Boolean Algebra Arithmetic SIG UNSIGN.ppt

DIGITAL SYSTEM DESIGN
Course Code: B000313(028)
Mr. Manjeet Singh Sonwani
Assistant Professor
Department of Electronics & Telecomm.
Government Engineering College Raipur

Course Contents
UNIT-I BOOLEAN ALGEBRA & MINIMIZATION TECHNIQUES
UNIT-II COMBINATIONAL CIRCUITS
UNIT-III SEQUENTIAL CIRCUITS
UNIT-IV FINITE STATE MACHINE
UNIT-V DIGITAL LOGIC FAMILIES
Text Books/Reference Books:
1. R.P. Jain, “Modern digital Electronics”, Tata McGraw Hill, 4th edition, 2009.
2. W.H. Gothmann, “Digital Electronics- An introduction to theory and practice”,
PHI, 2nd Edition ,2006.
3. D.V. Hall, “Digital Circuits and Systems”, Tata McGraw Hill, 1989
4. Digital Fundamentals: Floyd & Jain: Pearson Education.
5. Digital Electronics: A. P. Malvino: Tata McGraw Hill.

Course Code B000313(028)
Course Title Digital System Design
Credits 4
Contact
Periods
(L-T-P)
3-1-0
Course
Objective
1.To Design, Analyze and Interpret Combinational Circuits
2. To Design, Analyze and Interpret Sequential Circuits
Course
Outcome
At the end of this course students will demonstrate the ability to
C 313.1 Employ Boolean algebra and circuit minimization techniques.
C313.2 Design combinational circuits such as adders, subtractors, multiplexers,
decoder
C313.3 Design & analyze modular combinational circuits with MUX/DEMUX,
Decoder, Encoder etc.
C313.4 Design & analyze synchronous sequential logic circuits .
C313.5 Gain knowledge about various logic families and select a suitable one for a
specific application.
SCHEME OF TEACHING

UNIT-I
BOOLEAN ALGEBRA & MINIMIZATION
TECHNIQUES
• Introduction
• Review of Number Systems:
Decimal,Binary,Octal,Hexadecimal
• Representation of Signed Numbers and Binary Arithmetic in
Computers.
• Logic Simplification and Combinational Logic Design,
Boolean Algebra: Logic Operations; Axioms and Laws of
Boolean Algebra: Complementation Laws, AND Laws, OR
Laws, Commutative Laws, Associative Laws, Distributive
Laws, Absorption Laws, Transposition Theorem, De
Morgan’s Theorem; Duality; Reducing Boolean Expressions;
Functionally Complete Sets of Operations; Boolean Functions
and their Representation.

UNIT-I
BOOLEAN ALGEBRA & MINIMIZATION
TECHNIQUES
• Minimization Techniques:
• Expansion of a Boolean expression to SOP form; Expansion
of a Boolean expression to POS form;
• Karnaugh maps up to 4 variables, Mapping and minimization
of SOP and POS expressions; Concept of Don't Care Terms;
Quine – McClusky Method (Up to 5 variable);
• Synthesis using AND-OR, NAND-NOR and XOR forms;
Design Examples;
• Binary codes and code conversion(BCD, Excess-3, Gray
code)

Introduction
Digital Electronics : Digital Electronics or digital circuits are the
branch of Electronics that operates on digital signal.
Analog Signal : It gives its value at any instant of time.
Digital Signal : A signal that is sampled at discrete points and the
sampled values are quantized into a set of discrete values.
At a given time it can only take on one of a finite no. of values.
Discrete Signal : It gives its value at a particular instant of time.

Digital Systems
Digital vs. Analog Waveforms
Analog:
values vary over a broad range
continuously
Digital:
only assumes discrete values
+5
V
–5
Ti me
+5
V
–5
1 0 1
Time

Binary numbers?
Digital Computers: Imply that the computer deals with digital
information, i.e., it deals with the information that is
represented by binary digits.
Why BINARY ? instead of Decimal or other number
system ?
Computers work only on two states
– On
– Off
• Basic memory elements(transistor/MOSFET) hold only two
states
– Zero / One
• Thus a number system with two elements {0,1}
• A binary digit – bit !

Review of Number Systems
Number Systems : In Digital Electronics , the number system is
used for representing the information.
The NS has different bases and the most common of them are
decimal ,binary,octal,Hexadecimal .
The base and radix of any NS is the total no. of digit used in that
NS.
Example: Four number system
Decimal (10): 0,1,2,3,4,5,6,7,8,9
Binary (2) : 0,1
Octal (8) : 0,1,2,3,4,5,6,7
Hexadecimal (16) : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Application: (i)To communicate (ii) To perform task (iii) To
quantify (iv) To measure

Characteristics of Numbering Systems
1) The digits are consecutive/Successive.
2) The number of digits is equal to the size of the
base.
3) Zero is always the first digit.
4) The base number is never a digit.
5) When 1 is added to the largest digit, a sum of
zero and a carry of one results.
6) Numeric values are determined by the implicit
positional values of the digits.

Binary Quantities and Variables
• A binary quantity is one that can take only 2
states
A simple binary arrangement
S L
OPEN OFF
CLOSED ON
S L
0 0
1 1
A truth table

• A binary arrangement with two switches in
series
L = S1 AND S2

• A binary arrangement with two switches in
parallel
L = S1 OR S2

• Three switches in series
L = S1 AND S2 AND S3

• Three switches in parallel
L = S1 OR S2 OR S3

• A series/parallel arrangement
L = S1 AND (S2 OR S3)

Natural Numbers
Zero and any number obtained by repeatedly adding one to it.
Examples: 100, 0, 45645, 32
Negative Numbers
A value less than 0, with a – sign
Examples: -24, -1, -45645, -32
Integers
A natural number, a negative number
Examples: 249, 0, - 45645, - 32
Rational Numbers
An integer or the quotient of two integers
Examples: -249, -1, 0, 3/7, -2/5
Numbers

dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R1 + d1 * R0
As a formula:
1439 = 1 x 103 + 4 x 102 + 3 x 101 + 9 x 100
642 = 6 * 102 + 4 * 10 + 2 * 1
R is the base
of the number
n is the number of
digits in the number
d is the digit in the
ith position
in the number
Positional Notation

Decimal numbers
(1439)10= 1 x 103 + 4 x 102 + 3 x 101 + 9 x 100
Thousands Hundreds Tens Ones
• Base / Radix = 10

Binary Decimal
1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1
= 8 + 4 + 0 + 1
(1101)2 = (13)10
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ….

Significant Digits
Binary: 11101101
Most significant digit Least significant digit
Hexadecimal: 1D63A7A
Most significant digit Least significant digit

Binary Number System
• Also called the “Base 2 system”
• The binary number system is used to model the
series of electrical signals computers use to
represent information
• 0 represents the no voltage or an off state
• 1 represents the presence of voltage or an
on state

Binary Numbering Scale
Base 2
Number
Base 10
Equivalent
Power
Positional
Value
000 0 20 1
001 1 21 2
010 2 22 4
011 3 23 8
100 4 24 16
101 5 25 32
110 6 26 64
111 7 27 128

Step 1. Divide the decimal number by the new base
Step 2. Make the remainder the next digit to the left in the answer
Step 3. Replace the original decimal number with the quotient While
Step 4.Repeat Step 1 to Step 3 till the quotient is not zero.
Algorithm for converting number in base 10 to
other bases
Converting Decimal to Other Bases

Division Algorithm
Decimal to Binary Conversion
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R 1 Divide 67 by 2. Record quotient in next row
Step 2: 33 / 2 = 16 R 1 Again divide by 2; record quotient in next row
Step 3: 16 / 2 = 8 R 0 Repeat again
Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0
1 0 0 0 0 1 12

Binary to Decimal Conversion
• The easiest method for converting a binary
number to its decimal equivalent is to use the
Multiplication Algorithm
• Step 1. Multiply the binary digits by increasing
powers of two, starting from the right(LSB)
• Step 2. Then, to find the decimal number
equivalent, sum those products

Multiplication Algorithm
Convert (10101101)2 to its decimal equivalent:
Binary 1 0 1 0 1 1 0 1
Positional Values 2
7
2
6
2
5
2
4
2
3
2
2
2
1
2
0
128 + 32 + 8 + 4 + 1
Products
17310
1x27 +0x26 +1x25 +0x24 + 1x 23 +1x22 +0x 21 +1x 20

Hexadecimal Number System
• Base 16 system
• Uses digits 0-9 &
letters A,B,C,D,E,F
• Groups of four bits
represent each
base 16 digit

Decimal to Hexadecimal Conversion
Convert 83010 to its hexadecimal equivalent:
830 / 16 = Q 51 R 14
51 / 16 = Q 3 R 3
3 / 16 = Q 0 R 3
33 E16
= E in
Hex

Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
3 B 4 F
x
x
x 163 162 161 160
12288 +2816 + 64 +15
15,18310
Positional Values
Products
x

Binary to Hexadecimal Conversion
• The easiest method for converting binary to
hexadecimal is to use a substitution code
• Each hex number converts to 4 binary digits

Decimal Binary
13
6
3
1
0
2
2
2
2
1
0
1
1
(13)10 = (1101)2
MSB
LSB

Octal Decimal
137 = 1 x 82 + 3 x 81 + 7 x 80
= 1 x 64 + 3 x 8 + 7 x 1
= 64 + 24 + 7
(137)8 = (95)10
• Digits used in Octal number system – 0 to 7

Decimal Octal
95
11
1
0
8
8
8
7
3
1
(95)10 = (137)8
MSP
LSP

Hex Decimal
BAD = 11 x 162 + 10 x 161 + 13 x 160
= 11 x 256 + 10 x 16 + 13 x 1
= 2816 + 160 + 13
(BAD)16 = (2989)10
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Decimal Hex
2989
186
11
0
16
16
16
13
10
11
(2989)10 = (BAD)16
MSP
LSP

Why octal or hex?
• Ease of use and conversion
• Three bits make one octal digit
111 010 110 101
7 2 6 5 => 7265 in octal
• Four bits make one hexadecimal digit
1110 1011 0101
E B 5 => EB5 in hex
4 bits = nibble

Boolean Algebra Arithmetic SIG UNSIGN.ppt

Representation of Signed Numbers
Negative numbers
Three representations
• Signed magnitude
• 1’s complement
• 2’s complement

Sign magnitude
• Make MSB represent sign
• Positive = 0
• Negative = 1
• E.g. for a 3 bit set
– “-2”
Sign Bit Bit
1 1 0
MSB LSB

DRC(Diminished Radix Complement):(r-1)’s
complement
1’s complement
• MSB as in sign magnitude
• Complement all the other bits
• Given a positive number complement all
bits to get negative equivalent
– “-2” Sign Bit Bit
1 0 1
0 1 0

RXC(Radix Complement): r ’s complement
2’s complement
If MSD is a 0 :
• The number is positive
• Remaining (n-1) bits directly indicate the magnitude
If the MSD is 1 :
• The number is negative
• Magnitude is obtained by complementing all the remaining (n-1) bits and
adding a 1
• 1’s complement plus one
– “-2”
Sign Bit Bit
1 1 0
0 1 0

Range of n-bit numbers
Two’s complement numbers:
0111111 + 63
0000110 + 6
0000000 + 0
1111010 – 6
1000001 - 63
1000000 - 64
0 is represented by 000.....0
7- bit number covers the range from +63 to -64.
n-bit number has a range from +(2n-1 - 1) to -(2n-1)

Decimal
number
Signed
magnitude
2’s complement 1’s complement
3 011 011 011
2 010 010 010
1 001 001 001
0 000 000 000
-0 100 --- 111
-1 101 111 110
-2 110 110 101
-3 111 101 100
-4 100
No matter which scheme is used we get an even set of numbers but we need one less
(odd: as we have a unique zero)

Binary Arithmetic in Computers
• Addition / subtraction
• Unsigned
• Signed
– Using negative numbers

Unsigned: Addition
Like normal decimal addition
B
A
The carry out of the MSB is neglected
+ 0 1
0 0 1
1 1 10
0101 (5)
+ 1001 (9)
1110 (14)

Unsigned: Subtraction
Like normal decimal subtraction
B
A
A borrow (shown in red) from the MSB implies a
negative
- 0 1
0 0 11
1 1 0
1001 (9)
- 0101 (5)
0100 (4)

Signed arithmetic
• Use a negative number representation
scheme
• Reduces subtraction to addition

2’s complement
Negative numbers in 2’s complement
001 ( 1)10
101 (-3)10
110 (-2)10
The carry out of the MSB is lost/Discard the carry

Overflow / Underflow
• Maximum value N bits can hold : 2n –1
• When addition result is bigger than the biggest
number of bits can hold.
– Overflow
• When addition result is smaller than the smallest
number the bits can hold.
– Underflow
• Addition of a positive and a negative number
cannot give an overflow or underflow.

Overflow example
011 (+3)10
011 (+3)10
110 (+6)10 ????
1’s complement computer interprets it as –1 !!
(+6)10 = (0110)2 requires four bits !

Underflow examples
Two’s complement addition
101 (-3)10
101 (-3)10
Carry 1 010 (-6)10 ????
The computer sees it as +2.
(-6)10 = (1010)2 again requires four bits !

Boolean Algebra Arithmetic SIG UNSIGN.ppt

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Boolean Algebra Arithmetic SIG UNSIGN.ppt