Approximate initial-value solutions to a nonlinear Fredholm integral equation with semidegenerate covariance kernels in optimum receiver design

The suboptimum design of optimum receivers in nonlinear no-memory modulation systems in the presence of white Gaussian measurement noises is considered. Approximate initial-value solutions to a nonlinear Fredholm integral equation of the Hammerstein type with a symmetric semidegenerate (or separable) covariance kernel are given with the aid of an invariant embedding and the first-order Taylor series expansion. They are new online approximate filters and fixed-point smoothers using the covariance information of the signals and noises in the optimum receiver. A simple numerical example for a phase modulation system is shown.