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
may be defined as the response of the
filter to the impulse-train signal
, which, by periodic extension, is
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 for purposes of DFT
analysis, we may refer to more simply as the impulse
signal, and as the impulse response (as opposed to
impulse-train response). However, because convolution as
defined here (for DFT signals) is cyclic, the corresponding filter
interpretation requires periodic extension of all input signals
. In contrast, for the DTFT (§B.1), in which the
discrete-time axis is infinitely long, the impulse signal
is defined by
and no periodic extension is needed.
For any input signal
, we define the filter output signal
as the cyclic convolution of and
As discussed below in §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).
The convolution representation of linear, time-invariant, digital
filters is fully discussed in Book II  of the music signal
processing book series (in which this is Book I).