Performance bounds of complex-valued nonlinear estimators in learning systems

The widely linear estimator offers theoretical and practical advantages over the strictly linear counterpart for the processing of the generality of complex signals. When applying the widely linear technique to a nonlinear estimator of learning systems, the resulting augmented nonlinear estimator also empirically produces performance advantages over the standard (strictly) nonlinear estimator. However, the theoretical justification for this observation is still lacking in the literature. To this end, we establish performance bounds of the augmented nonlinear estimator over the strictly nonlinear one in this paper. By expanding the analytic nonlinear function with the first-order Taylor formula, we obtain the closed form of the MSE difference ? MSE and that of the complementary MSE difference CMSE for the two estimators. By jointly considering the MSE and the complementary MSE, it is theoretically established that ? MSE > 0 always holds for noncircular input signals, and MSE performance advantages always exist in both the real and imaginary channels over its strictly nonlinear counterpart except for the jointly circular case. The simulation results validate our theoretical analysis.