1. Sequential Multipliers
Lecture 8
Sequential
Multipliers

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

3. a Multiplicand ak-1ak-2 . . . a1 a0 x Multiplier xk-1xk-2 . . . x1 x0
Notation
a Multiplicand ak-1ak a1 a0
x Multiplier xk-1xk x1 x0
p Product (a  x) p2k-1p2k p2 p1 p0
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
k-1
x =  xi  2i
p = a  x
i=0
k-1
p = a  x =  a  xi  2i =
= x0a20 + x1a21 + x2a22 + … + xk-1a2k-1
i=0

6. Shift/Add Algorithm
Right-shift version

7. Right-shift algorithm
Shift/Add Algorithms
Right-shift algorithm
p = a  x = x0a20 + x1a21 + x2a22 + … + xk-1a2k-1 =
= (...((0 + x0a2k)/2 + x1a2k)/ xk-1a2k)/2 =
k times
p(0) = 0
p(j+1) = (p(j) + xj a 2k) / 2
j=0..k-1
p = p(k)

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. Right-shift algorithm: multiply-add
Shift/Add Algorithms
Right-shift algorithm: multiply-add
p(0) = y2k
p(j+1) = (p(j) + xj a 2k) / 2
j=0..k-1
p = p(k)
= (...((y2k + x0a2k)/2 + x1a2k)/ xk-1a2k)/2 =
k times
= y + x0a20 + x1a21 + x2a22 + … + xk-1a2k-1 = y + a  x

12. Signed Multiplication
Previous sequential multipliers are for unsigned multiplication
For signed multiplication:
assume sign-extended operation for p(j) + xja
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 xk-1 and subtract xk-1a 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. (positive multiplier)
Sequential
multiplication
of 2’s-complement
numbers
with right shifts
(positive multiplier)

14. (negative multiplier)
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 = x0a20 + x1a21 + x2a22 + … + xk-1a2k-1 =
= (...((02 + xk-1a)2 + xk-2a) x1a)2 + x0a=
k times
p(0) = 0
p(j+1) = (p(j) 2 + xk-1-ja)
j=0..k-1
p = p(k)

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. Left-shift algorithm: multiply-add
Shift/Add Algorithms
Left-shift algorithm: multiply-add
p(0) = y2-k
p(j+1) = (p(j) 2 + xk-(j+1)a)
j=0..k-1
p = p(k)
= (...((y2-k 2 + xk-1a)2 + xk-2a) x1a)2 + x0a =
k times
= y + xk-1a2k-1 + xk-2a2k-2 + … + x1a21 + x0a = 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 (an-1an a1 a0)r
x Multiplier (xn-1xn x1 x0)r
p Product (a  x) (p2n-1p2n p2 p1 p0)r

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

25. Basic Multiplication Equations
x =  xi  ri
p = a  x
i=0
n-1
p = a  x =  a  xi  ri =
= x0ar0 + x1a r1 + x2a r2 + … + xn-1a rn-1
i=0

26. High-Radix Shift/Add Algorithms Right-shift high-radix algorithm
p = a  x = x0ar0 + x1ar1 + x2ar2 + … + xn-1arn-1 =
= (...((0 + x0arn)/r + x1arn)/r xn-1arn)/r =
n times
p(0) = 0
p(j+1) = (p(j) + xj a rn) / r
j=0..n-1
p = p(n)

27. High-Radix Shift/Add Algorithms Left-shift high-radix algorithm
p = a  x = x0ar0 + x1ar1 + x2ar2 + … + xn-1arn-1 =
= (...((0r + xn-1a)r + xn-2a)r x1a)r + x0a=
n times
p(0) = 0
p(j+1) = (p(j)  r + xn-1-ja)
j=0..n-1
p = p(n)

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
j+1 j i

33. Radix-2 Booth Recoding
yi = -xi + xi-1

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

35. Y Multiplicand ym-1ym-2 . . . y1 y0 X Multiplier xm-1xm-2 . . . x1 x0
Notation
Y Multiplicand ym-1ym y1 y0
X Multiplier xm-1xm x1 x0
P Product (Y  X ) p2m-1p2m p2 p1 p0
If multiplicand and multiplier are of different sizes,
usually multiplier has the smaller size

36. Radix-4 Booth Recoding
(1)

37. zi/2 = -2xi+1 + xi + xi-1

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

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

40. High-Radix Multipliers
with Carry-Save Adder

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

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

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

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

45. Radix-16 multiplier with carry-save adders

46. Bit-Serial Multipliers

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

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

49. Semisystolic Bit-Serial Multiplier (1)

50. Semisystolic Bit-Serial Multiplier (2)
a3x a2x a1x a0x0
a3x a2x a1x a0x1 p0
a3x a2x a1x a0x2 p1
a3x a2x a1x a0x3 p2
a a a a0 0 p3
a a a a0 0 p4
a a a a0 0 p5
a a a a0 0 p6 p7

51. Retiming k k d
k+n+d
k+n
k+d d k
k+d+n
k+d+n

52. Bit-Serial Multiplier (1)
Retimed Semisystolic
Bit-Serial Multiplier (1)

53. Bit-Serial Multiplier (2)
Retimed Semisystolic
Bit-Serial Multiplier (2)
a a a a0x0 p0
a a a1x a0x1 p1
a a2x a1x a0x2 p2
a3x a2x a1x a0x3 p3
a3 x a2x a1x a0 0 p4
a3 x a2x a a0 0 p5
a3x a a a0 0 p6
a a a a0 0 p7

54. Systolic Bit-Serial Multiplier

55. Modular Multipliers

56. Modular Multiplication
Special Cases
k bits a a
a x
= p = pH 2k + pL x x pH pL p
a x mod 2k = pL
a x mod 2k-1 = pL + pH + carry
a x mod 2k+1 = pL - pH - borrow

57. Modular Multiplication
Special Case (1)
a x mod 2k-1 = (pH 2k + pL) mod (2k-1) =
= (pH (2k mod (2k-1)) + pL) mod (2k-1) =
= pH + pL mod (2k-1) = =
pH + pL if pH + pL < 2k - 1
pH + pL - (2k-1) if pH + pL  2k - 1
pL + pH + carry =
carry = carry from addition pL + pH

58. Modular Multiplication
Special Case (2)
a x mod 2k+1 = (pH 2k + pL) mod (2k+1) =
= (pH (2k+1-1) + pL) mod (2k+1) =
= pL - pH mod (2k+1) = =
pL - pH if pL - pH  0
pL - pH + (2k+1) if pL - pH < 0
pL - pH + borrow =
borrow = borrow from subtraction pL + pH

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

60. 4 x 4 Modulo 15 Multiplier

61. 4 x 4 Modulo 13 Multiplier