Abstract
We consider a model of the form y = Ax + n, where x isin CM is sparse with at most L nonzero coefficients in unknown locations, y isin CN is the observation vector, A isin CN times M is the measurement matrix and n isin CN 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) .
Highlights
We consider the problem of estimating a sparse vector based on noisy observations
We construct an estimator based on Shannon theory and the notion of typicality [4] that asymptotically achieves the Cramér–Rao bound on the estimation error of the genie-aided estimator (GAE) without the knowledge of the locations of the nonzero elements of, for Gaussian measurement matrices
The problem of finding efficient and low-complexity estimators that achieve the Cramér–Rao bound for noisy compressive sampling still remains open
Summary
We consider the problem of estimating a sparse vector based on noisy observations. Suppose that we have a compressive sampling (Please see [2] and [5]) model of the form (1). Each iteration requires solving an system of linear equations and the iterations are repeated until convergence is attained In this correspondence, we construct an estimator based on Shannon theory and the notion of typicality [4] that asymptotically achieves the Cramér–Rao bound on the estimation error of the GAE without the knowledge of the locations of the nonzero elements of , for Gaussian measurement matrices. The problem of finding efficient and low-complexity estimators that achieve the Cramér–Rao bound for noisy compressive sampling still remains open. The outline of this correspondence follows next.
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