On the residual correlation of finite-dimensional discrete Fourier transforms of stationary signals (Corresp.)

Abstract

The covariance matrix of the Fourier coefficients of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> - sampled stationary random signals is studied. Three theorems are established. 1) If the covariance sequence is summable, the magnitude of every off-diagonal covariance element converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . 2) If the covariance sequence is only square summable, the magnitude of the covariance elements sufficiently far from the diagonal converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . 3) If the covariance sequence is square summable, the weak norm of the matrix containing only the off-diagonal elements converges to zero as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N \rightarrow \infty</tex> . The rates of convergence are also determined when the covariance sequence satisfies additional conditions.