DCT AND ITS RELATIONS TO THE KARHUNEN-LOEVE TRANSFORM

This chapter discusses discrete cosine transform (DCT) and its relation to the Karhunen-Loeve transform. DCT is not simply discretized versions of the Fourier cosine transform properties. The transforms were obtained originally to diagonalize certain matrices. The chapter presents the Karhunen-Loeve transform (KLT), first discussed by Karhunen and later by Loeve. This is a series representation of a given random function. The orthogonal basis functions are obtained as the eigenvectors of the corresponding auto-covariance matrix. This transform is optimal in that it completely decorrelates the random function (that is, the signal sequence) in the transform domain. The chapter discusses the formal treatment of asymptotic equivalence between classes of matrices and their orthonormal representations. Such considerations are then extended by using quadrature approximation to the actual generation of discrete unitary transforms. The outcome of this development of transforms based on fast algorithms leads to a rather general and interesting procedure for generating certain discrete unitary transforms for a given class of signal covariance matrices.