Sign in

Not a member? | Forgot your Password?

Search Online Books

Search tips

Free Online Books

Free PDF Downloads

A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

C++ Tutorial

Introduction of C Programming for DSP Applications

Fixed-Point Arithmetic: An Introduction

Cascaded Integrator-Comb (CIC) Filter Introduction


FFT Spectral Analysis Software

See Also

Embedded SystemsFPGA
Chapter Contents:

Search Mathematics of the DFT


Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?


A First Look at Taylor Series

Most ``smooth'' functions $ f(x)$ can be expanded in the form of a Taylor series expansion:

$\displaystyle f(x) = f(x_0) + \frac{f^\prime(x_0)}{1}(x-x_0)
+ \frac{f^{\prim...
+ \frac{f^{\prime\prime\prime}(x_0)}{1\cdot 2\cdot 3}(x-x_0)^3
+ \cdots .

This can be written more compactly as

$\displaystyle f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n,

where `$ n!$' is pronounced ``$ n$ factorial''. An informal derivation of this formula for $ x_0=0$ 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 $ 20$ kHz, and the audible spectrum of any sum of sinusoids is infinitely differentiable. (Recall that $ \sin^\prime(x)=\cos(x)$ and $ \cos^\prime(x)=-\sin(x)$, etc.). See §E.6 for more about this point.

Previous: Real Exponents
Next: Imaginary Exponents

Order a Hardcopy of Mathematics of the DFT

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


No comments yet for this page

Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )