Design Methods for Digital Systems

This book constitutes an introduction to the theory of binary switch­ ing networks (binary logic circuits) such as are encountered in industrial automatic systems, in communications networks and, more particularly, in digital computers. These logic circuits, with or without memory, (sequential circuits, combinational circuits) play an increasing part in many sectors of in­ dustry. They are, naturally, to be found in digital computers where, by means of an assembly (often complex) of elerpentary circuits, the func­ tions of computation and decision which are basic to the treatment of information, are performed. In their turn these computers form the heart of an increasing number of digital systems to which they are coupled by interface units which, themselves, fulfil complex functions of information processing. Thus the digital techniques penetrate ever more deeply into industrial and scientific activities in the form of systems with varying degrees of specialization, from the wired-in device with fixed structure to those systems centered on a general-purpose programmable com­ puter. In addition, the present possibility of mass producing microminiaturi­ sed logic circuits (integrated circuits, etc. ) gives a foretaste of the intro­ duction of these techniques into the more familiar aspects of everyday life. The present work is devoted to an exposition of the algebraic techni­ ques nesessary for the study and synthesis of such logic networks. No previous knowledge of this field of activity is necessary: any technician or engineer possessing an elementary knowledge of mathematics and electronics can undertake its reading.

Table of Contents:
1 General Concepts.- 1.1 Digital Systems.- 1.2 Sets.- 1.3 Relations.- 1.4 Univocal Relations or Functions.- 1.5 Concept of an Algebraic Structure.- 2 Numeration Systems. Binary Numeration.- 2.1 Numbers and Numeration Systems.- 2.2 Decimal System of Numeration.- 2.3 Numeration System on an Integer Positive Base.- 2.4 Binary Numeration System.- 2.5 Passage from One Numeration System to Another.- 2.6 Decimal-Binary and Binary-Decimal Conversions.- 3 Codes.- 3.1 Codes.- 3.2 The Coding of Numbers.- 3.3 Unit Distance Codes.- 3.4 Residue checks.- 3.5 The Choice of a Code.- 4 Algebra of Contacts.- 4.1 Electromagnetic Contact Relays.- 4.2 Binary Variables Associated with a Relay.- 4.3 Complement of a Variable.- 4.4. Transmission Function of a Dipole Contact Network.- 4.5 Operations on Dipoles.- 5 Algebra of Classes. Algebra of Logic.- 5.1 Algebra of Classes.- 5.2 Algebra of Logic (Calculus of Propositions).- 5.3 Algebra of Contacts and Algebra of Logic.- 5.4 Algebra of Classes and Algebra of Contacts.- 5.5 Concept of Boolean Algebra.- 6 Boolean Algebra.- 6.1 General. Axiomatic Definitions.- 6.2 Boolean Algebra.- 6.3 Fundamental Relations in Boolean Algebra.- 6.4 Dual Expressions. Principle of Duality.- 6.5 Examples of Boolean Algebra.- 6.6 Boolean Variables and Expressions.- 7 Boolean Functions.- 7.1 Binary Variables.- 7.2 Boolean Functions.- 7.3 Operations on Boolean Functions.- 7.4 The Algebra of Boolean Functions of n Variables.- 7.5 Boolean Expressions and Boolean Functions.- 7.6 Dual Functions.- 7.7 Some Distinguished Functions.- 7.8 Threshold Functions.- 7.9 Functionally Complete Set of Operators.- 7.12 Incompletely Specified Functions.- 7.13 Characteristic Function for a Set.- 7.14 Characteristic Set for a Function.- 8 Geometric Representations of Boolean Functions.- 8.1The n-Dimensional Cube.- 8.2 Venn Diagrams.- 8.3 The Karnaugh Diagram.- 8.4 The Simplification of Algebraic Expressions by the Karnaugh Diagram Method.- 9 Applications and Examples.- 9.1 Switching Networks. Switching Elements.- 9.2 Electronic Logic Circuits. Gates.- 9.3 Combinational Networks. Function and Performance. Expressions and Structure.- 9.4 Examples.- 10 The Simplification of Combinational Networks.- 10.1 General.- 10.2 Simplification Criterion. Cost Function.- 10.3 General Methods for Simplification.- 10.4 The Quine-McCluskey Algorithm.- 10.5 Functional Decompositions.- 11 Concept of the Sequential Network.- 11.1 Elementary Example. The Ferrite Core.- 11.2 Ecoles-Jordan Flip-Flop.- 11.3 Dynamic Type Flip-Flop.- 11.4 Some Elementary Counters.- 12 Sequential Networks. Definitions and Representations.- 12.1 Quantization of Physical Parameters and Time in Sequential Logic Networks.- 12.2 Binary Sequential Networks. General Model.- 12.3 Sequential Networks and Associated Representations.- 12.4 Study of the Operation of Sequential Networks.- 12.5 Adaptation of the Theoretical Model to the Physical Circuit.- 12.6 Some Supplementary Physical Considerations.- 12.7 Incompletely Specified Circuits.- 12.8 Sequential Networks and Combinatorial Networks.- 13 Regular Expressions and Regular Events.- 13.1 Events.- 13.2 Regular Expressions. Regular Events.- 13.3 Regular Expressions Associated with a States Diagram.- 14 The Simplification of Sequential Networks and Minimisation of Transition Tables.- 14.1 Introduction.- 14.2 Minimisation of the Number of States for a Completely Specified Table.- 14.3 Minimisation of the Number of Internal States for an Incompletely Specified Table.- 15 The Synthesis of Synchronous Sequential Networks.- 15.1 General.- 15.2 Direct Method. Examples.- 15.3 State Diagram Method.- 15.4 Diagrams Associated with a Regular Expression.- 15.5 Coding of Initial States. Memory Control Circuits.- 16 counters.- 16.1 Introduction.- 16.2 Pure Binary Counters.- 16.3 Decimal Counters.- 16.4 Reflected Binary Code Counters.