Study of the detection and resolution capabilities of a one-dimensional array of sensors by using differential geometry

The paper is concerned with the investigation of the detection and resolution capabilities inherent to the manifold of a linear (1-dimensional) array of omnidirectional sensors, not necessarily uniform. An initial discussion, associated with the shape of the manifold of a linear array, is presented by the application of differential geometry in complex space. The shape of the manifold, which for the case of linear arrays is a hyperhelix, can be fully characterised by its length and curvatures. In this paper it is shown that the array manifold may be locally approximated by a circular arc whose radius is the inverse of the manifold's first curvature. By using the above approximation, expressions for the thresholds of resolution and detection are derived as a function of the manifold length and first curvature even for sources of unequal powers. Two laws are established: the fourth-root law which is related to the resolution problem and the square-root law which is related to the detection problem. The overall conclusion is that the thresholds predicted via differential geometry concepts represent basic performance measures for assessing and comparing the performance of different linear array geometries irrespective of the algorithm used.