Dolph-Chebyshev Window

Minimize the Chebyshev norm of the side lobes, e.g.,

$$\begin{eqnarray*} \displaystyle && \mbox{min}_{w,\sum w=1} \left\Vert\,\hbox{sidelobes($W$)}\,\right\Vert _\infty\\ [5pt] &\equiv& \mbox{min}_{w,\sum w=1} \left\{\mbox{max}_{\omega>\omega_c} \left\vert W(\omega)\right\vert\right\} \end{eqnarray*}$$

Alternatively, minimize main lobe width subject to a sidelobe spec:

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

Closed-Form Window Transform (Dolph):

$$\begin{eqnarray*} W(\omega_k) &=& \frac{\cos\left\{M\cos^{-1}\left[\Gamma\cos\left(\frac{\pi k}{M}\right) \right]\right\}}{\cosh\left[M\cosh^{-1} (\Gamma)\right]}, \quad (\vert k\vert\leq M-1) \\ \Gamma &=& \cosh \left[\frac{1}{M}\cosh^{-1}(10^\alpha)\right] \;\ge\;1, \qquad (\alpha\approx 2,3,4) \end{eqnarray*}$$

• Window $w = \hbox{\sc IDFT}(W)$ [zero-centered case]
  or IDFT of $(-1)^k W(\omega_k)$ for causal case
• $\alpha$ controls sidelobe level (``stopband ripple''):
     Side-Lobe Level in dB$$= -20\alpha.$$

• smaller ripple $\Rightarrow$ larger $\omega_c$
• see matlab function ``chebwin(M,ripple)''