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

Abstract

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 boundedN times NToeplitz covariance matrixT_Nis asymptotically equivalent to a nonstandard circulant matrixC_Nderived from the DFT ofT_N. We then derive bounds on a normed distance betweenT_NandC_Nfor finiteN, and show thatmid T_N - C_N mid ^ 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 tomid T_N - C_N midand therefore vanishes as the inverse square root of the block sizeNwhenN rightarrow infty.