Most ``smooth'' functions can be expanded in the form of a
Taylor series expansion:

This can be written more compactly as

where `' is pronounced `` factorial''.
An informal derivation of this formula for is given in
Appendix E. 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 are in that category,
because hearing is bandlimited to below kHz, and the audible
spectrum of any sum of sinusoids is infinitely differentiable. (Recall
that
and
,
etc.). See §E.6 for more about this point.

Detailed derivation of the Discrete Fourier Transform (DFT) and its associated mathematics, including elementary audio signal processing applications and matlab programming examples.