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Dolph-Chebyshev Window Theory

In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.

Chebyshev Polynomials

Figure 3.34:
\includegraphics[width=\twidth]{eps/first-even-chebs-c}

The $ n$ th Chebyshev polynomial may be defined by

$\displaystyle T_n(x) = \left\{\begin{array}{ll} \cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt] \cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\ \end{array} \right..$ (4.46)

The first three even-order cases are plotted in Fig.3.35. (We will only need the even orders for making Chebyshev windows, as only they are symmetric about time 0.) Clearly, $ T_0(x)=1$ and $ T_1(x)=x$ . Using the double-angle trig formula $ \cos(2\theta)=2\cos^2(\theta)-1$ , it can be verified that

$\displaystyle T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)$ (4.47)

for $ n\ge 2$ . The following properties of the Chebyshev polynomials are well known:
  • $ T_n(x)$ is an $ n$ th-order polynomial in $ x$ .
  • $ T_n(x)$ is an even function when $ n$ is an even integer, and odd when $ n$ is odd.
  • $ T_n(x)$ has $ n$ zeros in the open interval $ (-1,1)$ , and $ n+1$ extrema in the closed interval $ [-1,1]$ .
  • $ T_n(x)>1$ for $ x>1$ .

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]

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

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

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

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

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

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

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

Dolph-Chebyshev Window Main-Lobe Width

Given the window length $ M$ and ripple magnitude $ r$ , the main-lobe width $ 2\omega_c$ may be computed as follows [155]:

\begin{eqnarray*} x_0 &=& \cosh\left[\frac{\cosh^{-1}\left(\frac{1}{r}\right)}{M-1}\right]\\ \omega_c &=& 2\cos^{-1}\left(\frac{1}{x_0}\right) \end{eqnarray*}

This is the smallest main-lobe width possible for the given window length and side-lobe spec.

Dolph-Chebyshev Window Length Computation

Given a prescribed side-lobe ripple-magnitude $ r$ and main-lobe width $ 2\omega_c$ , the required window length $ M$ is given by [155]

$\displaystyle M = 1 + \frac{\cosh^{-1}(1/r)}{\cosh^{-1}[\sec(\omega_c/2)]}.$ (4.52)

For $ \omega_c\ll\pi$ (the typical case), the denominator is close to $ \omega_c/2$ , and we have

$\displaystyle M \approx 1 + \frac{2}{\omega_c}\cosh^{-1}\left(\frac{1}{r}\right)$ (4.53)

Thus, half the time-bandwidth product in radians is approximately

$\displaystyle \beta \isdefs (M-1) \omega_c\approx 2\cosh^{-1}\left(\frac{1}{r}\right),$ (4.54)

where $ \beta $ is the parameter often used to design Kaiser windows (§3.9).

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