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Booth Multiplier

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Sudhir Kumar
Sudhir Kumar
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Multiplication
Multiplier Notation Partial Products Logical-AND
Shift and Add Paradigm
Shift and Add Examples
Programmed Multiplication
Programmed Multiplication (cont.)
Hardware Shift and Add (right)
Hardware Shift and Add
Hardware Shift and Add (left)
Signed Number Multiplication (positive case)
Signed Number Multiplication (negative case)
Booth’s Recoding (or encoding) • Developed for Speeding Up Multiplication in Early Computers • When a Partial Product of 0 Occurs, Can Skip Addition and Just Shift • Doesn’t 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 One’s in • Booth Observed that a String of 1’s May be Replaced as: j −1 i +1 j +1 2 +2 j +L+ 2 +2 =2 i −2 i
Booth’s Recoding Example xn xn-1 ... xi xi-1 ... x0 (0) yi=xi-1 - xi yn ... yi ... y0 xi xi-1 Operation Comments yi 0 0 shift only string of zeros 0 1 1 shift only string of ones 0 1 0 subtract shift beg. string of ones -1 0 1 addition shift end string of ones 1 EXAMPLE 0011110011(0) 0100010101
Booth’s Recoding • Maps Words With Digit Set [0,1] to Those With [-1,1]
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)
Booth Multiplier Example
Booth’s Recoding Drawbacks • Number of add/sub Operations are Variable • Some Inefficiencies EXAMPLE 001010101(0) 011111111 • Can Use Modified Booth’s Recoding to Prevent • Will Look at This in Later Class
Sign Extension • Consider 6-bit 2’s Complement Number s=0 Positive Value; s=1 Negative Value • Show Sign Extension Works: s s s s s p4 p3 p2 p1 p0 = − s ×29 + s ×28 + s ×27 + s ×26 + s ×25 + p4 ×24 + p3 ×23 + p2 ×2 2 + p1 ×21 + p0 ×20 4 = − s ×2 + s ×(2 + 2 + 2 + 2 ) + ∑ pi ×2i 9 8 7 6 5 i =0 4 = − s ×2 + s ×(2 − 2 ) + ∑ pi ×2i 9 9 5 i =0 4 = − s ×2 + ∑ pi ×2i 5 i =0 • Definition of 2’s Complement
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)
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)
Methods for Fast Multiplication • Reduce Number of Partial Products to be Added – Group Multiplier Bits Together – Higher Radix Multiplier • Add the Partial Products Faster
Radix-r Shift and Add
Radix-4 Multiplication • Shifter is Multi-bit • No Longer a Simple AND of xi with a • Need 4:1 MUX with 0, a, 2a, 3a as Inputs
Partial Product Selection • 0, a and 2a are easy • 3a=a+2a → Requies an Adder! • Need a Way to Compute 3a Efficiently
Example With 3a Availability
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
Example With 3a Availability
Using Radices >4 • 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
Booth Recoding • Modern Arithmetic Circuits DO NOT Apply Booth Recoding Directly • Useful in Understanding Higher-radix Versions of Booth Recoding • No Consecutive 1’s or –1’s 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
Modified Booth Recoding • Booth Recoding Results From xi and xi-1 • Radix-4 Multiplier Digits Implies Booth Recoding Based on xi+1, xi and xi-1 • Similar to Classical Booth Recoding, Modified Booth Recoding Encodes Multipliers into [-2,2]
Modified Booth Recoding
Example Modified Booth Recoding
Example Multiplication with MBR
Hardware MBR Example

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Booth Multiplier

  • 3. Shift and Add Paradigm
  • 4. Shift and Add Examples
  • 7. Hardware Shift and Add (right)
  • 9. Hardware Shift and Add (left)
  • 10. Signed Number Multiplication (positive case)
  • 11. Signed Number Multiplication (negative case)
  • 12. Booth’s Recoding (or encoding) • Developed for Speeding Up Multiplication in Early Computers • When a Partial Product of 0 Occurs, Can Skip Addition and Just Shift • Doesn’t 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 One’s in • Booth Observed that a String of 1’s May be Replaced as: j −1 i +1 j +1 2 +2 j +L+ 2 +2 =2 i −2 i
  • 13. Booth’s Recoding Example xn xn-1 ... xi xi-1 ... x0 (0) yi=xi-1 - xi yn ... yi ... y0 xi xi-1 Operation Comments yi 0 0 shift only string of zeros 0 1 1 shift only string of ones 0 1 0 subtract shift beg. string of ones -1 0 1 addition shift end string of ones 1 EXAMPLE 0011110011(0) 0100010101
  • 14. Booth’s Recoding • Maps Words With Digit Set [0,1] to Those With [-1,1]
  • 15. 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)
  • 17. Booth’s Recoding Drawbacks • Number of add/sub Operations are Variable • Some Inefficiencies EXAMPLE 001010101(0) 011111111 • Can Use Modified Booth’s Recoding to Prevent • Will Look at This in Later Class
  • 18. Sign Extension • Consider 6-bit 2’s Complement Number s=0 Positive Value; s=1 Negative Value • Show Sign Extension Works: s s s s s p4 p3 p2 p1 p0 = − s ×29 + s ×28 + s ×27 + s ×26 + s ×25 + p4 ×24 + p3 ×23 + p2 ×2 2 + p1 ×21 + p0 ×20 4 = − s ×2 + s ×(2 + 2 + 2 + 2 ) + ∑ pi ×2i 9 8 7 6 5 i =0 4 = − s ×2 + s ×(2 − 2 ) + ∑ pi ×2i 9 9 5 i =0 4 = − s ×2 + ∑ pi ×2i 5 i =0 • Definition of 2’s 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)
  • 20. 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)
  • 21. Methods for Fast Multiplication • Reduce Number of Partial Products to be Added – Group Multiplier Bits Together – Higher Radix Multiplier • Add the Partial Products Faster
  • 23. Radix-4 Multiplication • Shifter is Multi-bit • No Longer a Simple AND of xi with a • Need 4:1 MUX with 0, a, 2a, 3a as Inputs
  • 24. Partial Product Selection • 0, a and 2a are easy • 3a=a+2a → Requies an Adder! • Need a Way to Compute 3a Efficiently
  • 25. Example With 3a Availability
  • 26. 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
  • 28. Using Radices >4 • 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
  • 29. Booth Recoding • Modern Arithmetic Circuits DO NOT Apply Booth Recoding Directly • Useful in Understanding Higher-radix Versions of Booth Recoding • No Consecutive 1’s or –1’s 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
  • 30. Modified Booth Recoding • Booth Recoding Results From xi and xi-1 • Radix-4 Multiplier Digits Implies Booth Recoding Based on xi+1, xi and xi-1 • Similar to Classical Booth Recoding, Modified Booth Recoding Encodes Multipliers into [-2,2]