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 \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
.