1. Boolean Expressions Lecture 3 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007

2. Logistics Do the reading Handout –Lecture Notes –On web: Lab 1 solutions, Lab 2

3. Overview Boolean Algebra K-maps X’s and Z’s

4. Boolean Expressions

5. 2.2.2 Sum-of-Products Form

6. Boolean Expressions

7. 2.2.3 Product-of-Sums Form

8. Boolean Equations You are going to the Hoch for lunch –You won’t eat lunch (E) if it’s not open (O) or –If they only serve corndogs (C) Write a truth table for determining if you will eat lunch (E).

9. SOP & POS Form SOP – sum-of-products POS – product-of-sums

10. SOP & POS Form SOP – sum-of-products POS – product-of-sums

11. 2.4 From Logic to Gates Fig. 2.23 shows Schematic of It is an example of Gate Array

12. 2.4 From Logic to Gates Two-level logic: 8 forms into 2 groups

13. Multiple Output Circuits

14. Priority Encoder Hardware

15. 2.6 Don’t Cares

16. 2.3.3 Boolean Axioms & Theorems

17. Boolean Axioms & Theorems 無法靠直覺

18. What is the Boolean expression for this circuit? Bubble Pushing

19. 2.7 Karnaugh Maps (K-Maps) Sum-of-products (SOP) form can be tedious to simplify using Boolean algebra K-maps allow us to do the same thing graphically m0 m3

20. 3-input K-map

21. A’B BC’

22. K-map Definitions Complement: variable with a bar over it Literal: variable or its complement Implicant: product of literals Prime implicant: implicant corresponding to the larges circle in the K-map

23. K-map Rules Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction Each circle must be as large as possible A circle may wrap around the edges of the K- map A one in a K-map may be circled multiple times A “don't care” (X) is circled only if it helps minimize the equation

24. 4-input K-map (m2)(m10) (m15)

25. 7-segment display See example 2.10 for it. It is useful in implementing clock

26. 2.6 Don’t Cares

27. Contention: X Not don’t care Not just 1’s and 0’s Contention: X

28. Floating: Z Tri-state Buffer

29. 2.8 Building Blocks Multiplexer vs. Demultiplexer Decoders vs. Encoder Priority Encoder

30. Next Time Timing Hazards Sequential Circuits