Discrete Chebyshev transform ~ INDAHNYA BERBAGI

Kamis, 12 Juni 2014

Discrete Chebyshev transform

In applied mathematics, the discrete Chebyshev transform (DCT), named after Pafnuty Chebyshev, is one of either of two main varieties of DCTs: the discrete Chebyshev transform on the 'roots' grid of the Chebyshev polynomials of the first kind

T_n (x)

, and the discrete Chebyshev transform on the 'extrema' grid of the Chebyshev polynomials of the first kind.

The discrete chebyshev transform of u(x) at the points

{x_n}

is given by:

a_m =\frac{p_m}{N}\sum_{n=0}^{N-1} u(x_n) T_m (x_n)

where:

x_n = -\cos\left(\frac{\pi}{N} (n+\frac{1}{2})\right)

a_m = \frac{p_m}{N} \sum_{n=0}^{N-1} u(x_n) \cos\left(m \cos^{-1}(x_n)\right)

where

p_m =1 \Leftrightarrow m=0

and

p_m = 2

otherwise.

Using the definition of

x_n

a_m =\frac{p_m}{N} \sum_{n=0}^{N-1} u(x_n) \cos\left(\frac{m\pi}{N}(N+n+\frac{1}{2}) \right)

a_m =\frac{p_m}{N} \sum_{n=0}^{N-1} u(x_n) (-1)^m\cos\left(\frac{m\pi}{N}(n+\frac{1}{2}) \right)

and its inverse transform:

u_n =\sum_{m=0}^{N-1} a_m T_m (x_n)

(This so happens to the standard Chebyshev series evaluated on the roots grid.)

u_n =\sum_{m=0}^{N-1} a_m \cos\left(\frac{m\pi}{N}(N+n+\frac{1}{2}) \right)

\therefore u_n =\sum_{m=0}^{N-1} a_m (-1)^m\cos\left(\frac{m\pi}{N}(n+\frac{1}{2}) \right)

This can readily be obtained by manipulating the input arguments to a discrete cosine transform.

Sumber : http://en.wikipedia.org/wiki/Discrete_Chebyshev_transform

INDAHNYA BERBAGI

Kamis, 12 Juni 2014