Digital Logic & Circuit

Unit-1
Binary Systems

Introduction

We are in “Information age” since digital systems have such a prominent and growing role in modern society. They are involved in our business transactions, communications, transportation, medical treatment and entertainment. In industrial world they are heavily employed in design, manufacturing, distribution and sales.

Analog System

Analog systems process analog signals (continuous time signals) which can take any value within a range, for example the output from a speaker or a microphone.

Digital System

Digital systems process digital signals which can take only a limited number of values (discrete steps), usually just two values are used: the positive supply voltage (+Vs) and zero volts (0V). Digital systems contain devices such as logic gates, flip-flops, shift registers and counters. The general purpose digital computer is a best known example of digital system.

Generic Digital computer structure

Working principle of generic digital computer:

Memory stores programs as well as input, output and intermediate data. The datapath performs arithmetic and other data-processing operations as specified by the program. The control unit supervises the flow of information between the various units. A datapath, when combined with the control unit, forms a component referred to as a central processing unit, or CPU. The program and data prepared by the user are transferred into memory by means of an input device such as a keyboard. An output device, such as a CRT (cathode-ray tube) monitor, displays the results of the computations and presents them to the user.

Advantages of digital system:

- Have made possible many scientific, industrial, and commercial advances that would have been unattainable otherwise.
- Less expensive
- More reliable
- Easy to manipulate
- Flexibility and Compatibility
- Information storage can be easier in digital computer systems than in analog ones.
- New features can often be added to a digital system more easily too.

Disadvantages of digital system:

- Use more energy than analog circuits to accomplish the same tasks, thus producing more heat as well.
- Digital circuits are often fragile, in that if a single piece of digital data is lost or misinterpreted, the meaning of large blocks of related data can completely change.
- Digital computer manipulates discrete elements of information by means of a binary code.
- Quantization error during analog signal sampling.

Information Representation

Signals

- Information variables represented by physical quantities. - For digital systems, the variables take on discrete values. - Two level or binary values are the most prevalent values in digital systems. - Binary values are represented abstractly by: i) digits 0 and 1 ii) words (symbols) False (F) and True (T) iii) words (symbols) Low (L) and High (H) iv) and words On and Off. - Binary values are represented by values or ranges of values of physical quantities

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What are other physical quantities representing 0 and 1?

CPU: Voltage
Disk: Magnetic Field Direction
CD: Surface Pits/Light
Dynamic RAM: Electrical Charge

Number Systems

Here we discuss positional number systems with Positive radix (or base) r. A number with radix r is represented by a string of digits as below i.e. wherever you guys see numbers of whatever bases, all numbers can be written in general as:

Most Significant Digit (MSD) Least Significant Digit (LSD)
Integer portion (n digits) Fractional portion (m digits)
in which 0 ≤ Ai < r (since each being a symbol for particular base system viz. for r = 10 (decimal number system) Ai will be one of 0,1,2,…,8,9). Subscript i gives the position of the coefficient and, hence, the weight ri by which the coefficient must be multiplied.

HEY! Confused? Don’t worry! I will describe specific number systems (r=2, 8, 10 and 16) used in digital computers later one by one, then the concept will be quite clear.

In general, a number in base r contains r digits, 0, l, 2... r- 1, and is expressed as a power series in r with the general form: (Number)r = An-1 rn-1 + An-2 rn-2 + … + A1 r1 + A0 r0 + A-1 r-1 + A-2 r-2 + … + A-m+1 r-m+1 + A-m r-m

Decimal Number System (Base-10 system)

Radix (r) = 10 Symbols = 0 through r-1 = 0 through 10-1 = {0, 1, 2... 8, 9}

I am starting from base-10 system since it is used vastly in everyday arithmetic besides computers to represent numbers by strings of digits or symbols defined above, possibly with a decimal point. Depending on its position in the string, each digit has an associated value of an integer raised to the power of 10.
Example:
decimal number 724.5 is interpreted to represent 7 hundreds plus 2 tens plus 4 units plus 5 tenths.
724.5 = 7 X 102 + 2 X 101 + 4 X 100 +5 X 10-1

Binary Number System (Base-2 system)

Radix (r) = 2 Symbols = 0 through r-1 = 0 through 2-1 = {0, 1}

A binary numbers are expressed with a string of 1'sand 0's and, possibly, a binary point within it. The decimal equivalent of a binary number can be found by expanding the number into a power series with a base of 2.
Example:
(11010.01)2 can be interpreted using power series as:
(11010.01)2 = 1 X 24 + 1 X 23 + 0 X 22 + 1 X 21 + 0 X 20 +0 X 2-1 + 1 X 2-2 = (26.25)10