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�Digital Electronics�

Computer Architecture V/S Computer Organization

• Switches and Transistors

• Boolean Algebra and Logic

• Binary Arithmetic and Number Systems

• Combinational Logic and Circuits

• Sequential Logic and Circuits

• Memory Logic Design

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Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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�Computer Architecture V/S Computer Organization�

• Computer Architecture is the programmer’s perspective on functional behavior of a computer for example 32 bits will be used to represent an integer value.

• Computer organization is the internal structural relationships not visible to a program for example physical memory

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Memory

CPU

I/O

I/O Processor

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Two Side View of Computer System

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I/O system

Instr. Set Proc.

Compiler

Operating

System

Application

Digital Design

Circuit Design

Instruction Set

Architecture

Firmware

Datapath & Control

Layout

Software

Hardware

Software/Hardware

Boundary

High-Level Language Programs

Assembly Language

Programs

Microprogram

Register Transfer

Notation (RTN)

Logic Diagrams

Circuit Diagrams

Machine Language

Program

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Basic Units

Architectural levels: Programs and applications
Electrical Signals: discrete, atomic elements of a digital system

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input

output

An ideal switch

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Analog and Digital Systems

Analog devices and systems process time-varying signals that can take on any value across a continuous range.

Digital systems use digital circuits that process digital signals which can take on one of two values, we call:

0 and 1 (digits of the binary number system= Bits conventionally 0=LOW or False and 1= HIGH or TRUE

Digital computers represent the most common digital systems.
Once-analog Systems that use digital systems today:

Audio recording (CDs, mp3)
Telephone system switching
Automobile engine control
Movie effects
Still and video cameras….

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High

Low

Digital circuit

inputs

outputs

:

Analog Signal

Digital Signal

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�Advantages of Digital Systems Over Analog Systems

Reproducibility of the results
Accuracy of results
More reliable than analog systems due to better immunity to noise.
Ease of design: No special math skills needed to visualize the behavior of small digital (logic) circuits.
Flexibility and functionality.
Programmability.
Speed: A digital logic element can produce an output in less than 10 nanoseconds (10^-8 seconds).
Economy: Due to the integration of millions of digital logic elements on a single miniature chip forming low cost integrated circuit (ICs).

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Number Systems

A number system consists of an order set of symbols (digits) with relations defined for +,-,*, /
The radix (or base) of the number system is the total number of digits allowed in the the number system.

Example, for the decimal number system:

Radix, r = 10, Digits allowed = 0,1, 2, 3, 4, 5, 6, 7, 8, 9

In positional number systems, a number is represented by a string of digits, where each digit position has an associated weight.
The value of a number is the weighted sum of the digits.
The general representation of an unsigned number D with whole and fraction portions number in a number system with radix r:

D_r = d _p-1 d _p-2 ….. d₁ d₀.d_-1 d_-2 …. D_-n

The number above has p digits to the left of the radix point and n fraction digits to the right.
A digit in position i has as associated weight rⁱ
The value of the number is the sum of the digits multiplied by the associated weight rⁱ :

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Number Systems Used in Computers

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Number System

Radix

Ordered Set Example

Decimal

r = 10

r = 2

r = 16

r = 8

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 55₁₀

Binary

{0, 1, 2, 3, 4, 5, 6, 7} 377₈

{0, 1} 11111111₂

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F} FF₁₆

Octal

Hexadecimal

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

Hex 0 1 2 3 4 5 6 7 8 9 A B C D E F

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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�Binary numbers�

• Bit a binary digit representing a 0 or a 1.

Base or Radix of Binary numbers are base 2.
Position weights of bits (Instead of decimal places such as 1s, 10s, 100s, 1000s, etc.), uses powers of two to have 1s, 2s, 4s, 8s, 16s, 32s, 64s, etc.

(101)₂= (1×2²)+(0×2¹)+(1×2⁰)=410 + 110 = 5₁₀

(10111)₂= (1×2⁴)+(0×2³)+(1×2²)+(1×2¹)+(1×2⁰)=23₁₀

(41)₁₀ = 41/2 + remainder = 1🡪1LSB

= 20/2 + remainder = 0 🡪2SB

= 10/2 + remainder = 0 🡪3SB

= 5/2 + remainder = 1 🡪 4SB

= 4/2 + remainder = 0 🡪 5SB

= 2/2 = 1

🡪 101001₂

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�Largest numbers�

The largest number of d digits in base R is = R^d- 1

Examples:

3 digits of base 10: 10³-1 = 999

2 digits of base 16: 16²-1 = 255

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Decimal-to-Binary Conversion

Separate the decimal number into whole and fraction portions.
To convert the whole number portion to binary, use successive division by 2 until the quotient is 0. The remainders form the answer, with the first remainder as the least significant bit (LSB) and the last as the most significant bit (MSB).
Example: Convert 179₁₀to binary:

179 / 2 = 89 remainder 1 (LSB)

/ 2 = 44 remainder 1

/ 2 = 22 remainder 0

/ 2 = 11 remainder 0

/ 2 = 5 remainder 1

/ 2 = 2 remainder 1

/ 2 = 1 remainder 0

/ 2 = 0 remainder 1 (MSB)

179₁₀ = 10110011₂

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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Solved Examples of Decimal-to-Binary

108/2 = 54

54 * 2 = 108, remainder 0

54 /2 = 27

27 * 2 = 54, remainder 0

27/2 = 13.5

13 * 2 = 26, remainder 1

13 /2 = 6.5

6 * 2 = 12, remainder 1

6/2 = 3

3 * 2 = 6, remainder 0

3/2 = 1

1 * 2 = 2, remainder 1

1101100₂

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11/2 = 5.5

5 * 2 = 10, remainder 1

5/2 = 2.5

2 * 2 = 4, remainder 1

2/2 = 1

1 * 2 = 2, remainder 0

1 / 2 = 0

0 * 2 = 0, remainder 1

1011₂

7/2 = 3.5

3 * 2 = 6, remainder 1

3/2 = 1

1 * 2 = 2, remainder 1

1/2 = 0

0 * 2 = 0, remainder 1

111₂

90/2 = 45

45 * 2 = 90, remainder 0

45/2 = 22.5

22 * 2 = 44, remainder 1

22 * 2 = 44, remainder 0

22/2 = 11

11 * 2 = 22, remainder 0

11/2 = 5.5

5 * 2 = 10, remainder 1

5/2 = 2.5

2 * 2 = 4, remainder 1

2/2 = 1

1 * 2 = 2, remainder 0

1 / 2 = 0

0 * 2 = 0, remainder 1

1011010₂

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Decimal-to-Hex examples

108/16 = 6.75

6 * 16 = 96, remainder 12

6 /16 = 0

0 * 16 = 0, remainder 6

6C₁₆

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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20/16 = 1

1 * 16 = 16, remainder 4

1/16 = 0

0 * 16 = 0, remainder 1

14₁₆

32/16 = 2

2 * 16 = 32, remainder 0

2 /16 = 0

0 * 16 = 0, remainder 2

20₁₆

90/16 = 5.625

5 * 16 = 80, remainder 10

5 / 16 = 0

0 * 16 = 0, remainder 5

5A₁₆

160/16 = 10

10 * 16 = 160, remainder 0

10/16 = 0

0 * 16 = 0, remainder 10

A0₁₆

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Decimal-to-Octal example

108/8 = 13.5

13 * 8 = 104, remainder 4

13/8 = 1

1 * 8 = 8, remainder 5

1 / 8 = 0

0 * 8 = 0, remainder 1

154₈

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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10/8 = 1

1 * 8 = 8, remainder 2

1/8 = 0

0 * 8 = 0, remainder 1

12₈

16/8 = 2

2 * 8 = 16, remainder 0

2/8 = 0

0 * 8 = 0, remainder 2

20₈

24/8 = 3

3 * 8 = 24, remainder 0

3/8 = 0

0 * 8 = 0, remainder 3

30₈

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Decimal-to-Binary Conversion

To convert decimal fractions to binary, repeated multiplication by 2 is used, until the fractional product is 0 (or until the desired number of binary places). The whole digits of the multiplication results produce the answer, with the first as the MSB, and the last as the LSB.
Example: Convert 0.3125₁₀to binary

Result Digit

.3125 × 2 = 0.625 0 (MSB)

.625 × 2 = 1.25 1

.25 × 2 = 0.50 0

.5 × 2 = 1.0 1 (LSB)

0.3125₁₀= .0101₂

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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Binary Arithmetic Operations - Addition

Similar to decimal number addition, two binary numbers are added by adding each pair of bits together with carry propagation.
Addition Example:

1 0 1 1 1 1 0 0 0 Carry

X 190 1 0 1 1 1 1 1 0

Y + 141 + 1 0 0 0 1 1 0 1

X + Y 331 1 0 1 0 0 1 0 1 1

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 with a carry of 1

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Binary Arithmetic- subtraction�

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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95 = 1011111

-16 = 0010000

79 = 1001111

0 – 0 = 0

1 – 0 = 1

1 – 1 = 0

0 – 1 = 1 with a borrow of 1

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Binary Arithmetic Operations: Subtraction

Two binary numbers are subtracted by subtracting each pair of bits together with borrowing, where needed.
Subtraction Example:

0 0 1 1 1 1 1 0 0 Borrow

X 229 1 1 1 0 0 1 0 1

Y - 46 - 0 0 1 0 1 1 1 0

183 1 0 1 1 0 1 1 1

Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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Binary Arithmetic - Multiplication�

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1011

*101

1011

0000

1011

110111

0 * 0 = 0

0 * 1 = 0

1 * 0 = 0

1 * 1 = 1

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Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/

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Negative Binary Number Representations

Signed-Magnitude Representation:

For an n-bit binary number:

Use the first bit (most significant bit, MSB) position to

represent the sign where 0 is positive and 1 is negative.

Ex. 1 1 1 1 1 1 1 1₂= - 127₁₀

Remaining n-1 bits represent the magnitude which may range from:

-2^(n-1) + 1 to 2^(n-1) - 1

This scheme has two representations for 0; i.e., both positive and negative 0: for 8 bits: 00000000, 10000000

Arithmetic under this scheme uses the sign bit to indicate the nature of the operation and the sign of the result, but the sign bit is not used as part of the arithmetic.

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Sign

Magnitude

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Parity bit�

Pad an extra bit to MSB side to make the number of 1’s to be even or odd.
Sender and receiver of messages make sure that even/odd transmission patterns match

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�Gray codes�

• In binary codes, number of bit changes are not constant,

000🡪001🡪010🡪011🡪100🡪101🡪110🡪111🡪1000…

bit changes in gray codes are constant
000🡪001🡪011🡪010🡪110🡪111🡪000…

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Alphanumeric Binary Codes: ASCII

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7 bit codes are used to represent all upper and lower case letters, numbers, punctuation and control characters

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Boolean Algebra

Boolean Algebra named after George Boole who used it to study human logical reasoning – calculus of proposition.
Elements : true or false ( 0, 1)
Operations: a OR b; a AND b, NOT a

e.g. 0 OR 1 = 1 0 OR 0 = 0

1 AND 1 = 1 1 AND 0 = 0

NOT 0 = 1 NOT 1 = 0

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What is an Algebra?

set of elements (e.g. 0,1,2,..) + set of operations (e.g. +, -, *,..) + postulates/axioms (e.g. +x=x,..)

For example : (e.g. algebra of integers)

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Digital (logic) Elements: Gates

Digital devices or gates have one or more inputs and produce an output that is a function of the current input value(s).

All inputs and outputs are binary and can only take the values 0 or 1

A gate is called a combinational circuit because the output only depends on the current input combination.

Digital circuits are created by using a number of connected gates such as the output of a gate is connected to to the input of one or more gates in such a way to achieve specific outputs for input values.

Digital or logic design is concerned with the design of such circuits.

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Boolean Algebra

Set of Elements: {0,1}
Set of Operations: {., + , ¬ }

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Signals: High = 5V = 1; Low = 0V = 0

x

y

x.y

x

y

x+y

x

x'

AND

OR

NOT

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Logic Gates

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EXCLUSIVE OR

a

b

a.b

a

b

a+b

a

a'

a

b

(a+b)'

a

b

(a.b)'

a

b

a ⊕ b

a

b

a.b

&

a

b

a+b

+

AND

a

a'

1

a

b

(a.b)'

&

a

b

(a+b)'

≥1

a

b

a ⊕ b

=1

OR

NOT

NAND

NOR

Symbol set 1

Symbol set 2

(ANSI/IEEE Standard 91-1984)

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Truth Tables

Provide a listing of every possible combination of values of binary inputs to a digital circuit and the corresponding outputs.

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x

y

x . y

x + y

0

1

0

1

0

1

Example (2 inputs, 2 outputs):

Digital circuit

inputs

outputs

x

y

inputs

outputs

x + y

x . y

Truth table

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Logic Gates: The AND Gate

The AND Gate

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A

B

A.B

Truth table

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Ground

Vcc

Top View of a TTL 74LS family 74LS08 Quad 2-input AND Gate IC Package

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Logic Gates: The OR Gate

The OR Gate

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A

B

A+B

Truth table

Top View of a TTL 74LS family 74LS08 Quad 2-input OR Gate IC Package

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Logic Gates: The NAND Gate

The NAND Gate

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A

B

(A.B)'

A

B

(A.B)'

≡

Truth table

Top View of a TTL 74LS family 74LS00 Quad 2-input NAND Gate IC Package

NAND gate is self-sufficient (can build any logic circuit with it).
Can be used to implement AND/OR/NOT.
Implementing an inverter using NAND gate:

x

x'

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Logic Gates: The NOR Gate

The NOR Gate

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≡

A

B

(A+B)'

A

B

(A+B)'

Truth table

Top View of a TTL 74LS family 74LS02 Quad 2-input NOR Gate IC Package

NOR gate is also self-sufficient (can build any logic circuit with it).
Can be used to implement AND/OR/NOT.
Implementing an inverter using NOR gate:

x

x'

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Logic Gates: The XOR Gate

The XOR Gate

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

Ground

Vcc

A

B

A ⊕ B

Truth table

Top View of a TTL 74LS family 74LS86 Quad 2-input XOR Gate IC Package

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Drawing Logic Circuits

When a Boolean expression is provided, we can easily draw the logic circuit.
Examples:

F1 = xyz'

(note the use of a 3-input AND gate)

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x

y

z

F1

z'

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Analyzing Logic Circuits

When a logic circuit is provided, we can analyze the circuit to obtain the logic expression.
Example: What is the Boolean expression of F4?

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A'B'

A'B'+C

(A'B'+C)'

A'

B'

C

F4

F4 = (A'B'+C)'

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Integrated Circuits

An Integrated circuit (IC) is a number of logic gated fabricated on a single silicon chip.
ICs can be classified according to how many gates they contain as follows:

Small-Scale Integration (SSI): Contain 1 to 20 gates.
Medium-Scale Integration (MSI): Contain 20 to 200 gates. Examples: Registers, decoders, counters.
Large-Scale Integration (LSI): Contain 200 to 200,000 gates. Include small memories, some microprocessors, programmable logic devices.
Very Large-Scale Integration (VLSI): Usually stated in terms of number of transistors contained usually over 1,000,000. Includes most microprocessors and memories.

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Computer Hardware Generations

The First Generation, 1946-59: Vacuum Tubes, Relays, Mercury Delay Lines:

ENIAC (Electronic Numerical Integrator and Computer): First electronic computer, 18000 vacuum tubes, 1500 relays, 5000 additions/sec.
First stored program computer: EDSAC (Electronic Delay Storage Automatic Calculator).

The Second Generation, 1959-64: Discrete Transistors.

(e.g IBM 7000 series, DEC PDP-1)

The Third Generation, 1964-75: Small and Medium-Scale Integrated (SSI, MSI) Circuits. (e.g. IBM 360 mainframe)

The Fourth Generation, 1975-Present: The Microcomputer. VLSI-based Microprocessors.

Assignment : Prepare Time line of Operating Systems.

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Assignment

Write pseudo code/Flowchart/Python Code for Conversion of information from one base to other
Write pseudo code/Flowchart/Python Code for Complement representation for negative integers
Develop a prototype for information processing calculator of base-r .and extend it to compute arithmatic from multiple base and produces results in a desired base.
Develop ANN for each Combinational Logic Circuits.
Develop ANN for each Sequential Logic Circuits.

Note :- Submit Video Links . And code links in Github.

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Dr. Shaligram Prajapat https://sites.google.com/site/shaligramiipsdavvindore/