Multiplication

1
Multiplication
2
Multiplier Notation
Partial Products Logical-AND
3
Shift and Add Paradigm
4
Shift and Add Examples
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Programmed Multiplication
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Programmed Multiplication (cont.)
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Hardware Shift and Add (right)
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Hardware Shift and Add
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Hardware Shift and Add (left)
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Signed Number Multiplication(positive case)
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Signed Number Multiplication(negative case)
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Booths Recoding (or encoding)

• Developed for Speeding Up Multiplication in
  Early Computers
• When a Partial Product of 0 Occurs, Can Skip
  Addition and Just Shift
• Doesnt Help Multipliers Where Datapaths Go
  Through Adder Such as Previous Examples
• Does Help Designs for Asynchronous
  Implementation or Microprogramming Since
  Shifting is Faster Than Addition
• Variable Delay Depends on Number of Ones in
• Booth Observed that a String of 1s May be
  Replaced as

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Booths Recoding Example
xn xn-1 ... xi xi-1 ... x0
(0)
yixi-1 - xi
yn ... yi ... y0
EXAMPLE
0011110011(0) 0100010101
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Booths Recoding

• Maps Words With Digit Set 0,1 to Those With
  -1,1

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Sequential Multiplication
A 1011 (-510) X
1101 (-310) Y 0111
(recoded) (-1) Add A 0101 Shift
00101 (1) Add A 1011
11011 Shift 111011 (-1) Add A
0101 001111 Shift
0001111 (1510)
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Booth Multiplier Example
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Booths Recoding Drawbacks

• Number of add/sub Operations are Variable
• Some Inefficiencies

EXAMPLE 001010101(0)
011111111

• Can Use Modified Booths Recoding to Prevent
• Will Look at This in Later Class

18
Sign Extension

• Consider 6-bit 2s Complement Number
• s0 Positive Value s1 Negative Value
• Show Sign Extension Works

• Definition of 2s Complement

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Sign Extension Example
A 010110 (2210) X 001011
(1110) Y 010101 (recoding)
11111101010 (neg. A) 0000000000 (0 A)
111101010 (neg. A) 00000000 (0 A)
0010110 (neg. A) 000000 (0 A)
00011110010 (24210)
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Sign Extension Example

• Same Trick as Before, Complement Original Sign
  Bit
• Add 1 to Column 5

1 001010 (neg. A) 100000
(0 A) 001010 (neg. A) 100000
(0 A) 110110 (neg. A) 100000
(0 A) 00011110010 (24210)