Appendix A: Linear Time-Invariant Filters and Convolution

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Appendix A: Linear Time-Invariant Filters and Convolution

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 Convolution

A reason for the importance of convolution is that every linear time-invariant system8.9 can be represented by a convolution. Thus, in the convolution equation

\

we may interpret $x$ as the input signal to a filter, $y$ as the output signal, and $h$ as the digital filter, as shown in Fig. 8.11.
Figure 8.11:The filter interpretation of convolution.
\

The impulse or “unit pulse” signal is defined by

\

For example, for $N=4$, $\. The impulse signal is the identity element under convolution, since
\

If we set $x=\ in the filter equation above, we get
\

Thus, $h$ is the impulse response of 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 $h(n)$, and implement the system by convolving the input signal $x$ with the impulse response $h$. In other words, every LTI system has a convolution representation in terms of its impulse response.



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