Ch 2 . Number Systems and Codes

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# Ch 2 . Number Systems and Codes - PowerPoint PPT Presentation

Ch 2 . Number Systems and Codes. 2.2 Octal and Hexadecimal Numbers. 10 ~ 15 : Alphabet . 2.3 General Positional-Number-System Conversions. p digit to the left of the point and n digits to the right of the point. Ex) A number D of the form has the value . p. n.

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Ch 2. Number Systems and Codes

10 ~ 15 : Alphabet

2.3 General Positional-Number-System

Conversions

• p digit to the left of the point and n digits to the right of the point

Ex) A number D of the form has the value

p

n

: Carry in

: Burrow in

: Input data 1

: Input data 2

: Carry out

: Sum

: Burrow out

: Difference

1 0

1

0

1

-

+

-

+

1

0

1

1

1

1

1

1

1

0

0

1

1

1

Burrow in

Carry in

Carry out

Carry out

Burrow out

Burrow out

2.5 Representation of Negative Numbers

• Signed-Magnitude System
• Magnitude and Symbol ( ‘+’, ‘-’ )
• Applied to binary number by using ‘sign bit’
• Ex)

Sign bit

• Complement System
• Negates a number by taking its complement
• More difficult than changing the sign bit
• Can be added or subtracted directly

:

:

Conversion example

Number :

Easy to complement

One’s complement

End-around carry

+6 (0110)

+

-3 (1100)

10010

1

0011

Binary to Gray Code

Gray Code to Binary

(0) 1 1 0

(0) 1 0 1

1

3

2

1

2

1 1 0

1 0 1

3

If different, ‘1’

else (same) ‘0’

If different, ‘1’

else (same) ‘0’

Hamming Distance

• Distance between two vertices, the number of difference bits in each position
• EX) D(010, 111) = 2

If minimum distance = 2C+1,

• up to C-bits can be corrected
• If 2C+D+1, then C-bits can be corrected,
• and d bits can be detected
• 4= 2C+D+1,
• C=1, D=1
• 1 bit can be corrected
• D=3, 3 bit errors can be detected

111

110

101

100

011

010

001

LSB is 1 if all 7 bits are odd

LSB is 0 if all 7 bits are even

k = # of parity bits

m = # of info bits

, m=4,3,2,1

, m=11,10,9,…,2,1