### Dolph-Chebyshev Window Theory

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

#### Chebyshev Polynomials

The th Chebyshev polynomial may be defined by

 (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, and . Using the double-angle trig formula , it can be verified that

 (4.47)

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

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

 (4.48)

where is defined by the desired ripple specification:

 (4.49)

where is the main lobe edge frequency'' defined by

 (4.50)

Expanding in terms of complex exponentials yields

 (4.51)

where . Thus, the coefficients give the length Dolph-Chebyshev window in zero-phase form.

#### Dolph-Chebyshev WindowMain-Lobe Width

Given the window length and ripple magnitude , the main-lobe width may be computed as follows [155]:

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 and main-lobe width , the required window length is given by [155]

 (4.52)

For (the typical case), the denominator is close to , and we have

 (4.53)

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

 (4.54)

where is the parameter often used to design Kaiser windows3.9).
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