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Convolution as a Filtering Operation

We may interpret either of the signals $ x$ or $ y$ as the impulse-train response of a linear, time-invariant, digital filter (see §8.3 for an introduction to the digital-filter point of view). To emphasize this interpretation, we use the notation $ h(n)$ to denote the impulse-train-response signal at time $ n$. More specifically, the impulse-train response $ h\in{\bf C}^N$ may be defined as the response of the filter to the impulse-train signal $ \delta
\isdeftext [1,0,\ldots,0]\in{\bf R}^N$, which, by periodic extension, is equal to

$\displaystyle \delta(n) = \left\{\begin{array}{ll}
1, & n=0\;\mbox{(mod $N$)} \\ [5pt]
0, & n\ne 0\;\mbox{(mod $N$)}. \\

Thus, $ N$ is the period of the impulse-train in samples--there is an ``impulse'' (a `$ 1$') every $ N$ samples. Neglecting the assumed periodic extension of all signals in $ {\bf C}^N$ for purposes of DFT analysis, we may refer to $ \delta$ more simply as the impulse signal, and $ h$ as the impulse response (as opposed to impulse-train response). However, because convolution as defined here (for DFT signals) is cyclic, the corresponding filter interpretation requires periodic extension of all input signals $ x\in{\bf C}^N$. In contrast, for the DTFTB.1), in which the discrete-time axis is infinitely long, the impulse signal $ \delta(n)$ is defined by

$\displaystyle \delta(n) \isdef \left\{\begin{array}{ll}
1, & n=0 \\ [5pt]
0, & n\ne 0 \\

and no periodic extension is needed.

For any input signal $ x\in{\bf C}^N$, we define the filter output signal $ y\in{\bf C}^N$ as the cyclic convolution of $ x$ and $ h\in{\bf C}^N$:

$\displaystyle y = h\circledast x = x \circledast h

As discussed below in §7.2.7, one may embed acyclic convolution within a larger cyclic convolution. In this way, real-world systems may be simulated using fast DFT convolutions (see Appendix A for more on fast convolution algorithms).

The convolution representation of linear, time-invariant, digital filters is fully discussed in Book II [66] of the music signal processing book series (in which this is Book I).

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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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