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
Kaiser and DPSS Windows Compared