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

2.2.2 Sum-of-Products Form

Boolean Expressions

2.2.3 Product-of-Sums Form

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).

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

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

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

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

Multiple Output Circuits

Priority Encoder Hardware

2.6 Don’t Cares

2.3.3 Boolean Axioms & Theorems

Boolean Axioms & Theorems 無法靠直覺

What is the Boolean expression for this circuit? Bubble Pushing

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

3-input K-map

A’B BC’

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

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

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

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

2.6 Don’t Cares

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

Floating: Z Tri-state Buffer

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

Next Time Timing Hazards Sequential Circuits