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### CPS-304DIGITAL LOGIC & DESIGN

Instructor : Ms. Saba Iqbal

- Digital Design by Morris Mano , 2 nd Edition/ 3rd Edition/Digital Fundamentals.

What’s Course About?

- Introduction to concepts of digital logic, gates, and the digital circuits
- Design and analysis of combinational and sequential circuits
- Basics of logic design of computer hardware

Course Outline

- Binary Systems
- Binary Algebra
- Simplification of Boolean Functions
- Combinational Logic
- Sequential Logic
- MSI Sequential Circuits

Digital Systems

- Digital Computer follow a sequence of instructions, Digital System play a prominent role in this digital age
- Communication, medical treatment, internet, DVD, CD, Space ,Programme.Scientific &Educational field ,ATC commercial etc.
- called programs, that operate on given data
- User can specify and change program or data according to needs
- Like Digital Computers, most digital devices are programmable
- Digital Systems have the ability to Manipulate discrete elements of information.
- Any set that is restricted to a finite number of elements contains discrete information
- 10 Decimal digits
- 26 Alphabet letters
- 52 Playing cards
- 64 squares of a chessboard

Digital Systems

- Digital Systems can do hundreds of millions of operations per second
- Extreme reliability due to error-correcting codes
- A Digital System is interconnection of digital modules
- To understand Digital module, we need to know about digital circuits and their logical functions
- Hardware Description Language (HDL) is a programming language that is suitable for describing digital circuit in a textual form
- Simulate a digital system to verify operation before it is built

COMPUTER

Analog Computer,. It responds to continuous signals.

Digital computer. It responds to 0 and 1. also called Binary.

Main Modules.

Memory Unit

Processor Unit

Control Unit

Input Device / Output Device

CPU Processor combined with Control Unit

Micro Processor. CPU in a Small integrated circuit

CPU combined with Memory and Interface control for i/p and o/p devices form a micro computer.

DATA FLOW

- Fetch Time. Getting data and instructions from ALU and then issue command Fix time
- Execute Time. ALU carries out execution Time is variable
- Master clock. It is in control unit and control all functions
- Memory
- RAM Semi conductor memory & Ferrite core memory
- Sequential Memory . Mag tape, mag disk, CD Floppy Mag Drum.
- each info has a location and an address.

DEFINATIONS MEMORY

- Random Access Memory,. Access time to a location is constant.
- Sequential access memory. Access time to all locations are different
- Main memory and Secondary memory. How we store
- Semi conductor Magnetic Material
- Binary Req. as material can store only 1 and 0
- Three things are stored, Instructions, Data, Address.

Decimal Number

- 7,392= 7x103 +3x102 + 9x101 + 2x100
- Thousands, hundreds, etc…power of 10 implied by position of coefficient
- Generally a decimal number is represented by a series of coefficients
- a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4
- aj cofficient are any of the 10 digit (0,1,2…9)
- Decimal number are base 10

Binary Number

- Digital Systems manipulate discrete quantities of information in binary form
- Operands in calculations
- Decimal Digits
- Results
- Strings of binary digits (“bits”)
- Two possible values 0 and 1

Binary Numbers

- Each digit represents a power of 2
- Coefficient have two possible values 0 and 1
- Strings of binary digits (“bits”)
- n bits can store numbers from 0 to 2n-1
- n bits can store 2n distinct combinations of 1’s and 0’s
- Each coefficient aj is multiplied by 2j
- So 101 binary is

1 x 22 + 0 x21 + 1 x 20

or

1 x 4 + 0 x 2 + 1 x 1 = 5

BITs & Bytes

- A bit (short for binary digit) is the smallest unit of data in a computer.
- A bit can hold only one of two values: 0 or 1, corresponding to the electrical values of off or on, respectively.
- Because bits are so small, you rarely work with information one bit at a time
- A byte is a unit of measure for digital information.
- A single byte contains eight consecutive bits
- Binary Arithmetic. Addition, Subtraction Multiplication
- Give example

Octal

- Octal is base 8
- A number is represented by a series of coefficients
- a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4
- aj cofficient are any of 8 digit (0,1,2…7)
- Need 3 bits for representation
- Example:

(127.4)8= 1 X 82 +2 X 81 +7 X 80 + 4 X 8-1

64+16+7+.5= (87.5)10

Hexadecimal

- Hexadecimal is base 16
- A number is represented by a series of coefficients
- a6 a5 a4 a3 a2 a1 a0.a-1 a-2 a-3 a-4
- aj cofficient are any of 16 digit (0,1,2,3,4,5,6,7,8, 9,A,B,C,D,E,F)
- Need 4 bits for representation
- (B65F)16

11 X 163 +6 X 162 + 5 X 161 +15 X 160

= 11x4096 + 6x256 +5x16 +15

= 45056 + 1536 + 80 +15 = 46,687

Converting Binary to Decimal

- Easy, just multiply digit by power of 2
- Just like a decimal number is represented
- Example follows

Decimal to Binary

- A little more work than binary to decimal
- Some examples
- 3 = 2 + 1 = 11 (that’s 1•21 + 1•20)
- 5 = 4 + 1 = 101 (that’s 1•22 + 0•21 + 1•20)

Algorithm – Decimal to Binary

- Find largest power-of-two smaller than decimal number
- Make the appropriate binary digit a ‘1’
- Subtract the power of 2 from decimal
- Do the same thing again

Decimal Binary Example

- Convert 28 decimal to binary

32 is too large, so use 16

Binary 10000

Decimal 28 – 16 = 12

Next is 8

Binary 11000

Decimal 12 – 8 = 4

Next is 4

Binary 11100

Decimal 4 – 4 = 0

Decimal Binary (Fraction)

- Convert decimal 0.6875 to binary

Integer Fraction Coefficient

0.6875 X 2= 1 0.3750 a-1=1

0.3750 X 2= 0 0.7500 a-2=0

0.7500 X 2= 1 0.5000 a-3=1

0.5000 X 2= 1 0.0000 a-4=1

(0.6875)10 = (0.1011)2

Decimal to Octal

Similar to decimal binary.

- Find largest power-of-8 smaller than decimal number
- Divide by power-of-8. The integer result is Octal digit.
- The remainder is new decimal number.
- Do the same thing again

Decimal Octal

- Convert decimal 153 to Octal

512 is too large, so use 64

Octal 200

Decimal 153 – 64X2 = 25

Next is 8

Decimal 25 – 8X3= 1

Octal 230

Decimal 1 – 1X1 = 0

Next is 1

Octal 231

Decimal Octal (Fraction)

- Convert decimal 0.513 to Octal

Integer Fraction Coefficient

0.513 X 8 = 4 + 0.104 a-1=4

0.104 X 8 = 0 + 0.832 a-2=0

0.832 X 8 = 6 + 0.656 a-3=6

0.656 X 8 = 5 + 0.248 a-4=5

0.248 X 8 = 1 + 0.984 a-5=1

0.984 X 8 = 7 + 0.872 a-5=7

(0.513)10= (0.406517)8

Binary to Octal

- Partition Binary number into group of three digits each
- The corresponding octal digit is then assigned to each group
- (10 110 001 101 011 . 111 100 000 100)2
- (10 110 001 101 011 . 111 100 000 100)2 = (26153.7460)8

Octal to Binary

- Each Octal digit is converted to its three digit binary equivalent
- (26153.7460)8 = (010 110 001 101 011 . 111 100 000 100)2

1010

1100

Hex to Binary- Convention – write 0x before number
- Hex to Binary – just convert digits

0x2ac

0x2ac = 001010101100

No magic – remember hex digit = 4 bits

3

7

b

Binary to Hex- Just convert groups of 4 bits

101001101111011

1011

0101

0011

0111

101001101111011 = 0x537b

Hex to Decimal

- Just multiply each hex digit by decimal value, and add the results.

0x2ac

2 • 256

+ 10 • 16

+ 12 • 1

= 684

Decimal to Hex

Similar to decimal binary.

- Find largest power-of-16 smaller than decimal number
- Divide by power-of-16. The integer result is hex digit.
- The remainder is new decimal number.
- Do the same thing again

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