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Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors

Liu, Chun-Lin and Vaidyanathan, P. P. (2017) Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors. Digital Signal Processing, 61 . pp. 43-61. ISSN 1051-2004. doi:10.1016/j.dsp.2016.04.011.

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The Cramér–Rao bound (CRB) offers a lower bound on the variances of unbiased estimates of parameters, e.g., directions of arrival (DOA) in array processing. While there exist landmark papers on the study of the CRB in the context of array processing, the closed-form expressions available in the literature are not easy to use in the context of sparse arrays (such as minimum redundancy arrays (MRAs), nested arrays, or coprime arrays) for which the number of identifiable sources D exceeds the number of sensors N . Under such situations, the existing literature does not spell out the conditions under which the Fisher information matrix is nonsingular, or the condition under which specific closed-form expressions for the CRB remain valid. This paper derives a new expression for the CRB to fill this gap. The conditions for validity of this expression are expressed as the rank condition of a matrix defined based on the difference coarray. The rank condition and the closed-form expression lead to a number of new insights. For example, it is possible to prove the previously known experimental observation that, when there are more sources than sensors, the CRB stagnates to a constant value as the SNR tends to infinity. It is also possible to precisely specify the relation between the number of sensors and the number of uncorrelated sources such that these conditions are valid. In particular, for nested arrays, coprime arrays, and MRAs, the new expressions remain valid for D=O(N^2), the precise detail depending on the specific array geometry.

Item Type:Article
Related URLs:
URLURL TypeDescription
Liu, Chun-Lin0000-0003-3135-9684
Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© 2016 Elsevier Inc. Available online 9 May 2016. This work was supported in parts by the ONR grant N00014-15-1-2118, and the California Institute of Technology.
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-15-1-2118
Subject Keywords:Cramér–Rao bounds; Fisher information matrix; Sparse arrays; Coprime arrays; Difference coarrays
Record Number:CaltechAUTHORS:20170216-084233440
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Official Citation:Chun-Lin Liu, P.P. Vaidyanathan, Cramér–Rao bounds for coprime and other sparse arrays, which find more sources than sensors, Digital Signal Processing, Volume 61, February 2017, Pages 43-61, ISSN 1051-2004, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:74364
Deposited By: Tony Diaz
Deposited On:16 Feb 2017 17:03
Last Modified:11 Nov 2021 05:26

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