**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## LTI Filters and the Convolution Theorem

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

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

where we used the fact that is zero for and 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 is being obtained using ''time-limited interpolation in the frequency domain'' based on the samples . This interpolation is possible only when the frequency samples are sufficiently dense: for a length finite-impulse-response (FIR) filter , we require at least samples around the unit circle (length DFT) in order that 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:Theamplitude responseof a filter is defined as themagnitudeof the frequency response

From the convolution theorem, we can see that the amplitude response is thegainof the filter at frequency , since

Definition:Thephase responseof a filter is defined as thephaseof 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].