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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
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