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Serial Multipliers Prawat Nagvajara

Serial Multipliers Prawat Nagvajara. Serial Bit-vector Multiplication Mapping Dependency Graph onto Signal Flow Graph (DG => SFG) A schedule that implies N-bit Adder Project Serial Multiplier Reading. Serial Bit-vector Multiplication. Two nested-loop Algorithm For I in 0 to n-1 loop

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Serial Multipliers Prawat Nagvajara

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  1. Serial MultipliersPrawat Nagvajara • Serial Bit-vector Multiplication • Mapping Dependency Graph onto Signal Flow Graph (DG => SFG) • A schedule that implies N-bit Adder • Project Serial Multiplier • Reading

  2. Serial Bit-vector Multiplication • Two nested-loop Algorithm For I in 0 to n-1 loop For J in 0 to n-1 loop … End loop; End loop; • Compute inner loop using combination N-bit adder and iterate Outer loop in time • It will take N clock cycles to complete • Array multiplier does not work with clock. It is a combinational circuit

  3. Dependency Graph equals Algorithm

  4. A Version of Serial Multiplier (a0,a1,a2,a3,a4) Partial Sum b0, b1, …, b4 Serially AND gates carry N-bit Adder P0, P1, …, P4 serially Register

  5. Mapping Dependency Graph Onto Signal Flow Graph (DG => SFG)

  6. DG => SFG • SFG dimension less than DG due to iteration in time • We often linear project DG to obtain SFG, e.g., a line to a point in the adder example • How do we compute the DG? • Hyper plane of computations done at each clock cycle • Schedule for the nodes. When and where they are computed

  7. Mapping Multiplication DG onto an SFG t = 0 t = 1 t = 2 t = 3 t = 4 carry b(t) D D D D D p(t)

  8. Processing Element X_j Y_i AND Full Adder C_out C_in DFF PS_in PS_out CK

  9. Another Version of Serial Multiplier x4 x3 x2 x1 x0 b0,…,b4 ‘0’ p0,p1,… Application Note: When t = 0, 1, 2, 3, 4 apply b0, b1, b2, b3, b4; When t = 5, 6, 7, 8, 9 apply ‘0’, to flush out p4, p5, …, p9

  10. A Pipeline Mapping Multiplication DG t = 0 t = 1 t = 2 carry b(t) D D D D D D D D D D p(t)

  11. Processing Element X_j DFF Y_i Y_out AND DFF C_out Full Adder C_in DFF PS_in PS_out

  12. A Pipelined Multiplier a2 a1 a0 0 0 0 0 0 0 0 0 0 • We will do an example of (111) * (111) = (110001)

  13. Snapshots at t= 0, 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0

  14. Snapshots at t= 2, 3 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 1 0

  15. Snapshots at t= 4, 5 1 1 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0

  16. Snapshots at t= 6, 7 1 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 1 0

  17. Snapshots at t= 8, 9 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0

  18. Snapshot at t= 10 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1

  19. The SchedulePartial Sum computed at t=6 is an output at t=8Carry computed at t=6 is an output at t=10 t=0 t=1 t=2 t=3 t=4 t=6 t=5

  20. Pipeline Structure • Temporal Parallelism • Schedule that infers delay at edges of the signal flow graph • Pipelining rate (bandwidth): The rate at which the data are piped into the array, e.g., the multiplier example has the rate ½ (every other clock cycle the multiplier bit is applied • Latency: The time it takes to complete an algorithm

  21. Is There Other Pipeline Multiplier?

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