A form of Barankin's greatest lower bound on estimation error [7] is obtained, which is easy to compute and easy to interpret. This gives a lower bound on estimation error for non-linear modulation systems in an additive Gaussian noise back-ground. Threshold effects are included. This bound is applied to a set of pulse-position modulation waveforms designed to reduce threshold effects. It is shown that these signals do, in fact, offer reduced threshold levels (e.g.,\approx 3.5dB) with very small (\approx \frac{1}{2}dB) degradation in large signal performance.

An upper bound is obtained on the probability density of the estimate of the parametermwhen a nonlinear functions(t, m)is transmitted over a channel that adds Gaussian noise, and maximum likelihood or maximum a posteriori estimation is used. If this bound is integrated with a loss function, an upper bound on the average error is obtained. Nonlinear (below threshold) effects are included. The problem is viewed in a Euclidean space. Evaluation of the probability density can be reduced to integrating the probability density of the observation over part of a hyperplane. By bounding the integrand, and using a larger part of the hyperplane, an upper bound is obtained. The resulting bound on mean-square error is quite close for the cases calculated.