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

Chapters

FFT Spectral Analysis Software

See Also

Embedded SystemsFPGA
Chapter Contents:

Search Spectral Audio Signal Processing

  

Book Index | Global Index


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

  

Chebyshev Polynomials

Figure 3.28:
\includegraphics[width=\twidth]{eps/first-even-chebs-c}

The $ n$th Chebyshev polynomial may be defined by

$\displaystyle T_n(x) = \left\{\begin{array}{ll}
\cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt]
\cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\
\end{array}\right..
$

The first three even-order cases are plotted in Fig.3.29. (We will only need the even orders for making Chebyshev windows.) Clearly, $ T_0(x)=1$ and $ T_1(x)=x$. Using the double-angle trig formula $ \cos(2\theta)=2\cos^2(\theta)-1$, it can be verified that

$\displaystyle T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)
$

for $ n\ge 2$. The following properties of the Chebyshev polynomials are well known:
  • $ T_n(x)$ is an $ n$th-order polynomial in $ x$.
  • $ T_n(x)$ is an even function when $ n$ is an even integer, and odd when $ n$ is odd.
  • $ T_n(x)$ has $ n$ zeros in the open interval $ (-1,1)$, and $ n+1$ extrema in the closed interval $ [-1,1]$.
  • $ T_n(x)>1$ for $ x>1$.


Previous: Dolph-Chebyshev Window Theory
Next: Dolph-Chebyshev Window Definition

Order a Hardcopy of Spectral Audio Signal Processing


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 Associate Professor (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 http://ccrma.stanford.edu/~jos/ for details.


Comments


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? )