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The Laplace Transform

The one-sided (unilateral) Laplace transform of a signal $ x(t)$ is defined as

$\displaystyle X(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}{\cal L}_s\{x\} \mathrel{\stackrel{\mathrm{\Delta}}{=}}\int_0^\infty x(t) e^{-st} dt

When evaluated along the $ j\omega$ axis (i.e., $ \sigma=0$ ), the Laplace Transform reduces to the unilateral Fourier transform:

$\displaystyle X(j\omega) = \int_0^\infty x(t) e^{-j\omega t} dt

Thus, the Laplace transform generalizes the Fourier transform from the real line (the frequency axis) to the entire complex plane.

$\textstyle \parbox{0.8\textwidth}{The Fourier transform equals the Laplace
transform evaluated along the $j\omega$\ axis in the complex $s$\ plane}$
The Laplace Transform can also be seen as the Fourier transform of an exponentially windowed causal signal $ x(t)$

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``The Laplace Transform'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-03-30 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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