ECE 3110: Introduction to Digital SystemsCombinational logic circuitCombinational-Circuit AnalysisSwitching algebraBoolean operatorsLogic symbolsSome definitionsAxioms (postulates)Theorems (Single variable)Two- and three- variable TheoremsSummaryNext…ECE 3110: Introduction to Digital SystemsCombinational Logic Design Principles2Combinational logic circuitOutputs depend only on the current inputs (Not on history)Contain an arbitrary number of logic gates and inverters, but NO feedback loops.3Combinational-Circuit AnalysisKinds of combinational analysis:exhaustive (truth table)algebraic (expressions)simulation / test bench ( not in this course)Write functional description in HDLDefine test conditions / test vectorsCompare circuit output with functional description (or known-good realization)Repeat for “random” test vectors4Switching algebra“Boolean algebra”deals with Boolean values -- 0, 1Positive-logic conventionanalog voltages LOW, HIGH --> 0, 1Negative logic -- seldom usedSignals denoted by symbolic variables(X, Y, FRED, etc.)5Boolean operatorsComplement: X (opposite of X)AND: X YOR: X + Ybinary operators, describedfunctionally by truth table.6Logic symbols7Some definitionsLiteral: a variable or its complementX, X, FRED, CS_LExpression: literals combined by AND, OR, parentheses, complementationX+YP Q RA + B C((FRED Z) + CS_L A B C + Q5) RESETEquation: Variable = expressionP = ((FRED Z) + CS_L A B C + Q5) RESET8Axioms (postulates)A1) X=0 if X‡1 A1’ ) X=1 if X‡0A2) if X=0, then X’=1 A2’ ) if X=1, then X’=0A3) 0 • 0=0 A3’ ) 1+1=1A4) 1 • 1=1 A4’ ) 0+0=0A5) 0 • 1= 1 • 0 =0 A5’ ) 1+0=0+1=1Logic multiplication and additionprecedence9Theorems (Single variable)Proofs by perfect induction10Two- and three- variable Theorems11SummaryVariables, expressions, equationsAxioms (A1-A5 pairs)Theorems (T1-T11 pairs)Single variable2- or 3- variablePrime, complement, logic multiplication/addition, precedence12Next…Chapter 4.1,4.2N-variable theorem, DeMorgan’s theoremsStandard representations of logic functionsHW
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