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Linearity

The Laplace transform is a linear operator. To show this, let $ w(t)$ denote a linear combination of signals $ x(t)$ and $ y(t)$ ,

$\displaystyle w(t) = \alpha x(t) + \beta y(t),
$

where $ \alpha$ and $ \beta$ are real or complex constants. Then we have

\begin{eqnarray*}
W(s) &\isdef & {\cal L}_s\{w\} \isdef {\cal L}_s\{\alpha x(t) + \beta y(t)\}\\
&\isdef & \int_0^\infty \left[\alpha x(t) + \beta y(t)\right] e^{-st} dt\\
&=& \alpha \int_0^\infty x(t) e^{-st} dt
+ \beta \int_0^\infty y(t) e^{-st} dt\\
&\isdef & \alpha X(s) + \beta Y(s).
\end{eqnarray*}

Thus, linearity of the Laplace transform follows immediately from the linearity of integration.


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA