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 N N Toeplitz covariance matrix TN is asymptotically equivalent to a nonstandard circulant matrix CN derived from the DFT of TN. We then derive bounds on a normed distance between TN and CN for finite N, and show that TN - CN ^ 2 = O (1/N) 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 TN - CN and therefore vanishes as the inverse square root of the block size N when N.