1. Sequential Multipliers Lecture 9

2. Required Reading Chapter 9, Basic Multiplication Scheme Chapter 10, High-Radix Multipliers Chapter 12.3, Bit-Serial Multipliers Chapter 12.4, Modular Multipliers Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design

3. Notation a Multiplicand a k-1 a k-2... a 1 a 0 x Multiplier x k-1 x k-2... x 1 x 0 p Product (a  x) p 2k-1 p 2k-2... p 2 p 1 p 0 If multiplicand and multiplier are of different sizes, usually multiplier has the smaller size

4. Multiplication of two 4-bit unsigned binary numbers in dot notation Partial Product 0 Partial Product 1 Partial Product 2 Partial Product 3 Number of partial products = number of bits in multiplier x Bit-width of each partial product = bit-width of multiplicand a

5. Basic Multiplication Equations x =  x i  2 i i=0 k-1 p = a  x p = a  x =  a  x i  2 i = = x 0 a2 0 + x 1 a2 1 + x 2 a2 2 + … + x k-1 a2 k-1 i=0 k-1

6. Shift/Add Algorithm Right-shift version

7. Shift/Add Algorithms Right-shift algorithm p = a  x = x 0 a2 0 + x 1 a2 1 + x 2 a2 2 + … + x k-1 a2 k-1 = (...((0 + x 0 a2 k )/2 + x 1 a2 k )/2 +... + x k-1 a2 k )/2 = k times = p (0) = 0 p = p (k) p (j+1) = (p (j) + x j a 2 k ) / 2 j=0..k-1

8. Sequential shift-and-add multiplier for right-shift algorithm

9. Right-shift multiplication algorithm: Example

10. Area optimization for the sequential shift-and-add multiplier with the right-shift algorithm

11. Shift/Add Algorithms Right-shift algorithm: multiply-add = (...((y2 k + x 0 a2 k )/2 + x 1 a2 k )/2 +... + x k-1 a2 k )/2 = k times p (0) = y2 k p = p (k) p (j+1) = (p (j) + x j a 2 k ) / 2 j=0..k-1 = y + x 0 a2 0 + x 1 a2 1 + x 2 a2 2 + … + x k-1 a2 k-1 = y + a  x

12. Signed Multiplication Previous sequential multipliers are for unsigned multiplication For signed multiplication: –assume sign-extended operation for p (j) + x j a – if 2's complement multiplier is POSITIVE right-shift sequential algorithms (shift-add) will work directly –if 2's complement multiplier is NEGATIVE than we must use "negative weight” for x k-1 and subtract x k-1 a in the last cycle Slight increase in area due to control and one-bit sign extension on inputs of adder –Unsigned: k bit number + k bit number  k+1 bit number –Signed: k+1 bit sign extended number + k+1 bit sign extended number  k+1 bit number

13. Sequential multiplication of 2’s-complement numbers with right shifts (positive multiplier)

14. Sequential multiplication of 2’s-complement numbers with right shifts (negative multiplier)

15. Shift/Add Algorithm Left-shift version

16. Shift/Add Algorithms Left-shift algorithm p = a  x = x 0 a2 0 + x 1 a2 1 + x 2 a2 2 + … + x k-1 a2 k-1 = (...((0  2 + x k-1 a)  2 + x k-2 a)  2 +... + x 1 a)  2 + x 0 a= k times = p (0) = 0 p = p (k) p (j+1) = (p (j)  2 + x k-1-j a) j=0..k-1

17. Sequential shift-and-add multiplier for left-shift algorithm Left shifts are not as efficient for two's complement because must sign extend multiplicand by k bits

18. Left-shift multiplication algorithm: Example

19. p (0) = y2 -k p = p (k) p (j+1) = (p (j)  2 + x k-(j+1) a) j=0..k-1 Shift/Add Algorithms Left-shift algorithm: multiply-add = (...((y2 -k  2 + x k-1 a)  2 + x k-2 a)  2 +... + x 1 a)  2 + x 0 a = k times = y + x k-1 a2 k-1 + x k-2 a2 k-2 + … + x 1 a2 1 + x 0 a = y + a  x

20. Shift/Add Algorithm Right-shift version with Carry-Save Adder

21. Sequential shift-and-add multiplier with a carry save adder

22. High-Radix Sequential Multipliers

23. High-Radix Notation a Multiplicand (a n-1 a n-2... a 1 a 0 ) r x Multiplier (x n-1 x n-2... x 1 x 0 ) r p Product (a  x) (p 2n-1 p 2n-2... p 2 p 1 p 0 ) r

24. Radix-4, or two-bit-at-a-time, multiplication in dot notation

25. Basic Multiplication Equations x =  x i  r i i=0 n-1 p = a  x p = a  x =  a  x i  r i = = x 0 ar 0 + x 1 a r 1 + x 2 a r 2 + … + x n-1 a r n-1 i=0 n-1

26. High-Radix Shift/Add Algorithms Right-shift high-radix algorithm p = a  x = x 0 ar 0 + x 1 ar 1 + x 2 ar 2 + … + x n-1 ar n-1 = (...((0 + x 0 ar n )/r + x 1 ar n )/r +... + x n-1 ar n )/r = n times = p (0) = 0 p = p (n) p (j+1) = (p (j) + x j a r n ) / r j=0..n-1

27. High-Radix Shift/Add Algorithms Left-shift high-radix algorithm p = a  x = x 0 ar 0 + x 1 ar 1 + x 2 ar 2 + … + x n-1 ar n-1 = (...((0  r + x n-1 a)  r + x n-2 a)  r +... + x 1 a)  r + x 0 a= n times = p (0) = 0 p = p (n) p (j+1) = (p (j)  r + x n-1-j a) j=0..n-1

28. The multiple generation part of a radix-4 multiplier with precomputation of 3a

29. Example of radix-4 multiplication using the 3a multiple

30. The multiple generation part of a radix-4 multiplier based on replacing 3a with 4a (carry into next higher radix-4 multiplier digit) and -a

31. Higher Radix Multiplication In radix-8, one must precompute 3a, 5a, 7a –Overhead becomes prohibitive and does not help However, when we discuss CSA this may be useful

32. Radix-2 Booth Recoding i j j+1

33. Radix-2 Booth Recoding y i = -x i + x i-1

34. Sequential multiplication of 2’s-complement numbers with right shifts using Booth’s recoding

35. Notation Y Multiplicand y m-1 y m-2... y 1 y 0 X Multiplier x m-1 x m-2... x 1 x 0 P Product (Y  X ) p 2m-1 p 2m-2... p 2 p 1 p 0 If multiplicand and multiplier are of different sizes, usually multiplier has the smaller size

36. Radix-2 Booth Multiplier Basic Step

37. Radix-2 Booth Multiplier Basic Step in Xilinx FPGAs

38. Radix-2 Booth Multiplier in Xilinx FPGAs

39. Radix-4 Booth Recoding (1) -1 0 1 0 0 -1 1 0 -1 1 -1 1 0 0 -1 0

40. z i/2 = -2x i+1 + x i + x i-1

41. Example radix-4 multiplication with modified Booth’s recoding of the 2’s-complement multiplier

42. The multiple generation part of a radix-4 multiplier based on Booth’s recoding

43. Notation Y Multiplicand y m-1 y m-2... y 1 y 0 X Multiplier x m-1 x m-2... x 1 x 0 P Product (Y  X ) p 2m-1 p 2m-2... p 2 p 1 p 0 If multiplicand and multiplier are of different sizes, usually multiplier has the smaller size

44. Radix-4 Booth Multiplier Basic Step

45. Radix-4 Booth Multiplier: Left Shifter & Control

46. High-Radix Multipliers with Carry-Save Adder

47. Radix-4 multiplication with a carry-save adder used to combine the cumulative partial product, x i a, and 2x i+1 a into two numbers

48. Radix-4 multiplier with a carry-save adder and Booth’s recoding

49. Booth recoding and multiple selection logic for high-radix multiplication

50. Radix-4 multiplier with two carry-save adders

51. Radix-16 multiplier with carry-save adders

52. Bit-Serial Multipliers

53. Bit Serial Multipliers Advantages small area reduced pin count reduced wire length high clock rate

54. Systolic Array Systolic array: synchronous arrays of processing elements that are interconnected by only short, local wires thus allowing very high clock rates

55. Semisystolic Bit-Serial Multiplier (1)

56. Semisystolic Bit-Serial Multiplier (2) a 3 x 0 a 2 x 0 a 1 x 0 a 0 x 0 a 3 x 1 a 2 x 1 a 1 x 1 a 0 x 1 a 3 x 2 a 2 x 2 a 1 x 2 a 0 x 2 a 3 x 3 a 2 x 3 a 1 x 3 a 0 x 3 a 3 0 a 2 0 a 1 0 a 0 0 p0p0 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7

57. Retiming d kk k+n k+n+d d k k+d k+d+n

58. Retimed Semisystolic Bit-Serial Multiplier (1)

59. Retimed Semisystolic Bit-Serial Multiplier (2) a 3 0 a 2 0 a 1 0 a 0 x 0 a 3 0 a 2 0 a 1 x 0 a 0 x 1 a 3 0 a 2 x 0 a 1 x 1 a 0 x 2 a 3 x 0 a 2 x 1 a 1 x 2 a 0 x 3 a 3 x 1 a 2 x 2 a 1 x 3 a 0 0 a 3 x 2 a 2 x 3 a 1 0 a 0 0 a 3 x 3 a 2 0 a 1 0 a 0 0 a 3 0 a 2 0 a 1 0 a 0 0 p0p0 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7

60. Systolic Bit-Serial Multiplier

61. Modular Multipliers

62. Modular Multiplication Special Cases a x pHpH pLpL a x = p = p H 2 k + p L k bits a x mod 2 k = p L a x mod 2 k -1 = p L + p H + carry p a x a x mod 2 k +1 = p L - p H - borrow

63. Modular Multiplication Special Case (1) a x mod 2 k -1 = (p H 2 k + p L ) mod (2 k -1) = = (p H (2 k mod (2 k -1)) + p L ) mod (2 k -1) = = p H + p L mod (2 k -1) = = p H + p L if p H + p L < 2 k - 1 p H + p L - (2 k -1) if p H + p L  2 k - 1 = p L + p H + carry carry = carry from addition p L + p H

64. Modular Multiplication Special Case (2) a x mod 2 k +1 = (p H 2 k + p L ) mod (2 k +1) = = (p H (2 k +1-1) + p L ) mod (2 k +1) = = p L - p H mod (2 k +1) = = p L - p H if p L - p H  0 p L - p H + (2 k +1) if p L - p H < 0 = p L - p H + borrow borrow = borrow from subtraction p L + p H

65. Modulo (2 b -1) Carry Save Adder

66. 4 x 4 Modulo 15 Multiplier

67. 4 x 4 Modulo 13 Multiplier