ME451 S07ME451 S0714Output Signal of a Linear SystemOutput Signal of a Linear SystemInput signalInput signalOutput signalOutput signalSuperposition!Superposition!def: def: impulse responseimpulse responsedef: def: convolutionconvolutiondef: def: causalitycausalityME451 S0715Impulse ResponseImpulse ResponseME451 S07ME451 S0716Causal Linear Time Invariant (LTI) SystemCausal Linear Time Invariant (LTI) SystemA A causal systemcausal system(a (a physicalphysicalor or nonanticipative systemnonanticipative system) is a ) is a system where the output only depends on the input valsystem where the output only depends on the input values ues Thus, the current output can be generated by the causThus, the current output can be generated by the causal al system with the system with the currentcurrentand and pastpastinput valuesinput valuesCausal LTI impulse responseCausal LTI impulse responseThus, we haveThus, we haveME451 S0717Causal System (Physically Realizable)Causal System (Physically Realizable)Systemfuturefuturepast pastcurrent currentME451 S07ME451 S0718Causal System?Causal System?Derivative operator (input: Derivative operator (input: positionposition, output: , output: velocityvelocity))Integral operator (input: Integral operator (input: velocityvelocity, output: , output: positionposition))ME451 S0719Complex NumbersComplex NumbersOrdered pair of two real numbersOrdered pair of two real numbersConjugateConjugateAdditionAdditionMultiplicationMultiplicationME451 S07ME451 S0720Complex NumbersComplex NumbersEulerEuler’’s identitys identityPolar formPolar formMagnitudeMagnitudePhasePhaseME451 S0721ME451 S07ME451 S0722Transfer Function: Laplace Transform of Unit Impulse Transfer Function: Laplace Transform of Unit Impulse Response of the SystemResponse of the SystemInput signal:Input signal:Output signal:Output signal:Take Take Laplace transform ofLaplace transform ofthe impulse responsethe impulse responsedef: def: Transfer FunctionTransfer
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