DSPRelated.com
Free Books
   Free Books    Spectral Audio Signal Processing  

Chebyshev Polynomials

Figure 3.34:
\includegraphics[width=\twidth]{eps/first-even-chebs-c}

The $ n$ th Chebyshev polynomial may be defined by

$\displaystyle T_n(x) = \left\{\begin{array}{ll} \cos[n\cos^{-1}(x)], & \vert x\vert\le1 \\ [5pt] \cosh[n\cosh^{-1}(x)], & \vert x\vert>1 \\ \end{array} \right..$ (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, $ T_0(x)=1$ and $ T_1(x)=x$ . Using the double-angle trig formula $ \cos(2\theta)=2\cos^2(\theta)-1$ , it can be verified that

$\displaystyle T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x)$ (4.47)

for $ n\ge 2$ . The following properties of the Chebyshev polynomials are well known:
  • $ T_n(x)$ is an $ n$ th-order polynomial in $ x$ .
  • $ T_n(x)$ is an even function when $ n$ is an even integer, and odd when $ n$ is odd.
  • $ T_n(x)$ has $ n$ zeros in the open interval $ (-1,1)$ , and $ n+1$ extrema in the closed interval $ [-1,1]$ .
  • $ T_n(x)>1$ for $ x>1$ .

Next Section:
Dolph-Chebyshev Window Definition
Previous Section:
Kaiser and DPSS Windows Compared