1. number systemsieee.daiict.ac.in/ifest2012/pdf/part02.pdf · common number systems system base...

1. Number Systems

Common Number Systems

System

Base

Symbols

Used by

humans?

Used in

computers?

Decimal 10 0, 1, … 9 Yes No

Binary 2 0, 1 No Yes

Octal 8 0, 1, … 7 No No

Hexa-

decimal

16 0, 1, … 9,

A, B, … F

No No

Quantities/Counting (1 of 3)

Decimal

Binary

Octal

Hexa-

decimal

0 0 0 0

1 1 1 1

2 10 2 2

3 11 3 3

4 100 4 4

5 101 5 5

6 110 6 6

7 111 7 7


Decimal

Binary

Octal

Hexa-

decimal

8 1000 10 8

9 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F


Decimal

Binary

Octal

Hexa-

decimal

16 10000 20 10

17 10001 21 11

18 10010 22 12

19 10011 23 13

20 10100 24 14

21 10101 25 15

22 10110 26 16

23 10111 27 17 Etc.

Conversion Among Bases

• The possibilities:

Hexadecimal

Decimal Octal

Binary

Quick Example

2510 = 110012 = 318 = 1916

Base

Decimal to Decimal (just for fun)

Hexadecimal

Decimal Octal

Binary

12510 => 5 x 100 = 5

2 x 101 = 20

1 x 102 = 100

125

Base

Weight

Binary to Decimal

Hexadecimal

Decimal Octal

Binary

Binary to Decimal

• Technique

– Multiply each bit by 2n, where n is the “weight”

of the bit

– The weight is the position of the bit, starting

from 0 on the right

– Add the results

Example

1010112 => 1 x 20 = 1

1 x 21 = 2

0 x 22 = 0

1 x 23 = 8

0 x 24 = 0

1 x 25 = 32

4310

Bit “0”

Octal to Decimal

Hexadecimal

Decimal Octal

Binary

Octal to Decimal

• Technique

– Multiply each bit by 8n, where n is the “weight”

of the bit


from 0 on the right

– Add the results

Example

7248 => 4 x 80 = 4

2 x 81 = 16

7 x 82 = 448

46810

Hexadecimal to Decimal

Hexadecimal

Decimal Octal

Binary

Hexadecimal to Decimal

• Technique

– Multiply each bit by 16n, where n is the

“weight” of the bit


from 0 on the right

– Add the results

Example

ABC16 => C x 160 = 12 x 1 = 12

B x 161 = 11 x 16 = 176

A x 162 = 10 x 256 = 2560

274810

Decimal to Binary

Hexadecimal

Decimal Octal

Binary

Decimal to Binary

• Technique

– Divide by two, keep track of the remainder

– First remainder is bit 0 (LSB, least-significant

bit)

– Second remainder is bit 1

– Etc.

Example

12510 = ?2 2 125

62 1 2

31 0 2

15 1 2

7 1 2

3 1 2

1 1 2

0 1

12510 = 11111012

Octal to Binary

Hexadecimal

Decimal Octal

Binary

Octal to Binary

• Technique

– Convert each octal digit to a 3-bit equivalent

binary representation

Example

7058 = ?2

7 0 5

111 000 101

7058 = 1110001012

Hexadecimal to Binary

Hexadecimal

Decimal Octal

Binary

Hexadecimal to Binary

• Technique

– Convert each hexadecimal digit to a 4-bit

equivalent binary representation

Example

10AF16 = ?2

1 0 A F

0001 0000 1010 1111

10AF16 = 00010000101011112

Decimal to Octal

Hexadecimal

Decimal Octal

Binary

Decimal to Octal

• Technique

– Divide by 8

– Keep track of the remainder

Example

123410 = ?8

8 1234

154 2 8

19 2 8

2 3 8

0 2

123410 = 23228

Decimal to Hexadecimal

Hexadecimal

Decimal Octal

Binary

Decimal to Hexadecimal

• Technique

– Divide by 16

– Keep track of the remainder

Example

123410 = ?16

123410 = 4D216

16 1234

77 2 16

4 13 = D 16

0 4

Binary to Octal

Hexadecimal

Decimal Octal

Binary

Binary to Octal

• Technique

– Group bits in threes, starting on right

– Convert to octal digits

Example

10110101112 = ?8

1 011 010 111

1 3 2 7

10110101112 = 13278

Binary to Hexadecimal

Hexadecimal

Decimal Octal

Binary

Binary to Hexadecimal

• Technique

– Group bits in fours, starting on right

– Convert to hexadecimal digits

Example

10101110112 = ?16

10 1011 1011

2 B B

10101110112 = 2BB16

Octal to Hexadecimal

Hexadecimal

Decimal Octal

Binary

Octal to Hexadecimal

• Technique

– Use binary as an intermediary

Example

10768 = ?16

1 0 7 6

001 000 111 110

2 3 E

10768 = 23E16

Hexadecimal to Octal

Hexadecimal

Decimal Octal

Binary

Hexadecimal to Octal

• Technique

– Use binary as an intermediary

Example

1F0C16 = ?8

1 F 0 C

0001 1111 0000 1100

1 7 4 1 4

1F0C16 = 174148

Exercise – Convert ...

Decimal

Binary

Octal

Hexa-

decimal

33

1110101

703

1AF

Exercise – Convert …

Decimal

Binary

Octal

Hexa-

decimal

33 100001 41 21

117 1110101 165 75

451 111000011 703 1C3

431 110101111 657 1AF

Answer

Common Powers (1 of 2)

• Base 10 Power Preface Symbol

10-12 pico p

10-9 nano n

10-6 micro

10-3 milli m

103 kilo k

106 mega M

109 giga G

1012 tera T

Value

.000000000001

.000000001

.000001

.001

1000

1000000

1000000000

1000000000000

Common Powers (2 of 2)

• Base 2 Power Preface Symbol

210 kilo k

220 mega M

230 Giga G

Value

1024

1048576

1073741824

• What is the value of “k”, “M”, and “G”?

• In computing, particularly w.r.t. memory,

the base-2 interpretation generally applies

Example

/ 230 =

In the lab…

1. Double click on My Computer

2. Right click on C:

3. Click on Properties

Exercise – Free Space

• Determine the “free space” on all drives on

a machine in the lab

Drive

Free space

Bytes GB

A:

C:

D:

E:

etc.

Review – multiplying powers

• For common bases, add powers

26 210 = 216 = 65,536

or…

26 210 = 64 210 = 64k

ab ac = ab+c

Binary Addition (1 of 2)

• Two 1-bit values

pp. 36-38

A B A + B

0 0 0

0 1 1

1 0 1

1 1 10 “two”

Binary Addition (2 of 2)

• Two n-bit values

– Add individual bits

– Propagate carries

– E.g.,

10101 21

+ 11001 + 25

101110 46

1 1

Multiplication (1 of 3)

• Decimal (just for fun)

pp. 39

35

x 105

175

000

35

3675


• Binary, two 1-bit values

A B A B

0 0 0

0 1 0

1 0 0

1 1 1


• Binary, two n-bit values

– As with decimal values

– E.g.,

1110

x 1011

1110

1110

0000

1110

10011010

Fractions

• Decimal to decimal (just for fun)

3.14 => 4 x 10-2 = 0.04

1 x 10-1 = 0.1

3 x 100 = 3

3.14

Fractions

• Binary to decimal

10.1011 => 1 x 2-4 = 0.0625

1 x 2-3 = 0.125

0 x 2-2 = 0.0

1 x 2-1 = 0.5

0 x 20 = 0.0

1 x 21 = 2.0

2.6875

Fractions

• Decimal to binary

3.14579

.14579

x 2

0.29158

x 2

0.58316

x 2

1.16632

x 2

0.33264

x 2

0.66528

x 2

1.33056

etc. 11.001001...

Exercise – Convert ...

Decimal

Binary

Octal

Hexa-

decimal

29.8

101.1101

3.07

C.82

Exercise – Convert …

Decimal

Binary

Octal

Hexa-

decimal

29.8 11101.110011… 35.63… 1D.CC…

5.8125 101.1101 5.64 5.D

3.109375 11.000111 3.07 3.1C

12.5078125 1100.10000010 14.404 C.82

Answer

Thank you

Basic Logic Gates

and Basic Digital Design

• NOT, AND, and OR Gates

• NAND and NOR Gates

• DeMorgan’s Theorem

• Exclusive-OR (XOR) Gate

• Multiple-input Gates

NOT Gate -- Inverter

X Y

0

1

1

0

X Y

Y

NOT

X Y

Y = ~X

NOT

• Y = ~X (Verilog)

• Y = !X (ABEL)

• Y = not X (VHDL)

• Y = X’

• Y = X

• Y = X (textook)

• not(Y,X) (Verilog)

NOT

NOT

X ~X ~~X = X

X ~X ~~X

0 1 0

1 0 1

AND Gate AND

X

Y

Z

Z = X & Y

X Y Z

0 0 0

0 1 0

1 0 0

1 1 1

• X & Y (Verilog and ABEL)

• X and Y (VHDL)

• X Y

• X Y

• X * Y

• XY (textbook)

• and(Z,X,Y) (Verilog)

AND

U

V

OR Gate

OR

X

Y Z

Z = X | Y

X Y Z

0 0 0

0 1 1

1 0 1

1 1 1

OR

• X | Y (Verilog)

• X # Y (ABEL)

• X or Y (VHDL)

• X + Y (textbook)

• X V Y

• X U Y

• or(Z,X,Y) (Verilog)

Basic Logic Gates







NAND Gate NAND

X

Y

Z

X Y Z

0 0 1

0 1 1

1 0 1

1 1 0

Z = ~(X & Y)

nand(Z,X,Y)

NAND Gate

NOT-AND

X

Y

Z

W = X & Y

Z = ~W = ~(X & Y)

X Y W Z

0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

W

NOR Gate

NOR

X

Y Z

X Y Z

0 0 1

0 1 0

1 0 0

1 1 0 Z = ~(X | Y)

nor(Z,X,Y)

NOR Gate

NOT-OR

X

Y

W = X | Y

Z = ~W = ~(X | Y)

X Y W Z

0 0 0 1

0 1 1 0

1 0 1 0

1 1 1 0

Z W

Basic Logic Gates







NAND Gate

X

Y

X

Y

Z Z

Z = ~(X & Y) Z = ~X | ~Y

=

X Y W Z

0 0 0 1

0 1 0 1

1 0 0 1

1 1 1 0

X Y ~X ~Y Z

0 0 1 1 1

0 1 1 0 1

1 0 0 1 1

1 1 0 0 0

De Morgan’s Theorem-1

~(X & Y) = ~X | ~Y

• NOT all variables

• Change & to | and | to &

• NOT the result

NOR Gate

X

Y Z

Z = ~(X | Y)

X Y Z

0 0 1

0 1 0

1 0 0

1 1 0

X

Y

Z

Z = ~X & ~Y

X Y ~X ~Y Z

0 0 1 1 1

0 1 1 0 0

1 0 0 1 0

1 1 0 0 0

De Morgan’s Theorem-2

~(X | Y) = ~X & ~Y



• NOT the result

De Morgan’s Theorem



• NOT the result

• --------------------------------------------

• ~X | ~Y = ~(~~X & ~~Y) = ~(X & Y)

• ~(X & Y) = ~~(~X | ~Y) = ~X | ~Y

• ~X & !Y = ~(~~X | ~~Y) = ~(X | Y)

• ~(X | Y) = ~~(~X & ~Y) = ~X & ~Y

Basic Logic Gates







Exclusive-OR Gate

X Y Z XOR

X

Y Z 0 0 0

0 1 1

1 0 1

1 1 0

Z = X ^ Y

xor(Z,X,Y)

XOR

• X ^ Y (Verilog)

• X $ Y (ABEL)

• X @ Y

• xor(Z,X,Y) (Verilog)

X Y (textbook)

Exclusive-NOR Gate

X Y Z XNOR

X

Y Z 0 0 1

0 1 0

1 0 0

1 1 1

Z = ~(X ^ Y)

Z = X ~^ Y

xnor(Z,X,Y)

XNOR

• X ~^ Y (Verilog)

• !(X $ Y) (ABEL)

• X @ Y

• xnor(Z,X,Y) (Verilog)

X Y

Basic Logic Gates







Multiple-input Gates

Z 1 2

3 4 Z Z

Z

Multiple-input AND Gate

Z 1

Output is HIGH only if all inputs are HIGH Z 1

An open input will float HIGH

Multiple-input OR Gate

Output is LOW only if all inputs are LOW Z 2

2 Z

Multiple-input NAND Gate

Output is LOW only if all inputs are HIGH Z 3

3 Z

Multiple-input NOR Gate

Output is HIGH only if all inputs are LOW Z 4

4 Z

Additional Gates and Circuits – 1

NAND Gate

The basic NAND gate has the following symbol

and truth table:

• AND-Invert (NAND) Symbol:

NAND represents NOT AND. The small “bubble”

circle represents the invert function

The NAND gate is implemented efficiently in

CMOS technology in terms of chip area and speed

X

Y X · Y

X Y NAND

0

0

1

1

0

1

0

1

1

1

1

0


NAND Gate: Invert-OR Symbol

Applying DeMorgan's Law: Invert-OR = NAND

This NAND symbol is called Invert-OR

• Since inputs are inverted and then ORed together

AND-Invert & Invert-OR both represent NAND gate

• Having both makes visualization of circuit function easier

Unlike AND, the NAND operation is NOT associative

(X NAND Y) NAND Z ≠ X NAND (Y NAND Z)

X

Y X + Y = X · Y = NAND


NAND gates can implement any Boolean function

NAND gates can be used as inverters, or to

implement AND / OR operations

A NAND gate with one input is an inverter

AND is equivalent to NAND with inverted output

OR is equivalent to NAND with inverted inputs

The NAND Gate is Universal

X

Y

X · Y X · Y X

Y

X · Y ≡

X · Y= X+Y X

Y

X + Y ≡

X

Y

X

Y


NOR Gate

The basic NOR gate has the following symbol and

truth table:

• OR-Invert (NOR) Symbol:

NOR represents NOT OR. The small “bubble”

circle represents the invert function.

The NOR gate is also implemented efficiently in

CMOS technology in terms of chip area and speed

X

Y

X + Y

X Y NOR

0

0

1

1

0

1

0

1

1

0

0

0


NOR Gate: Invert-AND Symbol

The Invert-AND symbol is also used for NOR

This NOR symbol is called Invert-AND, since inputs

are inverted and then ANDed together

OR-Invert & Invert-AND both represent NOR gate

• Having both makes visualization of circuit function easier

Unlike OR, the NOR operation is NOT associative

(X NOR Y) NOR Z ≠ X NOR (Y NOR Z)

X

Y X · Y = X + Y = NOR


The NOR Gate is also Universal

NOR gates can implement any Boolean function

NOR gates can be used as inverters, or to

implement AND / OR operations

A NOR gate with one input is an inverter

OR is equivalent to NOR with inverted output

AND is equivalent to NOR with inverted inputs

X

Y

X + Y X + Y X

Y

X + Y ≡

X

Y

X

Y

X + Y= X · Y X

Y

X · Y ≡


NAND–NAND Implementation

Consider the Following SOP Expression:

A 2-level AND-OR circuit can be converted

easily to a NAND-NAND implementation

ZYWXZF

X

Z

W

Z Y

F

X

Z

W

Z Y

F

Two successive bubbles

on the same line cancel

each other

X

Z

W

Z Y

F


NOR–NOR Implementation

Consider the Following POS Expression:

A 2-level OR-AND circuit can be converted

easily to a NOR-NOR implementation

))(( ZYWZXF

Two successive bubbles

on the same line cancel

each other

X

Z

W

Z Y

F

X

Z

W

Z Y

F

X

Z

W

Z Y

F


Other Types of 2-Level Circuits

Other useful types of 2-level circuits:

• AND-NOR and NAND-AND

• OR-NAND and NOR-OR

AND-NOR Function:

Similarly, OR-NAND circuits can be converted to NOR-OR

ZXWXYF

X

Y

W

Z X

F

X

Y

W

Z X

F

NAND-AND

X

Y

W

Z X

F

AND-NOR


Exclusive OR / Exclusive NOR

The eXclusive-OR (XOR) function is an important

Boolean function used extensively in logic circuits

The XOR function may be:

• Implemented directly as an electronic circuit (true gate)

• Implemented by interconnecting other gate types (XOR

is used as a convenient representation)

The eXclusive-NOR (XNOR) function is the

complement of the XOR function

XOR and XNOR gates are complex gates


XOR / XNOR Tables and Symbols

The XNOR is also denoted as equivalence

XOR XNOR

X Y X Y

0 0 0

0 1 1

1 0 1

1 1 0

X Y X Y

0 0 1

0 1 0

1 0 0

1 1 1

XOR Symbol XNOR Symbol


Uses for XOR / XNOR

SOP Expressions for XOR/XNOR:

• The XOR function is:

• The eXclusive NOR (XNOR) function, know also as

equivalence is:

Uses for the XOR and XNORs gate include:

• Adders/subtractors/multipliers

• Counters/incrementers/decrementers

• Parity generators/checkers

Strictly speaking, XOR and XNOR gates do no

exist for more that two inputs. Instead, they are

replaced by odd and even functions.

Y X Y X Y X

Y X Y X Y X


XOR Implementations

X Y

X

Y

X

Y

X Y

SOP implementation

for XOR:

X Y = X Y + X Y

NAND only

implementation

for XOR:


XOR / XNOR Identities

XOR and XNOR are associative operations

X 0 X X 1 X

1 X X 0 X X

X Y Y X

) = X Y Z Z Y ( X Z ) Y X (

Y = X Y X Y X

) = X Y Z Z Y ( X Z ) Y X (


Odd Function

The XOR function can be extended to 3 or more variables

For 3 or more variables, XOR is called an odd function

• The function is 1 if the total number of 1’s in the inputs is odd

Z Y X Z Y X Z Y X Z Y X Z Y X

1 1

1 1

YZ

X 00 01 11 10

0

1

1 1

1 1

YZ

WX 00 01 11 10

00

01

1 1

1 1

11

10

X Y Z

W X Y Z


Odd and Even Functions

The 1s of an odd function correspond to inputs

with an odd number of 1s

The complement of an odd function is called an

even function

The 1s of an even function correspond to inputs

with an even number of 1s

Implementation of odd and even functions use

trees made up of 2-input XOR or XNOR gates


Odd/Even Function Implementation

Design a 3-input odd function with 2-input XOR:

3-input odd function:

F = (X Y) Z

Design a 4-input even function with 2-input XOR

and XNOR gates:

4-input even function:

F = (W X) (Y Z)

X Y

Z F

W

X

Y F

Z


Buffer

A buffer is a gate with the function F = X

In terms of Boolean function,

a buffer is the same as a connection!

So why use it?

• A buffer is used to amplify an input signal

• Permits more gates to be attached to output

• Also, increases the speed of circuit operation

X F X F

0

1

0

1

SOP and POS Implementation

1

2

Karnaugh Map (K-Map)

• A graphical map method to simplify

Boolean function up to 6 variables

• A diagram made up of squares

• Each square represents one minterm (or

maxterm) of a given Boolean function

3

Karnaugh Map Examples

Note that the Hamming Distance between adjacent columns or adjacent rows (including cyclic ones) must be 1 for simplification purposes

4

Karnaugh Map

Adjacent columns or rows allow grouping of minterms (maxterms) for simplification

5

Implicant

• Definition

– A product term is an Implicant

of a Boolean function if the

function has an output 1 for all

minterms of the product term.

• In K-map, an Implicant is

– bubble covers only 1 (bubble

size must be a power of 2)

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

AB CD

6

Prime Implicant

• Definition

– If the removal of any literal from an

implicant I results in a product term that is

not an implicant of the Boolean function,

then I is an Prime Implicant.

– Examples

• BCD is an implicant, but CD or BD or BC do

not imply a 1 in this function; BCD is a PI

• B’C’D is an implicant, but B’C’ is not an

implicant, thus B’C’D is not a PI

• In K-map, a Prime Implicant (PI) is

– bubble that is expanded as big as possible

(bubble size must be a power of 2)

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

AB CD

7

Essential Prime Implicant

• Definition

– If a minterm of a Boolean

function is included in only one

PI, then this PI is an Essential

Prime Implicant.

• In K-map, an Essential

Prime Implicant is

– Bubble that contains a 1

covered only by itself and no

other PI bubbles

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

AB CD

8

Non-Essential Prime Implicant

• Definition

– A Non-Essential Prime

Implicant is a PI that is not an

Essential PI.

• In K-map, an Non-Essential

Prime Implicant is

– A 1 covered by more than one

PI bubble

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

AB CD

9

Simplification for SOP • Form K-Map for the given Boolean function

• Identify all Essential Prime Implicants for 1’s in the K-map

• Identify non-Essential Prime Implicants in the K-map for the 1’s which are not covered by the Essential Prime Implicants

• Form a sum-of-products (SOP) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 1’s

10

Example for SOP

• Identify all the

essential PIs for 1’s

• Identify the non-

essential PIs to cover

1’s

• Form an SOP based

on the selected PIs

7) 6, 4, 1, m(0,F

00 01 11 10

0 1 1 0 0

1 1 0 1 1

A BC

CAABBAF

or

CBABBAF

11

Example for SOP





1’s


on the selected PIs

15) 14, 13, 9, 8, 7, 1, m(0,F

00 01 11 10

00 1 1 0 0

01 0 0 1 0

11 0 1 1 1

10 1 1 0 0

AB CD

ABDBCDABCCBF

or

DCABCDABCCBF

12

Example for SOP





1’s


on the selected PIs

12) 11, 6, 4, 3, M(1,F

00 01 11 10

00 1 0 0 1

01 0 1 1 0

11 0 1 1 1

10 1 1 0 1

AB CD

CBAABCBDDBF

or

DCAABCBDDBF

13

Prime Implicants

• All the prior definitions apply to ‘0’ (or

maxterm) as well

• Consider these implicants imply a ‘0’ output

14

Simplification for POS • Form K-Map for the given Boolean function

• Identify all Essential Prime Implicants for 0’s in the K-map

• Identify non-Essential Prime Implicants in the K-map for the 0’s which are not covered by the Essential Prime Implicants

• Form a product-of-sums (POS) with all Essential Prime Implicants and the necessary non-Essential Prime Implicants to cover all 0’s

15

Example for POS





0’s

• Form an POS based

on the selected PIs

)CBA)(B(AF

00 01 11 10

0 1 1 0 0

1 1 0 1 1

A BC

5) 3, M(2,F

16

Example for POS





0’s

• Form an POS based

on the selected PIs

D)B)(ADC)(BDBD)(ACB(F

00 01 11 10

00 1 0 0 1

01 0 1 1 0

11 0 1 1 1

10 1 1 0 1

AB CD

12) 11, 6, 4, 3, M(1,F

17

Don’t Care Condition X

• Don’t care (X)

– Those input combinations which are irrelevant to the target function (i.e. If the input combination signals can be guaranteed never occur)

– Can be used to simplify Boolean equations, thus simply logic design

• In K-map

– Use X to express Don’t Care in the map

– Don’t care can be bubbled as 1 or 0 depending on SOP or POS simplification to result into bigger bubble

18

Karnaugh Map Example in B5

27) 24, 20, 19, 18, 16, 11, 8, 7, 6, 4, 3, 2, m(0,D)C,B,A, F(E,

00 01 11 10

00 1 0 1 1

01 1 0 1 1

11 0 0 0 0

10 1 0 1 0

AB CD

00 01 11 10

00 1 0 1 1

01 1 0 0 0

11 0 0 0 0

10 1 0 1 0

AB CD

E = 0 E = 1

CDBDCBDCACBAECAF

19

Karnaugh Map Example in B6

)45(61) 53, 37, 31, 29, 23, 21, 13, 9, 5, m(1,Z)Y,X,W,V,F(U, d

00 01 11 10

00 1

01 1

11 1 1

10 1

WX YZ

U,V=0,0

00 01 11 10

00

01 1

11 X

10

WX YZ

U,V=1,0

00 01 11 10

00

01 1 1

11 1 1

10

WX YZ

U,V=0,1

00 01 11 10

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1. number systemsieee.daiict.ac.in/ifest2012/pdf/part02.pdf · common number systems system base...

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