ECE 2110: Introduction to Digital Systems Combinational Logic Design Principles

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.

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

Switching algebra “Boolean algebra” Positive-logic convention deals with Boolean values -- 0, 1 Positive-logic convention analog voltages LOW, HIGH --> 0, 1 Negative logic -- seldom used Signals denoted by symbolic variables (X, Y, FRED, etc.)

Boolean operators Complement: X¢ (opposite of X) AND: X × Y OR: X + Y binary operators, described functionally by truth table.

Logic symbols

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¢

Axioms (postulates) Logic multiplication and addition precedence 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

Theorems (Single variable) Proofs by perfect induction

Two- and three- variable Theorems

Summary Variables, expressions, equations Axioms (A1-A5 pairs) Theorems (T1-T11 pairs) Single variable 2- or 3- variable Prime, complement, logic multiplication/addition, precedence

Next… Chapter 4.1,4.2 N-variable theorem, DeMorgan’s theorems Standard representations of logic functions HW #4