**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),

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

where is the th sample of the DFT of the input signal , and is the DFT of the output signal .

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University