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

The covariance matrix of the Fourier coefficients of
N
- 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
N \rightarrow \infty
. 2) If the covariance sequence is only square summable, the magnitude of the covariance elements sufficiently far from the diagonal converges to zero as
N \rightarrow \infty
. 3) If the covariance sequence is square summable, the weak norm of the matrix containing only the off-diagonal elements converges to zero as
N \rightarrow \infty
. The rates of convergence are also determined when the covariance sequence satisfies additional conditions.