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MSU ME 451 - Transfer Function: Laplace Transform of Unit Impulse Response of the System

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ME451 S07ME451 S0722Transfer Function: Laplace Transform of Unit Impulse Transfer Function: Laplace Transform of Unit Impulse Response of the SystemResponse of the SystemInput signal:Input signal:Output signal:Output signal:Take Take Laplace transform ofLaplace transform ofthe impulse responsethe impulse responsedef: def: Transfer FunctionTransfer FunctionME451 S0723Frequency ResponseFrequency ResponseInputInputWe knowWe knowComplex numbers Complex numbers Phase shiftPhase shiftMagnitudeMagnitudeME451 S07ME451 S0724Frequency ResponseFrequency ResponseME451 S0725The Laplace Transform (Appendix B)The Laplace Transform (Appendix B)Laplace transform converts a calculus problem (the linear Laplace transform converts a calculus problem (the linear differential equation) to an algebra problemdifferential equation) to an algebra problemHow to Use it: How to Use it: Take the Take the Laplace transformLaplace transformof a linear differential equationof a linear differential equationSolve the algebra problemSolve the algebra problemTake the Take the Inverse Laplace transform Inverse Laplace transform to obtain the solution to the to obtain the solution to the original differential equationoriginal differential equationdef: Laplace transformdef: Laplace transformdef: Inverse Laplace transformdef: Inverse Laplace transformME451 S07ME451 S0726The Laplace Transform (Appendix B)The Laplace Transform (Appendix B)Laplace Transform of a function Laplace Transform of a function f(tf(t))Convolution integralConvolution integralME451 S0727Properties of Laplace Transforms (page 641Properties of Laplace Transforms (page 641--643)643)LinearityLinearityTime DelayTime DelayNonNon--rational functionrational functionME451 S07ME451 S0728Properties of Laplace TransformsProperties of Laplace TransformsShift in FrequencyShift in FrequencyDifferentiationDifferentiationME451 S0729Properties of Laplace TransformsProperties of Laplace TransformsDifferentiation ( in time domain Differentiation ( in time domain ⇔⇔s in Laplace domain)s in Laplace domain)Integration (Integration (in time domain in time domain ⇔⇔1/s in Laplace domain)1/s in Laplace domain)ME451 S07ME451 S0730Laplace Transform of Impulse and Unit StepLaplace Transform of Impulse and Unit StepImpulseImpulseUnit StepUnit StepME451 S0731Unit RampUnit RampME451 S07ME451 S0732Exponential FunctionExponential FunctionME451 S0733Sinusoidal FunctionsSinusoidal FunctionsME451 S07ME451 S0734PartialPartial--fraction Expansion (Text, page 637fraction Expansion (Text, page 637--641)641)F(sF(s) is rational, ) is rational, realizable condition (realizable condition (d/dtd/dtis not is not


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MSU ME 451 - Transfer Function: Laplace Transform of Unit Impulse Response of the System

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