ECE 2110: Introduction to Digital Systems

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” • 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.)

Complement: X¢ (opposite of X) AND: X × Y OR: X + Y Boolean operators binary operators, describedfunctionally 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) • A1) X=0 if X‡1 A1’ ) X=1 if X‡0 • A2) if X=0, then X’=1A2’ ) 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

ECE 2110: Introduction to Digital Systems