On the approximation of the discrete Karhunen-Loeve transform for stationary processes

Abstract

Suboptimal fast transforms are useful substitutes to the optimal Karhunen-Loève transform (KLT). The selection of an efficient approximation for the KLT must be done with respect to some performance criterion that might differ from one application to another. A general class of criterion functions including most of the commonly used performance measures is introduced. They are shown to be optimized by the KLT. Various properties of the eigenvectors of the symmetric Toeplitz covariance matrix of a wide sense stationary process are reviewed. Several transforms such as the complex or real, odd and even Fourier transforms (DFT, DOFT, DREFT, DROFT), the cosine and even sine transforms (DCT, DEST) are obtained from the decomposition of a symmetric Toeplitz matrix in the sum of a circulant and a skew circulant matrix. These transforms are compared on the basis of a general performance criterion and appear to be good substitutes for the optimal KLT. Finally, it is shown that these transforms are asymptotically equivalent in performances to the KLT of an arbitrary wide sense stationary process.