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Convolution as a Filtering Operation

In a convolution of two signals $ x\circledast y$ , where both $ x$ and $ y$ are signals of length $ N$ (real or complex), we may interpret either $ x$ or $ y$ as a filter that operates on the other signal which is in turn interpreted as the filter's ``input signal''. 7.5 Let $ h\in\mathbb{C}^N$ denote a length $ N$ signal that is interpreted as a filter. Then given any input signal $ x\in\mathbb{C}^N$ , the filter output signal $ y\in\mathbb{C}^N$ may be defined as the cyclic convolution of $ x$ and $ h$ :

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

Because the convolution is cyclic, with $ x$ and $ h$ chosen from the set of (periodically extended) vectors of length $ N$ , $ h(n)$ is most precisely viewed as the impulse-train-response of the associated filter at time $ n$ . Specifically, the impulse-train response $ h\in\mathbb{C}^N$ is the response of the filter to the impulse-train signal $ \delta\isdeftext [1,0,\ldots,0]\in\mathbb{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 $N$)}. \\
\end{array} \right.

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 $ \mathbb{C}^N$ , we may refer to $ \delta$ more simply as the impulse signal, and $ h$ as the impulse response (as opposed to impulse-train response). In contrast, for the DTFTB.1), in which the discrete-time axis is infinitely long, the impulse signal $ \delta(n)$ is defined as

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

and no periodic extension arises.

As discussed below (§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).

Note that only linear, time-invariant (LTI) filters can be completely represented by their impulse response (the filter output in response to an impulse at time 0 ). The convolution representation of LTI digital filters is fully discussed in Book II [71] of the music signal processing book series (in which this is Book I).

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University