Dolph-Chebyshev Window Theory
In this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.
The th Chebyshev polynomial may be defined by
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, and . Using the double-angle trig formula , it can be verified that
for . The following properties of the Chebyshev polynomials are well known:
- is an th-order polynomial in .
- is an even function when is an even integer, and odd when is odd.
- has zeros in the open interval , and extrema in the closed interval .
- for .
Dolph-Chebyshev Window Definition
where is defined by the desired ripple specification:
where is the ``main lobe edge frequency'' defined by
Expanding in terms of complex exponentials yields
where . Thus, the coefficients give the length Dolph-Chebyshev window in zero-phase form.
Dolph-Chebyshev Window Main-Lobe Width
Given the window length and ripple magnitude , the main-lobe width may be computed as follows :
This is the smallest main-lobe width possible for the given window length and side-lobe spec.
Dolph-Chebyshev Window Length Computation
For (the typical case), the denominator is close to , and we have
Thus, half the time-bandwidth product in radians is approximately
where is the parameter often used to design Kaiser windows (§3.9).
Matlab for the Gaussian Window
Chebyshev and Hamming Windows Compared