Booths Compact Signed Multiplication Algorithm

1
Booths Compact Signed Multiplication Algorithm

• Salient Features
• 1. Applicable for signed binary integers only.
• 2. Employs 2s complement scheme to represent
  ALL signed binary integers.
• 3. Reduces number of multiplication steps.
• 4. Uses both ADD Subtract as well as right
  shift arithmetic.

2
The Task at Hand

• Given Operand A n bit 2s complement with
  Leftmost (n-1)th bit Representing Sign
• ( The Multiplicand ( MPD ) .
• Given Operand B n bit 2s complement with
  Leftmost (n-1)th bit Representing Sign
• ( The Multiplier ( MPR ) .
• Produce the 2n bit Result C ? A X B also in 2s
  complement with leftmost bit representing sign.

3
Booths Algorithm Essential Facts

• Ability to ADD SUBTRACT using complemented
  number
• There exists multiple ways to compute a product.
• Example 1
• MPD A 2(Decimal) 0010 4 bits 2s
  complement
• MPR B 6 ( Decimal) 0110 4 bits 2s
  complement
• Product C A X B (0010) X (0110)
• A X 6(Decimal)
• A X 8 2 A X 23
  22
• A X 01000 00100 A X
  ( 1 0 1 0 )

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Booths Algorithm Essential Facts - 2

• Example 2
• MPR B 30 ( Decimal)
• 00011110 8 bits 2s complement
• Product C A X B A X (00011110)
• A X 30(Decimal)
• A X ( 24232221) A X (
  25-21)
• Since 2n2(n-1) .2(n-k) 2(n1) 2(n-k)
• A X ( 0 0 1 0 0 0 1 0 )
  

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Booths Algorithm Essential Facts - 3

• Result Higher Order n bits Initialized to ALL
  0s Lower n bits (Previous MPR) A 0 to the
  Right
• Multiplier (MPR) Bits
• a) Run of 0s OR Run of 1s ? Multiply with 0
  i.e. only Right Shift Arithmetic the Partial
  Result.
• b) Start of 1 i.e. 1 appearing to the left of 0
  ( 10 ) Subtract MPD from higher order n bits of
  the partial Result i.e. ADD 2s complement of MPD
  to the higher order n bits of the partial result
  followed by Right Shift Arithmetic the Partial
  Result .
• c) Start of 0 i.e. 0 appearing to the left of
  1( 01 ) ADD MPD to the higher order n bits of
  the partial Result followed by Right Shift
  Arithmetic the Partial Result .

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Booths Multiplication Algorithm - 1

• Step 0(Pre Processing) Equate the sizes of
  Multiplicand (MPD) A as well as Multiplier (MPR)
  B both to n bits by sign Extension.
• Step 1(Initialization) Set up higher order n
  bits of the Partial Result C All Zeros 0s
  Lower Order n Bits Multiplier (MPR) B.
• Operation Counter n (For n bit Case).
  Set up 2s Complement of MPD A D.
• A dummy multiplier bit b_-1 0 to the
  Right of the Result (MPR B).
• Copy MPR B in some other place for
  future reference E .
• Step 2 Consider the value of the bit pair
  consisting of current MPR ( Rightmost n bits of
  the Result) lsbit (LSB) b_0 together with the
  dummy multiplier bit b_-1.
• Step 3 WHILE (Operation Counter gt 0 ) DO

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Booths Multiplication Algorithm - 2

• Step 4a. If b0 b-1 0 0 OR 1 1
• Right Shift Arithmetic the partial Result (C)
  MPR (B)
• So that c0 comes to b(n-1) , b0
  goes to b-1
• Step 4b. If b0 b-1 0 1 Then C ? C PLUS
  MPD (A) Ignore Carry
• Right Shift Arithmetic the partial Result (C)
  MPR (B)
• So that c0 comes to b(n-1) , b0
  goes to b-1
• Step 4c. If b0 b-1 1 0 Then
• C ? C PLUS 2s Complement of MPD
  (D) Ignore Carry
• Right Shift Arithmetic the partial Result (C)
  MPR (B)
• So that c0 comes to b(n-1) , b0
  goes to b-1
• Step 5 Decrement Operation Counter . Go To
  Step 3.

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Booths Multiplication Example - 1

• Size of MPD (A) MPR (B) 4 bits.
• Hence size of Result (C ) 8 bits.

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Booths Multiplication Example - 2

• Multiplicand (MPD) A 1110 -2
  (Decimal)
• Multiplier (MPR) B 0101 5
  (Decimal)
• Result C A X B 10110 -10 (Decimal)
• After Sign Extension
  
• Expected 8 bit Result C 1111 0 110
  

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Booths Multiplication Algorithm Steps - 1

• MPD A 1110 , MPR B 0101
• Step 1 Initialize Partial Result C 0 0 0 0
  4 bits
• Set up Operation Counter 4
• Set dummy Multiplier Bit b-1 0
• Step 2 Consider the two rightmost multiplier
  bit pair
• b0b-1 1 0 here
• Step 3 C ? ( C ) PLUS 2s Complement of MPD A
  0010
• 0 0 1 0 0 1 0 1 0
• Step 4 Right Shift Arithmetic 0 0 0 1 0 0 1
  0 1
• Step 5 Decrement Operation Counter 4/3

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Booths Multiplication Algorithm Steps - 2

• Step 6 Consider the two rightmost multiplier
  bit pair b0b-1 0 1 here
• Step 7 C ? ( C ) PLUS MPD A 1110
• 1 1 1 1 0 0 1 0 1
• Step 8 Right Shift Arithmetic 1 1 1 1 1 0 0
  1 0
• Step 9 Decrement Operation Counter 3/2

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Booths Multiplication Algorithm Steps - 3

• Step 10 Consider the two rightmost multiplier
  bit pair b0b-1 1 0 here
• Step 11 C ? ( C ) PLUS 2s Comp of MPD A
  0010
• 0 0 0 1 1 0 0 1 0
• Step 12 Right Shift Arithmetic 0 0 0 0 1 1
  0 0 1
• Step 13 Decrement Operation Counter 2/1

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Booths Multiplication Algorithm Steps - 4

• Step 14 Consider the two rightmost multiplier
  bit pair b0b-1 0 1 here
• Step 15 C ? ( C ) PLUS MPD A 1110
• 1 1 1 0 1 1 0 0 1
• Step 16 Right Shift Arithmetic 1 1 1 1 0 1
  1 0 0
• Step 17 Decrement Operation Counter 1/0
• 8 bit Result 1 1 1 1 0 1 1 0