### 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 Window Main-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 windows (§3.9).

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Chebyshev and Hamming Windows Compared