On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities

Abstract

The problem of reconstructing the normalized covariance function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(t)</tex> of a zero-mean stationary Gaussian process observed through a zero-memory nonlinearity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x)</tex> is considered, when the nonlinearity and the correlation function or the second-order distribution of the output process are known. Three kinds of results are established. (i) Arbitrary covariances can be reconstructed for certain nonlinearities, including monotonic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> , appropriate interval windows, and certain quite general <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> . (ii) Certain covariances can be reconstructed for arbitrary nonlinearities: included here are positive covariances <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\geq 0)</tex> , covariances with rational spectral densities, and bandlimited covariances. (iii) Certain covariances, satisfying rather weak conditions, that can easily be checked in terms of the output correlation function, can be reconstructed for certain nonlinearities that include symmetric as well as nonsymmetric <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</tex> .