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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{\bf C}^N$ denote a length $ N$ signal that is interpreted as a filter. Then given any input signal $ x\in{\bf C}^N$, the filter output signal $ y\in{\bf 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{\bf C}^N$ is 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 $N$)}. \\ \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 $ {\bf 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 DTFT (§B.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 [68] of the music signal processing book series (in which this is Book I).
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Convolution Example 1: Smoothing a Rectangular Pulse
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Commutativity of Convolution