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

Thus, is the

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 [70] of the music signal processing book series (in which this is Book I).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University