Convolution as a Filtering Operation

We may interpret either of the signals or as the
*impulse-train response* of a *linear, time-invariant,
digital filter* (see
§8.3 for an introduction to the digital-filter point of view).
To emphasize this interpretation, we use the notation to denote
the impulse-train-response signal at time . More specifically, the
impulse-train response
may be defined as the response of the
filter to the *impulse-train signal*
, which, by periodic extension, is
equal to

For any input signal
, we define the filter output signal
as the *cyclic convolution* of and
:

The convolution representation of linear, time-invariant, digital filters is fully discussed in Book II [66] of the music signal processing book series (in which this is Book I).

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