Asymptotic Achievability of the CramÉr–Rao Bound for Noisy Compressive Sampling

We consider a model of the form y = Ax + n, where x isin C^M is sparse with at most L nonzero coefficients in unknown locations, y isin C^N is the observation vector, A isin C^N times M is the measurement matrix and n isin C^N is the Gaussian noise. We develop a Cramer-Rao bound on the mean squared estimation error of the nonzero elements of x, corresponding to the genie-aided estimator (GAE) which is provided with the locations of the nonzero elements of x. Intuitively, the mean squared estimation error of any estimator without the knowledge of the locations of the nonzero elements of x is no less than that of the GAE. Assuming that L/N is fixed, we establish the existence of an estimator that asymptotically achieves the Cramer-Rao bound without any knowledge of the locations of the nonzero elements of x as N rarr infin , for A a random Gaussian matrix whose elements are drawn i.i.d. according to N (0,1) .