1. COMP541 Arithmetic Circuits
Montek Singh
Mar 26, 2012

2. Test #1: Take Home Will assign it Wednesday, 3/28
Give you five days to work on it (due 4/2)
Covers topics up to Lecture 15 (Memories II)

3. Today’s Topics Adder circuits How to subtract Overflow
Why complemented representation works out so well
Overflow

4. Iterative Circuit
Like a hierachy, except functional blocks per bit

5. 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

6. Half Adder
S = X  Y
C = XY

7. Full Adder
Three inputs. Third is Cin
Two outputs: sum and carry

8. Two Half Adders (and an OR)

9. Ripple-Carry Adder Straightforward – connect full adders
Carry-out to carry-in chain
C0 in case this is part of larger chain, or just ‘0’

10. Hierarchical 4-Bit Adder
We can easily use hierarchy here
Design half adder
Use in full adder
Use full adder in 4-bit adder

11. Behavioral Verilog // 4-bit Adder: Behavioral Verilog
module adder_4_b_v(A, B, C0, S, C4);
input[3:0] A, B;
input C0;
output[3:0] S;
output C4;
assign {C4, S} = A + B + C0;
endmodule
Addition (unsigned)
Concatenation operation

12. What’s the Problem with this Design?
Delay
Approx how much?
Imagine a 64-bit adder
Look at carry chain

13. Delays (Post Place and Route)
Odd delays caused by placement

14. 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)

15. Carry Lookahead Adder Note that add itself just 2 level
Idea is to separate carry from adder function
Then make carry approx 2-level all way across larger adder

16. 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.

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

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

19. 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

20. Ripple Carry Delay: 8 Gates

21. Turn Into Two Gate Delays
Design changed from deep (in delay) to wide

22. C1 Just Like Ripple Carry

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

24. 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

25. What Happens as Scale Up?
Can I realistically make 64-bit adder like this?
Have to AND 63 propagates and Cin!
Compromise
Hierarchical design
More levels of gates

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

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

28. Group Generate
Does G created upstream pass on because of string of Ps (also G3)?
Indicates carry generated in block

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

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

31. 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
We might cover later in semester, time permitting

32. On to Subtraction First, look at unsigned numbers
Motivates why we typically use complemented representation
Then look at 2’s complement
Imagine a subtractor circuit (next)

33. One-bit Subtractor Inputs: Borrow in, minuend and subtrahend
Review: subtrahend is subtracted from minuend
Outputs: Difference, borrow out
Could use like adder
One per bit
1-bit sub M S
Bout D
Bin

34. Example Borrow 1 Minuend Subtrahend Difference Correct Diff -
If no borrow, then result is non-negative (minuend >= subtrahend).
Borrow 1
Minuend
Subtrahend
Difference
Correct Diff -
Since there is borrow, result must be negative.
The magnitude must be corrected.
Next slide.

35. Correcting Result What, mathematically, does it mean to borrow?
If borrowing at digit i-1 you are adding 2i
Next Slide

36. Correcting Result 2
If M is minuend and N subtrahend of numbers length n, difference was
2n + M – N
What we want is magnitude of N-M (with minus sign in front)
Can get by subtracting previous result from 2n
N - M = 2n – (M – N + 2n)
This is called
2’s complement

37. Put Another Way
This is equivalent to how we do subtraction in our heads
Decide which is greater
Swap if necessary
Subtract
Could build a circuit this way…
Or just look at borrow bit

38. Algorithm Subtract N from M If no borrow, then M  N and result is OK
Otherwise, N > M so result must be subtracted from 2n (and minus sign prepended)

39. Pretty Expensive Hardware!

40. That Complex Design not Used
That’s why people use complemented interpretation for numbers
2’s complement
1’s complement

41. 1’s Complement Given: binary number N with n digits
1’s complement defined as
(2n – 1) - N
2n - 1 1
- N
1’s Compl.

42. 2’s Complement Given: binary number N with n digits
2’s complement defined as
2n – N for N  0
0 for N = 0
Exception is so result will always have n bits
2’s complement is just a 1 added to 1’s complement

43. Observations 1’s C and Signed Mag have two zeros
2’s C has more negative than positive
All negative numbers have 1 in high-order

44. Adder-Subtractor
Need only adder and complementer for input to subtract
Need selective complementer to make negative output back from 2’s complement
Or go through adder again. See next slide

45. Advantages/Disadvantages
Signed magnitude has problem that we need to correct after subtraction
One’s complement has a positive and negative zero
Two’s complement is most popular
Arithmetic operations easy

46. Conclusion: 2’s Complement
Addition easy on any combination of positive and negative numbers
To subtract
Take 2’s complement of subtrahend
Add
This performs A + ( -B), same as A – B

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

48. 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 can be OK

49. 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

50. 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

51. Summary Today Next class: adders and subtractors overflow
full processor datapath
Test #1 released