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?

  

Dolph-Chebyshev Window

The Dolph-Chebyshev Window (or Chebyshev window, or Dolph window) minimizes the Chebyshev norm of the side lobes for a given main lobe width $ 2\omega_c$ [60,97], [212, p. 94]:

$\displaystyle \displaystyle
\hbox{min}_{w,\sum w=1} \left\Vert\,\hbox{sidelobes...
...1} \left\{\hbox{max}_{\omega>\omega_c} \left\vert W(\omega)\right\vert\right\}
$

The Chebyshev norm is also called the $ L-infinity$ norm, uniform norm, minimax norm, or simply the maximum absolute value.

An equivalent formulation is to minimize main lobe width subject to a side-lobe specification:

$\displaystyle \displaystyle
\left. \min_{w,W(0)=1}(\omega_c) \right\vert _{\,\l...
...(\omega)\,\right\vert \leq\, c_\alpha,\; \forall \vert\omega\vert\geq\omega_c}
$

The optimal Dolph-Chebyshev window transform can be written in closed form [60,97,100,147]:

\begin{eqnarray*}
W(\omega_k) &=& \frac{\cos\left\{M\cos^{-1}\left[\beta\cos\le...
...1}{M}\cosh^{-1}(10^\alpha)\right], \qquad (\alpha\approx 2,3,4).
\end{eqnarray*}

The zero-phase Dolph-Chebyshev window, $ w(n)$, is then computed as the inverse DFT of $ W(\omega_k)$.4.9 The $ \alpha $ parameter controls the side-lobe level via the formula [147]

   Side-Lobe Level in dB$\displaystyle = -20\alpha.
$

Thus, $ \alpha=2$ gives side-lobes which are $ 40$ dB below the main-lobe peak. Since the side lobes of the Dolph-Chebyshev window transform are equal height, they are often called ``ripple in the stop-band'' (thinking now of the window transform as a low-pass filter frequency response). The smaller the ripple specification, the larger $ \omega_c$ has to become to satisfy it, for a given window length $ M$.

The Chebysev window can be regarded as the impulse response of an optimal Chebyshev lowpass filter having a zero-width passband (i.e., the main lobe consists of two ``transition bands''--see Appendix E regarding FIR filter design more generally).



Subsections
Previous: Kaiser and DPSS Windows Compared
Next: Matlab for the Dolph-Chebyshev Window

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? )