Nonlinear & Neural Networks LAB. CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION 1.1Digital Systems and Switching Circuits 1.2Number Systems and Conversion

Nonlinear & Neural Networks LAB.

CHAPTER 1

INTRODUCTION NUMBER SYSTEMS AND CONVERSION

1.1 Digital Systems and Switching Circuits1.2 Number Systems and Conversion1.3 Binary Arithmetic1.4 Representation of Negative Numbers1.5 Binary Codes


Objectives

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


1.1 Digital Systems and Switching Circuits

• Digital systems: computation, data processing, control,

communication, measurement

- Reliable, Integration• 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


Binary Digit?

• Binary:- Two values(0, 1)

- Each digit is called as a “bit”

- 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


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” function


1.2 Number Systems and Conversion

33

22

11

00

11

22

33

44

32101234

).(

RaRaRa

RaRaRaRaRa

aaaaaaaaN R

10

10128

375.1038

373264838784813.147

2101210 10810710310510978.953

10

2101232

75.114

311

4

1

2

11208

21212121202111.1011

Decimal:

Binary:

Radix(Base):

Example:

10012

16 260715322560161516216102 FAHexa-Decimal:



01

12

21

10121 )( aRaRaRaRaaaaaaN nn

nnRnn

0111

22

11 remainder , aQaRaRaRa

R

N nn

nn

1221

33

121 remainder , aQaRaRaRa

R

Q nn

nn

2334

132 remainder , aQaRaRa

R

Q nn

nn

Conversion of Decimal to Base-R



Example: Decimal to Binary Conversion

532

262

132

62

32

12

rem. = 1 = a0

rem. = 0 = a1

rem. = 1 = a2

rem. = 0 = a3

rem. = 1 = a4

0 rem. = 1 = a5

210 11010153



mmRm RaRaRaRaaaaaF

33

22

11321 )(.

1112

31

21 FaRaRaRaaFR mm

2221

321 FaRaRaaRF mm

333

32 FaRaaRF mm

)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:



Example: Convert 0.7 to binary

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 previouslyobtained

210 0110 0110 0110 1.07.0



1654

2 5.41100 0101 1101 0100010111.1001101 CDCD

Example: Convert 231.3 to base-7

104 75.454

31431623.231

457

67

0 rem.6

rem.3

75).1(

7

25).5(

7

75).1(

7

25).5(

7

75.

710 5151.6375.45



Conversion of Binary to Octal, Hexa-decimal

(101011010111 )2 = ( )8, octal

(10111011)2 = ( )8, octal

(1010111100100101)2 = ( )16, Hexadecimal

(1101101000)2 = ( )16, Hexadecimal


1.3 Binary Arithmetic

10

10

10

24 11000

1011 11

1011 13

1111

carries

Addition

Example:

01 1

10 1

11 0

00 0

and carry 1 to the next column



01 1

10 1

11 0

00 0

and borrow 1 from the next column

1010

00111

11101

1

(indicatesa borrowFrom the3rd column)

1101

11

10000

11 1 1

borrows

101110

0111

111001

11 1

borrows

Subtraction

Example:



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

187

18

205

column 2 column 1



111

001

010

000

1014310001111

1101

0000

1101

1101

1011

1101

11000011

1111

)1001011(

1111

)01111(

0000

1111

1011

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 prodoc

t)

Multiplication Multiply: 13 x11(10)



Division

10

1011

1101

1011

1110

1011

10010001 1011

1101

The quotient is 1101 with a remainderof 10.


1.4 Representation of Negative Numbers

bn 1– b1 b0

Magnitude

MSB

(a) Unsigned number

bn 1– b1 b0

MagnitudeSign

(b) Signed number

bn 2–

0 denotes1 denotes

+

–

MSB



NN n 2*

2’s complement representation for Negative Numbers

1111

1110

1101

1100

1011

1010

1001

1000

-

-

1111

1110

1101

1100

1011

1010

1001

1000

1000

1001

1010

1011

1100

1101

1110

1111

-

-0

-1

-2

-3

-4

-5

-6

-7

-8

0000

0001

0010

0100

0101

0110

0111

+0

+1

+2

+3

+4

+5

+6

+7

1’s complement 2’s complement N*

Sign and

magnitude

Negative integers

-N

Positive

integers

(all systems)+N N



101010

010101

11111112

N

N

n

11)12(2* NNNN nn

NNNN nn )12( and *2

11 222 nnn

1’s complement representation for Negative Numbers

NN n )12(

Example:

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


1.4 Representation of Negative Number

6

5

1011

0110

0101

wrong answer because of overflow (+11 requires5 bits including sign)

6

5

1111

1010

0101

(correct answer)

6

5

0001)1(

0110

1011

correct answer when the carry from the sign bitis ignored (this is not an overflow)

Case 1

Case 2

Case 3

Case 4

Addition of 2’s complement Numbers

7

4

3

0111

0100

0011

(correct answer)



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




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)

6

5

1000

1

0111 )1(

1011

1100

Case 3

Case 4

Case 5




(end-around carry)(wrong answer because of overflow)

6

5

0100

1

0011 )1(

1001

1010

1)(2)12(

) (where :4

ABBABA

ABBACasenn

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

)2( :5 1

BABABA

BABACasennnn

n

Case 6




)20(

)11(

3011100000

1

11011111 )1(

11101011

11110100

(end-around carry)

11

19

)8(

00010110)1(

00010011

11110001

(end-around carry)




1.5 Binary Codes

Decimal

Digit

8-4-2-1

Code

(BCD)

6-3-1-1

Code

Excees-3

Code

2-out-of-5

Code

Gray

Code

0

1

2

3

4

5

6

7

8

9

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

0000

0001

0011

0100

0101

0111

1000

1001

1011

1100

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

00011

00101

00110

01001

01010

01100

10001

10010

10100

11000

0000

0001

0011

0010

0110

1110

1010

1011

1001

1000

5 2 . 7 3 9

0101 0010 . 0111 0011 1001


1.5 Binary Codes

00112233 awawawawN

811110316 N

1010011 1110100 1100001 1110010 1110100

S t a r t

6-3-1-1 Code:

ASCII Code

Documents

Nonlinear & Neural Networks LAB. CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION 1.1Digital Systems and Switching Circuits 1.2Number Systems and Conversion