On coding and filtering stationary signals by discrete Fourier transforms (Corresp.)
Jacob Pearl · IEEE Transactions on Information Theory · 1973
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 that\mid 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 to\mid T_N - C_N \midand therefore vanishes as the inverse square root of the block sizeNwhenN \rightarrow \infty.