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

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1 Author(s)

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 \times N Toeplitz covariance matrix T_N is asymptotically equivalent to a nonstandard circulant matrix C_N derived from the DFT of T_N . We then derive bounds on a normed distance between T_N and C_N for finite N , 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 \mid and therefore vanishes as the inverse square root of the block size N when N \rightarrow \infty .

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IEEE Transactions on Information Theory  (Volume:19 ,  Issue: 2 )