#### Convolution as a Filtering Operation

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

*impulse-train-response*of the associated filter at time . Specifically, the impulse-train response is the response of the filter to the

*impulse-train signal*, which, by periodic extension, is equal to

*period*of the impulse-train in samples--there is an ``impulse'' (a `') every samples. Neglecting the assumed periodic extension of all signals in , we may refer to more simply as the

*impulse signal*, and 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 is defined as

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