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Convolution of Short Signals
figure[htbp]
Figure 7.1 illustrates the conceptual operation of filtering an input
signal by a filter with impulseresponse to produce an
output signal . By the convolution theorem for DTFTs
(§2.3.5),
or,
where
and
are arbitrary real or complex sequences, and
and
are the DTFTs of
and
, respectively. The
convolution
of
and
is defined by
In practice, we always use the DFT (preferably the FFT) in place of the
DTFT, in which case we may write
where now
(length
complex sequences). It is
important to remember that the specific form of convolution implied in
the DFT case is
cyclic (also called
circular)
convolution [
248]:

(8.2) 
where
means ``
modulo
.''
Another way to look at convolution is as the inner product of , and
, where
, i.e.,
This form describes
graphical convolution in which the output
sample at time
is computed as an inner product of the
impulse
response after flipping it about time 0 and shifting time 0 to time
. See [
248, p. 105] for an illustration of graphical
convolution.
Subsections
Previous: OverlapAdd (OLA)
STFT ProcessingNext: Cyclic FFT Convolution
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.