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ECE

PREPARED BY,

LAVANYA. M

ECE

DIGITAL PRINCIPLES AND

SYSTEM DESIGN


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AIM:

To provide an in-depth knowledge of the design of digital circuits and the use of Hardware

Description Language in digital system design.

OBJECTIVES :

To understand different methods used for the simplification of Boolean functions To design and implement combinational circuits To design and implement synchronous

sequential circuits

To design and implement asynchronous sequential circuits To study the fundamentals ofVHDL / Verilog HDL

UNIT I BOOLEAN ALGEBRA AND LOGIC GATES

Review of binary number systems - Binary arithmetic Binary codes Boolean algebra andtheorems - Boolean functions Simplifications of Boolean functions using Karnaugh map and

tabulation methods Implementation of Boolean functions using logic gates

UNIT II COMBINATIONAL LOGIC

Combinational circuits Analysis and design procedures - Circuits for arithmetic operations -

Code conversion Introduction to Hardware Description Language (HDL)

UNIT III DESIGN WITH MSI DEVICES

Decoders and encoders - Multiplexers and demultiplexers - Memory and programmable logic -HDL for combinational circuits

UNIT IV SYNCHRONOUS SEQUENTIAL LOGIC

Sequential circuits Flip flops Analysis and design procedures - State reduction and state

assignment - Shift registers Counters HDL for Sequential Circuits.

UNIT V ASYNCHRONOUS SEQUENTIAL LOGIC Analysis and design of asynchronous

sequential circuits - Reduction of state and flow tables Race-free state assignment Hazards,

ASM Chart.

TEXT BOOKS

1. M.Morris Mano, Digital Design, 3rd edition, Pearson Education, 2007.REFERENCES 1.

Charles H.Roth, Jr. Fundamentals of Logic Design, 4th Edition, Jaico Publishing House,

Cengage Earning, 5th ed, 2005. 2. Donald D.Givone, Digital Principles and Design, Tata

McGraw-Hill, 2007


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BOOLEAN ALGEBRA AND LOGIC GATES

INTRODUCTION:

OBJECTIVES :

To understand basic number systems and complements, and also number systemconversion.

Review of binary number systems:

The term digital refers to any process that is accomplished using discrete units Digital computer is the best example of a digital system.

Basically deal with two types of signals in electronics

i) Analog

ii) Digital


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Types of Number Systems are

i) Decimal Number system

ii) Binary Number system

iii) Octal Number system

iv) Hexadecimal Number system

Complements :

Complements are used in digital computers for simplifying the subtraction operation and for

logical manipulation. There are two types of complements

i) rs complement

ii) (r-1)s complement.


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NUMBER SYSTEM FORMAT:

Dec Hex Oct Bin0

1

23

4

56

7

89

10

11

12

1314

15

0

1

23

4

56

7

89

A

B

C

DE

F

000

001

002003

004

005006

007

010011

012

013

014

015016

017

00000000

00000001

0000001000000011

00000100

0000010100000110

00000111

0000100000001001

00001010

00001011

00001100

0000110100001110

00001111

Binary to decimal conversion:

Step1: Assigning position to Binary number

Step 2:Draw lines, starting from the right, connecting each consecutive digit of the binary

number to the power of two that is next in the list above it.


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Step 3:Move through each digit of the binary number. If the digit is a 1, write its

corresponding power of two below the line, under the digit. If the digit is a 0, write a 0 below theline, under the digit.

Step4:

Add the numbers written below the line. The sum should be 155. This is the decimal

equivalent of the binary number 10011011. Or, written with base subscripts:

Step5:Repetition of this method will result in memorization of the powers of two, whichwill allow you to skip step 1.

Binary to octal(vive versa):

Every octal digit can be re-written as three binary bits and vice versa.octal binary octal binary

octal binary octal binary

0 0 = 022+0210 000 4 4 = 122+0210100


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1 1 = 022+0211 001 5 5 = 122+0211 101

2 2 = 022+1210 010 6 6 = 122+1210 110

3 3 = 022+1211 011 7 7 = 122+1211 111

Example: Convert375 (octal) to binary

3/7/5 =011/111/101 binary

Example: Convert 10110100 (binary) to octal

10 /110/100= 264octal

Hexadecimal to octal conversion

Hexadecimal to octal conversion proceeds by first converting the hexadecimal digits to 4-bit

binary values, then regrouping the binary bits into 3-bit octal digits.

For example, to convert 3FA516:

To binary:

3 F A 5

0011 1111 1010 0101

then to octal:

0 011 111 110 100 101

0 3 7 6 4 5

Therefore, 3FA516 = 376458.

Octal to hexadecimal conversion

The conversion is made in two steps using binary as an intermediate base. Octal is converted to

binary and then binary to hexadecimal, grouping digits by fours, which correspond each to a

hexadecimal digit.

For instance, convert octal 1057 to hexadecimal:

To binary:

1 0 5 7

001 000 101 111

then to hexadecimal:

0010 0010 1111


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2 2 F

Therefore, 10578 = 22F16.

Binary to octal conversion

The process is the reverse of the previous algorithm. The binary digits are grouped by threes,

starting from the decimal point and proceeding to the left and to the right. Add leading 0s (ortrailing zeros to the right of decimal point) to fill out the last group of three if necessary. Then

replace each trio with the equivalent octal digit.

For instance, convert binary 1010111100 to octal:

001 010 111 100

1 2 7 4

Therefore, 10101111002 = 12748.

Convert binary 11100.01001 to octal:

011 100 . 010 010

3 4 . 2 2

Therefore, 11100.010012 = 34.22

Octal to decimal conversion

To convert a number kto decimal, use the formula that defines its base-8 representation:

In this formula, ai is an individual octal digit being converted, where i is the position of the digit

(counting from 0 for the right-most digit).

Example: Convert 7648 to decimal:

7648 = 782 + 681 + 480 = 448 + 48 + 4 = 50010


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For double-digit octal numbers this method amounts to multiplying the lead digit by 8 and

adding the second digit to get the total.

Example: 658 = 68 + 5 = 5310Binary to hexadecimal conversion:

Conversion between hex and binary is easy. Simply substitute four-bit groups for the hex digit of

the same value. Specifically:

Hex Digit: 0 1 2 3 4 5 6 7

Bit Group: 0000 0001 0010 0011 0100 0101 0110 0111

Hex Digit: 8 9 a b c d e f

Bit Group: 1000 1001 1010 1011 1100 1101 1110 1111

For conversion from hex to binary, simply string together the bits for each hex digit. For

instance, 0x509d7a is binary 10100001001110101111010. To wit:

Hex Number: 5 0 9 d 7 a

Binary Number: 0101 0000 1001 1101 0111 1010

To convert the other way, break the binary number into groups of four, then replace each one

with its hex digit. Group the digits starting from the right. If you don't have a complete group of

four when you reach the left,pad with zero bits on the leftto fill the last group. For instance,

binary 111011011111110001 is 0x3b7f1:

Binary Groups: 0011 1011 0111 1111 0001

Hex Digits: 3 b 7 f 1

Because this conversion is so easy, the easiest way to convert between binary and decimal isusually to go through hex. It generally requires fewer operations,

Questions:

1.What is meant by radix?

2. What is the base of hexadecimal?

3.What is the use of Number syatem?


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Boolean Algebra and Theorems

Objective:

To know about Boolean Basics and its uses in number system.

Boolean algebra is a deductive mathematical system closed over the values zero andone (false and true).

A binary operator defined over this set of values accepts a pair of boolean inputs andproduces a single boolean value. For example, the boolean AND operatoraccepts two

boolean inputs and produces a single boolean output (the logical AND ofthe two inputs).

Postulates:

For any given algebra system, there are some initial assumptions, orpostulates, thatthe system follows. You can deduce additional rules, theorems, and other properties of the

system from this basic set of postulates

Closure:

The boolean system is closedwith respect to a binary operator if for everypair of boolean values, it produces a boolean result. For example, logical AND is

closed in the boolean system because it accepts only boolean operands and produces

only boolean results.


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LAWS AND THEOREMS OF BOOLEAN ALGEBRA

Identity Dual

Operations with 0 and 1:1. X + 0 = X (identity)

3. X + 1 = 1 (null element)

2. X.1 = X4. X.0 = 0

Idempotency theorem:5. X + X = X 6. X.X = X

Complementarity:7. X + X = 1 8. X.X = 0

Involution theorem:9. (X) = X

Identities for multiple variables

Cummutative law:10. X + Y = Y + X 11. X.Y = Y X

Associative law:12. (X + Y) + Z = X + (Y + Z)

= X + Y + Z

13. (XY)Z = X(YZ)

= XYZ

Distributive law:

14. X(Y + Z) = XY + XZ 15. X + (YZ) = (X + Y)(X + Z)

DeMorgans theorem:16. (X + Y + Z + ...) = XYZ...

or {f(X1,X2,...,Xn,0,1,+,.)}= {f(X1,X2,...,Xn,1,0,.,+)}

17. (XYZ...) = X + Y + Z + ...

Simplification theorems:18. XY + XY = X (uniting)20. X + XY = X (absorption)

22. (X + Y)Y = XY (adsorption)

19. (X + Y)(X + Y) = X21. X(X + Y) = X

23. XY + Y = X + Y

Consensus theorem:24. XY + XZ + YZ = XY + XZ

25. (X + Y)(X + Z)(Y + Z)

= (X + Y)(X + Z)

Duality:26. (X + Y + Z + ...)D = XYZ...

or {f(X1,X2,...,Xn,0,1,+,.)}D

27. (XYZ ...)D = X + Y + Z + ...


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= f(X1,X2,...,Xn,1,0,.,+)

Demorgans law:

A mathematician named DeMorgan developed a pair of important rules regarding group

complementation in Boolean algebra. By group complementation, I'm referring to the

complement of a group of terms, represented by a long bar over more than one variable.

(X + Y)' = X' . Y', These can be proved by the use of truth tables.


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Proof of (X + Y)' = X' . Y'

The two truth tables are identical, and so the two expressions are identical.

(X.Y) = X' + Y', These can be proved by the use of truth tables

Proof of (X.Y) = X' + Y'

DeMorgans Laws are applicable for any number of variables.

X Y X+Y (X+Y)'

0 0 0 1

0 1 1 0

1 0 1 01 1 1 0

X Y X' Y' X'.Y'

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 0 0 0

X Y X.Y (X.Y)'

0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

X Y X' Y' X'+Y'

0 0 1 1 1

0 1 1 0 1

1 0 0 1 1

1 1 0 0 0


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Questions:

1. What is the use of Boolean function?2. What is meant y Duality?3. Why DEmorgans law is adopted for all variables.

Boolean function:

A Boolean function describes how to determine a Boolean value output based on somelogical calculation from Boolean inputs. Such functions play a basic role in questions of

complexity theory as well as the design of circuits and chips for digital computers.

K map:

Objective:

To know the method to simplify the Digital expression

Introduction:

A Karnaugh map provides a pictorial method of grouping together expressions withcommon factors and therefore eliminating unwanted variables.

The Karnaugh map can also be described as a special arrangement of a truth table. The diagram below illustrates the correspondence between the Karnaugh map and the

truth table for the general case of a two variable problem.
http://en.wikipedia.org/wiki/Boolean_datatypehttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Digital_computerhttp://en.wikipedia.org/wiki/Digital_computerhttp://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Boolean_logichttp://en.wikipedia.org/wiki/Boolean_datatype