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4x4 Signed Multiplication 2’s Complement: X= –x n–1 2 n–1 +Unsigned Y*X=Z

Multiplications. 4x4 Signed Multiplication 2’s Complement: X= –x n–1 2 n–1 +Unsigned Y*X=Z. 2’s C Sequential MUL 1-1. ADD Y if x i =1( i=0 to n–2); Shift SUB Y if x n-1 =1; Shift EX:Y= – 5, X=3. 2’s C Sequential MUL 1-2. ADD Y if x i =1( i=0 to n–2); Shift SUB Y if x n-1 =1; Shift.

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4x4 Signed Multiplication 2’s Complement: X= –x n–1 2 n–1 +Unsigned Y*X=Z

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  1. Multiplications • 4x4 Signed Multiplication • 2’s Complement: X= –xn–12n–1+Unsigned • Y*X=Z

  2. 2’s C Sequential MUL 1-1 • ADD Y if xi =1( i=0 to n–2); Shift • SUB Y if xn-1 =1; Shift • EX:Y= – 5, X=3

  3. 2’s C Sequential MUL 1-2 • ADD Y if xi =1( i=0 to n–2); Shift • SUB Y if xn-1 =1; Shift

  4. 1’s C Sequential MUL 1-1 • 1’s C: X= –xn–1(2n–1–ulp) +Unsigned • A(0)= Y(ulp) if xn–1 =1 else A(0)=0 • ADD Y if xi =1( i=0 to n–2); Shift • SUB Y if xn-1 =1; Shift • EX1: Y= 5, X= – 3

  5. 1’s C Sequential MUL 1-2 • 5* – 3 = – 22 ???

  6. 1’s C Sequential MUL 1-3 • 5* – 3 = – 15

  7. 1’s C Sequential MUL 1-4 • – 5* – 3 = 10 !!! What’s wrong???

  8. 1’s C Sequential MUL 1-5 • End around carry: – 5* – 3 = 12!!!???

  9. 1’s C Sequential MUL 1-6 • Add 1s instead of 0’s: – 5* – 3 = 15 O.K.

  10. Compare 2’sC & 1’sC Seq. MUL • 2’C : n-bit adder & 2n-bit Shift Reg. • 1’C : 2n-bit adder & 2n-bit Shift Reg.

  11. Multiplications(2) • Two SUB; (n-2) ADD • X*Y=Z=(z1, z2, …zn)

  12. Multiplications(3) • 4x4 Signed Multiplication • Sign extension • One SUB; (n-2) ADD

  13. Multiplications(4) • 4x4 Signed Multiplication • all ADD (2n)

  14. Multiplications(5) • 4x4 Signed Multiplication • all ADD (2n–1)

  15. Serial Multiplications (1)

  16. Serial Multiplications(2) • Serial Multiplication Scheme X0Y0

  17. Serial Multiplications(3) • Serial Multiplication Scheme Z0 X1Y1 W10 +W01 W11

  18. Serial Multiplications(4) • Serial Multiplication Scheme Z1 X2Y2 W21 +W12 V2 +W20 +W02 W22

  19. Serial Multiplications(5) • Serial Multiplication Scheme Z2 X3Y3 V3 +W30 +W03 V4 +W31 +W13 W32 +W23 W33

  20. Serial Multiplications(6) • Serial Multiplication Scheme Z3 V4 +W30 +W03 V5 +W31 +W13 V6 +W32 +W23 W33 +W33 +W33

  21. Serial Multiplications(7) • Serial Multiplication Scheme Z4 V5 +W30 +W03 V6 +W31 +W13 V7 +W32 +W23 +W33 +W33

  22. Serial Multiplications(7) • Serial Multiplication Scheme Z5 V6 +W30 +W03 V7 +W31 +W13 +W32 +W23 +W33 +W33

  23. Serial Multiplications(7) • Serial Multiplication Scheme Z6 V7 +W30 +W03

  24. Serial/Parallel Multiplier(1-1)

  25. Serial/Parallel Multiplier(1-2)

  26. Serial/Parallel Multiplier(2)

  27. Serial/Parallel Multiplier(3-1)

  28. Serial/Parallel Multiplier(3-2)

  29. Serial/Parallel Multiplier(4-1)

  30. Serial/Parallel Multiplier(4-2)

  31. Serial/Parallel Multiplier(4-3)

  32. Serial/Parallel Multiplier(5)

  33. High-Speed Multiplications • Reduce Partial Product terms • Accelerate Addition • 3 Types of Multiplication • Parallel MUL • HS Seq. MUL • Array MUL

  34. Reduce Partial Product terms • Booth Algorithm • Concept: A* 0011...110= A*0100…0(–1)0 • Ex. Old PP#=4, New PP# =2

  35. Booth Algorithm • Booth Algorithm • A*X A*Y • Conversion Table (right) • Start from LSB (add a 0) • Overlap 1 bit • EX: A*01110011  A*0 1110011 (0)  A* 100(–1)010(–1)

  36. Booth Algorithm(2) • 0100*0111  0100*100(–1)

  37. Booth Algorithm(3) • 1011*1101  1011*0(–1)1 (–1)

  38. Modified Booth Algorithm • Drawbacks • ADD/SUB Variable • Inefficient for isolate 1s • Modified Booth Alg. • Scan 3-bit at a time • Overlap 1-bit • If n= Even, it can handle 2’s C #

  39. Modified Booth Algorithm(2) • Original • 1011*1101  1011*0(–1)1 (–1) • Now • 1011*110 1(0)  1011*0(–1)01

  40. Canonical Recoding • Find min. +/– for MUL: •  Find min. SD representation •  Find min.  |yi| • Z=(z1, z2, …zn) is min. if • zizi+1 =0 • (111) is min. but zizi+1 0 • Canonical Recoding • Find such Z • Using sequence step

  41. Canonical Recoding(2) • EX:Assume C0=0 • X=011001  z0=1, c1=0 • X=01100  z1=0, c2=0 • X=0110  z2=0, c3=0 • X=011  z3= –1, c4=1 • X=01  z4=0, c5=1 • X=(0)0  z5=1, c6=0 • Z= 10(–1)001

  42. Array Multipliers • Conventional CSM

  43. Array Multipliers(2) • Conventional CSM

  44. Array Multipliers (3-0) • Pezaris Array Multiplier • – 1 = – 2•1+1 • 0 = – 2•0+ 0

  45. Array Multipliers(3) • Pezaris Array Multiplier

  46. Array Multipliers(4) • Modified Pezaris Array Multiplier

  47. Array Multipliers (5-01) • Baugh-Wooley Array Multiplier • –24x4 •yi2i (i=0 to 3)= –(0,0, x4y3,…, x4y0) 24 •  ADD (1, 1, y’3,…, y’0) 24 ADD 24 if x4=1 • or ADD 0 if x4=0 • [(1, x’4, x4y’3,…, x4y’0) ADD (0,1,0…, x4)]24 • So is –24y4 •xi2i (i=0 to 3) • {(1, x’4+y’4, [x4y’3+x’3y4],…, [ x4y’0+ y4x’0 + x4+y4]} 24

  48. Array Multipliers (5-02) • Baugh-Wooley Array Multiplier

  49. Array Multipliers(5) • Baugh-Wooley Array Multiplier

  50. Array Multipliers(6) • The On-the-fly CSM

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