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

The problem of reconstructing the normalized covariance function
R(t)
of a zero-mean stationary Gaussian process observed through a zero-memory nonlinearity
f(x)
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
f
, appropriate interval windows, and certain quite general
f
. (ii) Certain covariances can be reconstructed for arbitrary nonlinearities: included here are positive covariances
(\geq 0)
, 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
f
.