Analysis of Fisher Information and the Cramér-Rao Bound for Nonlinear Parameter Estimation after Random Compression

In this paper, we analyze the impact of compression with complex random matrices on Fisher information and the Cramér-Rao Bound (CRB) for estimating unknown parameters in the mean value function of a complex multivariate normal distribution. We consider the class of random compression matrices whose distribution is right-unitarily invariant. The compression matrix whose elements are i.i.d. standard complex normal random variables is one such matrix. We show that for all such compression matrices, the Fisher information matrix has a complex matrix beta distribution. We also derive the distribution of CRB. These distributions can be used to quantify the loss in CRB as a function of the Fisher information of the noncompressed data. In our numerical examples, we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in CRB.