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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$ [61,101], [224, p. 94]:

$\displaystyle \displaystyle \hbox{min}_{w,\sum w=1} \left\Vert\,\hbox{sidelobes($W$)}\,\right\Vert _\infty \isdefs \hbox{min}_{w,\sum w=1} \left\{\hbox{max}_{\omega>\omega_c} \left\vert W(\omega)\right\vert\right\}$ (4.43)

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:

$\displaystyle \displaystyle \left. \min_{w,W(0)=1}(\omega_c) \right\vert _{\,\left\vert W(\omega)\,\right\vert \leq\, c_\alpha,\; \forall \vert\omega\vert\geq\omega_c}$ (4.44)

The optimal Dolph-Chebyshev window transform can be written in closed form [61,101,105,156]:

W(\omega_k) &=& \frac{\cos\left\{M\cos^{-1}\left[\beta\cos\left(\frac{\pi k}{M}\right)
\right]\right\}}{\cosh\left[M\cosh^{-1} (\beta)\right]},
\qquad k=0,1,2,\ldots,M-1 \\
\beta &=& \cosh \left[\frac{1}{M}\cosh^{-1}(10^\alpha)\right], \qquad (\alpha\approx 2,3,4).

The zero-phase Dolph-Chebyshev window, $ w(n)$ , is then computed as the inverse DFT of $ W(\omega_k)$ .4.14 The $ \alpha $ parameter controls the side-lobe level via the formula [156]

Side-Lobe Level in dB$\displaystyle = -20\alpha.$ (4.45)

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 lowpass 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 Chebyshev window can be regarded as the impulse response of an optimal Chebyshev lowpass filter having a zero-width pass-band (i.e., the main lobe consists of two ``transition bands''--see Chapter 4 regarding FIR filter design more generally).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University