NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Note that the output of the th delay element in Fig. B.1 is, , where is the input signalamplitude at time . The output signal is therefore
where we have used the convolution operator '''', defined in general for any two signals , as
An FIR filter thus operates by convolving the input signal with the filter's impulse response .