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
![$ [0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1128.png)
because
![$ h(n)$](http://www.dsprelated.com/josimages_new/mdft/img46.png)
is zero for
![$ n<0$](http://www.dsprelated.com/josimages_new/mdft/img1542.png)
and
![$ n>N-1$](http://www.dsprelated.com/josimages_new/mdft/img1543.png)
. Thus, the DTFT can be obtained from
the
DFT by simply replacing
![$ \omega_k$](http://www.dsprelated.com/josimages_new/mdft/img677.png)
by
![$ \omega$](http://www.dsprelated.com/josimages_new/mdft/img368.png)
, which corresponds
to infinite
zero-padding in the time domain. Recall from
§
7.2.10 that zero-padding in the time domain gives
ideal interpolation of the
frequency-domain samples
![$ H(\omega_k)$](http://www.dsprelated.com/josimages_new/mdft/img1544.png)
(assuming the original DFT included all nonzero samples of
![$ h$](http://www.dsprelated.com/josimages_new/mdft/img1167.png)
).
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