Dolph-Chebyshev Window TheoryIn this section, the main elements of the theory behind the Dolph-Chebyshev window are summarized.
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 DefinitionLet denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by 
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 WidthGiven the window length and ripple magnitude , the main-lobe width may be computed as follows :
Dolph-Chebyshev Window Length ComputationGiven a prescribed side-lobe ripple-magnitude and main-lobe width , the required window length is given by 
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