Optimization of Spanning Tree Adders

1. Optimization of Spanning Tree Adders 報告者 曾俊雄  賴韋志  巫祈賢

2. Overview • Compare designs of spanning tree adders in terms of • delay • complexity • power consumption • Spanning tree adders is an existing category of adders, this paper does not focus on inventing new spanning tree adders but performing experiments on several possible configuration of spanning tree adders.

3. Introduction to several adders • Why spanning tree adders are needed? • Experimental result • Conclusion & future work

4. Agenda • Introduction to Adders & Problem Statement • Spanning Tree Adders • Result & Conclusion

5. Introduction to Adders & Problem Statement • Spanning Tree Adders • Result & Conclusion

6. Introduction to Adders & Problem Statement source: http://en.wikipedia.org/wiki/Adder_(electronics)

7. Very brief introduction on several existing adders: • ripple carry adders • carry skip adders • carry look ahead adders • tree structured adders • Brent-Kung • Spanning Tree • Ling • Han-Carlson • Kogge-Stone

8. Very brief introduction on several existing adders: • ripple carry adders • carry skip adders • carry look ahead adders • tree structured adders • Brent-Kung • Spanning Tree • Ling • Han-Carlson • Kogge-Stone

9. Propagation Delay • Basically, the performance of adders are evaluated with propagation delay (in this paper)

10. Propagation Delay: also called gate delay, the time difference between the change of input and output signals. source: http://www.seas.upenn.edu/~ese201/lab/CarryLookAhead/CarryLookAheadF01.html

11. Ripple Carry Adders • Goal: the adder should be able to compute the sum of more than 1 bits • Idea: just chain several adders together

12. source: http://en.wikipedia.org/wiki/Adder_(electronics)

13. Pros & Cons • Pros: • straight forward, easy to implement • low power consumption, low resource source: D.J.Kinniment, J.D.Garside, B.Gao, A comparison of Power Consumption in Some CMOS Adder Circuits. In: Proceedings of the Fifth International Workshop PATMOS-‘95, Oldenburg, Germany, ISBN 3-8142- 0526-X, pp. 106-118.

14. Cons: • slow (high propagation delay) • a 32-bits adder requires 31 carry computations source: http://en.wikipedia.org/wiki/Adder_(electronics)

15. Carry look ahead adders • proposed by Weinberger and Smith in 1958 source: http://en.wikipedia.org/wiki/Carry_look-ahead_adder

16. carry look ahead adders work by creating “Propagate” and “Generate” signals for each bit • Propagate: controls whether a carry is propagated from lower bits to higher bits • Generate: controls whether a carry is generated

17. a 4-bits carry look ahead adder source: http://www.seas.upenn.edu/~ese201/lab/CarryLookAhead/CarryLookAheadF01.html

18. Pros & Cons • Pros: • faster than ripple-carry adders, since some computation can be done in advance for example, part of C4 can be pre-computed after C0, P0, G0 are known

19. Cons: • much more complicated when > 4 bits (http://www.seas.upenn.edu/~ese201/lab/CarryLookAhead/CarryLookAheadF01.html) • the carry look ahead block requires gates with more than 2 input, which may cause efficiency problems (according to Megha Ladha; Earl E. Swartzlander, Jr )

20. Alternatives • Alternatives: • improved carry look ahead adders: Ling adders • tree-style adders to decrease input numbers for each level: • Kogge-Stone Adder • Brent-Kung Adder • Han-Carlson Adder • Spanning Tree Adder

21. Kogge-Stone Adder Brent-Kung Adder source: http://www.acsel-lab.com/Projects/fast_adder/adder_designs.htm

22. Introduction to Adders & Problem Statement • Spanning Tree Adders • Result & Conclusion

23. Spanning Tree Adder (STA) • Spanning tree adder • use minimum number of multi-input gates • Hybrid adder • Manchester Carry Chain (MCC) • use ripple carries • Propagate rapidly • Size of MCC is a trade of • # of inputs • # of levels MCC

24. 32-bit STA

25. STA based on MCC • Gi:j=MCC2(Gi:k-1, Gk:j, Pi:k-1, Pk:j) • Gi:j=MCC3(Gi:k-1, Gk:m-1, Gm:j, Pi:k-1, Pk:m-1, Pk:j) • Ci=G0:i-1

26. Carry Select Adder (CSA) • Two sum outputs: • Assumed carry input as 0 • Assumed carry input as 1 • # of MCC level is decided by the size of final stage CSA • m-bits wide CSA • Require Cm, C2m, C3m… carry select adder

27. Final stage adder: if m is large  carry select adder takes more time if m is mall  need more MCC levels 8-bits wide carry select adder 4 sum outputs

28. Introduction to Adders & Problem Statement • Spanning Tree Adders • Result & Conclusion

29. Main Comparion • 16-bit vs. 32-bit adders • CLA vs. Spanning tree adders • 4-bit, 8-bit, 16-bit RCA vs. CSA • Measure: Delay, complexity, power consumption • Delay and power consumption are performed with Synopsys and Cadence

30. 16-bit vs. 32-bit Adders 16-bit spanning tree 32-bit spanning tree Size of CSA ↑: Delay of MCC ↓, CSA ↑

31. 4-bit, 8-bit, 16-bit RCA vs. CSA - Delay 16 and 32-bit spanning tree with CSA and RCA in final stage Delay: Except 4-bit module, CSA is better

32. 4-bit, 8-bit, 16-bit RCA vs. CSA - Complexity 16 and 32-bit spanning tree with CSA and RCA in final stage Complexity: Almost the same

33. 4-bit, 8-bit, 16-bit RCA vs. CSA - Power Consumption 16 and 32-bit spanning tree with CSA and RCA in final stage Size of Final Adders ↑: Power Consumption ↓

34. CLA vs. Spanning tree adders - Delay 16 and 32-bit CLA and Optimum Spanning Tree(OST) Ratio of Delay: 32bit decreases more

35. CLA vs. Spanning tree adders - Complexity 16 and 32-bit CLA and Optimum Spanning Tree(OST) Ratio of Complexity: 32bit increases less

36. CLA vs. Spanning tree adders - Power Consumption 16 and 32-bit CLA and Optimum Spanning Tree(OST) Ratio of Power Consumption: Almost the same

37. Conclusion • Performance: • In 16-bit Spanning tree adder, 4-bit, RCA and CSA are the same • In 32-bit Spanning tree adder, 8-bit above, RCA is better • Spanning tree is more ideal for larger size.

38. Future Work • Use transistor to design final adders and MCC blocks • Sized for optimum performance • Use transistor multiplexer instead of gate level multiplexer

39. Some Others – IC Design • Language level: Like C language • Gate level: Assembly language • Transistor level: Machine code