1. Arithmetic Circuits

2. Programmable Logic Device
Outline
Adders
Multipliers
ALU Design
CS Digital Logic
Programmable Logic Device

3. Programmable Logic Device
Adders
Half adder: without carry in
Two inputs
Two outputs – sum and carry out
Full adder: with carry in
Three inputs – include carry in
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Programmable Logic Device

4. Programmable Logic Device
Half Adder
S = AB
Cout = AB A B S
Cout 1
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Programmable Logic Device

5. Programmable Logic Device
Full Adder
S = ABCin
Cout = AB+BCin+ ACin A B
Cin S
Cout 1
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Programmable Logic Device

6. Programmable Logic Device
Full Adder
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Programmable Logic Device

7. Programmable Logic Device
4-bit Parallel Adder
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Programmable Logic Device

8. 4-Bit Adder-Subtractor
CS Digital Logic
Programmable Logic Device

9. 4-Bit Adder-Subtractor
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Programmable Logic Device

10. Carry Look-Ahead Logic
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Programmable Logic Device

11. Carry Look-Ahead Logic
Pi = Ai  Bi (carry Propagate)
Gi = AiBi (carry generate)
Si = Ai  Bi  Ci = Pi  Ci
Ci+1 = AiBi + BiCi + AiCi
= Gi + (Ai + Bi)Ci
= Gi + (Ai  Bi)Ci
= Gi + PiCi
Cin A B
Cout S 1
CS Digital Logic
Programmable Logic Device

12. Carry Look-Ahead Logic
C1 = G0 + P0C0
C2 = G1 + P1C1 = G1 + P1 (G0 + P0C0) = G1 + P1G0 + P1P0C0
C3 = G2 + P2C2 = G2 + P2G1 + P2P1G0 + P2P1P0C0
C4 = G3 + P3C3 = G3 + P3G2 + P3P2G1 + P3P2P1G0 + P3P2P1P0C0
S0 = A0  B0  C0
S1 = A1  B1  C1
S2 = A2  B2  C2
S3 = A3  B3  C3
CS Digital Logic
Programmable Logic Device

13. Carry Look-Ahead Logic
CS Digital Logic
Programmable Logic Device

14. Programmable Logic Device
BCD Addition
A BCD adder is a circuit that adds two BCD digits in parallel and produces a sum digit also in BCD. Consider the sum in BCD and binary representations in 5 bits:
CS Digital Logic
Programmable Logic Device

15. Programmable Logic Device
Decimal
Cout B8 B4 B2 B1 S8 S4 S2 S1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
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Programmable Logic Device

16. Programmable Logic Device
BCD Addition
CS Digital Logic
Programmable Logic Device

17. Programmable Logic Device
BCD Addition
CS Digital Logic
Programmable Logic Device

18. Programmable Logic Device
Magnitude Comparator
1-bit comparator
F= = AiBi + Ai' Bi' = (Ai  Bi)'
F< = Ai'Bi
F> = AiBi' A B F=
F<
F> 1
CS Digital Logic
Programmable Logic Device

19. Programmable Logic Device
2-bit Comparator
Suppose N1 = A1A0, N0 = B1B0
Case 1: N1 = N0 -- A1= B1 and A0 = B0
F= = (A1  B1)' (A0  B0)'
Case 2: N1 < N0 -- A1< B1 or A1= B1 and A0 < B0
F< = A1'B1 + (A1  B1)' A0'B0
Case 3: N1 < N0 -- A1> B1 or A1= B1 and A0 > B0
F> = A1B1' + (A1  B1)' A0B0'
CS Digital Logic
Programmable Logic Device

20. Programmable Logic Device
Multiplier
Partial product accumulation
1001 (9)
1101 (13)
----
1001
0000
(117)
= 117
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Programmable Logic Device

21. Programmable Logic Device
2-Bit Multiplier
A1 A0
B1 B0
A1•B0 A0•B0
A1•B1 A0•B1
S3 S2 S1 S0
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Programmable Logic Device

22. Programmable Logic Device
4-Bit Multiplier
A3 A2 A1 A0
B3 B2 B1 B0
A3•B0 A2•B0 A1•B0 A0•B0
A3•B1 A2•B1 A1•B1 A0•B1
A3•B2 A2•B2 A1•B2 A0•B2
A3•B3 A2•B3 A1•B3 A0•B3
S7 S6 S5 S4 S3 S2 S1 S0
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Programmable Logic Device

23. Programmable Logic Device
CS Digital Logic
Programmable Logic Device

24. Arithmetic Logic Unit (ALU) Design
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Programmable Logic Device

25. Programmable Logic Device
ALU S1 S0
Cin Yi F
F = A 1
F = A + 1 B
F = A + B
F = A + B + 1 B'
F = A + B'
F = A + B' + 1
F = A -- 1 S1 S0 Yi 1 B B'
Xi = Ai
Yi = s0Bi + s1Bi'
Cin = Cin
CS Digital Logic
Programmable Logic Device

26. Programmable Logic Device
Example
Design one bit slice for an ALU unit using a full adder block to perform the following: M S1 S0
Function Name F Xi Yi
Cin
Complement A'
Ai' 1
AND
A AND B
Ai AND Bi
Identity A Ai OR
A OR B
Ai OR Bi
Decrement
A - 1
Add
A + B Bi
Subtract
A + B'
Bi'
Increment
A + 1
CS Digital Logic
Programmable Logic Device

27. Programmable Logic Device
Example
Determine Yi M S1 S0 Yi 1 Bi
Bi'
Yi = MS1'Bi + MS0'Bi'
Determine Xi M S1 S0 Xi
Ai' 1
Ai Bi Ai
Ai + Bi X
Xi = M'S1'S0'Ai' + M'S1S0Bi + S0AiBi + S1Ai + MAi
Cin = MS1
CS Digital Logic
Programmable Logic Device

28. Programmable Logic Device
CS Digital Logic
Programmable Logic Device

29. Programmable Logic Device
Example
Design one bit slice for an ALU unit using a full adder block to perform the following: M S1 S0
Function Name F Xi Yi
Cin
Add
A + B Ai Bi 1
Subtract
A - B
Bi'
Increment
A + 1
Decrement
A - 1
AND
A AND B
Ai AND Bi OR
A OR B
Ai OR Bi
Complement A'
Ai'
XOR
A  B
Ai Bi
CS Digital Logic
Programmable Logic Device

30. Programmable Logic Device
Example
Determine Yi M S1 S0 Yi Bi 1
Bi'
Yi = M'S1S0 + M'S1S0Bi + M'S0'Bi'
Determine Xi M S1 S0 Xi 1
Ai Bi
Ai + Bi
Ai'
Ai Bi X Ai
Xi = MS1S0'Ai' + MS0Ai'Bi + MS1'AiBi + MS0AiBi' + M'Ai
Cin = M' (S1 S0)
CS Digital Logic
Programmable Logic Device

31. Programmable Logic Device
Exercises
p. 325: 4.32, 4.33, 4.34, 4.36, 4.37, 4.39, 4.40, 4.43, 4.44
CS Digital Logic
Programmable Logic Device