Implements the Fast Cosine Transform for transformation of one-dimensional real data sets. For reference, see James S. Walker,
Fast Fourier Transforms, chapter 3 (ISBN 0849371635).
There are several variants of the discrete cosine transform. The present implementation corresponds to DCT-I, with various normalization conventions, which are specified by the parameter {@link DctNormalization}.
DCT-I is equivalent to DFT of an even extension of the data series. More precisely, if x0, …, xN-1 is the data set to be cosine transformed, the extended data set x0#, …, x2N-3# is defined as follows
- xk# = xk if 0 ≤ k < N,
- xk# = x2N-2-k if N ≤ k < 2N - 2.
Then, the standard DCT-I y0, …, yN-1 of the real data set x0, …, xN-1 is equal to half of the N first elements of the DFT of the extended data set x0#, …, x2N-3#
yn = (1 / 2) ∑k=02N-3 xk# exp[-2πi nk / (2N - 2)] k = 0, …, N-1.
The present implementation of the discrete cosine transform as a fast cosine transform requires the length of the data set to be a power of two plus one (N = 2n + 1). Besides, it implicitly assumes that the sampled function is even.
@since 1.2