Digital Logic & Design
Lecture 01

Analogue Quantities
Continuous Quantity
 Intensity of Light
 Temperature
 Velocity

Digital Values
 Discrete set of values

Continuous Signal
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time
temperature
0
C

Continuous Signal
1 2
4
7
34
25
23
37
29
42 41
25
22
18
35
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
time
temperature
0
C

Digital Representation
1 2
4
7
18
34
25
23
35
37
29
42 41
25
22
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
samples
temperature
0
C

Under Sampling
0
5
10
15
20
25
30
35
40
45
1 3 5 7 9 11 13 15
samples
temperature
0
C

Electronic Processing
 Analogue Systems
 Digital Systems
 Representing quantities in Digital Systems

Representing Digital Values
39 0C ?
a1
1
a2
2
3
a3
4
a4
b1
b2
b3
b4
5
6
7
8
Vcc1
0
GND
0
1mV = 1
39mV
6.25 x 1015 V !!
Digital
System
6.25 x 1018 ?

Digital Systems
 Two Voltage Levels
 Two States
 On/Off
 Black/White
 Hot/Cold
 Stationary/Moving

Binary Number System
 Binary Numbers
 Representing Multiple Values
 Combination of 0v & 5v

Merits of Digital Systems
 Efficient Processing & Data Storage
 Efficient & Reliable Transmission
 Detection and Correction of Errors
 Precise & Accurate Reproduction
 Easy Design and Implementation
 Occupy minimum space

Information Processing
 Numbers
 Text
 Formula and Equations
 Drawings and Pictures
 Sound and Music

Logic Gates
 Building Blocks
 AND, OR and NOT Gates
 NAND, NOR, XOR and XNOR Gates
 Integrated Circuits (ICs)

Logic Gate Symbol and ICs
AND Gate OR Gate NOT Gate
1
2
3
4
5
6
GND
Vcc
13
12
11
10
9
8
7400
NAND Gate NOR Gate XOR Gate XNOR Gate
NAND Gate IC

Combinational Circuits
 Combination of Logic Gates
 Adder Combinational Circuit

Adder Combinational Circuit
Sum
Carry

Functional Devices
 Functional Devices
 Adders
 Comparators
 Encoders/Decoders
 Multiplexers/Demultiplexers

Sequential Circuits
 Memory Element
 Current & Previous State
 Flip-Flops
 Counters & Registers

Block Diagram of a Sequential Circuit
a1
1
a2
2
b1
b2
5
6
Combinational
Logic Circuit
Output
Input
a1
1
b1
5
Memory Element

Programmable Logic Devices (PLDs)
 Configurable Hardware
 Combinational Circuits
 Sequential Circuits
 Low chip count
 Lower Cost
 Short development time

Memory
 Storage
 RAM (Random Access Memory)
 Read-Write
 Volatile
 ROM (Read-Only Memory)
 Read-Only
 Non-Volatile

A/D & D/A Converters
 Processing of Continuous values
 Conversion
 Analogue to Digital A/D
 Digital to Analogue D/A
 Industrial Control Application

Digital Industrial Control
Digital
Controller
Thermocouple
A/D
Converter
u1
x1
* / *
D/A
Converter
u1
x1
* / *
Reaction
Vessel
Heater
Control

Summary
 Continuous Signals
 Digital Representation in Binary
 Information Processing
 Logic Gates

 Combinational & Sequential Circuits
 Programmable Logic Devices (PLDs)
 Memory (RAM & ROM)
 A/D & D/A Converters
Summary

Number Systems and Codes
 Decimal Number System
 Caveman Number System
 Binary Number System
 Hexadecimal Number System
 Octal Number System

Decimal Number System
 Ten unique numbers 0,1..9
 Combination of digits
 Positional Number System
 275 = 2 x 102 + 7 x 101 + 5 x 100
 Base or Radix 10
 Weight 1, 10, 100, 1000 ….

Representing Fractions
 Fractions can be represented in decimal number
system in a manner
= 3 x 102 + 8 x 101 + 2 x 100 + 9 x 10-1
+ 1 x 10-2
= 300 + 80 + 2 + 0.9 + 0.01
= 382.91

Caveman Number System
 ∑, ∆, >, Ω and ↑
 Base – 5 Number System
 ∆Ω↑∑ = 220

Caveman Number System
Decimal Number Caveman Number Decimal Number Caveman Number
0 ∑ 10 >∑
1 ∆ 11 >∆
2 > 12 >>
3 Ω 13 >Ω
4 ↑ 14 >↑
5 ∆∑ 15 Ω∑
6 ∆∆ 16 Ω∆
7 ∆> 17 Ω>
8 ∆Ω 18 ΩΩ
9 ∆↑ 19 Ω↑

Caveman Number System
 Mr. Caveman is using a base 5 number system.
Thus the number ∆Ω↑∑ in decimal is
= ∆ x 53 + Ω x 52 + ↑ x 51 + ∑ x 50
= ∆ x 125 + Ω x 25 + ↑ x 5 + ∑ x 1
= (1) x 125 + (3) x 25 + (4) x 5 + (0) x 1
= 125 + 75 + 20 + 0 = 220

Binary Number System
 Two unique numbers 0 and 1
 Base – 2
 A binary digit is a bit
 Combination of bits to represent larger values

Binary Number System
Decimal Number Binary Number Decimal Number Binary Number
0 0 10 1010
1 1 11 1011
2 10 12 1100
3 11 13 1101
4 100 14 1110
5 101 15 1111
6 110 16 10000
7 111 17 10001
8 1000 18 10010
9 1001 19 10011

Combination of Binary Bits
 Combination of Bits
 100112 = 1910
= (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21)
+ (1 x 20)
= (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2)
+ (1 x 1)
= 16 + 0 + 0 + 2 + 1
= 19

Fractions in Binary
 Fractions in Binary
 1011.1012 = 11.625
= (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
+ (1 x 2-1) + (0 x 2-2) + (1 x 2-3)
= (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
+ (1 x 1/2) + (0 x 1/4) + (1 x 1/8)
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125
= 11.625
 Floating Point Notations

Decimal-Binary Conversion
 Binary to Decimal Conversion
 Sum-of-Weights
 Adding weights of non-zero terms
 Decimal to Binary Conversion
 Sum-of-Weights (in reverse)
 Repeated Division by 2

Number Weight Result after subtraction Binary
392 256 392-256=136 1
136 128 136-128=8 1
8 54 0
8 32 0
8 16 0
8 8 8-8=0 1
0 4 0
0 2 0
0 1 0
Decimal to binary conversion using
Sum of weight

Decimal-Binary Conversion
2
4 3 2 1
0
10011
(1 2 ) (0 2 ) (0 2 ) (1 2 )
(1 2 )
      
 
Terms 16,0,0.2 and 1
19
 Binary to Decimal Conversion
 Sum-of-Weights
 Adding weights of non-zero terms

Decimal-Binary Conversion
 Binary to Decimal Conversion
 Sum-of-Weights
 Adding weights of non-zero terms

 Binary to Decimal Conversion
 Sum-of-Weights
 Adding weights of non-zero terms
Decimal-Binary Conversion
2
2
10011 16 2 1 19
1 1
1011.101 8 2
2 8
5
11
8
11.625
   
   
 


Lecture No. 1
Number Systems
A Summary