DIAGONALIZING PROPERTIES OF THE DISCRETE COSINE TRANSFORMS

There exist eight types of discrete cosine transforms (DCT's). In this paper, we obtain the eight types of DCT's as the complete orthonormal set of eigenvectors generated by a general form of matrices in the sa me way as the discrete Fourier transform (DFT) can be obtained as the eigenvectors of an arbitrary circulant matrix, These matrices can be d ecomposed as the sum of a symmetric Toeplitz matrix plus a Hankel or c lose to Hankel matrix scaled by some constant factors. We also show th at all the previously proposed,generating matrices for the DCT's are s imply particular cases of these general matrix forms. Using these matr ices, we obtain, for each DCT, a class of stationary processes verifyi ng certain conditions with respect to which the corresponding DCT has a good asymptotic behavior in the sense that it approaches Karhunen-Lo eve transform performance as block size N tends to infinity. As a part icular result, we prove that the eight types of DCT's are asymptotical ly optimal for all finite-order Markov processes, We finally study the decorrelating power of the DCT's, obtaining expressions that show the decorrelating behavior of each DCT with respect to any stationary pro cesses.