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]:

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

The Chebyshev norm is also called the $\ensuremath{L_\infty}$ 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:

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

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

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

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]

$$= -20\alpha.$$ (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).