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Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Introduction to Laplace Transform Analysis
       Laplace Transform Theorems
          Linearity

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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) ... ..._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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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