Digital Systems NV Conversions

7/29/2019 Digital Systems NV Conversions

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Digital SystemsUNIT I

8/23/2012

BEEE101 UNIT III Digital Systems

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8/23/2012


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Most natural quantities that we see are analog and vary

continuously. Analog systems can generally handle higher

power than digital systems.

Digital systems can process, store, and transmit data more

efficiently but can only assign discrete values to each point.

Analog Quantities

1

100

A.M .

95

90

85

80

75

2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

P.M.

Temperature

(F)

70

Time of day


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Many systems use a mix of analog and digital electronics to

take advantage of each technology. A typical CD player

accepts digital data from the CD drive and converts it to an

analog signal for amplification.

Analog and Digital Systems

Digital data

CD drive

10110011101

Analog

reproduction

of music audio

signalSpeaker

Sound

waves

Digital-to-analog

converterLinear amplifier


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Digital electronics uses circuits that have two states, which

are represented by two different voltage levels called HIGH

and LOW. The voltages represent numbers in the binary

system.

Binary Digits and Logic Levels

In binary, a single number is

called a bit (forbinary digit). A

bit can have the value of either

a 0 or a 1, depending on if thevoltage is HIGH or LOW.

HIGH

LOW

VH(max)

VH(min)

VL(max)

VL(min)

Invalid


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Digital waveforms change between the LOW and HIGH

levels. A positive going pulse is one that goes from a

normally LOW logic level to a HIGH level and then back

again. Digital waveforms are made up of a series of pulses.

Digital Waveforms

Falling orleading edge

(b) Negativegoing pulse

HIGH

Rising ortrailing edge

LOW

(a) Posit ivegoing pulse

HIGH

Rising orleading edge

Falling ortrailing edge

LOW t0

t1

t0

t1


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A timing diagram is used to show the relationship between

two or more digital waveforms,

Timing Diagrams

Clock

A

B

C

A diagram like this can be observed

directly on a logic analyzer.


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Data can be transmitted by either serial transfer or parallel

transfer.

Serial and Parallel Data

Computer Modem

1 0 1 1 0 0 1 0

t0 t1 t2 t3 t4 t5 t6 t7

Com puter Printer

0

t0 t1

1

0

0

1

1

0

1


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General statement

In general ,

Analog is Continuous

Digital is Discrete or step by step

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Advantages of Digital Systems

Easier to design. Exact values of voltage

or current are not important, only the

range (HIGH or LOW) in which they fall.Information storage is easy.

Accuracy is good ( Like 4.95789 V)

Precision is good.

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Advantage contd.,

Operation can be programmed easily.

Digital circuits are less affected by noise.

More digital circuitry can be fabricated

on IC chips so that cost is less

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Types of Digital Systems

Basically four types , they are

1. Decimal (0,1,2,3,9 )

2. Binary(0 and 1 only )

3. Octal (0,1,2,3,7)

4. Hexa Decimal( 0, 1,2,3,..F)

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8/23/2012BEEE101 UNIT III Digital Systems

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The position of each digit in a weighted number system is

assigned a weight based on the base orradix of the system.

The radix of decimal numbers is ten, because only ten

symbols (0 through 9) are used to represent any number.

The column weights of decimal numbers are powers

of ten that increase from right to left beginning with 100 =1:

Decimal Numbers

105 104 103 102 101 100.

For fractional decimal numbers, the column weightsare negative powers of ten that decrease from left to right:

102 101 100. 10-1 10-2 10-3 10-4


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Decimal Numbers

Express the number 480.52 as the sum of values of each

digit.

(9 x 103) + (2 x 102) + (4 x 101) + (0 x 100)

or9 x 1,000 + 2 x 100 + 4 x 10 + 0 x 1

Decimal numbers can be expressed as the sum of the

products of each digit times the column value for that digit.

Thus, the number 9240 can be expressed as

480.52 = (4 x 102) + (8 x 101) + (0 x 100) + (5 x 10-1) +(2 x 10-2)


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Binary Systems

In this system any number is represented using digits: 0

and 1 only ,

Ex : In computers all calculations are done internally

using binary

Combinations of 0 and 1 only like 110102 , 11100.102Base 2 , Powers of 2n

It can be called as High or Low , True or False ,Yes or No , On and Off and 1 and 0.


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Binary Numbers

A binary counting sequence for numbers

from zero to fifteen is shown.

0 0 0 0 01 0 0 0 1

2 0 0 1 0

3 0 0 1 1

4 0 1 0 0

5 0 1 0 1

6 0 1 1 07 0 1 1 1

8 1 0 0 0

9 1 0 0 1

10 1 0 1 0

11 1 0 1 1

12 1 1 0 0

13 1 1 0 1

14 1 1 1 0

15 1 1 1 1

Decimal

Number

Binary

Number

Notice the pattern of zeros and ones in

each column.

Counter Decoder1 0 1 0 1 0 1 00 1

0 1 1 0 0 1 1 00 0

0 0 0 1 1 1 1 00 0

0 0 0 0 0 0 0 10 1

Digital counters frequently have this

same pattern of digits:


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Octal Numbers

Octal is also a weighted number

system. The column weights are

powers of 8, which increase from right

to left.

.

3 7 0 28

198610

Column weights 83 82 81 80

512 64 8 1 .{

Express 37028 in decimal.

Start by writing the column weights:512 64 8 1

3(512) + 7(64) +0(8) +2(1) =

01

2

3

4

5

6

7

8

9

10

1112

13

14

15

01

2

3

4

5

6

7

10

11

12

1314

15

16

17

00000001

0010

0011

0100

0101

0110

0111

1000

1001

1010

10111100

1101

1110

1111

Decimal Octal Binary


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Hexadecimal

NumbersHexadecimal uses sixteen characters to

represent numbers: the numbers 0 through

9 and the alphabetic characters A through

F.

01

2

3

4

5

6

7

8

9

10

1112

13

14

15

01

2

3

4

5

6

7

8

9

A

BC

D

E

F

00000001

0010

0011

0100

0101

0110

0111

1000

1001

1010

10111100

1101

1110

1111

Decimal Hexadecimal Binary

Large binary number can easily be

converted to hexadecimal by grouping bits

4 at a time and writing the equivalent

hexadecimal character.

Express 1001 0110 0000 11102 in hexadecimal:

Group the binary number by 4-bits starting from

the right. Thus, 960E


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Hexadecimal Numbers

Hexadecimal is a weighted number

system. The column weights are

powers of 16, which increase from

right to left.

.

1 A 2 F16

670310

Column weights 163 162 161 160

4096 256 16 1 .{

Express 1A2F16 in decimal.

Start by writing the column weights:4096 256 16 1

1(4096) + 10(256) +2(16) +15(1) =

01

2

3

4

5

6

7

8

9

10

1112

13

14

15

01

2

3

4

5

6

7

8

9

A

BC

D

E

F

00000001

0010

0011

0100

0101

0110

0111

1000

1001

1010

10111100

1101

1110

1111

Decimal Hexadecimal Binary


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Conversion Decimal to Binary

Successive Division Method

Example 1 : 34


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34

17 - 0

8 - 1

4 - 0

2- 0

1 - 0

2

2

2

2

2

(34)10 = (100010)2MSB LSB


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Conversion Decimal to Binary contd.,

Successive Division Method :

Example 2 : (0. 625 ) 10 = ( ? )2 carry

Step 1 : 0.625 X 2 = 1. 250Step 2 : 0.250 X 2 = 0 .500

Step 3 : 0.500 X 2 = 1 .000

(0.625 ) 10 = ( 0.101)2


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1

0

1


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Decimal to Octal Conversion

Example 1 : 3737


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8

4 - 5

(37)10 = ( 45)8


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Decimal to Octal Conversion Contd.,

Example 2 : (0. 220 )10

0. 220 x 8 = 1.76

0.76 x 8 = 6.080.08 x 8 = 0.64


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(0.220)10 = ( 0.160)8

1

6

0


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Decimal System Contd.,

Decimal to Hexadecimal Conversion :

Example 1 : ( 3580 ) 10 = ( ? )16

3580


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16

223 - 1216

13 - 1516

13 = D

15 = F

12 = C

Ans ( 3580 )10 = ( DFC ) 16

Problem 1. ( 457 ) 10 = ( ? ) 16

Problem 2 .( 3208 ) 10 = ( ? )16


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Decimal System Contd.,

Example 2 : (0. 452 ) 10 = ( ? )160. 452 x 16 = 7.232

0.232 x 16 = 3.712

0.712 x 16 = 11.392

(0. 452 ) 10 = ( 73B )16


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7

3

B


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Binary to Other Conversions

Binary to Decimal :

Ex : 1 1 0 0 0 1 0


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0 X 2 0 = 0

1 X 2 1 = 2

0 X 2 2 = 0

0 X 2 3 = 0

0 X 2 4 = 0

1 X 2 5 = 32

34


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Conversion contd.,

Ex 2 :

1 0 0 0 1 0 . 1 0 1


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0 X 20 = 0

1 X 2

1

= 20 X 2 2 = 00 X 2 3 = 00 X 2 4 = 01 X 2 5 = 32

34

1 X 2 -3 = 0.125

0 X 2 -2 = 0

1 X 2 -1 = 0.5

(100010.101)2= (34.625)10


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Binary to Octal Conversion

Convert (1111011110101)2 = (? )8

Step 1 : Divide into 3 Groups from the LSB

11 ,111,011,110, 101Step 2 : Express each

group in decimal 3 7 3 6 5

Step 3 : Therefore (1111011110101)2 = ( 37 , 365 ) 8


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Binary to Hexa Conversion :

Example 1 : (10 0101. 0111 1000) 2 = ( ? ) 16

Step 1 : Given is 10 0101 . 0111 1000

Step 2 : Group into 2 5 7 8

4- bit from LSB

Step 3: Ans : (100101.0111 1000) 2 = ( 25 . 78 ) 16


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Octal to Decimal Conversion

Example 1 : (45)8 = ( ? ) 10

Step 1 : Convert into Binary 4 5

100 101

Step 2 : Convert Binary into decimal ,1 0 0 1 0 1


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1 X 2 0 = 1

0 X 2 1 = 0

1X 2 2 = 4

0 X 2 3 = 0

0 X 2 4 = 0

1 X 2 5 = 32

37


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Octal to Binary Conversion

Convert ( 3764) 8 = ( ? )2

Step 1 : Copy the Octal number 3 6 7 4

Step 2 : Convert each to Binary 011 110 111 100

Step 3 : Therefore , (3674 ) 8 = ( 11,110,111,100)2


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Hexa to Decimal System

Hexadecimal to Decimal Conversion:

Example 1 : C F 3 D . 2 4 1


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1 X 16-3

= 0.00024 X 16 -2= 0.0156

2 X 16 -1 = 0.124

D X 160 = 13

3 X 161 = 48

F X 162 = 3840

C X 163 = 49152

53053.1398


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Hexa to Binary Conversion

Hexadecimal to Binary :

Example 1 : A 3 F E

Step 1 : A 3 F E

Step 2 : Directly 1010 0011 1111 1110

write the 4 bit form

( A 3 F E) 16 = ( 1010 0011 1111 1110)2


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Hexa to Binary Conversion

Hexadecimal to Binary :

Example 2 : A 3 F E . FC

Ans : A 3 F E . F C

1010 0011 1111 1110 1111 1100

(A 3 F E . FC )16 = (1010 00111111 1110 . 11111100)2

Problems :

1. (C5F9 )16 = ( ? )2

2. (25D . 7B)16 = ( ? ) 2


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