Contravariant adaptation on structured matrix spaces
Signal Processing
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The manifold of a general two- or three-dimensional (2-D or 3-D) array of sensors is studied using differential geometry. By considering the azimuth and elevation angles as the parameters of interest, a manifold surface is formed embedded in a multidimensional complex space, initially, this surface is investigated by establishing a number of differential geometry parameters associated with the array manifold. Then, the concept of development is proposed for mapping manifold surfaces (embedded in a multidimensional complex space) on the real 2-D parameter plane. The proposed mapping preserves certain differential geometry characteristics of the manifold surface and provides a simplified representation for the analysis. The potential benefits of this mapping, as well as of the proposed parameters, are demonstrated in the analysis of 3-D and planar arrays of omnidirectional sensors and in a number of potential applications, varying from array design to handling the array ambiguity problem