1. Kuliah Rangkaian Digital Kuliah 7: Unit Aritmatika
Teknik Komputer
Universitas Gunadarma

2. Topic #7 – Arithmetic Units
Comparator
Adders
Half-adder & Full-adder
Carry-ripple adder & carry-look-ahead adder
Overflow detection
Subtractor
Multiplier

3. (Equality) Comparators (using XOR)
1-bit comparator
4-bit comparator

4. Half adder Adds two 1-bit input to produce a sum and a carry-out
Does not account for carry-in
S = X’·Y + X·Y
= XY
Cout = X·Y
Inputs
Outputs Y 1 S X
Cout X Y S
Cout

5. Full adder Building block to realize binary arithmetic operations
1-bit-wide adder with carry-in, produces sum and carry-out
Truth table:
X Y Cin S Cout

6. Designing full adder 1 00 01 11 10 Cout S 1 00 01 11 10
Cin X 1 Y XY 2 3 6 7 4 5
Cout S
Cin X 1 Y XY 2 3 6 7 4 5
S = X’·Y’·(Cin) + X·Y’·(Cin)’
+ X·Y’·(Cin)’ + X·Y·(Cin)
= X Å Y Å (Cin)
Cout = XY + X(Cin) + Y(Cin)

7. Resulting circuit

8. Alternative: full adder using AND-OR
X’Y’C-in
XY’C-in’
Sum S
X’YC-in’
XYC-in X’ X Y’ Y
C-in
C-in’ X X’ Y Y’
C-in
C-in’
Full
Adder X Y S
C-in
C-out XY
YC-in
C-out
XC-in X Y
C-in

9. Q: Do we need C4 for a 4-bit 2’s complement addition?
Ripple adder
Speed limited by carry chain
2 per full adder  2n for an n-bit adder
Approach: eliminate or reduce carry chain
Carry look-ahead: compute Cin directly from external inputs
Q: Do we need C4 for a 4-bit 2’s complement addition?

10. Carry look-ahead adder
Let Ci+1 = (Xi·Yi)+ (Xi+Yi)· Ci = Gi + Pi · Ci
For a 4-bit adder …
C1 = G0 + P0·C0
C2 = G1 + P1·C1 = G1 + P1·G0 + P1·P0·C0
C3 = G2 + P2·G1 + P2·P1·G0 + P2·P1·P0·C0
C4 = G3 + P3·G2 + P3·P2·G1 + P3·P2·P1·G0 + P3·P2·P1·P0·C0
where Gi = Xi · Yi Pi = Xi + Yi
This is a 3 level circuit including generating the Gs and Ps
Rule of thumb: one carry look-ahead circuit every 4-bit

11. Carry look-ahead circuit
Gs and Ps are also useful
for generating the sums

12. 16-bit carry
ripple adder

13. 16-bit carry
look-ahead
adder

14. Subtraction
Recall our discussion on subtraction for 2’s complement …
X – Y = X + Y + 1
Invert all bits of Y and set Cin to 1
Example: 4-bit subtractor using 4-bit adder
Add a control circuit in
ALU s.t. same circuit can
be used for both addition
and subtraction
4-bit
Adder
X3 X2 X1 X0
D3 D2 D1 D0
C-in
C-out C4
Y3 Y2 Y1 Y0
C0 = 1
S3 S2 S1 S0

15. Multipliers
8x8 multiplier

16. Full-adder array

17. Faster carry chain

18. Binary Multiplication
An n-bit X n-bit multiplier can be realized in combinational circuitry by using an array of n-1 n-bit adders where is adder is shifted by one position.
For each adder one input is the multiplied by 0 or 1 (using AND gates) depending on the multiplier bit, the other input is n partial product bits.
X3 X2 X1 X0
x Y3 Y2 Y1 Y0
__________________________
X3.Y0 X2.Y0 X1.Y0 X0.Y0
X3.Y1 X2.Y1 X1.Y1 X0.Y1
X3.Y2 X2.Y2 X1.Y2 X0.Y2
X3.Y3 X2.Y3 X1.Y3 X0.Y3
_______________________________________________________________________________________________________________________________________________
P P P P P P P P0

19. 4x4 Array Multiplier