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Embedded SystemsFPGAElectronics

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

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Differentiation

The differentiation theorem for Laplace transforms states that

$\displaystyle {\dot x}(t) \leftrightarrow s X(s) - x(0), $

where $ {\dot x}(t) \isdef \frac{d}{dt}x(t)$, and $ x(t)$ is any differentiable function that approaches zero as $ t$ goes to infinity. In operator notation,

$\displaystyle \zbox {{\cal L}_{s}\{{\dot x}\} = s X(s) - x(0).} $



Proof: This follows immediately from integration by parts:

\begin{eqnarray*} {\cal L}_{s}\{{\dot x}\} &\isdef & \int_{0}^\infty {\dot x}(t)... ...y} - \int_{0}^\infty x(t) (-s)e^{-s t} dt\\ &=& s X(s) - x(0) \end{eqnarray*}

since $ x(\infty)=0$ by assumption.



Corollary: Integration Theorem

$\displaystyle \zbox {{\cal L}_{s}\left\{\int_0^t x(\tau)d\tau\right\} = \frac{X(s)}{s}} $

Thus, successive time derivatives correspond to successively higher powers of $ s$, and successive integrals with respect to time correspond to successively higher powers of $ 1/s$.

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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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