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
Thus,
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).