Chun-Lin Liu – Topics – Cramér-Rao Bounds for Sparse Arrays

Cramér-Rao Bounds for Sparse Arrays


The Cramér-Rao bound as a function of the number of sources with the coprime array [1]. (View larger)

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.

Our Papers

  1. C.-L. Liu and P. P. Vaidyanathan, ‘‘Cramér-Rao Bounds for Coprime and Other Sparse Arrays, which Find More Sources than Sensors,’’ Digital Signal Processing, vol. 61, pp. 43-61, Feb. 2017, Special Issue on Coprime Sampling and Arrays.

    1. DOI and Full text

    2. Sample code for Fig. 3 Please execute main_CRB_vs_SNR.m for the plots.

  2. C.-L. Liu and P. P. Vaidyanathan, ‘‘New Cramér-Rao Bound Expressions for Coprime and Other Sparse Arrays,’’ in Proc. of the Ninth IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM 2016), pp. 1-5, Rio de Janeiro, Brazil, July 2016.

    1. DOI and Full text

    2. Slides