Chapter 1 Ppt

A VERY

GOOD

MORNING TO

ALL

OF

U

BINARY SYSTEMS

CHAPTER 1

DIGITAL LOGIC DESIGN

TOPICS

• DIGITAL SYSTEMS• NUMBER SYSTEMS• NUMBER BASE CONVERSIONS• OCTAL AND HEXADECIMAL NUMBERS• COMPLEMENTS• SIGNED BINARY NUMBERS• BINARY CODES• BINARY STORAGE AND REGISTERS• BINARY LOGIC

DIGITAL SYTEMS

• SYSTEM• TYPES OF SYSTEMS• CLASSIFICATION OF SYSTEMS• APPLICATIONS OF DIGITAL SYSTEMS• ADVANTAGES OF DIGITAL SYSTEMS• DISADVANTAGES OF DIGITAL SYSTEMS• DIGITAL COMPUTER• DESIGN OF DIGITAL SYSTEMS• SWITCHING CIRCUITS• WHY BINARY IN DIGITAL SYSTEMS?

SYSTEM

Accepts various Inputs Performs a particular task Generates output

Classification of SystemsAnalog Systems

(Continuous)Digital Systems(Discrete step

by step)

Physical quantities or signals may vary continuously over a specified range.

Physical quantities or signals can assume only discrete values .

Hence represented by continuously variable indicator

Represented by symbols called digits

Thermometer Digital Clock

Applications of Digital Systems

• Communication


• Business Transactions


• Traffic Control


• Space Guidance• Weather

Monitoring


• Medicine


• Internet


• Commercial


• Commercial


• Commercial


Commercial


Industry


• Scientific Enterprises


• Military

Advantages of Digital Systems

• Easier to Design• Information storage is easy• Accuracy and Precision

through out the system• Operations can be

programmed• Digital Circuits are less prone

to noise

Disadvantages of Digital Systems

• Real world is analog• Digitization of information is a

time consuming process.

Digital Computer• It can follow a sequence of instructions

called a program, that operates on given data.

• Program and data can be varied according to the user’s needs.

• Hence, it can perform various information processing tasks that fulfill several applications.

Digital Computer

• One important characteristic of digital computer is the ability to manipulate discrete elements of information.

• Discrete information must contain finite number of elements.

• Eg: 10 Decimal digits, 26 alphabets, 52 playing cards, 64 chess squares

Digital Computer• The name Digital Computer emerged

from an application.• Early computers are used for numeric

computations in which discrete elements are digits.

• Physical quantities used to represent discrete information are signals.

• Some of the signals are voltage and current signals (Implemented by transistors)

Digital Computer• But signals used in digital systems have 2

discrete values 0 & 1 ( binary)• Binary Digit- Bit 0 or 1• Group of bits – Binary Codes• Hence using various techniques, groups of bits

can represent discrete symbols.• These symbols are again used to develop

system in digital format.• So, we can say that digital system manipulates

discrete elements of information which is again represented internally in binary form

Design of Digital SystemsSystem Design

Logic Design

Circuit Design

System Design •Breaking overall system into subsystems & specifying chcs. of each sub-systems

Logic Design •Determines how to interconnect and control these sub-systems

Circuit Design •Specifying the interconnection of specific components like resistors, diodes and transistors to form gates, flip flops, or other logic building blocks

Switching Circuits

• Many of sub systems of digital systems take form of a switching circuit.

• It consists of one or more inputs and outputs having discrete values.

Switching Circuit

x₁

x₂..

xm

Z₁

Z₂..

Zm

Classification of Switching Circuits

Combinational Circuits Sequential Circuits

Output depends on present on input only

Output depends on both past and present inputs

Building blocks are logic gates

Building blocks are logic gates and flip flops

No memory is required as no storage is necessary

Memory is required as past inputs are to be stored

Eg: Multiplexers, Decoders, Encoders, PLDs, PLAs, PALs, CPLDs, FPGAs etc.

Eg: Ring Counter, Synchronous Counter, Ripple Counter

Why binary in Digital Systems

• In general, switching devices used in digital systems are generally two-state devices.

• So, output can assume only two discrete values.

Switching devices

Relay

On

Diode Transistor

Number Systems

• Decimal Number System (10)• Binary Number System ( 2 )• Octal Number System ( 8 )• Hexadecimal Number System (16)

Decimal Number System

Representation

=5*10^2 + 0*10^1 + 1*10^0 + 6*10^-1 + 8*10^-2

(501.68)10

Decimal Number System

• Numbers have positional importance• 349.2510

In the binary system, positional importance follows powers of 2

3 x 102 = 3 x 100 = 300

4 x 101 = 4 x 10 = 40

9 x 100 = 9 x 1 = 9 2 x 10-1 = 2/10

5 x 10-2 = 5/100

Conversion from Binary to Decimal

• (11001.11)₂

= 1*2^4 + 1*2^3 + 0*2^2 + 0* 2^1 + 1*2^0 + 1*2^-1 + 1*2^-2 = 16 + 8 + 0 + 0 + 1 + 0.5 + 0.25

= ( 25.75 )10

Conversion from Octal to Decimal

( 347.205)₈

= 3 * 8^2 + 4 * 8^1 + 7 * 8^0 + 2 * 8^-1 + 0 * 8^-2 + 5 * 8^-3

= 192 + 32 + 7 + 0.25 + 0 + 0.01

= ( 231.26)10

Conversion from Hexadecimal to Decimal

• ( 23A4.EC)16

= 2 * 16^3 + 3 * 16^2 + A * 16^1 + 4 * 16^0 + E * 16^-1 + C * 16^-2= 8192 + 768 + 160 + 4 + 0.875 + 0.0468

= (9214.9218 )10

Conversion from Decimal to Binary

( 61 )10

Conversion from Decimal to Binary

( 61 )10

Decimal value Integer Fraction Coefficient

0001

Conversion from Decimal to Octal

(247.6875)10

= (367.54)₈

Octal Number SystemOctal Numbers Binary Equivalents

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

Hexadecimal Number SystemDecimal Values Hexadecimal

RepresentationBinary Equivalents

0 0 0000

1 1 0001

2 2 0010

3 3 0011

4 4 0100

5 5 0101

6 6 0110

7 7 0111

8 8 1000

9 9 1001

10 A 1010

11 B 1011

12 C 1100

13 D 1101

14 E 1110

15 F 1111

Binary Arithmetic

• Binary Addition• Binary Subtraction• Binary Multiplication• Binary Division

Binary Addition

1 0 1 1 0 1 0 1 + 1 1 0 0 + 1 1 1 1 _ _ _ _ _ _ _ _ _ _ 1 0 1 1 1 1 0 1 0 0

Addition of two binary numbers

1410 = 011102

+2510 = 1100121

0

1

0

1

0

0

1

0

1

1

0

= 32 + 7 = 39

1 0 1 1 0 1

+ 0 1 1 1 0 1

1

0

0

1

1

0

1

1

1

0

1

0

1

Carry

Sum

Check your work

45

+ 29

= 74

Binary Subtraction

1 0 1 1 Minuend11101

- 0 1 1 0 Subtrahend - 10011

_ _ _ _ _ _ _ _ _ _ _ _ 0 1 0 1 Difference

01010

Binary Multiplication

1001 Mul t ip l icand *1101 Mul t ip l ier _____

1001 0000 1001

1001 _________

1110101

Partial Products

Final Product

Binary Division 1101 1001 1110101 1001 - - - - - - - - - 1011 1001 - - - - - - - - - - - 1001 1001

- - - - - - - - - - - 0000

Complements

Used in digital computers • for simplifying the subtraction

operation and• for logical manipulation.

Diminished Radix Complement

• Also called (R-1)’s complement• (R-1)’s complement of any number system

can be defined as ( Rⁿ-1 )-N• R = Base or Radix of a given number system• N = given number• n = no. of digits present in the given number• For Decimal Number System, R-1’s

complement is (10ⁿ-1) –N• For Binary Number System, R-1’s

complement is (2ⁿ-1) –N

For Decimal Number System

• R = 10 ==> (R-1) = 9• ( R-1 )’s complement = 9’s complement = (10ⁿ-1) –N (R-1)’s complement of 546700 is 9’s complement of 546700 = (10⁶-1)-

(546700)

= 99999-546700

= 453299

For Binary Number System

• R = 2 ==> (R-1) = 1• ( R-1 )’s complement = 9’s complement = (2ⁿ-1) –N (R-1)’s complement of 1011000 is 1’s complement of 1011000 =(2⁷-1)-

(1011000)

= 1111111-1011000

= ( 0100111 )₂

Radix Complement

• Also called R’s complement• R ’s complement of any number system can

be defined as [ ( Rⁿ-1 )-N ]+ 1• R = Base or Radix of a given number system• N = given number• n = no. of digits present in the given number• For Decimal Number System, R-1’s

complement is [ (10ⁿ-1) –N ] + 1• For Binary Number System, R ’s complement

is [ (2ⁿ-1) –N ] + 1


• R = 10 R’s complement = 9’s complement

= [ (10ⁿ-1) –N ] + 1 R’s complement of 546700 is 10’s complement of 546700 = 9’s

complement + 1

9’s complement = (10⁶-1)- (546700) = 999999-546700

= 45329910’s complement = 453299 + 1 = 453300


• R = 2 ==> R’s complement = 2’s complement

= (2ⁿ-1) –N R’s complement of 1011000 is 1’s

complement + 1

1’s complement of 1011000 =(2⁷-1)- (1011000)

= 1111111-1011000

= ( 0100111 )₂

2’s complement = 0100111 + 1 = (01001000)₂

Subtraction with Complements(Unsigned Numbers)

• During subtraction of two n- digit unsigned numbers M & N of same base R, there occurs two cases

• Case (i): M>N• Case (ii): M<N• This operation can be applied for

any number system


• M= minuend = 72532

• N = subtrahend = 3250• Perform M-N• Case (i) M>N is to be applied

M= 72532 M= 72532N = 03250 10’s complement N = 96750

--_______ +_______

69282 169282


So case (ii) : M<NM = 03250 M = 03250N = 72532 N = 27468

-______ +_____

-69282 30718


• M= minuend = 1010100• N = subtrahend = 1000011• Perform M-N• Case (i) M>N is to be applied

M= 1010100 M= 1010100N = 1000011 2’s complement N = 0111101

-- _________ +_________ 0010001 10010001


So case (ii) : M<NM = 1000011 M = 1000011N = 1010100 N = 0101100

-_______ +________

- 0010001 1 1 0 1 1 1 1

Final Answer = - ( 2’s complement of 1101111) = - ( 0010001)₂

Signed- Binary Numbers

• “+ve” sign indicates a positive number• “-ve” sign indicates a negative number• Digital Circuits can understand only two

numbers 0 & 1• Hence to indicate the sign, an additional

bit is placed as the most significant bit.• 0 represents a +ve number• 1 represents a –ve number

• “+ve” sign indicates a positive number• “-ve” sign indicates a negative number• Digital Circuits can understand only two

numbers 0 & 1• Hence to indicate the sign, an additional

bit is placed as the most significant bit.• 0 represents a +ve number• 1 represents a –ve number


Consider an 8 bit number (01000100)₂

MSB of this no. is 0 which represents a +ve sign i.e., a positive number

Its equivalent is (01000100)₂ = (+68)10

Consider another number (11000100)₂

MSB of this no. is 1 which represents –ve sign i.e., a negative number

Its equivalent is (11000100)₂ = (-68)10


(101100)₂ =

MSB = 1 -ve number Magnitude =

(01100) = (12)₂ 10

Answer = Sign & Magnitude = (-12)10


(0111)₂ =

MSB = 0 +ve number Magnitude =

(111) = ( 7 )₂ 10

Answer = Sign & Magnitude = ( +7 )10

Signed- Binary Numbers1’s Complement representation

Also called Signed Complement representation

i.e., Sign + Complement

( 0101 )₂ = (+5)10

( 1010 )₂ = ( -5 )10 in 1’s complement form

( 01000 )₂ = (+8)10


Signed- Binary Numbers2’s Complement representation

Also called Signed 2’s Complement representation

i.e., Sign + 2’s Complement of magnitude

( 0101 )₂ = (+5)10


( 01000 )₂ = (+8)10

( 11000)₂ = ( -8 )10 in 2’s complement form

Signed Decimal Numbers

Signed Magnitude Signed 1’s complement

Signed 2’s complement

(+7) 0111 0111 0111

(+6) 0110 0110 0110

(+5) 0101 0101 0101

(+4) 0100 0100 0100

(+3) 0011 0011 0011

(+2) 0010 0010 0010

(+1) 0001 0001 0001

(+0) 0000 0000 0000

(-0) 1000 1111 --

(-1) 1001 1110 1111

(-2) 1010 1101 1110

(-3) 1011 1100 1101

(-4) 1100 1011 1100

(-5) 1101 1010 1011

(-6) 1110 1001 1010

(-7) 1111 1000 1001

(-8) -- -- 1000

Addition of 2 signed numbers

+ 6 00000110 +13 00001101 - - - - - -

- - - - - - - - - - - - - - - - - - - +19 00010011

- 6 11111010 +13 00001101 ------ ------------------- +19 00000111

- 6 11111010 -13 11110011 ------ ------------------- +19 11101101

+ 6 00000110 -13 11110011 ------ ------------------- +19 11111001

Subtraction of 2 signed numbers

• (+ A) – (+B) = (+ A) + (-B)• (+ A) – (-B) = (+ A) + (+B)

Overflow, Signed Integers

• As has been shown, when numbers are treated as signed integers, a “carry” of 1 from the addition of the most significant bits DOES NOT indicate an overflow, 3 00011+ (-3) +11101= 0 = 00000, with a carry of “1”

• For signed integers, overflow occurs when:

• The addition of two positive numbers results in a negative number, orThe addition of two negative numbers results in a positive number

Overflow Examples• In a 6-bit register

+ 17 = 010001+ 16 = +010000

=100001100001 = - (011110 + 1) = - 011111 = -31

• In an 8-bit register- 100 = - (0110 0100) = 1001 1011 +1 = 1001 1100- 50 = - (0011 0010) = 1100 1101 +1 = 1100 1110

= 0110 10100110 1010 = 6A16=6*16 + 10 = +106

Range of a numberOverflow during addition

• A fixed-length register can only hold a Range of numbers

• For a 4-bit device, the range of positive integers is 0 - 15

• For an 8-bit device the range of positive integers is 0 - 255

• When adding positive integers, Overflow occurs when the sum falls outside the range of the register

Overflow Summary

• For positive integers, overflow occurs when the carry from addition of the leftmost bits is a “1”

• For signed integers, overflow occurs when either

The addition of two negative numbers gives a positive number, or

The addition of two positive numbers gives a negative number.

Binary Codes

Codes

Weighted

Non-weighted

Weighted Codes

• BCD (8421) • (2421)• (5421)• (63-1-1)• (7421)

• N= w₃a₃ + w₂a₂ + w₁a₁

BINARY CODES

• Most compatible system for a computer or a digital system is binary system

• Most of the users are accustomed to decimal number system

• To reduce this gap, decimal numbers are converted to binary, arithmetic calculations are performed in binary, and then converted back to decimal.

BINARY CODES• Code is a symbolic representation of an

information transform

• During this process, we need to store decimal numbers in computer for performing conversion

• But computers accept only digits 0s & 1s

• So, we must represent these decimal digits by means of a code consisting of 0s & 1s.

• Arithmetic operations can be directly performed with decimal numbers when they are stored in computer in coded from.

BCD Code

• In simplest form of binary code, each decimal digit is represented by its binary equivalent.

8 5 4 . 7 9 2

1000 0101 0100 0111 1001 0010

Representation of BCD Code

( 3 4 5 )10 = ( 0011 0100 0101)BCD

= ( 101011001 )₂

( 1 5 7 )10 = ( 0001 0101 0111 )₂

BCD representation – 12 bits – denotes a decimal number

Binary value – 8 bits – denotes binary value itself

BCD Addition

1 8 4 0001 1000 0100 + 5 7 6 0101 01 1 1 01 10 ---------- ---------------------- 7 6 0 0 111 10000 1010 0 11 0 0110

----------------------- 0 111 0110 0000

Non-Weighted Codes

• Excess – 3 code• Gray code• 2 out of 5 code• Biquinary code

Excess-3 code

Gray Codes

The property of this code is that the successive decimal digits differ in exactly one bit.

Conversion from Binary to Gray

Conversion of Gray to Binary

Alpha numeric codes

• ASCII code• EBCDIC code

Binary Storage & Registers

Register Transfer

Logical Operation in Registers

Binary Logic