An upper bound for the directivity index of superdirective acoustic vector sensor arrays

Approximate analytical expressions of the white noise gain (WNG) for two superdirective acoustic vector sensor arrays are provided, which disclose the strong dependence of the tradeoff between the WNG and the directivity index (DI) on the highest order of the modes for the pattern synthesis. The considered arrays are a uniform linear array and a uniform circular array. A condition on the WNG that ensures a high array gain in the two-dimensional homogeneous and isotropic noise field is deduced. Using this condition, an upper bound on the highest order of the modes for the pattern synthesis can be derived, and hence the maximum DI can be determined. The presented results are not strictly limited to the two array geometries considered herein, and can be extended to other superdirective acoustic array designs.