# Asymptotic Achievability of the Cramér–Rao Bound For Noisy Compressive Sampling

 Title: Asymptotic Achievability of the Cramér–Rao Bound For Noisy Compressive Sampling Author: Kalouptsidis, Nicholas; Tarokh, Vahid; Babadi, Behtash Note: Order does not necessarily reflect citation order of authors. Citation: Babadi, Behtash, Nicholas Kalouptsidis, and Vahid Tarokh. 2009. Asymptotic achievability of the Cramér–Rao Bound for noisy compressive sampling. IEEE Transactions on Signal Processing 57(3): 1233-1236. Full Text & Related Files: Babadi_Asymptotic.pdf (835.9Kb; PDF) Abstract: We consider a model of the form ${bf y}={bf Ax}+{bf n}$, where ${bf x}inBBC^{M}$ is sparse with at most $L$ nonzero coefficients in unknown locations, ${bf y}inBBC^{N}$ is the observation vector, ${bf A}inBBC^{Ntimes M}$ is the measurement matrix and ${bf n}inBBC^{N}$ is the Gaussian noise. We develop a CramÉr–Rao bound on the mean squared estimation error of the nonzero elements of ${bf x}$, corresponding to the genie-aided estimator (GAE) which is provided with the locations of the nonzero elements of ${bf x}$. Intuitively, the mean squared estimation error of any estimator without the knowledge of the locations of the nonzero elements of ${bf x}$ 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 CramÉr–Rao bound without any knowledge of the locations of the nonzero elements of ${bf x}$ as $Nrightarrowinfty$ , for ${bf A}$ a random Gaussian matrix whose elements are drawn i.i.d. according to ${cal N}(0,1)$ . Published Version: http://dx.doi.org/10.1109/TSP.2008.2010379 Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2757546 Downloads of this work: