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 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
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