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Figure 8.1 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),

(9.9) 
or,

(9.10) 
where
and
are arbitrary real or complex sequences, and
and
are the DTFTs of
and
, respectively. The convolution
of
and
is defined by

(9.11) 
In practice, we always use the DFT (preferably an FFT) in place of the
DTFT, in which case we may write

(9.12) 
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 [264]:

(9.13) 
where
means ``
modulo
.''
Another way to look at convolution is as the inner product of
, and
, where
, i.e.,

(9.14) 
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 [264, p. 105] for an illustration of graphical
convolution.
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