Diagonalizing properties of the discrete cosine transforms

Since its introduction in 1974 by Ahmed et al., the discrete cosine transform (DCT) has become a significant tool in many areas of digital signal processing, especially in signal compression. There exist eight types of discrete cosine transforms (DCTs). We obtain the eight types of DCTs 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 DCTs 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 the block size N tends to infinity. As a particular result, we prove that the eight types of DCTs are asymptotically optimal for all finite-order Markov processes. We finally study the decorrelating power of the DCTs, obtaining expressions that show the decorrelating behavior of each DCT with respect to any stationary processes.