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Introduction to Digital Filters
Introduction to Laplace Transform Analysis
Relation to the z Transform

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Relation to the z Transform

The Laplace transform is used to analyze continuous-time systems. Its discrete-time counterpart is the transform:

$\displaystyle X_d(z) \isdef \sum_{n=0}^\infty x_d(nT) z^{-n}$

If we define $z=e^{sT}$ , the

transform becomes proportional to the Laplace transform of a sampled continuous-time signal:

$\displaystyle X_d(e^{sT}) = \sum_{n=0}^\infty x_d(nT) e^{-snT}$

As the sampling interval

goes to zero, we have

$\displaystyle \lim_{T\to 0} X_d(e^{sT})T = \lim_{\Delta t\to 0} \sum_{n=0}^\infty x_d(t_n) e^{-st_n} \Delta t = \int_{0}^\infty x_d(t) e^{-st} dt \isdef X(s)$

where $t_n\isdef nT$ and $\Delta t \isdef t_{n+1} - t_n = T$ .

In summary,

$\textstyle \parbox{0.8\textwidth}{the {\it z} transform\ (times the sampling in... ...o0$, the Laplace transform\ of the underlying continuous-time signal $x_d(t)$.}$

Note that the plane and plane are generally related by

$\displaystyle \zbox {z = e^{sT}.}$

In particular, the discrete-time frequency axis $\omega_d \in(-\pi/T,\pi/T)$ and continuous-time frequency axis $\omega_a \in(-\infty,\infty)$ are related by

$\displaystyle \zbox {e^{j\omega_d T} = e^{j\omega_a T}.}$

For the mapping $z=e^{sT}$ from the

plane to the

plane to be invertible, it is necessary that $X(j\omega_a )$ be zero for all $\vert\omega_a \vert\geq \pi/T$ . If this is true, we say

is bandlimited to half the sampling rate. As is well known, this condition is necessary to prevent aliasing when sampling the continuous-time signal

at the rate

to produce

, $n=0,1,2,\ldots\,$ (see [84, Appendix G]).

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Next: Laplace Transform Theorems

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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