ECE 3110: Introduction to Digital Systems

Combinational Logic Design Principles

2

Combinational logic circuit

Outputs depend only on the current inputs (Not on history)

Contain an arbitrary number of logic gates and inverters, but NO feedback loops.

3

Combinational-Circuit Analysis

Kinds of combinational analysis: exhaustive (truth table) algebraic (expressions) simulation / test bench ( not in this course)

Write functional description in HDL Define test conditions / test vectors Compare circuit output with functional description (or

known-good realization) Repeat for “random” test vectors

4

Switching algebra

“Boolean algebra” deals with Boolean values -- 0, 1

Positive-logic convention analog voltages LOW, HIGH --> 0, 1

Negative logic -- seldom usedSignals denoted by symbolic variables

(X, Y, FRED, etc.)

5

Boolean operators

Complement: X (opposite of X)AND: X YOR: X + Y

binary operators, describedfunctionally by truth table.

6

Logic symbols

7

Some definitions

Literal: a variable or its complement X, X, FRED, CS_L

Expression: literals combined by AND, OR, parentheses, complementation X+Y P Q R A + B C ((FRED Z) + CS_L A B C + Q5) RESET

Equation: Variable = expression P = ((FRED Z) + CS_L A B C + Q5) RESET

8

Axioms (postulates)

A1) X=0 if X‡1 A1’ ) X=1 if X‡0 A2) if X=0, then X’=1 A2’ ) if X=1, then X’=0 A3) 0 • 0=0 A3’ ) 1+1=1 A4) 1 • 1=1 A4’ ) 0+0=0 A5) 0 • 1= 1 • 0 =0 A5’ ) 1+0=0+1=1

Logic multiplication and addition

precedence

9

Theorems (Single variable)

Proofs by perfect induction

10

Two- and three- variable Theorems

11

Summary

Variables, expressions, equationsAxioms (A1-A5 pairs)Theorems (T1-T11 pairs)

Single variable 2- or 3- variable

Prime, complement, logic multiplication/addition, precedence

12

Next…

Chapter 4.1,4.2N-variable theorem, DeMorgan’s theoremsStandard representations of logic functions

HW #4