Optimal Signal Design for Multi-Parameter Estimation Problems

In this paper, the optimal stochastic design of multiple parameters is investigated for an array of fixed estimators both in the absence and presence of an average power constraint. Two different performance criteria are considered: the total Bayes risk criterion and the maximum Bayes risk criterion. It is obtained that in the presence of K parameters and the average power constraint, the optimal stochastic parameter design results in randomization (time sharing) among at most two and (K+1) different signals for the total Bayes risk and the maximum Bayes risk criteria, respectively. The average transmitted signal powers corresponding to the optimal parameter design approaches are specified, and the characterization of the optimal approaches is provided in various scenarios. In addition, sufficient conditions are derived to specify when the stochastic parameter design or the deterministic parameter design is optimal. Finally, numerical examples are presented to investigate the theoretical results, and to illustrate performance improvements achieved via the proposed approaches.