M. MORRIS MANO
`
`mama???
`Design
`
`
`
`Skyworks EX. 2006
`Kinetic V. Skyworks
`Case IPR2014—00530
`
`Page 1
`
`Page 1
`
`Skyworks Ex. 2006
` Kinetic v. Skyworks
` Case IPR2014-00530
`
`

`

`
`
`M. Morris Mano
`
`Profifssor of Engineering
`Cahfornm Starr; Universny, L03 Angela:
`
`”ll”||
`
`Premiue Hall, Englewood Cliffs, New Jersey 07632
`
`Page 2
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`Page 2
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`

`

`Llhrarv af Cnngres: Catalaging-In-FuhlIcat1un Dara
`
`Hauu, M. Mchr1s,
`U‘gfiTal des1gn H H. MUFPIS Harz. —- 2nd ed.
`p.
`cm.
`Inc‘udes Index.
`ESBN EriB-2‘293?—x
`
`rr
`2.
`. E'act"on1c d 911a? canputar5——C1rCuiTs.
`d. -ou‘: castyn.
`4. Digital
`nregrateu c1r:ult§.
`TH?BBB.3.H343
`1331
`521.39'5--n:2fl
`
`-ugic Circuirs.
`J TITIE_
`
`99-37650
`CIP
`
`Editorialipmductinn supervisiorj: Jennifer WBH'LE]
`Interior and cover design: Jerry Votta
`Manufacturing buyer: Lori Bulwin
`
`__‘
`
`'5‘ 199] , 1984 by Prentice-Hall, Inc.
`A Diviainn 0f Simon EL Schustcr
`Englewood Cliflk. New Jerscy 07"632
`
`AH rigitrs reserved. No par: (if this book may be
`reproduced. in any form or by any means.
`without permission in writing from the pubiisizer.
`
`Printed in {he United States of Amen’ca
`
`10
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`9
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`S
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`7
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`6
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`5
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`4
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`3
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`2
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`1
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`ISBN D-lEPElE‘iEW-X
`
`Prentice-Hall Intcmaljonal (UK) Limiicd, Landon
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`Prentica—llflj] of Australia Pry. Limilcd, Sydney
`Prentice-Hall Canada 1:10., Tamara
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`Prentice-Hall of India Private Limited. New Deihi
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`Editura Prenlice-Hall do Bmsil, Lida... Rio dc Janeiro
`
`Page 3
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`Page 3
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`

`

`
`
`PREFACE
`
`1 BINARY SYSTEMS
`
`Ix
`
`1-1
`
`1-2
`
`1—3
`
`1-4
`
`1-5
`
`1-6
`
`1-?
`
`1-8
`
`L9
`
`Digital Computers and Digital Systems
`
`Binary Numbers
`
`4
`
`Number Base Conversions
`
`6
`
`Octal and l-Iexadccima] Numbcrs
`
`9
`
`Complements
`
`10
`
`Signed Binary Numbers
`
`14‘
`
`Binary Codes
`
`1?
`
`Binary Sterage and Registers
`
`25
`
`Binary Logic
`
`28
`
`References
`
`32
`
`Problems
`
`33
`
`36
`2 BOOLEAN ALGEBRA AND LOGIC GATES
`
`2-]
`
`2-2
`
`2-3
`
`Basic Definitions
`
`.36
`
`Axiomalic Definition of Boolean Algabra
`
`38
`
`Basic Theorems and Prupcrtics of Boolean Algebra
`
`41
`
`III
`
`Page 4
`
`Page 4
`
`

`

`iv
`
`Contents
`
`2-4
`
`2-5
`
`2-6
`
`2-3
`
`Boolean Functions
`
`45
`
`Canonical and Standard Forms
`
`49
`
`Other Logic Operations
`
`56
`
`Digital logic Gates
`
`Integrated Circuits
`
`References
`
`69
`
`53
`
`62
`
`Problems
`
`69
`
`3 SlMPLlFlCATlON OF BOOLEAN FUNCTIONS
`
`f2
`
`3-]
`
`3-2
`
`3-3
`
`3-4
`
`3—5
`
`3-6
`
`3-?
`
`3—8
`
`33
`
`3-10
`
`3-11
`
`3—12
`
`The Map Method
`
`72
`
`Two- and 'I‘hree-Variable Maps
`
`73
`
`Four-Variable Map
`
`FivevVariablc Map
`
`73
`
`82
`
`Product of Sums Simplification
`
`34
`
`NAND and NOR. Implementation
`
`Other Two-Level Implementations
`
`Don’t-Care Conditions
`
`98
`
`88
`
`94
`
`The Tabulation Method
`
`10!
`
`Determination of Prime [niplieants
`
`10f
`
`Selection of Prime lmplieanls
`
`106
`
`Concluding Remarks
`
`108
`
`References
`
`I I 0
`
`Problems
`
`“'1
`
`4 COMBINATIONAL LOGIC
`
`”4
`
`4-1
`
`4-2
`
`4-3
`
`4.4
`
`4-5
`
`4-6
`
`4-7
`
`4-3
`
`introduction
`
`U4
`
`Design Procedure
`
`”5
`
`Adders
`
`116
`
`Subtractors
`
`121
`
`Code Conversion
`
`1'24
`
`Analysis Procedure
`
`126
`
`Multilevel NAND Circuits
`
`[30
`
`Multilevel NOR Circuits
`
`138
`
`Page 5
`
`Page 5
`
`

`

`4-9
`
`Exclusive-OR Functions
`
`142
`
`References
`
`143
`
`Problems
`
`1'49
`
`Contents
`
`V
`
`5 MSI AND PLD COMPONENTS
`152
`
`
`5—1
`
`5-2
`
`5-3
`
`5-4
`
`5-5
`
`5-6
`
`5-7
`
`5-8
`
`5-9
`
`Introduction
`
`152
`
`Binary Adder and Subtractor
`
`I54
`
`Decimal Adder
`
`I60
`
`Magnitude Comparator
`
`Decoders and Encoders
`
`163
`
`I66
`
`Mtfltiplcxers
`
`I 23
`
`Read—Only Memory (ROM)
`
`180
`
`Programmable Logic Array {FLA}
`
`Programmable Array.r Logic [PAL]
`
`187
`
`192
`
`References
`
`I97
`
`Problems
`
`19?
`
`6 SYNCHRONOUS SEQUENTIAL LOGIC
`
`202
`
`6-1
`
`6-2
`
`6-3
`
`6-4
`
`6-5
`
`6-6
`
`6*?
`
`6-8
`
`Introduction
`
`202
`
`Flip-Flops
`
`204
`
`Triggering of Flip—Flops
`
`210
`
`Analysis of Clockcd Sequential Circuits
`
`218
`
`State Reduction and Assignment
`
`228
`
`Flip-Flop Excitation Tables
`
`231'
`
`Design Procedure
`
`236
`
`Design of Counters
`
`247
`
`References
`
`25}
`
`Problems
`
`25!
`
`7 REGISTERS. COUNTERS, AND THE MEMOITVr UNIT
`
`25?
`
`7-1
`
`7-2
`
`Introduction
`
`25?
`
`Registcrs
`
`258
`
`Page 6
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`Page 6
`
`

`

`Vi
`
`Content:
`
`7-3
`
`7-4
`
`7-5
`
`7-6
`
`7-7
`
`7-3
`
`7-9
`
`Shift Registers
`
`264
`
`Ripple Counters
`
`2?2
`
`Synchronous Counters
`
`277
`
`Timing Sequences
`
`2'85
`
`RandomeAccess Memory (RA-M]
`
`289
`
`Memory Decoding
`
`293
`
`Error—Correcting Codes
`
`299
`
`References
`
`302
`
`Problems
`
`303
`
`8 ALGORITHMIC STATE MACHINES {ASMI
`
`307
`
`8-1
`
`8-2
`
`8-3
`
`3-4
`
`8-5
`
`8-6
`
`Introduction
`
`ASM Chart
`
`307
`
`309
`
`Timing Considerations
`
`312
`
`Control implementation
`
`31?
`
`Design with Multiplexers
`
`323
`
`PLA Control
`
`330
`
`References
`
`336
`
`Problems
`
`337
`
`3"”
`9 ASYNCHRONOUS SEQUENTIAL LOGIC
`—-——_.______________________________
`
`9-1
`
`9-2
`
`9-3
`
`9-4
`
`9—5
`
`9-6
`9-7
`
`9—8
`
`Introduction
`
`341
`
`Analysis Procedure
`
`343
`
`Circuits with Latches
`
`352
`
`Design Proocdure
`
`359
`
`Reduction 01‘ State and Flow Tables
`
`366
`
`Race—Free State Assignment
`Hazards
`379
`
`374
`
`Design Example
`
`385
`
`References
`
`391'
`
`Problems
`
`392
`
`Page 7
`
`Page 7
`
`

`

`ID DIGITAL INTEGRATED CIRCUITS
`399
`
`
`Contents
`
`VII
`
`13-1
`
`Introduetion
`
`399
`
`10-2
`
`10-3
`
`10-4
`
`10-5
`
`10-6
`
`Special Characteristics
`
`401
`
`Bipolar-Transistor Characteristics
`
`406
`
`RTL and DTL Circuits
`
`409
`
`Transistor-Transistor Logic (TTL)
`
`412
`
`Emmitter-Coupled Logic [Hill
`
`422
`
`10-? Metal-Oxide Semiconductor {M05}
`
`424
`
`I'll-8
`
`10-9
`
`Complementary MOS (CMOS)
`
`427
`
`CMOS Transmission Gate Circuits
`
`430
`
`References
`
`433
`
`Problems
`
`434
`
`'I ‘I LABORATORY EXPERIMENTS
`436
`
`
`”-0
`
`11-1
`
`11-2
`
`11-3
`
`11-4
`
`Introduction to Experiments
`
`436
`
`Binary and Decimal Numbers
`
`441
`
`Digital Logic Gates
`
`444
`
`Simplification of Boolean Functions
`
`446
`
`Combinational Circuits
`
`447
`
`11-5
`
`Code Converters
`
`449
`
`11-6
`
`11-7
`
`11-3
`
`11-9
`
`Design with Multiplexers
`
`451
`
`Adders and Subtractors
`
`452
`
`Flip-Flops
`
`455
`
`Sequential Circuits
`
`453
`
`11-10 Counters
`
`459
`
`11-11
`
`Shift Registers
`
`11-12
`
`Serial Addition
`
`46]
`
`464
`
`11-13
`
`tricolor).r Unit
`
`465
`
`11-14 Lamp Handball
`
`46?
`
`11-15 C lock-Pulse Generator
`
`4?}
`
`11-16
`
`Parallel Adder
`
`4?}
`
`11-17 Binary Multiplier
`
`475
`
`11-18 Asynchronous Sequential Circuits
`
`477
`
`Page 8
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`Page 8
`
`

`

`Vi"
`
`Contents
`
`
`
` 12 STANDARD GRAPHIC SYMBOLS 479
`
`12-1
`
`12—2
`
`12-3
`
`12-4
`
`12-5
`
`12-6
`
`12-7
`
`12-3
`
`RnslangulanShape Symbols
`
`479
`
`Qualifying Symbols
`
`482
`
`Dependency Notation
`
`484
`
`Symbols for Combinational Elements
`
`4815
`
`Symbols for Flip-Flops
`
`439
`
`Symbols for Registers
`
`Symbols for Counters
`
`491
`
`494
`
`Symbnl for RAM
`
`496
`
`Referfinccs
`
`497
`
`Problems
`
`497
`
`APPENDIX: ANSWERS TO SELECTED PROBLEMS
`
`INDEX
`
`499
`
`.512
`
`Page 9
`
`Page 9
`
`

`

`
`
`Digital Design is concerned with the design of digital electronic circuits. The subject
`is also known by other names such as logic design. digital logic, switching circuits. and
`digital systems. Digital circuits are employed in the design of systems such as digital
`computers, control systems, data communications, and many other applications that re-
`quire electronic digital hardware. This book presents the basic tools for the design of
`digital circuits and provides methods and procedures suitable for a variety of digital de-
`sign applications.
`Many features of the second edition remain the same as those of the first edition.
`The material is still organized in the same manner. The first five chapters cover combir
`national circuits. The next three chapters deal with synchronous clocked sequential cir—
`cuits. Asynchronous sequential circuits are introduced next. The last three chapters
`deal with various aspects of commercially available integrated circuits.
`The second edition, however, offers several improvements over the first edition.
`Many sections have been rewritten to clarify the presentation. Chapters 1
`through 7
`and Chapter 10 have been revised by adding new upito~datc material and deleting ob-
`solete subjects. New problems have been formulated for the first seven chapters- These
`replace the problem set from the first edition. Three new experiments have been added
`in Chapter I
`l
`. Chapter 12, a new chapter, presents the JEEE standard graphic symbols
`for logic elements.
`The following is a brief description of the subjects that are covered in each chapter
`with an Emphasis on the revisions that were made in the second edition.
`
`Page 10
`
`Page 10
`
`

`

`,X
`
`Preface
`
`Chapter 1 presents the various binary systems suitable for representing information
`in digital systems. The binary number system is explained and binary codes are illus—
`trated. A new section has been added on signed binary numbers.
`Chapter 2 introduces the basic postulates of Boolean algebra and shovvs the correla+
`tion between Boolean expressions and their corresponding logic diagrams. All possible
`logic operations for two variables are investigated and from that. the most useful logic
`gates used in the design of digital systems are determined. The characteristics of inte-
`grated circuit gates are mentioned in this chapter but a more detailed analysis of the
`electronic circuits of the gates is done in Chapter 1!.
`Chapter 3 covers the map and tabulation methods for simplifying Boolean expres-
`sions. The map method is also used to simplify digital circuits constructed with AND—
`OR. NAND. or NOR gates. All other possible two-level gate circuits are considered
`and their method of implementation is summarised in tabular form for easy reference.
`Chapter 4 outlines the formal procedures for the analysis and design of combina-
`tional circuits. Some basic components used in the design of digital systems, such as
`adders and code converters. are introduced as design examples. The sections on multi-
`level NAND and NOR implementation have been revised to show a simpler procedure
`for converting AND-OR diagrams to NAND or NOR diagrams.
`Chapter 5 presents various medium scale integration [MSU circuits and pron
`grammable logic device (FLU) components. Frequently used digital
`logic functions
`such as parallel adders and subtractors, decoders. encoders. and multiplexers. are ex-
`plained. and their use in the design of combinational circuits is illustrated with exam—
`ples- in addition to the programmable read only memory (PROM) and programmable
`logic array (FLA) the book now shows the internal construction of the programmable
`array logic (PALJ. These three PLD components are extensively used in the design and
`implementation of complex digital circuits.
`Chapter 6 outlines the formal procedures for the analysis and design of clocked syn-
`chronous sequential circuits. The gate structure of several types of flip-flops is pre-
`sented together with a discussion on the difference between pulse level and pulse tran-
`sition triggering. Specific examples are used to show the derivation of the state table
`and state diagram when analyzing a sequential circuit. A number of design examples
`are presented with added emphasis on sequential circuits that use D~type liip-Ilops.
`Chapter 7 presents various sequential digital components such as registers. shift
`registers, and counters- These digital components are the basic building blocks from
`which more complex digital systems are constructed. The sections on the random ac-
`cess memory (RAM) have been completely revised and a new section deals with the
`Hamming error correcting code.
`Chapter 8 presents the algorithmic state machine (ABM) method of digital design.
`The ASM chart is a special llow chart suitable for describing both sequential and paral«
`lel operations with digital hardware. A number of design examples demonstrate the use
`of the ASM chart in the design of state machines.
`Chapter 9 presents formal procedures for the analysis and design of asynchronous
`sequential circuits. Methods are outlined to show how an asynchronous sequential cir—
`
`Page 11
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`Page 11
`
`

`

`Pretzel:
`
`XI
`
`suit can be implemented as a combinational circuit with feedback. An alternate imple»
`tnentation is also described that uses SR latches as the storage elements in an asyn—
`chronous scquantial circuit.
`Chapter [0 presents the most common integrated circuit digital logic feelings, The
`electronic circuits of the common gate in each family is analyzed using electrical circuit
`theory. A basic knOchdge of electronic circuits is necessary to fully understand the
`material in this chapter. Taro new sections are included in the second edition. One sec—
`tion shows how to evaluate the numerical values of four electrical characteristics of a
`
`gate. The other section introduces the CMOS transmission gate. and gives a few exam—
`ples of its usefulness in the construction of digital circuits.
`Chapter 11 outlines 18 experiments that can he performed in the laboratory with
`hardware that is readily and inexpensively available commercially. These experiments
`use standard integrated circuits of the 'l‘TL type. The operation of the integrated cir—
`cuits is explained by referring to diagrams in previous chapters where similar compo-
`nents are originally introduced. Each experiment is presented informally rather than in
`a steprby—step fashion so that the student is expected to prodiJCe the details of the cir-
`cuit diagram and formulate a procedure for checking the operation of the circuit in the
`laboratory.
`Chapter 12 presents the standard graphic symbols for logic functions recommended
`by ANSIIIEEF. standard 91—1984. These graphic symbols have been dchloped for 551
`and MSI components so that the user can recognize each function from the unique
`graphic symbol assigned to ill The best time to learn the standard symbols is while
`learning about digital systems. Chapter 12 shoWs the. standard graphic symbols of all
`the integrated circuits used in the laboratory experiments of Chapter 11.
`The various digital componets that are represented throughout the book are similar
`to commercial MS] circuits. However, the text does not mention specific integrated cir—
`cuits except. in Chapters 1] and 12. The practical application of digital design will be
`enhanced by doing the suggested experiments in Chapter 1] while studying the theory
`presented in the test.
`Each chapter in the book has a list of references and a set of problems. Answers to
`most of the problems appear in the Appendix to aid the student and to help the inde-
`pendent reader. A solutions manual is available for the. instructor from the publisher.
`
`M. Morris Mono
`
`Page 12
`
`Page 12
`
`

`

`Page 13
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`Page 13
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`

`

`Binary Systems
`
`I-‘I
`DIGITAL COMPUTERS AND DIGITAL SYSTEMS
`
`Digital computers have made possible many scientific, industrial, and commercial at]—
`vances that would have been unattainable otherwise. Our space program would have
`been impossible without real—time, continuous computer monitoring, and many busi-
`ness enterprises function efficiently only with the aid of automatic data processing.
`Computers are used in scientific calculations, commercial and business data processing,
`air traffic control, space guidance, the educational field, and many other areas. The
`most striking property of a digital computer is its generality. it can follow a sequence
`of instructions, called a program, that operates on given data. The user can specify and
`change programs auditor data according to the specific need. As a result of this
`flexibility, general-purpose digital computers can perform a wide variety of informa-
`tion-processing tasks.
`The general—purpose digital computer is the hestuknown example of a digital system.
`Other examples include telephone switching exchanges, digital vottnteters, digital
`counters, electronic calculators, and digital displays. Characteristic of a digital system
`is its manipulation of discrete elements of information. Such discrete elements may he
`electric impulses. the decimal digits, the letters of an alphabet, arithmetic operations,
`punctuation marks, or any other set of meaningful symbols. The juxtaposition of dis-
`crete elements of information represents a quantity of information. For example, the
`letters d. o. and g form the word dog. The digits 237' form a number. Thus, a sequence
`of discrete elements forms a language, that is, a discipline that conveys information.
`Early digital computers were used mostly for numerical computations. In this case, the
`
`Page 14
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`Page 14
`
`

`

`2
`
`Chapter 1 Blnary Systems
`
`discrete elements used are the digits. From this application, the term digital computer
`has emerged. A more appropriate name for a digital computer would be a “discrete in-
`formation-processing system.”
`Discrete elements of information are represented in a digital system by physical
`quantities called signals. Electrical signals such as voltages and currents are the most
`common. The signals in all present-day electronic digital systems have only two dis-
`crete values and are said to be binary. The digital—system designer is restricted to the
`use of binary signals because of the lower reliability of many-valued electronic circuits.
`In other words, a circuit with ten states, using one discrete voltage value for each state,
`can be designed, but it would possess a very low reliability of operation. In contrast, a
`transistor circuit that is either on or off has two possible signal values and can be con-
`structed to be extremely reliable. Because of this physical restriction of components,
`and because human logic tends to be binary, digital systems that are constrained to take
`discrete values are further constrained to take binary values.
`Discrete quantities of information arise either from the nature of the process or may
`be quantized from a continuous process. For example, a payroll schedule is an inher-
`ently discrete process that contains employee names, social security numbers, weekly
`salaries, income taxes, etc. An employee’s paycheck is processed using discrete data
`values such as letters of the alphabet (names), digits (salary), and special symbols such
`as 35. On the other hand, a research scientist may observe a continuous process but
`record only Specific quantities in tabular form. The scientist is thus quantizing his con-
`tinuous data. Each number in his table is a discrete element of information.
`
`Many physical systems can be described mathematically by differential equations
`whose solutions as a function of time give the complete mathematical behavior of the
`process. An analog computer performs a direct simulation of a physical system. Each
`section of the computer is the analog of some particular portion of the process under
`study. The variables in the analog computer are represented by continuous signals, usu-
`ally electric voltages that vary with time. The signal variables are considered analogous
`to those of the process and behave in the same manner. Thus, measurements of the
`analog voltage can be substituted for variables of the process. The term analog signal is
`sometimes substituted for continuous signal because “analog computer” has come to
`mean a computer that manipulates continuous variables.
`To simulate a physical process in a digital computer, the quantities must be quan-
`tized. When the variables of the process are presented by real-time continuous signals,
`the latter are quantized by an analog-to-digital conversion device. A physical system
`whose behavior is described by mathematical equations is simulated in a digital com-
`puter by means of numerical methods. When the problem to be processed is inherently
`discrete, as in commercial applications, the digital computer manipulates the variables
`in their natural form.
`
`A block diagram of the digital computer is shown in Fig. 1-1. The memory unit
`stores programs as well as input, output, and intermediate data. The processor unit per-
`forms arithmetic and other data-processing tasks as specified by a program. The con-
`trol unit supervises the flow of information between the various units. The control unit
`retrieves the instructions, one by one, from the program that is stored in memory. For
`
`Page 15
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`

`

`Sectlon 1-1 Dlgltal Computers and Digital Systems
`
`3
`
`Control
`unit
`
`
`
`Processor,
`or
`arithmetic unit
`
`
`Storage,
`or
`memory unit
`
`
`
`lnput
`devices
`and control
`
`Output
`devices
`and control
`
`FIGURE 1-1
`
`Block diagram of a digital computer
`
`informs the processor to execute the operation
`the control unit
`each instruction,
`specified by the instruction. Both program and data are stored in memory. The control
`unit supervises the program instructions, and the processor manipulates the data as
`Specified by the program.
`The program and data prepared by the user are transferred into the memory unit by
`means of an input device such as a keyboard. An output device, such as a printer, re-
`ceives the result of the computations and the printed results are presented to the user.
`The input and output devices are special digital systems driven by electromechanical
`parts and controlled by electronic digital circuits.
`An electronic calculator is a digital system similar to a digital computer, with the in-
`put device being a keyboard and the output device a numerical diSplay. Instructions are
`entered in the calculator by means of the function keys, such as plus and minus. Data
`are entered through the numeric keys. Results are displayed directly in numeric form.
`Some calculators come close to resembling a digital computer by having printing capa—
`bilities and programmable facilities. A digital computer, however, is a more powerful
`device than a calculator. A digital computer can accommodate many other input and
`output devices; it can perform not only arithmetic computations, but logical operations
`as well and can be programmed to make decisions based on internal and external con-
`ditions.
`
`A digital computer is an interconnection of digital modules. To understand the oper-
`ation of each digital module, it is necessary to have a basic knowledge of digital sys—
`tems and their general behavior. The first four chapters of the book introduce the basic
`tools of digital design such as binary numbers and codes, Boolean algebra, and the bas-
`ic building blocks from which electronic digital circuits are constructed. Chapters 5
`and 7 present the basic components found in the processor unit of a digital computer.
`
`Page 16
`
`Page 16
`
`

`

`4
`
`Chapter ‘I Binary Systems
`
`The operational characteristics of the memory unit are explained at the end of Chapter
`7. The design of the control unit is discussed in Chapter 8 using the basic principles of
`sequential circuits from Chapter 6.
`It has already been mentioned that a digital computer manipulates discrete elements
`of information and that these elements are represented in the binary form. Operands
`used for calculations may be expressed in the binary number system. Other discrete ele-
`ments, including the decimal digits, are represented in binary codes. Data processing is
`carried out by means of binary logic elements using binary signals. Quantities are
`stored in binary storage elements. The purpose of this chapter is to introduce the vari-
`ous binary concepts as a frame of reference for further detailed study in the succeeding
`chapters.
`
`1-2 BINARY NUMB_ERS
`
`A decimal number such as 7392 represents a quantity equa‘I‘to 7 thousands plus 3 hun-
`dreds, plus 9 tens, plus 2 units. The thousands, hundreds, etc. are powers of 10 implied
`by the position of the coefficients. To be more exact, 7392 should be written as
`
`7><103+3><102+9><10‘+2><100
`
`However, the convention is to write only the coefficients and from their position de-
`duce the necessary powers of 10. In general, a number with a decimal point is repre-
`sented by a series of coefficients as follows:
`
`05046130261100 .d—la—Za—B
`
`, 9), and the subscript value j
`.
`.
`.
`The a,- coefficients are one of the ten digits (0, l, 2,
`gives the place value and, hence, the power of 10 by which the coefficient must be mul-
`tiplied.
`
`10545 + 10404 + 10303 + 102(12 + 1010] + 10000 + 10_l0_1 + 10—20—2 + 10—30—3
`
`The decimal number system is said to be of base, or radix, 10 because it uses ten digits
`and the coefficients are multiplied by powers of 10. The binary system is a different
`number system. The coefficients of the binary numbers system have two possible val-
`ues: 0 and 1. Each coefficient ai is multiplied by 2’. For example, the decimal equiva-
`lent of the binary number 11010.11 is 26.75, as shown from the multiplication of the
`coefficients by powers of 2:
`
`lX24+1X23+0><22+lx2‘+0><2°+1X2"+l><2‘2=26.75
`
`In general, a number expressed in base-r system has coefficients multiplied by powers
`of r:
`
`a,,-r" +a,,_,‘r"_l + .
`
`-
`
`- +a2'r2+a1'r +a0
`
`+ 0-1-r
`
`—l + a_2,r"2 + .
`
`.
`
`. + a_m‘r-m
`
`Page 17
`
`Page 17
`
`

`

`Sectlon 1-2 Binary Numbers
`
`5
`
`The coefficients at] range in value from 0 to r — 1. To distinguish between numbers of
`different bases, we enclose the coefficients in parentheses and write a subscript equal to
`the base used (except sometimes for decimal numbers, where the content makes it ob-
`vious that it is decimal). An example of a base-5 number is
`
`(4021.2)5=4><53+0>< 52+2><5l +1 ><5°+2><5*'=(511.4)l0
`
`Note that coefficient values for base 5 can be only 0, l, 2, 3, and 4.
`It is customary to borrow the needed r digits for the coefficients from the decimal
`system when the base of the number is less than 10. The letters of the alphabet are used
`to supplement the ten decimal digits when the base of the number is greater than 10.
`For example, in the hexadecimal (base 16) number system, the first ten digits are bor-
`rowed from the decimal system. The letters A, B, C, D, E, and F are used for digits
`10, 11, 12, 13, 14, and 15, respectively. An example of a hexadecimal number is
`
`(3651?).6 = 11 x 163 + 6 x 162 + 5 x 16 + 15 =(46687)10
`
`The first 16 numbers in the decimal. binary, octal, and hexadecimal systems are listed
`in Table 1-1.
`
`TABLE H
`Numbers with Different Bases
`
`Decimal
`(base 10)
`
`Binary
`(base 2)
`
`Octal
`Hexadecimal
`(base 8)
`(base 16)
`
`
`00
`01
`02
`03
`04
`05
`O6
`07
`08
`09
`10
`11
`12
`13
`14
`15
`
`0000
`0001
`0010
`0011
`0100
`0101
`0110
`01] 1
`1000
`1001
`1010
`1011
`1 100
`1101
`1 1 10
`1 11 1
`
`0
`00
`l
`01
`2
`02
`3
`03
`4
`04
`5
`05
`6
`06
`7
`07
`8
`10
`9
`1 1
`A
`12
`B
`13
`C
`14
`D
`15
`E
`16
`
`17 F
`
`Arithmetic operations with numbers in base r follow the same rules as for decimal
`numbers. When other than the familiar base 10 is used, one must be careful to use only
`the r allowable digits. Examples of addition, subtraction, and multiplication of two bi-
`nary numbers are as follows:
`
`Page 18
`
`Page 18
`
`

`

`6
`
`Chapter 1 Binary Systems
`
`augend:
`
`101101
`
`minuend:
`
`101101
`
`multiplicand:
`
`addend:
`
`+100111
`
`subtrahend: — 100111
`
`multiplier:
`
`sum:
`
`1010100
`
`difference:
`
`000110
`
`1011
`
`X 101
`
`1011
`
`0000
`
`
`101 1
`
`product:
`
`1 101 11
`
`The sum of two binary numbers is calculated by the same rules as in decimal, except
`that the digits of the sum in any significant position can be only 0 or 1. Any carry ob-
`tained in a given significant position is used by the pair of digits one significant position
`higher. The subtraction is slightly more complicated. The rules are still the same as in
`decimal, except that the borrow in a given significant position adds 2 to a minuend
`digit. (A borrow in the decimal system adds 10 to a minuend digit.) Multiplication is
`very simple. The multiplier digits are always 1 or 0. Therefore, the partial products are
`equal either to the multiplicand or to 0.
`
`41k
`
`‘l-3 NUMBER BASE CONVERSIONS
`
`A binary number can be converted to decimal by forming the sum of the powers of 2
`of those coefficients whose value is 1. For example
`
`(1010.011)2 = 23 + 21 + 2'2 + 2‘3 = (10.375)1o
`
`The binary number has four 1’s and the decimal equivalent is found from the sum of
`four powers of 2. Similarly, a number expressed in base r can be converted to its deci-
`mal equivalent by multiplying each coefficient with the corresponding power of r and
`adding. The following is an example of octal—to-decimal conversion:
`
`(630.4)3 = 6 X 82 + 3 X 8 + 4 X 8'1 = (408.5)10
`
`The conversion from decimal to binary or to any other base-r system is more con-
`venient if the number is separated into an integer part and a fraction part and the
`conversion of each part done separately. The conversion of an integer from decimal to
`binary is best explained by example.
`
`Example
`1-1
`
`Convert decimal 41 to binary. First, 41 is divided by 2 to give an integer quotient of 20
`and a remainder of i- The quotient is again divided by 2 to give a new quotient and
`remainder. This process is continued until the integer quotient becomes 0. The coef-
`ficients of the desired binary number are obtained from the reminders as follows:
`
`Page 19
`
`Page 19
`
`

`

`Section 1-3 Number Base Conversion!
`
`7
`
`Integer
`quotient W Coefiiclem
`20
`+
`%
`do = 1
`10
`+
`0
`0| = 0
`
`5
`
`2
`l
`
`0
`
`+
`
`+
`+
`
`+
`
`0
`
`é
`0
`
`%
`
`a2 = 0
`
`a; = 1
`a4 = 0
`
`a5 = 1
`
`% =
`% =
`
`% =
`
`g =
`g =
`
`i =
`
`answer: (41)“) = (asa4a3azalaoh = (1010002
`
`The arithmetic process can be manipulated more conveniently as follows:
`Integer Remainder
`
`
`
`41
`
`20
`
`10
`
`5
`
`2
`
`1
`
`0
`
`1
`
`0
`
`0
`
`1
`
`0
`
`1
`
`101001 = answer
`
`I
`
`The conversion from decimal integers to any base-r system is similar to the exam-
`ple, except that division is done by r instead of 2.
`
`Example
`1-2
`
`Convert decimal 153 to octal. The required base r is 8. First, 153 is divided by 8 to
`give an integer quotient of 19 and a remainder of 1. Then 19 is divided by 8 to give an
`integer quotient of 2 and a remainder of 3. Finally, 2 is divided by 8 to give a quotient
`of O and a remainder of 2. This process can be conveniently manipulated as follows:
`
`153
`
`19
`
`2
`
`o
`
`1
`
`3
`
`2 T_ =(231)a
`
`I
`
`Page 20
`
`Page 20
`
`

`

`8
`
`Chapter 1 Binary Systems
`
`The conversion of a decimal fraction to binary is accomplished by a method similar
`to that used for integers. However, multiplication is used instead of division, and in-
`tegers are accumulated instead of remainders. Again, the method is best explained by
`example.
`
`
`
`Example
`1-3
`
`Convert (0.6875)io to binary. First, 0.6875 is multiplied by 2 to give an integer and a
`fraction. The new fraction is multiplied by 2 to give a new integer and a new fraction.
`This process is continued until
`the fraction becomes 0 or until the number of digits
`have sufficient accuracy. The coefficients of the binary number are obtained from the
`integers as follows:
`
` Integer
`
`Fraction Coefficlent \‘
`
`0.6875 X 2 =
`
`0.3750 X 2 =
`
`0.7500 X 2 =
`
`0.5000 X 2 =
`
`l
`
`0
`
`l
`
`l
`
`+
`
`+
`
`+
`
`+
`
`0.3750
`
`(1.) = 1
`
`0.7500
`
`(L2 = 0
`
`0.5000
`
`(L3 — 1
`
`0.0000
`
`(1-4 = 1
`
`Answer: (0.6875)10 = (0.0 - 1a -20 .— 30 — 4)), = (01011)),
`
`I
`
`To convert a decimal fraction to a number expressed in base r, a similar procedure is
`used. Multiplication is by r instead of 2, and the coefficients found from the integers
`may range in value from 0 to r — 1 instead of 0 and l.
`
`
`
`Example
`"4
`
`Convert (0.513) ,0 to octal.
`
`0.513 x 8 = 4.104
`
`0.104 X 8 = 0.832
`
`0.832 X 8 = 6.656
`
`0.656 X 8 = 5.248
`
`0.248 X 8 = 1.984
`
`0.984 X 8 = 7.872
`
`The answer, to seven significant figures, is obtained from the integer part of the prod—
`ucts:
`
`(0.513)10 = (0.406517 .
`
`.
`
`.
`
`)3
`
`I
`
`The conversion of decimal numbers with both integer and fraction parts is done by
`converting the integer and fraction separately and then combining the two answers. Us-
`ing the results of Examples 1-1 and 1-3, we obtain
`
`Page 21
`
`Page 21
`
`

`

`Section 1-4 Octal and Hexadecimal Numbers
`
`9
`
`From Examples 1—2 and 1—4, we have
`
`(416875)“, = (101001.1011)2
`
`(153513)“; = (231.406517)3
`
`1-4 OCTAL AND HEXADECIMAL NUMBERS
`
`The conversion from and to binary, octal, and hexadecimal plays an important part in
`digital computers. Since 23 = 8 and 2“ = 16, each octal digit corresponds to three bi-
`nary digits and each hexadecimal digit corresponds to four binary digits. The conver-
`sion from binary to octal is easily accomplished by partitioning the binary number into
`groups of three digits each, starting from the binary point and proceeding to the left
`and to the right. The corresponding octal digit is then assigned to each group. The fol-
`lowing example illustrates the procedure:
`
`( 10
`|_l
`
`001
`110
`l_l L_l
`
`011
`101
`|__.l L_l
`
`.
`
`100
`111
`|_l I_|
`
`000
`L.__.l
`
`110 )2 = (261517460);
`l_l
`
`2
`
`6
`
`1
`
`5
`
`3
`
`7
`
`4
`
`O
`
`6
`
`Conversion from binary to hexadecimal is similar, except that the binary number is di-
`vided into groups of four digits:
`
`1011
`0110
`(10 1100
`I__lI___lL_ll_l
`
`.
`
`1111
`I_l
`
`0010 )2: (2C6B.F2).6
`I__i
`
`2
`
`C
`
`6
`
`B
`
`F
`
`2
`
`The corresponding hexadecimal {or octal) digit for each group of binary digits is easily
`remembered after studying the values listed in Table 1-1.
`Conversion from octal or hexadecimal to binary is done by a procedure reverse to
`the above. Each octal digit is converted to its three-digit binary equivalent. Similarly,
`each hexadecimal digit is converted to its four-digit binary equivalent. This is illus-
`trated in the following examples:
`
`|_1 L_l
`
`|_1
`
`|_: L.__.J
`
`|_:
`
`6
`
`7
`
`3
`
`1
`
`2
`
`4
`
`0110
`0000
`(306.D)1s = ( 0011
`1_1 I_l 1_—1
`
`.
`
`1101 )2
`1__1
`
`3
`
`0
`
`6
`
`D
`
`to work with because they require three or four
`Binary numbers are difficult
`times as many digits as their decimal equivalent. For example,
`the binary number
`111111111111 is equivalent to decimal 4095. However, digital computers use binary
`numbers and it is sometimes necessary for th