Convolution Theorem

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Convolution Theorem

The convolution theorem for z transforms states that for any (real or) complex causal signals and , convolution in the time domain is multiplication in the domain, i.e.,

$\displaystyle \zbox {x\ast y \;\leftrightarrow\; X\cdot Y}$

or, using operator notation,

$\displaystyle {\cal Z}_z\{x \ast y\} \;=\; X(z)Y(z),$

where $X(z)\isdef {\cal Z}_z(x)$ , and $Y(z)\isdef {\cal Z}_z(y)$ . (See [84] for a development of the convolution theorem for discrete Fourier transforms.)

Proof:

$\begin{eqnarray*} {\cal Z}_z(x\ast y) &\isdef & \sum_{n=0}^{\infty}(x\ast y)_n z... ...(by the Shift Theorem)}\\ &\isdef & X(z)Y(z) % \quad\pfendmath \end{eqnarray*}$

The convolution theorem provides a major cornerstone of linear systems theory. It implies, for example, that any stable causal LTI filter (recursive or nonrecursive) can be implemented by convolving the input signal with the impulse response of the filter, as shown in the next section.

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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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Chapters

Introduction to Digital Filters
    Transfer Function Analysis
       Shift and Convolution Theorems
          Convolution Theorem