**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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Appendix A: Linear Time-Invariant Filters and ConvolutionA reason for the importance of convolution is that

every linear time-invariant system. Thus, in the convolution equation^{8.9}can be represented by a convolution

we may interpret as theinputsignal to a filter, as theoutputsignal, and as thedigital filter, as shown in Fig. 8.11.The

impulseor ''unit pulse'' signal is defined by

For example, for , . The impulse signal is theidentity elementunder convolution, since

If we set in the filter equation above, we get

Thus, is theimpulse responseof the filter.It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response. No matter what the LTI system is, we can give it an impulse, record what comes out, call it , and implement the system by convolving the input signal with the impulse response . In other words, every LTI system has a

convolution representationin terms of its impulse response.

Subsections