### Frequency Response

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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