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Definition: The frequency response of an LTI filter may be defined
as the Fourier transform of its impulse response. In particular, for
finite, discrete-time signals
, the sampled frequency
response may be defined as
The complete (continuous) frequency response is defined using the DTFT (see
§B.1), i.e.,
where the summation limits are truncated to
because
is zero for
and
. Thus, the DTFT can be obtained from
the DFT by simply replacing
by
, which corresponds
to infinite zero-padding in the time domain. Recall from
§7.2.10 that zero-padding in the time domain gives
ideal interpolation of the frequency-domain samples
(assuming the original DFT included all nonzero samples of
).
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