1.1 Introduction
The vast majority of the devices and consumer instruments that we encounter in industries, work places, homes, or laboratories are electronic in nature. While some of these are analog in type many others are based on digital principles. Analog systems typically process information that is represented by continuous-time electrical signals. Digital systems, on the other hand, are characterized by discrete-time electrical signals. Discrete implies distinct or separate as opposed to continuous. The words digit and digital were coined to refer to counting discrete numbers. Digital systems are thus systems that process discrete information. Such systems receive input, process or control data manipulation, and output information in a discrete format. It is understood that analog devices and systems have played a significant role in the advancement of the electronics industry; however, most of the newer and more sophisticated electronic systems are turning out to be digital. This is in part because digital devices are inexpensive, reliable, and flexible. The ongoing development and continuous refinement of silicon technology, in particular, have opened up so many previously unthought-of application areas that we may hardly think of a man-made machine now that has not incorporated one or more digital components in it.
A digital system is often a combination of circuit components interconnected in a manner such that it is capable of performing a specific function when all its input variables are allowed only a finite number of discrete states. Two-valued discrete systems, in particular, are referred to as binary systems. Both the inputs and outputs in binary systems can assume either of two allowable discrete values. By combining such discrete states, one may represent numbers, characters, codes, and other pertinent information. There are several advantages that binary systems have over the corresponding analog systems. The electronic devices used in digital circuits are extremely reliable, inexpensive, and remarkably consistent in their performances so long as they are maintained in either of two logical states. Also, because binary circuits are maintained in either of two allowable states, they are much less susceptible to variations of environment and have tolerable accuracy.
Number systems provide the basis for quantifying information for operations in digital processing systems. The binary (two-valued) number system, in particular, serves as the most important basis for understanding digital systems since the electronic devices involved can assume only two output values. In this chapter, we shall study the binary number system, its relationship with other number systems, and then show how they can be represented in binary coded form. Many of these other number systems are of relevance to optical and quantum computing systems that are currently under development.
In Chapter 5, an introduction to computer-aided design (CAD) systems is provided so that it can be used by students. Before that, in Chapter 2, we shall describe interactions of various system variables in terms of logical operations. Logical operations describing the desired outputs of the to-be-designed system in terms of the inputs allow us to explore the various design strategies that can be employed for designing the system in question.
1.2 Positional Number Systems
The number system that is routinely used by us is the base-10 or decimal system. It uses positional number representation, which implies that the value of each digit in a multibit number is a function of its relative position in the number. The term decimal comes from the Latin word for âten.â Counting in decimal numbers and the numbers of fingers or toes are probably interrelated. We have become so used to this number system that we seldom consider the fact that even though we use only ten Arabic digits 0,1,2,3,4,5,6, 7,8, and 9, we are able to represent numbers in excess of nine. To understand its relevance, consider expressing 1998, for example, in the roman number system. The roman equivalent of 1998 is given by MCMXCVIII requiring nine digits! The roman number XI, for example, represents deci...
