NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

As mentioned earlier, every audio signal can be regarded as infinitely differentiable due to the finite bandwidth of human hearing. One of the Fourier properties we will learn later in this course is thata signal cannot be both time limited and frequency limited.Therefore, by conceptually ''lowpass filtering'' every audio signal to reject all frequencies above kHz, we implicitly make every audio signal last forever! Another way of saying this is that the ''ideal lowpass filter `rings' forever''. Such fine points do not concern us in practice, but they are important for fully understanding the underlying theory. Since, in reality, signals can be said to have a true beginning and end, we must admit in practice that all signals we work with have infinite-bandwidth at turn-on and turn-off transients.^3.1

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