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The Laplace Transform - CCRMA

Music 420: The Laplace TransformJulius O. Smith III for Computer Research in Music and Acoustics ( CCRMA )Department of Music, Stanford UniversityStanford, California 94305 February 5, 2019 Outline Definition Linearity and Differentiation Theorem Examples of Mass-Spring system analysis1 The Laplace TransformThe one-sided (unilateral) Laplace transformof a signalx(t)is defined asX(s) =Ls{x} = 0x(t)e stdt t=time in seconds s= +j is a complex variable Appropriate forcausalsignalsWhen evaluated along thej axis ( , = 0), theLaplace Transform reduces to the unilateralFouriertransform:X(j ) = 0x(t)e j tdtThus, the Laplace Transform generalizes the Fouriertransform from the real line (the frequency axis) to theentire complex Fourier Transform equals the Laplacetransform evaluated along thej axis in thecomplexsplaneThe Laplace Transform can also be seen as the Fouriertransform of anexponentially windowedcausal signalx(t)2 Relation to the z TransformThe Laplace Transform is used to analyzecontinuous-timesystems.

The Laplace Transform can also be seen as the Fourier transform of an exponentially windowed causal signal x(t) 2 Relation to the z Transform The Laplace transform is used to analyze continuous-time systems. Its discrete-time counterpart is the z transform:

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