Boolean Expressions Lecture 3 Digital Design and Computer

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

2. 4 From Logic to Gates • Fig. 2. 23 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

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

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 m 0 m 3

3 -input K-map

3 -input K-map BC’ A’B

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 Kmap • 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 (m 15) (m 2) (m 10)

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