1. 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

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;
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
AND gates
b0, b1, …, b4
Serially
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
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_outCK

9. Another Version of Serial Multiplierx4 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
carry
b(t) D D D D D D D D D D
p(t)

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

12. A Pipelined Multiplier
We will do an example of (111) * (111) = (110001)

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

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

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

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

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

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

19. The Schedule Partial Sum computed at t=6 is an output at t=8 Carry 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?