Geometric Aspects in Array Processing

Abstract

The theory of array processing is usually directed to nonmoving, particular array shapes, like the linear, rectangular, disc, crossed arrays. These are geometric figures whose description is incorporated into the problem in an expedite way. In practice, due to causes beyond the control of the designer, the actual array departs from these simple geometries. Colinearity is difficult of maintaining in very long arrays. Uncertainty may arise concerning the exact location of the array sensors. If the array is a towed flexible aggregate, its steady state shape may significantly depart from a closed type description. The array may move, through irregular paths, eg. random accelerating. These motions are best modeled as stochastic dynamics.

The paper addresses the above questions, proposing and developing a new point of view for the description of the array shape. This is then coupled to the general situation of moving arrays and moving sources. The description uses ordinary differential equations, or more generally, stochastic differential equations. Instead of describing the geometry in an integrated way, like saying that it is linear, the array is assumed to be a given curve in space. Curves are studied in geometry, which provides parameterized descriptions. For example, trajectories of moving objects are normally parameterized by time. Curves may admit alternative parameter descriptions. The work concentrates on using the arc length ℓ as the description parameter for the curve. Specific applications might advantageously use others. The ℓ-description dualizes the t-description of point trajectories. Also, it provides an interpretation whose generalization to models with random uncertainty on the sensors location, remains intuitive. Because the description is in terms of stochastic differential equations, it is recursive. It is emphasized that this recursiveness is both with respect to the ℓ and t parameters.

To extend the array processing techniques to moving line arrays, a rigidity assumption is made. The array is considered nondeformable. The dynamics are then the motions of a rigid body — a problem of Classical Mechanics. The full complete model presented encompasses the following: i) irregular array shapes; ii) stochastic uncertainty on the sensors’ locations; iii) randomly moving arrays; iv) stochastic source motions.

The paper presents examples that illustrate the techniques studied. Array processing and localization problems are cast in the framework of recursive estimation theory. For a specific problem the receiver is discussed, one of its blocks designed and its performance studied.