Lecture 8. Sequential Multipliers

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1 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 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 p = a x k-1 x = x i 2 i i=0 k-1 p = a x = a x i 2 i = i=0 = x 0 a2 0 + x 1 a2 1 + x 2 a x k-1 a2 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 a x k-1 a2 k-1 = = (...((0 + x 0 a2 k )/2 + x 1 a2 k )/ x k-1 a2 k )/2 = p (0) = 0 k times p (j+1) = (p (j) + x j a 2 k ) / 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 p (0) = y2 k Shift/Add Algorithms Right-shift algorithm: multiply-add p (j+1) = (p (j) + x j a 2 k ) / 2 j=0..k-1 p = p (k) = (...((y2 k + x 0 a2 k )/2 + x 1 a2 k )/ x k-1 a2 k )/2 = k times = y + x 0 a2 0 + x 1 a2 1 + x 2 a 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 à k+1 bit number Signed: k+1 bit sign extended number + k+1 bit sign extended number à 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 a x k-1 a2 k-1 = = (...((0 2 + x k-1 a) 2 + x k-2 a) x 1 a) 2 + x 0 a= p (0) = 0 k times p (j+1) = (p (j) 2 + x k-1-j a) 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 p (0) = y2 -k Shift/Add Algorithms Left-shift algorithm: multiply-add p (j+1) = (p (j) 2 + x k-(j+1) a) j=0..k-1 p = p (k) = (...((y2 -k 2 + x k-1 a) 2 + x k-2 a) x 1 a) 2 + x 0 a = k times = y + x k-1 a2 k-1 + x k-2 a2 k 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 x Multiplier p Product (a x) (a n-1 a n-2... a 1 a 0 ) r (x n-1 x n-2... x 1 x 0 ) r (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 p = a x n-1 x = x i r i i=0 n-1 p = a x = a x i r i = i=0 = x 0 ar 0 + x 1 a r 1 + x 2 a r x n-1 a r 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 x n-1 ar n-1 = = (...((0 + x 0 ar n )/r + x 1 ar n )/r x n-1 ar n )/r = p (0) = 0 n times p (j+1) = (p (j) + x j a r n ) / r j=0..n-1 p = p (n)

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 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= p (0) = 0 n times p (j+1) = (p (j) r + x n-1-j a) 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 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-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, x i a, and 2x i+1 a 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) 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 a 3 0 a 2 0 a 1 0 a 0 0 a 3 0 a 2 0 a 1 0 a 0 0 a 3 0 a 2 0 a 1 0 a 0 0 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7

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

52 Retimed Semisystolic Bit-Serial Multiplier (1)

53 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 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7

54 Systolic Bit-Serial Multiplier

55 Modular Multipliers

56 Modular Multiplication k bits Special Cases a x a x a x = p = p H 2 k + p L p H p L p a x mod 2 k = p L a x mod 2 k -1 = p L + p H + carry a x mod 2 k +1 = p L - p H - borrow

57 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 - (2k -1) if p H + p L 2 k - 1 = p L + p H + carry carry = carry from addition p L + p H

58 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 + (2k +1) if p L - p H < 0 = p L - p H + borrow borrow = borrow from subtraction p L + p H

59 Modulo (2 b -1) Carry Save Adder

60 4 x 4 Modulo 15 Multiplier

61 4 x 4 Modulo 13 Multiplier

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