1.2 Binary Numbers

1.3 Number‐Base Conversions

1.4 Octal and Hexadecimal Numbers

1.5 Complements of Numbers

1.6 Signed Binary Numbers

1.7 Binary Codes

1.8 Binary Storage and Registers

1.9 Binary Logic

**1.1 D I G I TA L S Y S T E M S :**

Digital systems have such a prominent role in everyday life that we refer to the present

technological period as the digital age. Digital systems are used in communication, busi-

ness transactions, traffic control, spacecraft guidance, medical treatment, weather mon-

itoring, the Internet, and many other commercial, industrial, and scientific enterprises.

We have digital telephones, digital televisions, digital versatile discs, digital cameras,

handheld devices, and, of course, digital computers. We enjoy music downloaded to our

portable media player (e.g., iPod Touch™) and other handheld devices having high‐

resolution displays. These devices have graphical user interfaces (GUIs), which enable

them to execute commands that appear to the user to be simple, but which, in fact,

involve precise execution of a sequence of complex internal instructions. Most, if not all,

of these devices have a special‐purpose digital computer embedded within them. The

most striking property of the 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 the program or the data according to the specific need. Because of this flexibil-

ity, general‐purpose digital computers can perform a variety of information‐processing

tasks that range over a wide spectrum of applications.

One characteristic of digital systems is their ability to represent and manipulate dis-

crete elements of information. Any set that is restricted to a finite number of elements

contains discrete information. Examples of discrete sets are the 10 decimal digits, the

26 letters of the alphabet, the 52 playing cards, and the 64 squares of a chessboard. Early

digital computers were used for numeric computations. In this case, the discrete ele-

ments were the digits. From this application, the term digital computer emerged. Dis-

crete 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.

Electronic devices called transistors predominate in the circuitry that implements these

signals. The signals in most present‐day electronic digital systems use just two discrete

values and are therefore said to be binary. A binary digit, called a bit, has two values: 0

and 1. Discrete elements of information are represented with groups of bits called binary

codes. For example, the decimal digits 0 through 9 are represented in a digital system

with a code of four bits (e.g., the number 7 is represented by 0111). How a pattern of

bits is interpreted as a number depends on the code system in which it resides. To make

this distinction, we could write (0111) 2 to indicate that the pattern 0111 is to be inter-

preted in a binary system, and (0111) 10 to indicate that the reference system is decimal.

Then 0111 2 = 7 10 , which is not the same as 0111 10, or one hundred eleven. The subscript

indicating the base for interpreting a pattern of bits will be used only when clarification

is needed. Through various techniques, groups of bits can be made to represent discrete

symbols, not necessarily numbers, which are then used to develop the system in a digital

format. Thus, a digital system is a system that manipulates discrete elements of informa-

tion represented internally in binary form. In today’s technology, binary systems are most

practical because, as we will see, they can be implemented with electronic components.

Discrete quantities of information either emerge from the nature of the data being

processed or may be quantized from a continuous process. On the one hand, a payroll

schedule is an inherently discrete process that contains employee names, social security

numbers, weekly salaries, income taxes, and so on. An employee’s paycheck is processed

by means of discrete data values such as letters of the alphabet (names), digits (salary),

and special symbols (such as $). 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 continuous data, making each number in his or her table a discrete

quantity. In many cases, the quantization of a process can be performed automatically

by an analog‐to‐digital converter, a device that forms a digital (discrete) representation

of a analog (continuous) quantity.

The general‐purpose digital computer is the best‐known example of a digital system.

The major parts of a computer are a memory unit, a central processing unit, and input–

output units. The memory unit stores programs as well as input, output, and intermedi-

ate data. The central processing unit performs arithmetic and other data‐processing

operations as specified by the program. The program and data prepared by a user are

transferred into memory by means of an input device such as a keyboard. An output

device, such as a printer, receives the results of the computations, and the printed results

are presented to the user. A digital computer can accommodate many input and output

devices. One very useful device is a communication unit that provides interaction with

other users through the Internet. A digital computer is a powerful instrument that can

perform not only arithmetic computations, but also logical operations. In addition, it can

be programmed to make decisions based on internal and external conditions.

There are fundamental reasons that commercial products are made with digital cir-

cuits. Like a digital computer, most digital devices are programmable. By changing the

program in a programmable device, the same underlying hardware can be used for many

different applications, thereby allowing its cost of development to be spread across a

wider customer base. Dramatic cost reductions in digital devices have come about because of advances in digital integrated circuit technology. As the number of transistors

that can be put on a piece of silicon increases to produce complex functions, the cost per

unit decreases and digital devices can be bought at an increasingly reduced price. Equip-

ment built with digital integrated circuits can perform at a speed of hundreds of millions

of operations per second. Digital systems can be made to operate with extreme reli-

ability by using error‐correcting codes. An example of this strategy is the digital versa-

tile disk (DVD), in which digital information representing video, audio, and other data

is recorded without the loss of a single item. Digital information on a DVD is recorded

in such a way that, by examining the code in each digital sample before it is played back,

any error can be automatically identified and corrected.

A digital system is an interconnection of digital modules. To understand the opera-

tion of each digital module, it is necessary to have a basic knowledge of digital circuits

and their logical function. The first seven chapters of this book present the basic tools

of digital design, such as logic gate structures, combinational and sequential circuits, and

programmable logic devices. Chapter 8 introduces digital design at the register transfer

level (RTL) using a modern hardware description language (HDL). Chapter 9 concludes

the text with laboratory exercises using digital circuits.

A major trend in digital design methodology is the use of a HDL to describe and simulate

the functionality of a digital circuit. An HDL resembles a programming language and is

suitable for describing digital circuits in textual form. It is used to simulate a digital system

to verify its operation before hardware is built. It is also used in conjunction with logic syn-

thesis tools to automate the design process. Because it is important that students become

familiar with an HDL‐based design methodology, HDL descriptions of digital circuits are

presented throughout the book. While these examples help illustrate the features of an HDL,

they also demonstrate the best practices used by industry to exploit HDLs. Ignorance of

these practices will lead to cute, but worthless, HDL models that may simulate a phenom-

enon, but that cannot be synthesized by design tools, or to models that waste silicon area or

synthesize to hardware that cannot operate correctly.

As previously stated, digital systems manipulate discrete quantities of information

that are represented in binary form. Operands used for calculations may be expressed

in the binary number system. Other discrete elements, including the decimal digits and

characters of the alphabet, are represented in binary codes. Digital circuits, also referred

to as logic circuits, process data by means of binary logic elements (logic gates) using

binary signals. Quantities are stored in binary (two‐valued) storage elements (flip‐flops).

The purpose of this chapter is to introduce the various binary concepts as a frame of

reference for further study in the succeeding chapters.