
3.2 Binary Subtraction 38
v
vi contents
3.2.1 Simple Manual Subtraction 38
3.2.2
Representing Negative Binary Numbers Using
Sign-and-Magnitude 39
3.2.3
Representing Negative Binary Numbers Using
Signed Complements 39
3.2.4
Subtracting Using the Diminished Radix Com-
plement 43
3.2.5
Subtracting Using the Radix Complement
45
3.2.6 Overflow 46
3.3 Binary Multiplication 48
3.3.1 Multiplying Unsigned Numbers 48
3.3.2 Multiplying Signed Numbers 49
3.4 Binary Division 49
3.5 Bitwise Operations 50
3.6 Codes 50
3.6.1 Introduction 50
3.6.2 Computer Codes 51
4 boolean functions 65
4.1 Introduction to Boolean Functions 65
4.2 Primary Logic Operations 67
4.2.1 AND 67
4.2.2 OR 69
4.2.3 NOT 71
4.3 Secondary Logic Functions 72
4.3.1 NAND 72
4.3.2 NOR 73
4.3.3 XOR 73
4.3.4 XNOR 74
4.3.5 Buffer 75
4.4 Univariate Boolean Algebra Properties 76
4.4.1 Introduction 76
4.4.2 Identity 76
4.4.3 Idempotence 77
4.4.4 Annihilator 78
4.4.5 Complement 79
4.4.6 Involution 80
4.5 Multivariate Boolean Algebra Properties 80
4.5.1 Introduction 80
4.5.2 Commutative 81
4.5.3 Associative 81
4.5.4 Distributive 82
4.5.5 Absorption 83
4.5.6 Adjacency 84
4.6 DeMorgan’s Theorem 85
4.6.1 Introduction 85
4.6.2 Applying DeMorgan’s Theorem 86
contents vii
4.6.3 Simple Example 86
4.6.4
Incorrect Application of DeMorgan’s Theorem
87
4.6.5 About Grouping 87
4.6.6 Summary 88
4.6.7 Example Problems 89
4.7 Boolean Functions 89
4.8 Functional Completeness 91
5 boolean expressions 93
5.1 Introduction 93
5.2 Creating Boolean Expressions 94
5.2.1 Example 94
5.3 Minterms and Maxterms 96
5.3.1 Introduction 96
5.3.2 Sum Of Products (SOP) Defined 96
5.3.3 Product of Sums (POS) Defined 97
5.3.4 About Minterms 97
5.3.5 About Maxterms 100
5.3.6 Minterm and Maxterm Relationships 102
5.3.7 Sum of Products Example 104
5.3.8 Product of Sums Example 105
5.3.9 Summary 106
5.4 Canonical Form 107
5.4.1 Introduction 107
5.4.2 Converting Terms Missing One Variable 109
5.4.3
Converting Terms Missing Two Variables
109
5.4.4 Summary 111
5.4.5 Practice Problems 112
5.5 Simplification Using Algebraic Methods 112
5.5.1 Introduction 112
5.5.2 Starting From a Circuit 112
5.5.3 Starting From a Boolean Equation 113
5.5.4 Practice Problems 115
6 karnaugh maps 117
6.1 Introduction 117
6.2 Reading Karnaugh maps 118
6.3 Drawing Two-Variable Karnaugh maps 119
6.4 Drawing Three-Variable Karnaugh maps 120
6.4.1 The Gray Code 121
6.5 Drawing Four-Variable Karnaugh maps 121
6.6 Simplifying Groups of Two 124
6.7 Simplifying Larger Groups 126
6.7.1 Groups of 16 126
6.7.2 Groups of Eight 127
6.7.3 Groups of Four 128
6.7.4 Groups of Two 129
6.8 Overlapping Groups 130
viii contents
6.9 Wrapping Groups 131
6.10 Karnaugh maps for Five-Variable Inputs 132
6.11 “Don’t Care” Terms 134
6.12 Karnaugh map Simplification Summary 135
6.13 Practice Problems 136
6.14 Reed-M
¨
uller Logic 137
6.15 Introduction 137
6.16 Zero In First Cell 138
6.16.1 Two-Variable Circuit 138
6.16.2 Three-Variable Circuit 138
6.16.3 Four-Variable Circuit 139
6.17 One In First Cell 142
7 advanced simplifying methods 145
7.1 Quine-McCluskey Simplification Method 145
7.1.1 Introduction 145
7.1.2 Example One 146
7.1.3 Example Two 151
7.1.4 Summary 157
7.1.5 Practice Problems 157
7.2 Automated Tools 159
7.2.1 KARMA 159
7.2.2 32x8 169
ii practice
8 arithmetic circuits 173
8.1 Adders and Subtractors 173
8.1.1 Introduction 173
8.1.2 Half Adder 173
8.1.3 Full Adder 174
8.1.4 Cascading Adders 176
8.1.5 Half Subtractor 177
8.1.6 Full Subtractor 179
8.1.7 Cascading Subtractors 180
8.1.8 Adder-Subtractor Circuit 181
8.1.9 Integrated Circuits 183
8.2 Arithmetic Logic Units 183
8.3 Comparators 184
9 encoder circuits 189
9.1 Multiplexers/Demultiplexers 189
9.1.1 Multiplexer 189
9.1.2 Demultiplexer 191
9.1.3 Minterm Generators 192
9.2 Encoders/Decoders 193
9.2.1 Introduction 193
9.2.2 Ten-Line Priority 194
9.2.3 Seven-Segment Display 195
contents ix
9.2.4 Function Generators 198
9.3 Error Detection 198
9.3.1 Introduction 198
9.3.2 Iterative Parity Checking 200
9.3.3 Hamming Code 202
9.3.4 Hamming Code Notes 208
9.3.5 Sample Problems 209
10 register circuits 211
10.1 Introduction 211
10.2 Timing Diagrams 211
10.3 Flip-Flops 213
10.3.1 Introduction 213
10.3.2 SR Latch 214
10.3.3 Data (D) Flip-Flop 216
10.3.4 JK Flip-Flop 217
10.3.5 Toggle (T) Flip-Flop 218
10.3.6 Master-Slave Flip-Flops 219
10.4 Registers 219
10.4.1 Introduction 219
10.4.2 Registers As Memory 220
10.4.3 Shift Registers 220
11 counters 223
11.1 Introduction 223
11.2 Counters 223
11.2.1 Introduction 223
11.2.2 Asynchronous Counters 224
11.2.3 Synchronous Counters 226
11.2.4 Ring Counters 228
11.2.5 Modulus Counters 230
11.2.6 Up-Down Counters 231
11.2.7 Frequency Divider 232
11.2.8 Counter Integrated Circuits (IC) 233
11.3 Memory 233
11.3.1 Read-Only Memory 233
11.3.2 Random Access Memory 234
12 finite state machines 235
12.1 Introduction 235
12.2 Finite State Machines 235
12.2.1 Introduction 235
12.2.2 Moore Finite State Machine 236
12.2.3 Mealy Finite State Machine 236
12.2.4 Finite State Machine Tables 237
12.3 Simulation 239
12.4 Elevator 239
13 central processing units 241
13.1 Introduction 241
L I S T O F F I G U R E S
Figure 1.1 Simple Lamp Circuit 6
Figure 1.2 Transistor-Controlled Lamp Circuit 7
Figure 1.3 Simple OR Gate Using Transistors 7
Figure 1.4 OR Gate 8
Figure 3.1 Optical Disc 60
Figure 4.1 AND Gate 69
Figure 4.2 AND Gate Using IEEE Symbols 69
Figure 4.3 OR Gate 70
Figure 4.4 NOT Gate 72
Figure 4.5 NAND Gate 72
Figure 4.6 NOR Gate 73
Figure 4.7 XOR Gate 74
Figure 4.8 XNOR Gate 75
Figure 4.9 Buffer 76
Figure 4.10 OR Identity Element 77
Figure 4.11 AND Identity Element 77
Figure 4.12 Idempotence Property for OR Gate 78
Figure 4.13 Idempotence Property for AND Gate 78
Figure 4.14 Annihilator For OR Gate 78
Figure 4.15 Annihilator For AND Gate 79
Figure 4.16 OR Complement 79
Figure 4.17 AND Complement 80
Figure 4.18 Involution Property 80
Figure 4.19 Commutative Property for OR 81
Figure 4.20 Commutative Property for AND 81
Figure 4.21 Associative Property for OR 82
Figure 4.22 Associative Property for AND 82
Figure 4.23 Distributive Property for AND over OR 83
Figure 4.24 Distributive Property for OR over AND 83
Figure 4.25 Absorption Property (Version 1) 83
Figure 4.26 Absorption Property (Version 2) 84
Figure 4.27 Adjacency Property 85
Figure 4.28 DeMorgan’s Theorem Defined 85
Figure 4.29 DeMorgan’s Theorem Example 1 86
Figure 4.30 DeMorgan’s Theorem Example 2 87
Figure 4.31
DeMorgan’s Theorem Example 2 Simplified
88
Figure 4.32 Generic Function 89
Figure 4.33 XOR Derived From AND/OR/NOT 91
Figure 5.1
Logic Diagram From Switching Equation
96
Figure 5.2 Logic Diagram For SOP Example 96
Figure 5.3 Logic Diagram For POS Example 97
xi
xii list of figures
Figure 5.4 Example Circuit 113
Figure 5.5 Example Circuit With Gate Outputs 113
Figure 6.1 Simple Circuit For K-Map 118
Figure 6.2 Karnaugh map for Simple Circuit 119
Figure 6.3 Karnaugh map With Greek Letters 119
Figure 6.4 Karnaugh map For Two-Input Circuit 120
Figure 6.5 Karnaugh map for Three-Input Circuit 121
Figure 6.6 K-Map For Four Input Circuit 122
Figure 6.7 K-Map For Sigma Notation 123
Figure 6.8 K-Map For PI Notation 123
Figure 6.9 K-Map for Groups of Two: Ex 1 124
Figure 6.10 K-Map for Groups of Two: Ex 1, Solved 124
Figure 6.11
K-Map Solving Groups of Two: Example 2
126
Figure 6.12 K-Map Solving Groups of 8 127
Figure 6.13
K-Map Solving Groups of Four, Example 1
128
Figure 6.14
K-Map Solving Groups of Four, Example 2
128
Figure 6.15
K-Map Solving Groups of Four, Example 3
129
Figure 6.16
K-Map Solving Groups of Two, Example 1
129
Figure 6.17
K-Map Solving Groups of Two, Example 2
130
Figure 6.18
K-Map Overlapping Groups, Example 1
130
Figure 6.19
K-Map Overlapping Groups, Example 2
131
Figure 6.20 K-Map Wrapping Groups Example 1 132
Figure 6.21 K-Map Wrapping Groups Example 2 132
Figure 6.22 K-Map for Five Variables, Example 1 133
Figure 6.23
K-Map Solving for Five Variables, Example
2 134
Figure 6.24
K-Map With “Don’t Care” Terms, Example 1
135
Figure 6.25
K-Map With “Don’t Care” Terms, Example 2
135
Figure 6.26 Reed-M
¨
uller Two-Variable Example 137
Figure 6.27 Reed-M
¨
uller Three-Variable Example 1 138
Figure 6.28 Reed-M
¨
uller Three-Variable Example 2 139
Figure 6.29 Reed-M
¨
uller Four-Variable Example 1 139
Figure 6.30 Reed-M
¨
uller Four-Variable Example 2 140
Figure 6.31 Reed-M
¨
uller Four-Variable Example 3 140
Figure 6.32 Reed-M
¨
uller Four-Variable Example 4 141
Figure 6.33 Reed-M
¨
uller Four-Variable Example 5 141
Figure 6.34 Reed-M
¨
uller Four-Variable Example 6 142
Figure 6.35 Reed-M
¨
uller Four-Variable Example 7 142
Figure 6.36 Reed-M
¨
uller Four-Variable Example 8 143
Figure 7.1 KARMA Start Screen 159
Figure 7.2 Karnaugh Map Screen 160
Figure 7.3 The Expression 1 Template 161
Figure 7.4 A KARMA Solution 165
Figure 7.5 The Minimized Boolean Expression 166
Figure 7.6 The BDDeiro Solution 166
Figure 7.7 Quine-McCluskey Solution 167
list of figures xiii
Figure 7.8 Selecting Implicants 168
Figure 8.1 Half-Adder 174
Figure 8.2 K-Map For The SUM Output 175
Figure 8.3 K-Map For The COut Output 175
Figure 8.4 Full Adder 176
Figure 8.5 4-Bit Adder 177
Figure 8.6 Half-Subtractor 178
Figure 8.7 K-Map For The Difference Output 179
Figure 8.8 K-Map For The BOut Output 180
Figure 8.9 Subtractor 180
Figure 8.10 4-Bit Subtractor 181
Figure 8.11 4-Bit Adder-Subtractor 182
Figure 8.12 One-Bit Comparator 184
Figure 8.13 K-Map For A > B 185
Figure 8.14 K-Map For A = B 186
Figure 8.15 K-Map For A = B 186
Figure 8.16 Two-Bit Comparator 187
Figure 9.1 Multiplexer Using Rotary Switches 190
Figure 9.2 Simple Mux 190
Figure 9.3 Simple Dmux 191
Figure 9.4 1-to-4 Dmux 192
Figure 9.5 1-to-4 Dmux As Minterm Generator 193
Figure 9.6 Three-line to 2-Bit Encoder 193
Figure 9.7 Four-Bit to 4-Line Decoder 194
Figure 9.8 Seven-Segment Display 195
Figure 9.9 7-Segment Decoder 197
Figure 9.10 Hex Decoder 197
Figure 9.11 Minterm Generator 198
Figure 9.12 Parity Generator 200
Figure 9.13 Hamming Distance 202
Figure 9.14 Generating Hamming Parity 206
Figure 9.15 Checking Hamming Parity 208
Figure 10.1 Example Timing Diagram 211
Figure 10.2 Example Propagation Delay 213
Figure 10.3 Capacitor Charge and Discharge 213
Figure 10.4 SR Latch Using NAND Gates 214
Figure 10.5 SR Latch Timing Diagram 215
Figure 10.6 SR Latch 215
Figure 10.7 D Flip-Flop Using SR Flip-Flop 216
Figure 10.8 D Flip-Flop 216
Figure 10.9 D Latch Timing Diagram 216
Figure 10.10 JK Flip-Flop 217
Figure 10.11 JK Flip-Flop Timing Diagram 217
Figure 10.12 T Flip-Flop 218
Figure 10.13 Toggle Flip-Flop Timing Diagram 218
Figure 10.14 Master-Slave Flip-Flop 219
Figure 10.15 4-Bit Register 220
Figure 10.16 Shift Register 221
Figure 11.1 Asynchronous 2-Bit Counter 224
Figure 11.2 Asynchronous 3-Bit Counter 225
Figure 11.3 Asynchronous 4-Bit Counter 225
Figure 11.4
4-Bit Asynchronous Counter Timing Diagram
226
Figure 11.5 Synchronous 2-Bit Counter 226
Figure 11.6 Synchronous 3-Bit Counter 227
Figure 11.7 Synchronous 4-Bit Up Counter 227
Figure 11.8
4-Bit Synchronous Counter Timing Diagram
227
Figure 11.9 Synchronous 4-Bit Down Counter 228
Figure 11.10
4-Bit Synchronous Down Counter Timing Dia-
gram 228
Figure 11.11 4-Bit Ring Counter 229
Figure 11.12 4-Bit Ring Counter Timing Diagram 229
Figure 11.13 4-Bit Johnson Counter 230
Figure 11.14 4-Bit Johnson Counter Timing Diagram 230
Figure 11.15 Decade Counter 231
Figure 11.16 4-Bit Decade Counter Timing Diagram 231
Figure 11.17 Up-Down Counter 232
Figure 11.18 Synchronous 2-Bit Counter 232
Figure 11.19 Frequency Divider 233
Figure 12.1 Moore Vending Machine FSM 236
Figure 12.2 Mealy Vending Machine FSM 237
Figure 12.3 Elevator 239
Figure 13.1 Simple Data Flow Control Circuit 242
Figure 13.2 Simplified CPU Block Diagram 243
Figure 13.3 CPU State Diagram 248
L I S T O F TA B L E S
Table 2.1 Roman Numerals 13
Table 2.2 Binary-Decimal Conversion 15
Table 2.3 Hexadecimal Numbers 16
Table 2.4 Counting To Twenty 17
Table 2.5 Expanding a Decimal Number 19
Table 2.6 Binary-Octal Conversion Examples 22
Table 2.7
Binary-Hexadecimal Conversion Examples
24
Table 2.8 Decimal to Binary 26
Table 2.9 Decimal to Octal 27
Table 2.10 Decimal to Hexadecimal 27
Table 2.11 Decimal to Binary Fraction 28
Table 2.12 Decimal to Binary Fraction Example 28
xiv
list of tables xv
Table 2.13 Decimal to Long Binary Fraction 29
Table 2.14 Decimal to Binary Mixed Integer 29
Table 2.15 Decimal to Binary Mixed Fraction 30
Table 2.16 Practice Problems 31
Table 2.17 Floating Point Examples 33
Table 3.1 Addition Table 36
Table 3.2 Binary Addition Problems 37
Table 3.3 Binary Subtraction Problems 39
Table 3.4 Ones Complement 41
Table 3.5 Twos Complement 42
Table 3.6 Example Twos Complement 43
Table 3.7 Binary Multiplication Table 48
Table 3.8 ASCII Table 52
Table 3.9 ASCII Symbols 52
Table 3.10 ASCII Practice 53
Table 3.11 BCD Systems 54
Table 3.12 Nines Complement for 127 56
Table 3.13 BCD Practice 56
Table 3.14 Gray Codes 63
Table 4.1 Truth Table for AND 68
Table 4.2 Truth Table for OR 70
Table 4.3 Truth Table for NOT 71
Table 4.4 Truth Table for NAND Gate 72
Table 4.5 Truth Table for NOR 73
Table 4.6 Truth Table for XOR 74
Table 4.7 Truth Table for XNOR 75
Table 4.8 Truth Table for a Buffer 75
Table 4.9 Truth Table for Absorption Property 84
Table 4.10 Truth Table for Generic Circuit One 89
Table 4.11 Truth Table for Generic Circuit Two 90
Table 4.12 Boolean Function Six 90
Table 4.13 Boolean Functions 91
Table 5.1 Truth Table for Example 95
Table 5.2 Truth Table for First Minterm Example 98
Table 5.3
Truth Table for Second Minterm Example
99
Table 5.4 Truth Table for First Maxterm Example 100
Table 5.5
Truth Table for Second Maxterm Example
101
Table 5.6 Minterm and Maxterm Relationships 102
Table 5.7 Minterm-Maxterm Relationships 104
Table 5.8 Truth Table for SOP Example 105
Table 5.9 Truth Table for POS Example 106
Table 5.10 Canonical Example Truth Table 107
Table 5.11
Truth Table for Standard Form Equation
108
Table 5.12
Truth Table for Standard Form Equation
110
Table 5.13 Canonical Form Practice Problems 112
Table 5.14 Simplifying Boolean Expressions 115
xvi list of tables
Table 6.1 Circuit Simplification Methods 118
Table 6.2 Truth Table for Simple Circuit 118
Table 6.3 Truth Table with Greek Letters 119
Table 6.4 Truth Table for Two-Input Circuit 120
Table 6.5 Truth Table for Three-Input Circuit 120
Table 6.6 Truth Table for Four-Input Circuit 122
Table 6.7 Karnaugh maps Practice Problems 136
Table 6.8 Truth Table for Checkerboard Pattern 137
Table 7.1 Quine-McCluskey Ex 1: Minterm Table 146
Table 7.2
Quine-McCluskey Ex 1: Rearranged Table
147
Table 7.3
Quine-McCluskey Ex 1: Size 2 Implicants
148
Table 7.4
Quine-McCluskey Ex 1: Size 4 Implicants
149
Table 7.5
Quine-McCluskey Ex 1: Prime Implicants
149
Table 7.6 Quine-McCluskey Ex 1: 1st Iteration 150
Table 7.7 Quine-McCluskey Ex 1: 2nd Iteration 150
Table 7.8 Quine-McCluskey Ex 1: 3rd Iteration 151
Table 7.9 Quine-McCluskey Ex 2: Minterm Table 152
Table 7.10
Quine-McCluskey Ex 2: Rearranged Table
153
Table 7.11
Quine-McCluskey Ex 2: Size Two Implicants
154
Table 7.12
Quine-McCluskey Ex 2: Size 4 Implicants
155
Table 7.13
Quine-McCluskey Ex 2: Prime Implicants
156
Table 7.14 Quine-McCluskey Ex 2: 1st Iteration 156
Table 7.15 Quine-McCluskey Ex 2: 2nd Iteration 157
Table 7.16 Quine-McCluskey Practice Problems 158
Table 7.17 Truth Table for E81A Output 164
Table 7.18 KARMA Practice Problems 168
Table 8.1 Truth Table for Half-Adder 174
Table 8.2 Truth Table for Full Adder 175
Table 8.3 Truth Table for Half-Subtractor 178
Table 8.4 Truth Table for Subtractor 179
Table 8.5 One-Bit Comparator Functions 184
Table 8.6 Truth Table for Two-Bit Comparator 185
Table 9.1 Truth Table for a Multiplexer 191
Table 9.2 Truth Table for a Demultiplexer 192
Table 9.3 Truth Table for Priority Encoder 195
Table 9.4 Truth Table for Seven-Segment Display 196
Table 9.5 Even Parity Examples 199
Table 9.6 Iterative Parity 201
Table 9.7 Iterative Parity With Error 201
Table 9.8 Hamming Parity Bits 203
Table 9.9 Hamming Parity Cover Table 203
Table 9.10 Hamming Example - Iteration 1 204
Table 9.11 Hamming Example - Iteration 2 204
Table 9.12 Hamming Example - Iteration 3 204
Table 9.13 Hamming Example - Iteration 4 204
Table 9.14 Hamming Example - Iteration 5 205
list of tables xvii
Table 9.15 Hamming Example - Iteration 6 205
Table 9.16
Hamming Parity Cover Table Reproduced
207
Table 9.17 Hamming Parity Examples 209
Table 9.18 Hamming Parity Errors 209
Table 10.1 Truth Table for SR Latch 214
Table 10.2 JK Flip-Flop Timing Table 218
Table 11.1 Counter IC’s 233
Table 12.1 Crosswalk State Table 238
Table a.1 Univariate Properties 251
Table a.2 Multivariate Properties 252
Table a.3 Boolean Functions 252
Part I
T H E O R Y
Digital Logic, as most computer science studies, depends
on a foundation of theory. This part of the book concerns
the theory of digital logic and includes binary mathematics,
gate-level logic, Boolean algebra, and simplifying Bool-
ean expressions. An understanding of these foundational
conceptsis essential before attempting to design complex
combinational and sequential logic circuits.
1
I N T R O D U C T I O N
What to Expect
This chapter introduces the key concepts of digital logic and
lays the foundation for the rest of the book. This chapter covers
the following topics.
• Define “digital logic”
•
Describe the relationship between digital logic and physi-
cal electronic circuits
• Outline a brief history of digital logic
• Introduce the basic tenants of Boolean equations
1.1 preface
1.1.1 Introduction to the Study of Digital Logic
Digital logic is the study of how electronic devices make decisions.
It functions at the lowest level of computer operations: bits that can
either be “on” or “off” and groups of bits that form “bytes” and
“words” that control physical devices. The language of digital logic is
Boolean algebra, which is a mathematical model used to describe the
logical function of a circuit; and that model can then be used to design
the most efficient device possible. Finally, various simple devices, such
as adders and registers, can be combined into increasingly complex
circuits designed to accomplish advanced decision-making tasks.
1.1.2 Introduction to the Author
I have worked with computers and computer controlled systems for
more than 30 years. I took my first programming class in 1976; and,
several years later, was introduced to digital logic while taking classes
to learn how to repair computer systems. For many years, my profes-
sion was to work on computer systems, both as a repair technician
and a programmer, where I used the principles of digital logic daily. I
then began teaching digital logic classes at Cochise College and was
able to share my enthusiasm for the subject with Computer Informa-
tion Systems students. Over the years, I have continued my studies
3
4 introduction
of digital logic in order to improve my understanding of the topic; I
also enjoy building logic circuits on a simulator to solve interesting
challenges. It is my goal to make digital logic understandable and to
also ignite a lifelong passion for the subject in students.
1.1.3 Introduction to This Book
This book has two goals:
1. Audience. Many, perhaps most, digital logic books are designed
for third or fourth year electronics engineering or computer science
students and presume a background that includes advanced mathe-
matics and various engineering classes. For example, it is possible to
find a digital logic book that discusses topics like physically building
circuits from discrete components and then calculating the heat rise
of those circuits while operating at maximum capacity. This book,
though, was written for students in their second year of a Computer
Information Systems program and makes no assumptions about prior
mathematics and engineering classes.
2. Cost. Most digital logic books are priced at
$
150 (and up) but
this book is published under a Creative Commons license and, though
only a tiny drop in the proverbial textbook ocean, is hoped to keep
the cost of books for at least one class as low as possible.
Following are the features for the various editions of this book:
1.
2012. Prior to 2012, handouts were given to students as they
were needed during class; however, it was in this year that the
numerous disparate documents were assembled into a cohesive
book and printed by a professional printing company.
2.
2013. A number of complex circuits were added to the book,
including a Hamming Code generator/checker, which is used
for error detection, and a Central Processing Unit (CPU) using
discrete logic gates.
3.
2014. New material on Mealy and Moore State Machines was
included, but the major change was in the laboratory exercises
where five Verilog labs were added to the ten gate-level simula-
tion labs.
4.
2015. New information was added about adding/subtracting
Binary Coded Decimal (BCD) numbers and representing floating
point numbers in binary form; and all of the laboratory exercises
were re-written in Verilog. Also, the book was reorganized and
converted to L
A
T
E
Xfor printing.
5.
2018. The labs were re-written using Logisim-evolution because
students find that system easier to understand than iVerilog.
1.2 about digital logic 5
This book was written with L
A
T
E
X using TeXstudio. The source for
this book is available at GITHUB,
http://bit.ly/2w6qU2C
, and anyone
is welcomed to fork the book and develop their own version.
DISCLAIMER
I wrote, edited, illustrated, and published this book myself. While I
did the best that I could, there are, no doubt, errors. I apologize in
advance if anything presented here is factually erroneous; I’ll correct
those errors as soon as they are discovered. I’ll also correct whatever
typos I overlooked, despite TeXstudio’s red squiggly underlines trying
to tell me to check my work. –George Self
1.1.4 About the Creative Commons License
This book is being released under the Creative Commons 0 license,
which is the same as public domain. That permits people to share,
remix, or even rewrite the work so it can be used to help educate
students wherever they are studying digital logic.
1.2 about digital logic
1.2.1 Introduction
Digital logic is the study of how logic is used in digital devices to
complete tasks in fields as diverse as communication, business, space
exploration, and medicine (not to mention everyday life). This defini-
tion has two main components: logic and digital. Logic is the branch
of philosophy that concerns making reasonable judgment based on
sound principles of inference. It is a method of problem solving based
on a linear, step-by-step procedure. Digital is a system of mathematics
where only two possible values exist: True and False (usually repre-
sented by 1 and 0). While this approach may seem limited, it actually
works quite nicely in computer circuits where True and False can be
easily represented by the presence or absence of voltage.
Digital logic is not the same as programming logic, though there
is some relationship between them. A programmer would use the
constructs of logic within a high-level language, like Java or C++, to
get a computer to complete some task. On the other hand, an engineer
would use digital logic with hardware devices to build a machine,
like an alarm clock or calculator, which executes some task. In broad
strokes, then, programming logic concerns writing software while
digital logic concerns building hardware.
Digital logic may be divided into two broad classes:
6 introduction
1.
Combinational Logic, in which the outputs are determined
solely by the input states at one particular moment with no
memory of prior states. An example of a combinational circuit
is a simple adder where the output is determined only by the
values of two inputs.
2.
Sequential Logic, in which the outputs depend on both current
and prior inputs, so some sort of memory is necessary. An
example of a sequential circuit is a counter where a sensed new
event is added to some total contained in memory.
Both combinational and sequential logic are developed in this book,
along with complex circuits that require both combinational and
sequential logic.
1.2.2 A Brief Electronics Primer
Electricity is nothing more than the flow of electrons from point A
to point B. Along the way, those electrons can be made to do work
as they flow through various devices. Electronics is the science (and,
occasionally, art) of using tiny quantities of electricity to do work. As
an example, consider the circuit schematic illustrated in Figure 1.1:
B
1
Lmp
1
Sw
1
Figure 1.1: Simple Lamp Circuit
In this diagram, battery B1 is connected to lamp Lmp1 through
switch Sw1. When the switch is closed, electrons will flow from the
negative battery terminal, through the switch and lamp, and back to
the positive battery terminal. As the electrons flow through the lamp’s
filament it will heat up and glow: light!
A slightly more complex circuit is illustrated in Figure 1.2. In this
case, a transistor, Q1, has been added to the circuit. When switch
Sw1 is closed, a tiny current can flow from the battery, through the
transistor’s emitter (the connection with the arrow) to its base, through
the switch, through a resistor R1, and back to the positive terminal
of the battery. However, that small current turns on transistor Q1 so
a much larger current can also flow from the battery, through the
collector port, to lamp Lmp1, and back to the battery. The final effect
is the same for both circuits: close a switch and the lamp turns on.
However, by using a transistor, the lamp can be controlled by applying
any sort of positive voltage to the transistor’s base; so the lamp could
1.2 about digital logic 7
Q
1
Sw
1
B
1
R
1
Lmp
1
Figure 1.2: Transistor-Controlled Lamp Circuit
be controlled by a switch, as illustrated, or by the output of some other
electronic process, like a photo-electric cell sensing that the room is
dark.
Using various electronic components, like transistors and resistors,
digital logic “gates” can be constructed, and these become the building
blocks for complex logic circuits. Logic gates are more thoroughly
covered in a later chapter, but one of the fundamental logic gates is
an OR gate, and a simplified schematic diagram for an OR gate is in
Figure 1.3. In this circuit, any voltage present at Input A or Input B will
activate the transistors and develop voltage at the output.
Q
1
Q
2
out
A
B
R
2
R
1
R
3
Figure 1.3: Simple OR Gate Using Transistors
Figure 1.4
1
shows a more complete OR gate created with what are
called “N-Channel metal-oxide semiconductor” (or nMOS) transistors.
The electronic functioning of this circuit is beyond the scope of this
book (and is in the domain of electronics engineers), but the important
1
This schematic diagram was created by Ram
´
on Jaramillo and found at
http://www.
texample.net/tikz/examples/or-gate/
8 introduction
point to keep in mind is that this book concerns building electronic
circuits designed to accomplish a physical task rather than write a
program to control a computer’s processes.
V
DD
A
B
V
DD
V
SS
V
SS
C = A + B
Figure 1.4: OR Gate
Early electronic switches took the form of vacuum tubes, but those
were replaced by transistors which were much more energy efficient
and physically smaller. Eventually, entire transistorized circuits, like
the OR gate illustrated in Figure 1.4, were miniaturized and placed
in a single Integrated Circuit (IC), sometimes called a “chip,” smaller
than a postage stamp.
ICs make it possible to produce smaller, faster, and more powerful
electronics devices. For example, the original ENIAC computer, built in
1946, occupied more than 1800 square feet of floor space and required
150KW of electricity, but by the 1980s integrated circuits in hand-held
calculators were more powerful than that early computer. Integrated
circuits are often divided into four classes:
1. Small-scale integration with fewer than 10 transistors
2. Medium-scale integration with 10-500 transistors
3. Large-scale integration with 500-20,000 transistors
4. Very large-scale integration with 20,000-1,000,000 transistors
Integrated circuits are designed by engineers who use software
written specifically for that purpose. While the intent of this book
is to afford students a foundation in digital logic, those who pursue
a degree in electronics engineering, software engineering, or some
related field, will need to study a digital logic language, like iVerilog.
1.3 boolean algebra 9
1.3 boolean algebra
1.3.1 History
The Greek philosopher Aristotle founded a system of logic based
on only two types of propositions: True and False. His bivalent (two-
mode) definition of truth led to four foundational laws of logic: the
Law of Identity (A is A); the Law of Non-contradiction (A is not non-A);
the Law of the Excluded Middle (either A or non-A); and the Law
of Rational Inference. These laws function within the scope of logic
where a proposition is limited to one of two possible values, like True
and False; but they do not apply in cases where propositions can hold
other values.
The English mathematician George Boole (1815-1864) sought to give
symbolic form to Aristotle’s system of logic. Boole wrote a treatise
on the subject in 1854, titled An Investigation of the Laws of Thought, on
Which Are Founded the Mathematical Theories of Logic and Probabilities,
which codified several rules of relationship between mathematical
quantities limited to one of two possible values: True or False,
1
or
0
.
His mathematical system became known as Boolean Algebra.
All arithmetic operations performed with Boolean quantities have
but one of two possible outcomes:
1
or
0
. There is no such thing as
2
or
−1
or
1
3
in the Boolean world and numbers other than
1
and
0
are
invalid by definition. Claude Shannon of MIT recognized how Boolean
algebra could be applied to on-and-off circuits, where all signals are
characterized as either high (
1
) or low (
0
), and his 1938 thesis, titled A
Symbolic Analysis of Relay and Switching Circuits, put Boole’s theoretical
work to use in land line telephone switching, a system Boole never
could have imagined.
While there are a number of similarities between Boolean algebra
and real-number algebra, it is important to bear in mind that the
system of numbers defining Boolean algebra is severely limited in
scope:
1
and
0
. Consequently, the “Laws” of Boolean algebra often
differ from the “Laws” of real-number algebra, making possible such
Boolean statements as
1 + 1 = 1
, which would be considered absurd
for real-number algebra.
1.3.2 Boolean Equations
Boolean algebra is a mathematical system that defines a series of
logical operations performed on a set of variables. The expression of
a single logical function is a Boolean Equation that uses standardized
Sometimes Boolean
Equations are called
Switching
Equations
symbols and rules. Boolean expressions are the foundation of digital
circuits.
Binary variables used in Boolean algebra are like variables in reg-
ular algebra except that they can only have two values: one or zero.
10 introduction
Boolean algebra includes three primary logical functions: AND, OR,
and NOT; and five secondary logical functions: NAND, NOR, XOR, and
XNOR is sometimes
called equivalence
and Buffer is
sometimes called
transfer.
XNOR, and Buffer. A Boolean equation defines an electronic circuit
that provides a relationship between input and output variables and
takes the form of:
C = A ∗ B (1.1)
where A and B are binary input variables that are related to the
output variable C by the function AND (denoted by an asterisk).
In Boolean algebra, it is common to speak of truth. This term does
not mean the same as it would to a philosopher, though its use is
based on Aristotelian philosophy where a statement was either True or
False. In Boolean algebra as used in electronics, True commonly means
“voltage present” (or “
1
”) while False commonly means “voltage absent”
(or “
0
”), and this can be applied to either input or output variables. It
is common to create a Truth Table for a Boolean equation to indicate
which combination of inputs should evaluate to a True output and
Truth Tables are used very frequently throughout this book.
1.4 about this book
This book is organized into two main parts:
1.
Theory. Chapters two through six cover the foundational theory
of digital logic. Included are chapters on binary mathematics,
Boolean algebra, and simplifying Boolean expressions using
tools like Karnaugh maps and the Quine-McCluskey method.
2.
Practice. Chapters seven through nine expand the theory of dig-
ital logic into practical applications. Covered are combinational
and sequential logic circuits, and then simulation of various
physical devices, like elevators.
There is also an accompanying lab manual where Logisim-evolution is
used to build digital logic circuits. By combining the theory presented
in this book along with the practical application presented in the lab
manual it is hoped that students gain a thorough understanding of
digital logic.
By combining the theoretical background of binary mathematics
and Boolean algebra with the practical application of building logic
devices, digital logic becomes understandable and useful.
2
F O U N D AT I O N S O F B I N A RY A R I T H M E T I C
What to Expect
The language of digital logic is the binary number system and
this chapter introduces that system. Included are these topics:
•
The various bases used in digital logic: binary (base 2),
octal (base 8), decimal (base 10), and hexadecimal (base
16)
• Converting numbers between the bases
• Representing floating point numbers in binary
2.1 introduction to number systems
2.1.1 Background
The expression of numerical quantities is often taken for granted,
which is both a good and a bad thing in the study of electronics. It
is good since the use and manipulation of numbers is familiar for
many calculations used in analyzing electronic circuits. On the other
hand, the particular system of notation that has been taught from
primary school onward is not the system used internally in modern
electronic computing devices and learning any different system of
notation requires some re-examination of assumptions.
It is important to distinguish the difference between numbers and
the symbols used to represent numbers. A number is a mathematical
quantity, usually correlated in electronics to a physical quantity such
as voltage, current, or resistance. There are many different types of
numbers, for example:
• Whole Numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9...
• Integers: −4, −3, −2, −1, 0, 1, 2, 3, 4...
• Rational Numbers: −5.3, 0,
1
3
, 6.7
•
Irrational Numbers:
π
(approx.
3
.
1416
), e (approx.
2
.
7183
), and
the square root of any prime number
•
Real Numbers: (combination of all rational and irrational num-
bers)
11
12 foundations of binary arithmetic
• Complex Numbers: 3 − j4
Different types of numbers are used for different applications in
electronics. As examples:
•
Whole numbers work well for counting discrete objects, such as
the number of resistors in a circuit.
• Integers are needed to express a negative voltage or current.
•
Irrational numbers are used to describe the charge/discharge
cycle of electronic objects like capacitors.
•
Real numbers, in either fractional or decimal form, are used
to express the non-integer quantities of voltage, current, and
resistance in circuits.
•
Complex numbers, in either rectangular or polar form, must be
used rather than real numbers to capture the dual essence of
the magnitude and phase angle of the current and voltage in
alternating current circuits.
There is a difference between the concept of a “number” as a mea-
sure of some quantity and “number” as a means used to express that
quantity in spoken or written communication. A way to symbolically
denote numbers had to be developed in order to use them to describe
processes in the physical world, make scientific predictions, or balance
a checkbook. The written symbol that represents some number, like
how many apples there are in a bin, is called a cipher and in western ,
the commonly-used ciphers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
2.1.2 Binary Mathematics
Binary mathematics is a specialized branch of mathematics that con-
cerns itself with a number system that contains only two ciphers: zero
and one. It would seem to be very limiting to use only two ciphers;
however, it is much easier to create electronic devices that can differ-
entiate between two voltage levels rather than the ten that would be
needed for a decimal system.
2.1.3 Systems Without Place Value
hash marks. One of the earliest cipher systems was to simply
use a hash mark to represent each quantity. For example, three apples
could be represented like this:
|||
. Often, five hash marks were “bun-
dled” to aid in the counting of large quantities, so eight apples would
be represented like this: |||| |||.
2.1 introduction to number systems 13
roman numerals. The Romans devised a system that was a
substantial improvement over hash marks, because it used a variety of
ciphers to represent increasingly large quantities. The notation for one
is the capital letter I. The notation for
5
is the capital letter V. Other
ciphers, as listed in Table 2.1, possess increasing values:
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
Table 2.1: Roman Numerals
If a cipher is accompanied by a second cipher of equal or lesser
value to its immediate right, with no ciphers greater than that second
cipher to its right, the second cipher’s value is added to the total
quantity. Thus, VIII symbolizes the number
8
, and CLVII symbolizes
the number
157
. On the other hand, if a cipher is accompanied by
another cipher of lesser value to its immediate left, that other cipher’s
value is subtracted from the first. In that way, IV symbolizes the
number
4
(V minus I), and CM symbolizes the number
900
(M minus
C). The ending credit sequences for most motion pictures contain
the date of production, often in Roman numerals. For the year
1987
,
it would read: MCMLXXXVII. To break this numeral down into its
constituent parts, from left to right:
(M = 1000) + (CM = 900) + (LXXX = 80) + (VII = 7)
Large numbers are very difficult to denote with Roman numerals;
and the left vs. right (or subtraction vs. addition) of values can be very
confusing. Adding and subtracting two Roman numerals is also very
challenging, to say the least. Finally, one other major problem with
this system is that there is no provision for representing the number
zero or negative numbers, and both are very important concepts
in mathematics. Roman culture, however, was more pragmatic with
respect to mathematics than most, choosing only to develop their
numeration system as far as it was necessary for use in daily life.
2.1.4 Systems With Place Value
decimal numeration. The Babylonians developed one of the
most important ideas in numeration: cipher position, or place value,
14 foundations of binary arithmetic
to represent larger numbers. Instead of inventing new ciphers to rep-
resent larger numbers, as the Romans had done, they re-used the
same ciphers, placing them in different positions from right to left to
represent increasing values. This system also required a cipher that
represents zero value, and the inclusion of zero in a numeric system
was one of the most important inventions in all of mathematics (many
would argue zero was the single most important human invention, pe-
riod). The decimal numeration system uses the concept of place value,
with only ten ciphers (
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
, and
9
) used in “weighted”
positions to symbolize numbers.
Each cipher represents an integer quantity, and each place from
right to left in the notation is a multiplying constant, or weight, for
the integer quantity. For example, the decimal notation “
1206
” may be
broken down into its constituent weight-products as such:
1206 = (1X1000) + (2X100) + (0X10) + (6X1)
Each cipher is called a “digit” in the decimal numeration system,
and each weight, or place value, is ten times that of the place to the
immediate right. So, working from right to left is a “ones” place, a
“tens” place, a “hundreds” place, a “thousands” place, and so on.
While the decimal numeration system uses ten ciphers, and place-
weights that are multiples of ten, it is possible to make a different
numeration system using the same strategy, except with fewer or more
ciphers.
binary numeration. The binary numeration system uses only
two ciphers and the weight for each place in a binary number is two
times as much as the place to its right. Contrast this to the decimal
numeration system that has ten different ciphers and the weight for
each place is ten times the place to its right. The two ciphers for
the binary system are zero and one, and these ciphers are arranged
right-to-left in a binary number, each place doubling the weight of the
previous place. The rightmost place is the “ones” place; and, moving
to the left, is the “twos” place, the “fours” place, the “eights” place, the
“sixteens” place, and so forth. For example, the binary number
11010
can be expressed as a sum of each cipher value times its respective
weight:
11010 = (1X16) + (1X8) + (0X4) + (1X2) + (0X1)
The primary reason that the binary system is popular in modern
electronics is because it is easy to represent the two cipher states
(zero and one) electronically; if no current is flowing in the circuit it
represents a binary zero while flowing current represents a binary
one. Binary numeration also lends itself to the storage and retrieval
of numerical information: as examples, magnetic tapes have spots
of iron oxide that are magnetized for a binary one or demagnetized
2.1 introduction to number systems 15
for a binary zero and optical disks have a laser-burned pit in the
aluminum substrate representing a binary one and an unburned spot
representing a binary zero.
Digital numbers require so many bits to represent relatively small
numbers that programming or analyzing electronic circuitry can be
a tedious task. However, anyone working with digital devices soon
learns to quickly count in binary to at least
11111
(that is decimal
31
). Any time spent practicing counting both up and down between
zero and
11111
will be rewarded while studying binary mathematics,
codes, and other digital logic topics. Table 2.2 will help in memorizing
binary numbers:
Bin Dec Bin Dec Bin Dec Bin Dec
0 0 1000 8 10000 16 11000 24
1 1 1001 9 10001 17 11001 25
10 2 1010 10 10010 18 11010 26
11 3 1011 11 10011 19 11011 27
100 4 1100 12 10100 20 11100 28
101 5 1101 13 10101 21 11101 29
110 6 1110 14 10110 22 11110 30
111 7 1111 15 10111 23 11111 31
Table 2.2: Binary-Decimal Conversion
octal numeration. The octal numeration system is place-weighted
with a base of eight. Valid ciphers include the symbols
0
,
1
,
2
,
3
,
4
,
5
,
6
,
and
7
. These ciphers are arranged right-to-left in an octal number, each
place being eight times the weight of the previous place. For example,
the octal number
4270
can be expressed, just like a decimal number,
as a sum of each cipher value times its respective weight:
4270 = (4X512) + (2X64) + (7X8) + (0X1)
hexadecimal numeration. The hexadecimal numeration sys-
tem is place-weighted with a base of sixteen. There needs to be ciphers
The word
“hexadecimal” is a
combination of “hex”
for six and “decimal”
for ten
for numbers greater than nine so English letters are used for those
values. Table 2.3 lists hexadecimal numbers up to decimal 15:
16 foundations of binary arithmetic
Hex Dec Hex Dec
0 0 8 8
1 1 9 9
2 2 A 10
3 3 B 11
4 4 C 12
5 5 D 13
6 6 E 14
7 7 F 15
Table 2.3: Hexadecimal Numbers
Hexadecimal ciphers are arranged right-to-left, each place being
16
times the weight of the previous place. For example, the hexadecimal
number 13A2 can be expressed, just like a decimal number, as a sum
of each cipher value times its respective weight:
13A2 = (1X4096) + (3X256) + (AX16) + (2X1)
Maximum Number Size
It is important to know the largest number that can be repre-
sented with a given number of cipher positions. For example,
if only four cipher positions are available then what is the
largest number that can be represented in each of the numer-
ation systems? With the crude hash-mark system, the number
of places IS the largest number that can be represented, since
one hash mark “place” is required for every integer step. For
place-weighted systems, however, the answer is found by taking
the number base of the numeration system (
10
for decimal,
2
for
binary) and raising that number to the power of the number of
desired places. For example, in the decimal system, a five-place
number can represent
10
5
, or
100
,
000
, with values from zero to
99
,
999
. Eight places in a binary numeration system, or
2
8
, can
represent 256 different values, 0 − 255.
2.1.5 Summary of Numeration Systems
Table 2.4 counts from zero to twenty using several different numeration
systems:
2.1 introduction to number systems 17
Text Hash Marks Roman Dec Bin Oct Hex
Zero n/a n/a 0 0 0 0
One | I 1 1 1 1
Two || II 2 10 2 2
Three ||| III 3 11 3 3
Four |||| IV 4 100 4 4
Five |||| V 5 101 5 5
Six |||| | VI 6 110 6 6
Seven |||| || VII 7 111 7 7
Eight |||| ||| VIII 8 1000 10 8
Nine |||| |||| IX 9 1001 11 9
Ten |||| |||| X 10 1010 12 A
Eleven |||| |||| | XI 11 1011 13 B
Twelve |||| |||| || XII 12 1100 14 C
Thirteen |||| |||| ||| XIII 13 1101 15 D
Fourteen |||| |||| |||| XIV 14 1110 16 E
Fifteen |||| |||| |||| XV 15 1111 17 F
Sixteen |||| |||| |||| | XVI 16 10000 20 10
Seventeen |||| |||| |||| || XVII 17 10001 21 11
Eighteen |||| |||| |||| ||| XVIII 18 10010 22 12
Nineteen |||| |||| |||| |||| XIX 19 10011 23 13
Twenty |||| |||| |||| |||| XX 20 10100 24 14
Table 2.4: Counting To Twenty
18 foundations of binary arithmetic
Numbers for Computer Systems
An interesting footnote for this topic concerns one of the first
electronic digital computers: ENIAC. The designers of the
ENIAC chose to work with decimal numbers rather than binary
in order to emulate a mechanical adding machine; unfortunately,
this approach turned out to be counter-productive and required
more circuitry (and maintenance nightmares) than if they had
they used binary numbers. “ENIAC contained
17
,
468
vacuum
tubes,
7
,
200
crystal diodes,
1
,
500
relays,
70
,
000
resistors,
10
,
000
capacitors and around
5
million hand-soldered joints”
a
. Today,
all digital devices use binary numbers for internal calculation
and storage and then convert those numbers to/from decimal
only when necessary to interface with human operators.
a http://en.wikipedia.org/wiki/Eniac
2.1.6 Conventions
Using different numeration systems can get confusing since many
ciphers, like “
1
,” are used in several different numeration systems.
Therefore, the numeration system being used is typically indicated
with a subscript following a number, like
11010
2
for a binary number
or
26
10
for a decimal number. The subscripts are not mathematical
operation symbols like superscripts, which are exponents; all they do
is indicate the system of numeration being used. By convention, if no
subscript is shown then the number is assumed to be decimal.In this book,
subscripts are
normally used to
make it clear whether
the number is binary
or some other system.
Another method used to represent hexadecimal numbers is the
prefix
0x
. This has been used for many years by programmers who
work with any of the languages descended from C, like C++, C#,
Java, JavaScript, and certain shell scripts. Thus,
0x1A
would be the
hexadecimal number 1A.
One other commonly used convention for hexadecimal numbers is
to add an h (for hexadecimal) after the number. This is used because that
is easier to enter with a keyboard than to use a subscript and is more
intuitive than using a
0x
prefix. Thus,
1A
16
would be written
1Ah
. In
this case, the h only indicates that the number
1A
is hexadecimal; it is
not some sort of mathematical operator.
Occasionally binary numbers are written with a
0b
prefix; thus
0b1010
would be
1010
2
, but this is a programmer’s convention not
often found elsewhere.
2.2 converting between radices 19
2.2 converting between radices
2.2.1 Introduction
The radix of a system
is also commonly
called its “base.”
The number of ciphers used by a number system (and therefore,
the place-value multiplier for that system) is called the radix for the
system. The binary system, with two ciphers (zero and one), is radix
two numeration, and each position in a binary number is a binary digit
(or bit). The decimal system, with ten ciphers, is radix-ten numeration,
and each position in a decimal number is a digit. When working with
various digital logic processes it is desirable to be able to convert
between binary/octal/decimal/hexadecimal radices.
2.2.2 Expanded Positional Notation
Expanded Positional Notation is a method of representing a num-
ber in such a way that each position is identified with both its cipher
symbol and its place-value multiplier. For example, consider the num-
ber 347
10
:
347
10
= (3X10
2
) + ( 4X10
1
) + ( 7X10
0
) (2.1)
The steps to use to expand a decimal number like
347
are found in
Table 2.5.
Step Result
Count the number of digits in the number. Three Digits
Create a series of
( X )
connected by plus
signs such that there is one set for each of
the digits in the original number.
( X ) + ( X ) + ( X )
Fill in the digits of the original number on
the left side of each set of parenthesis.
(3X ) + (4X ) + (7X )
Fill in the radix (or base number) on the
right side of each parenthesis.
(3X10) + (4X10) + (7X10)
Starting on the far right side of the expres-
sion, add an exponent (power) for each of
the base numbers. The powers start at zero
and increase to the left.
(3X10
2
) + ( 4X10
1
) + ( 7X10
0
)
Table 2.5: Expanding a Decimal Number
Additional examples of expanded positional notation are:
2413
10
= (2x10
3
) + ( 4X10
2
) + ( 1X10
1
) + ( 3X10
0
) (2.2)
20 foundations of binary arithmetic
1052
8
= (1X8
3
) + ( 0X8
2
) + ( 5X8
1
) + ( 2X8
0
) (2.3)
139
16
= (1X16
2
) + ( 3X16
1
) + ( 9X16
0
) (2.4)
The above examples are for positive decimal integers; but a num-
ber with any radix can also have a fractional part. In that case, the
number’s integer component is to the left of the radix point (called
the “decimal point” in the decimal system), while the fractional part
is to the right of the radix point. For example, in the number
139
.
25
10
,
139
is the integer component while
25
is the fractional component. If a
number includes a fractional component, then the expanded positional
notation uses increasingly negative powers of the radix for numbers
to the right of the radix point. Consider this binary example:
101
.
011
2
.
The expanded positional notation for this number is:
101.011
2
= (1X2
2
) + (0X2
1
) + (1X2
0
) + (0X2
−1
) + (1X2
−2
) + (1X2
−3
)
(2.5)
Other examples are:
526.14
10
= (5X10
2
) + (2X10
1
) + (6X10
0
) + (1X10
−1
) + (4X10
−2
) (2.6)
65.147
8
= (6X8
1
) + (5X8
0
) + (1X8
−1
) + (4X8
−2
) + (7X8
−3
) (2.7)
D5.3A
16
= (13X16
1
) + ( 5X16
0
) + ( 3X16
−1
) + ( 10X16
−2
) (2.8)
When a number in expanded positional notation includes one or
more negative radix powers, the radix point is assumed to be to the
immediate right of the “zero” exponent term, but it is not actually
written into the notation. Expanded positional notation is useful in
converting a number from one base to another.
2.2.3 Binary to Decimal
To convert a number in binary form to decimal, start by writing the
binary number in expanded positional notation, calculate the values
for each of the sets of parenthesis in decimal, and then add all of the
values. For example, convert 1101
2
to decimal:
2.2 converting between radices 21
1101
2
= (1X2
3
) + ( 1X2
2
) + ( 0X2
1
) + ( 1X2
0
) (2.9)
= (8) + (4) + (0) + (1)
= 13
10
Binary numbers with a fractional component are converted to dec-
imal in exactly the same way, but the fractional parts use negative
powers of two. Convert binary 10.11
2
to decimal:
10.11
2
= (1X2
1
) + ( 0X2
0
) + ( 1X2
−1
) + ( 1X2
−2
) (2.10)
= (2) + (0) + (
1
2
) + (
1
4
)
= 2 + .5 + .25
= 2.75
10
Most technicians who work with digital circuits learn to quickly
convert simple binary integers to decimal in their heads. However,
for longer numbers, it may be useful to write down the various place
weights and add them up; in other words, a shortcut way of writing
expanded positional notation. For example, convert the binary number
11001101
2
to decimal:
Binary Number: 1 1 0 0 1 1 0 1
- - - - - - - -
(Read Down) 1 6 3 1 8 4 2 1
2 4 2 6
8
A bit value of one in the original number means that the respective
place weight gets added to the total value, while a bit value of zero
means that the respective place weight does not get added to the
total value. Thus, using the example above this paragraph, the binary
number 11001101
2
is converted to: 128 + 64 + 8 + 4 + 1, or 205
10
.
Naming Conventions
The bit on the right end of any binary number is the Least
Significant Bit (LSB) because it has the least weight (the ones
place) while the bit on the left end is the Most Significant Bit
(MSB) because it has the greatest weight. Also, groups of bits
are normally referred to as words, so engineers would speak
of
16
-bit or
32
-bit words. As exceptions, an eight-bit group is
commonly called a byte and a four-bit group is called a nibble
(occasionally spelled nybble).
22 foundations of binary arithmetic
2.2.4 Binary to Octal
The octal numeration system serves as a “shorthand” method of
denoting a large binary number. Technicians find it easier to discuss a
number like 57
8
rather than 101111
2
.“Five Seven Octal”
is not pronounced
“Fifty Seven” since
“fifty” is a decimal
number.
Because octal is a base eight system, and eight is
2
3
, binary num-
bers can be converted to octal by creating groups of three and then
simplifying each group. As an example, convert 101111
2
to octal:
101 111
5 7
Thus, 101111
2
is equal to 57
8
.
If a binary integer cannot be grouped into an even grouping of three,
it is padded on the left with zeros. For example, to convert
11011101
2
to octal, the most significant bit must be padded with a zero:
011 011 101
3 3 5
Thus, 11011101
2
is equal to 335
8
.
A binary fraction may need to be padded on the right with zeros in
order to create even groups of three before it is converted into octal.
For example, convert 0.1101101
2
to octal:
0 . 110 110 100
0 . 6 6 4
Thus, 0.1101101
2
is equal to .664
8
.
A binary mixed number may need to be padded on both the left
and right with zeros in order to create even groups of three before
it can be converted into octal. For example, convert
10101
.
00101
2
to
octal:
010 101 . 001 010
2 5 . 1 2
Thus, 10101.00101
2
is equal to 25.12
8
.
Table 2.6 lists additional examples of binary/octal conversion:
Binary Octal
100 101.011 45.3
1 100 010.1101 142.64
100 101 011.110 100 1 453.644
1 110 010 011 101.000 110 10 16 235.064
110 011 010 100 111.011 101 63 247.35
Table 2.6: Binary-Octal Conversion Examples
2.2 converting between radices 23
While it is easy to convert between binary and octal, the octal system
is not frequently used in electronics since computers store and transmit
binary numbers in words of 16, 32, or 64 bits, which are multiples of
four rather than three.
2.2.5 Binary to Hexadecimal
The hexadecimal numeration system serves as a “shorthand” method
of denoting a large binary number. Technicians find it easier to discuss
a number like
2F
16
rather than
101111
2
. Because hexadecimal is a
base
16
system, and
16
is
2
4
; binary numbers can be converted to
hexadecimal by creating groups of four and then simplifying each
group. As an example, convert 10010111
2
to hexadecimal:
1001 0111
9 7
Thus, 10010111
2
is equal to 97
16
. “Nine Seven
Hexadecimal,” or,
commonly, “Nine
Seven Hex,” is not
pronounced “Ninety
Seven” since “ninety”
is a decimal number.
A binary integer may need to be padded on the left with zeros in
order to create even groups of four before it can be converted into
hexadecimal. For example, convert 1001010110
2
to hexadecimal:
0010 0101 0110
2 5 6
Thus, 1001010110
2
is equal to 256
16
.
A binary fraction may need to be padded on the right with zeros
in order to create even groups of four before it can be converted into
hexadecimal. For example, convert 0.1001010110
2
to hexadecimal:
0 . 1001 0101 1000
. 9 5 8
Thus, 0.1001010110
2
is equal to 0.958
16
.
A binary mixed number may need to be padded on both the left
and right with zeros in order to create even groups of four before it
can be converted into hexadecimal. For example, convert
11101
.
10101
2
to hexadecimal:
0001 1101 . 1010 1000
1 D . A 8
Thus, 11101.10101
2
is equal to 1D.A8
16
.
Table 2.7 lists additional examples of binary/hexadecimal conver-
sion:
24 foundations of binary arithmetic
Binary Hexadecimal
100 101.011 25.6
1 100 010.1101 62.D
100 101 011.110 100 1 12B.D2
1 110 010 011 101.000 110 10 1C9D.1A
110 011 010 100 111.011 101 66A7.74
Table 2.7: Binary-Hexadecimal Conversion Examples
2.2.6 Octal to Decimal
The simplest way to convert an octal number to decimal is to write
the octal number in expanded positional notation, calculate the values
for each of the sets of parenthesis, and then add all of the values. For
example, to convert 245
8
to decimal:
245
8
= (2X8
2
) + ( 4X8
1
) + ( 5X8
0
) (2.11)
= (2X64) + ( 4X8) + (5X1)
= (128) + ( 32) + (5)
= 165
10
If the octal number has a fractional component, then that part would
be converted using negative powers of eight. As an example, convert
25.71
8
to decimal:
25.71
8
= (2X8
1
) + ( 5X8
0
) + ( 7X8
−1
) + ( 1X8
−2
) (2.12)
= (2X8) + (5X1) + (7X0.125) + (1X0.015625)
= (16) + ( 5) + (0.875) + (0.015625)
= 21.890625
10
Other examples are:
42.6
8
= (4X8
1
) + ( 2X8
0
) + ( 6X8
−1
) (2.13)
= (4X8) + ( 2X1) + (6X0.125)
= (32) + ( 2) + (0.75)
= 34.75
10
32.54
8
= (3X8
1
) + ( 2X8
0
) + ( 5X8
−1
) + ( 4X8
−2
) (2.14)
= (3X8) + (2X1) + (5X0.125) + (4X0.015625)
= (24) + ( 2) + (0.625) + (0.0625)
= 26.6875
10
2.2 converting between radices 25
436.27
8
= (4X8
2
) + ( 3X8
1
) + ( 6X8
0
) + ( 2X8
−1
) + ( 7X8
−2
) (2.15)
= (4X64) + ( 3X8) + (6X1) + (2X0.125) + (7X0.015625)
= (256) + ( 24) + (6) + (0.25) + (0.109375)
= 286.359375
10
2.2.7 Hexadecimal to Decimal
The simplest way to convert a hexadecimal number to decimal is
to write the hexadecimal number in expanded positional notation,
calculate the values for each of the sets of parenthesis, and then add
all of the values. For example, to convert 2A6
16
to decimal:
2A6
16
= (2X16
2
) + ( AX16
1
) + ( 6X16
0
) (2.16)
= (2X256) + (10X16) + (6X1)
= (512) + ( 160) + (6)
= 678
10
If the hexadecimal number has a fractional component, then that
part would be converted using negative powers of
16
. As an example,
convert 1B.36
16
to decimal:
1B.36
16
= (1X16
1
) + ( 11X16
0
) + ( 3X16
−1
) + ( 6X16
−2
) (2.17)
= (16) + ( 11) + (3X
1
16
) + ( 6X
1
256
)
= 16 + 11 + 0.1875 + 0.0234375
= 27.2109375
10
Other examples are:
A32.1C
16
= (AX16
2
) + ( 3X16
1
) + ( 2X16
0
) + ( 1X16
−1
) + ( CX16
−2
)
(2.18)
= (10X256) + (3X16) + (2X1) + (1X
1
16
) + ( 12X
1
256
)
= 2560 + 48 + 2 + 0.0625 + 0.046875
= 6300.109375
10
439.A
16
= (4X16
2
) + ( 3X16
1
) + ( 9X16
0
) + ( AX16
−1
) (2.19)
= (4X256) + (3X16) + (9X1) + (10X
1
16
)
= 1024 + 48 + 9 + 0.625
= 1081.625
10
26 foundations of binary arithmetic
2.2.8 Decimal to Binary
2.2.8.1 Integers
Note: Converting
decimal fractions is a
bit different and is
covered on page 27.
Converting decimal integers to binary (indeed, any other radix) in-
volves repeated cycles of division. In the first cycle of division, the
original decimal integer is divided by the base of the target numera-
tion system (binary=
2
, octal=
8
, hex=
16
), and then the whole-number
portion of the quotient is divided by the base value again. This process
continues until the quotient is less than one. Finally, the binary, octal,
or hexadecimal digits are determined by the “remainders” left over at
each division step.After a decimal
number is converted
to binary it can be
easily converted to
either octal or
hexadecimal.
Table 2.8 shows how to convert
87
10
to binary by repeatedly divid-
ing
87
by
2
(the radix for binary) until reaching zero. The number in
column one is divided by two and that quotient is placed on the next
row in column one with the remainder in column two. For example,
when
87
is divided by
2
, the quotient is
43
with a remainder of one.
This division process is continued until the quotient is less than one.
When the division process is completed, the binary number is found
by using the remainders, reading from the bottom to top. Thus
87
10
is
1010111
2
.
Integer Remainder
87
43 1
21 1
10 1
5 0
2 1
1 0
0 1
Table 2.8: Decimal to Binary
This repeat-division technique will also work for numeration sys-
tems other than binary. To convert a decimal integer to octal, for
example, divide each line by
8
; but follow the process as described
above. As an example, Table 2.9 shows how to convert 87
10
to 127
8
.
2.2 converting between radices 27
Integer Remainder
87
10 7
1 2
0 1
Table 2.9: Decimal to Octal
The same process can be used to convert a decimal integer to
hexadecimal; except, of course, the divisor would be
16
. Also, some of
the remainders could be greater than
10
, so these are written as letters.
For example, to convert
678
10
to
2A6
16
use the process illustrated in
Table 2.10.
Integer Remainder
678
42 6
2 A
0 2
Table 2.10: Decimal to Hexadecimal
2.2.8.2 Fractions
Converting decimal fractions to binary is a repeating operation similar
to converting decimal integers, but each step repeats multiplication
rather than division. To convert
0
.
8215
10
to binary, repeatedly multiply
the fractional part of the number by two until the fractional part is
zero (or whatever degree of precision is desired). As an example, in
Table 2.11, the number in column two,
8215
, is multiplied by two and
the integer part of the product is placed in column one on the next
row while the fractional part in column two. Keep in mind that the
“Remainder” is a decimal fraction with an assumed leading decimal
point. That process continues until the fractional part reaches zero.
28 foundations of binary arithmetic
Integer Remainder
8125
1 625
1 25
0 5
1 0
Table 2.11: Decimal to Binary Fraction
When the multiplication process is completed, the binary number is
found by using the integer parts and reading from the top to the bottom.
Thus 0.8125
10
is 0.1101
2
.
As another example, Table 2.12 converts
0
.
78125
10
to
0
.
11001
2
. The
solution was carried out to full precision (that is, the last multiplication
yielded a fractional part of zero).
Integer Remainder
78125
1 5625
1 125
0 25
0 5
1 0
Table 2.12: Decimal to Binary Fraction Example
Often, a decimal fraction will create a huge binary fraction. In that
case, continue the multiplication until the desired number of binary
places are achieved. As an example, in Table 2.13, the fraction
0
.
1437
10
was converted to binary, but the process stopped after 10 bits.
2.2 converting between radices 29
Integer Remainder
1437
0 2874
0 5748
1 1496
0 2992
0 5984
1 1968
0 3936
0 7872
1 5744
1 1488
Table 2.13: Decimal to Long Binary Fraction
Thus, 0.1437
10
= 0.0010010011
2
(with 10 bits of precision). To calculate this to
full precision
requires 6617 bits;
thus, it is normally
wise to specify the
desired precision.
Converting decimal fractions to any other base would involve the
same process, but the base is used as a multiplier. Thus, to convert a
decimal fraction to hexadecimal multiply each line by
16
rather than
2.
2.2.8.3 Mixed Numbers
To convert a mixed decimal number (one that contains both an integer
and fraction part) to binary, treat each component as a separate prob-
lem and then combine the result. As an example, Table 2.14 and Table
2.15 show how to convert 375.125
10
to 101110111.001
2
.
Integer Remainder
375
187 1
93 1
46 1
23 0
11 1
5 1
2 1
1 0
0 1
Table 2.14: Decimal to Binary Mixed Integer
30 foundations of binary arithmetic
Integer Remainder
125
0 25
0 5
1 0
Table 2.15: Decimal to Binary Mixed Fraction
A similar process could be used to convert decimal numbers into
octal or hexadecimal, but those radix numbers would be used instead
of two.
2.2.9 Calculators
For the most part, converting numbers between the various “computer”
bases (binary, octal, or hexadecimal) is done with a calculator. Using a
calculator is quick and error-free. However, for the sake of applying
digital logic to a mathematical problem, it is essential to understand
the theory behind converting bases. It will not be possible to construct
a digital circuit where one step is “calculate the next answer on a
hand-held calculator.” Conversion circuits (like all circuits) need to be
designed with simple gate logic, and an understanding of the theory
behind the conversion process is important for that type of problem.
Online Conversion Tool
Excel will convert between decimal/binary/octal/hexadecimal
integers (including negative integers), but cannot handle frac-
tions; however, the following website has a conversion tool that
can convert between common bases, both integer and fraction:
http://baseconvert.com/
. An added benefit for this site is con-
version with twos complement, which is how negative binary
numbers are represented and is covered on page 40.
2.2.10 Practice Problems
Table 2.16 lists several numbers in decimal, binary, octal, and hexadec-
imal form. To practice converting between numbers, select a number
on any row and then covert it to the other bases.
2.3 floating point numbers 31
Decimal Binary Octal Hexadecimal
13.0 1101.0 15.0 D.0
1872.0 11 101 010 000.0 3520.0 750.0
0.0625 0.0001 0.04 0.1
0.457 031 25 0.011 101 01 0.352 0.75
43.125 101 011.001 53.1 2B.2
108.718 75 1 101 100.101 11 154.56 6C.B8
Table 2.16: Practice Problems
2.3 floating point numbers
Numbers can take two forms: Fixed Point and Floating Point. A fixed
point number generally has no fractional component and is used for
integer operations (though it is possible to design a system with a
fixed fractional width). On the other hand, a floating point number
has a fractional component with a variable number of places.
Before considering how floating point numbers are stored in mem-
ory and manipulated, it is important to recall that any number can be
represented using scientific notation. Thus,
123
.
45
10
can be represented
as
1
.
2345X10
2
and
0
.
0012345
10
can be represented as
1
.
2345X10
−3
.
Numbers in scientific notation with one place to the left of the radix
point, as illustrated in the previous sentence, are considered normalized.
While most people are familiar with normalized decimal numbers, the
same process can be used for any other base, including binary. Thus,
1101
.
101
2
can be written as
1
.
101101X2
3
. Notice that for normalized
binary numbers the radix is two rather than ten since binary is a radix
two system, also there is only one bit to the left of the radix point. IEEE Standard 754
defines Floating
Point Numbers
By definition, floating point numbers are stored and manipulated in
a computer using a 32-bit word (64 bits for “double precision” num-
bers). For this discussion, imagine that the number
10010011
.
0010
2
is to be stored as a floating point number. That number would first
be normalized to
1
.
00100110010X2
7
and then placed into the floating
point format:
x xxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx
sign exponent mantissa
•
Sign: The sign field is a single bit that is zero for a positive num-
ber and one for a negative number. Since the example number is
positive the sign bit is zero.
•
Exponent: This is an eight-bit field containing the radix’s expo-
nent, or
7
in the example. However, the field must be able to
contain both positive and negative exponents, so it is offset by
32 foundations of binary arithmetic
127
. The exponent of the example,
7
, is stored as
7 + 127
, or
134
;
therefore, the exponent field contains 10000110.
•
Mantissa (sometimes called significand): This 23-bit field con-
tains the number that is being stored, or
100100110010
in the
example. While it is tempting to just place that entire number
in the mantissa field, it is possible to squeeze one more bit of
precision from this number with a simple adjustment. A nor-
malized binary number will always have a one to the left of
the radix point since in scientific notation a significant bit must
appear in that position and one is the only possible significant
bit in a binary number. Since the first bit of the stored number
is assumed to be one it is dropped. Thus, the mantissa for the
example number is 00100110010000000000000.
Here is the example floating point number (the spaces have been
added for clarity):
10010011.0010 = 0 10000110 00100110010000000000000
A few floating point special cases have been defined:
•
zero: The exponent and mantissa are both zero and it does not
matter whether the sign bit is one or zero.
•
infinity: The exponent is all ones and the mantissa is all zeros.
The sign bit is used to represent either positive or negative
infinity.
•
Not a Number (NaN): The exponent is all ones and the mantissa
has at least one one (it does not matter how many or where).
NaN is returned as the result of an illegal operation, like an
attempted division by zero.
Two specific problems may show up with floating point calculations:
•
Overflow. If the result of a floating point operation creates a
positive number that is greater than
(2 − 2
−23
)X2
127
it is a posi-
tive overflow or a negative number less than
−(2 − 2
−23
)X2
127
it is a negative overflow. These types of numbers cannot be con-
tained in a 32-bit floating point number; however, the designer
could opt to increase the circuit to 64-bit numbers (called “dou-
ble precision” floating point) in order to work with these large
numbers.
•
Underflow. If the result of a floating point operation creates a
positive number that is less than
2
−149
it is a positive underflow
or a negative number greater than
−2
−149
it is a negative under-
flow. These numbers are vanishingly small and are sometimes
2.3 floating point numbers 33
simply rounded to zero. However, in certain applications, such
as multiplication problems, even a tiny fraction is important and
there are ways to use “denormalized numbers” that will sacrifice
precision in order to permit smaller numbers. Of course, the
circuit designer can always opt to use 64-bit (“double precision”)
floating point numbers which would permit negative exponents
about twice as large as 32-bit numbers.
Table 2.17 contains a few example floating point numbers.
Decimal Binary Normalized Floating Point
34.5 100010.1 1.000101X2
5
0 10000100 00010100000000000000000
324.75 101000100.11 1.0100010011X2
8
0 10000111 01000100010000111101100
-147.25 10010011.01 1.001001101X2
7
1 10000110 00100110100000000000000
0.0625 0.0001 1.0X2
−4
0 01111011 00000000000000000000000
Table 2.17: Floating Point Examples
3
B I N A RY A R I T H M E T I C O P E R AT I O N S
What to Expect
This chapter develops a number of different binary arithmetic
operations. These operations are fundamental in understanding
and constructing digital logic circuits using components like
adders and logic gates like NOT. Included in this chapter are
the following topics.
•
Calculating addition/subtraction/multiplication/division
problems with binary numbers
• Countering overflow in arithmetic operations
• Representing binary negative numbers
•
Contrasting the use of ones-complement and twos-
complement numbers
•
Developing circuits where bitwise operations like masking
are required
• Using and converting codes like ASCII/BCD/Gray
3.1 binary addition
Adding binary numbers is a simple task similar to the longhand
addition of decimal numbers. As with decimal numbers, the bits
are added one column at a time, from right to left. Unlike decimal
addition, there is little to memorize in the way of an “Addition Table,”
as seen in Table 3.1
35
36 binary arithmetic operations
Inputs Outputs
Carry In Augend Addend Sum Carry Out
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Table 3.1: Addition Table
Just as with decimal addition, two binary integers are added one
column at a time, starting from the LSB (the right-most bit in the
integer):
1001101
+0010010
1011111
When the sum in one column includes a carry out, it is added to
the next column to the left (again, like decimal addition). Consider the
following examples:
11 1 <--Carry Bits
1001001
+0011001
1100010
11 <--Carry Bits
1000111
+0010110
1011101
The “ripple-carry” process is simple for humans to understand,
but it causes a significant problem for designers of digital circuits.
Consequently, ways were developed to carry a bit to the left in an
electronic adder circuit and that is covered in Section 8.1, page 173.
Binary numbers that include a fractional component are added
just like binary integers; however, the radix points must align so the
augend and addend may need to be padded with zeroes on either the
left or the right. Here is an example:
111 1 <--Carry Bits
1010.0100
3.1 binary addition 37
+0011.1101
1110.0001
3.1.1 Overflow Error
One problem circuit designers must consider is a carry out bit in the
MSB (left-most bit) in the answer. Consider the following:
11 11 <--Carry Bits
10101110
+11101101
110011011
This example illustrates a significant problem for circuit designers.
Suppose the above calculation was done with a circuit that could only
accommodate eight data bits. The augend and addend are both eight
bits wide, so they are fine; however, the sum is nine bits wide due to
the carry out in the MSB. In an eight-bit circuit (that is, a circuit where
the devices and data lines can only accommodate eight bits of data),
the carry out bit would be dropped since there is not enough room to
accommodate it.
The result of a dropped bit cannot be ignored. The example prob-
lem above, when calculated in decimal, is
174
10
+ 237
10
= 411
10
. If,
though, the MSB carry out is dropped, then the answer becomes
155
10
,
which is, of course, incorrect. This type of error is called an Overflow
Error, and a circuit designer must find a way to correct overflow. One
typical solution is to simply alert the user that there was an overflow
error. For example, on a handheld calculator, the display may change
to something like -E- if there is an error of any sort, including overflow.
3.1.2 Sample Binary Addition Problems
The following table lists several binary addition problems that can be
used for practice.
Augend Addend Sum
10 110.0 11 101.0 110 011.0
111 010.0 110 011.0 1 101 101.0
1011.0 111 000.0 1 000 011.0
1 101 001.0 11 010.0 10 000 011.0
1010.111 1100.001 10 111.000
101.01 1001.001 1110.011
Table 3.2: Binary Addition Problems
38 binary arithmetic operations
3.2 binary subtraction
3.2.1 Simple Manual Subtraction
Subtracting binary numbers is similar to subtracting decimal numbers
and uses the same process children learn in primary school. The
minuend and subtrahend are aligned on the radix point, and then
columns are subtracted one at a time, starting with the least significant
place and moving to the left. If the subtrahend is larger than the
minuend for any one column, an amount is “borrowed” from the
column to the immediate left. Binary numbers are subtracted in the
same way, but it is important to keep in mind that binary numbers
have only two possible values: zero and one. Consider the following
problem:
10.1
-01.0
01.1
In this problem, the LSB column is
1 − 0
, and that equals one. The
middle column, though, is
0 − 1
, and one cannot be subtracted from
zero. Therefore, one is borrowed from the most significant bit, so the
problem in middle column becomes
10 − 1
. (Note: do not think of this
as “ten minus one” - remember that this is binary so this problem
is “one-zero minus one,” or two minus one in decimal) The middle
column is 10 − 1 = 1, and the MSB column then becomes 0 − 0 = 0.
The radix point must be kept in alignment throughout the problem,
so if one of the two operands has too few places it is padded on the
left or right (or both) to make both operands the same length. As an
example, subtract: 101101.01 − 1110.1:
101101.01
-001110.10
11110.11
There is no difference between decimal and binary as far as the
subtraction process is concerned. In each of the problems in this section
the minuend is greater than the subtrahend, leading to a positive
difference; however, if the minuend is less than the subtrahend, the
result is a negative number and negative numbers are developed in
the next section of this chapter.
Table 3.3 includes some subtraction problems for practice:
3.2 binary subtraction 39
Minuend Subtrahend Difference
1 001 011.0 111 010.0 10 001.0
100 010.0 10 010.0 10 000.0
101 110 110.0 11 001 010.0 10 101 100.0
1 110 101.0 111 010.0 111 011.0
11 011 010.1101 101 101.1 10 101 101.0101
10 101 101.1 1 101 101.101 111 111.111
Table 3.3: Binary Subtraction Problems
3.2.2
Representing Negative Binary Numbers Using Sign-and-Magnitude
Sign-and-magnitude
was used in early
computers since it
mimics real number
arithmetic, but has
been replaced by
more efficient
negative number
systems in modern
computers.
Binary numbers, like decimal numbers, can be both positive and nega-
tive. While there are several methods of representing negative binary
numbers; one of the most intuitive is using sign-and-magnitude, which
is essentially the same as placing a “–” in front of a decimal number.
With the sign-and-magnitude system, the circuit designer simply des-
ignates the MSB as the sign bit and all others as the magnitude of the
number. When the sign bit is one the number is negative, and when it
is zero the number is positive. Thus,
−5
10
would be written as
1101
2
.
Unfortunately, despite the simplicity of the sign-and-magnitude
approach, it is not very practical for binary arithmetic, especially
when done by a computer. For instance, negative five (
1101
2
) cannot be
added to any other binary number using standard addition technique
since the sign bit would interfere. As a general rule, errors can easily
occur when bits are used for any purpose other than standard place-
weighted values; for example,
1101
2
could be misinterpreted as the
number
13
10
when, in fact, it is meant to represent
−5
. To keep things
straight, the circuit designer must first decide how many bits are going
to be used to represent the largest numbers in the circuit, add one
more bit for the sign, and then be sure to never exceed that bit field
length in arithmetic operations. For the above example, three data bits
plus a sign bit would limit arithmetic operations to numbers from
negative seven (1111
2
) to positive seven (0111
2
), and no more.
This system also has the quaint property of having two values for
zero. If using three magnitude bits, these two numbers are both zero:
0000
2
(positive zero) and 1000
2
(negative zero).
3.2.3
Representing Negative Binary Numbers Using Signed Complements
3.2.3.1 About Complementation
Before discussing negative binary numbers, it is important to under-
stand the concept of complementation. To start, recall that the radix
40 binary arithmetic operations
(or base) of any number system is the number of ciphers available for
counting; the decimal (or base-ten) number system has ten ciphers
(
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
) while the binary (or base-two) number system
has two ciphers (
0
,
1
). By definition, a number plus its complement
equals the radix (this is frequently called the radix complement). For
example, in the decimal system four is the radix complement of six
since
4 + 6 = 10
. Another type of complement is the diminished radix
complement, which is the complement of the radix minus one. For ex-
ample, in the decimal system six is the diminished radix complement
of three since 6 + 3 = 9 and nine is equal to the radix minus one.
decimal. In the decimal system the radix complement is usually
called the tens complement since the radix of the decimal system is
ten. Thus, the tens complement of eight is two since
8 + 2 = 10
. The
diminished radix complement is called the nines complement in the
decimal system. As an example, the nines complement of decimal
eight is one since
8 + 1 = 9
and nine is the diminished radix of the
decimal system.
To find the nines complement for a number larger than one place,
the nines complement must be found for each place in the number.
For example, to find the nines complement for
538
10
, find the nines
complement for each of those three digits, or
461
. The easiest way
to find the tens complement for a large decimal number is to first
find the nines complement and then add one. For example, the tens
complement of
283
is
717
, which is calculated by finding the nines
complement, 716, and then adding one.
binary. Since the radix for a binary number is
10
2
, (be careful!
this is not ten, it is one-zero in binary) the diminished radix is
1
2
.
The diminished radix complement is normally called the ones comple-
ment and is obtained by reversing (or “flipping”) each bit in a binary
number; so the ones complement of 100101
2
is 011010
2
.
The radix complement (or twos complement) of a binary number is
found by first calculating the ones complement and then adding one
to that number. The ones complement of
101101
2
is
010010
2
, so the
twos complement is 010010
2
+ 1
2
= 010011
2
.
3.2.3.2 Signed Complements
In circuits that use binary arithmetic, a circuit designer can opt to use
ones complement for negative numbers and designate the most signif-
icant bit as the sign bit; and, if so, the other bits are the magnitude of
the number. This is similar to the sign-and-magnitude system discussed
on page 39. By definition, when using ones complement negative
numbers, if the most significant bit is zero, then the number is positive
and the magnitude of the number is determined by the remaining bits;
but if the most significant bit is one, then the number is negative and
3.2 binary subtraction 41
the magnitude of the number is determined by calculating the ones
complement of the number. Thus:
0111
2
= +7
10
, and
1000
2
= −7
10
(the ones complement for
1000
2
is
0111
2
). Table 3.4 may help to clarify
this concept:
Decimal Positive Negative
0 0000 1111
1 0001 1110
2 0010 1101
3 0011 1100
4 0100 1011
5 0101 1010
6 0110 1001
7 0111 1000
Table 3.4: Ones Complement
In a four-bit binary number, any decimal number from
−7
to
+7
can
be represented; but, notice that, like the sign-and-magnitude system,
there are two values for zero, one positive and one negative. This
requires extra circuitry to test for both values of zero after subtraction
operations.
To simplify circuit design, a designer can opt to use twos comple-
ment negative numbers and designate the most significant bit as the
sign bit so the other bits are the number’s magnitude. To use twos
complement numbers, if the most significant bit is zero, then the num-
ber is positive and the magnitude of the number is determined by the
remaining bits; but if the most significant bit is one, then the number
is negative and the magnitude of the number is determined by taking
the twos complement of the number (that is, the ones complement
plus one). Thus:
0111 = 7
, and
1001 = −7
(the ones complement of
1001
is
0110
, and
0110 + 1 = 0111
). Table 3.5 may help to clarify this
concept:
42 binary arithmetic operations
Decimal Positive Negative
0 0000 10000
1 0001 1111
2 0010 1110
3 0011 1101
4 0100 1100
5 0101 1011
6 0110 1010
7 0111 1001
8 N/A 1000
Table 3.5: Twos Complement
The twos complement removes that quirk of having two values for
zero. Table 3.5 shows that zero is either
0000
or
10000
; but since this
is a four-bit number the initial one is discarded, leaving
0000
for zero
whether the number is positive or negative. Also,
0000
is considered a
positive number since the sign bit is zero. Finally, notice that
1000
is
−8
(ones complement of
1000
is
0111
, and
0111 + 1 = 1000
). This means
Programmers
reading this book
may have wondered
why the
maximum/minimum
values for various
types of variables is
asymmetrical.
that binary number systems that use a twos complement method
of designating negative numbers will be asymmetrical; running, for
example, from
−8
to
+7
. A twos complement system still has the same
number of positive and negative numbers, but zero is considered
positive, not neutral.
All modern computer
systems use radix (or
twos) complements
to represent negative
numbers.
One other quirk about the twos complement system is that the
decimal value of the binary number can be quickly calculated by
assuming the sign bit has a negative place value and all other places
are added to it. For example, in the negative number
1010
2
, if the
sign bit is assumed to be worth
−8
and the other places are added to
that, the result is
−8 + 2
, or
−6
; and
−6
is the value of
1010
2
in a twos
complement system.
3.2.3.3 About Calculating the Twos Complement
In the above section, the twos (or radix) complement is calculated by
finding the ones complement of a number and then adding one. For
machines, this is the most efficient method of calculating the twos
complement; but there is a method that is much easier for humans
to use to find the twos complement of a number. Start with the LSB
(the right-most bit) and then read the number from right to left. Look
for the first one and then invert every bit to the left of that one. As an
example, the twos complement for
101010
2
is formed by starting with
the least significant bit (the zero on the right), and working to the left,
looking for the first one, which is in the second place from the right.
3.2 binary subtraction 43
Then, every bit to the left of that one is inverted, ending with:
010110
2
(the two LSBs are underlined to show that they are the same in both
the original and twos complement number).
Table 3.6 displays a few examples:
Number Twos Complement
0110100 1001100
11010 00110
001010 110110
1001011 0110101
111010111 000101001
Table 3.6: Example Twos Complement
3.2.4 Subtracting Using the Diminished Radix Complement
When thinking about subtraction, it is helpful to remember that
A − B
is the same as
A + (−B)
. Computers can find the complement of
a particular number and add it to another number much faster and
easier than attempting to create separate subtraction circuits. Therefore,
subtraction is normally carried out by adding the complement of the
subtrahend to the minuend.
3.2.4.1 Decimal
This method is
commonly used by
stage performers who
can subtract large
numbers in their
heads. While it seems
somewhat
convoluted, it is
fairly easy to master.
It is possible to subtract two decimal numbers by adding the nines
complement, as in the following example:
735
-142
Calculate the nines complement of the subtrahend:
857
(that is
9 − 1
,
9 − 4, and 9 − 2). Then, add that nines complement to the minuend:
735
+857
1592
The initial one in the sum (the thousands place) is dropped so the
number of places in the answer is the same as for the two addends,
leaving
592
. Because the diminished radix used to create the subtra-
hend is one less than the radix, one must be added to the answer;
giving 593, which is the correct answer for 735 − 142.
44 binary arithmetic operations
3.2.4.2 Binary
The diminished radix complement (or ones complement) of a binary
number is found by simply “flipping” each bit. Thus, the ones com-
plement of
11010
is
00101
. Just as in decimal, a binary number can be
subtracted from another by adding the diminished radix complement
of the subtrahend to the minuend, and then adding one to the sum.
Here is an example:
101001
-011011
Add the ones complement of the subtrahend:
101001
+100100
1001101
The most significant bit is discarded so the solution has the same
number of bits as for the two addends. This leaves
001101
2
and adding
one to that number (because the diminished radix is one less than the
radix) leaves 1110
2
. In decimal, the problem is 41 − 27 = 14.
Often, diminished radix subtraction circuits are created such that
they use end around carry bits. In this case, the most significant bit is
carried around and added to the final sum. If that bit is one, then that
increases the final answer by one, and the answer is a positive number.
If, though, the most significant bit is zero, then there is no end around
carry so the answer is negative and must be complemented to find the
true value. Either way, the correct answer is found.
Here is an example:
0110 (6)
-0010 (2)
Solution:
0110 (6)
+1101 (-2 in ones complement)
10011
1 (End-around carry the MSB)
=0100 (4)
Answer: 4 (since there was an end-around carry the solution is a
positive number). Here is a second example:
0010 (2)
-0110 (6)
Solution:
3.2 binary subtraction 45
0010 (2)
+1001 (-6 in ones complement)
1011 (No end-around carry, so ones complement)
=0100 (-4: no end-around carry so negative answer)
Because the diminished radix (or ones) complement of a binary
number includes that awkward problem of having two representations
for zero, this form of subtraction is not used in digital circuits; instead,
the radix (or twos) complement is used (this process is discussed next).
It is worth noting that subtracting by adding the diminished radix
of the subtrahend and then adding one is awkward for humans, but
complementing and adding is a snap for digital circuits. In fact, many
early mechanical calculators used a system of adding complements
rather than having to turn gears backwards for subtraction.
3.2.5 Subtracting Using the Radix Complement
3.2.5.1 Decimal
The radix (or tens) complement of a decimal number is the nines
complement plus one. Thus, the tens complement of
7
is
3
; or (
(9 − 7) +
1
) and the tens complement of
248
is
752
(find the nines complement
of each place and then add one to the complete number:
751 + 1
). It is
possible to subtract two decimal numbers using the tens complement,
as in the following example:
735
-142
Calculate the tens complement of the subtrahend,
142
, by finding the
nines complement for each digit and then adding one to the complete
number:
858
(that is
9 − 1
,
9 − 4
, and
9 − 2 + 1
). Then, add that tens
complement number to the original minuend:
735
+858
1593
The initial one in the answer (the thousands place) is dropped so
the answer has the same number of decimal places as the addends,
leaving 593, which is the correct answer for 735 − 142.
3.2.5.2 Binary
To find the radix (or twos) complement of a binary number, each bit
in the number is “flipped” (making the ones complement) and then
one is added to the result. Thus, the twos complement of
11010
2
is
00110
2
(or
(00101
2
) + 1
2
). Just as in decimal, a binary number can
46 binary arithmetic operations
be subtracted from another by adding the radix complement of the
subtrahend to the minuend. Here’s an example:
101001
-011011
Add the twos complement of the subtrahend:
101001
+011011
1001110
The most significant bit is discarded so the solution has the same
number of bits as for the two addends. This leaves
001110
2
(or
14
10
).
Converting all of this to decimal, the original problem is
41 − 27 = 14
.
Here are two worked out examples:
Calculate 0110
2
− 0010
2
(or 6
10
− 2
10
):
0110 (6)
-0010 (2)
Solution:
0110 (6)
+1110 (-2 in twos complement)
10100 (Discard the MSB, the answer is 4)
Calculate 0010
2
− 0110
2
(or 2
10
− 6
10
)
0010 (2)
-0110 (6)
Solution:
0010 (2)
+1010 (-6 in twos complement)
1100
0100 (Twos complement of the sum, -4)
3.2.6 Overflow
One caveat with signed binary numbers is that of overflow, where the
answer to an addition or subtraction problem exceeds the magnitude
which can be represented with the allotted number of bits. Remember
that the sign bit is defined as the most significant bit in the number.
For example, with a six-bit number, five bits are used for magnitude,
so there is a range from
00000
2
to
11111
2
, or
0
10
to
31
10
. If a sign bit
is included, and using twos complement, numbers as high as
011111
2
(
+31
10
) or as low as
100000
2
(
−32
10
) are possible. However, an addi-
tion problem with two signed six-bit numbers that results in a sum
greater than
+31
10
or less than
−32
10
will yield an incorrect answer.
As an example, add 17
10
and 19
10
with signed six-bit numbers:
3.2 binary subtraction 47
010001 (17)
+010011 (19)
100100
The answer (
100100
2
), interpreted with the most significant bit as a
sign, is equal to
−28
10
, not
+36
10
as expected. Obviously, this is not
correct. The problem lies in the restrictions of a six-bit number field.
Since the true sum (
36
) exceeds the allowable limit for our designated
bit field (five magnitude bits, or
+31
), it produces what is called an
overflow error. Simply put, six places is not large enough to represent
the correct sum if the MSB is being used as a sign bit, so whatever
sum is obtained will be incorrect. A similar error will occur if two
negative numbers are added together to produce a sum that is too
small for a six-bit binary field. As an example, add
−17
10
and
−19
10
:
-17 = 101111
-19 = 101101
101111 (-17)
+101101 (-19)
1011100
The solution as shown:
011100
2
=
+28
10
. (Remember that the most
significant bit is dropped in order for the answer to have the same
number of places as the two addends.) The calculated (incorrect)
answer for this addition problem is
28
because true sum of
−17 + −19
was too small to be properly represented with a five bit magnitude
field.
Here is the same overflow problem again, but expanding the bit
field to six magnitude bits plus a seventh sign bit. In the following
example, both
17 + 19
and
(−17) + (−19)
are calculated to show that
both can be solved using a seven-bit field rather than six-bits.
Add 17 + 19:
0010001 (17)
+0010011 (19)
0100100 (36)
Add (-17) + (-19):
-17 = 1101111
-19 = 1101101
1101111 (-17)
+1101101 (-19)
11011100 (-36)
The correct answer is only found by using bit fields sufficiently large
to handle the magnitude and sign bits in the sum.
48 binary arithmetic operations
3.2.6.1 Error Detection
Overflow errors in the above problems were detected by checking the
problem in decimal form and then comparing the results with the
binary answers calculated. For example, when adding
+17
and
+19
,
the answer was supposed to be
+36
, so when the binary sum was
−28
,
something had to be wrong. Although this is a valid way of detecting
overflow errors, it is not very efficient, especially for computers. After
all, the whole idea is to reliably add binary numbers together and not
have to double-check the result by adding the same numbers together
in decimal form. This is especially true when building logic circuits
to add binary quantities: the circuit must detect an overflow error
without the supervision of a human who already knows the correct
answer.
The simplest way to detect overflow errors is to check the sign of
the sum and compare it to the signs of the addends. Obviously, two
positive numbers added together will give a positive sum and two
negative numbers added together will give a negative sum. With an
overflow error, however, the sign of the sum is always opposite that of
the two addends:
(+17) + (+19) = −28
and
(−17) + (−19) = +28
. By
checking the sign bits an overflow error can be detected.
It is not possible to generate an overflow error when the two ad-
dends have opposite signs. The reason for this is apparent when the
nature of overflow is considered. Overflow occurs when the magni-
tude of a number exceeds the range allowed by the size of the bit field.
If a positive number is added to a negative number then the sum will
always be closer to zero than either of the two added numbers; its
magnitude must be less than the magnitude of either original number,
so overflow is impossible.
3.3 binary multiplication
3.3.1 Multiplying Unsigned Numbers
Multiplying binary numbers is very similar to multiplying decimal
numbers. There are only four entries in the Binary Multiplication
Table:
0X0 = 0
0X1 = 0
1X0 = 0
1X1 = 1
Table 3.7: Binary Multiplication Table
3.4 binary division 49
To multiply two binary numbers, work through the multiplier one
number at a time (right-to-left) and if that number is one, then shift
left and copy the multiplicand as a partial product; if that number is
zero, then shift left but do not copy the multiplicand (zeros can be
used as placeholders if desired). When the multiplying is completed
add all partial products. This sounds much more complicated than it
actually is in practice and is the same process that is used to multiply
two decimal numbers. Here is an example problem.
1011 (11)
X 1101 (13)
1011
0000
1011
1011
10001111 (143)
3.3.2 Multiplying Signed Numbers
The simplest method used to multiply two numbers where one or
both are negative is to use the same technique that is used for decimal
numbers: multiply the two numbers and then determine the sign from
the signs of the original numbers: if those signs are the same then
the result is positive, if they are different then the result is negative.
Multiplication by zero is a special case where the result is always zero.
The multiplication method discussed above works fine for paper-
and-pencil; but is not appropriate for designing binary circuits. Unfor-
tunately, the mathematics for binary multiplication using an algorithm
that can become an electronic circuit is beyond the scope of this book.
Fortunately, though, ICs already exist that carry out multiplication
of both signed and floating-point numbers, so a circuit designer can
use a pre-designed circuit and not worry about the complexity of the
multiplication process.
3.4 binary division
Binary division is accomplished by repeated subtraction and a right
shift function; the reverse of multiplication. The actual process is rather
convoluted and complex and is not covered in this book. Fortunately,
though, ICs already exist that carry out division of both signed and
floating-point numbers, so a circuit designer can use a pre-designed
circuit and not worry about the complexity of the division process.
50 binary arithmetic operations
3.5 bitwise operations
It is sometimes desirable to find the value of a given bit in a byte. For
example, if the LSB is zero then the number is even, but if it is one
then the number is odd. To determine the “evenness” of a number, a
bitwise mask is multiplied with the original number. As an example,
imagine that it is desired to know if 1001010
2
is even, then:
1001010 <- Original Number
BitX 0000001 <- "Evenness" Mask
0000000
The bits are multiplied one position at a time, from left-to-right.
Any time a zero appears in the mask that bit position in the product
will be zero since any number multiplied by zero yields zero. When
a one appears in the mask, then the bit in the original number will
be copied to the solution. In the given example, the zero in the least
significant bit of the top number is multiplied with one and the result
is zero. If that LSB in the top number had been one then the LSB in
the result would have also been one. Therefore, an “even” original
number would yield a result of all zeros while an odd number would
yield a one.
3.6 codes
3.6.1 Introduction
Codes are nothing more than using one system of symbols to represent
another system of symbols or information. Humans have used codes
to encrypt secret information from ancient times. However, digital
logic codes have nothing to do with secrets; rather, they are only
concerned with the efficient storage, retrieval, and use of information.
3.6.1.1 Morse Code
As an example of a familiar code, Morse code changes letters to electric
pulses that can be easily transmitted over a radio or telegraph wire.
Samuel Morse’s code uses a series of dots and dashes to represent
letters so an operator at one end of the wire can use electromagnetic
pulses to send a message to some receiver at a distant end. Most
people are familiar with at least one phrase in Morse code: SOS. Here
is a short sentence in Morse:
-.-. --- -.. . ... .- .-. . ..-. ..- -.
c o d e s a r e f u n
3.6 codes 51
3.6.1.2 Braille Alphabet
As one other example of a commonly-used code, in
1834
Louis Braille,
at the age of
15
, created a code of raised dots that enable blind people
to read books. For those interested in this code, the Braille alphabet
can be found at http://braillebug.afb.org/braille print.asp.
3.6.2 Computer Codes
The fact is, computers can only work with binary numbers; that is
how information is stored in memory, how it is processed by the CPU,
how it is transmitted over a network, and how it is manipulated in
any of a hundred different ways. It all boils down to binary numbers.
However, humans generally want a computer to work with words
(such as email or a word processor), ciphers (such as a spreadsheet),
or graphics (such as photos). All of that information must be encoded
into binary numbers for the computer and then decoded back into
understandable information for humans. Thus, binary numbers stored
in a computer are often codes used to represent letters, programming
steps, or other non-numeric information.
3.6.2.1 ASCII
Computers must be able to store and process letters, like those on this
page. At first, it would seem easiest to create a code by simply making
A=1
,
B=2
, and so forth. While this simple code does not work for a
number of reasons, the idea is on the right track and the code that is
actually used for letters is similar to this simple example.
In the early
1960
s, computer scientists came up with a code they
named American Standard Code for Information Interchange (ASCII)
and this is still among the most common ways to encode letters and
other symbols for a computer. If the computer program knows that a
particular spot in memory contains binary numbers that are actually
ASCII-coded letters, it is a fairly easy job to convert those codes to
letters for a screen display. For simplicity, ASCII is usually represented
by hexadecimal numbers rather than binary. For example, the word
Hello in ASCII is: 048 065 06C 06C 06F.
ASCII code also has a predictable relationship between letters. For
example, capital letters are exactly
20
16
higher in ASCII than their
lower-case version. Thus, to change a letter from lower-case to upper-
case, a programmer can add
20
16
to the ASCII code for the lower-case
letter. This can be done in a single processing step by using what is
known as a bit-wise AND on the bit representing
20
16
in the ASCII
code’s binary number.
An ASCII chart using hexadecimal values is presented in Table 3.8.
The most significant digit is read across the top row and the least
52 binary arithmetic operations
significant digit is read down the left column. For example, the letter
A is 41
16
and the number 6 is 36
16
.
0 1 2 3 4 5 6 7
0 NUL DLE 0 @ P ’ p
1 SOH DC1 ! 1 A Q a q
2 STX DC2 ” 2 B R b r
3 ETX DC3 # 3 C S c s
4 EOT DC4 4 D T d t
5 ENQ NAK % 5 E U e u
6 ACK SYN & 6 F V f v
7 BEL ETB ’ 7 G W g w
8 BS CAN ( 8 H X h x
9 HT EM ) 9 I Y i y
A LF SUB * : J Z j z
B VT ESC + ; K [ k {
C FF FS , < L \ l |
D CR GS - = M ] m }
E SO RS . > N ∧ n ∼
F SI US / ? O o DEL
Table 3.8: ASCII Table
Teletype operators
from decades past tell
stories of sending 25
or more 07
16
codes
(ring the bell) to a
receiving terminal
just to irritate
another operator in
the middle of the
night.
ASCII
20
16
is a space character used to separate words in a message
and the first two columns of ASCII codes (where the high-order nibble
are zero and one) were codes essential for teletype machines, which
were common from the
1920
s until the
1970
s. The meanings of a few
of those special codes are given in Table 3.9.
NUL All Zeros (a “null” byte)
SOH Start of Header
STX Start of Text
ETX End of Text
EOT End of Transmission
ENQ Enquire (is the remote station on?)
ACK Acknowledge (the station is on)
BEL Ring the terminal bell (get the operator’s attention)
Table 3.9: ASCII Symbols
3.6 codes 53
Table 3.10 contains a few phrases in both plain text and ASCII for
practice.
Plain Text ASCII
codes are fun 63 6f 64 65 73 20 61 72 65 20 66 75 6e
This is ASCII 54 68 69 73 20 69 73 20 41 53 43 49 49
365.25 days 33 36 35 2e 32 35 20 64 61 79 73
It’s a gr8 day! 49 74 27 73 20 61 20 67 72 38 20 64 61 79 21
Table 3.10: ASCII Practice
While the ASCII code is the most commonly used text representa-
tion, it is certainly not the only way to encode words. Another popular
code is Extended Binary Coded Decimal Interchange Code (EBCDIC)
(pronounced like “Eb See Deck”), which was invented by IBM in
1963
and has been used in most of their computers ever since.
Since the early 2000’s, computer programs have begun to use Uni-
code character sets, which are similar to ASCII but multiple bytes
are combined to expand the number of characters available for non-
English languages like Cyrillic.
3.6.2.2 Binary Coded Decimal (BCD)
It is often desirable to have numbers coded in such a way that they
can be easily translated back and forth between decimal (which is easy
for humans to manipulate) and binary (which is easy for computers
to manipulate). BCD is the code used to represent decimal numbers
in binary systems. BCD is useful when working with decimal input
(keypads or transducers) and output (displays) devices.
There are, in general, two types of BCD systems: non-weighted and
weighted. Non-weighted codes are special codes devised for a single
purpose where there is no implied relationship between one value
and the next. As an example,
1001
could mean one and
1100
could
mean two in some device. The circuit designer would create whatever
code meaning is desired for the application.
Weighted BCD is a more generalized system where each bit position
is assigned a “weight,” or value. These types of BCD systems are
far more common than non-weighted and are found in all sorts of
applications. Weighted BCD codes can be converted to decimal by
adding the place value for each position in exactly the same way that
Expanded Positional Notation is used for to covert between decimal
and binary numbers. As an example, the weights for the Natural BCD
system are
8 − 4 − 2 − 1
. (These are the same weights used for binary
numbers; thus the name “natural” for this system.) The code
1001
BCD
is converted to decimal like this:
54 binary arithmetic operations
1001
BCD
= (1X8) + (0X4) + (0X2) + (1X1) (3.1)
= (8) + (0) + (0) + (1)
= 9
10
Because there are ten decimal ciphers (
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
), it re-
quires four bits to represent all decimal digits; so most BCD code
systems are four bits wide. In practice, only a few different weighted
BCD code systems are commonly used and the most common are
shown in Table 3.11.
Decimal 8421 (Natural) 2421 Ex3 5421
0 0000 0000 0011 0000
1 0001 0001 0100 0001
2 0010 0010 0101 0010
3 0011 0011 0110 0011
4 0100 0100 0111 0100
5 0101 1011 1000 1000
6 0110 1100 1001 1001
7 0111 1101 1010 1010
8 1000 1110 1011 1011
9 1001 1111 1100 1100
Table 3.11: BCD Systems
Remember that BCD
is a code system, not
a number system; so
the meaning of each
combination of
four-bit codes is up
to the designer and
will not necessarily
follow any sort of
binary numbering
sequence.
The name of each type of BCD code indicates the various place
values. Thus, the
2421
BCD system gives the most significant bit of
the number a value of two, not eight as in the natural code. The Ex3
code (for “Excess 3”) is the same as the natural code, but each value is
increased by three (that is, three is added to the natural code).
In each of the BCD code systems in Table 3.11 there are six unused
four-bit combinations; for example, in the Natural system the unused
codes are:
1010
,
1011
,
1100
,
1101
,
1110
, and
1111
. Thus, any circuit de-
signed to use BCD must include some sort of check to ensure that if
unused binary values are accidentally input into a circuit it does not
create an undefined outcome.
Normally, two BCD codes, each of which are four bits wide, are
packed into an eight-bit byte in order to reduce wasted computer
memory. Thus, the packed BCD
01110010
contains two BCD numbers:
72
. In fact, a single 32-bit word, which is common in many computers,
can contain
8
BCD codes. It is a trivial matter for software to either
pack or unpack BCD codes from a longer word.
It is natural to wonder why there are so many different ways to
code decimal numbers. Each of the BCD systems shown in Table 3.11
3.6 codes 55
has certain strengths and weaknesses and a circuit designer would
choose a specific system based upon those characteristics.
converting between bcd and other systems. One thing
that makes BCD so useful is the ease of converting from BCD to
decimal. Each decimal digit is converted into a four-bit BCD code, one
at a time. Here is 37
10
in Natural BCD:
0011 0111
3 7
It is, generally, very easy to convert Natural BCD to decimal since
the BCD codes are the same as binary numbers. Other BCD systems
use different place values, and those require more thought to convert
(though the process is the same). The place values for BCD systems
other than Natural are indicated in the name of the system; so, for
example, the
5421
system would interpret the number
1001
BCD5421
as:
1001
BCD5421
= (1X5) + (0X4) + (0X2) + (1X1) (3.2)
= (5) + (0) + (0) + (1)
= 6
10
Converting from decimal to BCD is also a rather simple process.
Each decimal digit is converted to a four-bit BCD equivalent. In the
case of Natural BCD the four-bit code is the binary equivalent to the
decimal number, other weighted BCD codes would be converted with
a similar process.
2 4 5
0010 0100 0101
To convert binary to BCD is no trivial exercise and is best done
with an automated process. The normal method used is called the
Shift Left and Add Three algorithm (or, frequently, Double-Dabble). The
process involves a number of steps where the binary number is shifted
left and occasionally three is added to the resulting shift. Wikipedia
(
https://en.wikipedia.org/wiki/Double dabble
) has a good explanation
of this process, along with some examples.
Converting BCD to any other system (like hexadecimal) is most
easily done by first converting to binary and then to whatever base is
desired. Unfortunately, converting BCD to binary is not as simple as
concatenating two BCD numbers; for example,
01000001
is 41 in BCD,
but those two BCD numbers concatenated,
01000001
, is
65
in binary.
One way to approach this type of problem is to use the reverse of the
Double-Dabble process: Shift Right and Subtract Three. As in converting
binary to BCD, this is most easily handled by an automated process.
56 binary arithmetic operations
self-complementing. The Excess-3 code (called Ex3 in the table)
is self-complementing; that is, the nines complement of any decimal
number is found by complementing each bit in the Ex3 code. As an
example, find the nines complement for 127
10
:
1 127
10
Original Number
2 0100 0101 1010
Ex3
Convert 127 to Excess 3
3 1011 1010 0101
Ex3
Ones Complement
4 872
10
Convert to Decimal
Table 3.12: Nines Complement for 127
Thus,
872
10
, is the nines complement of
127
10
. It is a powerful
feature to be able to find the nines complement of a decimal number
by simply complementing each bit of its Ex3 BCD representation.
reflexive. Some BCD codes exhibit a reflexive property where
each of the upper five codes are complementary reflections of the
lower five codes. For example,
0111
Ex3
(4) and
1000
Ex3
(5) are com-
plements,
0110
Ex3
(3) and
1001
Ex3
(6) are complements, and so forth.
The reflexive property for the
5421
code is different. Notice that the
codes for zero through four are the same as those for five through
nine, except for the MSB (zero for the lower codes, one for the upper
codes). Thus,
0000
5421
(zero) is the same as
1000
5421
(five) except for
the first bit,
0001
5421
(one) is the same as
1001
5421
(six) except for the
first bit, and so forth. Studying Table 3.11 should reveal the various
reflexive patterns found in these codes.
practice. Table 3.13 shows several decimal numbers in various
BCD systems which can be used for practice in converting between
these number systems.
Dec 8421 2421 Ex3 5421
57 0101 1110 10111 1101 1000 1010 1000 1010
79 0111 1001 1101 1111 1010 1100 1010 1100
28 0010 1000 0010 1110 0101 1011 0010 1011
421 0100 0010 0001 0100 0010 0001 0111 0101 0100 0100 0010 0001
903 1001 0000 0011 1111 0000 0011 1100 0011 0110 1100 0000 0011
Table 3.13: BCD Practice
adding bcd numbers. BCD numbers can be added in either of
two ways. Probably the simplest is to convert the BCD numbers to
binary, add them as binary numbers, and then convert the sum back
3.6 codes 57
to BCD. However, it is possible to add two BCD numbers without
converting. When two BCD numbers are added such that the result is
less than ten, then the addition is the same as for binary numbers:
0101 (5)
+0010 (2)
0111 (7)
However, four-bit binary numbers greater than
1001
2
(that is:
9
10
)
are invalid BCD codes, so adding two BCD numbers where the result
is greater than nine requires a bit more effort:
0111 (7)
+0101 (5)
1100 (12 -- not valid in BCD)
When the sum is greater than nine, then six must be added to that
result since there are six invalid binary codes in BCD.
1100 (12 -- from previous addition)
+0110 (6)
1 0010 (12 in BCD)
When adding two-digit BCD numbers, start with the Least Signifi-
cant Nibble (LSN), the right-most nibble, then add the nibbles with
carry bits from the right. Here are some examples to help clarify this
concept:
0101 0010 (52)
+0011 0110 (36)
1000 1000 (88 in BCD)
0101 0010 (52)
+0101 0110 (56)
1010 1000 (1010, MSN, invalid BCD code)
+0110 0000 (Add 6 to invalid code)
1 0000 1000 (108 in BCD)
0101 0101 (55)
+0101 0110 (56)
1010 1011 (both nibbles invalid BCD code)
+0000 0110 (Add 6 to LSN)
1011 0001 (1 carried over to MSN)
+0110 0000 (Add 6 to MSN)
1 0001 0001 (111 in BSD)
58 binary arithmetic operations
negative numbers. BCD codes do not have any way to store
negative numbers, so a sign nibble must be used. One approach to
this problem is to use a sign-and-magnitude value where a sign nibble
is prefixed onto the BCD value. By convention, a sign nibble of
0000
makes the BCD number positive while
1001
makes it negative. Thus,
the BCD number 000000100111 is 27, but 100100100111 is −27.
A more mathematically rigorous, and useful, method of indicating
negative BCD numbers is use the tens complement of the BCD num-
ber since BCD is a code for decimal numbers, exactly like the twos
complement is used for binary numbers. It may be useful to review
Section 3.2.3.1 on page 39 for information about the tens complement.
In the Natural BCD system the tens complement is found by adding
one to the nines complement, which is found by subtracting each digit
of the original BCD number from nine. Here are some examples to
clarify this concept:
0111 (7 in BCD)
0010 (2, the 9’s complement of 7 since 9-7=2)
0011 (3, the 10’s complement of 7, or 2+1)
0010 0100 (24 in BCD)
0111 0101 (75, the 9’s complement of 24)
0111 0110 (76, the 10’s complement of 24)
Also, some BCD code systems are designed to easily create the
tens complement of a number. For example, in the
2421
BCD system
the tens complement is found by nothing more than inverting the
MSB. Thus, three is the tens complement of seven and in the
2421
BCD system
0011
BCD2421
is the tens complement of
1101
BCD2421
, a
difference of only the MSB. Therefore, designers creating circuits that
must work with negative BCD numbers may opt to use the
2421
BCD
system.
subtracting bcd numbers. A BCD number can be subtracted
from another by changing it to a negative number and adding. Just
like in decimal,
5 − 2
is the same as
5 + (−2)
. Either a nines or tens
complement can be used to change a BCD number to its negative, but
for this book, the tens complement will be used. If there is a carry-out
bit then it can be ignored and the result is positive, but if there is
no carry-out bit then answer is negative so the magnitude must be
found by finding the tens complement of the calculated sum. Compare
this process with subtracting regular binary numbers. Here are a few
examples:
3.6 codes 59
7 - 3 = 4
0111 (7 in BCD)
+0111 (add the 10’s complement of 3)
1110 (invalid BCD code)
+0110 (add 6 to invalid BCD code)
1 0100 (4 - drop the carry bit)
7 - 9 = -2
0111 (7 in BCD)
+0001 (10’s complement of 9)
1000 (valid BCD code)
0010 (10’s complement)
32 - 15 = 17
0011 0010 (32 in BCD)
+1000 0101 (10’s complement of 15)
1011 0111 (MSB is invalid BCD code)
+0110 0000 (add 6 to MSB)
1 0001 0111 (17, drop the carry bit)
427 - 640 = -213
0100 0010 0111 (427 in BCD)
+0011 0110 0000 (10’s complement of 640)
0111 1000 0111 (no invalid BCD code)
0010 0001 0011 (10’s complement)
369 - 532 = -163
0011 0110 1001 (369 in BCD)
+0100 0110 1000 (10’s complement of 532)
0111 1100 0001 (two invalid BCD codes)
+0000 0110 0110 (add 6 to invalid codes)
1000 0011 0111 (sum)
0001 0110 0011 (10’s complement)
Here are some notes on the last example: adding the LSN yields
1001 + 1000 = 10001
. The initial one is ignored, but this is an invalid
BCD code so this byte needs to be corrected by adding six to it. Then
the result of that addition includes an understood carry into the next
nibble after the correction is applied. In the same way, the middle
nibble was corrected:
1100 + 0110 = 10011
but the initial one in this
60 binary arithmetic operations
answer is carried to the Most Significant Nibble (MSN) and added
there.
3.6.2.3 Gray Code
Figure 3.1:
Optical
Disc
It is often desirable to use a wheel to encode
digital input for a circuit. As an example, con-
sider the tuning knob on a radio. The knob is
attached to a shaft that has a small, clear disk
which is etched with a code, similar to Figure
3.1. As the disk turns, the etched patterns pass
or block a laser beam from reaching an optical
sensor, and that pass/block pattern is encoded
into binary input.
One of the most challenging aspects of using
a mechanical device to encode binary is ensur-
ing that the input is stable. As the wheel turns past the light beam, if
two of the etched areas change at the same time (thus, changing two
bits at once), it is certain that the input will fluctuate between those two
values for a tiny, but significant, period of time. For example, imagine
that the encoded circuit changes from
1111
to
0000
at one time. Since
it is impossible to create a mechanical wheel precise enough to change
those bits at exactly the same moment in time (remember that the light
sensors will “see” an input several million times a second), as the bits
change from
1111
to
0000
they may also change to
1000
or
0100
or any
of dozens of other possible combinations for a few microseconds. The
entire change may form a pattern like
1111 − 0110 − 0010 − 0000
and
that instability would be enough to create havoc in a digital circuit.
The solution to the stability problem is to etch the disk with a
code designed in such a way that only one bit changes at a time. The
code used for that task is the Gray code. Additionally, a Gray code
is cyclic, so when it reaches its maximum value it can cycle back to
its minimum value by changing only a single bit. In Figure 3.1, each
of the concentric rings encodes one bit in a four-bit number. Imagine
that the disk is rotating past the fixed laser beam reader —the black
areas (“blocked light beam”) change only one bit at a time, which is
characteristic of a Gray code pattern.
It is fairly easy to create a Gray code from scratch. Start by writing
two bits, a zero and one:
0
1
Then, reflect those bits by writing them in reverse order underneath
the original bits:
0
1
3.6 codes 61
-----
1
0
Next, prefix the top half of the group with a zero and the bottom
half with a one to get a two-bit Gray code.
00
01
11
10 (2-bit Gray code)
Now, reflect all four values of the two-bit Gray code.
00
01
11
10
------
10
11
01
00
Next, prefix the top half of the group with a zero and the bottom
half with a one to get a three-bit Gray code.
000
001
011
010
110
111
101
100 (3-bit Gray code)
Now, reflect all eight values of a three-bit Gray code.
000
001
011
010
110
111
101
100
-------
100
62 binary arithmetic operations
101
111
110
010
011
001
000
Finally, prefix the top half of the group with a zero and the bottom
half with a one to get a four-bit Gray code.
0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000 (four-bit Gray code)
The process of reflecting and prefixing can continue indefinitely
to create a Gray code of any desired bit length. Of course, Gray
code tables are also available in many different bit lengths. Table 3.14
contains a two-bit, three-bit, and four-bit Gray code:
3.6 codes 63
2-Bit Code 3-Bit Code 4-Bit Code
00 000 0000
01 001 0001
11 011 0011
10 010 0010
110 0110
111 0111
101 0101
100 0100
1100
1101
1111
1110
1010
1011
1001
1000
Table 3.14: Gray Codes
4
B O O L E A N F U N C T I O N S
What To Expect
In 1854, George Boole introduced an algebra system designed
to work with binary (or base 2) numbers, but he could not have
foreseen the immense impact his work would have on systems
like telephony and computer science. This chapter includes the
following topics.
•
Developing and using the primary Logic Functions: AND,
OR, NOT
•
Developing and using the secondary Logic Functions:
XOR, XNOR, NAND, NOR, Buffer
•
Developing and using the univariate Boolean Algebra
Properties: Identity, Idempotence, Annihilator, Comple-
ment, Involution
•
Developing and using the multivariate Boolean Algebra
Properties: Commutative, Associative, Distributive, Ab-
sorption, Adjacency
• Analyzing circuits using DeMorgan’s Theorem
4.1 introduction to boolean functions
Before starting a study of Boolean functions, it is important to keep in
mind that this mathematical system concerns electronic components
that are capable of only two states: True and False (sometimes called
High-Low or
1
-
0
). Boolean functions are based on evaluating a series
of True-False statements to determine the output of a circuit.
For example, a Boolean expression could be created that would
describe “If the floor is dirty OR company is coming THEN I will
mop.” (The things that I do for visiting company!) These types of logic
statements are often represented symbolically using the symbols
1
and
0
, where
1
stands for True and
0
stands for False. So, let “Floor is dirty”
equal
1
and “Floor is not dirty” equal
0
. Also, let “Company is coming”
equal
1
and “Company is not coming” equal
0
. Then, “Floor is dirty OR
company is coming” can be symbolically represented by
1
OR
1
. Within
the discipline of Boolean algebra, common mathematical symbols are
65
66 boolean functions
used to represent Boolean expressions; for example, Boolean OR is
frequently represented by a mathematics plus sign, as shown below.
0 + 0 = 0 (4.1)
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
These look like addition problems, but they are not (as evidenced
by the last line). It is essential to keep in mind that these are merely
symbolic representations of True-False statements. The first three lines
make perfect sense and look like elementary addition. The last line,
though, violates the principles of addition for real numbers; but it
is a perfectly valid Boolean expression. Remember, in the world of
Boolean algebra, there are only two possible values for any quantity:
1
or
0
; and that last line is actually saying True OR True is True. To
use the dirty floor example from above, “The floor is dirty” (True)
OR “Company is coming” (True) SO “I will mop the floor” (True) is
symbolized by:
1 + 1 = 1
. This could be expressed as T OR T SO T;
but the convention is to use common mathematical symbols; thus:
1 + 1 = 1.
Moreover, it does not matter how many or few terms are OR ed
together; if just one is True, then the output is True, as illustrated below:
1 + 1 + 1 = 1 (4.2)
1 + 0 + 0 + 0 + 0 = 1
1 + 1 + 1 + 1 + 1 + 1 = 1
Next, consider a very simple electronic sensor in an automobile:
IF the headlights are on AND the driver’s door is open THEN a
buzzer will sound. In the same way that the plus sign is used to
mathematically represent OR , a times sign is used to represent AND
. Therefore, using common mathematical symbols, this automobile
alarm circuit would be represented by
1X1 = 1
. The following list
shows all possible states of the headlights and door:
0X0 = 0 (4.3)
0X1 = 0
1X0 = 0
1X1 = 1
The first row above shows “False (the lights are not on) AND False
(the door is not open) results in False (the alarm does not sound)”. For
AND logic, the only time the output is True is when all inputs are also
4.2 primary logic operations 67
True; therefore:
1X1X1X1X0 = 0
. In this way, Boolean AND behaves
somewhat like algebraic multiplication.
Within Boolean algebra’s simple True-False system, there is no equiv-
alent for subtraction or division, so those mathematical symbols are
not used. Like real-number algebra, Boolean algebra uses alphabeti-
cal letters to denote variables; however, Boolean variables are always
CAPITAL letters, never lower-case, and normally only a single letter.
Thus, a Boolean equation would look something like this:
A + B = Y (4.4)
As Boolean expressions are realized (that is, turned into a real, or
physical, circuit), the various operators become gates. For example, the
above equation would be realized using an OR gate with two inputs
(labeled A and B) and one output (labeled Y).
Boolean algebra includes three primary and four secondary logic
operations (plus a Buffer, which has no logic value), six univariate,
and six multivariate properties. All of these will be explored in this
chapter.
4.2 primary logic operations
4.2.1 AND
An AND gate is a Boolean operation that will output a logical one,
or True, only if all of the inputs are True. As an example, consider
this statement: “If I have ten bucks AND there is a good movie at the
cinema, then I will go see the movie.” In this statement, “If I have
ten bucks” is one variable and “there is a good movie at the cinema”
is another variable. If both of these inputs are True, then the output
variable (“I will go see the movie”) will also be True. However, if either
of the two inputs is False, then the output will also be False (or, “I
will not go see the movie”). Of course, if I want popcorn, I would
need another ten spot, but that is not germane to this example. When
written in an equation, the Boolean AND term is represented a number
of different ways. One method is to use the logic AND symbol as found
in Equation ??.
A ∧ B = Y (4.5)
One other method is to use the same symbols that are used for
multiplication in traditional algebra; that is, by writing the variables
next to each other, with parenthesis, or, sometimes, with an asterisk
between them, as in Equation 4.6.
68 boolean functions
AB = Y (4.6)
(A)(B) = Y
A ∗ B = Y
The multiplication
symbols X and •
(dot) are not
commonly used in
digital logic
equations.
Logic AND is normally represented in equations by using an algebra
multiplication symbol since it is easy to type; however, if there is any
chance for ambiguity, then the Logic AND symbol (
∧
) can be used to
differentiate between multiplication and a logic AND function.
Following is the truth table for the AND operator.
Inputs Output
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
Table 4.1: Truth Table for AND
A truth table is used to record all possible inputs and the output
for each combination of inputs. For example, in the first line of Table
4.1, if input
A
is
0
and input
B
is
0
then the output,
Y
, will be
0
. All
possible input combinations are normally formed in a truth table by
counting in binary starting with all variables having a
0
value to all
variables having a
1
value. Thus, in Table 4.1, the inputs are
00
,
01
,
10
,
11
. Notice that the output for the AND operator is False (that is,
0
)
until the last row, when both inputs are True. Therefore, it could be
said that just one False input would inactivate a physical AND gate.
For that reason, an AND operation is sometimes called an inhibitor.
AND Gate Switches
Because a single False input can turn an AND gate off, these
types of gates are frequently used as a switch in a logic circuit.
As a simple example, imagine an assembly line where there are
four different safety sensors of some sort. The sensor outputs
could be routed to a single four-input AND gate and then as
long as all sensors are True the assembly line motor will run.
If, however, any one of those sensors goes False due to some
unsafe condition, then the AND gate would also output a False
and cause the motor to stop.
Logic gates are realized (or created) in electronic circuits by using
transistors, resistors, and other components. These components are
4.2 primary logic operations 69
normally packaged into a single IC “chip,” so the logic circuit designer
does not need to know all of the details of the electronics in order to
use the gate. In logic diagrams, an AND gate is represented by a shape
that looks like a capital D. In Figure 4.1, the input variables
A
and
B
are wired to an AND gate and the output from that gate goes to Y.
Figure 4.1: AND Gate
Notice that each input and output is named in order to make it
easier to describe the circuit algebraically. In reality, AND gates are not
packaged or sold one at a time; rather, several would be placed on a
single IC, like the 7408 Quad AND Gate. The designer would design
the circuit card to use whichever of the four gates are needed while
leaving the unused gates unconnected.
There are two common sets of symbols used to represent the var-
ious elements in logic diagrams, and whichever is used is of little
consequence since the logic is the same. Shaped symbols, as used in
Figure 4.1, are more common in the United States; but the IEEE has its
own symbols which are sometimes used, especially in Europe. Figure
4.2 illustrates the circuit in Figure 4.1 using IEEE symbols: In this book, only
shaped symbols will
be used.
Figure 4.2: AND Gate Using IEEE Symbols
As an example of an AND gate at work, consider an elevator: if the
door is closed (logic
1
) AND someone in the elevator car presses a
floor button (logic
1
), THEN the elevator will move (logic
1
). If both
sensors (door and button) are input to an AND gate, then the elevator
motor will only operate if the door is closed AND someone presses a
floor button.
4.2.2 OR
An OR gate is a Boolean operation that will output a logical one, or
True, if any or all of the inputs are True. As an example, consider this
statement: “If my dog needs a bath OR I am going swimming, then I
will put on a bathing suit.” In this statement, “if my dog needs a bath”
is one input variable and “I am going swimming” is another input
70 boolean functions
variable. If either of these is True, then the output variable, “I will put
on a bathing suit,” will also be True. However, if both of the inputs
are False, then the output will also be False (or, “I will not put on a
bathing suit”). If you think it odd that I would wear a bathing suit to
bathe my dog then you have obviously never met my dog.
When written in an equation, the Boolean OR term is represented a
number of different ways. One method is to use the logic OR symbol,
as found in Equation 4.7.
A ∨ B = Y (4.7)
A more common method is to use the plus sign that is used for
addition in traditional algebra, as in Equation 4.8.
A + B = Y (4.8)
For simplicity, the mathematical plus symbol is normally used to
indicate OR in printed material since it is easy to enter with a keyboard;
however, if there is any chance for ambiguity, then the logic OR symbol
(∨) is used to differentiate between addition and logic OR.
Table 4.2 is the truth table for an OR operation.
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
Table 4.2: Truth Table for OR
Notice for the OR truth table that the output is True (
1
) whenever
at least one input is True. Therefore, it could be said that one True
input would activate an OR Gate. In the following diagram, the input
variables
A
and
B
are wired to an OR gate, and the output from that
gate goes to Y.
Figure 4.3: OR Gate
As an example of an OR gate at work, consider a traffic signal.
Suppose an intersection is set up such that the light for the main road
4.2 primary logic operations 71
is normally green; however, if a car pulls up to the intersection from
the crossroad, or if a pedestrian presses the “cross” button, then the
main light is changed to red to stop traffic. This could be done with a
simple OR gate. An automobile sensor on the crossroad would be one
input and the pedestrian “cross” button would be the other input; the
output of the OR gate connecting these two inputs would change the
light to red when either input is activated.
4.2.3 NOT
NOT (or inverter) is a Boolean operation that inverts the input. That
is, if the input is True then the output will be False or if the input is
False then the output will be True. When written in an equation, the
Boolean NOT operator is represented in many ways, though two are
most popular. The older method is to overline (that is, a line above) a
term, or group of terms, that are to be inverted, as in Equation 4.9. Equation 4.9 is read
A OR B NOT = Q
(notice that when
spoken, the word not
follows the term that
is inverted).
A + B = Y (4.9)
Another method of indicating NOT is to use the algebra prime
indicator, an apostrophe, as in Equation 4.10.
A + B
0
= Y (4.10)
The reason that NOT is most commonly indicated with an apostro-
phe is because that is easier to enter on a computer keyboard. There are
many other ways authors use to represent NOT in a formula, but none
are considered standardized. For example, some authors use an excla-
mation point:
A+!B = Q
, others use a broken line:
A + ¬B = Q
, others
use a backslash:
A + \B = Q
, and still others use a tilde:
A+ ∼ B = Q
.
However, only the apostrophe and overline are consistently used to
indicate NOT. Table 4.3 is the truth table for NOT:
Input Output
0 1
1 0
Table 4.3: Truth Table for NOT
In a logic diagram, NOT is represented by a small triangle with a
“bubble” on the output. In Figure 4.4, the input variable
A
is inverted
by a NOT gate and then sent to output Y.
72 boolean functions
Figure 4.4: NOT Gate
4.3 secondary logic functions
4.3.1 NAND
NAND is a Boolean operation that outputs the opposite of AND, that
is, NOT AND; thus, it will output a logic False only if all of the inputs
are True. The NAND operation is not often used in Boolean equations,
but when necessary it is represented by a vertical line. Equation 4.11
shows a NAND operation.
A|B = Y (4.11)
Table 4.4 is the Truth Table for a NAND gate.
Inputs Output
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
Table 4.4: Truth Table for NAND Gate
In Figure 4.5, the input variables
A
and
B
are wired to a NAND
gate, and the output from that gate goes to Y.
Figure 4.5: NAND Gate
Inverting bubbles are
never found by
themselves on a wire;
they are always
associated with either
the inputs or output
of a logic gate. To
invert a signal on a
wire, a NOT gate is
used.
The logic diagram symbol for a NAND gate looks like an AND
gate, but with a small bubble on the output port. A bubble in a logic
diagram always represents some sort of signal inversion, and it can
appear at the inputs or outputs of nearly any logic gate. For example,
the bubble on a NAND gate could be interpreted as “take whatever
the output would be generated by an AND gate—then invert it.”
4.3 secondary logic functions 73
4.3.2 NOR
NOR is a Boolean operation that is the opposite of OR, that is, NOT
OR; thus, it will output a logic True only if all of the inputs are False.
The NOR operation is not often used in Boolean equations, but when
necessary it is represented by a downward-pointing arrow. Equation
4.12 shows a NOR operation.
A ↓ B = Y (4.12)
Table 4.5 is the truth table for NOR .
Inputs Output
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
Table 4.5: Truth Table for NOR
In Figure 4.6, the input variables
A
and
B
are wired to a NOR gate,
and the output from that gate goes to Y.
Figure 4.6: NOR Gate
4.3.3 XOR
XOR (Exclusive OR) is a Boolean operation that outputs a logical one, or
True, only if the two inputs are different. This is useful for circuits that
compare inputs; if they are different then the output is True, otherwise
it is False. Because of this, an XOR gate is sometimes referred to as
a Difference Gate. The XOR operation is not often used in Boolean
equations, but when necessary it is represented by a plus sign (like the
OR function) inside a circle. Equation 4.13 shows an XOR operation.
A ⊕ B = Y (4.13)
Table 4.6 is the truth table for an XOR gate.
74 boolean functions
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
Table 4.6: Truth Table for XOR
In Figure 4.7, the input variables
A
and
B
are wired to an XOR gate,
and the output from that gate goes to Y.
Figure 4.7: XOR Gate
There is some debate about the proper behavior of an XOR gate
that has more than two inputs. Some experts believe that an XOR gate
should output a True if one, and only one, input is True regardless of
the number of inputs. This would seem to be in keeping with the rules
of digital logic developed by George Boole and other early logisticians
and is the strict definition of XOR promulgated by the IEEE. This
is also the behavior of the XOR gate found in Logisim-evolution, the
digital logic simulator used in the lab manual accompanying this text.
Others believe, though, that an XOR gate should output a True if an
odd number of inputs is True. in Logisim-evolution this type of behavior
is found in a device called a “parity gate” and is covered in more
detail elsewhere in this book.
4.3.4 XNOR
XNOR is a Boolean operation that will output a logical one, or True,
only if the two inputs are the same; thus, an XNOR gate is often
referred to as an Equivalence Gate. The XNOR operation is not often
used in Boolean equations, but when necessary it is represented by a
dot inside a circle. Equation 4.14 shows an XNOR operation.
A B = Y (4.14)
Table 4.7 is the truth table for XNOR .
4.3 secondary logic functions 75
Inputs Output
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
Table 4.7: Truth Table for XNOR
In Figure 4.8, the input variables
A
and
B
are wired to an XNOR
gate, and the output from that gate goes to Y.
Figure 4.8: XNOR Gate
4.3.5 Buffer
A buffer (sometimes called Transfer) is a Boolean operation that trans-
fers the input to the output without change. If the input is True, then
the output will be True and if the input is False, then the output will
be False. It may seem to be an odd function since this operation does
not change anything, but it has an important use in a circuit. As logic
circuits become more complex, the signal from input to output may
become weak and no longer able to drive (or activate) additional gates.
A buffer is used to boost (and stabilize) a logic level so it is more
dependable. Another important function for a buffer is to clean up
an input signal. As an example, when an electronic circuit interacts
with the physical world (such as a user pushing a button), there is
often a very brief period when the signal from that physical device
waivers between high and low unpredictably. A buffer can smooth
out that signal so it is a constant high or low without voltage spikes
in between.
Table 4.8 is the truth table for buffer.
Input Output
0 0
1 1
Table 4.8: Truth Table for a Buffer
76 boolean functions
Buffers are rarely used in schematic diagrams since they do not
actually change a signal; however, Figure 4.9, illustrates a buffer.
Figure 4.9: Buffer
4.4 univariate boolean algebra properties
4.4.1 Introduction
Boolean Algebra, like real number algebra, includes a number of
properties. This unit introduces the univariate Boolean properties, or
those properties that involve only one input variable. These properties
permit Boolean expressions to be simplified, and circuit designers
are interested in simplifying circuits to reduce construction expense,
power consumption, heat loss (wasted energy), and troubleshooting
time.
4.4.2 Identity
In mathematics, an identity is an equality where the left and right
members are the same regardless of the values of the variables present.
As an example, Equation 4.15 is an identity since the two members
are identical regardless of the value of α:
α
2
= 0.5α (4.15)
An Identity Element is a special member of a set such that when
that element is used in a binary operation the other element in that
operation is not changed. This is sometimes called the Neutral Element
since it has no effect on binary operations. As an example, in Equation
4.16 the two members of the equation are always identical. Therefore,
zero is the identity element for addition since anything added to zero
remains unchanged.
a + 0 = a (4.16)
In a logic circuit, combining any logic input with a logic zero
through an OR gate yields the original input. Logic zero, then, is
considered the OR identity element because it causes the input of the
gate to be copied to the output unchanged. Because OR is represented
by a plus sign when written in a Boolean equation, and the identity
element for OR is zero, Equation 4.17 is True.
4.4 univariate boolean algebra properties 77
A + 0 = A (4.17)
The bottom input to the OR gate in 4.10 is a constant logic zero,
or False. The output for this circuit,
Y
, will be the same as input
A
;
therefore, the identity element for OR is zero.
Figure 4.10: OR Identity Element
In the same way, combining any logic input with a logic one through
an AND gate yields the original input. Logic one, then, is considered
the AND identity element because it causes the input of the gate to
be copied to the output unchanged. Because AND is represented by
a multiplication sign when written in a Boolean equation, and the
identity element for AND is one, Equation 4.18 is True.
A ∗ 1 = A (4.18)
The bottom input to the AND gate in 4.11 is a constant logic one,
or True. The output for this circuit,
Y
, will be the same as input
A
;
therefore, the identity element for AND is one.
Figure 4.11: AND Identity Element
4.4.3 Idempotence
If the two inputs of either an OR or AND gate are tied together, then
the same signal will be applied to both inputs. This results in the
output of either of those gates being the same as the input; and this
is called the idempotence property. An electronic gate wired in this
manner performs the same function as a buffer. Remember that in
Boolean expressions
a plus sign
represents an OR
gate, not
mathematical
addition.
A + A = A (4.19)
78 boolean functions
Figure 4.12: Idempotence Property for OR Gate
Figure 4.12 illustrates the idempotence property for an AND gate.Remember that in a
Boolean expression a
multiplication sign
represents an AND
gate, not
mathematical
multiplying.
A ∗ A = A (4.20)
Figure 4.13: Idempotence Property for AND Gate
4.4.4 Annihilator
Combining any data and a logic one through an OR gate yields a
constant output of one. This property is called the annihilator since
the OR gate outputs a constant one; in other words, whatever other
data were input are lost. Because OR is represented by a plus sign
when written in a Boolean equation, and the annihilator for OR is one,
the following is true:
A + 1 = 1 (4.21)
Figure 4.14: Annihilator For OR Gate
The bottom input for the OR gate in Figure 4.14 is a constant logic
one, or True. The output for this circuit will be True (or
1
) no matter
whether input A is True or False (1 or 0).
Combining any data and a logic zero with an AND gate yields a
constant output of zero. This property is called the annihilator since the
AND gate outputs a constant zero; in other words, whatever logic data
were input are lost. Because AND is represented by a multiplication
sign when written in a Boolean equation, and the annihilator for AND
is zero, the following is true:
4.4 univariate boolean algebra properties 79
A ∗ 0 = 0 (4.22)
Figure 4.15: Annihilator For AND Gate
The bottom input for the AND gate in Figure 4.15 is a constant logic
zero, or False. The output for this circuit will be False (or
0
) no matter
whether input A is True or False (1 or 0).
4.4.5 Complement
In Boolean logic there are only two possible values for variables:
0
and
1
. Since either a variable or its complement must be one, and since
combining any data with one through an OR gate yields one (see the
Annihilator in Equation 4.21), then the following is true:
A + A
0
= 1 (4.23)
Figure 4.16: OR Complement
In Figure 4.16, the output (
Y
) will always equal one, regardless of
the value of input
A
. This leads to the general property that when a
variable and its complement are combined through an OR gate the
output will always be one.
In the same way, since either a variable or its complement must be
zero, and since combining any data with zero through an AND gate
yields zero (see the Annihilator in Equation 4.22), then the following
is true:
A ∗ A
0
= 0 (4.24)
80 boolean functions
Figure 4.17: AND Complement
4.4.6 Involution
The Involution
Property is
sometimes called the
“Double Complement”
Property.
Another law having to do with complementation is that of Involution.
Complementing a Boolean variable two times (or any even number of
times) results in the original Boolean value.
(A
0
)
0
= A (4.25)
Figure 4.18: Involution Property
In the circuit illustrated in Figure 4.18, the output (
Y
) will always be
the same as the input (A).
Propagation Delay
It takes the two NOT gates a short period of time to pass a
signal from input to output, which is known as “propagation
delay.” A designer occasionally needs to build an intentional
signal delay into a circuit for some reason and two (or any even
number of) consecutive NOT gates would be one option.
4.5 multivariate boolean algebra properties
4.5.1 Introduction
Boolean Algebra, like real number algebra, includes a number of
properties. This unit introduces the multivariate Boolean properties,
or those properties that involve more than one input variable. These
properties permit Boolean expressions to be simplified, and circuit
designers are interested in simplifying circuits to reduce construc-
tion expense, power consumption, heat loss (wasted energy), and
troubleshooting time.
4.5 multivariate boolean algebra properties 81
4.5.2 Commutative
The examples here
show only two
variables, but this
property is true for
any number of
variables.
In essence, the commutative property indicates that the order of the
input variables can be reversed in either OR or AND gates without
changing the truth of the expression. Equation 4.26 expresses this
property algebraically.
A + B = B + A (4.26)
A ∗ B = B ∗ A
Figure 4.19: Commutative Property for OR
XOR and XNOR are
also commutative;
but for only two
variables, not three
or more.
Figure 4.20: Commutative Property for AND
In Figures 4.19 and 4.20 the inputs are reversed for the two gates, but
the outputs are the same. For example,
A
is entering the top input for
the upper OR gate, but the bottom input for the lower gate; however,
Y1 is always equal to Y2.
4.5.3 Associative
The examples here
show only three
variables, but this
property is true for
any number of
variables.
This property indicates that groups of variables in an OR or AND gate
can be associated in various ways without altering the truth of the
equations. Equation 4.27 expresses this property algebraically:
(A + B) + C = A + (B + C) (4.27)
(A ∗ B) ∗ C = A ∗ (B ∗ C)
XOR and XNOR are
also associative; but
for only two
variables, not three
or more.
In the circuits in Figure 4.21 and 4.22 , notice that
A
and
B
are
associated together in the first gate, and then
C
is associated with the
82 boolean functions
output of that gate. Then, in the lower half of the circuit,
B
and
C
are
associated together in the first gate, and then
A
is associated with the
output of that gate. Since
Y1
is always equal to
Y2
for any combination
of inputs, it does not matter which of the two variables are associated
together in a group of gates.
Figure 4.21: Associative Property for OR
Figure 4.22: Associative Property for AND
4.5.4 Distributive
The distributive property of real number algebra permits certain vari-
ables to be “distributed” to other variables. This operation is frequently
used to create groups of variables that can be simplified; thus, simplify-
ing the entire expression. Boolean algebra also includes a distributive
property, and that can be used to combine OR or AND gates in various
ways that make it easier to simplify the circuit. Equation 4.28 expresses
this property algebraically:
A(B + C) = AB + AC (4.28)
A + ( BC) = (A + B)(A + C)
In the circuits illustrated in Figures 4.23 and 4.24, notice that input
A
in the top half of the circuit is distributed to inputs
B
and
C
in
the bottom half. However, output
Y1
is always equal to output
Y2
regardless of how the inputs are set. These two circuits illustrate
Distributive of AND over OR and Distributive of OR over AND .
4.5 multivariate boolean algebra properties 83
Figure 4.23: Distributive Property for AND over OR
Figure 4.24: Distributive Property for OR over AND
4.5.5 Absorption
The absorption property is used to remove logic gates from a circuit if
those gates have no effect on the output. In essence, a gate is “absorbed”
if it is not needed. There are two different absorption properties:
A + ( AB) = A (4.29)
A(A + B) = A
The best way to think about why these properties are true is to
imagine a circuit that contains them. The first circuit below illustrates
the top equation.
Figure 4.25: Absorption Property (Version 1)
Table 4.9 is the truth table for the circuit in Figure 4.25.
84 boolean functions
Inputs Output
A B Y
0 0 0
0 1 0
1 0 1
1 1 1
Table 4.9: Truth Table for Absorption Property
Notice that the output,
Y
, is always the same as input
A
. This means
that input
B
has no bearing on the output of the circuit; therefore, the
circuit could be replaced by a piece of wire from input
A
to output
Y
. Another way to state that is to say that input
B
is absorbed by the
circuit.
The circuit illustrated in Figure 4.26 is the second version of the
Absorption Property. Like the first Absorption Property circuit, a truth
table would demonstrate that input B is absorbed by the circuit.
Figure 4.26: Absorption Property (Version 2)
4.5.6 Adjacency
The adjacency property simplifies a circuit by removing unnecessary
gates.
AB + AB
0
= A (4.30)
This property can be proven by simple algebraic manipulation:
AB + AB
0
Original Expression (4.31)
A(B + B
0
) Distributive Property
A1 Complement Property
A Identity Element
The circuit in Figure 4.27 illustrates the adjacency property. If this
circuit were constructed it would be seen that the output,
Y
, is always
the same as input
A
; therefore, this entire circuit could be replaced by
a single wire from input A to output Y.
4.6 demorgan’s theorem 85
Figure 4.27: Adjacency Property
4.6 demorgan’s theorem
4.6.1 Introduction
A mathematician named Augustus DeMorgan developed a pair of
important theorems regarding the complementation of groups in
Boolean algebra. DeMorgan found that an OR gate with all inputs
inverted (a Negative-OR gate) behaves the same as a NAND gate
with non-inverted inputs; and an AND gate with all inputs inverted
(a Negative-AND gate) behaves the same as a NOR gate with non-
inverted inputs. DeMorgan’s theorem states that inverting the output
of any gate is the same as using the opposite type of gate with inverted
inputs. Figure 4.28 illustrates this in circuit terms: the NAND gate with
normal inputs and the OR gate with inverted inputs are functionally
equivalent; that is,
Y1
will always equal
Y2
, regardless of the values of
input A or B.
Figure 4.28: DeMorgan’s Theorem Defined
The NOT function is commonly represented in an equation as an
apostrophe because it is easy to enter with a keyboard, like:
(AB)
0
for A
AND B NOT. However, it is easiest to work with DeMorgan’s theorem
if NOT is represented by an overline rather than an apostrophe, so it
would be written as
AB
rather than
(AB)
0
. Remember that an overline
is a grouping symbol (like parenthesis) and it means that everything
under that bar would first be combined (using an AND or OR gate)
and then the output of the combination would be complemented.
86 boolean functions
4.6.2 Applying DeMorgan’s Theorem
Applying DeMorgan’s theorem to a Boolean expression may be thought
of in terms of breaking the bar. When applying DeMorgan’s theorem to
a Boolean expression:
1. A complement bar is broken over a group of variables.
2.
The operation (AND or OR ) directly underneath the broken bar
changes.
3. Pieces of the broken bar remain over the individual variables.
To illustrate:
A ∗ B ↔ A + B (4.32)
A + B ↔ A ∗ B (4.33)
Equation 4.32 shows how a two-input NAND gate is “broken” to
form an OR gate with two inverted inputs and equation 4.33 shows
how a two-input NOR gate is “broken” to form an AND gate with two
complemented inputs.
4.6.3 Simple Example
When multiple “layers” of bars exist in an expression, only one bar is
broken at a time, and the longest, or uppermost, bar is broken first.
As an example, consider the circuit in Figure 4.29:
Figure 4.29: DeMorgan’s Theorem Example 1
By writing the output at each gate (as illustrated in Figure 4.29),
it is easy to determine the Boolean expression for the circuit. Note:
all circuit diagrams in this book are generated with Logisim-evolution
and the text tool in that software does not permit drawing overbars.
Therefore, the circuit diagram will use the apostrophe method of
indicating NOT but overbars will be used in the text.
A + BC (4.34)
4.6 demorgan’s theorem 87
To simplify the circuit, break the bar covering the entire expression
(the “longest bar”), and then simplify the resulting expression.
A + BC Original Expression (4.35)
A BC ”Break” the longer bar
ABC Involution Property
As a result, the original circuit is reduced to a three-input AND gate
with one inverted input.
4.6.4 Incorrect Application of DeMorgan’s Theorem
More than one bar is never broken in a single step, as illustrated in
Equation 4.36:
A + BC Original Expression (4.36)
AB + C Improperly Breaking Two Bars
AB + C Incorrect Solution
Thus, as tempting as it may be to take a shortcut and break more
than one bar at a time, it often leads to an incorrect result. Also,
while it is possible to properly reduce an expression by breaking the
short bar first; more steps are usually required and that process is not
recommended.
4.6.5 About Grouping
An important, but easily neglected, aspect of DeMorgan’s theorem
concerns grouping. Since a bar functions as a grouping symbol, the
variables formerly grouped by a broken bar must remain grouped or
else proper precedence (order of operation) will be lost. Therefore,
after simplifying a large grouping of variables, it is a good practice to
place them in parentheses in order to keep the order of operation the
same.
Consider the circuit in Figure 4.30.
Figure 4.30: DeMorgan’s Theorem Example 2
88 boolean functions
As always, the first step in simplifying this circuit is to generate
the Boolean expression for the circuit, which is done by writing the
sub-expression at the output of each gate. That results in Expression
4.37, which is then simplified.
A + BC + AB Original Expression (4.37)
(A + BC)(AB) Breaking the Longest Bar
(A + BC)(AB) Involution
(AAB)(BCAB) Distribute AB to (A + BC)
(AB) + (BCAB) Idempotence: AA = A
(AB) + (0CA)) Complement: BB = 0
(AB) + 0 Annihilator: 0CA = 0
AB Identity: A + 0 = A
The equivalent gate circuit for this much-simplified expression is as
follows:
Figure 4.31: DeMorgan’s Theorem Example 2 Simplified
4.6.6 Summary
Here are the important points to remember about DeMorgan’s Theo-
rem:
•
It describes the equivalence between gates with inverted inputs
and gates with inverted outputs.
•
When ”breaking” a complementation (or NOT) bar in a Boolean
expression, the operation directly underneath the break (AND or
OR) reverses and the broken bar pieces remain over the respec-
tive terms.
•
It is normally easiest to approach a problem by breaking the
longest (uppermost) bar before breaking any bars under it.
• Two complementation bars are never broken in one step.
•
Complementation bars function as grouping symbols. Therefore,
when a bar is broken, the terms underneath it must remain
grouped. Parentheses may be placed around these grouped
terms as a help to avoid changing precedence.
4.7 boolean functions 89
4.6.7 Example Problems
The following examples use DeMorgan’s Theorem to simplify a Boolean
expression.
Original Expression Simplified
1 (A + B)(ABC)(AC) A B C
2 (AB + BC) + (BC + AB) B C
3 (AB + BC)(AC + A C) A + C
4.7 boolean functions
Consider Figure 4.32, which is a generic circuit with two inputs and
one output.
Figure 4.32: Generic Function
Without knowing anything about what is in the unlabeled box at
the center of the circuit, there are a number of possible truth tables
which could describe the circuit’s output. Two possibilities are shown
in Truth Table 4.10 and Truth Table 4.11.
Inputs Output
A B Y
0 0 0
0 1 1
1 0 0
1 1 1
Table 4.10: Truth Table for Generic Circuit One
90 boolean functions
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
Table 4.11: Truth Table for Generic Circuit Two
In fact, there are
16
possible truth tables for this circuit. Each of those
truth tables reflect a single potential function of the circuit by setting
various combinations of input/output. Therefore, any two-input, one-
output circuit has
16
possible functions. It is easiest to visualize all
16
combinations of inputs/outputs by using an odd-looking truth
table. Consider only one of those
16
functions, the one for the generic
circuit described by Truth Table 4.11. That function is also found
in the Boolean Functions Table 4.13 and one row from that table is
reproduced in Table 4.12.
A 0 0 1 1
B 0 1 0 1
F
6
0 1 1 0 Exclusive Or (XOR): A ⊕ B
Table 4.12: Boolean Function Six
The line shown in Table 4.12 is for Function 6, or
F
6
(note that the
pattern of the outputs is
0110
, which is binary
6
). Inputs
A
and
B
are
listed at the top of the table. For example, the highlighted column of
the table shows that when
A
is zero and
B
is one the output is one.
Therefore, on the line that defines
F
6
, the output is True when
[(A = 0
AND
B = 1)
OR
(A = 1
AND
B = 0)]
This is an XOR function, and the
last column of the table verbally describes that function.
Table 4.13 is the complete Boolean Function table.
4.8 functional completeness 91
A 0 0 1 1
B 0 1 0 1
F
0
0 0 0 0 Zero or Clear. Always zero (Annihilation)
F
1
0 0 0 1 Logical AND: A ∗ B
F
2
0 0 1 0 Inhibition: AB
0
or A > B
F
3
0 0 1 1 Transfer A to Output, Ignore B
F
4
0 1 0 0 Inhibition: A
0
B or B > A
F
5
0 1 0 1 Transfer B to Output, Ignore A
F
6
0 1 1 0 Difference, XOR: A ⊕ B
F
7
0 1 1 1 Logical OR: A + B
F
8
1 0 0 0 Logical NOR: (A + B)
0
F
9
1 0 0 1 Equivalence, XNOR: (A = B)
0
F
10
1 0 1 0 Not B and ignore A, B Complement
F
11
1 0 1 1 Implication, A + B
0
, B >= A
F
12
1 1 0 0 Not A and ignore B, A Complement
F
13
1 1 0 1 Implication, A
0
+ B, A >= B
F
14
1 1 1 0 Logical NAND: (A ∗ B)
0
F
15
1 1 1 1 One or Set. Always one (Identity)
Table 4.13: Boolean Functions
4.8 functional completeness
A set of Boolean operations is said to be functionally complete if every
possible Boolean function can be derived from that set. The Primary
Logic Operations (page
??
) are functionally complete since the Sec-
ondary Logic Functions (page 72) can be derived from them. As an
example, Equation 4.38 and Figure 4.33 shows how an XOR function
can be derived from only AND, OR, and NOT gates.
((A ∗ B)
0
∗ (A + B)) = A ⊕ B (4.38)
Figure 4.33: XOR Derived From AND/OR/NOT
92 boolean functions
While the Primary Operations are functionally complete, it is pos-
sible to define other functionally complete sets of operations. For
example, using DeMorgan’s Theorem, the set of
{
AND, NOT
}
is also
functionally complete since the OR operation can be defined as
(A
0
B
0
)
0
.
In fact, both
{
NAND
}
and
{
NOR
}
operations are functionally com-
plete by themselves. As an example, the NOT operation can be derived
using only NAND gates:
(A|A)
. Because all Boolean functions can be
derived from either NAND or NOR operations, these are sometimes
considered universal operations and it is a common challenge for stu-
dents to create some complex Boolean function using only one of these
two types of operations.
5
B O O L E A N E X P R E S S I O N S
What to Expect
A Boolean expression uses the various Boolean functions to
create a mathematical model of a digital logic circuit. That
model can then be used to build a circuit in a simulator like
Logisim-Evolution. The following topics are included in this
chapter.
• Creating a Boolean expression from a description
•
Analyzing a circuit’s properties to develop a minterm or
maxterm expression
• Determining if a Boolean expression is in canonical form
•
Converting a Boolean expression with missing terms to
its canonical equivalent
•
Simplifying a complex Boolean expression using algebraic
methods
5.1 introduction
Electronic circuits that do not require any memory devices (like flip-
flops or registers) are created using what is called “Combinational
Logic.” These systems can be quite complex, but all outputs are de-
termined solely by input signals that are processed through a series
of logic gates. Combinational circuits can be reduced to a Boolean
Algebra expression, though it may be quite complex; and that expres-
sion can be simplified using methods developed in this chapter and
Chapter 6, Practice Problems, page 117, and Chapter 7, One In First
Cell, page 145. Combinational circuits are covered in Chapter 8, 32x8,
page 173.
In contrast, electronic circuits that require memory devices (like
flip-flops or registers) use what is called “Sequential Logic.” Those
circuits often include feedback loops, so the final output is determined
by input signals plus the feedback loops that are processed through a
series of logic gates. This makes sequential logic circuits much more
complex than combinational and the simplification of those circuits is
covered in Chapter 9, Comparators, page 189.
93
94 boolean expressions
Finally, most complex circuits include both combinational and se-
quential sub-circuits. In that case, the various sub-circuits would be
independently simplified using appropriate methods. Several exam-
ples of these types of circuits are analyzed in Chapter 10, Sample
Problems, page 211.
5.2 creating boolean expressions
A circuit designer is often only given a written (or oral) description of
a circuit and then asked to build that device. Too often, the designer
may receive notes scribbled on the back of a dinner napkin, along
with some verbal description of the desired output, and be expected
to build a circuit to accomplish that task. Regardless of the form of
the request, the process that the designer follows, in general, is:
1.
Write The Problem Statement. The problem to be solved is
written in a clear, concise statement. The better this statement
is written the easier each of the following steps will be, so time
spent polishing the problem statement is worthwhile.
2.
Construct A Truth Table. Once the problem is clearly defined,
the circuit designer constructs a truth table where all input-
s/outputs are included. It is essential that all possible input
combinations that lead to a True output are identified.
3.
Write A Boolean Expression. When the truth table is com-
pleted, it is easy to create a Boolean expression from that table
as covered in Example 5.2.1.
4.
Simplify the Boolean Expression. The expression should be
simplified as much as possible, and that process is covered in
this chapter, Chapter 6, Practice Problems, page 117, and Chapter
7, One In First Cell, page 145.
5.
Draw The Logic Diagram. The logic diagram for a circuit is
constructed from the simplified Boolean expression.
6.
Build The Circuit. If desired, a physical circuit can be built
using the logic diagram.
5.2.1 Example
A machine is to be programmed to help pack shipping boxes for the
ABC Novelty Company. They are running a promotion so if a customer
purchases any two of the following items, but not all three, a free poster
will be added to the purchase: joy buzzer, fake blood, itching powder.
Design the logic needed to add the poster to appropriate orders.
5.2 creating boolean expressions 95
1.
Problem Statement. The problem is already fairly well stated.
A circuit is needed that will activate the “drop poster” machine
when any two of three inputs (joy buzzer, fake blood, itching
powder), but not all three, are True.
2.
Truth Table. Let
J
be the Joy Buzzer,
B
be the Fake Blood, and
P
be the Itching Powder; and let the truth table inputs be True
(or
1
) when any of those items are present in the shipping box.
Let the output
D
be for “Drop Poster” and when it is True (or
1
)
then a poster will be dropped into the shipping box. The Truth
table is illustrated in Table 5.1.
Inputs Output
J B P D
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
Table 5.1: Truth Table for Example
3.
Write Boolean Expression. According to the Truth Table, the
poster will be dropped into the shipping box in only three cases
(when output
D
is True). Equation 5.1 was generated from the
truth table.
BP + JP + JB = D (5.1)
4.
Simplify Boolean Expression. The Boolean expression for this
problem is already as simple as possible so no further simplifica-
tion is needed.
5.
Draw Logic Diagram. Figure 5.1 was drawn from the switching
equation.
96 boolean expressions
Figure 5.1: Logic Diagram From Switching Equation
6.
Build The Circuit. This circuit could be built (or “realized”)
with three AND gates and one 3-input OR gate.
5.3 minterms and maxterms
5.3.1 Introduction
The solution to a Boolean equation is normally expressed in one of
two formats: Sum of Products (SOP) or Product of Sums (POS).
5.3.2 Sum Of Products (SOP) Defined
Equation 5.2 is an example of a SOP expression. Notice that the ex-
pression describes four inputs (
A
,
B
,
C
,
D
) that are combined through
two AND gates and then the output of those AND gates are combined
through an OR gate.
(A
0
BC
0
D) + ( AB
0
CD) = Y (5.2)
Each of the two terms in this expression is a minterm. Minterms can
be identified in a Boolean expression as a group of inputs joined by an
AND gate and then two or more minterms are combined with an OR
gate. The circuit illustrated in Figure 5.2 would realize Equation 5.2.Notice the inverting
bubble on three of the
AND gate inputs.
Figure 5.2: Logic Diagram For SOP Example
5.3 minterms and maxterms 97
5.3.3 Product of Sums (POS) Defined
Equation 5.3 is an example of a POS expression. Notice that the ex-
pression describes four inputs (
A
,
B
,
C
,
D
) that are combined through
two OR gates and then the output of those OR gates are combined
through an AND gate.
(A
0
+ B + C +
0
D)(A + B
0
+ C + D) = Y (5.3)
Each term in this expression is called a maxterm. Maxterms can be
identified in a Boolean expression as a group of inputs joined by an
OR gate; and then two or more maxterms are combined with an AND
gate. The circuit illustrated in Figure 5.3 would realize Equation 5.3. Notice the inverting
bubble on three of the
OR gate inputs.
Figure 5.3: Logic Diagram For POS Example
5.3.4 About Minterms
A term that contains all of the input variables in one row of a truth
table joined with an AND gate is called a minterm. Consider truth
table 5.2 which is for a circuit with three inputs (
A
,
B
, and
C
) and
one output ( Q).
98 boolean expressions
Inputs Outputs
A B C Q m
0 0 0 0 0
0 0 1 0 1
0 1 0 0 2
0 1 1 1 3
1 0 0 0 4
1 0 1 1 5
1 1 0 0 6
1 1 1 0 7
Table 5.2: Truth Table for First Minterm Example
The circuit that is represented by this truth table would output a
True in only two cases, when the inputs are
A
0
BC
or
AB
0
C
. Equation
5.4 describes this circuit.
(A
0
BC) + ( AB
0
C) = Q (5.4)
The terms
A
0
BC
and
AB
0
C
are called minterms and they contain
every combination of input variables that outputs a True when the
three inputs are joined by an AND gate. Minterms are most often used
to describe circuits that have fewer True outputs than False (that is,
there are fewer
1
’s than
0
’s in the output column). In the example
above, there are only two True outputs with six False outputs, so
minterms describe the circuit most efficiently.
Minterms are frequently abbreviated with a lower-case
m
along
with a subscript that indicates the decimal value of the variables. For
example,
A
0
BC
, the first of the True outputs in the truth table above,
has a binary value of
011
, which is a decimal value of
3
; thus, the
minterm is
m
3
. The other minterm in this equation is
m
5
since its
binary value is
101
, which equals decimal
5
. It is possible to verbally
describe the entire circuit as:
m
3
OR
m
5
. For convenience, each of the
minterm numbers are indicated in the last column of the truth table.
As another example, consider truth table 5.3.
5.3 minterms and maxterms 99
Inputs Outputs
A B C D Q m
0 0 0 0 1 0
0 0 0 1 0 1
0 0 1 0 0 2
0 0 1 1 0 3
0 1 0 0 0 4
0 1 0 1 1 5
0 1 1 0 1 6
0 1 1 1 0 7
1 0 0 0 0 8
1 0 0 1 0 9
1 0 1 0 0 10
1 0 1 1 0 11
1 1 0 0 0 12
1 1 0 1 0 13
1 1 1 0 1 14
1 1 1 1 0 15
Table 5.3: Truth Table for Second Minterm Example
Equation 5.5 describes this circuit because these are the rows where
the output is True.
(A
0
B
0
C
0
D
0
) + ( A
0
BC
0
D) + ( A
0
BCD
0
) + ( ABCD
0
) = Q (5.5)
These would be minterms
m
0
,
m
5
,
m
6
, and
m
14
. Equation 5.6
shows a commonly used, more compact way to express this result.
Z
(A, B, C, D) =
X
(0, 5, 6, 14) (5.6)
Equation 5.6 would read: “For the function of inputs
A
,
B
,
C
, and
D
,
the output is True for minterms
0
,
5
,
6
, and
14
when they are combined
with an OR gate.” This format is called Sigma Notation, and it is easy
to derive the full Boolean equation from it by remembering that
m
0
is
0000
, or
A
0
B
0
C
0
D
0
;
m
5
is
0101
, or
A
0
BC
0
D
;
m
6
is
0110
, or
A
0
BCD
0
;
and
m
14
is
1110
, or
ABCD
0
. Therefore, the Boolean equation can be
quickly created from the Sigma Notation.
A logic equation that is created using minterms is often called
the Sum of Products (or SOP) since each term is composed of inputs
ANDed together (“products”) and the terms are then joined by OR
gates (“sums”).
100 boolean expressions
5.3.5 About Maxterms
A term that contains all of the input variables joined with an OR gate
(“added together”) for a False output is called a maxterm. Consider
Truth Table 5.4, which has three inputs (
A
,
B
, and
C
) and one output
( Q):
Inputs Outputs
A B C Q M
0 0 0 1 0
0 0 1 1 1
0 1 0 0 2
0 1 1 1 3
1 0 0 1 4
1 0 1 1 5
1 1 0 0 6
1 1 1 1 7
Table 5.4: Truth Table for First Maxterm Example
The circuit that is represented by this truth table would output a
False in only two cases. Since there are fewer False outputs than True,
it is easier to create a Boolean equation that would generate the False
outputs. Because the equation describes the False outputs, each term
is built by complementing the inputs for each of the False output lines.
After the output groups are defined, they are joined with an AND gate.
Equation 5.7 is the Boolean equation for Truth Table 5.4.
(A + B
0
+ C)(A
0
+ B
0
+ C) = Q (5.7)
The terms
A + B
0
+ C
and
A
0
+ B
0
+ C
are called
maxterms
, and
they contain the complement of the input variables for each of the
False output lines. Maxterms are most often used to describe circuits
that have fewer False outputs than True. In Truth Table 5.4, there are
only two False outputs with six True outputs, so maxterms describe
the circuit most efficiently.
Maxterms are frequently abbreviated with an upper-case
M
along
with a subscript that indicates the decimal value of the complements
of the variables. For example, the complement of
A + B
0
+ C
, the first
of the False outputs in Truth Table 5.4, is
010
, which is a decimal value
of 2; thus, the maxterm would be
M
2
. This can be confusing, but
remember that the complements of the inputs are used to form the
expression. Thus,
A + B
0
+ C
is
010
, not
101
. The other maxterm in
this equation is
M
6
since the binary value of its complement is
110
,
5.3 minterms and maxterms 101
which equals decimal 6. It is possible to describe the entire circuit as a
the product of two groups of maxterms: M
2
and M
6
.
As another example, consider Truth Table 5.5.
Inputs Outputs
A B C D Q M
0 0 0 0 1 0
0 0 0 1 1 1
0 0 1 0 1 2
0 0 1 1 0 3
0 1 0 0 1 4
0 1 0 1 1 5
0 1 1 0 1 6
0 1 1 1 1 7
1 0 0 0 1 8
1 0 0 1 0 9
1 0 1 0 1 10
1 0 1 1 1 11
1 1 0 0 0 12
1 1 0 1 1 13
1 1 1 0 1 14
1 1 1 1 1 15
Table 5.5: Truth Table for Second Maxterm Example
Equation 5.8 describes this circuit.
(A + B + C
0
+ D
0
)(A
0
+ B + C + D
0
)(A
0
+ B
0
+ C + D) = Q
(5.8)
These would be maxterms
M
3
,
M
9
, and
M
12
. Equation 5.9 shows a
commonly used, more compact way to express this result.
Z
(A, B, C, D) =
Y
(3, 9, 12) (5.9)
Equation 5.9 would read: “For the function of
A
,
B
,
C
,
D
, the output
is False for maxterms
3
,
9
, and
12
when they are combined with an
AND gate.” This format is called Pi Notation, and it is easy to derive the
Boolean equation from it. Remember that
M
3
is
0011
, or
A + B + C
0
+
D
0
(the complement of the inputs),
M
9
is
1001
, or
A
0
+ B + C + D
0
,
and
M
12
is
1100
, or
A
0
+ B
0
+ C + D
. The original equation can be
quickly created from the Pi Notation.
102 boolean expressions
A logic equation that is created using maxterms is often called the
Product of Sums (or POS) since each term is composed of inputs OR
ed together (“sums”) and the terms are then joined by AND gates
(“products”).
5.3.6 Minterm and Maxterm Relationships
The minterms and maxterms of a circuit have three interesting rela-
tionships: equivalence, duality, and inverse. To define and understand
these terms, consider Truth Table 5.6 for some unspecified “black box”
circuit:
Inputs Outputs Terms
A B C Q Q’ minterm Maxterm
0 0 0 0 1 A
0
B
0
C
0
(m
0
) A + B + C (M
0
)
0 0 1 1 0 A
0
B
0
C (m
1
) A + B + C
0
(M
1
)
0 1 0 0 1 A
0
BC
0
(m
2
) A + B
0
+ C (M
2
)
0 1 1 0 1 A
0
BC (m
3
) A + B
0
+ C
0
(M
3
)
1 0 0 0 1 AB
0
C
0
(m
4
) A
0
+ B + C (M
4
)
1 0 1 1 0 AB
0
C (m
5
) A
0
+ B + C
0
(M
5
)
1 1 0 0 1 ABC
0
(m
6
) A
0
+ B
0
+ C (M
6
)
1 1 1 1 0 ABC (m
7
) A
0
+ B
0
+ C
0
(M
7
)
Table 5.6: Minterm and Maxterm Relationships
5.3.6.1 Equivalence
The minterms and the maxterms for a given circuit are considered
equivalent ways to describe that circuit. For example, the circuit de-
scribed by Truth Table 5.6 could be defined using minterms (Equation
5.10).
Z
(A, B, C) =
X
(1, 5, 7) (5.10)
However, that same circuit could also be defined using maxterms
(Equation 5.11).
Z
(A, B, C) =
Y
(0, 2, 3, 4, 6) (5.11)
These two functions describe the same circuit and are, consequently,
equivalent. The Sigma Function includes the terms
1
,
5
, and
7
while
the Pi Function includes all other terms in the truth table (
0
,
2
,
3
,
4
,
and
6
). To put it a slightly different way, the Sigma Function describes
5.3 minterms and maxterms 103
the truth table rows where Q =
1
(minterms) while the Pi Function
describes the rows in the same truth table where Q =
0
(maxterms).
Therefore, Equation 5.12 can be derived.
X
(1, 5, 7) ≡
Y
(0, 2, 3, 4, 6) (5.12)
5.3.6.2 Duality
Each row in Truth Table 5.6 describes two terms that are considered
duals. For example, minterm
m
5
(
AB
0
C
) and maxterm
M
5
(
A
0
+ B +
C
0
) are duals. Terms that are duals are complements of each other (
Q
vs.
Q
0
) and the input variables are also complements of each other;
moreover, the inputs for the three minterms are combined with an
AND while the maxterms are combined with an OR. The output of the
circuit described by Truth Table 5.6 could be defined using minterms
(Equation 5.13).
Q =
X
(1, 5, 7) (5.13)
The dual of the circuit would be defined by using the maxterms for
the same output rows. However, those rows are the complement of the
circuit (Equation 5.14).
Q
0
=
Y
(1, 5, 7) (5.14)
This leads to the conclusion that the complement of a Sigma Function
is the Pi Function with the same inputs, as in Equation 5.15 (the
overline was used to emphasize the the fact that the PI Function is
complemented).
X
(1, 5, 7) =
Y
(1, 5, 7) (5.15)
5.3.6.3 Inverse
The complement of a function yields the opposite output. For example
the following functions are inverses because one defines
Q
while the
other defines
Q
0
using only minterms of the same circuit (or truth
table).
Q =
X
(1, 5, 7) (5.16)
Q
0
=
X
(0, 2, 3, 4, 6) (5.17)
104 boolean expressions
5.3.6.4 Summary
These three relationships are summarized in the following table. Imag-
ine a circuit with two or more inputs and an output of
Q
. Table 5.7
summarizes the various relationships in the Truth Table for that circuit.
Minterms where Q is 1 Minterms where Q’ is 1
Maxterms where Q is 1 Maxterms where Q’ is 1
Table 5.7: Minterm-Maxterm Relationships
The adjacent items in a single column are equivalent (that is,
Q
Minterms are equivalent to
Q
Maxterms), items that are diagonal are
duals (
Q
Minterms and
Q
0
Maxterms are duals), and items that are
adjacent in a single row are inverses (
Q
Minterms and
Q
0
Minterms
are inverses).
5.3.7 Sum of Products Example
5.3.7.1 Given
A “vote-counter” machine is designed to turn on a light if any two or
more of three inputs are True. Create a circuit to realize this machine.
5.3.7.2 Truth Table
When realizing a circuit from a verbal description, the best place
to start is constructing a truth table. This will make the Boolean
expression easy to write and then make the circuit easy to realize. For
the “vote-counter” problem, start by defining variables for the truth
table: Inputs A, B, C and Output Q.
Next, construct the truth table by identifying columns for each of the
three input variables and then indicate the output for every possible
input condition (Table 5.8).
5.3 minterms and maxterms 105
Inputs Outputs
A B C Q m
0 0 0 0 0
0 0 1 0 1
0 1 0 0 2
0 1 1 1 3
1 0 0 0 4
1 0 1 1 5
1 1 0 1 6
1 1 1 1 7
Table 5.8: Truth Table for SOP Example
5.3.7.3 Boolean Equation
In a truth table, if there are fewer True outputs than False, then it
is easiest to construct a Sum-of-Products equation. In that case, a
Boolean expression can be derived by creating the minterms for all
True outputs and combining those minterms with OR gates. Equation
5.18 is the Sigma Function of this circuit.
Z
(A, B, C) =
X
(3, 5, 6, 7) (5.18)
It may be possible to
simplify the Boolean
expression, but that
process is covered
elsewhere in this
book.
At this point, a circuit could be created with four 3-input AND gates
combined into one 4-input OR gate.
5.3.8 Product of Sums Example
5.3.8.1 Given
A local supply company is designing a machine to sort packages for
shipping. All packages go to the post office except packages going to
the local ZIP code containing chemicals (they are shipped by courier)
and packages going to a distant ZIP code containing only perishables
(they are shipped via air freight).
5.3.8.2 Truth Table
When realizing a circuit from a verbal description, the best place
to start is constructing a truth table. This will make the Boolean
expression easy to write and then make the circuit easy to realize. For
the sorting machine problem, start by defining variables for the truth
table:
106 boolean expressions
•
Ship via post office (the Output):
O
is True if ship by Post Office
• Zip Code: Z is True if the zip code is local
• Chemicals: C is True if the package contains chemicals
• Perishable: P is True if the package contains perishables
Next, construct the truth table by identifying columns for each of the
three input variables and then indicate the output for every possible
input condition (Table 5.8).
Inputs Outputs
Z C P O M
0 0 0 1 0
0 0 1 0 1
0 1 0 1 2
0 1 1 0 3
1 0 0 1 4
1 0 1 1 5
1 1 0 0 6
1 1 1 0 7
Table 5.9: Truth Table for POS Example
5.3.8.3 Boolean Equation
In the truth table, if there are fewer False outputs than True then it
is easiest to construct a Products-of-Sums equation. In that case, a
Boolean expression can be derived by creating the maxterms for all
False outputs and combining the complement those maxterms with
AND gates. Equation 5.19 is the Pi Expression of this circuit.
Z
(Z, C, P) =
Y
(1, 3, 6, 7) (5.19)
It may be possible to
simplify the Boolean
expression, but that
process is covered
elsewhere in this
book.
At this point, a circuit could be created with four 3-input OR gates
combined into one 4-input AND gate.
5.3.9 Summary
SOP Boolean expressions may be generated from truth tables quite
easily, by determining which rows of the table have an output of
True, writing one minterm for each of those rows, and then summing
all of the minterms. The resulting expression will lend itself well to
5.4 canonical form 107
implementation as a set of AND gates (products) feeding into a single
OR gate (sum).
POS Boolean expressions may be generated from truth tables quite
easily, by determining which rows of the table have an output of False,
writing one maxterm for each of those rows, and then multiplying
all of the maxterms. The resulting expression will lend itself well to
implementation as a set of OR gates (sums) feeding into a single AND
gate (product).
5.4 canonical form
5.4.1 Introduction
The word “canonical” simply means “standard” and it is used through-
out mathematics and science to denote some standard form for equa-
tions. In digital electronics, Boolean equations are considered to be
in canonical form when each of the terms in the equation includes
all of the possible inputs and those terms appear in the same order
as in the truth table. Using the canonical form is important when
simplifying a Boolean equation. For example, imagine the solution to
a given problem generated table 5.10.
Inputs Outputs
A B C Q m
0 0 0 0 0
0 0 1 1 1
0 1 0 0 2
0 1 1 1 3
1 0 0 0 4
1 0 1 0 5
1 1 0 0 6
1 1 1 1 7
Table 5.10: Canonical Example Truth Table
Minterm equation 5.20 is derived from the truth table and is pre-
sented in canonical form. Notice that each term includes all possible
inputs (
A
,
B
, and
C
), and that the terms are in the same order as they
appear in the truth table.
(A
0
B
0
C) + (A
0
BC) + ( ABC) = Q (5.20)
Frequently, though, a Boolean equation is expressed in standard
form, which is not the same as canonical form. Standard form means
108 boolean expressions
that some of the terms have been simplified and not all of the inputs
will appear in all of the terms. For example, consider Equation 5.21,
which is the solution for a 4-input circuit.
(A
0
C) + (B
0
CD) = Q (5.21)
This equation is in standard form so the first term,
A
0
C
, does not
include inputs
B
or
D
and the second term,
B
0
CD
, does not include
input
A
. However, all inputs must be present in every term for an
equation to be in canonical form.
Building a truth table for an equation in standard form raises an
important question. Consider the truth table for Equation 5.21.
Inputs Outputs
A B C D Q m
0 0 0 0 0 0
0 0 0 1 0 1
0 0 1 0 0 2
0 0 1 1 ? 3
0 1 0 0 0 4
0 1 0 1 0 5
0 1 1 0 0 6
0 1 1 1 0 7
1 0 0 0 0 8
1 0 0 1 0 9
1 0 1 0 0 10
1 0 1 1 ? 11
1 1 0 0 0 12
1 1 0 1 0 13
1 1 1 0 0 14
1 1 1 1 0 15
Table 5.11: Truth Table for Standard Form Equation
In what row would
B
0
CD
, the second term in Equation 5.21, be
placed?
B
0
CD
is
011
(in binary), but since the
A
term is missing
would it be a
0
or
1
; in other words, would
B
0
CD
generate an output
of
1
for row
0011 (m
3
)
or
1011 (m
11
)
? (The output for these two
rows are marked with a question mark in Table 5.11.) In fact, the
output for both of these rows must be considered True in order to
ensure that all possible combinations of input are covered. Thus, the
final equation for this circuit must include at least these two terms:
5.4 canonical form 109
(A
0
B
0
CD) + ( AB
0
CD)
. In the same way, the term
A
0
C
means that the
output is True for
m
2
,
m
3
,
m
6
, and
m
7
since input
A
0
C
is True and
any minterm that contains those two value is also considered True.
Thus, the final equation for this circuit must include at least these four
terms: (A
0
B
0
CD
0
) + ( A
0
B
0
CD) + ( A
0
BCD
0
) + ( A
0
BCD).
5.4.2 Converting Terms Missing One Variable
To change a standard Boolean expression that is missing one input
term into a canonical Boolean expression, insert both True and False for
the missing term into the original standard expression. As an example,
consider the term
B
0
CD
. Since term
A
is missing, both
A
and
A
0
must be included in the converted canonical expression. Equation 5.22
proves that
B
0
CD
can be expanded to include both possible values for
A by using the Adjacency Property (page 84).
(B
0
CD) → (AB
0
CD) + ( A
0
B
0
CD) (5.22)
A term that is missing one input variable will expand into two
terms that include all variables. For example, in a system with four
input variables (as above), any standard term with only three variables
will expand to a canonical expression containing two groups of four
variables.
Expanding a standard term that is missing one variable can also be
done with a truth table. To do that, fill in an output of
1
for every line
where the True inputs are found while ignoring all missing variables.
As an example, consider Truth Table 5.11 where the outputs for
m
3
and
m
1
1
are marked with a question mark. However, the output for
both of these lines should be marked as True because
B
0
CD
is True
(input
A
is ignored). Then, those two minterms lead to the Boolean
expression AB
0
CD + A
0
B
0
CD.
5.4.3 Converting Terms Missing Two Variables
It is easiest to expand a standard expression that is missing two terms
by first inserting one of the missing variables and then inserting the
other missing variable in two distinct steps. The process for inserting
a single missing variable is found in Section 5.4.2. Consider the term
A
0
C
in a four-variable system. It is missing both the
B
and
D
variables.
To expand that term to its canonical form, start by inserting either
of the two missing variables. For example, Equation 5.23 illustrates
entering B and B
0
into the expression.
(A
0
C) → (A
0
BC) + ( A
0
B
0
C) (5.23)
110 boolean expressions
Then, Equation 5.24 illustrates inserting
D
and
D
0
into the expres-
sion.
(A
0
BC) → (A
0
BCD) + (A
0
BCD
0
) (5.24)
(A
0
B
0
C) → (A
0
B
0
CD) + ( A
0
B
0
CD
0
)
In the end, A
0
C expands to Equation 5.25:
(A
0
C) →(A
0
BCD) + (A
0
BCD
0
) (5.25)
+ (A
0
B
0
CD) + ( A
0
B
0
CD
0
)
Thus, in a four-variable system, any standard term with only two
variables will expand to a canonical expression with four groups of
four variables.
Expanding a standard term that is missing two variables can also be
done with a truth table. To do that, fill in an output of
1
for every line
where the True inputs are found while ignoring all missing variables.
As an example, consider a Table 5.12, where A
0
C is marked as True:
Inputs Outputs
A B C D Q m
0 0 0 0 0 0
0 0 0 1 0 1
0 0 1 0 1 2
0 0 1 1 1 3
0 1 0 0 0 4
0 1 0 1 0 5
0 1 1 0 1 6
0 1 1 1 1 7
1 0 0 0 0 8
1 0 0 1 0 9
1 0 1 0 0 10
1 0 1 1 0 11
1 1 0 0 0 12
1 1 0 1 0 13
1 1 1 0 0 14
1 1 1 1 0 15
Table 5.12: Truth Table for Standard Form Equation
Notice that outputs for
m
2
,
m
3
,
m
6
, and
m
7
are True because for
each of those minterms
A
0
C
is True (inputs
B
and
D
are ignored).
5.4 canonical form 111
Then, those four minterms lead to the Boolean expression
A
0
B
0
CD
0
+
A
0
B
0
CD + A
0
BCD
0
+ A
0
BCD.
5.4.4 Summary
This discussion started with Equation 5.26, which is in standard form.
(A
0
C) + (B
0
CD) = Q (5.26)
After expanding both terms, Equation 5.27 is generated.
(A
0
BCD) + (A
0
BCD
0
) + ( A
0
B
0
CD) (5.27)
+(A
0
B
0
CD
0
) + ( AB
0
CD) + ( A
0
B
0
CD) = Q
Notice, though, that the term
A
0
B
0
CD
appears two times, so one
of those can be eliminated by the Idempotence property (page 77),
leaving Equation 5.28.
(A
0
BCD) + (A
0
BCD
0
) (5.28)
+(A
0
B
0
CD
0
) + ( AB
0
CD) + ( A
0
B
0
CD) = Q
To put the equation in canonical form, which is important for sim-
plification; all that remains is to rearrange the terms so they are in the
same order as they would appear in a truth table, which results in
Equation 5.29.
(A
0
B
0
CD
0
) + ( A
0
B
0
CD) (5.29)
+(A
0
BCD
0
) + ( A
0
BCD) + (AB
0
CD) = Q
112 boolean expressions
5.4.5 Practice Problems
Parenthesis were not
used in order to save
space; however, the
variable groups are
evident.
1
Standard (A,B,C) A
0
B + C + AB
0
Cannonical A
0
B
0
C + A
0
BC
0
+ A
0
BC + AB
0
C
0
+
AB
0
C + ABC
2
Standard (A,B,C,D) A
0
BC + B
0
D
Cannonical A
0
B
0
C
0
D
0
+ A
0
B
0
CD
0
+ A
0
BCD
0
+
A
0
BCD + AB
0
C
0
D
0
+ AB
0
CD
0
3
Standard (A,B,C,D) A
0
+ D
Cannonical A
0
B
0
C
0
D
0
+ A
0
B
0
C
0
D + A
0
B
0
CD
0
+
A
0
B
0
CD + A
0
BC
0
D
0
+ A
0
BC
0
D +
A
0
BCD
0
+ A
0
BCD + AB
0
C
0
D +
AB
0
CD + ABC
0
D + ABCD
4
Standard (A,B,C) A(B
0
+ C)
Cannonical AB
0
C
0
+ AB
0
C + ABC
Table 5.13: Canonical Form Practice Problems
5.5 simplification using algebraic methods
5.5.1 Introduction
One method of simplifying a Boolean equation is to use common
algebraic processes. It is possible to reduce an equation step-by-step
using the various properties of Boolean algebra in the same way that
real-number equations can be simplified.
5.5.2 Starting From a Circuit
Occasionally, the circuit designer is faced with an existing circuit and
must attempt to simplify it. In that case, the first step is to find the
Boolean equation for the circuit and then simplify that equation.
5.5.2.1 Generate a Boolean Equation
In the circuit illustrated in Figure 5.4, the
A
,
B
, and
C
input signals
are assumed to be provided from switches, sensors, or perhaps other
sub-circuits. Where these signals originate is of no concern in the task
of gate reduction.
5.5 simplification using algebraic methods 113
Figure 5.4: Example Circuit
To generate the Boolean equation for a circuit, write the output of
each gate as determined by the input signals and type of gate, working
from the inputs to the final output. Figure 5.5 illustrates the result of
this process.
Figure 5.5: Example Circuit With Gate Outputs
This process leads to Equation 5.30.
AB + BC(B + C) = Y (5.30)
5.5.3 Starting From a Boolean Equation
If a logic circuit’s function is expressed as a Boolean equation, then
algebraic methods can be applied to reduce the number of logic gates,
resulting in a circuit that performs the same function with fewer
components. As an example, Equation 5.31 simplifies the circuit found
in Figure ??.
AB + BC(B + C) Original Expression (5.31)
AB + BBC + BCC Distribute BC
AB + BC + BC Idempotence: BB=B and CC=C
AB + BC Idempotence: BC+BC=BC
B(A + C) Factor
114 boolean expressions
The final expression,
B(A + C)
, requires only two gates, and is
much simpler than the original, yet performs the same function. Such
component reduction results in higher operating speed (less gate
propagation delay), less power consumption, less cost to manufacture,
and greater reliability.
As a second example, consider Equation 5.32.
A + AB = Y (5.32)
This is simplified below.
A + AB Original Expression (5.33)
A(1 + B) Factor
A(1) Annihilation (1+B=1)
A Identity A1=A
The original expression,
A + AB
has been reduced to
A
so the
original circuit could be replaced by a wire directly from input
A
to
output
Y
. Equation 5.34 looks similar to Equation 5.32, but is quite
different and requires a more clever simplification.
A + A
0
B = Y (5.34)
This is simplified below.
A + A
0
B Original Expression (5.35)
A + AB + A
0
B Expand A to A+AB (Absorption)
A + B(A + A
0
) Factor B out of the last two terms
A + B(1) Complement Property
A + B Identityt: B(1)=B
Note how the Absorption Property (
A + AB = A
) is used to “un-
simplify” the first
A
term, changing
A
into
A + AB
. While this may
seem like a backward step, it ultimately helped to reduce the expres-
sion to something simpler. Sometimes “backward” steps must be taken
to achieve the most elegant solution. Knowing when to take such a
step is part of the art of algebra.
As another example, simplify this POS expression equation:
(A + B)(A + C) = Y (5.36)
This is simplified below.
5.5 simplification using algebraic methods 115
(A + B)(A + C) Original Expression (5.37)
AA + AC + AB + BC Distribute A+B
A + AC + AB + BC Idempotence: AA=A
A + AB + BC Absorption: A+AC=A
A + BC Absorption: A+AB=A
In each of the examples in this section, a Boolean expression was
simplified using algebraic methods, which led to a reduction in the
number of gates needed and made the final circuit more economical
to construct and reliable to operate.
5.5.4 Practice Problems
Table 5.14 shows a Boolean expression on the left and its simplified
version on the right. This is provided for practice in simplifying
expressions using algebraic methods.
Original Expression Simplified
A(A
0
+ B) AB
A + A
0
B A + B
(A + B)(A + B
0
) A
AB + A
0
C + BC AB + A
0
C
Table 5.14: Simplifying Boolean Expressions
6
K A R N AU G H M A P S
What to Expect
This chapter introduces a graphic tool that is used to simplify
Boolean expressions: Karnaugh Maps. A Karnaugh Map plots
a circuit’s output on a matrix and then use a straightforward
technique to combine those outputs to create a simplified circuit.
This chapter includes the following topics.
• Drawing a Karnaugh map for two/three/four variables
• Simplifying groups of two/four/eight/sixteen
•
Simplifying a Karnaugh Map that includes overlapping
groups
•
Wrapping groups “around the edge” of a Karnaugh Map
• Analyzing circuits that include “Don’t Care” terms
• Applying Reed-Muller logic to a Karnaugh Map
6.1 introduction
Maurice Karnaugh
developed this
process at Bell Labs
in 1953 while
designing switching
circuits for landline
telephones.
A Karnaugh map, like Boolean algebra, is a tool used to simplify
a digital circuit. Keep in mind that “simplify” means reducing the
number of gates and inputs for each gate and as components are
eliminated, not only does the manufacturing cost go down, but the
circuit becomes simpler, more stable, and more energy efficient.
In general, using Boolean Algebra is the easiest way to simplify a
circuit involving one to three input variables. For four input variables,
Boolean algebra becomes tedious and Karnaugh maps are both faster
and easier (and are less prone to error). However, Karnaugh maps
become rather complex with five input variables and are generally
too difficult to use above six variables (with more than four input
variables, a Karnaugh map uses multiple dimensions that become
very challenging to manipulate). For five or more input variables,
circuit simplification should be done by Quine-McClusky methods
(page ??) or the use of Computer-Aided Tools (CAT) (page 159).
In theory, any of the methods will work for any number of variables;
however, as a practical matter, the guidelines presented in Table 6.1
work well. Normally, there is no need to resort to CAT to simplify
117
118 karnaugh maps
a simple equation involving two or three variables since it is much
quickly to use either Boolean Algebra or Karnaugh maps. However, for
more complex input/output combinations, then CAT become essential
to both speed the process and improve accuracy.
Variables Algebra K-Map Quine-McClusky CAT
1-2 X
3 X X
4 X X
5-6 X X
7-8 X X
¿8 X
Table 6.1: Circuit Simplification Methods
6.2 reading karnaugh maps
Following are four different ways to represent the same thing, a two-
input digital logic function.
Figure 6.1: Simple Circuit For K-Map
AB + A
0
B = Y (6.1)
Inputs Output
A B Y
0 0 0
0 1 1
1 0 0
1 1 1
Table 6.2: Truth Table for Simple Circuit
6.3 drawing two-variable karnaugh maps 119
0
1
0
1
B
A
0 1
0
1
00 02
01 03
Figure 6.2: Karnaugh map for Simple Circuit
Karnaugh maps
frequently include
the minterm
numbers in each cell
to aid in placing
variables.
First is a circuit diagram, followed by its Boolean equation, truth
table, and, finally, Karnaugh map. Think of a Karnaugh map as simply
a rearranged truth table; but simplifying a three or four input circuit
using a Karnaugh map is much easier and more accurate than with
either a truth table or Boolean equation.
6.3 drawing two-variable karnaugh maps
Truth Table 6.3 and Karnaugh map 6.3 illustrate the relationship
between these two representations of the same circuit using Greek
symbols.
Inputs Output
A B Y
0 0 α
0 1 β
1 0 γ
1 1 δ
Table 6.3: Truth Table with Greek Letters
α
β
γ
δ
B
A
0 1
0
1
00 02
01 03
Figure 6.3: Karnaugh map With Greek Letters
On the Karnaugh map, all of the possible values for input
A
are
listed across the top of the map and values for input
B
are listed down
the left side. Thus, to find the output for
A = 0
;
B = 0
, look for the cell
where those two quantities intersect; which is output
α
in the example.
It should be clear how all four data squares on the Karnaugh map
correspond to their equivalent rows (the minterms) in the Truth Table.
120 karnaugh maps
Truth Table 6.4 and Karnaugh map 6.4 illustrate another example of
this relationship.
Inputs Output
A B Y
0 0 1
0 1 1
1 0 0
1 1 1
Table 6.4: Truth Table for Two-Input Circuit
1
1 1
B
A
0 1
0
1
00 02
01 03
Figure 6.4: Karnaugh map For Two-Input Circuit
Karnaugh maps
usually do not
include zeros to
decrease the chance
for error.
6.4 drawing three-variable karnaugh maps
Consider Equation 6.2.
ABC
0
+ AB
0
C + A
0
B
0
C = Y (6.2)
Table 6.5 and Karnaugh map 6.5 represent this equation.
Inputs Output
A B C Y minterm
0 0 0 0 0
0 0 1 1 1
0 1 0 0 2
0 1 1 0 3
1 0 0 0 4
1 0 1 1 5
1 1 0 1 6
1 1 1 0 7
Table 6.5: Truth Table for Three-Input Circuit
6.5 drawing four-variable karnaugh maps 121
1 1
1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 6.5: Karnaugh map for Three-Input Circuit
In Karnaugh map 6.5 all possible values for inputs
A
and
B
are
listed across the top of the map while input
C
is listed down the
left side. Therefore, minterm
m
05
, in the lower right corner of the
Karnaugh map, is for
A = 1
;
B = 0
;
C = 1
, (
AB
0
C
), one of the True
terms in the original equation.
6.4.1 The Gray Code
It should be noted that the values across the top of the Karnaugh
map are not in binary order. Instead, those values are in “Gray Code”
order. Gray code is essential for a Karnaugh map since the values for
adjacent cells must change by only one bit. Constructing the Gray
Code for three, four, and five variables is covered on page 60; however,
for the Karnaugh maps used in this chapter, it is enough to know the
two-bit Gray code: 00, 01, 11, 10.
6.5 drawing four-variable karna ugh maps
Consider Equation 6.3.
ABCD
0
+ AB
0
CD + A
0
B
0
CD = Y (6.3)
Table 6.6 and the Karnaugh map in Figure 6.6 illustrate this equation.
122 karnaugh maps
Inputs Output
A B C D Y minterm
0 0 0 0 0 0
0 0 0 1 0 1
0 0 1 0 0 2
0 0 1 1 1 3
0 1 0 0 0 4
0 1 0 1 0 5
0 1 1 0 0 6
1 1 1 1 0 7
1 0 0 0 0 8
1 0 0 1 0 9
1 0 1 0 0 10
1 0 1 1 1 11
1 1 0 0 0 12
1 1 0 1 0 13
1 1 1 0 1 14
1 1 1 1 0 15
Table 6.6: Truth Table for Four-Input Circuit
1 1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.6: K-Map For Four Input Circuit
This Karnaugh map is similar to those for two and three variables,
but the top row is for the
A
and
B
inputs while the left column is for
the
C
and
D
inputs. Notice that the values in both the top row and
left column use Gray Code sequencing rather than binary counting.
It is easy to indicate the minterms that are True on a Karnaugh
map if Sigma Notation is available since the numbers following the
Sigma sign are the minterms. As an example, Equation 6.4 creates the
Karnaugh map in Figure 6.7.
6.5 drawing four-variable karnaugh maps 123
Z
(A, B, C, D) =
X
(0, 1, 2, 4, 5) (6.4)
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.7: K-Map For Sigma Notation
It is also possible to map values if the circuit is represented in Pi
notation; but remember that maxterms indicate where zeros are placed
on the Karnaugh map and the simplified circuit would actually be the
inverse of the needed circuit. As an example, Equation 6.5 creates the
Karnaugh map at Figure 6.8.
Z
(A, B, C, D) =
Y
(8, 9, 12, 13) (6.5)
0
0
0
0
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.8: K-Map For PI Notation
To simplify this map, the designer could place ones in all of the
empty cells and then simplify the “ones” circuit using the techniques
explained below. Alternatively, the designer could also simplify the
map by combining the zeros as if they were ones, and then finding
the DeMorgan inverse (page 85) of that simplification. As an example,
the maxterm Karnaugh map above would simplify to
AC
0
, and the
DeMorgan equivalent for that is
A
0
+ C
, which is the minterm version
of the simplified circuit.
124 karnaugh maps
6.6 simplifying groups of two
To simplify a Boolean equation using a Karnaugh map, start by cre-
ating the Karnaugh map, indicating the input variable combinations
that lead to a True output for the circuit. Equations 6.6 and 6.7 are for
the same circuit and the Karnaugh map in Figure 6.9 was built from
these equations.
A
0
B
0
C
0
D
0
+ A
0
BC
0
D
0
+ A
0
BCD + AB
0
CD
0
+ AB
0
CD (6.6)
Z
(A, B, C, D) =
X
(0, 4, 7, 10, 11) (6.7)
1 1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.9: K-Map for Groups of Two: Ex 1
Once the True outputs are indicated on the Karnaugh map, mark
any groups of ones that are adjacent to each other, either horizontally
or vertically (but not diagonally). Also, mark any ones that are “left
over” and are not adjacent to any other ones, as illustrated in the
Karnaugh map in Figure 6.16.
1 1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.10: K-Map for Groups of Two: Ex 1, Solved
6.6 simplifying groups of two 125
Notice the group in the top-left corner (minterms
00
and
04
). This
group includes the following two input combinations:
A
0
B
0
C
0
D
0
+
A
0
BC
0
D
0
. In this expression, the
B
and
B
0
terms can be removed by
the Complement Property (page 79); so this group reduces to
A
0
C
0
D
0
.
To simplify this expression by inspecting the Karnaugh map, notice
that the variable
A
is zero for both of these minterms; therefore,
A
0
must be part of the final expression. In the same way, variables
C
and
D
are zero for both terms; therefore,
C
0
D
0
must be part of the final
expression. Since variable B changes it can be ignored when forming
the simplified expression.
The group in the lower-right corner (minterms
10
and
11
) includes
the following two input variable combinations:
AB
0
CD + AB
0
CD
0
.
The
D
and
D
0
terms can be removed by the Complement Property;
so this group simplifies to
AB
0
C
. Again, inspecting these two terms
would reveal that the variables
AB
0
C
do not change between the two
terms, so they must appear in the final expression.
Minterm
07
, the lone term indicated in column two, cannot be
reduced since it is not adjacent to any other ones. Therefore, it must
go into the simplified equation unchanged: A
0
BCD.
When finished, the original equation reduces to Equation 6.8.
A
0
C
0
D
0
+ AB
0
C + A
0
BCD (6.8)
Using a Karnaugh map, the circuit was simplified from four four-
input AND gates to two three-input AND gates and one four-input
AND gate.
The various ones on a Karnaugh map are called the Implicants of
the solution. These are the algebraic products that are necessary to
“imply” (or bring about) the final simplification of the circuit. When an
implicant cannot be grouped with any others, or when two or more
implicants are grouped together, they are called Prime Implicants. The
three groups (
A
0
C
0
D
0
,
AB
0
C
, and
A
0
BCD
) found by analyzing the
Karnaugh map above are the prime implicants for this equation. When
prime implicants are a necessary part of the final simplified equation,
and they are not subsumed by any other implicants, they are called
Essential Prime Implicants. For the simple example given above, all of
the prime implicants are essential; however, more complex Karnaugh
maps may have numerous prime implicants that are subsumed by
other implicants; thus, are not essential. There are examples of these
types of maps later in this chapter.
A second example is illustrated in Equation 6.9 and the Karnaugh
map in Figure 6.17.
A
0
B
0
C
0
D + A
0
B
0
CD + A
0
BCD
0
+ (6.9)
ABC
0
D + ABCD
0
+ AB
0
C
0
D
0
= Y
126 karnaugh maps
Z
(A, B, C, D) =
X
(1, 3, 6, 8, 13, 14) (6.10)
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.11: K-Map Solving Groups of Two: Example 2
All groups of adjacent ones have been marked, so this circuit can
be simplified by looking for groups of two. Starting with minterms
01
and
03
,
A
0
B
0
C
0
D + A
0
B
0
CD
simplifies to
A
0
B
0
D
. Minterms
06
and
14
simplifies to
BCD
0
. The other two marked minterms are not adjacent
to any others, so they cannot be simplified. Each of the marked terms
are prime implicants; and since they are not subsumed by any other
implicants, they are essential prime implicants. Equation 6.17 is the
simplified solution.
ABC
0
D + AB
0
C
0
D
0
+ A
0
B
0
D + BCD
0
= Y (6.11)
6.7 simplifying larger groups
When simplifying Karnaugh maps, it is most efficient to find groups
of
16
,
8
,
4
, and
2
adjacent ones, in that order. In general, the larger
the group, the simpler the expression becomes; so one large group is
preferable to two smaller groups. However, remember that any group
can only use ones that are adjacent along a horizontal or vertical line,
not diagonal.
6.7.1 Groups of 16
Groups of
16
reduce to a constant output of one. This is because if
a circuit is built such that every possible combination of four inputs
yields a True output, then the circuit is unnecessary and can be replaced
by a wire. There is no example Karnaugh map posted here to illustrate
a circuit like this because if every cell in a Karnaugh map contains a
one, then the circuit is unnecessary. By the same token, any Karnaugh
6.7 simplifying larger groups 127
map that contains only zeros indicates that the circuit would never
output a True condition so the circuit is unnecessary.
6.7.2 Groups of Eight
Groups of eight simplifies to a single output variable. Consider the
Karnaugh map in Figure 6.12.
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.12: K-Map Solving Groups of 8
The expression for row two is:
A
0
B
0
C
0
D + A
0
BC
0
D + ABC
0
D +
AB
0
C
0
D
. The term
C
0
D
is constant in this group, while
A
and
B
change, so this one line would simplify to
C
0
D
. The expression for
row three is:
A
0
B
0
CD + A
0
BCD + ABCD + AB
0
CD
. The term
CD
is
constant in this group, so this one line would simplify to
CD
. Then,
if the two rows are combined:
C
0
D + CD
,
C
and
C
0
are dropped by
the complement property and the circuit simplifies to
D
. To put this
another way, since the only term in this group of eight that never
changes is D, then Equation 6.12 is the simplified solution.
D = Y (6.12)
This Karnaugh map also provides a good example of prime impli-
cants that are not essential. Consider row two of the map. Minterms
01
and
05
form a Prime Implicant for this circuit since it is a group of two;
however, that group was subsumed by the group of eight that was
formed with the next row. Since every cell in the group of two is also
present in the group of eight, then the group of two is not essential
to the final circuit simplification. While this may seem to be rather
obvious, it is important to remember that frequently implicants are
formed that are not essential and they can be ignored. This concept
will come up again when using the Quine-McCluskey Simplification
method on page ??.
128 karnaugh maps
6.7.3 Groups of Four
Groups of four can form as a single row, a single column, or a square.
In any case, the four cells will simplify to a two-variable expression.
Consider the Karnaugh map in Figure 6.13
1 1 11
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.13: K-Map Solving Groups of Four, Example 1
Since the
A
and
B
variables can be removed due to the Complement
Property, Equation 6.13 shows the simplified solution.
C
0
D = Y (6.13)
The Karnaugh map in Figure 6.14 is a second example.
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.14: K-Map Solving Groups of Four, Example 2
Since the
C
and
D
variables can be removed due to the Complement
Property, Equation 6.14 shows the simplified circuit.
AB = Y (6.14)
The Karnaugh map in Figure 6.15 is an example of a group of four
that forms a square.
6.7 simplifying larger groups 129
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.15: K-Map Solving Groups of Four, Example 3
Since the
B
and
C
variables can be removed due to the Complement
Property, Equation 6.15 shows the simplified circuit.
A
0
D = Y (6.15)
6.7.4 Groups of Two
Groups of two will simplify to a three-variable expression. The Kar-
naugh map in Figure 6.16 is one example of a group of two.
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.16: K-Map Solving Groups of Two, Example 1
Since
C
is the only variable that can be removed due to the Comple-
ment Property, the above circuit simplifies to Equation 6.16.
AB
0
D = Y (6.16)
As a second example, consider the Karnaugh map in Figure 6.17.
130 karnaugh maps
1 1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.17: K-Map Solving Groups of Two, Example 2
Equation 6.17 is the simplified equation for the Karnaugh map in
6.17.
BCD = Y (6.17)
6.8 overlapping groups
Frequently, groups overlap to create numerous patterns on the Kar-
naugh map. Consider the following two examples.
1
1 1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.18: K-Map Overlapping Groups, Example 1
The one in cell
A
0
BCD
(minterm
07
) can be grouped with either the
horizontal or vertical group (or both). This creates the following three
potential simplified circuits:
•
Group minterms
05
-
07
with a separate minterm (
15
):
Q =
A
0
BD + ABCD
•
Group minterms
07
-
15
with a separate minterm (
05
):
Q = BCD +
A
0
BC
0
D
•
Two groups of two minterms (
05
-
07
and
07
-
15
):
Q = A
0
BD +
BCD
6.9 wrapping groups 131
In general, it would be considered simpler to have two three-input
AND gates rather than one three-input AND gate and one four-input
AND gate, so the last grouping option would be chosen. The designer
always chooses whatever grouping yields the smallest number of
gates and the smallest number of inputs per gate. The equation for
the simplified circuit is:
A
0
BD + BCD = Y (6.18)
Karnaugh map 6.19 is a more complex example.
1
1
1
1
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.19: K-Map Overlapping Groups, Example 2
The circuit represented by this Karnaugh map 6.19 would simplify
to:
A + A
0
B
0
C
0
+ BCD + A
0
B
0
CD
0
= Y (6.19)
6.9 wrapping groups
A Karnaugh map also “wraps” around the edges (top/bottom and
left/right), so groups can be formed around the borders. It is almost
like the Karnaugh map is on some sort of weird sphere where every
edge touches the edge across from it (but not diagonal corners). The
Karnaugh map in Figure 6.20 is an example.
132 karnaugh maps
1 1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.20: K-Map Wrapping Groups Example 1
The two ones on this map can be grouped “around the edge” to
form Equation 6.20.
B
0
C
0
D = Y (6.20)
The Karnaugh map in Figure 6.21 is another example of wrapping.
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.21: K-Map Wrapping Groups Example 2
The ones on the above map can be formed into a group of four and
simplify into Equation 6.21.
B
0
D
0
= Y (6.21)
6.10 karnaugh maps for five-variable inputs
It is possible to create a Karnaugh map for circuits with five input
variables; however, the map must be simplified as a three-dimensional
object so it is more complex than the maps described above. As an
example, imagine a circuit that is defined by Equation 6.22.
Z
(A, B, C, D, E) =
X
(0, 9, 13, 16, 25, 29) (6.22)
6.10 karnaugh maps for five-variable inputs 133
The Karnaugh map in Figure 6.22 would be used to simplify that
circuit.
1 1
1 1 1 1
DE
ABC
000 001 011 010 100 101 111 110
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
16 20 28 24
17 21 29 25
19 23 31 27
18 22 30 26
Figure 6.22: K-Map for Five Variables, Example 1
This map lists variables
A
,
B
, and
C
across the top row with
D
and
E
down the left column. Variables
B
and
C
are in Gray Code
order, while variable
A
is zero on the left side of the map and one on
the right. To simplify the circuit, the map must be imagined to be a
three-dimensional item; so the map would be cut along the heavy line
between minterms
08
and
16
(where variables
ABC
change from
010
to
100
) and then the right half would slide under the left half in such
a way that minterm 16 ends up directly under minterm 00.
By arranging the map in three dimensions there is only one bit
different between minterms
00
and
16
: bit
A
changes from zero to one
while the bits
B
and
C
remain at zero. Those two cells then form a
group of two and would be
A
0
B
0
C
0
D
0
E
0
+ AB
0
C
0
D
0
E
0
. Since variable
A
and
A
0
are both present in this expression, and none of the other
variables change, it can be simplified to:
B
0
C
0
D
0
E
0
. This process is
exactly the same as for a two-dimension Karnaugh map, except that
adjacent cells may include those above or below each other (though
diagonals are still not simplified).
Next, consider the group formed by minterms
09
,
13
,
25
, and
29
.
The expression for that group is
(A
0
BC
0
D
0
E + A
0
BCD
0
E + ABC
0
D
0
E +
ABCD
0
E)
. The variables
A
and
C
can be removed from the simplified
expression by the Complement Property, leaving:
BD
0
E
for this group.
Equation 6.23 is the final simplified expression for this circuit.
(B
0
C
0
D
0
E
0
) + ( BD
0
E) = Y (6.23)
The Karnaugh map in Figure 6.23 is a more complex example.
134 karnaugh maps
1 1
1 1 1
1 1 1
1
DE
ABC
000 001 011 010 100 101 111 110
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
16 20 28 24
17 21 29 25
19 23 31 27
18 22 30 26
Figure 6.23: K-Map Solving for Five Variables, Example 2
It is possible to
simplify a six-input
circuit with a
Karnaugh map, but
that becomes quite
challenging since the
map must be
simplified in four
dimensions.
On the map in Figure 6.23, the largest group would be minterms
13
-
15
-
29
-
31
, so they should be combined first. Minterms
05
-
13
,
27
-
31
, and
00
-
02
would form groups of two. Finally, even though Karnaugh maps
wrap around the edges, minterm
00
will not group with minterm
24
since they are on different layers (notice that two bits,
A
and
B
, change
between those two minterms, so they are not adjacent); therefore,
minterms
00
and
24
will not group with any other minterms. Equation
6.24 is the simplified expression for this circuit.
(A
0
B
0
C
0
E
0
) + ( A
0
CD
0
E) + ( BCD) + (ABDE) + (ABC
0
D
0
E
0
) = Y
(6.24)
6.11 “don’t care” terms
Occasionally, a circuit designer will run across a situation where the
output for a particular minterm makes no difference in the circuit;
so that minterm is considered “don’t care;” that is, it can be either
one or zero without having any effect on the entire circuit. As an
example, consider a circuit that it designed to work with BCD (page 53)
values. In that system, minterms
10
-
15
do not exist, so they would be
considered “don’t care.” On a Karnaugh map, “don’t care” terms are
indicated using several different methods, but the two most common
are a dash or an “x”. When simplifying a Karnaugh map that contains
“don’t care” values, the designer can choose to consider those values
as either zero or one, whichever makes simplifying the map easier.
Consider the following Karnaugh map:
6.12 karnaugh map simplification summary 135
X
1
1
1
X
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.24: K-Map With “Don’t Care” Terms, Example 1
On this map, minterm
13
is “don’t care.” Since it could form a group
of four with minterms
08
,
09
, and
12
, and since groups of four are
preferable to two groups of two, then minterm
13
should be considered
a one and grouped with the other three minterms. However, minterm
02
does not group with anything, so it should be considered a zero and
it would then be removed from the simplified expression altogether.
This Karnaugh map would simplify to AC
0
.
The Karnaugh map in Figure 6.25 is another example circuit with
“don’t care” terms:
1
1
1
1
1X
X
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.25: K-Map With “Don’t Care” Terms, Example 2
This circuit would simplify to Equation 6.25.
ACD
0
+ BC + A
0
BC + A
0
CD = Y (6.25)
6.12 karnaugh map simplification summary
Here are the rules for simplifying a Boolean equation using a four-
variable Karnaugh map:
1. Create the map and plot all ones from the truth table output.
136 karnaugh maps
2.
Circle all groups of
16
. These will reduce to a constant output of
one and the entire circuit is unnecessary.
3.
Circle all groups of eight. These will reduce to a one-variable
expression.
4.
Circle all groups of four. These will reduce to a two-variable
expression. The ones can be either horizontal, vertical, or in a
square.
5.
Circle all groups of two. These will reduce to a three-variable
expression. The ones can be either horizontal or vertical.
6.
Circle all ones that are not in any other group. These do not
reduce and will result in a four-variable expression.
7. All ones must be circled at least one time.
8. Groups can overlap.
9.
If ones are in more than one group, they can be considered part
of either, or both, groups.
10.
Groups can wrap around the edges to the other edge of the map.
11.
“Don’t Care” terms can be considered either one or zero, whichever
makes the map simpler.
6.13 practice problems
1
Expression (A,B) A
0
B + AB
0
+ AB
Simplified A + B
2
Exression (A,B,C) A
0
BC + AB
0
C
0
+ ABC
0
+ ABC
Simplified BC + AC
0
3
Exression (A,B,C) A
0
BC
0
D + AB
0
CD
Simplified C + A
0
B
4
Exression (A,B,C,D) A
0
B
0
C
0
+ B
0
CD
0
+ A
0
BCD
0
+
AB
0
C
0
Simplified B
0
D
0
+ B
0
C
0
+ A
0
CD
0
5
Exression
R
(A, B, C, D) =
P
(0, 1, 6, 7, 12, 13)
Simplified A
0
B
0
C
0
+ ABC
0
+ A
0
BC
6
Exression
R
(A, B, C, D) =
Q
(0, 2, 4, 10)
Simplified D + BC + AC
0
Table 6.7: Karnaugh maps Practice Problems
6.14 reed-m
¨
uller logic 137
6.14 reed-m
¨
uller logic
6.15 introduction
Irving Reed and D.E. M
¨
uller are noted for inventing various codes
that self-correct transmission errors in the field of digital communica-
tions. However, they also formulated ways of simplifying digital logic
expressions that do not easily yield to traditional methods, such as a
Karnaugh map where the ones form a checkerboard pattern.
Consider the Truth Table 6.8.
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
Table 6.8: Truth Table for Checkerboard Pattern
This pattern is easy to recognize as an XOR gate. The Karnaugh map
for Truth Table 6.8 is in Figure 6.26 (the zeros have been omitted and
the cells with ones have been shaded to emphasize the checkerboard
pattern):
1
1
B
A
0 1
0
1
00 02
01 03
Figure 6.26: Reed-M
¨
uller Two-Variable Example
Equation 6.26 describes this circuit.
AB
0
+ A
0
B = Y (6.26)
This equation cannot be simplified using common Karnaugh map
simplification techniques since none of the ones are adjacent verti-
cally or horizontally. However, whenever a Karnaugh map displays
a checkerboard pattern, the circuit can be simplified using XOR or
XNOR gates.
138 karnaugh maps
6.16 zero in first cell
Karnaugh maps with a zero in the first cell (that is, in a four-variable
map,
A
0
B
0
C
0
D
0
is False) are simplified in a slightly different manner
than those with a one in that cell. This section describes the technique
used for maps with a zero in the first cell and Section 6.17 (page 142)
describes the technique for maps with a one in the first cell.
With a zero in the first cell, the equation for the Karnaugh map is
generated by:
1.
Grouping the ones into horizontal, vertical, or square groups if
possible
2.
Identifying the variable in each group that is True and does not
change
3. Combining those groups with an XOR gate
6.16.1 Two-Variable Circuit
In Karnaugh map 6.26, it is not possible to group the ones. Since both
A and B are True in at least one cell, the equation for that circuit is:
A ⊕ B = Y (6.27)
6.16.2 Three-Variable Circuit
The Karnaugh map in Figure 6.27 is for a circuit containing three
inputs.
1
1 1
1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 6.27: Reed-M
¨
uller Three-Variable Example 1
In this Karnaugh map, it is not possible to group the ones. Since
A
,
B
, and
C
are all True in at least one cell, the equation for the circuit is:
A ⊕ B ⊕ C = Y (6.28)
The Karnaugh map in Figure 6.28 is more interesting.
6.16 zero in first cell 139
1 1
11
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 6.28: Reed-M
¨
uller Three-Variable Example 2
In this Karnaugh map, the ones form two groups in a checkerboard
pattern. If this map were to be simplified using common techniques it
would form Equation 6.29.
AC
0
+ A
0
C = Y (6.29)
If realized, this circuit would contain two two-input AND gates
joined by a two-input OR gate. However, this equation can be sim-
plified using the Reed-M
¨
uller technique. For the group in the upper
right corner
A
is a constant one and for the group in the lower left
corner C is a constant one. The equation simplifies to 6.30.
A ⊕ C = Y (6.30)
If realized, this circuit would contain nothing more than a two-input
XOR gate and is much simpler than the first attempt.
6.16.3 Four-Variable Circuit
The Karnaugh map in Figure 6.29 is for a circuit containing four
variable inputs.
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.29: Reed-M
¨
uller Four-Variable Example 1
In this Karnaugh map, it is not possible to group the ones. Since
A
,
B
,
C
, and
D
are all True in at least one cell, Equation 6.31 would
describe the circuit.
140 karnaugh maps
A ⊕ B ⊕ C ⊕ D = Y (6.31)
Following are more interesting four-variable Karnaugh maps with
groups of ones:
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.30: Reed-M
¨
uller Four-Variable Example 2
In the Karnaugh map in Figure 6.30, the group of ones in the upper
right corner has a constant one for
A
and the group in the lower left
corner has a constant one for
C
, so Equation 6.32 would describe the
circuit.
A ⊕ C = Y (6.32)
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.31: Reed-M
¨
uller Four-Variable Example 3
In the Karnaugh map in Figure 6.31, the group of ones in the first
shaded column has a constant one for
B
and in the second shaded
column have a constant one for
A
, so Equation 6.33 describes this
circuit.
A ⊕ B = Y (6.33)
6.16 zero in first cell 141
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.32: Reed-M
¨
uller Four-Variable Example 4
Keep in mind that groups can wrap around the edges in a Karnaugh
map. The group of ones in columns 1 and 4 combine and has a constant
one for
D
. The group of ones in rows 1 and 4 combine and has a
constant one for B, so Equation 6.34 describes this circuit.
B ⊕ D = Y (6.34)
It is interesting that all of the above examples used XOR gates to
combine the constant True variable found in groups of ones; however,
it would yield the same result if XOR gates combined the constant
False variable found in groups of ones. The designer could choose
either, but must be consistent. For example, consider the Karnaugh
map in Figure 6.33.
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.33: Reed-M
¨
uller Four-Variable Example 5
To avoid confusion
when simplifying
these maps it is
probably best to
always use the
constant True terms.
When this Karnaugh map was simplified earlier (Figure 6.30), the
constant True for the group in the upper right corner,
A
, and lower left
corner,
C
, was used; however, the constant False in the lower left corner,
A
, and upper right corner,
C
, would give the same result. Either way
yields Equation 6.35.
A ⊕ C = Y (6.35)
142 karnaugh maps
6.17 one in first cell
Karnaugh maps with a one in the first Cell (that is, in a four-variable
map,
A
0
B
0
C
0
D
0
is True) are simplified in a slightly different manner
than those with a zero in that cell. When a one is present in the first
cell, two of the terms must be combined with an XNOR rather than
an XOR gate (though it does not matter which two are combined). To
simplify these circuits, use the same technique presented above; but
then select any two of the terms and change the gate from XOR to
XNOR. Following are some examples.
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.34: Reed-M
¨
uller Four-Variable Example 6
In the Karnaugh map in Figure 6.34 it is not possible to group the
ones. Since
A
,
B
,
C
, and
D
are all True in at least one cell, they would
all appear in the final equation; however, since there is a one in the
first cell, then two of the terms must be combined with an XNOR gate.
Here is one possible solution:
A B ⊕ C ⊕ D = Y (6.36)
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.35: Reed-M
¨
uller Four-Variable Example 7
Remember that it does not matter if constant zeros or ones are
used to simplify any given group, and when there is a one in the
6.17 one in first cell 143
top left square it is usually easiest to look for constant zeros for that
group (since that square is for input
0000
). Row one of this map has
a constant zero for
C
and
D
, and row three has a constant one for
C
and
D
. Since there is a one in the first cell, then the two terms must
be combined with an XNOR gate:
C D = Y (6.37)
1
1
1
1
1
1
1
1
CD
AB
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 6.36: Reed-M
¨
uller Four-Variable Example 8
The upper left corner of this map has a constant zero for
A
and
C
,
and the lower right corner has a constant one for
A
and
C
. Since there
is a one in the first cell, then the two variables must be combined with
an XNOR gate:
A C = Y (6.38)
7
A D VA N C E D S I M P L I F Y I N G M E T H O D S
What to Expect
Boolean expressions are used to describe logic circuits and by
simplifying those expressions the circuits can also be simplified.
This chapter introduces both the Quine-McCluskey technique
and the Karma Automated Simplification Tool. These are methods
that are useful for simplifying complex Boolean expressions
that include five or more input variables. This chapter includes
the following topics.
•
Simplifying a complex Boolean expression using the
Quine-McCluskey method
•
Creating the implicants and prime implicants of a Boolean
expression
•
Combining prime implicants to create a simplified
Boolean expression
•
Simplifying a complex Boolean expression using the
KARMA automated tool
7.1 quine-mccluskey simplification method
7.1.1 Introduction
This method was
developed by W.V.
Quine and Edward J.
McCluskey and is
sometimes called the
method of prime
implicants.
When a Boolean equation involves five or more variables it becomes
very difficult to solve using standard algebra techniques or Karnaugh
maps; however, the Quine-McCluskey algorithm can be used to solve
these types of Boolean equations.
The Quine-McCluskey method is based upon a simple Boolean
algebra principle: if two expressions differ by only a single variable
and its complement then those two expressions can be combined:
ABC + ABC
0
= AB (7.1)
The Quine-McCluskey method looks for expressions that differ
by only a single variable and combines them. Then it looks at the
combined expressions to find those that differ by a single variable and
145
146 advanced simplifying methods
combines them. The process continues until there are no expressions
remaining to be combined.
7.1.2 Example One
7.1.2.1 Step 1: Create the Implicants
Equation 7.2 is the Sigma representation of a Boolean equation.
Z
(A, B, C, D) =
X
(0, 1, 2, 5, 6, 7, 9, 10, 11, 14) (7.2)
Truth Table 7.1 shows the input variables for the True minterm
values.
Minterm A B C D
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
9 1 0 0 1
10 1 0 1 0
11 1 0 1 1
14 1 1 1 0
Table 7.1: Quine-McCluskey Ex 1: Minterm Table
To simplify this equation, the minterms that evaluate to True (as
listed above) are first placed in a minterm table so that they form
sections that are easy to combine. Each section contains only the
minterms that have the same number of ones. Thus, the first section
contains all minterms with zero ones, the second section contains
the minterms with one one, and so forth. Truth Table 7.2 shows the
minterms rearranged appropriately.
7.1 quine-mccluskey simplification method 147
Number of 1’s Minterm Binary
0 0 0000
1
1 0001
2 0010
2
5 0101
6 0110
9 1001
10 1010
3
7 0111
11 1011
14 1110
Table 7.2: Quine-McCluskey Ex 1: Rearranged Table
Start combining minterms with other minterms to create Size Two
Implicants (called that since each implicant combines two minterms),
but only those terms that vary by a single binary digit can be combined.
When two minterms are combined, the binary digit that is different
between the minterms is replaced by a dash, indicating that the digit
does not matter. For example,
0000
and
0001
can be combined to form
000−
. The table is modified to add a Size Two Implicant column that
indicates all of the combined terms. Note that every minterm must
be compared to every other minterm so all possible implicants are
formed. This is easier than it sounds, though, since terms in section
one must be compared only with section two, then those in section
two are compared with section three, and so forth, since each section
differs from the next by a single binary digit. The Size Two Implicant
column contains the combined binary form along with the numbers
of the minterms used to create that implicant. It is also important to
mark all minterms that are used to create the Size Two Implicants
since allowance must be made for any not combined. Therefore, in the
following table, as a minterm is used it is also struck through. Table
7.3 shows the Size Two Implicants that were found.
148 advanced simplifying methods
1’s Mntrm Bin Size 2
0 0 0000 000- (0,1)
1
1 0001 00-0 (0,2)
2 0010 0-01 (1,5)
2
5 0101 -001 (1,9)
6 0110 0-10 (2,6)
9 1001 -010 (2,10)
10 1010 01-1 (5,7)
3
7 0111 011- (6,7)
11 1011 -110 (6,14)
14 1110 10-1 (9,11)
101- (10,11)
1-10 (10,14)
Table 7.3: Quine-McCluskey Ex 1: Size 2 Implicants
All of the Size Two Implicants can now be combined to form Size
Four Implicants (those that combine a total of four minterms). Again,
it is essential to only combine those with only a single binary digit
difference. For this step, the dash can be considered the same as a
single binary digit, as long as it is in the same place for both implicants.
Thus,
−010
and
−110
can be combined to
− − 10
, but
−010
and
0 − 00
cannot be combined since the dash is in different places in those
numbers. It helps to match up the dashes first and then look at the
binary digits. Again, as the various size-two implicants are used
they are marked; but notice that a single size-four implicant actually
combines four size-two implicants. Table 7.4 shows the Size Four
Implicants.
7.1 quine-mccluskey simplification method 149
1’s Mntrm Bin Size 2 Size 4
0 0 0000 000- (0,1) –10 (2,10,6,14)
1
1 0001 00-0 (0,2)
2 0010 0-01 (1,5)
2
5 0101 -001 (1,9)
6 0110 0-10 (2,6)
9 1001 -010 (2,10)
10 1010 01-1 (5,7)
3
7 0111 011- (6,7)
11 1011 -110 (6,14)
14 1110 10-1 (9,11)
101- (10,11)
1-10 (10,14)
Table 7.4: Quine-McCluskey Ex 1: Size 4 Implicants
None of the terms can be combined any further. All of the minterms
or implicants that are not marked are Prime Implicants. In the table
above, for example, the Size Two Implicant
000−
is a Prime Implicant.
The Prime Implicants will be placed in a chart and further processed
in the next step.
7.1.2.2 Step 2: The Prime Implicant Table
A Prime Implicant Table can now be constructed, as in Table 7.5. The
prime implicants are listed down the left side of the table, the decimal
equivalent of the minterms goes across the top, and the Boolean
representation of the prime implicants is listed down the right side of
the table.
0 1 2 5 6 7 9 10 11 14
000 − (0, 1) X X A
0
B
0
C
0
00 − 0 (0, 2) X X A
0
B
0
D
0
0 − 01 (1, 5) X X A
0
C
0
D
−001 (1, 9) X X B
0
C
0
D
01 − 1 (5, 7) X X A
0
BD
011 − (6, 7) X X A
0
BC
10 − 1 (9, 11) X X AB
0
D
101 − (10, 11) X X AB
0
C
− − 10 (2, 10, 6, 14) X X X X CD
0
Table 7.5: Quine-McCluskey Ex 1: Prime Implicants
150 advanced simplifying methods
An X marks the intersection where each minterm (on the top row)
is used to form one of the prime implicants (in the left column). Thus,
minterm
0
(or
0000
) is used to form the prime implicant
000 − (0
,
1)
in row one and 00 − 0(0, 2) in row two.
The Essential Prime Implicants can be found by looking for columns
that contain only one X. The column for minterm
14
has only one X, in
the last row,
− − 10(2
,
10
,
6
,
14)
; thus, it is an Essential Prime Implicant.
That means that the term in the right column for the last row,
CD
0
,
must appear in the final simplified equation. However, that term also
covers the columns for
2
,
6
, and
10
; so they can be removed from the
table. The Prime Implicant table is then simplified to 7.6.
0 1 5 7 9 11
000 − (0, 1) X X A
0
B
0
C
0
00 − 0 ( 0, 2) X A
0
B
0
D
0
0 − 01 (1, 5) X X A
0
C
0
D
−001 (1, 9) X X B
0
C
0
D
01 − 1 ( 5, 7) X X A
0
BD
011 − (6, 7) X A
0
BC
10 − 1 ( 9, 11) X X AB
0
D
101 − (10, 11) X AB
0
C
Table 7.6: Quine-McCluskey Ex 1: 1st Iteration
The various rows can now be combined in any order the designer
desires. For example, if row
10 − 1(9
,
11)
, is selected as a required
implicant in the solution, then minterms
9
and
11
are accounted for in
the final equation, which means that all X marked in those columns
can be removed. When that is done, then, rows
101 − (10
,
11)
and
10 − 1(9
,
11)
no longer have any marks in the table, and they can be
removed. Table 7.7 shows the last iteration of this solution.
0 1 5 7
000 − (0, 1) X X A
0
B
0
C
0
00 − 0 ( 0, 2) X A
0
B
0
D
0
0 − 01 (1, 5) X X A
0
C
0
D
−001 (1, 9) X B
0
C
0
D
01 − 1 ( 5, 7) X X A
0
BD
011 − (6, 7) X A
0
BC
Table 7.7: Quine-McCluskey Ex 1: 2nd Iteration
7.1 quine-mccluskey simplification method 151
The designer next decided to select
01 − 1(5
,
7)
,
A
0
BD
, as a required
implicant. That will include minterms
5
and
7
, and those columns
may be removed along with rows
01 − 1( 5
,
7)
,
A
0
BD
, and
011 − ( 6
,
7)
,
A
0
BC, as shown in Table 7.8.
0 1
000 − (0, 1) X X A
0
B
0
C
0
00 − 0 ( 0, 2) X A
0
B
0
D
0
0 − 01 (1, 5) X A
0
C
0
D
−001 (1, 9) X B
0
C
0
D
Table 7.8: Quine-McCluskey Ex 1: 3rd Iteration
The last two minterms (
0
and
1
) can be covered by the implicant
000 − (0, 1), and that also eliminates the last three rows in the chart.
The original Boolean expression, then, has been simplified from ten
minterms to Equation 7.3.
A
0
B
0
C
0
+ A
0
BD + AB
0
D + CD
0
= Y (7.3)
7.1.3 Example Two
7.1.3.1 Step 1: Create the Implicants
Given Equation 7.4, which is a Sigma representation of a Boolean
equation.
Z
(A, B, C, D, E, F) =
X
(0, 1, 8, 9, 12, 13, 14, 15, 32, 33, 37, 39, 48, 56)
(7.4)
Truth Table 7.9 shows the True minterm values.
152 advanced simplifying methods
Minterm A B C D E F
0 0 0 0 0 0 0
1 0 0 0 0 0 1
8 0 0 1 0 0 0
9 0 0 1 0 0 1
12 0 0 1 1 0 0
13 0 0 1 1 0 1
14 0 0 1 1 1 0
15 0 0 1 1 1 1
32 1 0 0 0 0 0
33 1 0 0 0 0 1
37 1 0 0 1 0 1
39 1 0 0 1 1 1
48 1 1 0 0 0 0
56 1 1 1 0 0 0
Table 7.9: Quine-McCluskey Ex 2: Minterm Table
To simplify this equation, the minterms that evaluate to True are
placed in a minterm table so that they form sections that are easy to
combine. Each section contains only the minterms that have the same
number of ones. Thus, the first section contains all minterms with zero
ones, the second section contains the minterms with one one, and so
forth. Table 7.10 shows the rearranged truth table.
7.1 quine-mccluskey simplification method 153
Number of 1’s Minterm Binary
0 0 000000
1
1 000001
8 001000
32 100000
2
9 001001
12 001100
33 100001
48 110000
3
13 001101
14 001110
37 100101
56 111000
4
15 001111
39 100111
Table 7.10: Quine-McCluskey Ex 2: Rearranged Table
Start combining minterms with other minterms to create Size Two
Implicants, as in Table 7.11.
154 advanced simplifying methods
1’s Mntrm Bin Size 2
0 0 000000 00000- (0,1)
1
1 000001 -000000 (0,32)
8 001000 00-000 (0,8)
32 100000 -00001 (1,33)
2
9 001001 00-001 (1,9)
12 001100 10000- (32,33)
33 100001 1-0000 (32,48)
48 110000 00100- (8,9)
3
13 001101 001-00 (8,12)
14 001110 100-01 (33,37)
37 100101 001-01 (9,13)
56 111000 00110- (12,13)
4
15 001111 0011-0 (12,14)
39 100111 11-000 (48,56)
1001-1 (37,39)
0011-1 (13,15)
00111- (14,15)
Table 7.11: Quine-McCluskey Ex 2: Size Two Implicants
All of the Size Two Implicants can now be combined to form Size
Four Implicants, as in Table 7.12.
7.1 quine-mccluskey simplification method 155
1’s Mntrm Bin Size 2 Size 4
0 0 000000 00000- (0,1) -0000- (0,1,32,33)
1
1 000001 -000000 (0,32) 00-00- (0,1,8,9)
8 001000 00-000 (0,8) 001-0- (8,9,12,13)
32 100000 -00001 (1,33) 0011– (12,13,14,15)
2
9 001001 00-001 (1,9)
12 001100 10000- (32,33)
33 100001 1-0000 (32,48)
48 110000 00100- (8,9)
3
13 001101 001-00 (8,12)
14 001110 100-01 (33,37)
37 100101 001-01 (9,13)
56 111000 00110- (12,13)
4
15 001111 0011-0 (12,14)
39 100111 11-000 (48,56)
1001-1 (37,39)
0011-1 (13,15)
00111- (14,15)
Table 7.12: Quine-McCluskey Ex 2: Size 4 Implicants
None of the terms can be combined any further. All of the minterms
or implicants that are not struck through are Prime Implicants. In the
table above, for example,
1 − 0000
is a Prime Implicant. The Prime
Implicants are next placed in a table and further processed.
7.1.3.2 Step 2: The Prime Implicant Table
A Prime Implicant Table can now be constructed, as in Table 7.13. The
prime implicants are listed down the left side of the table, the decimal
equivalent of the minterms goes across the top, and the Boolean
representation of the prime implicants is listed down the right side of
the table.
156 advanced simplifying methods
0 1 8 9 12 13 14 15 32 33 37 39 48 56
11 − 000 (48, 56) X X ABD
0
D
0
F
0
00 − 00 − (0, 1, 8, 9) X X X X A
0
B
0
D
0
E
0
1001 − 1 (37, 39) X X AB
0
C
0
DF
1 − 0000 (32, 48) X X AC
0
D
0
E
0
F
0
0011 − − (12, 13, 14, 15) X X X X A
0
B
0
CD
−0000 − (0, 1, 32, 33) X X X X B
0
C
0
D
0
E
0
001 − 0 − (8, 9, 12, 13) X X X X A
0
B
0
CE
0
100 − 01 (33, 37) X X AB
0
C
0
E
0
F
Table 7.13: Quine-McCluskey Ex 2: Prime Implicants
In the above table, there are four columns that contain only one X:
14
,
15
,
39
, and
56
. The rows that intersect the columns at that mark are
Essential Prime Inplicants, and their Boolean Expressions must appear
in the final equation. Therefore, the final equation will contain, at
a minimum:
A
0
B
0
CD
(row
5
, covers minterms
14
and
15
),
AB
0
C
0
DF
(row
3
, covers minterm
39
), and
ABD
0
E
0
F
0
(row
1
, covers minterm
56
).
Since those expressions are in the final equation, the rows that contain
those expressions can be removed from the chart in order to make
further analysis less confusing.
Also, because the rows with Essential Prime Implicants are con-
tained in the final equation, other minterms marked by those rows
are covered and need no further consideration. For example, minterm
48
is covered by row one (used for minterm
56
), so column
48
can be
removed from the table. In a similar fashion, columns
12
,
13
, and
37
are covered by other minterms, so they can be removed from the table.
Table 7.14 shows the next iteration of this process.
0 1 8 9 32 33
00 − 00 − (0, 1, 8, 9) X X X X A
0
B
0
D
0
E
0
1 − 0000 (32, 48) X AC
0
D
0
E
0
F
0
−0000 − (0, 1, 32, 33) X X X X B
0
C
0
D
0
E
0
001 − 0 − (8, 9, 12, 13) X X A
0
B
0
CE
0
100 − 01 (33, 37) X AB
0
C
0
E
0
F
Table 7.14: Quine-McCluskey Ex 2: 1st Iteration
The circuit designer can select the next term to include in the final
equation from any of the five rows still remaining in the chart; however,
the first term (00 − 00−, or A
0
B
0
D
0
E
0
) would eliminate four columns,
so that would be a logical next choice. When that term is selected for
the final equation, then row one,
00 − 00−
, can be removed from the
chart; and columns
0
,
1
,
8
, and
9
can be removed since those minterms
are covered.
7.1 quine-mccluskey simplification method 157
The minterms marked for row
001 − 0 − (8
,
9
,
12
,
13)
are also covered,
so this row can be removed. Table 7.15 shows the next iteration.
32 33
1 − 0000 (32, 48) X AC
0
D
0
E
0
F
0
−0000 − (0, 1, 32, 33) X X B
0
C
0
D
0
E
0
100 − 01 (33, 37) X AB
0
C
0
E
0
F
Table 7.15: Quine-McCluskey Ex 2: 2nd Iteration
For the next simplification, row
−0000−
is selected since that would
also cover the minterms that are marked for all remaining rows. Thus,
the expression B
0
C
0
D
0
E
0
will become part of the final equation.
When the analysis is completed, the original equation (7.4), which
contained
14
minterms, is simplified into Equation 7.5, which contains
only five terms.
ABD
0
E
0
F
0
+ A
0
B
0
D
0
E
0
+ AB
0
C
0
DF + A
0
B
0
CD + B
0
C
0
D
0
E
0
= Y
(7.5)
7.1.4 Summary
While the Quine–McCluskey method is useful for large Boolean ex-
pressions containing multiple inputs, it is also tedious and prone to
error when done by hand. Also, there are some Boolean expressions
(called “Cyclic” and “Semi-Cyclic” Primes) that do not reduce us-
ing this method. Finally, both Karnaugh maps and Quine-McCluskey
methods become very complex when more than one output is required
of a circuit. Fortunately, many automated tools are available to simplify
Boolean expressions using advanced mathematical techniques.
7.1.5 Practice Problems
The following problems are presented as practice for using the Quine-
McClusky method to simplify a Boolean expression. Note: designers
can select different Prime Implicants so the simplified expression could
vary from what is presented below.
158 advanced simplifying methods
1
Expression
R
(A
,
B
,
C
,
D) =
P
(0, 1, 2, 5, 6, 7, 9, 10, 11, 14)
Simplified A
0
B
0
C
0
+ A
0
BD + AB
0
D + CD
0
2
Exression
R
(A
,
B
,
C
,
D) =
P
(0, 1, 2, 3, 6, 7, 8, 9, 14, 15)
Simplified A
0
C + BC + B
0
C
0
3
Exression
R
(A
,
B
,
C
,
D) =
P
(1, 5, 7, 8, 9, 10, 11, 13, 15)
Simplified C
0
D + AB
0
+ BD
3
Exression
R
(A
,
B
,
C
,
D
,
E) =
P
(0
,
4
,
8
,
9
,
10
,
11
,
12
,
13
,
14
,
15
,
16
,
20
,
24
,
28)
Simplified A
0
B + D
0
E
0
Table 7.16: Quine-McCluskey Practice Problems
7.2 automated tools 159
7.2 automated tools
7.2.1 KARMA
There are numerous automated tools available to aid in simplifying
complex Boolean equations. Many of the tools are quite expensive
and intended for professionals working full time in large companies;
but others are inexpensive, or even free of charge, and are more than
adequate for student use. One free tool, a Java application named
KARnaugh MAp simplifier (KARMA), can be downloaded from
http://bit.ly/2mcXVp9
and installed on a local computer, though
the website also has a free online version that can be used without
installation. KARMA helps to simplify complex Boolean expressions
using both Karnaugh Maps and Quine-McCluskey methods. KARMA
is a good program with numerous benefits.
When first started, KARMA looks like Figure 7.1.
Figure 7.1: KARMA Start Screen
The right side of the screen contains a row of tools available in
KARMA and the main part of the screen is a canvas where most of
the work is done. The following tools are available:
• Logic2Logic
. Converts between two different logical represen-
tations of data; for example, a Truth Table can be converted to
Boolean expressions.
• Logic Equivalence
. Compares two functions and determines if
they are equivalent; for example, a truth table can be compared
with a SOP expression to see if they are the same.
160 advanced simplifying methods
• Logic Probability
. Calculates the probability of any one outcome
for a given Boolean expression.
• Karnaugh Map
. Analyzes a Karnaugh map and returns the
Minimized Expression.
• KM Teaching Mode
. Provides drill and practice with Karnaugh
maps; for example, finding adjacent minterms on a 6-variable
map.
• SOP and POS
. Finds the SOP and POS expressions for a given
function.
• Exclusive-OR. Uses XOR gates to simplify an expression.
• Multiplexer-Based. Realizes a function using multiplexers.
• Factorization. Factors Boolean expressions.
• About. Information about Karma.
For this lesson, only the Karnaugh Map analyzer will be used, and
the initial screen for that function is illustrated in Figure 7.2.
Figure 7.2: Karnaugh Map Screen
7.2.1.1 Data Entry
When using KARMA, the first step is to input some sort of information
about the circuit to be analyzed. That information can be entered in
several different formats, but the most common for this class would
be either a truth table or a Boolean expression.
To enter the initial data, click the Load Function button at the top of
the canvas.
7.2 automated tools 161
By default, the Load Function screen opens with a blank screen. In
the lower left corner of the Load Function window, the Source Format
for the input data can be selected. There is a template available for
each of the different source formats; and that template can be used
to help with data entry. The best way to work with KARMA is to
click the “Templates” button and select the data format being used. In
Figure 7.3, the “Expression 1” template was selected:
Figure 7.3: The Expression 1 Template
The designer would replace the “inputs” and “onset” lines with
information for the circuit being simplified. Once the source data are
entered into this window, click the “Load” button at the bottom of the
window to load the data into KARMA.
7.2.1.2 Data Source Formats
Karma works with input data in any of six different formats: Boolean
Expression, Truth Table, Integer, Minterms, Berkeley Logic Interchange
Format (BLIF), and Binary Decision Diagram (BDD). BLIF and BDD
are programming tools that are beyond the scope of this lesson and
will not be covered.
expression Boolean expressions can be defined in Karma using
the following format.
#Sample Expression
162 advanced simplifying methods
(!x1*!x2*!x4)+(!x1*x2*!x3)+(x1*!x4*!x5)+(x1*x3*x4)
Notes:
•
Any line that starts with a hash tag (“#”) is a comment and will
be ignored by Karma.
•
“Not” is indicated by a leading exclamation mark. Thus “!x1” is
the same as X1’.
•
All operations are explicit. In real-number algebra the phrase
“AB” is understood to be “A*B”. However, in KARMA, since vari-
able names can be more than one character long, all operations
must be explicitly stated. AND is indicated by an asterisk and
OR is indicated by a plus sign.
• No space is left between operations.
truth table A truth table can be defined in KARMA using the
following format.
#Sample Truth Table
inputs -> X, Y, Z
000 : 1
001 : 1
010 : 0
011 : 0
100 : 0
101 : 1
110 : 0
111 : 1
Notes:
•
Any line that starts with a hash tag (“#”) is a comment and will
be ignored by KARMA.
•
The various inputs are named before they are used. In the exam-
ple, there are three inputs: X, Y, and Z.
•
Each row in the truth table is shown, along with the output
required. So, in the example above, an input of 000 should yield
an output of 1.
• An output of “-” is permitted and means “don’t care.”
integer In Karma, an integer can be used to define the outputs
of the truth table, so it is “shorthand” for an entire truth table input.
Following is the example of the “integer” type input.
7.2 automated tools 163
#Sample Integer Input
inputs -> A, B, C, D
onset -> E81A base 16
Notes:
•
Any line that starts with a hash mark (“#”) is a comment and
will be ignored by Karma.
•
Input variables are defined first. In this example, there are four
inputs: A, B, C, and D.
•
The “onset” line indicates what combinations of inputs should
yield a True on a truth table.
In the example, the number E81A is a hexadecimal number that is
written like this in binary:
1110 1000 0001 1010
E 8 1 A
The least significant bit of the binary number, 0 in this example,
corresponds to the output of the first row in the truth table; thus, it
is false. Each bit to the left of the least significant bit corresponds to
the next row, counting from 0000 to 1111. Following is the truth table
generated by the hexadecimal integer E81A.
164 advanced simplifying methods
Inputs Output
A B C D Q
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 1
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 0
1 0 0 0 0
1 0 0 1 0
1 0 1 0 0
1 0 1 1 1
1 1 0 0 0
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1
Table 7.17: Truth Table for E81A Output
The “Output” column contains the binary integer
1110 1000 0001
1010 (or E81Ah) from bottom to top.
terms Data input can be defined by using the minterms for the
Boolean expression. Following is an example minterm input.
#Sample Minterms
inputs -> A, B, C, D
onset -> 0, 1, 2, 3, 5, 10
Notes:
•
Any line that starts with a hash mark (“#”) is a comment and
will be ignored by KARMA.
• The inputs, A, B, C, and D, are defined first.
•
The “onset” line indicates the minterms that yield a “true” out-
put.
•
This is similar to a SOP Sigma expression, and the digits in
that expression could be directly entered on the onset line. For
example, the onset line above would have been generated from
this Sigma expression:
7.2 automated tools 165
R
(A, B, C, D) =
P
(0, 1, 2, 3, 5, 10)
7.2.1.3 Truth Table and Karnaugh Map Input
While Karma will accept a number of different input methods, as
described above, one of the easiest to use is the Truth Table and its
related Karnaugh Map, and these are displayed by default when the
Karnaugh Map function is selected. The value of any of the cells in the
Out column in the Truth Table, or cells in the Karnaugh Map, and can
be cycled through 0, 1, and “don’t care” (indicated by a dash) on each
click of the mouse in the cell. The Truth Table and Karnaugh Map
are synchronized as cells are clicked. The number of input variables
can be adjusted by changing the Var setting at the top of the screen.
Also, the placement of those variables on the Karnaugh Map can be
adjusted as desired.
7.2.1.4 Solution
To simplify the Karnaugh Map, click the Minimize button. A number
of windows will pop up (illustrated in Figure 7.4), each showing the
circuit simplification in a slightly different way. Note: this Boolean
expression was entered to generate the following illustrations:
A
0
C +
A
0
B + AB
0
C
0
+ B
0
C
0
D
0
.
Figure 7.4: A KARMA Solution
boolean expression The minimized Boolean expression is shown
in Figure 7.5.
166 advanced simplifying methods
Figure 7.5: The Minimized Boolean Expression
In this solution, a NOT term is identified by a leading exclamation
point; thus, the minimized expression is:
A
0
D
0
+ AB
0
C
0
+ A
0
C + A
0
B
.
bddeiro The BDDeiro window is a form of Binary Decision Dia-
gram (BDD). The decision tree for the minimized expression is repre-
sented graphically in Figure 7.6.
Figure 7.6: The BDDeiro Solution
The top node represents input a, which can either be true or false. If
false, then follow the blue dotted line down to node d, which can also
be either true or false. If d is false, follow the blue dotted line down to
the 1 output. This would mean that one True output for this circuit is
A
0
D
0
, which can be verified as one of the outputs in the minimized
expression.
Starting at the top again, if a is true, then follow the solid red line
down to b. It that is true, then follow the red solid line down to
the 0 output. The expression
AB
is false and does not appear in the
minimized solution. In a similar way, all four True outputs, and three
false outputs, can be traced from the top to bottom of the diagram.
7.2 automated tools 167
quine-mccluskey Karma includes complete Quine-McCluskey
solution data. Several tables display the various implicants and show
how they are derived.
Figure 7.7: Quine-McCluskey Solution
Karma also displays the Covering Table for a Quine-McCluskey
solution. Each of the minterms (down the left column) can be turned
on or off by clicking on it. The smaller blue balls in the table indicate
prime implicants and the larger red balls (if any) indicate essential
prime implicants. Because this table is interactive, various different
solutions can be attempted by clicking some of the colored markers to
achieve the best possible simplification.
168 advanced simplifying methods
Figure 7.8: Selecting Implicants
7.2.1.5 Practice Problems
The following problems are presented as practice for usingKarma to
simplify a Boolean expression. Note: designers can select different
Prime Implicants so the simplified expression could vary from what
is presented below.
1
Expression
R
(A, B, C, D) =
P
(5, 6, 7, 9, 10, 11, 13, 14)
Simplified BC
0
D + A
0
BC + ACD
0
+ AB
0
D
2
Exression A
0
BC
0
D + A
0
BCD
0
+ A
0
BCD + AB
0
C
0
D +
AB
0
CD
0
+ AB
0
CD + AB
0
CD + ABC
0
D +
ABCD
0
Simplified BC
0
D + A
0
BC + ACD
0
+ AB
0
D
3
Exression
4-variable Karnaugh Map where cells 5,6,7,9,10 are
True and 13,14 are Don’t Care
Simplified BC
0
D + AC
0
D + A
0
BC + ACD
0
3
Exression
R
(A
,
B
,
C
,
D
,
E) =
P
(0, 3, 4, 12, 13, 14, 15, 24, 25, 28, 29, 30)
Simplified ABD
0
+ A
0
B
0
C
0
DE + BCE
0
+ A
0
BC + A
0
B
0
D
0
E
0
Table 7.18: KARMA Practice Problems
Part II
P R A C T I C E
Once the foundations of digital logic are mastered it is time
to consider creating circuits that accomplish some practical
task. This part of the book begins with the simplest of
combinational logic circuits and progresses to sequential
logic circuits and, finally, to complex circuits that combine
both combinational and sequential elements.
8
A R I T H M E T I C C I R C U I T S
What to Expect
This chapter develops various types of digital arithmetic circuits,
like adders and subtractors. It also introduces Arithmetic-Logic
Units (ALUs), which combine both arithmetic and logic func-
tions in a single integrated circuit. The following topics are
included in this chapter.
•
Developing and using half-adders, full adders, and cas-
cading adders
•
Developing and using half-subtractors, full subtractors,
and cascading subtractors
• Combining an adder and subtractor in a single IC
• Describing ALUs
• Creating a comparator
8.1 adders and subtractors
8.1.1 Introduction
In the Binary Mathematics chapter, the concept of adding two binary
numbers was developed and the process of adding binary numbers
can be easily implemented in hardware. If two one-bit numbers are
added, they will produce a sum and an optional carry-out bit and
the circuit that performs this is called a “half adder.” If two one-bit
numbers along with a carry-in bit from another stage are added they
will produce a sum and an optional carry-out, and the circuit that
performs this is function called a “full adder.”
Subtractors are similar to adders but they must be able to signal a
“borrow” bit rather than send a carry-out bit. Like adders, subtractors
are developed from half-subtractor circuits. This unit develops both
adders and subtractors.
8.1.2 Half Adder
Following is the truth table for a half adder.
173
174 arithmetic circuits
Inputs Output
A B Sum COut
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
Table 8.1: Truth Table for Half-Adder
The Sum column in this truth table is the same pattern as an XOR) so
the easiest way to create a half adder is to use an XOR gate. However,
if both input bits are high a half adder will also generate a carry out
(COut) bit, so the half adder circuit should be designed to provide that
carry out. The circuit in Figure 8.1 meets those requirements. In this
circuit, A and B are connected to XOR gate U1 and that is connected
to output Sum. A and B are also connected to AND gate U2 and that is
connected to carry-out bit COut.
Figure 8.1: Half-Adder
8.1.3 Full Adder
A full adder sums two one-bit numbers along with a carry-in bit and
produces a sum with a carry-out bit. Truth table 8.2 defines a full
adder.
8.1 adders and subtractors 175
Inputs Output
A B CIn Sum COut
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Table 8.2: Truth Table for Full Adder
Following are Karnaugh Maps for both outputs.
1
1 1
1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 8.2: K-Map For The SUM Output
1 1
1
1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 8.3: K-Map For The COut Output
Karnaugh map 8.2 is a Reed-Muller pattern that is typical of an XOR
gate. Karnaugh map 8.3 can be reduced to three Boolean expressions.
The full adder circuit is, therefore, defined by the following Boolean
equations.
176 arithmetic circuits
A ⊕ B ⊕ CIn = Sum (8.1)
(A ∗ B) + (A ∗ CIn) + (B ∗ CIn) = COut
Figure 8.4 is a full adder. In essence, this circuit combines two half-
adders such that U1 and U2 are one half-adder that sums A and B
while U3 and U4 are the other half-adder that sums the output of the
first half-adder and CIn.
Figure 8.4: Full Adder
8.1.4 Cascading Adders
The full adder developed above will only add two one-bit numbers
along with an optional carry-in bit; however, those adders can be
cascaded such that an adder of any bit width can be easily created.
Figure 8.5 shows a four-bit adder created by cascading four one-bit
adders.
8.1 adders and subtractors 177
Figure 8.5: 4-Bit Adder
This circuit would add two four-bit inputs, A and B. Stage zero, at
the top of the stack, adds bit zero from both inputs and then outputs
bit zero of the sum, Y0, along with a carry-out bit. The carry-out bit
from stage zero is wired directly into the stage one’s carry-in port.
That adder then adds bit one from both inputs along with the carry-
in bit to create bit one of the sum, Y1, along with a carry-out bit.
This process continues until all for bits have been added. In the end,
outputs Y0 - Y3 are combined to create a four-bit sum. If there is a
carry-out bit from the last stage it could be used as the carry-in bit for
another device or could be used to signal an overflow error.
8.1.5 Half Subtractor
To understand a binary subtraction circuit, it is helpful begin with
subtraction in a base-10 system.
83
-65
18
Since
5
cannot be subtracted from
3
(the least significant digits),
10
must be borrowed from
8
. This is simple elementary-school arithmetic
178 arithmetic circuits
but the principle is important for base-2 subtraction. There are only
four possible one-bit subtraction problems.
0 1 1 10
-0 -0 -1 -1
0 1 0 1
The first three examples above need no explanation, but the fourth
only makes sense when it is understood that it is impossible to subtract
1
from
0
so
10
2
was borrowed from the next most significant bit
position. The problems above were used to generate the following
half-subtractor truth table.
Inputs Outputs
A B Diff BOut
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0
Table 8.3: Truth Table for Half-Subtractor
Diff is the difference of A minus B. BOut (“Borrow Out”) is a signal
that a borrow is necessary from the next most significant bit position
when B is greater than A. The following Boolean equations define the
calculations needed for a half-subtractor.
A ⊕ B = Diff (8.2)
A
0
∗ B = BOut
The pattern for Diff is the same as an XOR gate so using an XOR
gate is the easiest way to generate the difference. BOut is only high
when A is low and B is high so a simple AND gate with one inverted
input can be used to generate BOut. The circuit in figure 8.6 realizes a
half-subtractor.
Figure 8.6: Half-Subtractor
8.1 adders and subtractors 179
8.1.6 Full Subtractor
A full subtractor produces a difference and borrow-out signal, just
like a half-subtractor, but also includes a borrow-in signal so they can
be cascaded to create a subtractor of any desired bit width.
Truth table 8.4 is for a full subtractor.
Inputs Output
A B BIn Diff BOut
0 0 0 0 0
0 0 1 1 1
0 1 0 1 1
0 1 1 0 1
1 0 0 1 0
1 0 1 0 0
1 1 0 0 0
1 1 1 1 1
Table 8.4: Truth Table for Subtractor
Diff is the difference of A minus B minus BIn. BOut (“Borrow Out”)
is a signal that a borrow is necessary from the next most significant
bit position when A is less than B plus BIn.
Following are Karnaugh Maps for both outputs.
1
1 1
1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 8.7: K-Map For The Difference Output
180 arithmetic circuits
1
1
1 1
C
AB
00 01 11 10
0
1
00 02 06 04
01 03 07 05
Figure 8.8: K-Map For The BOut Output
Karnaugh map 8.7 is a Reed-Muller pattern that is typical of an XOR
gate. Karnaugh map 8.8 can be reduced to three Boolean expressions.
The full subtactor circuit is, therefore, defined by the following Boolean
equations.
A ⊕ B ⊕ BIn = Diff (8.3)
A
0
B + (A
0
∗ BIn) + (B ∗ BIn) = BOut
The circuit in figure 8.9 realizes a subtractor.
Figure 8.9: Subtractor
8.1.7 Cascading Subtractors
The full subtractor developed above will only subtract two one-bit
numbers along with an optional borrow bit; however, those subtractors
can be cascaded such that a subtractor of any bit width can be easily
created. Figure 8.10 shows a four-bit subtractor created by cascading
four one-bit subtractors.
8.1 adders and subtractors 181
Figure 8.10: 4-Bit Subtractor
This circuit would subtract a four bit number B from A. The sub-
tractor is set up to solve
1110
2
− 0011
2
= 1011
2
. Stage zero, at the top
of the stack, subtracts bit zero of input B from bit zero of input A and
then outputs bit zero of the difference, Y0, along with a borrow-out
bit. The borrow-out bit from stage zero is wired directly into the stage
one’s borrow-in port. That stage then subtracts bit one of input B from
bit one of input A along with the borrow-in bit to create bit one of the
sum, Y1, along with a borrow-out bit. This process continues until all
for bits have been subtracted. In the end, outputs Y0 - Y3 are combined
to create a four-bit difference. The borrow-out bit of the last stage is
not connected to anything but it could be used as the borrow-in bit
for another device.
8.1.8 Adder-Subtractor Circuit
It is remarkably easy to create a device that both adds and subtracts
based on a single-bit control signal. Figure 8.11 is a 4-bit adder that
was modified to become both an adder and subtractor. The circuit has
been set up with this problem: 0101
2
− 0011
2
= 0010
2
.
182 arithmetic circuits
Figure 8.11: 4-Bit Adder-Subtractor
To change an adder to an adder-subtractor makes use of the binary
mathematics concept of subtracting by adding the twos complement
(see Section 3.2.5.2 on page 45). The “trick” is to use the XOR gates on
input B to convert that input to its complement then the adder will
subtract B from A instead of add.
To create the twos complement of a binary number each of the bits
are complemented and then one is added to the result (again, this
process is described in Section 3.2.5.2). Each of the B input bits are
wired through one input of an XOR gate. The other input of that gate
is a Ctrl (“Control”) bit. When Ctrl is low then each of the B inputs
are transmitted through an XOR gate without change and the adder
works as an adder. When Ctrl is high then each of the B inputs are
complemented by an XOR gate such that the ones complement is
created. However, Ctrl is also wired to the CIn input of the first stage
which has the effect of adding one to the result and turn input B into a
twos complement number. Now the adder will subtract input B from
input A.
In the end, the designer only needs to set Ctrl to zero to make the
circuit add or one to make the circuit subtract.
8.2 arithmetic logic units 183
8.1.9 Integrated Circuits
In practice, circuit designers rarely build adders or subtractors. There
are many different types of manufactured low-cost adders, subtractors,
and adder/subtractor combinations available and designers usually
find it easiest to use one of those circuits rather than re-invent the
proverbial wheel. A quick look at Wikipedia
1
found this list of adders:
• 7480, gated full adder
• 7482, two-bit binary full adder
• 7483, four-bit binary full adder
• 74183, dual carry-save full adder
• 74283, four-bit binary full adder
• 74385, quad four-bit adder/subtractor
• 74456, BCD adder
In addition to adder circuits, designers can also opt to use an ALU
IC.
8.2 arithmetic logic units
An ALU is a specialized IC that performs all arithmetic and logic
functions needed in a device. Most ALUs will carry out dozens of
different functions like the following few examples from a
74181
ALU
(assume that the ALU has two inputs, A and B, and one output, F):
• F = NOT A
• F = A NAND B
• F = (NOT A) OR B
• F = B
• F = (NOT A) AND B
• F = A − 1
• F = A − B
• F = AB − 1
• F = −1
1 https://www.wikipedia.com/en/List of 7400 series integrated circuits
184 arithmetic circuits
ALUs are very important in many devices, in fact, they are at the
core of a CPU. Because they are readily available at low cost, most
designers will use a commercially-produced ALU in a project rather
than try to create their own.
A quick look at Wikipedia
2
found this list of ALUs:
• 74181, four-bit arithmetic logic unit and function generator
• 74381
, four-bit arithmetic logic unit/function generator with
generate and propagate outputs
• 74382
, four-bit arithmetic logic unit/function generator with
ripple carry and overflow outputs
• 74881, Arithmetic logic unit
8.3 comparators
A comparator compares two binary numbers, A and B. One of three
outputs is generated by the comparison:
A = B, A > B, A < B
. A one-
bit comparator uses a combination of AND gates, NOT gates, and an
XNOR gate to generate a True output for each of the three comparisons:
A = B (A B)
0
A > B AB
0
A < B A
0
B
Table 8.5: One-Bit Comparator Functions
Figure 8.12 is the logic diagram for a one-bit comparator.
Figure 8.12: One-Bit Comparator
To compare numbers larger than one bit requires a more involved
analysis of the problem. First, a truth table is developed for every
possible combination of two 2-bit numbers, A and B.
2 https://www.wikipedia.com/en/List of 7400 series integrated circuits
8.3 comparators 185
Inputs Outputs
A1 A0 B1 B0 A < B A = B A > B
0 0 0 0 0 1 0
0 0 0 1 1 0 0
0 0 1 0 1 0 0
0 0 1 1 1 0 0
0 1 0 0 0 0 1
0 1 0 1 0 1 0
0 1 1 0 1 0 0
0 1 1 1 1 0 0
1 0 0 0 0 0 1
1 0 0 1 0 0 1
1 0 1 0 0 1 0
1 0 1 1 1 0 0
1 1 0 0 0 0 1
1 1 0 1 0 0 1
1 1 1 0 0 0 1
1 1 1 1 0 1 0
Table 8.6: Truth Table for Two-Bit Comparator
Next, Karnaugh Maps are developed for each of the three outputs.
1
1 1 1
1 1
A1A0
B1B0
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 8.13: K-Map For A > B
186 arithmetic circuits
1
1
1
1
A1A0
B1B0
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 8.14: K-Map For A = B
1 1 1
1 1
1
A1A0
B1B0
00 01 11 10
00
01
11
10
00 04 12 08
01 05 13 09
03 07 15 11
02 06 14 10
Figure 8.15: K-Map For A = B
Given the above K-Maps, the following Boolean Equations can be
derived.
A < B : A1
0
B1 + A0
0
B1B0 + A1
0
A0
0
B0 (8.4)
A = B : ( A0 B0)(A1 B1)
A > B : A1B1
0
+ A0B1
0
B0
0
+ A1A0B0
0
The above Boolean expressions can be used to create the circuit in
Figure 8.16.
8.3 comparators 187
Figure 8.16: Two-Bit Comparator
9
E N C O D E R C I R C U I T S
What to Expect
Encoders and decoders are used to change a coded byte from
one form to another. For example, a binary-to-BCD encoder
changes a binary byte to its BCD equivalent. Encoders and
decoders are very common in digital circuits and are used, for
example, to change a binary number to a visual display that
uses an Light Emitting Diode (LED). The following topics are
included in this chapter.
•
Developing circuits the use multiplexers and demultiplex-
ers
• Creating a minterm generator using a multiplexers
• Creating a ten-line priority encoder
•
Using a seven-segment display for a decoded binary num-
ber
• Employing a decoder as a function generator
•
Explaining the theory and process of error detection and
correction
•
Detecting errors in a transmitted byte using the Hamming
Code
9.1 multiplexers/demultiplexers
9.1.1 Multiplexer
A multiplexer is
usually called a
“mux” and a
demultiplexer is
called a “dmux.”
A multiplexer is used to connect one of several input lines to a sin-
gle output line. Thus, it selects which input to pass to the output.
This function is similar to a rotary switch where several potential
inputs to the switch can be sent to a single output. A demultiplexer
is a multiplexer in reverse, so a single input can be routed to any
of several outputs. While mux/dmux circuits were originally built
for transmission systems (like using a single copper wire to carry
several different telephone calls simultaneously), today they are used
189
190 encoder circuits
as “decision-makers” in virtually every digital logic system and are,
therefore, one of the most important devices for circuit designers.
To help clarify this concept, Figure 9.1 is a simple schematic diagram
that shows two rotary switches set up as a mux/dmux pair. As the
switches are set in the diagram, a signal would travel from INPUT B
to OUTPUT 2 through a connecting wire.
A
B
C
D
E
1
2
3
4
5
Figure 9.1: Multiplexer Using Rotary Switches
Imagine that the switches could somehow be synchronized so they
rotated among the setting together; that is, INPUT A would always
connect to OUTPUT 1 and so forth. That would mean a single wire
could carry five different signals. For example, imagine that the inputs
were connected to five different intrusion sensors in a building and
the five outputs were connected to lamps on a guard’s console in a
remote building. If something triggered sensor A then as soon as the
mux/dmux pair rotated to that position it would light lamp one on
the console. Carrying all of these signals on a single wire saves a lot of
expense. Of course, a true alarm system would be more complex than
this, but this example is only designed to illustrate how a mux/dmux
pair works in a transmission system.
Figure 9.2 is the logic diagram for a simple one-bit two-to-one
multiplexer. In this circuit, an input is applied to input ports A and
B. Port Sel is the selector and if that signal is zero then Port A will be
routed to output Y; if, though, Sel is one then Port B will be routed to
Y.
Figure 9.2: Simple Mux
Truth Table 9.1, below, is for a multiplexer:
9.1 multiplexers/demultiplexers 191
Inputs Output
A B Sel Y
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
Table 9.1: Truth Table for a Multiplexer
In this multiplexer, the data input ports are only a single bit wide;
however, in a normal circuit those ports could be a full
32
-bit or
64
-
bit word and the selected word would be passed to the output port.
Moreover, a multiplexer can have more than two input ports so a
very versatile switch can be built to handle switching full words from
one of eight or even sixteen different inputs. Because of its ability to
channel a selected data stream to a single bus line from many different
sub-circuits, the multiplexer is one of the workhorses for digital logic
circuits and is frequently found in complex devices like CPUs.
9.1.2 Demultiplexer
A demultiplexer is the functional opposite of a multiplexer: a single
input is routed to one of several potential outputs. Figure 9.3 is the
logic diagram for a one-bit one-to-two demultiplexer. In this circuit,
an input is applied to input port A. Port Sel is a control signal and if
that signal is zero then input A will be routed to output Y, but if the
control signal is one then input A will be routed to output Z.
Figure 9.3: Simple Dmux
Truth Table 9.2, below, is for a demultiplexer.
192 encoder circuits
Inputs Outputs
A Sel Y Z
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 0
Table 9.2: Truth Table for a Demultiplexer
In this demultiplexer the data input port is only a single bit wide;
however, in a normal circuit that port could be a full
32
-bit or
64
-bit
word and that entire word would be passed to the selected output
port. Moreover, a demultiplexer can have more than two outputs so
a very versatile switch can be built to handle switching full words to
one of eight or even sixteen different outputs. Because of its ability to
switch a data stream to different sub-circuits, the demultiplexer is one
of the workhorses for digital logic circuits and is frequently found in
complex devices like CPUs.
9.1.3 Minterm Generators
Demultiplexers can be combined with an OR gate and be used as a
minterm generator. Consider the circuit for this two-variable equation.
Z
(A, B) =
X
(1, 2) (9.1)
Since there are two input variables, A and B, the dmux needs to
have two select bits, one for each variable, and that would generate
four potential dmux outputs W, X, Y, and Z. This circuit could be
constructed using a four-output dmux with a two-bit control signal.
Figure 9.4: 1-to-4 Dmux
In Figure 9.4, a four-bit input (A) is routed to one of four output
ports: W, X, Y, or Z, depending on the setting of the select, Sel. Figure
9.4 shows the data input of 1010 being routed to Y by a select of 10.
However, the equation specifies that the only outputs that would
be used are when Sel is
01
or
10
. Thus, output ports X and Y must be
9.2 encoders/decoders 193
sent to an OR gate and the other two outputs ignored. The output of
the OR gate would only activate when Sel is set to
01
or
10
, as shown
in Figure 9.5.
Figure 9.5: 1-to-4 Dmux As Minterm Generator
9.2 encoders/decoders
9.2.1 Introduction
Encoders and decoders are used to convert some sort of coded data
into a different code. For example, it may be necessary to convert the
code created by a keyboard into ASCII for use in a word processor.
By definition, the difference between an encoder and a decoder is
the number of inputs and outputs: Encoders have more inputs than
outputs, while decoders have more outputs than inputs.
As an introduction to encoders, consider Figure 9.6, which is de-
signed to encode three single line inputs (maybe three different push
buttons on a control panel) into a binary number for further process-
ing by the computer. In this circuit, the junction between the two OR
gates and the output (
Y
) is a joiner that combines two bit streams into
a single bus. Physically, two wires (one from U1 and one from U2)
would be spliced together into a single cable (a bus) that contains two
strands. The figure shows that when input C is active the output of
the encoder is 11.
Figure 9.6: Three-line to 2-Bit Encoder
As an introduction to decoders, consider Figure 9.7, which is de-
signed to decode a two-bit binary input and drive a single output line
high. A circuit like this may be used to light an LED used as a warning
on a console if a particular binary code is generated elsewhere in a
circuit. In this circuit, the input is a two-bit number (
10
in the illustra-
tion), but those two bits are separated through a splitter and each is
applied to one of the inputs of a series of four AND gates. Imagine
194 encoder circuits
that the MSB was placed on the wire on the left of the grid and wired
to the bottom input of each of the AND gates. If
A = 10
, then AND
gate three would activate and output X would go high.
Figure 9.7: Four-Bit to 4-Line Decoder
Both encoders and decoders are quite common and are used in
many electronic devices. However, it is not very common to build these
circuits out of discrete components (like in the circuits above). Rather,
inexpensive integrated circuits are available for most encoder/decoder
operations and these are much easier, and more reliable, to use.
9.2.2 Ten-Line Priority
This encoder is
sometimes called
Ten-Line to
Four-Line.
Consider a ten-key keypad containing the numbers zero through
nine, like a keypad that could be used for numeric input from some
hand-held device. In order to be useful, a key press would need to be
encoded to a binary number for further processing by a logic circuit.
The keypad outputs a nine-bit number such that a single bit goes
high to indicate which key was. For example, when key number two
is pressed,
0 0000 0010
is output from the device. A priority encoder
would accept that nine-bit number and output a binary number that
could be used in computer circuits. Truth Table 9.3 is for the Priority
Encoder that meets the specification. The device is called a “priority”
encoder since it will respond to only the highest value key press. For
example, if someone pressed the three and five keys simultaneously
the encoder would ignore the three key press and transmit
0101
, or
binary five.Dashes are normally
used to indicate
“don’t care” on a
Truth Table.
9.2 encoders/decoders 195
Inputs Outputs
9 8 7 6 5 4 3 2 1 Y1 Y2 Y3 Y4
0 0 0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 0 1 - 0 0 1 0
0 0 0 0 0 0 1 - - 0 0 1 1
0 0 0 0 0 1 - - - 0 1 0 0
0 0 0 0 1 - - - - 0 1 0 1
0 0 0 1 - - - - - 0 1 1 0
0 0 1 - - - - - - 0 1 1 1
0 1 - - - - - - - 1 0 0 0
1 - - - - - - - - 1 0 0 1
Table 9.3: Truth Table for Priority Encoder
This circuit can be realized by using a grid input and routing the
various lines to an appropriate AND gate. This is one of the circuits
built in the lab manual that accompanies this text.
9.2.3 Seven-Segment Display
A seven-segment display is commonly used in calculators and other
devices to show hexadecimal numbers. To create the numeric shapes,
various segments are activated while others remain off, so binary
numbers must be decoded to turn on the various segments for any
given combination of inputs. A seven-segment display has eight input
ports and, when high, each of those ports will activate one segment of
the display.
a
b
c
d
e
f
g
Figure 9.8: Seven-Segment Display
Usually an eighth
“segment” is available
—a decimal point in
the lower right
corner.
In Figure 9.8 the seven segments are labeled and it shows that the
number “
2
” for example, is made by activating segments a, b, g, e, and
d. Table 9.4 shows the various segments that must be activated for
each of the 16 possible input values.
196 encoder circuits
Hex — Binary — Display
— 3 2 1 0 — a b c d e f g
0 — 0 0 0 0 — 1 1 1 1 1 1 0
1 — 0 0 0 1 — 0 1 1 0 0 0 0
2 — 0 0 1 0 — 1 1 0 1 1 0 1
3 — 0 0 1 1 — 1 1 1 1 0 0 1
4 — 0 1 0 0 — 0 1 1 0 0 1 1
5 — 0 1 0 1 — 1 0 1 1 0 1 1
6 — 0 1 1 0 — 1 0 1 1 1 1 1
7 — 0 1 1 1 — 1 1 1 0 0 0 0
8 — 1 0 0 0 — 1 1 1 1 1 1 1
9 — 1 0 0 1 — 1 1 1 1 0 1 1
A — 1 0 1 0 — 1 1 1 0 1 1 1
B — 1 0 1 1 — 0 0 1 1 1 1 1
C — 1 1 0 0 — 1 0 0 1 1 1 0
D — 1 1 0 1 — 0 1 1 1 1 0 1
E — 1 1 1 0 — 1 0 0 1 1 1 1
F — 1 1 1 1 — 1 0 0 0 1 1 1
Table 9.4: Truth Table for Seven-Segment Display
Notice that to display the number “
2
” (in bold font), segments a, b,
d, e, and g must be activated.
A decoder circuit, as in Figure 9.9, uses a demultiplexer to activate
a one-bit line based on the value of the binary input. Note: to save
space, two parallel decoders are used in this circuit.
9.2 encoders/decoders 197
Figure 9.9: 7-Segment Decoder
Figure 9.9 shows an input of
1111
2
so the last line on each demul-
tiplexer is activated. Those lines are used to activate the necessary
inputs on the seven-segment display to create an output of “F”.
A Hex Digit Display has a single port that accepts a four-bit binary
number and that number is decoded into a digital display. Figure 9.10
shows a hex digit display.
Figure 9.10: Hex Decoder
The Logisim-evolution simulator used in this class includes both a 7-
Segment Display and a Hex Digit Display. One of the strengths of using
a seven-segment display rather than a hex digit display is that the
circuit designer has total control over which segments are displayed.
It is common, for example, to activate each of the outer segments in
198 encoder circuits
a rapid sequence to give the illusion of a rotating circle. As opposed
to a seven-segment display, a hex digit display is very simple to wire
and use, as Figures 9.9 and 9.10 make clear. Both of these two types
of displays are available on the market and a designer would chose
whichever type meets the needs.
9.2.4 Function Generators
Decoders provide an easy way to create a circuit when given a minterm
function. Imagine that a circuit is needed for the function defined in
Equation 9.2.
Z
(A, B, C) =
X
(0, 2, 7) (9.2)
Whatever this circuit is designed to do, it should activate an output
only when the input is zero, two, or seven. The circuit in Figure 9.11
illustrates a simple minterm generator using a demultiplexer and an
OR gate. When input A is zero, two, or seven then output Y will go
high, otherwise it is low.
Figure 9.11: Minterm Generator
9.3 error detection
9.3.1 Introduction
Whenever a byte (or any other group of bits) is transmitted or stored,
there is always the possibility that one or more bits will be accidentally
complemented. Consider these two binary numbers:
0110 1010 0011 1010
0110 1010 0111 1010
They differ by only one bit (notice Group Three). If the top number
is what is supposed to be in a memory location but the bottom number
is what is actually there then this would, obviously, create a problem.
There could be any number of reasons why a bit would be wrong,
9.3 error detection 199
but the most common is some sort of error that creeps in while the
byte is being transmitted between two stores, like between a Universal
Synchronous Bus (USB) drive and memory or between two computers
sharing a network. It is desirable to be able to detect that a byte
contains a bad bit and, ideally, even know which bit is wrong so it can
be corrected.
Parity is a common method used to check data for errors and it can
be used to check data that has been transmitted, held in memory, or
stored on a hard drive. The concept of parity is fairly simple: A bit
(called the parity bit) is added to each data byte and that extra bit is
either set to zero or one in order to make the bit-count of that byte
contain an even or odd number of ones. For example, consider this
binary number:
1101
There are three ones in this number, which is an odd number. If odd
parity is being used in this circuit, then the parity bit would be zero
so there would be an odd number of ones in the number. However, if
the circuit is using even parity, then the parity bit would be set to one
in order to have four ones in the number, which is an even number.
Following is the above number with both even and odd parity bits
(those parity bits are in the least significant position and are separated
from the original number by a space for clarity):
1101 0 (Odd Parity)
1101 1 (Even Parity)
Table 9.5 shows several examples that may help to clarify this con-
cept. In each case, a parity bit is used to make the data byte even
parity (spaces were left in the data byte for clarity).
Data Byte Parity Bit
0000 0000 0
0000 0001 1
0000 0011 0
0000 0100 1
1111 1110 1
1111 1111 0
Table 9.5: Even Parity Examples
Generating a parity bit can be done with a series of cascading XOR
gates but Logisim-evolution had two parity gates, one that outputs high
when the inputs have an odd number of ones and the other when
there are an even number of ones. Figure 9.12 illustrates using an odd
200 encoder circuits
parity gate (labeled “2K+1”). In this circuit, if input A has an odd
number of ones, as illustrated, then the parity generator will output a
one to indicate input A has an odd number of ones. That parity bit
is added as the most significant bit to output Y. Since output Y will
always have an even number of bits this is an even parity circuit.
Figure 9.12: Parity Generator
Parity is a simple concept and is the foundation for one of the
most basic methods of error checking. As an example, if some byte is
transmitted using even parity but the data arrives with an odd number
of ones then one of the bits was changed during transmission.
9.3.2 Iterative Parity Checking
One of the problems with using parity for error detection is that while
it may be known that something is wrong, there is no way to know
which of the bits is wrong. For example, imagine an eight-bit system
is using even parity and receives this data and parity bit:
1001 1110 PARITY: 0
There is something wrong with the byte. It is indicating even parity
but has an odd number of ones in the byte. It is impossible to know
which bit changed during transmission. In fact, it may be that the byte
is correct but the parity bit itself changed (a false error). It would be
nice if the parity error detector would not only indicate that there was
an error, but could also determine which bit changed so it could be
corrected.
One method of error correction is what is known as Iterative Parity
Checking. Imagine that a series of eight-bit bytes were being transmitted.
Each byte would have a parity bit attached; however, there would also
be a parity byte that contains a parity bit for each bit in the preceding
five bytes. It is easiest to understand this by using a table (even parity
is being used):
9.3 error detection 201
Byte Data Parity
1 0 0 0 0 0 0 0 0 0
2 1 0 1 1 0 0 0 0 1
3 1 0 1 1 0 0 1 1 1
4 1 1 1 0 1 0 1 0 1
5 0 1 0 0 0 0 0 0 1
P 1 0 1 0 1 0 0 1 0
Table 9.6: Iterative Parity
In Table 9.6, Byte one is
0000 0000
. Since the system is set for even
parity, and it is assumed that a byte with all zeros is even, then the
parity bit is zero. Each of the five bytes has a parity bit that is properly
set such that each byte (with the parity bit) includes an even number
of bits. Then, after a group of five bytes a parity byte is inserted into
the data stream so that each column of five bits also has a parity check;
and that parity bit is found in row
P
on the table. Thus, the parity bit
at the bottom of the first column is one since that column has three
other ones. As a final check, the parity byte itself also has a parity bit
added.
Table 9.7 is the same as Table 9.6, but Bit Zero, the least significant
bit, in Byte One has been changed from a zero to a one (that number
is highlighted).
Byte Data Parity
1 0 0 0 0 0 0 0 1 0
2 1 0 1 1 0 0 0 0 1
3 1 0 1 1 0 0 1 1 1
4 1 1 1 0 1 0 1 0 1
5 0 1 0 0 0 0 0 0 1
P 1 0 1 0 1 0 0 1 0
Table 9.7: Iterative Parity With Error
In Table 9.7 the parity for Byte One is wrong, and the parity for Bit
Zero in the parity byte is wrong; therefore, Bit Zero in Byte One needs
to be changed. If the parity bit for a row is wrong, but no column
parity bits are wrong, or a column is wrong but no rows are wrong,
then the parity bit itself is incorrect. This is one simple way to not only
detect data errors, but correct those errors.
There are two weaknesses with iterative parity checking. First, it is
restricted to only single-bit errors. If more than one bit is changed in a
group then the system fails. This, though, is a general weakness for
202 encoder circuits
most parity checking schemes. The second weakness is that a parity
byte must be generated and transmitted for every few data bytes (five
in the example). This increases the transmission time dramatically and
normally makes the system unacceptably slow.
9.3.3 Hamming Code
9.3.3.1 Introduction
Richard Hamming worked at Bell labs in the
1940
s and he devised
a way to not only detect that a transmitted byte had changed, but
exactly which bit had changed by interspersing parity bits within the
data itself. Hamming first defined the “distance” between any two
binary words as the number of bits that were different between them.
As an example, the two binary numbers
1010
and
1010
has a distance
of zero between them since there are no different bits, but
1010
and
1011
has a distance of one since one bit is different. This concept is
called the Hamming Distance in honor of his work.
The circuit illustrated in Figure 9.13 calculates the Hamming dis-
tance between two four-bit numbers. In the illustration,
0100
and
1101
are compared and two bits difference in those two numbers is
reported.
Figure 9.13: Hamming Distance
The four bits for input A and input B are wired to four XOR gates
then the output of those gates is wired to a Bit Adder device. The XOR
gates will output a one if the two input bits are different then the bit
adder will total how many ones are present at its input. The output of
the bit adder is a three-bit number but to make it easier to read that
number it is wired to a hex digit display. Since that display needs a
four-bit input a constant zero is wired to the most significant bit of
the input of the hex digit display.
9.3 error detection 203
9.3.3.2 Generating Hamming Code
Hamming parity is designed so a parity bit is generated for various
combinations of bits within a byte in such a way that every data bit
is linked to at least three different parity bits. This system can then
determine not only that the parity is wrong but which bit is wrong.
The cost of a Hamming system is that it adds five parity bits to an
eight-bit byte to create a
13
-bit word. Consider the bits for the
13
-bit
word in Table 9.8.
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 0 0 0 0 0 0 0 0 0 0 0
Table 9.8: Hamming Parity Bits
The bits numbered
P
0
to
P
4
are Hamming parity bits and the bits
numbered
d
0
to
d
7
are the data bits. The Hamming parity bits are
interspersed with the data but they occur in positions zero, one, three,
and seven (counting right to left). The following chart shows the data
bits that are used to create each parity bit:
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 X 0 X 0 X 0 X 0 X 0 P
0 0 X X 0 0 X X 0 0 X P 0
0 X 0 0 0 0 X X X P 0 0 0
0 X X X X P 0 0 0 0 0 0 0
P X 0 X X 0 0 X X 0 X 0 0
Table 9.9: Hamming Parity Cover Table
From Table 9.9, line one shows the data bits that are used to set
parity bit zero (
P
0
). If data bits
d0
,
d1
,
d3
,
d4
, and
d6
are all one then
P
0
would be one (even parity is assumed). The data bits needed to
create the Hamming parity bit are marked in all five lines. A note is
necessary about parity bit
P
4
. In order to detect transmission errors
that are two bits large (that is, two bits were flipped), each data bit
needs to be covered by three parity bits. Parity bit
P
4
is designed to
provide the third parity bit for any data bits that have only two others.
For example, look down the column containing data bit
d
0
and notice
that it has only two parity bits (
P
0
and
P
1
) before
P
4
. By adding
P
4
to
the circuit that data bit gets a third parity bit.
As an example of a Hamming code, imagine that this byte needed
to be transmitted:
0110 1001
. This number could be placed in the data
bit positions of the Hamming table.
204 encoder circuits
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 0 1 0 0
Table 9.10: Hamming Example - Iteration 1
Bit zero,
P0
, is designed to generate even parity for data bits
d
0
,
d
1
,
d
3
,
d
4
, and
d
6
. Since there are three ones in that group, then
P
0
must
be one. That has been filled in below (for convenience, the Hamming
parity bit pattern for P
0
is included in the last row of the table).
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 0 1 0 1
0 0 X 0 X 0 X 0 X 0 X 0 P
Table 9.11: Hamming Example - Iteration 2
Bit one,
P
1
, is designed to generate even parity for data bits
d
0
,
d
2
,
d
3
,
d
5
, and
d
6
. Since there are four ones in that group, then
P
1
must
be zero. That has been filled in below.
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 0 1 0 1
0 0 X X 0 0 X X 0 0 X P 0
Table 9.12: Hamming Example - Iteration 3
Bit three,
P
2
, is designed to generate even parity for data bits
d
1
,
d
2
,
d
3
, and
d
7
. Since there is one one in that group, then
P
2
must be one.
That has been filled in below.
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 1 1 0 1
0 X 0 0 0 0 X X X P 0 0 0
Table 9.13: Hamming Example - Iteration 4
Bit seven,
P
3
, is designed to generate even parity for data bits
d
4
,
d
5
,
d
6
, and
d
7
. Since there are two ones in that group, then
P
3
must
be zero. That has been filled in below.
9.3 error detection 205
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 1 1 0 1
0 X X X X P 0 0 0 0 0 0 0
Table 9.14: Hamming Example - Iteration 5
Bit eight,
P
4
, is designed to generate even parity for data bits
d
0
,
d
1
,
d
2
,
d
4
,
d
5
, and
d
7
. Since there are two ones in that group, then
P
4
must be zero. That has been filled in below.
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 1 1 0 0 1 0 0 1 1 0 1
P X 0 X X 0 0 X X 0 X 0 0
Table 9.15: Hamming Example - Iteration 6
When including Hamming parity, the byte
0110 1001
is converted
to: 0 0110 0100 1101.
In Figure 9.14, a 11-bit input, A, is used to create a 16-bit word that
includes Hamming parity bits. In the illustration, input
010 0111 0111
is converted to 1010 0100 1110 1111.
206 encoder circuits
Figure 9.14: Generating Hamming Parity
The process used by the circuit in Figure 9.14 is to wire each of the
input bits to various parity generators and then combine the outputs
of those parity generators, along with the original bits, into a single 16-
bit word. While the circuit has a lot of wired connections the concept
is fairly simple. P0 calculates the parity for input bits
0
,
1
,
3
,
4
,
6
,
8
,
10
.
That is then wired to the least significant bit of output Y.
9.3.3.3 Checking Hamming Code
To check the accuracy of the data bits in a word that contains Hamming
parity bits the following general process is used:
1.
Calculate the Hamming Parity Bit for each of the bit groups
exactly like when the parity was first calculated.
9.3 error detection 207
2.
Compare the calculated Hamming Parity bits with the parity
bits found in the original binary word.
3.
If the parity bits match then there is no error. If the parity bits do
not match then the bad bit can be corrected by using the pattern
of parity bits that do not match.
As an example, imagine that bit eight (last bit in the first group of
four) in the Hamming code created above was changed from zero to
one:
0 0111 0100 1101
(this is bit
d
4
). Table 9.16 shows that Hamming
Bits
P
0
,
P
3
, and
P
4
would now be incorrect since
d
4
is used to create
those parity bits.
P
4
d
7
d
6
d
5
d
4
P
3
d
3
d
2
d
1
P
2
d
0
P
1
P
0
0 0 X 0 X 0 X 0 X 0 X 0 P
0 0 X X 0 0 X X 0 0 X P 0
0 X 0 0 0 0 X X X P 0 0 0
0 X X X X P 0 0 0 0 0 0 0
P X 0 X X 0 0 X X 0 X 0 0
Table 9.16: Hamming Parity Cover Table Reproduced
Since the only data bit that uses these three parity bits is
d
4
then
that one bit can be inverted to correct the data in the eight-bit byte.
The circuit illusted in Figure 9.15 realizes a Hamming parity check.
Notice that the input is the same 16-bit word generated in the circuit
in Figure 9.14 except bit eight (the last bit on the top row of the input)
has been complemented. The circuit reports that bit eight is in err so
it would not only alert an operator that something is wrong with this
data but it would also be able to automatically correct the wrong bit.
208 encoder circuits
Figure 9.15: Checking Hamming Parity
9.3.4 Hamming Code Notes
•
When a binary word that includes Hamming parity is checked to
verify the accuracy of the data bits using three overlapping parity
bits, as developed in this book, one-bit errors can be corrected
and two-bit errors can be detected. This type of system is often
called Single Error Correction, Double Error Detection (SECDED)
and is commonly used in computer memories to ensure data
integrity.
•
While it seems wasteful to add five Hamming bits to an eight-bit
byte (a 62.5% increase in length), the number of bits needed for
longer words does not continue to increase at that rate. Hamming
bits are added to a binary word in multiples of powers of two.
For example, to cover a 32-bit word only seven Hamming bits
are needed, an increase of only about 22.%; and to cover a 256-
9.3 error detection 209
bit word only
10
Hamming bits are needed, an increase of just
under 4%.
•
This lesson counts bits from right-to-left and considers the first
position as bit zero, which matches with the bit counting pattern
used throughout the book. However, many authors and online
resources count Hamming bits from left-to-right and consider
the left-most position as bit one because that is a natural way to
count.
9.3.5 Sample Problems
The following problems are provided for practice.
8-Bit Byte With Hamming
11001010 1110011011001
10001111 1100001110111
01101101 0011011100111
11100010 0111000010000
10011011 1100111011100
Table 9.17: Hamming Parity Examples
Hamming With Error Error Bit
0110101011001 1
1100000100110 3
0000100001100 5
1110111011010 9
1110010100110 12
Table 9.18: Hamming Parity Errors
10
R E G I S T E R C I R C U I T S
10.1 introduction
What to Expect
Flip-flops are the smallest possible memory available for a
digital device. A flip-flop can store a single bit of data and
“remembers” if that bit is on or off. Flip-flops are combined in
groups of eight or more to create a register, which is a memory
device for a byte or word of data in a system. The following
topics are included in this chapter.
• Analyzing a sequential circuit using a timing diagram
• Developing and using an SR latch
•
Developing and using D, JK, Toggle, and Master-Slave
flip-flops
• Designing memory components with registers
•
Converting data between serial and parallel formats using
registers
10.2 timing diagrams
Unlike combinational logic circuits, timing is essential in sequential
circuits. Normally, a device will include a clock IC that generates a
square wave that is used to control the sequencing of various activities
throughout the device. In order to design and troubleshoot these types
of circuits, a timing diagram is developed that shows the relationship
between the various input, intermediate, and output signals. Figure
10.1 is an example timing diagram.
#
1 2 3 4 5 6 7 8
Clk
In
Out
Figure 10.1: Example Timing Diagram
211
212 register circuits
Figure 10.1 is a timing diagram for a device that has three signals, a
clock, an Input, and an Output.Signals in a timing
diagram can be
described as
low//high, 0//1,
on//off, or any other
number of ways.
•
#: The first line of the timing diagram is only a counter that
indicates the number of times the system clock has cycled from
Low to High and back again. For example, the first time the
clock changes from Low to High is the beginning of the first
cycle. This counter is only used to facilitate a discussion about
the circuit timing.
•
Clk: A clock signal regularly varies between Low and High in
a predictable pattern, normally such that the amount of Low
time is the same as the High time. In a circuit, a clock pulse
is frequently generated by applying voltage to a crystal since
the oscillation cycle of a crystal is well known and extremely
stable. In Figure 10.1 the clock cycle is said have a
50
% duty
cycle, that is, half-low and half-high. The exact length of a single
cycle would be measured in micro- or nano-seconds but for the
purposes of this book all that matters is the relationship between
the various signals, not the specific length of a clock cycle.
•
In: The input is Low until Cycle #
1
, then goes High for one cycle.
It then toggles between High and Low at irregular intervals.
•
Out: The output goes High at the beginning of cycle #
1
. It then
follows the input, but only at the start of a clock cycle. Notice
that the input goes high halfway through cycle #
2
but the output
does not go high until the start of cycle #
3
. Most devices are
manufactured to be edge triggered and will only change their
output at either the positive or negative edge of the clock cycle
(this example is positive edge triggered). Thus, notice that no
matter when the input toggles the output only changes when
the clock transitions from Low to High.
Given the timing diagram in Figure 10.1 it would be relatively easy
to build a circuit to match the timing requirements. It would have a
single input and output port and the output would match the input
on the positive edge of a clock cycle.
Whenever the input for any device changes it takes a tiny, but mea-
surable, amount of time for the output to change since the various
transistors, capacitors, and other electronic elements must reach sat-
uration and begin to conduct current. This lag in response is known
as propagation delay and that is important for an engineer to consider
when building a circuit. It is possible to have a circuit that is so com-
Ideal circuits with no
propagation delay are
assumed in this book
in order to focus on
digital logic rather
than engineering.
plex that the output has still not changed from a previous input signal
before a new input signal arrives and starts the process over. In Figure
10.2 notice that the output goes High when the input goes Low, but
only after a tiny propagation delay.
10.3 flip-flops 213
#
1 2 3 4 5 6 7 8
Clk
In
Out
Figure 10.2: Example Propagation Delay
It is also true that a square wave is not exactly square. Due to the
presence of capacitors and inductors in circuits, a square wave will
actually build up and decay over time rather than instantly change. Fig-
ure 10.3 shows a typical charge/discharge cycle for a capacitance cir-
cuit and the resulting deformation of a square wave. It should be kept
in mind, though, that the times involved for capacitance charge/dis-
charge are quite small (measured in nanoseconds or smaller); so for
all but the most sensitive of applications, square waves are assumed
to be truly square.
Figure 10.3: Capacitor Charge and Discharge
All devices are manufactured with certain tolerance levels built-in
such that when the voltage gets “close” to the required level then the
device will react, thus mitigating the effect of the deformed square
wave. Also, for applications where speed is critical, such as space
exploration or medical, high-speed devices are available that both
sharpen the edges of the square wave and reduce propagation delay
significantly.
10.3 flip-flops
10.3.1 Introduction
Flip-flops are digital circuits that can maintain an electronic state even
after the initial signal is removed. They are, then, the simplest of
memory devices. Flip-flops are often called latches since they are able
to “latch” and hold some electronic state. In general, devices are called
flip-flops when sequential logic is used and the output only changes
on a clock pulse and they are called latches when combinational logic
is used and the output constantly reacts to the input. Commonly, flip-
flops are used for clocked devices, like counters, while latches are used
for storage. However, these terms are often considered synonymous
and are used interchangeably.
214 register circuits
10.3.2 SR Latch
One of the simplest of the flip-flops is the SR (for Set-Reset) latch.
Figure 10.4 illustrates a logic diagram for an SR latch built with NAND
This latch is often
also called an RS
Latch.
gates.
Figure 10.4: SR Latch Using NAND Gates
Table 10.2 is the truth table for an SR Latch. Notice that unlike truth
tables used earlier in this book, some of the outputs are listed as “Last
Q” since they do not change from the previous state.
Inputs Outputs
E S R Q Q’
0 0 0 Last Q Last Q’
0 0 1 Last Q Last Q’
0 1 0 Last Q Last Q’
0 1 1 Last Q Last Q’
1 0 0 Last Q Last Q’
1 0 1 0 1
1 1 0 1 0
1 1 1 Not Allowed Not Allowed
Table 10.1: Truth Table for SR Latch
In Figure 10.4, input E is an enable and must be high for the latch to
work; when it is low then the output state remains constant regardless
of how inputs S or R change. When the latch is enabled, if
S = R = 0
then the values of Q and Q’ remain fixed at their last value; or, the
circuit is “remembering” how they were previously set. When input
S goes high, then Q goes high (the latch’s output, Q, is “set”). When
input R goes high, then Q’ goes high (the latch’s output, Q, is “reset”).
Finally, it is important that in this latch inputs R and S cannot both
be high when the latch is enabled or the circuit becomes unstable and
output Q will oscillate between high and low as fast as the NAND
gates can change; thus, input
111
2
must be avoided. Normally, there
10.3 flip-flops 215
is an additional bit of circuit prior to the inputs to ensure S and R will
always be different (an XOR gate could do that job).
An SR Latch may or may not include a clock input; if not, then
the outputs change immediately when any of the inputs change, like
in a combinational circuit. If the designer needs a clocked type of
memory device, which is routine, then the typical choice would be a
JK Flip-Flop, covered later. Figure 10.5 is a timing diagram for the SR
Latch in Listing 10.4.
#
1 2 3 4 5 6 7 8
Ena
S
R
Q
Qn
Figure 10.5: SR Latch Timing Diagram
At the start of Figure 10.5 Enable is low and there is no change in
Q
and
Q
0
until frame 3, when Enable goes high. At that point,
S
is high
and
R
is low so
Q
is high and
Q
0
is low. At frame
4
both
S
and
R
are
low so there is no change in
Q
or
Q
0
. At frame 5
R
goes high so
Q
goes low and Q
0
goes high. In frame 6 both S and R are high so both
Q
and
Q
0
go low. Finally, in frame
7 S
stays high and
R
goes low so
Q
goes high and Q
0
stays low.
Logisim-evolution includes an S-R Flip-Flop device, as illustrated in
Figure 10.6.
Figure 10.6: SR Latch
The S-R Flip-Flop has S and R input ports and Q and Q’ output ports.
Also notice that there is no Enable input but there is a Clock input.
On a logic diagram a
clock input is
indicated with a
triangle symbol.
Because this is a clocked device the output would only change on the
edge of a clock pulse and that makes the device a flip-flop rather than
a latch. As shown, S is high and the clock has pulsed so Q is high, or
the flip-flop is “set.” Also notice that on the top and bottom of the
device there is an S and R input port that are not connected. These are
“preset” inputs that let the designer hard-set output Q at either one
or zero, which is useful during a power-up routine. Since this device
has no Enable input it is possible to use the R preset port as a type of
216 register circuits
enable. If a high is present on the R preset then output Q will go low
and stay there until the R preset returns to a low state.
10.3.3 Data (D) Flip-Flop
A Data Flip-Flop (or D Flip-Flop) is formed when the inputs for an SR
Flip-Flop are tied together through an inverter (which also means that
S and R cannot be high at the same time, which corrects the potential
problem with two high inputs in an SR Flip-Flop). Figure 10.7 illustrates
an SR Flip-Flop being used as a D Flip-Flop in Logisim-evolution .
Figure 10.7: D Flip-Flop Using SR Flip-Flop
In Figure 10.7, the D input (for “Data”) is latched and held on each
clock cycle. Even though Figure 10.7 shows the inverter external to the
latch circuit, in reality, a D Flip-Flop device bundles everything into a
single package with only D and clock inputs and Q and Q’ outputs,
as in Figure 10.8. Like the RS Flip-Flop, D Flip-Flops also have “preset”
inputs on the top and bottom that lets the designer hard-set output Q
at either one or zero, which is useful during a power-up routine.
Figure 10.8: D Flip-Flop
Figure 10.9 is the timing diagram for a Data Flip-Flop.
#
1 2 3 4 5 6 7 8
Clk
D
Q
Figure 10.9: D Latch Timing Diagram
In Figure 10.9 it is evident that output Q follows input D but only
on a positive clock edge. The latch “remembers” the value of D until
the next clock pulse no matter how it changes between pulses.
10.3 flip-flops 217
10.3.4 JK Flip-Flop
The JK flip-flop is the “workhorse” of the flip-flop family. Figure 10.10
is the logic diagram for a JK flip-flop.
Figure 10.10: JK Flip-Flop
Internally, a JK flip-flop is similar to an RS Latch. However, the
outputs to the circuit (Q and Q’) are connected to the inputs (J and K)
in such a way that the unstable input condition (both R and S high)
of the RS Latch is corrected. If both inputs, J and K, are high then the
outputs are toggled (they are “flip-flopped”) on the next clock pulse.
This toggle feature makes the JK flip-flop extremely useful in many
logic circuits.
Figure 10.11 is the timing diagram for a JK flip-flop.
#
1
2 3 4 5 6 7 8
Clk
J
K
Q
Q’
Figure 10.11: JK Flip-Flop Timing Diagram
Table 10.2 summarizes timing diagram 10.11. Notice that on clock
pulse five both J and K are high so Q toggles. Also notice that Q’ is
not indicated in the table since it is always just the complement of Q.
218 register circuits
Clock Pulse J K Q
1 H L H
2 L H L
3 L H L
4 L L L
5 H H H
6 L L H
7 L H L
8 L L L
Table 10.2: JK Flip-Flop Timing Table
10.3.5 Toggle (T) Flip-Flop
If the J and K inputs to a JK Flip-Flop are tied together, then when the
input is high the output will toggle on every clock pulse but when
the input is Low then the output remains in the previous state. This is
often referred to as a Toggle Flip-Flop (or T Flip-Flop). T Flip-Flops are
not usually found in circuits as separate ICs since they are so easily
created by soldering together the inputs of a standard JK Flip-Flop.
Figure 10.12 is a T Flip-Flop in Logisim-evolution .
Figure 10.12: T Flip-Flop
Figure 10.13 is the timing diagram for a T flip-flop.
#
1 2 3 4 5 6 7 8
Clk
D
Q
Q’
Figure 10.13: Toggle Flip-Flop Timing Diagram
In Figure 10.13 when D, the data input, is high then Q toggles on
the positive edge of every clock cycle.
10.4 registers 219
10.3.6 Master-Slave Flip-Flops
Master-Slave Flip-Flops are two flip-flops connected in a cascade and
operating from the same clock pulse. These flip-flops tend to stabilize
an input circuit and are used where the inputs may have voltage
glitches (such as from a push button). Figure 10.14 is the logic diagram
for two JK flip-flops set up as a Master-Slave Flip-Flop.
Figure 10.14: Master-Slave Flip-Flop
Because of the NOT gate on the clock signal these two flip-flops will
activate on opposite clock pulses, which means the flip-flop one will
enable first and read the JK inputs, then flip-flop two will enable and
read the output of flip-flop one. The result is that any glitches on the
input signals will tend to be eliminated.
Often, only one physical IC is needed to provide the two flip-flops
for a master-slave circuit because dual JK flip-flops are frequently
found on a single IC. Thus, the output pins for one flip-flop could be
connected directly to the input pins for the second flip-flop on the
same IC. By combining two flip-flops into a single IC package, circuit
design can be simplified and fewer components need to be purchased
and mounted on a circuit board.
10.4 registers
10.4.1 Introduction
A register is a simple memory device that is composed of a series
of flip-flops wired together such that they share a common clock
pulse. Registers come in various sizes and types and are often used as
“scratch pads” for devices. For example, a register can hold a number
entered on a keypad until the calculation circuit is ready do something
with that number or a register can hold a byte of data coming from a
hard drive until the CPU is ready to move that data someplace else.
220 register circuits
10.4.2 Registers As Memory
Internally, a register is constructed from D Flip-Flops as illustrated in
Figure 10.15.
Figure 10.15: 4-Bit Register
Data are moved from the input port, in the upper left corner, into
the register; one bit goes into each of the four D Flip-Flops. Because
each latch constantly outputs whatever it contains, the output port,
in the lower right corner, combines the four data bits and displays
the contents of the register. Thus, a four-bit register can “remember”
some number until it is needed, it is a four-bit memory device.
10.4.2.1 74x273 Eight-Bit Register
In reality, designers do not build memory from independent flip-flops,
as shown in Figure 10.15. Instead, they use a memory IC that contains
the amount of memory needed. In general, purchasing a memory IC is
cheaper and more reliable than attempting to build a memory device.
One such memory device is a 74x273 eight-bit register. This device
is designed to load and store a single eight-bit number, but other
memory devices are much larger, including Random Access Memory
(RAM) that can store and retrieve millions of eight-bit bytes.
10.4.3 Shift Registers
Registers have an important function in changing a stream of data
from serial to parallel or parallel to serial; a function that is called
“shifting” data. For example, data entering a computer from a network
or USB port is in serial form; that is, one bit at a time is streamed into
or out of the computer. However, data inside the computer are always
moved in parallel; that is, all of the bits in a word are placed on a bus
and moved through the system simultaneously. Changing data from
serial to parallel or vice-verse is an essential function and enables a
computer to communicate over a serial device.
Figure 10.16 illustrates a four-bit parallel-in/serial-out shift register.
The four-bit data into the register is placed on D0-D3, the Shift Write
bit is set to
1
and the clock is pulsed to write the data into the register.
10.4 registers 221
Then the Shift Write bit is set to
0
and on each clock pulse the four bits
are shifted right to the SOUT (for “serial out”) port.
Figure 10.16: Shift Register
It is possible to also build other shift register configurations: serial-
in/serial-out, parallel-in/parallel-out, and serial-in/parallel-out. How-
ever, universal shift registers are commonly used since they can be
easily configured to work with data in either serial or parallel formats.
11
C O U N T E R S
11.1 introduction
What to Expect
Counters are a type of sequential circuit that are designed to
count input pulses (normally from a clock) and then activate
some output when a specific count is reached. Counters are
commonly used as timers; thus, all digital clocks, microwave
ovens, and other digital timing displays use some sort of count-
ing circuit. Counters, however, have many other uses including
devices as diverse as speedometers and frequency dividers. The
following topics are included in this chapter.
• Comparing synchronous and asynchronous counters
•
Developing and using up, down, ring, and modulus coun-
ters
• Creating a frequency divider using a counter
• Listing some of the more common counter ICs
• Describing the common types of read-only memory ICs
•
Describing the function and use of random access memory
11.2 counters
11.2.1 Introduction
Counters are a type of sequential circuit designed to count input
pulses (normally from a clock) and then activate some output when
a specific count is reached. Counters are commonly used as timers;
thus, all digital clocks, microwave ovens, and other digital timing
displays use some sort of counting circuit. However, counters can be
used in many diverse applications. For example, a speed gauge is
a counter. By attaching some sort of sensor to a rotating shaft and
counting the number of revolutions for
60
seconds the Rotations Per
Minute (RPM) is determined. Counters are also commonly used as
frequency dividers. If a high frequency is applied to the input of a
223
224 counters
counter, but only the “tens” count is output, then the input frequency
will be divided by ten. As one final example, counters can also be
used to control sequential circuits or processes; each count can either
activate or deactivate some part of a circuit that controls one of the
sequential processes.
11.2.2 Asynchronous Counters
One of the simplest counters possible is an asynchronous two-bit
counter. This can be built with a two JK flip-flops in sequence, as shown
in Figure 11.1.
Figure 11.1: Asynchronous 2-Bit Counter
The J-K inputs on both flip-flops are tied high and the clock is wired
into FF00. On every positive-going clock pulse the Q output for FF00
toggles and that is wired to the clock input of FF01, toggling that
output. This type of circuit is frequently called a “ripple” counter
since the clock pulse must ripple through all of the flip-flops.
In Figure 11.1, assume both flip-flops start with the output low (or
00
out). On the first clock pulse, Q FF00 will go high, and that will
send a high signal to the clock input of FF01 which activates Q FF01.
At this point, the Q outputs for both flip-flops are high (or
11
out). On
the next clock pulse, Q FF00 will go low, but Q FF01 will not change
since it only toggles when the clock goes from low to high. At this
point, the output is
01
. On the next clock pulse, Q FF00 will go high
and Q FF01 will toggle low:
10
. The next clock pulse toggles Q FF00
to low but Q FF01 does not change:
00
. Then the cycle repeats. This
simple circuit counts:
00
,
11
,
10
,
01
. (Note, Q FF00 is the low-order
bit.) This counter is counting backwards, but it is a trivial exercise to
add the functionality needed to reverse that count.
An asynchronous three-bit counter looks much like the asynchronous
two-bit counter, except that a third stage is added.
11.2 counters 225
Figure 11.2: Asynchronous 3-Bit Counter
In Figure 11.2, the ripple of the clock pulse from one flip-flop to the
next is more evident than in the two-bit counter. However, the overall
operation of this counter is very similar to the two-bit counter.
More stages can be added so to any desired number of outputs.
Asynchronous (or ripple) counters are very easy to build and require
very few parts. Unfortunately, they suffer from two rather important
flaws:
•
Propagation Delay. As the clock pulse ripples through the
various flip-flops, it is slightly delayed by each due to the simple
physical switching of the circuitry within the flip-flop. Propa-
gation delay cannot be prevented and as the number of stages
increases the delay becomes more pronounced. At some point,
one clock pulse will still be winding its way through all of the
flip-flops when the next clock pulse hits the first stage and this
makes the counter unstable.
•
Glitches. If a three-bit counter is needed, there will be a very
brief moment while the clock pulse ripples through the flip-flops
that the output will be wrong. For example, the circuit should
go from
111
to
000
, but it will actually go from
111
to
110
then
100
then
000
as the “low” ripples through the flip-flops. These
glitches are very short, but they may be enough to introduce
errors into a circuit.
Figure 11.3 is a four-bit asynchronous counter.
Figure 11.3: Asynchronous 4-Bit Counter
Figure 11.4 is the timing diagram obtained when the counter in
Figure 11.3 is executing.
226 counters
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Clk
Y0
Y1
Y2
Y3
Figure 11.4: 4-Bit Asynchronous Counter Timing Diagram
Notice that Y0 counts at half the frequency of the clock and then Y1
counts at half that frequency and so forth. Each stage that is added
will count at half the frequency of the previous stage.
11.2.3 Synchronous Counters
Both problems with ripple counters can be corrected with a syn-
chronous counter, where the same clock pulse is applied to every
flip-flop at one time. Here is the logic diagram for a synchronous
two-bit counter:
Figure 11.5: Synchronous 2-Bit Counter
Notice in this circuit, the clock is applied to both flip-flops and
control is exercised by applying the output of one stage to both
J
and
K
inputs of the next stage, which effectively enables/disables that
stage. When Q FF00 is high, for example, then the next clock pulse
will make FF01 change states.
11.2 counters 227
Figure 11.6: Synchronous 3-Bit Counter
A three-bit synchronous counter applies the clock pulse to each
flip-flop. Notice, though, that the output from the first two stages
must routed through an AND gate so FF03 will only change when
Q FF00 and Q FF01 are high. This circuit would then count properly
from 000 to 111.
In general, synchronous counters become more complex as the
number of stages increases since it must include logic for every stage
to determine when that stage should activate. This complexity re-
sults in greater power usage (every additional gate requires power)
and heat generation; however, synchronous counters do not have the
propagation delay problems found in asynchronous counters.
Figure 11.7: Synchronous 4-Bit Up Counter
Figure 11.8 is the timing diagram for the circuit illustrated in Figure
11.7.
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Clk
Y0
Y1
Y2
Y3
Figure 11.8: 4-Bit Synchronous Counter Timing Diagram
228 counters
The timing diagram for the asynchronous counter in Figure 11.4
is the same as that for an synchronous counter in Figure 11.8 since
the output of the two counters are identical. The only difference in
the counters is in how the count is obtained, and designers would
normally opt for a synchronous IC since that is a more efficient circuit.
11.2.3.1 Synchronous Down Counters
It is possible to create a counter that counts down rather than up by
using the Q’ outputs of the flip-flops to trigger the next stage. Figure
11.9 illustrates a four-bit down counter.
Figure 11.9: Synchronous 4-Bit Down Counter
Figure 11.10 is the timing diagram for the circuit illustrated in Figure
11.9.
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Clk
Y0
Y1
Y2
Y3
Figure 11.10: 4-Bit Synchronous Down Counter Timing Diagram
11.2.4 Ring Counters
A ring counter is a special kind of counter where only one output is
active at a time. As an example, a four-bit ring counter may output
this type of pattern:
1000 − 0100 − 0010 − 0001
Notice the high output cycles through each of the bit positions and
then recycles. Ring counters are useful as controllers for processes
where one process must follow another in a sequential manner. Each
11.2 counters 229
of the bits from the ring counter can be used to activate a different
part of the overall process; thus ensuring the process runs in proper
sequence. The circuit illustrated in Figure 11.11 is a 4-bit ring counter.
Figure 11.11: 4-Bit Ring Counter
When the circuit is initialized none of the flip-flops are active. Be-
cause Q’ is high for all flip-flops, AND gate U2 is active and that sends
a high through U1 and into the data port of FF0. The purpose of U2 is
to initialize FF0 for the first count and then U2 is never activated again.
On each clock pulse the next flip-flop in sequence is activated. When
FF3 is active that output is fed back through U1 to FF0, completing
the ring and re-starting the process.
Figure 11.12 is the timing diagram for the ring counter illustrated
in Figure 11.11.
#
1 2 3 4 5 6 7 8
Clk
Y0
Y1
Y2
Y3
Figure 11.12: 4-Bit Ring Counter Timing Diagram
On each clock pulse a different bit is toggled high, proceeding
around the four-bit nibble in a ring pattern.
11.2.4.1 Johnson Counters
One common modification of a ring counter is called a Johnson, or
“Twisted Tail,” ring counter. In this case, the counter outputs this type
of pattern.
1000 − 1100 − 1110 − 1111 − 0111 − 0011 − 0001 − 0000
230 counters
The circuit illustrated in Figure 11.13 is a 4-bit Johnson counter.
Figure 11.13: 4-Bit Johnson Counter
This is very similar to the ring counter except the feedback loop
is from Q’ rather than Q of FF3 and because of that, the AND gate
initialization is no longer needed.
Figure 11.14 is the timing diagram for the Johnson counter illus-
trated in Figure 11.13.
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Clk
Y0
Y1
Y2
Y3
Figure 11.14: 4-Bit Johnson Counter Timing Diagram
11.2.5 Modulus Counters
Each of the counters created so far have had one significant flaw, they
only count up to a number that is a power of two. A two-bit counter
counts from zero to three, a three-bit counter counts from zero to
seven, a four-bit counter counts from zero to
15
, and so forth. With
a bit more work a counter can be created that stops at some other
number. These types of counters are called “modulus counters” and
an example of one of the most common modulus counters is a decade
counter that counts from zero to nine. The circuit illusted in Figure
11.15 is a decade counter.
11.2 counters 231
Figure 11.15: Decade Counter
This is only a four-bit counter (found in Figure 11.7) but with an
additional AND gate (U3). The inputs for that AND gate are set such
that when Y0-Y3 are
1010
then the gate will activate and reset all four
flip-flops.
Figure 11.16 is the timing diagram obtained from the decade counter
illustrated in Figure 11.15.
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Clk
Y0
Y1
Y2
Y3
Figure 11.16: 4-Bit Decade Counter Timing Diagram
The timing diagram shows the count increasing with each clock
pulse (for example, at
8
the outputs are
1000
) until pulse number ten,
when the bits reset to
0000
and the count starts over. A vertical line
was added at count ten for reference.
11.2.6 Up-Down Counters
In this chapter both up and down counters have been considered;
however counters can also be designed to count both up and down.
These counters are, of course, more complex than the simple counters
encountered so far and Figure 11.17 illustrates an up-down counter
circuit.
232 counters
Figure 11.17: Up-Down Counter
When the Up Dn bit is high the counter counts up but when that bit
is low then the counter counts down. This counter is little more than
a merging of the up counter in Figure 11.7 and the down counter in
Figure 11.9. U1-U3 transmit the Q outputs from each flip-flop to the
next stage while U7-U9 transmit the Q’ outputs from each flip-flop
to the next stage. The E bit activates one of those two banks of AND
gates.
No timing diagram is provided for this circuit since that diagram
would be the same as for the up counter (Figure 11.8 or the down
counter (Figure 11.10), depending on the setting of the Up Dn bit.
11.2.7 Frequency Divider
Often, a designer creates a system that may need various clock fre-
quencies throughout its subsystems. In this case, it is desirable to have
only one main pulse generator, but divide the frequency from that
generator so other frequencies are available where they are needed.
Also, by using a common clock source all of the frequencies are easier
to synchronize when needed.
The circuit illustrated in Figure 11.18 is the same synchronous two-
bit counter first considered in Figure 11.5.
Figure 11.18: Synchronous 2-Bit Counter
The circuit in Figure 11.18 produces the timing diagram seen in
Figure 11.19
11.3 memory 233
#
1 2 3 4 5 6 7 8
Clk
Y0
Y1
Figure 11.19: Frequency Divider
Notice that Y0 is half of the clock’s frequency and Y1 is one-quarter
of the clock’s frequency. If the circuit designer used Y1 as a timer for
some subcircuit then that circuit would operate at one-quarter of the
main clock frequency. It is possible to use only one output port from a
modulus counter, like the decade counter found in Figure 11.15, and
get any sort of division of the main clock desired.
11.2.8 Counter Integrated Circuits (IC)
In practice, most designers do not build counter circuits since there
are so many types already commercially available. When a counter
is needed, a designer can select an appropriate counter from those
available on the market. Here are just a few as examples of what is
available:
IC Function
74x68 dual four-bit decade counter
74x69 dual four-bit binary counter
74x90 decade counter (divide by 2 and divide by 5 sections)
74x143 decade counter/latch/decoder/seven-segment driver
74x163 synchronous four-bit binary counter
74x168 synchronous four-bit up/down decade counter
74x177 presettable binary counter/latch
74x291 four-bit universal shift register and up/down counter
Table 11.1: Counter IC’s
11.3 memory
11.3.1 Read-Only Memory
Read Only Memory (ROM) is an IC that contains tens-of-millions
of registers (memory locations) to store information. Typically, ROM
stores microcode (like bootup routines) and computer settings that do
not change. There are several types of ROM: mask, programmable,
and erasable.
234 counters
•
Mask ROMs are manufactured such that memory locations
already filled with data and that cannot be altered by the end
user. They are called “mask” ROMs since the manufacturing
process includes applying a mask to the circuit as it is being
created.
•
Programmable Read-Only Memorys (PROMs) are a type of
ROM that can be programmed by the end user, but it cannot be
altered after that initial programming. This permits a designer
to distribute some sort of program on a chip that is designed to
be installed in some device and then never changed.
•
Erasable PROMs can be programmed by the end user and then
be later altered. An example of where an erasable PROM would
be used is in a computer’s Basic Input/Output System (BIOS)
(the operating system that boots up a computer), where the user
can alter certain “persistent” computer specifications, like the
device boot order.
Two major differences between ROM and RAM are 1) ROM’s ability
to hold data when the computer is powered off and 2) altering the
data in RAM is much faster than in an erasable PROM.
11.3.2 Random Access Memory
RAM is an integrated circuit that contains tens-of-millions of registers
(memory locations) to store information. The RAM IC is designed
to quickly store data found at its input port and then look-up and
return stored data requested by the circuit. In a computer operation,
RAM contains both program code and user input (for example, the
LibreOffice Writer program along with whatever document the user
is working on). There are two types of RAM: dynamic and static.
Dynamic Random Access Memory (DRAM) stores bits in such a way
that it must be “refreshed” (or “strobed”) every few milliseconds.
Static Random Access Memory (SRAM) uses flip-flops to store data so
it does not need to be refreshed. One enhancement to DRAMs was to
package several in a single IC and then synchronize them so they act
like a single larger memory; these are called Synchronized Dynamic
Random Access Memorys (SDRAMs) and they are very popular for
camera and cell phone memory.
Both ROM and RAM circuits use registers, flip-flops, and other
components already considered in this chapter, but by the millions
rather than just four or so at a time. There are no example circuits
or timing diagrams for these devices in this book; however, the lab
manual that accompanies this book includes an activity for a ROM
device.
12
F I N I T E S TAT E M A C H I N E S
12.1 introduction
What to Expect
Finite State Machines (FSMs) are a model of a real-world ap-
plication. System designers often start with a FSM in order to
determine what a digital logic system must do and then use
that model to design the system. This chapter introduces the
two most common forms of FSMs: Moore and Mealy. It includes
the following topics.
• Analyzing a circuit using a Moore FSM
• Analyzing a circuit using a Mealy FSM
• Developing state tables for a circuit
• Creating a FSM for an elevator simulation
12.2 finite state machines
12.2.1 Introduction
Sequential logic circuits are dynamic and the combined inputs and
outputs of the circuit at any given stable moment is called a state. Over
time, a circuit changes states as triggering events take place. As an
example, a traffic signal may be green at some point but change to
red because a pedestrian presses the “cross” button. The current state
of the system would be “green” but the triggering event (the “cross”
button) changes the state to “red.”
The mathematical model of a sequential circuit is called a FSM. A
FSM is an abstract model of a sequential circuit where each state of
the circuit is indicated by circles and various triggering events are
used to sequence a process from one state to the next. The behavior of
many devices can be represented by a FSM model, including traffic
signals, elevators, vending machines, and robotic devices. FSMs are
an analysis tool that can help to simplify sequential circuits.
There are two fundamental FSM models: Moore and Mealy. These
two models are generalizations of a state machine and differ only in
the way that the outputs are generated. A Moore machine generates an
235
236 finite state machines
output as a function of only the current state while a Mealy machine
generates an output as a function of the current state plus the inputs
into that state. Moore machines tend to be safer since they are only
activated on a clock pulse and are less likely to create unwanted
feedback when two different modules are interconnected; however,
Mealy machines tend to be simpler and have fewer states than a Moore
machine. Despite the strengths and weaknesses for each of these two
FSMs, in reality, they are so similar that either can be effectively used
to model any given circuit and the actual FSM chosen by the designer
is often little more than personal preference.
12.2.2 Moore Finite State Machine
The Moore FSM is named after Edward F. Moore, who presented
the concept in a
1956
paper, Gedanken-experiments on Sequential Ma-
chines. The output of a Moore FSM depends only on its current state.
The Moore FSM is typically simpler than a Mealy FSM so modeling
hardware systems is usually best done using a Moore FSM.
As an example of a Moore FSM imagine a simple candy vending
machine that accepts either five cents or ten cents at a time and vends
a handful of product when
15
cents has been deposited (no change
is returned). Figure 12.1 is a Moore FSM diagram for this vending
machine.
0.00
0
0.05
0
0.10
0
0.15
1
N N N-D
D D
Figure 12.1: Moore Vending Machine FSM
In Figure 12.1, imagine that five cents is deposited between each
state circle (that action is indicated by the arrows labeled with an
N
,
for Nickel). The output at each state is zero (printed at the bottom of
each circle) until the state reaches
0
.
15
in the last circle, then the output
changes to one (the product is vended). After that state is reached the
system resets to state
0
.
00
and the entire process starts over. If a user
deposits ten cents, a Dime, then one of the nickel states is skipped.
12.2.3 Mealy Finite State Machine
The Mealy machine is named after George H. Mealy, who presented
the concept in a
1955
paper, A Method for Synthesizing Sequential Circuits.
The Mealy FSM output depends on both its current state and the
12.2 finite state machines 237
inputs. Typically a Mealy machine will have fewer states than a Moore
machine, but the logic to move from state to state is more complex.
As an example of a Mealy FSM, the simple candy vending machine
introduced in Figure 12.1 can be redesigned. Recall that the machine
accepts either five cents or ten cents at a time and vends a handful
of product when
15
cents has been deposited (no change is returned).
Figure 12.2 is a Mealy FSM diagram for this vending machine.
0.00 0.05 0.10
N/0 N/0
D/0
D/1
N-D/1
Figure 12.2: Mealy Vending Machine FSM
In Figure 12.2 the states are identified by the amount of money
that has been deposited, so the first state on the left (
0
.
00
) is when
no money has been deposited. Following the path directly to the
right of the first state, if five cents (indicated by “N” for a nickel) is
deposited, then the output is zero (no candy is dispensed) and the
state is changed to
0
.
05
. If another five cents is deposited, the output
remains zero and the state changes to
0
.
10
. At that point, if either five
or ten cents (“N” or “D”) is deposited, then the output changes to
1
(candy is dispensed) and the state resets to
0
.
00
. By following the
transition arrows various combinations of inputs and their resulting
output can be traced.
Because the Mealy FSM reacts immediately to any input it requires
one less state than the Moore machine (compare Figures 12.1 and
12.2). However, since Moore machines only change states on a clock
pulse they tend to be more predictable (and safer) when integrated
into other modules. Finally, Mealy machines tend to react faster than
Moore machines since they do not need to wait for a clock pulse and,
generally, are implemented with fewer gates. In the end, whether to
design with a Mealy or a Moore machine is left to the designer’s
discretion and, practically speaking, most designers tend to favor one
type of FSM over the other. The simulations in this book use Moore
machines because they tend to be easier to understand and more
stable in operation.
12.2.4 Finite State Machine Tables
Many designers enjoy using Moore and Mealy FSM diagrams as
presented in Figures 12.1 and 12.2; however, others prefer to design
with a finite state machine table that lists all states, inputs, and outputs.
238 finite state machines
As an example, imagine a pedestrian crosswalk in the middle of a
downtown block. The crosswalk has a traffic signal to stop traffic and
a Cross / Don’t Cross light for pedestrians. It also has a button that
pedestrians can press to make the light change so it is safe to cross.
This system has several states which can be represented in a State
Table. The traffic signal can be Red, Yellow, or Green and the pedestrian
signal can be Walk or Don’t Walk. Each of these signals can be either
zero (for Off ) or one (for On). Also, there are two triggers for the
circuit; a push button (the Cross button that a pedestrian presses) and
a timer (so the walk light will eventually change to don’t walk). If
the button is assumed to be one when it is pressed and zero when
not pressed, and the timer is assumed to be one when a certain time
interval has expired but zero otherwise, then State Table 12.1 can be
created.
Current State Trigger Next State
R Y G W D Btn Tmr R Y G W D
1 0 0 1 0 1 0 0 0 0 1 0 1
2 0 0 1 0 1 1 0 0 1 0 0 1
3 0 1 0 0 1 X 0 1 0 0 1 0
4 1 0 0 1 0 X 1 0 0 1 0 1
Table 12.1: Crosswalk State Table
The various states are
R
(for Red),
Y
(for Yellow), and
G
(for Green)
traffic lights, and
W
(for Walk) and
D
(for Don’t Walk) pedestrian lights.
The Btn (for Cross Button) and Tmr (for Timer) triggers can, potentially,
change the state of the system.
Row One on this table shows that the traffic light is Green and the
pedestrian light is Don’t Walk. If the button is not pressed (it is zero)
and the timer is not active (it is zero), then the next state is still a
Green traffic light and Don’t Walk pedestrian light; in other words, the
system is quiescent. In Row Two, the button was pressed (Btn is one);
notice that the traffic light changes state to Yellow, but the pedestrian
light is still Don’t Walk. In Row Three, the current state is a Yellow
traffic light with Don’t Walk pedestrian light (in other words, the Next
State from Row Two), the
X
for the button means it does not matter if
it is pressed or not, and the next state is a Red traffic light and Walk
pedestrian light. In Row Four, the timer expires (it changes to one
at the end of the timing cycle), and the traffic light changes back to
Green while the pedestrian light changes to Don’t Walk.
While Table 12.1 represents a simplified traffic light system, it could
be extended to cover all possible states. Since Red, Yellow, and Green
can never all be one at one time, nor could Walk and Don’t Walk,
the designer must specifically define all of the states rather than use
12.3 simulation 239
a simple binary count from
00000
to
11111
. Also, the designer must
be certain that some combinations never happen, like a Green traffic
light and a Walk pedestrian light at the same time, so those must be
carefully avoided.
State tables can be used with either Mealy or Moore machines and a
designer could create a circuit that would meet all of the requirements
from the state table and then realize that circuit to actually build a
traffic light system.
12.3 simulation
One of the important benefits of using a simulator like Logisim-evolution
is that a circuit designer can simulate digital systems to determine
logic flaws or weaknesses that must be addressed before a physical IC
is manufactured.
12.4 elevator
As a simple example of a IC, imagine an elevator control circuit. For
simplicity, this elevator is in a five story building and buttons like
“door open” will be ignored. There are two ways the elevator can be
called to a given floor: someone could push the floor button in the
elevator car to ride to that floor or someone could push the call button
beside the elevator door on a floor.
Figure 12.3 is the Moore FSM for this circuit:
S1
1
S2
2
S3
3
S4
4
S5
5
1
2
34
5
Rst
Figure 12.3: Elevator
In Figure 12.3 the various floors are represented by the five states
(S1-S5). While the elevator is at a floor then the output from the circuit
is the floor number. The elevator is quiescent while in any one state
and if a user pushes the button for that same floor then nothing will
happen, which is illustrated by the loops starting and coming back to
the same state. The star pattern in the middle of the FSM illustrates
that the elevator can move directly from one state to any other state.
Finally, a Reset signal will automatically move the elevator to state S1.
13
C E N T R A L P R O C E S S I N G U N I T S
13.1 introduction
What to Expect
A CPU is one of the most important components in a computer,
cell phone, or other “smart” digital device. CPUs provide the
interface between software and hardware and can be found in
any device that processes data, even automobiles and household
appliances often contain a CPU. This chapter introduces CPUs
from a digital logic perspective and touches on how those
devices function.
• Describing the function of a CPU
• Designing a CPU
• Creating a CPU instruction set
•
Deriving assembly and programming languages from an
instruction set
• Describing CPU states
13.2 central processing unit
13.2.1 Introduction
The CPU is the core of any computer, tablet, phone, or other computer-
like device. While many definitions of CPU have been offered, many
quite poetic (like the “heart” or “brain” of a computer); probably
the best definition is that the CPU is the intersection of software
and hardware. The CPU contains circuitry that converts the ones
and zeros that are stored in memory (a “program”) into controlling
signals for hardware devices. The CPU retrieves and analyzes bytes
that are contained in a program, turns on or off multiplexers and
control buffers so data are moved to or from various devices in the
computer, and permits humans to use a computer for intellectual
work. In its simplest form, a CPU does nothing more than fetch a list
of instructions from memory and then execute those instructions one
at a time. This chapter explores CPUs from a theoretical perspective.
241
242 central processing units
13.2.1.1 Concepts
A CPU processes a string of ones and zeros and uses the information
in that binary code to enable/disable circuits or hardware devices in a
computer system. For example, a CPU may execute a binary code and
create electronic signals that first place data on the computer’s data
bus and then spins up the hard drive to store that data. As another
example, perhaps the CPU detects a key press on a keyboard, transfers
that key’s code to memory and also sends a code to the monitor where
specific pixels are activated to display the letter pressed.
Figure 13.1 illustrates a very simple circuit used to control the flow
of data on a data bus.
Keyboard
Monitor
Memory
1
Din
A
B
S
Data
A
B
Figure 13.1: Simple Data Flow Control Circuit
In Figure 13.1, the demultiplexer at the bottom left corner of the
circuit controls four control buffers that, in turn, control access to the
data bus. When output
A
is active then input from the Keyboard is
stored in Memory; but when output
B
is active then output from mem-
ory is sent to the monitor. By setting the select bit in the demultiplexer
the circuit’s function can be changed from reading the keyboard to
writing to the monitor using a single data bus.
In a true CPU, of course, there are many more peripheral devices,
along with an ALU, registers, and other internal resources to control.
Figure 13.2 is a block diagram for a simplified CPU.
13.2 central processing unit 243
Control
ALU
General Regs
Program Counter
Address Reg
RAM
Peripherals
Control
Data
Address
Figure 13.2: Simplified CPU Block Diagram
The CPU in Figure 13.2 has three bus lines:
•
Control. This bus contains all of the signals needed to activate
control buffers, multiplexers, and demultiplexers in order to
move data.
• Data. This contains the data being manipulated by the CPU.
•
Address. This is the address for the next instruction to fetch
from RAM.
There are several blocks in the CPU in Figure 13.2:
• Control. This block contains the circuitry necessary to decode
an instruction that was fetched from RAM and then activate the
various devices needed to control the flow of data within the
CPU.
•
ALU. This is an ALU designed for the application that is using
this CPU.
•
General Registers. Most CPUs include a number of general
registers that temporarily hold binary numbers or instructions.
•
Program Counter. This is a register that contains the address
of the next instruction to fetch from RAM.
•
Address Register. This contains the address for the current
RAM operation.
•
RAM. While most computers and other devices have a large
amount of RAM outside the CPU, many CPUs are constructed
with a small amount of internal RAM for increased operational
efficiency. This type of high-speed RAM is usually called cache
(pronounced “cash”).
244 central processing units
In operation, the CPU moves the value of the Program Counter to
the Address Register and then fetches the instruction contained at that
RAM address. That instruction is then sent to the Control circuit where
it is decoded. The Control circuit activates the appropriate control
devices to execute the instruction. This process is repeated millions of
times every second while the CPU executes a program.
13.2.1.2 History
CPUs from the early days of computing (circa
1950
) were custom made
for the computer on which they were found. Computers in those early
days were rather rare and there was no need for a general-purpose
CPU that could function on multiple platforms. By the
1960
s IBM
designed a family of computers based on the design of the System/
360
,
or S/
360
. The goal was to have a number of computers use the same
CPU so programs written for one computer could be executed on
another in the same family. By doing this, IBM hoped to increase their
customer loyalty.
CPUs today can be divided into two broad groups: Complex In-
struction Set Computer (CISC) (pronounced like “sisk”) and Reduced
Instruction Set Computer (RISC). Early computers, like the IBM S/
360
had a large, and constantly growing, set of instructions, and these
types of CPUs were referred to as CISC. However, building the circuits
needed to execute all of those instructions became ever more challeng-
ing until, in the late
1980
s, computer scientists began to design RISC
CPUs with fewer instructions. Because there were fewer CPU instruc-
tions circuits could be smaller and faster, but the trade off was that
occasionally a desired instruction had to be simulated by combining
two or more other instructions, and that creates longer, more complex
computer programs. Nearly all computers, cell phones, tablets, and
other computing devices in use today use a RISC architecture.
More recent developments in CPUs include “pipelining” where
the CPU can execute two or more instructions simultaneously by
overlapping them, that is, fetching and starting an instruction while
concurrently finishing the previous instruction. Another innovation
changed compilers such that they can create efficient Very Long In-
struction Word (VLIW) codes that combine several instructions into a
single step. Multi-threading CPUs permit multiple programs to exe-
cute simultaneously and multi-core CPUs use multiple CPU cores on
the same substrate so programs can execute in parallel.
CPUs, indeed, all hardware devices, are normally designed using
an Hardware Description Language (HDL) like Verilog to speed de-
velopment and ensure high quality by using peer review before an
IC is manufactured. It is possible to find Open Source Verilog scripts
for many devices, including CPU cores
1
, so designers can begin with
1 http://www.opencores.org
13.2 central processing unit 245
mature, working code and then “tweak” it as necessary to match their
particular project.
13.2.1.3 CPU Design Principles
CPU design commonly follows these steps:
analysis of intended use Designing a CPU starts with an
analysis of its intended use since that will determine the type of CPU
that must be built. A simple four-bit CPU — that is, all instructions
are only four-bits wide — is more than adequate for a simple device
like a microwave oven, but for a cell phone or other more complex
device, a 16-bit, or larger, CPU is needed.
the instruction set After the purpose of the CPU is defined
then an instruction set is created. CPU instructions control the physical
flow of bits through the CPU and various components of a computer.
While an instruction set looks something like a programming language,
it is important to keep in mind that an instruction set is very different
from the more familiar higher-level programming languages like Java
and C++.
CPU instructions are 16-bit (or larger) words that are conceptually
divided into two sections: the operational code (opcode) and data. There
are several classes of instructions, but three are the most common:
•
R (Register). These instructions involve some sort of register
activity. The quintessential R-Type instruction is
ADD
, where the
contents of two registers are added together and the sum is
placed in another register.
•
I (Immediate). These instructions include data as part of the
instruction word and something happens immediately with that
data. As an example, the
LDI
instruction immediately loads
a number contained in the instruction word into a specified
register.
•
J (Jump). These instructions cause the program flow to jump to a
different location. In older procedural programming languages
(like C and Basic) these were often called GOTO statements.
assembly language A computer program is nothing more than
a series of ones and zeros organized into words the bit-width of
the instruction set, commonly
32
-bit or
64
-bit. Each word is a single
instruction and a series of instructions forms a program in machine
code that looks something like this:
246 central processing units
0000000000000000 (13.1)
1001000100001010
1001001000001001
0111001100011000
0110000000100011
A CPU fetches and executes one
16
-bit word of the machine code at
a time. If a programmer could write machine code directly then the
CPU could execute it without needing to compile it first. Of course, as
it is easy to imagine, no one actually writes machine code due to its
complexity.
The next level higher than machine code is called Assembly, which
uses easy-to-remember abbreviations (called “mnemonics”) to repre-
sent the available instructions. Following is the assembly language
program for the machine code listed above:
Label Mnemonic Operands Comment
START NOP 0 0 0 No Operation
LDI 1 0 a R1 ¡- 0ah
LDI 2 0 9 R2 ¡- 09h
SHL 3 1 4 R3 ¡- R1 ¡¡ 8
XOR 0 2 3 Acc ¡- R2 XOR R3
Each line of assembly has four parts. First is an optional Label that
can be used to indicate various sections of the program and facilitates
“jumps” around the program; second is the mnemonic for the code
being executed; third are one or more operands; and fourth is an
optional comment field.Machine code is
CPU specific so code
written for one type
of computer could
not be used on any
other type of
computer.
Once the program has been written in Assembly, it must be “as-
sembled” into machine code before it can be executed. An assembler
is a fairly simply program that does little more than convert a file
containing assembly code into instructions that can be executed by
the CPU. The assembly program presented above would be assembled
into Machine Code 13.1.
programming languages Many high level programming lan-
guages have been developed, for example Java and C++. These lan-
guages tend to be easy to learn and can enable a programmer to
quickly create very complex programs without digging into the com-
plexity of machine code.
Programs written in any of these high-level languages must be either
interpreted or compiled before they can be executed. Interpreters are
only available for “scripting” languages like PERL and Python and
they execute the source code one line at a time. In general, interpreters
cannot optimize the code so they are not efficient; but they enable a
13.2 central processing unit 247
programmer to quickly “try out” some bit of code without having to
compile it. A compiler, on the other hand, converts the entire program
to machine code (normally referred to as “object” code by a compiler)
and then creates an executable file. A compiler also optimizes the code
so it executes as efficiently as possible.
In the end, there are dozens of different programming languages,
but they all eventually reduce programming instructions to a series of
ones and zeros which the CPU can execute.
13.2.1.4 State Machine
Because a CPU is little more than a complex FSM, the next step in
the design process is to define the various states and the operations
that take place within each state. In general, an operating CPU cycles
endlessly through three primary states:
•
Fetch an instruction from memory. Instructions take the form
of 16-bit numbers similar in appearance to 1001000100001010.
•
Decode the instruction. The instruction fetched from memory
must be decoded into something like “Add the contents of
Register 2 to Register 3 and save the results in Register 1.”
• Execute the instruction.
In general, the way that a CPU functions is to fetch a single instruc-
tion from glsram and then decode and execute that instruction. As an
example, consider the Assembly example introduced above:
Label Mnemonic Operands Comment
START NOP 0 0 0 No Operation
LDI 1 0 a R1 <- 0ah
LDI 2 0 9 R2 <- 09h
SHL 3 1 4 R3 <- R1 << 8
XOR 0 2 3 Acc <- R2 XOR R3
Each line is an instruction and the CPU would fetch and execute
each instruction from RAM in order. The purpose of this short code
snip is to load a
16
-bit register with a number when only
8
bits are
available in the opcode. The eight high order bits are loaded into
Register one and the eight low order bits are loaded into Register two.
The high order bits are shifted to the left eight places and then the
two registers are XOR’d together:
1. NOP
: This is a “no operation” instruction so the CPU does noth-
ing.
2. LDI
: The number
0A
16
is loaded into register one. This is the
value of the high-order bits desired in the 16-bit number.
248 central processing units
3. LDI
: The number
09
16
is loaded into register two. This is the
value of the low-order bits desired in the 16-bit number.
4. SHL
: Register three is loaded with the value of register one
shifted left eight places.
5. XOR
: The Accumulator is loaded with the value of register two
XOR
’d with 4egister three. This leaves the accumulator with a
16-bit number, 0A09
16
that was loaded eight bits at a time.
13.2.2 CPU States
The first task that a CPU must accomplish is to fetch and decode an
instruction held in memory. Figure 13.3 is a simplified state diagram
that shows the first two CPU states as Fetch and Decode. After that, all
of the different instructions would create their own state (for simplicity,
only three instructions are shown in the Figure 13.3).
Fetch
Decode
NOP
ADD ADI
...
Figure 13.3: CPU State Diagram
The CPU designer would continue to design states for all instruc-
tions and end each state with a loop back to Fetch in order to get the
next instruction out of RAM.
Designing a CPU is no insignificant task and is well beyond the
scope of this book. However, one of the labs in the accompanying lab
manual design a very simple processor that demonstrates how bits
are moved around a circuit based upon control codes.
Part III
A P P E N D I X
a
B O O L E A N P R O P E R T I E S A N D F U N C T I O N S
AND Truth Table
Inputs Output
A B Y
0 0 0
0 1 0
1 0 0
1 1 1
OR Truth Table
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
XOR Truth Table
Inputs Output
A B Y
0 0 0
0 1 1
1 0 1
1 1 0
NAND Truth Table
Inputs Output
A B Y
0 0 1
0 1 1
1 0 1
1 1 0
NOR Truth Table
Inputs Output
A B Y
0 0 1
0 1 0
1 0 0
1 1 0
XNOR Truth Table
Inputs Output
A B Y
0 0 1
0 1 0
1 0 0
1 1 1
NOT Truth Table
Input Output
0 1
1 0
Buffer Truth Table
Input Output
0 0
1 1
Table a.1: Univariate Properties
Property OR AND
Identity A + 0 = A 1A = A
Idempotence A + A = A AA = A
Annihilator A + 1 = 1 0A = 0
Complement A + A
0
= 1 AA
0
= 0
Involution (A
0
)
0
= A
251
252 boolean properties and functions
Table a.2: Multivariate Properties
Property OR AND
Commutative A + B = B + A AB = BA
Associative (A + B) + C = A + (B + C) (AB)C = A(BC)
Distributive A + (BC) = (A + B)(A + C) A(B + C) = AB + AC
Absorption A + AB = A A(A + B) = A
DeMorgan A + B = A B AB = A + B
Adjacency AB + AB
0
= A
Table a.3: Boolean Functions
A 0 0 1 1
B 0 1 0 1
F
0
0 0 0 0 Zero or Clear. Always zero (Annihilation)
F
1
0 0 0 1 Logical AND: A ∗ B
F
2
0 0 1 0 Inhibition: AB
0
or A > B
F
3
0 0 1 1 Transfer A to Output, Ignore B
F
4
0 1 0 0 Inhibition: A
0
B or B > A
F
5
0 1 0 1 Transfer B to Output, Ignore A
F
6
0 1 1 0 Exclusive Or (XOR): A ⊕ B
F
7
0 1 1 1 Logical OR: A + B
F
8
1 0 0 0 Logical NOR: (A + B)
0
F
9
1 0 0 1 Equivalence: (A = B)
0
F
10
1 0 1 0 Not B and ignore A, B Complement
F
11
1 0 1 1 Implication, A + B
0
, B >= A
F
12
1 1 0 0 Not A and ignore B, A Complement
F
13
1 1 0 1 Implication, A
0
+ B, A >= B
F
14
1 1 1 0 Logical NAND: (A ∗ B)
0
F
15
1 1 1 1 One or Set. Always one (Identity)
G L O S S A RY
ALU Arithmetic-Logic Unit. 177, 178, 236, 237
ASCII
American Standard Code for Information Inter-
change. 50, 51, 187
BCD Binary Coded Decimal. 4, 52–58, 130
BIOS Basic Input/Output System. 228
CAT Computer-Aided Tools. 113
CISC Complex Instruction Set Computer. 237, 238
CPU
Central Processing Unit. 4, 50, 178, 185, 186, 213,
235–242
DRAM Dynamic Random Access Memory. 228
EBCDIC
Extended Binary Coded Decimal Interchange Code.
52
FSM Finite State Machine. 229–231, 233, 240
HDL Hardware Description Language. 238
IC
Integrated Circuit. 7, 48, 66, 177, 205, 212–214, 221,
227, 228, 233, 235, 238
IEEE
Institute of Electrical and Electronics Engineers. 31,
67, 72
KARMA KARnaugh MAp simplifier. 154–157, 159
LED Light Emitting Diode. 187
LSB Least Significant Bit. 21, 35, 37, 41, 42, 49
LSN Least Significant Nibble. 56, 58
MSB Most Significant Bit. 21, 36–38, 46, 55, 57, 188
MSN Most Significant Nibble. 59
253
254 glossary
NaN Not a Number. 32
POS Product of Sums. 94, 104, 111
PROM Programmable Read-Only Memory. 228
RAM Random Access Memory. 214, 228, 237, 241, 242
RISC Reduced Instruction Set Computer. 237, 238
ROM Read Only Memory. 227, 228
RPM Rotations Per Minute. 217
SDRAM
Synchronized Dynamic Random Access Memory.
228
SECDED
Single Error Correction, Double Error Detection.
202
SOP Sum of Products. 94, 103
SRAM Static Random Access Memory. 228
USB Universal Synchronous Bus. 193, 214
VLIW Very Long Instruction Word. 238
B I B L I O G R A P H Y
[1]
Digital Circuits. Aug. 4, 2018. url:
https://en.wikibooks.org/
w/index.php?title=Digital Circuits&oldid=3448235
(visited on
02/14/2019).
[2]
John Gregg. Ones and zeros: understanding boolean algebra, digital
circuits, and the logic of sets. Wiley-IEEE Press, 1998.
[3]
Brian Holdsworth and Clive Woods. Digital logic design. Elsevier,
2002.
[4]
Gideon Langholz, Abraham Kandel, and Joe L Mott. Foundations
of digital logic design. World Scientific Publishing Company, 1998.
[5] M Morris Mano. Digital design. EBSCO Publishing, Inc., 2002.
[6] Clive Maxfield. Designus Maximus Unleashed! Newnes, 1998.
[7]
Clive Maxfield. “Bebop to the Boolean Boogie: An Unconven-
tional Guide to Electronics (with CD-ROM).” In: (2003).
[8]
Noam Nisan and Shimon Schocken. The elements of computing
systems: building a modern computer from first principles. MIT press,
2005.
[9]
Charles Petzold. Code: The hidden language of computer hardware
and software. Microsoft Press, 2000.
[10]
Andrew Phelps. Constructing an Error Correcting Code. Univer-
sity of Wisconsin at Madison. 2006. url:
http://pages.cs.wisc.
edu/
∼
markhill/cs552/Fall2006/handouts/ConstructingECC.pdf
(visited on 02/14/2019).
[11]
Myke Predko. Digital electronics demystified. McGraw-Hill, Inc.,
2004.
[12]
Mohamed Rafiquzzaman. Fundamentals of digital logic and micro-
computer design. John Wiley & Sons, 2005.
[13]
Arijit Saha and Nilotpal Manna. Digital principles and logic design.
Jones & Bartlett Learning, 2009.
[14] Roger L Tokheim. Digital electronics. Glencoe, 1994.
[15]
John M Yarbrough and John M Yarbrough. Digital logic: Ap-
plications and design. West Publishing Company St. Paul, MN,
1997.
255
colophon
The logic diagrams in this book are screen captures of Logisim-evolution
circuits.
This book was typeset using the typographical look-and-feel
classicthesis
developed by Andr
´
e Miede. The style was inspired by Robert Bringhurst’s
seminal book on typography “The Elements of Typographic Style”.
classicthesis
is available for both L
A
T
E
X and L
Y
X:
https://bitbucket.org/amiede/classicthesis/
Happy users of
classicthesis
usually send a real postcard to the author,
a collection of postcards received so far is featured here:
http://postcards.miede.de/
Final Version as of September 30, 2020 (Edition 7.0).
Hermann Zapf’s Palatino and Euler type faces (Type 1 PostScript
fonts URW Palladio L and FPL) are used. The “typewriter” text is type-
set in Bera Mono, originally developed by Bitstream, Inc. as “Bitstream
Vera”. (Type 1 PostScript fonts were made available by Malte Rosenau
and Ulrich Dirr.)
