NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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A First Look at Taylor Series
Any ''smooth'' function can be expanded in the form of a Taylor series:
This can be written more compactly as
An informal derivation of this formula for is given in §3.2 and §3.3. Clearly, since many derivatives are involved, a Taylor series expansion is only possible when the function is so smooth that it can be differentiated again and again. Fortunately for us, all audio signals can be defined so as to be in that category. This is because hearing is bandlimited to kHz, and any sum of sinusoids up to some maximum frequency, i.e., any audible signal, is infinitely differentiable. (Recall that and , etc.). See §3.6 for more on this topic.