1. Discrete Systems I Lecture 10 Adder and ALU Profs. Koike and Yukita

2. More Digital Circuits Multiplexer, Adder, Comparator, and a simple ALU are explained. These circuits are some of the building blocks of complex digital systems including computers. 2

3. The Multiplexer (MUX) Data versus Control 3

4. The Truth Table 4

5. Realization of MUX via Decoder Feed the multiplexer’s control signals into a decoder. Each output line from the decoder is ANDed with a different data bit. 5

6. Realization of MUX via Decoder 6

7. 7

8. Quiz: Add negations to complete the decoder. 8

9. Embedding of Decoder 9

10. Sum-of-products expression 10

11. Problems 1 and 2 Problem 1: Draw a circuit diagram and write the corresponding algebraic expression for a three-control, eight-data multiplexer. Problem 2: Derive a full maxterm and minterm realizations of a one-control, two- data multiplexer. 11

12. Vectors and Parallel Operations Instead of dealing one bit at a time, we may handle several bits, for example 8 bits, at a time. Values consisting of many bits taken in parallel are known as vectors. 12

13. 4-data, 2-control multiplexer in which data paths are eight-bit wide vectors 13

14. Bit Slicing The multiplexer in the previous slide consists of eight distinct one-bit wide multiplexers: MUX0, MUX1, …, and MUX7, all of them operate in parallel. 14

15. 4-bit vector AND 15 =

16. n-bit Adder an example of a cascading circuit 16 In column k, a k and b k are bits to be added and c k is a carry bit from the previous column.

17. Cascading Circuits and Parallel Circuits Note: Cascading is different from parallel construction in that it takes for signals to ripple their way through the circuit before a final, accurate output is produced. 17

18. Addition and Carry 18

19. The Comparator Another example of a cascading circuit. It compares two binary numbers and determines which one is larger, or whether or not they are equal. 19

20. Do it from the most significant bit. 20 e k means “A and B are equal so far.” a g k means “A has already been determined to be greater than B.” 1010

21. The function that produces e o 21

22. The function that produces a go 22

23. Problem 3 Write algebraic expression for the (a) minterm and (b) maxterm realizations of the single-bit sum and carry function. 23

24. The ALU A simple arithmetic logic unit. It deals with 6-bit numbers. It produces a 6-bit number as output. It uses 3-bit control signal C to determine what operation to perform on its two 6-bit data inputs, A and B. For simplicity, we will not worry about overflows in the arithmetic operation. 24

25. The ALU 25

26. Opcodes Table 26

27. Summary of the behavior 27

28. Block diagram of a 6-bit ALU 28

29. Problem 4 29

30. Problem 5 30