Estimating the manifold parameters of one-dimensional arrays of sensors

Abstract

In the direction finding problem successful application of the signal-subspace type algorithms requires the use of the concept of the array manifold. However, despite the importance of this concept, signal-subspace algorithms use the array manifold in a primitive way, since its properties and parameters remain unknown. In this paper an attempt is made to identify, estimate and study the most important parameters of the manifold of a general linear array of N sensors by using differential geometry. Initially the general theory is developed and expressions are derived for the curvatures and coordinate vectors of the complex manifold-curve. Then the shape and orientation of the manifold of linear arrays are investigated. It is shown that the manifold-curve of a linear array of omnidirectional sensors has the shape of a complex circular hyperhelix on a sphere in the complex space C N while a measure of its orientation can be provided by introducing the concept of the inclination angle. Analytical expressions for the curvatures, manifold radii, number of windings etc. are also provided. Finally, it is shown that, if the array is symmetrical, i.e. all sensors are symmetrically located with respect to the array centroid, the complex hyperhelix manifold is equivalent to a real hyperhelix on a sphere in R N .