COMP541 Arithmetic Circuits

1. COMP541Arithmetic Circuits Montek Singh (Not covered)

2. Today’s Topics • Adder circuits • ripple-carry adder (revisited) • more advanced: carry-lookahead adder • Subtraction • by adding the negative • Overflow

3. Iterative Circuit • Like a hierarchy, except functional blocks per bit

4. Adders • Great example of this type of design • Design 1-bit circuit, then expand • Let’s look at • Half adder – 2-bit adder, no carry in • Inputs are bits to be added • Outputs: result and possible carry • Full adder – includes carry in, really a 3-bit adder • We have already studied adder in Lab 2/Comp411 • here we look at it from a different angle • modify it to be faster

5. Half Adder • Produces carry out • does not handle carry in

6. Full Adder • Three inputs • third is carry in • Two outputs • sum and carry out

7. Two Half Adders (and an OR)

8. Ripple-Carry Adder • Straightforward – connect full adders • Carry-out to carry-in chain • Cinin case this is part of larger chain, or just ‘0’ 32-bit ripple-carry adder

9. Lab 2: Hierarchical 4-Bit Adder • We used hierarchy in Lab 2 • Design full adder • Used 4 of them to make a 4-bit adder • Used two 4-bit adders to make an 8-bit adder • …

10. Specifying Addition in Behavioral Verilog // 4-bit Adder: Behavioral Verilog module adder_4_behav(A, B, C0, S, C4); input wire[3:0] A, B; input wire C0; output logic[3:0] S; output logic C4; assign {C4, S} = A + B + C0; endmodule Addition (unsigned) Concatenation operation

11. What’s the problem with this design? • Delay • Approx how much? • Imagine a 64-bit adder • Look at carry chain

12. Delays (post-synthesis simulation) • Delays are generally higher for more significant bits

13. Multibit Adders • Several types of carry propagate adders (CPAs) are: • Ripple-carry adders (slow) • Carry-lookahead adders (fast) • Prefix adders (faster) • Carry-lookahead and prefix adders are faster for large adders but require more hardware. • Adder symbol (right)

14. Carry Lookahead Adder • Note that add itself just 2 level • sum is produced with a delay of only two XOR gates • carry takes three gates, though • Idea is to separate carry from adder function • then make carry faster • actually, we will make carry have a 2-gate delay total, for all the bits of the adder! • these two gates might be huge though

15. Four-bit Ripple Carry Reference Adder function separated from carry Notice adder has A, B, C in and S out, as well as G,P out.

16. Propagate • The P signal is called propagate • P = A  B • Means to propagate incoming carry

17. Generate • The G is generate • G = AB, so new carry created • So it’s ORed with incoming carry

18. Said Differently • If A  B and there’s incoming carry, carry will be propagated • And S will be 0, of course • If AB, then will create carry • Incoming will determine whether S is 0 or 1

19. Ripple Carry Delay: 8 Gates • Key observation: • G and P are produced by each adder stage • without needing carry from the right! • need only 2 gate delays for all G’s and P’s to be generated! • critical path is the carry logic at the bottom • the G’s and P’s are “off the critical path”

20. Turn Into Two Gate Delays • Refactor the logic • changed from deep (in delay) to wide • for each stage, gather and squish together all the logic to the right

21. C1 Just Like Ripple Carry

22. C2 Circuit Two Levels G from before and P to pass on This checks two propagates and a carry in

23. C3 Circuit Two Levels Generate from level 0 and two propagates G from before and P to pass on This checks three propagates and a carry in

24. What happens as scaled up? • Can I realistically make 64-bit adder like this? • Have to AND 63 propagates and Cin! • Compromise • Hierarchical design • More levels of gates • use a tree of AND’s • delay grows only logarithmically

25. Making 4-Bit Adder Module • Create propagate and generate signals for whole module

26. Group Propagate • Make propagate of whole 4-bit block • P0-3 = P3P2P1P0

27. Group Generate • Indicates carry generated within this block

28. Hierarchical Carry A B A B 4-bit adder 4-bit adder S G P Cin S G P Cin C4 C8 Look Ahead C0 Left lookahead block is exercise for you

29. Practical Matters • FPGAs like ours have limited inputs per gate • Instead they have special circuits to make adders • So don’t expect to see same results as theory would suggest

30. Other Adder Circuits • What if hierarchical lookahead too slow • Other styles exist • Prefix adder (explained in text) had a tree to computer generate and propagate • Pipelined arithmetic units – multicycle but enable faster clock speed • These are for self-study

31. Adder-Subtractor • Need only adder and complementer for input to subtract • Need selective complementer to make negative output back from 2’s complement

32. Design of Adder/Subtractor Output is 2’s complement if B > A S low for add, high for subtract Inverts each bit of B if S is 1 Adds 1 to make 2’s complement

33. Overflow • Two cases of overflow for addition of signed numbers • Two large positive numbers overflow into sign bit • Not enough room for result • Two large negative numbers added • Same – not enough bits • Carry out by itself doesn’t indicate overflow

34. Overflow Examples 4-bit signed numbers: • Sometimes a leftmost carry is generated without overflow: • -7 + 7 • 5 + (-3) • Sometimes a leftmost carry is not generated, but overflow occurs: • 4 + 4

35. Overflow Detection • Basic condition: • if two +ve numbers are added and sum is –ve • if two -ve numbers are added and sum is +ve • Can be simplified to the following check: • either Cn-1 or Cn is high, but not both

36. Summary • Today • adders and subtractors • overflow