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 DTFT (§B.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  of the music signal processing book series (in which this is Book I).