Multiplication

1
Multiplication
2
Multiplier Notation
Partial Products Logical-AND
3
Shift and Add Paradigm
4
Shift and Add Examples
5
Programmed Multiplication
6
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

19
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)
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Methods for Fast Multiplication

• Reduce Number of Partial Products to be Added
• Group Multiplier Bits Together
• Higher Radix Multiplier
• Add the Partial Products Faster

22
Radix-r Shift and Add
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Radix-4 Multiplication

• Shifter is Multi-bit
• No Longer a Simple AND of xi with a
• Need 41 MUX with 0, a, 2a, 3a as Inputs

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Partial Product Selection

• 0, a and 2a are easy
• 3aa2a ? Requies an Adder!
• Need a Way to Compute 3a Efficiently

25
Example With 3a Availability
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Computing 3a

• One Way is to Precompute 3a and Store in
  Register Initially
• Another Way is When 3a Occurs Add -a
• Send Carry of 1 to Next into Next Radix-4 Digit
  of Multiplier
• Causes Incoming Multiple to be 0,4 Versus
  0,3
• 4 Because incoming carry to 112 Causes Digit
  1002
• Multiples 0, 1, 2 Handled Easily
• Multiple 3 Converted to 1 With Outgoing Carry
  of 1
• Multiple 4 Converted to 0 With Outgoing Carry of
  1
• Requires Extra Cycle of Computation Since MSD
  May Have Carry

27
Example With 3a Availability
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Using Radices gt4

• Could Also Use Radices of 8, 16, ...
• Bit Groupings of Size 3, 4, ...
• Multiple Generation Hardware Becomes More
  Complex
• Must Precompute 3a, 5a, 7a, ....
• Or Use 3a With a Carry Scheme
• Carry Scheme Converts Multipliers 5a, 6a, 7a
  to 3a, -2a, -a, etc.
• Carry Digit in This Form Becomes a 1

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Booth Recoding

• Modern Arithmetic Circuits DO NOT Apply Booth
  Recoding Directly
• Useful in Understanding Higher-radix Versions of
  Booth Recoding
• No Consecutive 1s or 1s Occur Using
  Previously Seen Booth Recoding
• Booth Recoding in Radix-4 Results in the
  Following
• Only Multiples of ?a or ?2a are Required
• These are Easily Obtained Using Shifting and
  Complementation

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Modified Booth Recoding

• Booth Recoding Results From xi and xi-1
• Radix-4 Multiplier Digits Implies Booth Recoding
  Based on xi1, xi and xi-1
• Similar to Classical Booth Recoding, Modified
  Booth Recoding Encodes Multipliers into -2,2

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Modified Booth Recoding
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Example Modified Booth Recoding
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Example Multiplication with MBR
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Hardware MBR Example