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Digital Logic & Design

Lecture 01

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Analogue Quantities

Continuous Quantity

Intensity of Light
Temperature
Velocity

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Digital Values

Discrete set of values

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Continuous Signal

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Continuous Signal

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Digital Representation

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Under Sampling

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Electronic Processing

Analogue Systems
Digital Systems
Representing quantities in Digital Systems

Electronic processing of these continuous and digital quantities requires that these quantities be converted into and represented in term of voltages.
Analogue Electronic Systems deal with electronic signals or voltages that are continuous and represent continuous quantities.
Thus a temperature transducer converts a continuous temperature of 39 ⁰C into 39 mVs and 42.75 ⁰C into 42.75 mVs.
Digital Electronic Systems on the other hand deal with discrete electronic signals or voltages that represent discrete or digital values.
How does a digital system such as a calculator process the number 39? Do the Digital systems represent discrete values in terms of voltages? Lets have a look.

One possible solution is to calibrate the electronic circuitry of the calculator to represent the number 1 by 1 mV. The number 10 is represented by 10 mV. The number 39 is represented by 39 mV.

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Representing Digital Values

39 ⁰C ?

39mV

6.25 x 10¹⁵ V !!

Digital System

6.25 x 10¹⁸ ?

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Digital Systems

Two Voltage Levels
Two States

On/Off
Black/White
Hot/Cold
Stationary/Moving

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Binary Number System

Binary Numbers
Representing Multiple Values
Combination of 0v & 5v

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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

Let us have a quick look at the advantages of a Digital System.
Processing and Storage of digital data is efficient.

Computers are very efficient at processing information that is in digital binary form.
A CD can store large number of digitized audio and video clips. Storing same number of audio or video clips in an analogue form requires a large number of audio or video cassettes.

Transmission of digital data is efficient and reliable and less prone to errors.
Even if an error occurs detection and correction of errors in digital data is easier.

We will be looking at a simple example of detecting errors using the parity bit method.

Digitally stored data can be precisely and accurately reproduced.

The picture quality and sound quality of digitized video or audio stored on CDs can be reproduced with a far superior quality as compared to analogue audio and video.

Digital circuits and systems are easier to design and implement.

We would be looking at some simple digital systems in the Digital Logic Design course.

Digital circuits in the form of Integrated circuits occupy very small space.

The PC has a motherboard which has an area less than 1 sq.ft but has all the important circuitry of the computer.
Digital memory implemented as an Integrated circuit small enough to fit in you hand can store an entire collection of books!

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Information Processing

Numbers
Text
Formula and Equations
Drawings and Pictures
Sound and Music

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Logic Gates

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

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Logic Gate Symbol and ICs

1

2

3

4

5

6

G

N

D

V

c

1

3

1

2

1

0

9

8

7400

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Combinational Circuits

Combination of Logic Gates
Adder Combinational Circuit

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Adder Combinational Circuit

Sum

Carry

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Functional Devices

Functional Devices

Adders
Comparators
Encoders/Decoders
Multiplexers/Demultiplexers

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Sequential Circuits

Memory Element
Current & Previous State
Flip-Flops
Counters & Registers

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Block Diagram of a Sequential Circuit

The Sequential circuit consists of a Combinational part and a memory element.
Consider the timer of a microwave oven. You key in the time to cook your favourite dish.
The microwave display unit displays the cooking time.
The memory element of the microwave oven sequential circuit stores the cooking time.
The cooking time is decremented by 1 after every second when a new input signal is received at the input of the combinational part of the Sequential circuit.
Ultimately, when the cooking time decrements to zero and the memory element stores zero , the next input signal sounds an alarm and turns the microwave off.
A traffic signal controller operates in a similar manner. It switches between the green, amber and red signal in a sequence on the basis of current and previous information.
The memory or storage element in the Sequential Circuit is implemented using a very simple digital circuit known as a flip-flop.
Examples of Sequential circuits are Counters and Registers. These Sequential circuits are available as MSI ICs.

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Programmable Logic Devices (PLDs)

Configurable Hardware
Combinational Circuits
Sequential Circuits
Low chip count
Lower Cost
Short development time

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Memory

Storage
RAM (Random Access Memory)

Read-Write
Volatile

ROM (Read-Only Memory)

Read-Only
Non-Volatile

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A/D & D/A Converters

Processing of Continuous values
Conversion

Analogue to Digital A/D
Digital to Analogue D/A

Industrial Control Application

Real-world quantities as mention earlier are continuous in nature and have widely varying ranges. Processing of real-world information can be efficiently and reliably done in the digital domain.
This requires real-world quantities to be read and converted into equivalent digital values which can be processed by a digital system. In most cases the processed output needs to be converted back into real-world quantities.
Two conversions are required, one from the real-world to the digital domain and then back from the digital domain to the real-world.
Modern digitally controlled industrial units extensively use Analogue to Digital (A/D) and Digital to Analogue (D/A) converters to covert quantities represented as an analogue voltage into an equivalent digital representation and vice versa.
The diagram shows a chemical reaction vessel being heated to expedite the chemical reaction.

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Digital Industrial Control

Digital

Controller

Thermocouple

A/D

Converter

u

1

x

1

* / *

D/A

Converter

u

1

x

1

* / *

Reaction

Vessel

Heater

Control

Temperature of the vessel is monitored to control the chemical reaction.
As the temperature of the vessel rises the heat has to be reduced by a proportional level.
An electronic temperature sensor converts the temperature into an equivalent voltage value. This voltage value is continuous and proportion to the temperature.
The voltage representing the temperature is converted into a digital representation which is fed to a digital controller circuit that generates a digital value corresponding to the desired amount of heat.
The digitized output representing the heat is converted back to a voltage value which is continuous and is used to control a valve that regulates the heat.
An A/D converter converts the analogue voltage value representing the temperature into a corresponding digital value for processing.
A D/A converter converts back the digital heat value to its corresponding continuous value for regulating the heater.
A/D and D/A converters and there interface with the real and digital word is an important aspect of digital systems.

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Summary

Continuous Signals
Digital Representation in Binary
Information Processing
Logic Gates

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Summary

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

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Number Systems and Codes

Decimal Number System
Caveman Number System
Binary Number System
Hexadecimal Number System
Octal Number System

Let us start our discussion with Number Systems.
Earlier we have discussed that Digital Systems can process different types of information such as numbers, text, pictures and sound which are represented in the form of Binary numbers.
For understanding the Digital System we need to be fully aware of the Binary Number System and should be able to perform all the functions and operations that we have been performing using the Decimal Number System.
We would be looking at five different Number Systems.
We will start our discussion with the Decimal Number system. We are already aware of the Decimal number system and have been using the decimal number system since our school days.
Next we would be discussing a hypothetical Caveman Number System. I am using the Caveman number system as an example to introduce a Number system which is different from the decimal number system.
The third Number System that I would be discussing is the Binary Number System used by digital systems
The last two number systems that I would be mentioning are the Hexadecimal and the Octal Number systems.

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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 ….

The decimal number system has ten unique digits 0, 1, 2, 3… 9
Using these single digits, ten different values can be represented
To represent values greater than ten, the digits have to be used in different combinations. Thus ten is represented by the number 10, two hundred seventy five is represented by 275 etc. Thus same set of numbers 0,1 2… 9 are repeated in a certain order.
The decimal number system is a positional number system as the position of a digit represents its true magnitude. For example 2 is less than 7, however 2 in 275 represents 200, whereas 7 represents 70. The left most digit has the highest weight and the right most digit has the lowest weight.
The weight of subsequent digits increases by a factor of 10 towards the left starting with 1 as the weight of the right most digit or the least significant digit.
275 can be written as 2 x 10² + 7 x 10¹ + 5 x 10⁰ = 200 + 70 + 5 = 275

The 10 represents the base or radix

10², 10¹, 10⁰ represent the weights 100, 10 and 1 of the numbers 2, 7 and 5

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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

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Caveman Number System

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

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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↑

The caveman can represent the first five decimal numbers (0-4) using a single digit of the caveman number system as shown in the table.
To represent higher values (5-9) a combination of two digits is used.
What should be the most significant digit?
Just think, how do you represent a value greater than 9 in the decimal system?
In a decimal number system after counting up to 9, two digits are used. The most significant digit is fixed at the next highest value 1, and the least significant digit varies between 0 and 9 (10 to19). After 19, the most significant digit is set to 2 and the least significant digit varies between 0 and 9 (20-29) and so on.
Thus in the caveman number system the most significant digit is ∆, and the least significant digit varies between ∑ to ↑. The caveman representation of decimal numbers 5 to 9 is shown in the table.
In a decimal number system how do we represent number after 19? The most significant digit is increased to the next value (from 1 to 2) and the least significant varies between 0 and 9.
Thus in the caveman number system same procedure is followed. The table shows the Caveman number system equivalents to the first 20 decimal values.

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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

Let us see how the caveman number ∆Ω↑∑ represent 220 in decimal.
The number ∆Ω↑∑ can be easily converted into decimal if we write an expression for the Caveman number ∆Ω↑∑ in terms of the base number and the weights.
What is the base number of the caveman number system? 5.
What is the weight of the most significant digit of the number ∆Ω↑∑? 5³
Similarly the weights of the digits Ω, ↑ and ∑ are 5², 5¹ and 5⁰ respectively.
The weights of the four digits starting from the most significant digit are 125, 25, 5 and 1
The weights change by a factor 5.
Thus writing the number ∆Ω↑∑ in terms of an expression is ∆ x 5³ + Ω x 5² + ↑ x 5¹ + ∑ x 5⁰
Replacing the caveman numbers by their decimal equivalents changes the expression to 1 x 5³ + 3 x 5² + 4 x 5¹ + 0 x 5⁰
Solving the expressiongives four terms 125, 75, 20 and 0. Adding the four terms gives the result 220.

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Binary Number System

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

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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

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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

Don’t worry, if you have not understood the binary representation. We would be discussing binary numbers through the course.
Let us find out what does the Binary combination 10011₂represent? It represents the decimal number 19.
Let us see how the Binary number 10011₂ represent decimal 19?
The decimal equivalent of a combination of binary number can easily be found by writing an expression in terms of the base value and weights as we did earlier with the caveman number system.
What is the base number of binary number system? 2
What are the weights of the 5 bits 10011? Starting from the most significant left most bit the weights are 2⁴, 2³, 2², 2¹ and 2⁰.
Writing an expression in terms of the binary bits and the respective weights results in the expression (1 x 2⁴) + (0 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰)
Solving the expression results in five terms 16, 0, 0, 2 and 1, which sum up to 19.
The subscript 2 indicates that the number 10011 is binary and not decimal ten thousand eleven.
The weights of the 5 bits starting from the left, most significant bit are 16, 8, 4, 2 and 1.
The weights in a Binary Number system change by a factor of two.
The most significant bit on the extreme left has the highest weight.
In the decimal number and the caveman number system the weights increased by a factor of 10 and 5 respectively.

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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

Both integers and fractions can be represented in the Binary number system just like in the decimal number system.
The number 1011.101₂ is equivalent to decimal 11.625
Let us see how?
Calculating the decimal equivalent of a binary number representing a fraction can easily be done by writing the expression in terms of the base value and weights as was done in the case of integer numbers.
The binary number 1011.101₂ has an integer part 1011 having the weights 2³, 2², 2¹ and 2⁰.
The fraction part 101 has the weights 2^-1, 2^-2 and 2^-3.
The weights of the fraction part of the binary number decrease by a factor of 2 starting from the bit immediately to the right of the decimal point.
(1 x 2³) + (0 x 2²) + (1 x 2¹) + (1 x 2⁰) + (1 x 2^-1) + (0 x 2^-2) + (1 x 2^-3)
Multiplying the weights with the corresponding decimal equivalents of the binary bits results in the terms 8, 0, 2, 1, 0.5, 0 and 0.125 which add up to give 11.625
One last word on binary representation of fractions. Computers do handle numbers such as 11.625 that have an integer part and a fraction part. However, it does not use the binary representation 1011.101. Such numbers are represented and used in Floating-Point Numbers notation which will be discussed latter

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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

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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

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Decimal-Binary Conversion

Binary to Decimal Conversion

Sum-of-Weights
Adding weights of non-zero terms

Terms 16,0,0.2 and 1

19

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Decimal-Binary Conversion

Binary to Decimal Conversion

Sum-of-Weights
Adding weights of non-zero terms

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Decimal-Binary Conversion

Binary to Decimal Conversion

Sum-of-Weights
Adding weights of non-zero terms