Analysis of Fisher Information and the Cram\'{e}r-Rao Bound for Nonlinear Parameter Estimation after Compressed Sensing

摘要

In this paper, we analyze the impact of compressed sensing with complexrandom matrices on Fisher information and the Cram\'{e}r-Rao Bound (CRB) forestimating unknown parameters in the mean value function of a complexmultivariate normal distribution. We consider the class of random compressionmatrices whose distribution is right-orthogonally invariant. The compressionmatrix whose elements are i.i.d. standard normal random variables is one suchmatrix. We show that for all such compression matrices, the Fisher informationmatrix has a complex matrix beta distribution. We also derive the distributionof CRB. These distributions can be used to quantify the loss in CRB as afunction of the Fisher information of the non-compressed data. In our numericalexamples, we consider a direction of arrival estimation problem and discuss theuse of these distributions as guidelines for choosing compression ratios basedon the resulting loss in CRB.