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 $h\in{\bf C}^N$, the sampled frequency response may be defined as

$$H(\omega_k) \isdef \hbox{\sc DFT}_k(h).$$

The complete (continuous) frequency response is defined using the DTFT (see §B.1), i.e.,

$$H(\omega) \isdef \hbox{\sc DTFT}_\omega(\hbox{\sc ZeroPad}_\infty(h)) \isdef \sum_{n=0}^{N-1}h(n) e^{-j\omega n}$$

where the summation limits are truncated to $[0,N-1]$ because $h(n)$ is zero for $n<0$ and $n>N-1$. Thus, the DTFT can be obtained from the DFT by simply replacing $\omega_k$ by $\omega$, which corresponds to infinite zero-padding in the time domain. Recall from §7.2.8 that zero-padding in the time domain gives ideal interpolation of the frequency-domain samples $H(\omega_k)$ (assuming the original DFT included all nonzero samples of $h$).