Diagonalizing Propexties of the
Victoria Shnchez, Antonio M. Peinado, Clara Segura, Antonio J. Rubio · 1995
Abstruct-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 same way as the discrete Fourier transform (DFT) can be obtained as the eigenvectors of an arbitrary circulant matrix. These matrices can be decomposed as the sum of a symmetric Toeplitz matrix plus a Hankel or close to Hankel matrix scaled by some constant factors. We also show that all the previously proposed generating matrices for the DCT’s are simply particular cases of these general matrix forms. Using these matrices, we obtain, for each DCT, a class of stationary processes verifying certain conditions with respect to which the corresponding DCT has a good asymptotic behavior in the sense that it approaches Karhunen-Loeve transform performance as block size N tends to infinity. As a particular result, we prove that the eight types of DCT’s are asymptotically optimal for all finiteorder 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 processes.