1. S. Reda EN160 SP’07 Design and Implementation of VLSI Systems (EN0160) Lecture 28: Datapath Subsystems 2/3 Prof. Sherief Reda Division of Engineering, Brown University Spring 2007 [sources: Weste/Addison Wesley – Rabaey/Pearson]

2. S. Reda EN160 SP’07 Outline Last lecture –Carry-ripple adder This lecture –More carry-ripple adders –Manchester carry chain adder –Carry-Skip adder –Carry-lookahead adder –Carry-select adder

3. S. Reda EN160 SP’07 Last lecture we designed a carry-ripple adder For a full adder, define what happens to carry –Generate: C out = 1 independent of C G = A B –Propagate: C out = C P = A  B –Kill: C out = 0 independent of C K = ~A ~B PG summary

4. S. Reda EN160 SP’07 Group carry calculations i j k k-1 The carry into bit i is the carry-out of bit i-1 The sum is equal to

5. S. Reda EN160 SP’07 Group generate i j k k-1

6. S. Reda EN160 SP’07 Carry-ripple adder revisited

7. S. Reda EN160 SP’07 Carry-ripple adder revisited The critical path now proceeds through a chain of AND-OR gates rather than a chain of majority gates

8. S. Reda EN160 SP’07 8-bit adder/subtractor 1-bit FA S0S0 C 0 =C in C1C1 1-bit FA S1S1 C2C2 S2S2 C3C3 C 8 =C out 1-bit FA S7S7 C7C7... A0A0 B0B0 A1A1 B1B1 A2A2 B2B2 A7A7 B7B7 add/subt  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)

9. S. Reda EN160 SP’07 Manchester carry adder Using transmission gates Using dynamic gates

10. S. Reda EN160 SP’07 Manchester carry chains Critical path involves a series propagate transistor for each bit  a significant over carry-ripple (which used majority or AND-OR gate)

11. S. Reda EN160 SP’07 Equivalence circuits for Manchester carry chain Do you remember this circuit?

12. S. Reda EN160 SP’07 Carry skip adder If (P 0 & P 1 & P 2 & P 3 = 1) then C o,3 = C i,0 otherwise the block itself kills or generates the carry internally A0A0 B0B0 S0S0 C i,0 FA A1A1 B1B1 S1S1 A2A2 B2B2 S2S2 A3A3 B3B3 S3S3 C o,3 BP = P 0 P 1 P 2 P 3 “Block Propagate”

13. S. Reda EN160 SP’07 Carry-skip adder Carry-ripple is slow through all N stages Carry-skip allows carry to skip over groups of n bits –Decision based on n-bit propagate signal Original design by Charles Babbage

14. S. Reda EN160 SP’07 Carry-lookahead adder Similar to the carry-skip adder, but computes generate signals as well as group propagate signals to avoid waiting for a ripple to determine if the group generates a carry.

15. S. Reda EN160 SP’07 Carry-select adder One adder calculates the sums assuming a carry-n of 0 while the other calculates the sums assuming a carry-in of 1. The actual carry triggers a multiplexer that chooses the appropriate sum

16. S. Reda EN160 SP’07 Next time Comparators/shifters/multipliers