A spatial smoothing formulation for location systems

Two extensions of the classic passive location problem are considered. The first examines the ranging ability of passive receivers. The second presents alternative ways of describing the geometric content in positioning problems. The implications of the different approaches on the structure and performance of the location receiver are discussed. Because range determination by passive means is missing, the classical formulation of passive location may be viewed as a local geometry demodulation problem. With the explicit consideration of range, passive location becomes a global problem. At stake is what may be gained by processing the small but valuable amount of information carried by the wavefront curvature of the signals. Relevant questions relate to the design of passive receivers that aptly demodulate the range and the remaining quantities defining the geometry. There are passive applications where models exhibiting a high degree of (geometric) regularity are viable from a practical point of view. These occur, for example, when one can assume that the array sensors are collinear and that the moving target follows a deterministic linear path. In these models, the geometry is completely determined by a finite set of (unknown) parameters (e.g., range, bearing, speed, etc.). Accordingly, it is said that the regular models use an integral or ensemble approach for the description of the geometry. In many other problems, the geometry is more adequately described by statistical processes. Examples arise when the source follows a disturbed path, or when, due to towing, the array shape deforms, acquiring a not-completely-known shape. The paper models these constraints via a set of stochastic differential equations. The resulting representation is termed a differential description. It is emphasized that the differential approach is not only applied to the time content (relative dynamics), but also to the spatial dimension (array shape). The technique dualizes the space and time aspects of the problem. It provides a more flexible framework than the previous one. More general motions and array shapes than the traditional collinear ones can be considered by the analysis, e.g., irregular line arrays or arrays where the sensors are located at positions with a certain degree of randomness. Each approach fits a different design framework. The ensemble description is associated with the maximum likelihood technique. The differential representation uses recursive estimation methods (as provided by the Kalman-Bucy filtering theory). The paper discusses the main aspects of the structure of the resulting receivers and the associated measures of error performance. A second advantage of the differential model is immediately apparent The recursiveness of the differential receiver reduces its computational load. The speed-up obtained is fully appreciated in tracking applications, where the observations are sequentially updated. Finally, it is interesting to note that the time/space duality provided by the differential approach exhibits a remarkable distinction: the location recursive receiver behaves in time as a filter, while it behaves in space as a smoother.