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Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Dolph-Chebyshev Window

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Dolph-Chebyshev Window

The Dolph-Chebyshev Window (or Dolph window) minimizes the Chebyshev norm of the side lobes for a given main lobe width $ 2\omega_c$:

$\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 [88,91,140]:

\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)$.4.8 The $ \alpha $ parameter controls the side-lobe level via the formula [140]

   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 B regarding FIR filter design more generally).


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written by 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.


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