1. Combinational Circuits

2. Decoder
Decoder :
Takes n inputs
Selects one of 2n output lines

3. Truth Table Expression for each output has one term
O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0

4. Decoder
Implementation
O0 = I1' I0' O1 = I1 I0' O2 = I1' I0 O3 = I1 I0

5. Sample Use
3  8 decoder used to pick instruction

6. Multiplexer Multiplexer : 2 input multiplexer 4 input multiplexer
N data lines lines
Log2(N) select lines
2 input multiplexer
4 input multiplexer

7. Decode to Mux Multiplexer selection logic = decoder
4 input multiplexer

8. Shifter
1 bit shifter
D4 goes to either S3 or S5

9. Shifter
1 bit shifter
If C = 1, goes to S5

10. Shifter
1 bit shifter
If C = 0, goes to S3

11. Shifter S = direction 0: left, 1: right Logical shift:
Left bit always 0 on right shift

12. Multibit Shift 4 bit wide 0-3bit shift S selects direction
Multiplexers select correct pattern

13. Half Adder
Half Adder : adds A + B, produce sum + carry 1 1 1

14. Half Adder Truth Table
Inputs a & b
Carry : c
Sum : s

15. First Pass

16. XOR Function XOR gate : exclusive OR Notation: A  B OR A ^ B
AB' + A'B
Notation: A  B OR A ^ B

17. Using XOR
Half Adder Final

18. Full Adder 1 1 1 Full Adder handles carry in Three inputs: Cin, A, B
Two out: Cout, S 1 1 1

19. Full Adder Circuit Made from two half adders Add A & B
Add sum to carry

20. Full Adder Abstracted

21. Ripple Carry Adder
Multibit adder : string of 1 bit full adders

22. Ripple Carry Adder
Implemented with carry out & signed overflow:

23. Ripple Carry Adder Adding subtraction to adder Select line 0 : Normal
1 : Invert one pattern, add extra 1 via carry

24. Ripple Carry Adder Fun Fact
Carry signal needs to propagate from one adder to next
3 gates delay per bit

25. Carry Select Fun Fact Cary Select Adder
Calculate second block with both 0 and 1 carry
Work in parallel with first
Pick right answer when carry available

26. ALU ALU : Arithmetic Logic Unit In: Out: 2 Operands Control signal
Result based on select operation

27. ALU Harware to implement each desired op
Multiplexor to select desired one

28. 1-Bit Simple ALU Processes two 1-bit inputs A, B
3 functions: AND (0), OR (1), Add (2)
Selected with function select bits

29. 1-Bit Simple ALU With selector to invert B input Can now also compute:
A OR ~B
A AND ~B
A + ~B

30. 4-Bit Simple ALU
Four 1-bit ALU's chained together

31. 4-Bit Simple ALU
AddOne allows for A + (~B) + 1 == A + (~B + 1) == A + (-B) == A - B