chp1 binary system

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1

INTRODUCTION

Zubair saif

DIGIAL LOGIC & DESIGN(EE-204)

Overview

Digital Systems, Computers, and BeyondInformation RepresentationNumber Systems [binary, octal and

hexadecimal]Arithmetic OperationsBase ConversionDecimal CodesAlphanumeric CodesParity BitGray Codes

DIGITAL & COMPUTER SYSTEMS - Digital System

Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs.

System State

DiscreteInformationProcessingSystem

DiscreteInputs Discrete

Outputs

Types of Digital Systems

No state present Combinational Logic System Output = Function(Input)

State present State updated at discrete times

Synchronous Sequential System State updated at any time

Asynchronous Sequential System State = Function (State, Input) Output = Function (State) or Function

(State, Input)

Digital System Example

A Digital Counter (e. g., odometer)

1 30 0 5 6 4Count Up

Reset

Inputs: Count Up, Reset

Outputs: Visual Display

State: "Value" of stored digits

Digital Computer Example

Inputs: keyboard, mouse, wireless, microphone

Outputs: LCD screen, wireless, speakers

Memory

Controlunit Datapath

Input/Output

CPU

And Beyond – Embedded Systems

Chapter 1 8

Computers as integral parts of other productsExamples of embedded computers

Microcomputers Microcontrollers Digital signal processors

Examples of Embedded Systems Applications Cell phones Automobiles Video games Copiers Dishwashers Flat Panel TVs Global Positioning Systems

INFORMATION REPRESENTATION - Signals

Chapter 1 9

Information variables represented by physical quantities. 

For digital systems, the variables take on discrete values.

Two level, or binary values are the most prevalent values in digital systems. 

Binary values are represented abstractly by: digits 0 and 1 words (symbols) False (F) and True (T) words (symbols) Low (L) and High (H) and words On and Off.

Binary values are represented by values or ranges of values of physical quantities

Signal Examples Over Time

Chapter 1 10

Analog

Asynchronous

Synchronous

Time

Continuous in value &

time

Discrete in value & continuous

in time

Discrete in value &

time

Digital

NUMBER SYSTEMS – Representation

Chapter 1 11

Positive radix, positional number systemsA number with radix r is represented by a

string of digits: An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m + 1 A- m

in which 0 £ Ai < r and . is the radix point.The string of digits represents the power

series:

( ) ( ) (Number)r = åå +

j = - m

jj

i

i = 0i rArA

(Integer Portion)

+ (Fraction Portion)

i = n - 1

j = - 1

Number Systems – Examples

Chapter 1 12

General Decimal Binary

Radix (Base) r 10 2

Digits 0 => r - 1 0 => 9 0 => 1

0123

Powers of 4

Radix 5-1-2-3-4-5

r0

r1

r2

r3

r4

r5

r -1

r -2

r -3

r -4

r -5

110

1001000

10,000100,000

0.10.01

0.0010.0001

0.00001

124816320.50.25

0.1250.0625

0.03125

Decimal Number System13 / 45

Base (also called radix) = 10 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Digit Position Integer & fraction

Digit Weight Weight = (Base) Position

Magnitude Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

5 1 2 7 4

10 1 0.1100 0.01

500 10 2 0.7

0.04

d2*B2+d1*B1+d0*B0+d-1*B-1+d-

2*B-2

(512.74)1

0

Octal Number System14 / 45

Base = 8 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }

Weights Weight = (Base) Position

Magnitude Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

8 1 1/864 1/64

5 1 2 7 4

5 *82+1 *81+2 *80+7 *8-

1+4 *8-2

=(330.9375)10

(512.74)8

Binary Number System15 / 45Base = 2

2 digits { 0, 1 }, called binary digits or “bits”

Weights Weight = (Base) Position

Magnitude Sum of “Bit x Weight”

Formal NotationGroups of bits 4 bits = Nibble

8 bits = Byte

1 0 -12 -2

2 1 1/24 1/4

1 0 1 0 1

1 *22+0 *21+1 *20+0 *2-

1+1 *2-2

=(5.25)10

(101.01)2

1 0 1 1

1 1 0 0 0 1 0 1

Hexadecimal Number System16 / 45Base = 16

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

Weight = (Base) Position

Magnitude Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

16 1 1/16256 1/256

1 E 5 7 A

1 *162+14 *161+5 *160+7 *16-

1+10 *16-2

=(485.4765625)10

(1E5.7A)16

The Power of 217 / 45

n 2n

0 20=1

1 21=2

2 22=4

3 23=8

4 24=16

5 25=32

6 26=64

7 27=128

n 2n

8 28=256

9 29=512

10 210=1024

11 211=2048

12 212=4096

20 220=1M

30 230=1G

40 240=1T

Mega

Giga

Tera

Kilo

Addition18 / 45

Decimal Addition

5 5

55+

011

= Ten ≥ Base Subtract a Base

11 Carry

Binary Addition19 / 45

Column Addition

1 0 1111

1111 0+

0000 1 11

≥ (2)10

111111= 61

= 23

= 84

Binary Subtraction20 / 45Borrow a “Base” when needed

0 0 1110

1111 0−

0101 1 10

= (10)2

2

2

2 2

1

000

1

= 77

= 23

= 54

Binary Multiplication21 / 45Bit by bit

01 1 1 1

01 1 0

00 0 0 0

01 1 1 1

01 1 1 1

0 0 000

0110111 0

x

Number Base Conversions22 / 45

Decimal(Base 10)

Octal(Base 8)

Binary(Base 2)

Hexadecimal(Base 16)

Evaluate Magnitude

Evaluate Magnitude

Evaluate Magnitude

Decimal (Integer) to Binary Conversion23 / 45

Divide the number by the ‘Base’ (=2)Take the remainder (either 0 or 1) as a coefficientTake the quotient and repeat the division

Example: (13)10

Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2

MSB LSB

Quotient Remainder Coefficient

13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1

Decimal (Fraction) to Binary Conversion24 / 45

Multiply the number by the ‘Base’ (=2)Take the integer (either 0 or 1) as a coefficientTake the resultant fraction and repeat the division

Example: (0.625)10

Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2

MSB LSB

Integer Fraction Coefficient

0.625× 2 = 1 . 250.25 × 2 = 0 . 5 a-2 = 00.5 × 2 = 1 . 0 a-3 = 1

a-1 = 1

Decimal to Octal Conversion25 / 45Example: (175)10

Quotient Remainder Coefficient

Answer: (175)10 = (a2 a1 a0)8 = (257)8

175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2

Example: (0.3125)10

Integer Fraction Coefficient

Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8

0.3125* 8 = 2 . 50.5 * 8 = 4 . 0 a-2 = 4

a-1 = 2

Binary − Octal Conversion26 / 458 = 23

Each group of 3 bits represents an octal digit

Octal Binary

0 0 0 0

1 0 0 1

2 0 1 0

3 0 1 1

4 1 0 0

5 1 0 1

6 1 1 0

7 1 1 1

Example:

( 1 0 1 1 0 . 0 1 )2

( 2 6 . 2 )8

Assume Zeros

Works both ways (Binary to Octal & Octal to Binary)

Binary − Hexadecimal Conversion27 / 4516 = 24

Each group of 4 bits represents a hexadecimal digit

Hex Binary0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1A 1 0 1 0B 1 0 1 1C 1 1 0 0D 1 1 0 1E 1 1 1 0F 1 1 1 1

Example:

( 1 0 1 1 0 . 0 1 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Binary to Hex & Hex to Binary)

Octal − Hexadecimal Conversion28 / 45

Example:

( 0 1 0 1 1 0 . 0 1 0 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Octal to Hex & Hex to Octal)

( 2 6 . 2 )8

Assume Zeros

Decimal, Binary, Octal and Hexadecimal29 / 45

Decimal Binary Octal Hex00 0000 00 001 0001 01 102 0010 02 203 0011 03 304 0100 04 405 0101 05 506 0110 06 607 0111 07 708 1000 10 809 1001 11 910 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F

Complements30 / 45

1’s Complement (Diminished Radix Complement) All ‘0’s become ‘1’s All ‘1’s become ‘0’sExample (10110000)2

(01001111)2

If you add a number and its 1’s complement …

1 0 1 1 0 0 0 0+ 0 1 0 0 1 1 1 1

1 1 1 1 1 1 1 1

Complements31 / 45

2’s Complement (Radix Complement) Take 1’s complement then add 1 Toggle all bits to the left of the first ‘1’ from the

rightExample:Number:1’s Comp.:

0 1 0 1 0 0 0 0

1 0 1 1 0 0 0 00 1 0 0 1 1 1 1

+ 1

OR

1 0 1 1 0 0 0 0

00001010

Negative Numbers32 / 45

Computers Represent Information in ‘0’s and ‘1’s ‘+’ and ‘−’ signs have to be represented in ‘0’s and

‘1’s3 Systems

Signed Magnitude 1’s Complement 2’s ComplementAll three use the left-most bit to represent the sign:

‘0’ positive ‘1’ negative

Signed Magnitude Representation33 / 45Magnitude is magnitude, does not change

with sign

(+3)10 ( 0 0 1 1 )2

(−3)10 ( 1 0 1 1 )2

Can’t include the sign bit in ‘Addition’

0 0 1 1 (+3)10

+ 1 0 1 1 (−3)10

1 1 1 0 (−6)10

Sign Magnitude

S Magnitude (Binary)

1’s Complement Representation34 / 45

Positive numbers are represented in “Binary”

Negative numbers are represented in “1’s Comp.”

(+3)10 (0 011)2

(−3)10 (1 100)2

There are 2 representations for ‘0’(+0)10 (0 000)2

(−0)10 (1 111)2

0 Magnitude (Binary)

1Code (1’s Comp.)

1’s Complement Range35 / 45

4-Bit Representation

24 = 16 Combinations − 7 ≤ Number ≤ + 7

−23+1 ≤ Number ≤ +23 − 1n-Bit Representation

−2n−1+1 ≤ Number ≤ +2n−1 − 1

Decimal 1’s Comp.+ 7 0 1 1 1+ 6 0 1 1 0+ 5 0 1 0 1+ 4 0 1 0 0+ 3 0 0 1 1+ 2 0 0 1 0+ 1 0 0 0 1+ 0 0 0 0 0− 0 1 1 1 1− 1 1 1 1 0− 2 1 1 0 1− 3 1 1 0 0− 4 1 0 1 1− 5 1 0 1 0− 6 1 0 0 1− 7 1 0 0 0

2’s Complement Representation36 / 45Positive numbers are represented in

“Binary”

Negative numbers are represented in “2’s Comp.”

(+3)10 (0 011)2

(−3)10 (1 101)2

There is 1 representation for ‘0’(+0)10 (0 000)2

(−0)10 (0 000)2

0 Magnitude (Binary)

1 Code (2’s Comp.)

1’s Comp. 1 1 1 1 + 1 1 0 0 0 0

2’s Complement Range37 / 45

4-Bit Representation

24 = 16 Combinations − 8 ≤ Number ≤ + 7

−23 ≤ Number ≤ + 23 − 1n-Bit Representation

−2n−1 ≤ Number ≤ + 2n−1 − 1

Decimal 2’s Comp.+ 7 0 1 1 1+ 6 0 1 1 0+ 5 0 1 0 1+ 4 0 1 0 0+ 3 0 0 1 1+ 2 0 0 1 0+ 1 0 0 0 1+ 0 0 0 0 0− 1 1 1 1 1− 2 1 1 1 0− 3 1 1 0 1− 4 1 1 0 0− 5 1 0 1 1− 6 1 0 1 0− 7 1 0 0 1− 8 1 0 0 0

Number Representations38 / 45

4-Bit Example

Unsigned Binary

Signed Magnitude 1’s Comp. 2’s Comp.

Range 0 ≤ N ≤ 15 -7 ≤ N ≤ +7 -7 ≤ N ≤ +7 -8 ≤ N ≤ +7

Positive

Binary Binary Binary Binary

Negative XBinary 1’s Comp. 2’s Comp.

0 0 0

1 1 1

Binary Subtraction Using 1’s Comp. Addition

39 / 45

Change “Subtraction” to “Addition”If “Carry” = 1

then add it to theLSB, and the resultis positive(in Binary)

If “Carry” = 0then the resultis negative(in 1’s Comp.)

0 1 0 1

+ 1 1 1 0

(5)10 – (1)10

(+5)10 + (-1)10

0 0 1 1+

0 1 0 0

0 1 0 1

+ 1 0 0 1

(5)10 – (6)10

(+5)10 + (-6)10

0 1 1 1 0

1 1 1 0

+ 4 − 1

1

Binary Subtraction Using 2’s Comp. Addition40 /

45Change “Subtraction” to “Addition”If “Carry” = 1

ignore it, and the result is positive(in Binary)

If “Carry” = 0then the resultis negative(in 2’s Comp.)

0 1 0 1

+ 1 1 1 1

(5)10 – (1)10

(+5)10 + (-1)10

1 0 1 0 0

0 1 0 1

+ 1 0 1 0

(5)10 – (6)10

(+5)10 + (-6)10

0 1 1 1 1

+ 4 − 1

Binary Numbers and Binary Coding

Chapter 1 41

Flexibility of representation Within constraints below, can assign any binary

combination (called a code word) to any data as long as data is uniquely encoded.

Information Types Numeric

Must represent range of data needed Very desirable to represent data such that simple,

straightforward computation for common arithmetic operations permitted

Tight relation to binary numbers Non-numeric

Greater flexibility since arithmetic operations not applied.

Not tied to binary numbers

Non-numeric Binary Codes

Chapter 1 42

Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers.

Example: Abinary codefor the sevencolors of therainbow

Code 100 is not used

Binary Number 000001010011101110111

ColorRedOrangeYellowGreenBlueIndigoViolet

Number of Elements Represented

Chapter 1 43

Given n digits in radix r, there are rn distinct elements that can be represented.

But, you can represent m elements, m < rn

Examples: You can represent 4 elements in radix r = 2 with

n = 2 digits: (00, 01, 10, 11). You can represent 4 elements in radix r = 2 with

n = 4 digits: (0001, 0010, 0100, 1000). This second code is called a "one hot" code.

DECIMAL CODES - Binary Codes for Decimal Digits

Chapter 1 44

Decimal 8,4,2,1 Excess3 8,4,-2,-1 Gray 0 0000 0011 0000 0000 1 0001 0100 0111 0100 2 0010 0101 0110 0101 3 0011 0110 0101 0111 4 0100 0111 0100 0110 5 0101 1000 1011 0010 6 0110 1001 1010 0011 7 0111 1010 1001 0001 8 1000 1011 1000 1001 9 1001 1100 1111 1000

There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits. A few are useful:

Binary Coded Decimal (BCD)

Chapter 1 45

The BCD code is the 8,4,2,1 code.

8, 4, 2, and 1 are weights

BCD is a weighted code

This code is the simplest, most intuitive binary code for decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9.

Example: 1001 (9) = 1000 (8) + 0001 (1)

How many “invalid” code words are there?

What are the “invalid” code words?

Excess 3 Code and 8, 4, –2, –1 Code

Chapter 1 46

What interesting property is common to these two codes?

Decimal Excess 3 8, 4, –2, –1

0 0011 0000

1 0100 0111

2 0101 0110

3 0110 0101

4 0111 0100

5 1000 1011

6 1001 1010

7 1010 1001

8 1011 1000

9 1100 1111

Warning: Conversion or Coding?

Chapter 1 47

Do NOT mix up conversion of a decimal number to a binary number

with coding a decimal number with a BINARY CODE. 

1310 = 11012 (This is conversion) 

13 0001|0011 (This iscoding)

ALPHANUMERIC CODES - ASCII Character Codes

Chapter 1 48

American Standard Code for Information Interchange

This code is a popular code used to represent information sent as character-based data. It uses 7-bits to represent:

94 Graphic printing characters. 34 Non-printing characters

Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return)

Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).

ASCII Table

49

ASCII Properties

Chapter 1 50

ASCII has some interesting properties: Digits 0 to 9 span Hexadecimal values 3016 to 3916.

Upper case A-Z span 4116 to 5A16. Lower case a-z span 6116 to 7A16 .

• Lower to upper case translation (and vice versa) occurs byflipping bit 6.

Delete (DEL) is all bits set, a carryover from when punched paper tape was used to store messages. Punching all holes in a row erased a mistake!

PARITY BIT Error-Detection Codes

Chapter 1 51

A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.

A code word has even parity if the number of 1’s in the code word is even.

A code word has odd parity if the number of 1’s in the code word is odd.

Error Detecting Codes52 / 45

ParityOne bit added to a group of bits to make the total number of ‘1’s (including the parity bit) even or odd

Even

Odd

Good for checking single-bit errors

1 0 1 1 1 1 0 0 0 0 0 10

0 0 1 1 1 1 0 0 0 0 0 11

4-bit Example 7-bit Example

Gray Code

One bit changes fromone code to the nextcode

Different than Binary

Binary0000000100100011010001010110011110001001101010111100110111101111

53 / 45

Decimal

Gray

00 000001 000102 001103 001004 011005 011106 010107 010008 110009 110110 111111 111012 101013 101114 100115 1000

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