CONE-ANGLE PARAMETRIZATION OF THE ARRAY MANIFOLD in DF SYSTEMS

The manifold of a planar array in a direction finding system may be considered as two families of azimuth and elevation curves, where the-parameter curves are as well as while the-parameter curves hyperhelical geodesic are. Since the-curves are not hyperhelical, their curvatures depend on and neither so analytical evaluation of curvatures of order greater than two can become exceedingly laborious and impractical. The advantages of having hyperhelical parameter curves are numerous. For one thing, all the curvatures of a hyperhelix may be evaluated recursively since they do not vary from point to point as a function of ² ³ lower-order curvatures. This has been demonstrated in [1] for the case of the single-parameter manifold of a linear array. Furthermore the convenient nature of a hyperhelix's geometry has proven invaluable in array design [2], in investigating the detection and resolution thresholds [3] and in identifying ambiguities inherent in array configurations [4]. In view of the above facts, it seems logical that an alternative parametrization of the manifold surface, which results in two sets of hyperhelical parameter curves, can provide a great deal of additional insight into the nature of planar array behaviour and design. In this investigation, such a parametrization is identified and its significance is demonstrated by a number of examples/applications. Furthermore properties, such as and , are defined and Gaussian geodesic curvatures their implications with regards to are discussed. isometric mappings Cone-Angle Parametrization of the Array Manifold in DF Systems 2 Nomenclature A, a Scalar [ ] Space spanned by columns of B [ [ A, a Vector a or Element by element exponential exp [ [, Matrix a Sum of vector elements A = sum² ³ | | Absolute value of scalar a Element by element power a k th | a | Euclidian norm of vector General bearing p parameter Re a a/ p. ² Î ³ Real part Derivative C C fix a a/ s ² Î ³ Integer part Derivative Z C C p Hadamard product operator