On coding and filtering stationary signals by discrete Fourier transforms (Corresp.)

This correspondence concerns real-time Fourier processing of stationary data and examines the widespread belief that coefficients of the discrete Fourier transform (DFT) are "almost" uncorrelated. We first show that any uniformly bounded Toeplitz covariance matrix is asymptotically equivalent to a nonstandard circulant matrix derived from the DFT of . We then derive bounds on a normed distance between and for finite , and show that for finite-order Markov processes. Finally we demonstrate that the performance degradation resulting from the use of DFT (as opposed to Karhunen-Loève expansion) in coding and filtering is proportional to and therefore vanishes as the inverse square root of the block size when .