Analysis of fisher information and the Cramer-Rao bound for nonlinear parameter estimation after compressed sensing

In this paper, we analyze the impact of compressed sensing with random matrices on Fisher information and the CRB for estimating unknown parameters in the mean value function of a multivariate normal distribution. We consider the class of random compression matrices that satisfy a version of the Johnson-Lindenstrauss lemma, and we derive analytical lower and upper bounds on the CRB for estimating parameters from randomly compressed data. These bounds quantify the potential loss in CRB as a function of Fisher information of the non-compressed data. In our numerical examples, we consider a direction of arrival estimation problem and compare the actual loss in CRB with our bounds.