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 ).
Definition: The amplitude response of a filter is defined as the magnitude of the frequency response
From the convolution theorem, we can see that the amplitude response is the gain of the filter at frequency , since
where is the th sample of the DFT of the input signal , and is the DFT of the output signal .