CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSIONelearning.kocw.net/KOCW/document/2015/sungkyunkwan/... · • Difference between Analog and Digital System • Difference between

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Fundamentals of Logic Design Chap. 1

INTRODUCTION

NUMBER SYSTEMS AND CONVERSION

This chapter in the book includes:

Objectives

Study Guide

1.1 Digital Systems and Switching Circuits

1.2 Number Systems and Conversion

1.3 Binary Arithmetic

1.4 Representation of Negative Numbers

1.5 Binary Codes

Problems

CHAPTER 1

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Topics introduced in this chapter:

• Difference between Analog and Digital System

• Difference between Combinational and Sequential Circuits

• Binary number and digital systems

• Number systems and Conversion

• Add, Subtract, Multiply, Divide Positive Binary Numbers

• 1’s Complement, 2’s Complement for Negative binary number

• BCD code, 6-3-1-1 code, excess-3 code

Objectives

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1.1 Digital Systems and Switching Circuits

computation, data processing, control, communication, measurement

Reliable, Integration

Digital systems

Analog – Continuous

- Natural Phenomena

(Pressure, Temperature, Speed…)

- Difficulty in realizing, processing using electronics

Digital – Discrete

- Binary Digit Signal Processing as Bit unit

- Easy in realizing, processing using electronics

- High performance due to Integrated Circuit Technology

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Binary Digit?

•Two values(0, 1)

•Each digit is called as a “bit”

Binary

•Number representation with only two values (0,1)

•Can be implemented with simple electronics devices

(ex: Voltage High(1), Low(0) ; Switch On (1) Off(0)…)

Good things in Binary Number

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Switching Circuit

•Combinational Circuit :

•outputs depend on only present inputs, not on past inputs

•Sequential Circuit:

•outputs depend on both present inputs and past inputs

•have “memory” elements

Figure 1-1: Switching circuit

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3

3

2

2

1

1

0

0

1

1

2

2

3

3

4

4

32101234

).(

RaRaRa

RaRaRaRaRa

aaaaaaaaN R

10

1012

8

375.103

8

373264838784813.147

21012

10 10810710310510978.953

10

210123

2

75.114

311

4

1

2

11208

21212121202111.1011

10

012

16 260715322560161516216102 FA

Decimal:

Binary:

Radix(Base):

Octal-Decimal:

Hexa-Decimal:

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0

1

1

2

2

1

10121 )( aRaRaRaRaaaaaaN n

n

n

nRnn

011

1

2

2

1

1 remainder , aQaRaRaRaR

N n

n

n

n

122

1

3

3

1

21 remainder , aQaRaRaRaR

Q n

n

n

n

233

4

1

32 remainder , aQaRaRaR

Q n

n

n

n

Conversion of Decimal to Base-R

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53 2

26 2

13 2

6 2

3 2

1 2

rem. = 1 = a0

rem. = 0 = a1

rem. = 1 = a2

rem. = 0 = a3

rem. = 1 = a4

0 rem. = 1 = a5

210 11010153

Example: Decimal to Binary Conversion

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m

mRm RaRaRaRaaaaaF

3

3

2

2

1

1321 )(.

11

12

3

1

21 FaRaRaRaaFR m

m

22

21

321 FaRaRaaRF m

m

33

3

32 FaRaaRF m

m

)1(

250.1

2

625.

1

a

F

)0(

500.0

2

250.

2

1

a

F

)1(

000.1

2

500.

3

2

a

F

210 101.625.

Conversion of a decimal fraction to Base-R

Example:

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8).0(

2

4).0(

2

2).1(

2

6).1(

2

8).0(

2

4).1(

2

7.

Process starts repeating here because .4 was previously

obtained

210 0110 0110 0110 1.07.0

Example: Convert 0.7 to binary

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104 75.454

31431623.231

45 7

6 7

0 rem.6

rem.3

75).1(

7

25).5(

7

75).1(

7

25).5(

7

75.

710 5151.6375.45

16

54

2 5.41100 0101 1101 0100010111.1001101 CDCD

Example: Convert 231.34 to base-7

Conversion of Binary to Hexa

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(101011010111 )2

= ( )8, octal

(10111011)2

= ( )8, octal

(1010111100100101)2

= ( )16, Hexadecimal

(1101101000)2

= ( )16, Hexadecimal

Conversion of Binary to Octal, Hexa-decimal

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01 1

10 1

11 0

00 0

and carry 1 to the next column

10

10

10

24 11000

1011 11

1011 13

1111

carries

Addition

Example:

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01 1

10 1

11 0

00 0

and borrow 1 from the next column

1010

00111

11101

1

(indicates

a borrow

From the

3rd column)

1101

11

10000

11 1 1

borrows

101110

0111

111001

11 1

borrows

Subtraction

Example:

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187

18

205

column 2 column 1

187]107 108 101[

]108 101) [

]101510)1010(10)12[(

]108 101 [

]10)510(10)10(102[

]108101 [

]105100102[18205

012

01

012

01

012

01

012

note borrow from column 1

note borrow from column 2

Subtraction Example with Decimal

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111

001

010

000

1014310001111

1101

0000

1101

1101

1011

1101

11000011

1111

)1001011(

1111

)01111(

0000

1111

1101

1111 multiplicand

multiplier

first partial product

second partial product

sum of first two partial products

third partial product

sum after adding third partial product

fourth partial product

final product (sum after adding fourth partial prodoct)

Multiplication Multiply: 13 x11(10)

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10

1011

1101

1011

1110

1011

10010001 1011

1101

The quotient is 1101 with a remainder

of 10.

Division

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b n 1 – b 1 b 0

Magnitude

MSB

(a) Unsigned number

b n 1 – b 1 b 0

Magnitude Sign

(b) Signed number

b n 2 –

0 denotes

1 denotes

+

–

MSB

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NN n 2*

2’s complement of a positive integer N

Table 1-1: Signed Binary Integers (word length n = 4)

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101010

010101

11111112

N

N

n

11)12(2* NNNN nn

NNNN nn )12( and *2

11 222 nnn

NN n )12(

== 2’s complement: 1’s complement + ‘1’

Example:

1’s complement of a positive integer N

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1.4 Representation of Negative Number

7

4

3

0111

0100

0011

(correct answer)

6

5

1011

0110

0101

wrong answer because of overflow (+11 requires

5 bits including sign)

6

5

1111

1010

0101

(correct answer)

6

5

0001)1(

0110

1011

correct answer when the carry from the sign bit

is ignored (this is not an overflow)

Addition of 2’s complement Numbers

Case 1

Case 2

Case 3

Case 4

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7

4

3

1001)1(

1100

1101

correct answer when the last carry is ignored

(this is not an overflow)

6

5

0101)1(

1010

1011

wrong answer because of overflow

(-11 requires 5 bits including sign)


Case 5

Case 6

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1

6

5

1110

1001

0101

(correct answer)

(end-around carry)

(correct answer, no overflow)

6

5

0001

1

0000 )1(

0110

1010

(end-around carry)

(correct answer, no overflow)

4

3

1000

1

0111 )1(

1011

1100


Case 3

Case 4

Case 5

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(end-around carry)

(wrong answer because of overflow)

6

5

0100

1

0011 )1(

1001

1010

1)(2)12(

) (where :4

ABBABA

ABBACase

nn

1)](12[2)12()12(

)2( :5 1

BABABA

BABACase

nnnn

n


Case 6

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)20(

)11(

3111100000

1

11011111 )1(

11101011

11110100

(end-around carry)

11

19

)8(

00010110)1(

00010011

11110001

(discard last carry)



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1.5 Binary Codes

5 2 . 7 3 9

0101 0010 . 0111 0011 1001

Decimal Digit

8-4-2-1 Code (BCD)

6-3-1-1 Code

Excess-3 Code

2-out-of-5 Code

Gray Code

0 0000 0000 0011 00011 0000

1 0001 0001 0100 00101 0001

2 0010 0011 0101 00110 0011

3 0011 0100 0110 01001 0010

4 0100 0101 0111 01010 0110

5 0101 0111 1000 01100 1110

6 0110 1000 1001 10001 1010

7 0111 1001 1010 10010 1011

8 1000 1011 1011 10100 1001

9 1001 1100 1100 11000 1000

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1.5 Binary Codes

6-3-1-1 Code:

00112233 awawawawN

811110316 N

ASCII Code

1010011 1110100 1100001 1110010 1110100

S t a r t

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1.5 Binary Codes

Table 1-3

ASCII code

(incomplete)

Documents

CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSIONelearning.kocw.net/KOCW/document/2015/sungkyunkwan/... · • Difference between Analog and Digital System • Difference between