1. Complex Combinational Circuits Binary Adders Key to enterprise: Addition table also a truth table S i = C i 'A i B i ' + C i 'A i 'B i + C i A i 'B i '+ C i A i B i C i+1 = C i B i + A i B i + C i A i

2. Complex Combinational Circuits Binary Adders Alternative representation S i = (C i XOR (A i XOR B i )) C i+1 = A i B i + C i (A i XOR B i ) Notes: i) Full adder (shown) composed of 2 half adders (dotted boxes) ii) Example of minimization for multiple outputs (XOR reused)

3. Complex Combinational Circuits Binary Adders Ripple-carry adder (for multiple column additions)

4. Complex Combinational Circuits Binary Adders Carry-look-ahead C i+1 = A i B i + C i (A i XOR B i ) C i+1 = G i + P i C i (recursion relation) where G i = A i B i and P i = A i XOR B i example of recursion calculation C i+2 = G i+1 + P i+1 C i+1 = G i+1 + P i+1 (G i + P i C i ) = G i+1 + P i+1 G i + P i+1 P i C i

5. Complex Combinational Circuits Binary Adders Two’s complement addition Notes: i) M indicates subtraction (flips B i ’s, and adds 1) ii) detects overflow when carry in does not match carry out for last column

6. Complex Combinational Circuits Decoders n-bit code to m-bit code, where m > n Binary decoders only active output line corresponds to binary quantity at input Example (2-to-4 binary decoder) Note: Enable turns circuit on.

7. Complex Combinational Circuits Decoders Binary decoders Example (2-to-4 binary decoder) circuitsymbol.

8. Complex Combinational Circuits Decoders Active high and active low Abstraction from 0’s and 1’s A circuit always does the same thing from the point of view of activity; however, input and outputs can change Example (active low enable; active low output).

9. Complex Combinational Circuits Decoders Cascading decoders.

10. Complex Combinational Circuits Decoders Applications Realizing an SOP PQ + QR ≡  PQR (3,6,7).

11. Complex Combinational Circuits Decoders Applications Driving a LED display.

12. Complex Combinational Circuits Encoders n-bit code to m-bit code, where n > m Binary encoders creates binary representation of active input.

13. Complex Combinational Circuits Encoders Problem: What if more than 1 input line is active? Solution: Use priority encoding.

14. Complex Combinational Circuits Multiplexers and Demultiplexers Multiplexers select from among multiple inputs Demultiplexers select from among multiple outputs Note: sources and destinations can be arbitrarily wide.

15. Complex Combinational Circuits Multiplexers Example (4-input 1-bit multiplexer).

16. Complex Combinational Circuits Multiplexers More examples 8-input 1-bit 4-input 2-bit.

17. Complex Combinational Circuits Multiplexers Applications Time Division Multiplexing 1100  01110010 0101 Realizing a function.

18. Complex Combinational Circuits Demultiplexers Decoders and demultiplexers identical! A 2-to-4 decoder recast as a 4-output 1-bit demultiplexer.

19. Complex Combinational Circuits Programmable logic devices (PLD’s) Compromise between gate-level design and application-specific integrated circuit (ASIC) 1) Programmable Read Only Memory (PROM).

20. Complex Combinational Circuits Programmable logic devices (PLD’s) Compromise between gate-level design and application-specific integrated circuit (ASIC) 2) Programmable Array Logic (PAL ® ) F1 = ABC + A'D' + B'CD' F2 = ACD' + B'D.

21. Complex Combinational Circuits Programmable logic devices (PLD’s) Compromise between gate-level design and application-specific integrated circuit (ASIC) 3) Programmable Logic Array (PLA) F1 = A'B' + ABC + A'CD F2 = A'B' + C'D' + AC' + CD' F3 = A'CD + AC'.

22. Complex Combinational Circuits Summary of Topics Adders Full adder Ripple-carry adder Carry-look-ahead Two’s complement Decoders Encoders Multiplexers Demultiplexer PLD’s PROM’s PAL’s PLA’s