Here, we will prove a Gabor L 2 (R)-frame-based KL-like expansion for a wide-sense stationary (WSS) random process X(t). The obtained random coefficients in this procedure are nearly orthogonal or uncorrelated in the sense that they constitute a Hilbert frame in the space spanned by the random process H(X). In some ways, these derivations resemble previous works using orthonormal bases, such as wavelets. A possible advantage of these kinds of results over the classic KL-expansion is that there is no need to diagonalize any covariance operator. The sufficient conditions are given in terms of the existence of the power spectral density of X(t) and the determination of certain whitening filters which are applied to the elements of the original frame. However, in contrast to other works, we will also prove that the proposed sufficient conditions are very nearly necessary.
Florentin et al. (Mon,) studied this question.