LTI Filters and the Convolution Theorem

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). LTI Filters and the Convolution Theorem

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LTI Filters and the Convolution Theorem



Definition: The frequency response of an LTI filter is defined as the Fourier transform of its impulse response. In particular, for finite, discrete-time signals $h\, the sampled frequency response is defined as

\

The complete frequency response is defined using the DTFT, i.e.,
\

where we used the fact that $h(n)$ is zero for $n<0$ and $n>N-1$ to truncate the summation limits. Thus, the infinitely zero-padded DTFT can be obtained from the DFT by simply replacing $\ by $\. In principle, the continuous frequency response $H(\ is being obtained using “time-limited interpolation in the frequency domain” based on the samples $H(\. This interpolation is possible only when the frequency samples $H(\ are sufficiently dense: for a length $N$finite-impulse-response (FIR) filter $h$, we require at least $N$ samples around the unit circle (length $N$ DFT) in order that $H(\ be sufficiently well sampled in the frequency domain. This is of course the dual of the usual sampling rate requirement in the time domain.8.10



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 $G(k)$is the gain of the filter at frequency $\, since
\



Definition: The phase response of a filter is defined as the phase of the frequency response

\

From the convolution theorem, we can see that the phase response$\ is the phase-shift added by the filter to an input sinusoidal component at frequency $\, since
\

The subject of this section is developed in detail in [1].

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