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
![$ \delta\isdeftext [1,0,\ldots,0]\in{\bf R}^N$](http://www.dsprelated.com/josimages_new/mdft/img1169.png)
,
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 [
68] of the
music signal processing book series (in which this is Book I).
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