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Multipliers in VLSI

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Kiranmai Sony
Kiranmai Sony

its a presentation on multipliers..which shows various types and its usage

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Sp09 CMPEN 411 L20 S.1 CMPEN 411 VLSI Digital Circuits Spring 2009 Lecture 20: Multiplier Design [Adapted from Rabaey’s Digital Integrated Circuits, Second Edition, ©2003 J. Rabaey, A. Chandrakasan, B. Nikolic]
Sp09 CMPEN 411 L20 S.2 Review: Basic Building Blocks  Datapath Execution units - Adder, multiplier, divider, shifter, etc. Register file and pipeline registers Multiplexers, decoders  Control Finite state machines (PLA, ROM, random logic)  Interconnect Switches, arbiters, buses  Memory Caches (SRAMs), TLBs, DRAMs, buffers
Sp09 CMPEN 411 L20 S.3 The Binary Multiplication x + Partial products Multiplicand Multiplier Result 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1
Sp09 CMPEN 411 L20 S.4 Multiply Operation  Multiplication is just a a lot of additions multiplicand multiplier partial product array double precision product N 2N N can be formed in parallel
Sp09 CMPEN 411 L20 S.5 Multiplication Approaches  Right shift and add Partial product array rows are accumulated from top to bottom on an N-bit adder - After each addition, right shift (by one bit) the accumulated partial product to align it with the next row to add Time for N bits Tserial_mult = O(NTadder) = O(N2 ) for a RCA  Making it faster Use a faster adder Use higher radix (e.g., base 4) multiplication – O(N/2 Tadder) - Use multiplier recoding to simplify multiple formation (booth) Form the partial product array in parallel and add it in parallel  Making it smaller (i.e., slower) Use serial-parallel mult Use an array multiplier - Very regular structure with only short wires to nearest neighbor cells. Thus, very simple and efficient layout in VLSI Can be easily and efficiently pipelined
Sp09 CMPEN 411 L20 S.6 Serial-parallel multiplier structure
Sp09 CMPEN 411 L20 S.7 The Array Multiplier Y0 Y1 X3 X2 X1 X0 X3 HA X2 FA X1 FA X0 HA Y2X3 FA X2 FA X1 FA X0 HA Z1 Z3Z6Z7 Z5 Z4 Y3X3 FA X2 FA X1 FA X0 HA
Sp09 CMPEN 411 L20 S.11 Booth multiplier  Encoding scheme to reduce number of stages in multiplication.  Performs two bits of multiplication at once—requires half the stages.  Each stage is slightly more complex than simple multiplier, but adder/subtracter is almost as small/fast as adder.
Sp09 CMPEN 411 L20 S.12 Booth encoding  Two’s-complement form of multiplier: y = -2n yn + 2n-1 yn-1 + 2n-2 yn-2 + ... (first bit is the sign bit) (example, y=18=010010 y= -18 = 101110 )  Rewrite using 2a = 2a+1 - 2a : y = 2n (yn-1-yn) + 2n-1 (yn-2 -yn-1) + 2n-2 (yn-3 -yn-2) + ...  Consider first two terms: by looking at three bits of y, we can determine whether to add x, 2x to partial product.
Sp09 CMPEN 411 L20 S.13 Booth actions yi yi-1 yi-2 increment 0 0 0 0 0 0 1 x 0 1 0 x 0 1 1 2x 1 0 0 -2x 1 0 1 -x 1 1 0 -x 1 1 1 0 y = 2n (yn-1-yn) + 2n-1 (yn-2 -yn-1) + 2n-2 (yn-3 -yn-2) + ...  Consider first two terms: by looking at three bits of y, we can determine whether to add x, 2x to partial product.
Sp09 CMPEN 411 L20 S.14 Booth example  x = 1001 (910), y = 0111 (710).  P0 = 00000000  y3y2y1=011 y1y0y-1=11(0)  y1y0y-1 = 110, P1 = P0 - (1001) = 11110111  x shift left for 2 bits to be 100100  y3y2y1 = 011, P2 = P1+ (10*100100) = 11110111+01001000 = 001111111 (6310)  An array multiplier needs N addtions, booth multiplier needs only N/2 additions
Sp09 CMPEN 411 L20 S.15 Review: A 64-bit Adder/Subtractor 1-bit FA S0 C0=Cin C1 1-bit FA S1 C2 1-bit FA S2 C3 C64=Cout 1-bit FA S63 C63 ...  Ripple Carry Adder (RCA) built out of 64 FAs  Subtraction – complement all subtrahend bits (xor gates) and set the low order carry-in  RCA  advantage: simple logic, so small (low cost)  disadvantage: slow (O(N) for N bits) and lots of glitching (so lots of energy consumption) A0 B0 A1 B1 A2 B2 A63 B63 add/subt
Sp09 CMPEN 411 L20 S.16 Booth structure
Sp09 CMPEN 411 L20 S.17 Wallace-Tree Multiplier 6 5 4 3 2 1 0 6 5 4 3 2 1 0 Partial products First stage Bit position 6 5 4 3 2 1 0 6 5 4 3 2 1 0 Second stage Final adder FA HA (a) (b) (c) (d)
Sp09 CMPEN 411 L20 S.18 Wallace-Tree Multiplier Partial products First stage Second stage Final adder FA FA FA HA HA FA x3y3 z7 z6 z5 z4 z3 z2 z1 z0 x3y2 x2y3 x1y1x3y0 x2y0 x0y1 x0y2 x2y2 x1y3 x1y2x3y1 x0y3 x1y0 x0y Full adder = (3,2) compressor
Sp09 CMPEN 411 L20 S.19 Making it Faster: Tree Multiplier Structure partial product array reduction tree fast carry propagate adder (CPA) P (product) mux + reduction tree (log N) + CPA (log N) Q (‘ier) D (‘icand) D D D 0 0 0 0 multiple forming circuits interconnect
Sp09 CMPEN 411 L20 S.20 (4,2) Counter  Built out of two (3,2) counters (just FA’s!) all of the inputs (4 external plus one internal) have the same weight (i.e., are in the same bit position) the internal carry output is fed to the next higher weight position (indicated by the ) (3,2) (3,2) Note: Two carry outs - one “internal” and one “external”
Sp09 CMPEN 411 L20 S.22 Tiling (4,2) Counters  Reduces columns four high to columns only two high Tiles with neighboring (4,2) counters Internal carry in at same “level” (i.e., bit position weight) as the internal carry out (3,2) (3,2) (3,2) (3,2) (3,2) (3,2)
Sp09 CMPEN 411 L20 S.24 4x4 Partial Product Array Reduction multiplicand multiplier partial product array reduced pp array (to CPA) double precision product  Fast 4x4 multiplication using (4,2) counters  How would you lay it out? five (4,2) counters 5-bit CPA multiplicand multiplier 8-bit product
Sp09 CMPEN 411 L20 S.25 8x8 Partial Product Array Reduction ‘icand ‘ier partial product array reduced partial product array  Wallace tree multiplier two rows of nine (4,2) counters one row of thirteen (4,2) counters to a 13-bit fast CPA
Sp09 CMPEN 411 L20 S.26 An 8x8 Multiplier Layout  How should it be laid out? multiplicand multiplier thirteen (4,2) counters nine (4,2) counters nine (4,2) counters 13-bit CPA
Sp09 CMPEN 411 L20 S.32 Multipliers —Summary • Optimization Goals Different Vs Binary Adder • Once Again: Identify Critical Path • Other possible techniques - Data encoding (Booth) - Pipelining FIRST GLIMPSE AT SYSTEM LEVEL OPTIMIZATION - Logarithmic versus Linear (Wallace Tree Mult)
Sp09 CMPEN 411 L20 S.33 Next Lecture and Reminders  Next lecture Shifters, decoders, and multiplexers - Reading assignment – Rabaey, et al, 11.5-11.6

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Multipliers in VLSI

  • 1. Sp09 CMPEN 411 L20 S.1 CMPEN 411 VLSI Digital Circuits Spring 2009 Lecture 20: Multiplier Design [Adapted from Rabaey’s Digital Integrated Circuits, Second Edition, ©2003 J. Rabaey, A. Chandrakasan, B. Nikolic]
  • 2. Sp09 CMPEN 411 L20 S.2 Review: Basic Building Blocks  Datapath Execution units - Adder, multiplier, divider, shifter, etc. Register file and pipeline registers Multiplexers, decoders  Control Finite state machines (PLA, ROM, random logic)  Interconnect Switches, arbiters, buses  Memory Caches (SRAMs), TLBs, DRAMs, buffers
  • 3. Sp09 CMPEN 411 L20 S.3 The Binary Multiplication x + Partial products Multiplicand Multiplier Result 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 1
  • 4. Sp09 CMPEN 411 L20 S.4 Multiply Operation  Multiplication is just a a lot of additions multiplicand multiplier partial product array double precision product N 2N N can be formed in parallel
  • 5. Sp09 CMPEN 411 L20 S.5 Multiplication Approaches  Right shift and add Partial product array rows are accumulated from top to bottom on an N-bit adder - After each addition, right shift (by one bit) the accumulated partial product to align it with the next row to add Time for N bits Tserial_mult = O(NTadder) = O(N2 ) for a RCA  Making it faster Use a faster adder Use higher radix (e.g., base 4) multiplication – O(N/2 Tadder) - Use multiplier recoding to simplify multiple formation (booth) Form the partial product array in parallel and add it in parallel  Making it smaller (i.e., slower) Use serial-parallel mult Use an array multiplier - Very regular structure with only short wires to nearest neighbor cells. Thus, very simple and efficient layout in VLSI Can be easily and efficiently pipelined
  • 6. Sp09 CMPEN 411 L20 S.6 Serial-parallel multiplier structure
  • 7. Sp09 CMPEN 411 L20 S.7 The Array Multiplier Y0 Y1 X3 X2 X1 X0 X3 HA X2 FA X1 FA X0 HA Y2X3 FA X2 FA X1 FA X0 HA Z1 Z3Z6Z7 Z5 Z4 Y3X3 FA X2 FA X1 FA X0 HA
  • 8. Sp09 CMPEN 411 L20 S.11 Booth multiplier  Encoding scheme to reduce number of stages in multiplication.  Performs two bits of multiplication at once—requires half the stages.  Each stage is slightly more complex than simple multiplier, but adder/subtracter is almost as small/fast as adder.
  • 9. Sp09 CMPEN 411 L20 S.12 Booth encoding  Two’s-complement form of multiplier: y = -2n yn + 2n-1 yn-1 + 2n-2 yn-2 + ... (first bit is the sign bit) (example, y=18=010010 y= -18 = 101110 )  Rewrite using 2a = 2a+1 - 2a : y = 2n (yn-1-yn) + 2n-1 (yn-2 -yn-1) + 2n-2 (yn-3 -yn-2) + ...  Consider first two terms: by looking at three bits of y, we can determine whether to add x, 2x to partial product.
  • 10. Sp09 CMPEN 411 L20 S.13 Booth actions yi yi-1 yi-2 increment 0 0 0 0 0 0 1 x 0 1 0 x 0 1 1 2x 1 0 0 -2x 1 0 1 -x 1 1 0 -x 1 1 1 0 y = 2n (yn-1-yn) + 2n-1 (yn-2 -yn-1) + 2n-2 (yn-3 -yn-2) + ...  Consider first two terms: by looking at three bits of y, we can determine whether to add x, 2x to partial product.
  • 11. Sp09 CMPEN 411 L20 S.14 Booth example  x = 1001 (910), y = 0111 (710).  P0 = 00000000  y3y2y1=011 y1y0y-1=11(0)  y1y0y-1 = 110, P1 = P0 - (1001) = 11110111  x shift left for 2 bits to be 100100  y3y2y1 = 011, P2 = P1+ (10*100100) = 11110111+01001000 = 001111111 (6310)  An array multiplier needs N addtions, booth multiplier needs only N/2 additions
  • 12. Sp09 CMPEN 411 L20 S.15 Review: A 64-bit Adder/Subtractor 1-bit FA S0 C0=Cin C1 1-bit FA S1 C2 1-bit FA S2 C3 C64=Cout 1-bit FA S63 C63 ...  Ripple Carry Adder (RCA) built out of 64 FAs  Subtraction – complement all subtrahend bits (xor gates) and set the low order carry-in  RCA  advantage: simple logic, so small (low cost)  disadvantage: slow (O(N) for N bits) and lots of glitching (so lots of energy consumption) A0 B0 A1 B1 A2 B2 A63 B63 add/subt
  • 13. Sp09 CMPEN 411 L20 S.16 Booth structure
  • 14. Sp09 CMPEN 411 L20 S.17 Wallace-Tree Multiplier 6 5 4 3 2 1 0 6 5 4 3 2 1 0 Partial products First stage Bit position 6 5 4 3 2 1 0 6 5 4 3 2 1 0 Second stage Final adder FA HA (a) (b) (c) (d)
  • 15. Sp09 CMPEN 411 L20 S.18 Wallace-Tree Multiplier Partial products First stage Second stage Final adder FA FA FA HA HA FA x3y3 z7 z6 z5 z4 z3 z2 z1 z0 x3y2 x2y3 x1y1x3y0 x2y0 x0y1 x0y2 x2y2 x1y3 x1y2x3y1 x0y3 x1y0 x0y Full adder = (3,2) compressor
  • 16. Sp09 CMPEN 411 L20 S.19 Making it Faster: Tree Multiplier Structure partial product array reduction tree fast carry propagate adder (CPA) P (product) mux + reduction tree (log N) + CPA (log N) Q (‘ier) D (‘icand) D D D 0 0 0 0 multiple forming circuits interconnect
  • 17. Sp09 CMPEN 411 L20 S.20 (4,2) Counter  Built out of two (3,2) counters (just FA’s!) all of the inputs (4 external plus one internal) have the same weight (i.e., are in the same bit position) the internal carry output is fed to the next higher weight position (indicated by the ) (3,2) (3,2) Note: Two carry outs - one “internal” and one “external”
  • 18. Sp09 CMPEN 411 L20 S.22 Tiling (4,2) Counters  Reduces columns four high to columns only two high Tiles with neighboring (4,2) counters Internal carry in at same “level” (i.e., bit position weight) as the internal carry out (3,2) (3,2) (3,2) (3,2) (3,2) (3,2)
  • 19. Sp09 CMPEN 411 L20 S.24 4x4 Partial Product Array Reduction multiplicand multiplier partial product array reduced pp array (to CPA) double precision product  Fast 4x4 multiplication using (4,2) counters  How would you lay it out? five (4,2) counters 5-bit CPA multiplicand multiplier 8-bit product
  • 20. Sp09 CMPEN 411 L20 S.25 8x8 Partial Product Array Reduction ‘icand ‘ier partial product array reduced partial product array  Wallace tree multiplier two rows of nine (4,2) counters one row of thirteen (4,2) counters to a 13-bit fast CPA
  • 21. Sp09 CMPEN 411 L20 S.26 An 8x8 Multiplier Layout  How should it be laid out? multiplicand multiplier thirteen (4,2) counters nine (4,2) counters nine (4,2) counters 13-bit CPA
  • 22. Sp09 CMPEN 411 L20 S.32 Multipliers —Summary • Optimization Goals Different Vs Binary Adder • Once Again: Identify Critical Path • Other possible techniques - Data encoding (Booth) - Pipelining FIRST GLIMPSE AT SYSTEM LEVEL OPTIMIZATION - Logarithmic versus Linear (Wallace Tree Mult)
  • 23. Sp09 CMPEN 411 L20 S.33 Next Lecture and Reminders  Next lecture Shifters, decoders, and multiplexers - Reading assignment – Rabaey, et al, 11.5-11.6

Hinweis der Redaktion

  1. Right shift approach (almost) always used because left shift requires 2n bit adder
  2. One simple,small way to implement is the serial-parallel multiplier. So called because the n-bit multiplier is fed in serially while the m bit multiplicand is held in parallel.
  3. See Wayne Wolf book the
  4. a balanced delay tree 2 csa delays total
  5. For class handout
  6. For lecture a balanced delay tree 2 csa delays total
  7. For class handout
  8. For lecture
  9. Completely populate (costs more in terms of (4,2) counters) – advantage is the CPA doesn’t have to be as wide, so the multiplier faster, and the reduction tree is more “regular”