Maximally economic sparse arrays and Cantor arrays

Abstract

Sparse arrays, where the sensors are properly placed with nonuniform spacing, are able to resolve more uncorrelated sources than sensors. This ability arises from the property that the difference coarray, defined as the differences between sensor locations, has many more consecutive integers (hole-free) than the number of sensors. In some implementations, it might be preferable that a) the arrays be symmetric, b) that the arrays be maximally economic, that is, each sensor be essential, and c) that the coarray be hole-free. The essentialness property of a sensor means that if it is deleted, then the difference coarray changes. Existing sparse arrays, such as minimum redundancy arrays (MRA), nested arrays, and coprime arrays do not satisfy these three criteria simultaneously. It will be shown in this paper that Cantor arrays meet all the desired properties mentioned above, based on a comprehensive study on the structure of the difference coarray. Even though Cantor arrays were previously proposed in fractal array design, their coarray properties have not been studied earlier. It will also be shown that the Cantor array has a hole-free difference coarray of size N^(log_2^3) ≈ N^(1.585) where N is the number of sensors. This is unlike the sizes of difference coarrays of the MRA, nested array, coprime array (all O(N^2)), and uniform linear arrays (O(N))^1.

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