ahmad almulhem, kfupm 2009 coe 202: digital logic design number systems part 4 dr. ahmad almulhem...

Ahmad Almulhem, KFUPM 2009

COE 202: Digital Logic DesignNumber Systems

Part 4

Dr. Ahmad AlmulhemEmail: ahmadsm AT kfupm

Phone: 860-7554Office: 22-324


Objectives

1. Overflow

2. Shift operations

3. Binary codes


Overflows

• Number’s sizes in computers are fixed• Overflows can occur when the result of an

operation does not fit.

• Q: When can an overflow occur?

Unsigned numbers Signed numbers

Subtracting two numbers Adding a positive number to a negative number

Adding two numbers Adding two negative numbers or two positive numbers


Overflow (2’s Complement)Q: Add +5 and +4 in 2’s complement.

A: An overflow happened. The correct answer +9 (1001) cannot be represented by 4 bits.

Detection: 1. Adding two positive numbers result in

a negative number!2. Carry in sign bit is different from carry

out of sign bit

Solution: Use one more bit, extend the sign bit

0101 + 510

+ 0100 + 410

0 1001 - 710 ??

0 1 0 0

00101 + 510

+ 00100 + 410

01001 + 910

0 0 1 0


Overflow (2’s Complement)Q: Add -5 and -4 in 2’s complement.

A: An overflow happened. The correct answer -9 (10111) cannot be represented by 4 bits.

Detection: 1. Adding two negative numbers result in

a positive number!2. Carry in sign bit is different from carry

out of sign bit

Solution: Use one more bit, extend the sign bit

1011 - 510

+ 1100 - 410

1 0111 + 710 ??

1 0 0 0

11011 - 510

+ 11100 - 410

10111 - 910

1 1 0 0


Range ExtensionsTo extend the representation of a 2’s complement number

possibly for storage and use in a larger-sized registerIf the number is positive, pad 0’s to the left of the integral number

(sign bit extension), and 0’s to the right of the fractional numberIf the number is negative, pad 1’s to the left of the integral number

(sign bit extension), and 0’s to the right of the fractional number

0 1 0 0 1 0 0 0 0 0 1 0 0 1

5-bit register (+9)

1 0 1 1 1 1 1 1 1 1 0 1 1 1

5-bit register (-9)

9-bit register

(-9) – sign bit extended

9-bit register

(+9) – sign bit extended


Arithmetic Shift• A binary number can be shifted right or left• To shift an unsigned numbers (right or left), pad with 0s.

• Example:

Left Shift

0001 110

0010 210

0100 410

1000 810

Q1: What is the effect of left-shifting?Q2: What is the effect of right-shifting?


Arithmetic Shift• To shift a signed number

– Left-Shift: pad with 0s– Right-Shift: Extend sign bit

• Example (2’s complement):

Right Shift

1000 - 810

1100 - 410

1110 - 210

1111 - 110

In General- left-shifting = multiply by r- right-shifting = divide by r


Arithmetic Shifts


Binary Codes

• Human communicating with computers– Humans understand decimal– Computers understands binary

• Communication mode is decimal• Binary codes are used to translate individual digits of a

given number (generally in base 10) to binary format based on given rules

Computers

01011 ….

Human

1234 ….


Binary Coded Decimal (BCD)

• BCD Code uses 4 bits to represent the 10 decimal digits {0 to 9}• 6 BCD codes unused• The weights of the individual positions of the bits of a BCD code are: 23=8, 22=4,

21=2, 20=1


BCD Addition


Gray Code

Gray code represents decimal numbers 0 to 15 using 16 4-bit codes

Gray codes of two adjacent decimal numbers differ by only one bit

Example:

(5)10 = 0111

(6)10 = 0101

(7)10 = 0100


ASCII Code (Alphanumeric)

American Standard Code for Information Interchange (ASCII): includes binary definitions for 128 characters that include the English alphabets, special symbols and control characters.

ASCII codes have a length of 7 bitsExamples:

A = (65)10 = 100 0001

B = (66)10 = 100 0010

Z = (90)10 = 101 1010

a = (97)10 = 110 0001

b = (98)10 = 110 0010

z = (122)10 = 111 1010


Parity Bits


Codes Summary

• Bits are bits– Modern digital devices represent everything as

collections of bits– A computer is one such digital device

• You can encode anything with sufficient 1’s and 0’s– Text (ASCII)– Computer programs (C code, assembly code,

machine code)– Sound (.wav, .mp3, ...)– Pictures (.jpg, .gif, .tiff)


• Numbers have fixed sizes in a computer, therefore overflow can occur in any representation.

• Detection of overflow by sign or carry-in carry-out of the sign bit.

• To extend a signed number, extend the sign

• Shift operations (right, left)

• Binary codes (BCD, gray code, ASCII)

Conclusions

ahmad almulhem, kfupm 2009 coe 202: digital logic design number systems part 4 dr. ahmad almulhem...

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