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Boolean Expressions

Boolean Expressions. Lecture 3 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007. Logistics. Do the reading Handout Lecture Notes On web: Lab 1 solutions, Lab 2. Overview. Boolean Algebra K-maps X’s and Z’s. Boolean Expressions.

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Boolean Expressions

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

  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. 3-input K-map BC’ A’B

  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 (m15) (m2) (m10)

  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

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