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#### 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 :

Because the convolution is cyclic, with and chosen from the set of (periodically extended) vectors of length , is most precisely viewed as the 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

Thus, is the 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 DTFTB.1), in which the discrete-time axis is infinitely long, the impulse signal is defined as

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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