Optimal Sequential Energy Allocation for Inverse Problems

This paper investigates the advantages of adaptive waveform amplitude design for estimating parameters of an unknown channel/medium under average energy constraints. We present a statistical framework for sequential design (e.g., design of waveforms in adaptive sensing) of experiments that improves parameter estimation (e.g., unknown channel parameters) performance in terms of reduction in mean-squared error (MSE). We treat an N time step design problem for a linear Gaussian model where the shape of the N input design vectors (one per time step) remains constant and their amplitudes are chosen as a function of past measurements to minimize MSE. For N=2, we derive the optimal energy allocation at the second step as a function of the first measurement. Our adaptive two-step strategy yields an MSE improvement of at least 1.65 dB relative to the optimal nonadaptive strategy, but is not implementable since it requires knowledge of the noise amplitude. We then present an implementable design for the two-step strategy which asymptotically achieves optimal performance. Motivated by the optimal two-step strategy, we propose a suboptimal adaptive N-step energy allocation strategy that can achieve an MSE improvement of more than 5 dB for N=50. We demonstrate our general approach in the context of MIMO channel estimation and inverse scattering problems