Dolph-Chebyshev Window Definition

Let $M$ denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by [155]

$$W(\omega) = \frac{T_{M-1}[x_0 \cos(\omega/2)]}{T_{M-1}(x_0)}$$ (4.48)

where $x_0>1$ is defined by the desired ripple specification:

$$\vert W(\omega)\vert \le r = \frac{1}{T_{M-1}(x_0)}, \quad \forall\vert\omega\vert\ge\omega_c,$$ (4.49)

where $\omega_c$ is the ``main lobe edge frequency'' defined by

$$\omega_c \isdefs 2\cos^{-1}\left[\frac{1}{x_0}\right].$$ (4.50)

Expanding the trigonometric polynomial $W(\omega)$ in terms of complex exponentials yields

$$W(\omega) = \sum_{n=-M_h}^{M_h} w(n) e^{-j \omega n}$$ (4.51)

where $M_h\isdef (M-1)/2$ . Thus, the coefficients $w(n)$ give the length $M$ Dolph-Chebyshev window in zero-phase form.