Optimal experimental design for an inverse scattering problem

We consider the problem of imaging a medium using an array of sensors. More specifically, we are interested in optimally designing a sequence of experiments for probing a medium in order to form an image of the scatterers present in the medium. We consider the case where the received signal is corrupted by noise. We derive an expression for the mean square error for estimating the scatter coefficients and find the optimal sequence scheme that minimizes this error. Using the expression for the minimum mean square error, we show that we can do better than any beamforming approach to imaging. In the process, we also find the optimal energy allocation between the sequence of experiments. Closed-form expressions for the optimal transmission scheme and the minimum mean square error are provided.