On Dimension Reduction in Sensor Array Signal Processing

Sammanfattning: During the last decades, sensor array signal processing has been a very active research area. More recently, relations between many of the proposed methods has been examined. The problem of assessing the estimation accuracy of these methods has also been addressed. Realworld applications of these techniques involves spatial distribution of several sensors to be used for collecting measurements of interesting emitted waveforms. From the measurements, detection and localization as well as estimation of the emitted waveforms can be accomplished. Common examples of applications are radar (electromagnetic waveforms) and sonar (acoustical underwater waveforms).Another aspect of array processing that recently has been addressed in the literature is that of dimension reduction, where the data vectors collected at the sensor outputs are reduced in size. This reduction is employed mainly in order to lower the amount of computations necessary for obtaining the parameter-estimates of interest; hut some other improvcments has also been observed. These include, e.g., lower sensitivity to sensor noise correlations and, for some estimation methods, higher resolution capability.In this thesis, it is demonstrated how to make the dimension reduction in an optimal fashion, where the optimality is with respect to estimation accuracy. More precisely, an expression to be satisfied by a transformation matrix acting on the sensor outputs is derived , that preserves the optimally achievable estimation accuracy (the Cramer-Rao bound) also in the reduced space. A transformation matrix design method that tries to reduce some unwanted properties of the optimal transformation is also outlined and examined. This method is based on numerical optimization of a particular performance mea.sure, motivated by the insight obtained in the process of finding the optimal transformation.l\foreover, an asymptotic analysis is performed, using the reduced data vectors, that examines the estimation accuracy of several estimation methods when a !arge number of sensor elements is used. This analysis is valid for a fairly general transformation matrix, and the methods considered are the Weighted Subspace Fitting (WSF) and Noise Subspace Fitting (NSF) methods, including MUSIC. By employing the optimal transformation matrix, the WSF method is shown to to be efficient, i.e., to attain the Cramer-Rao bound. An examination of the estimation accuracy, compared to that optimally attainable, is performed for the case when the transformation matrix differs from the optimal one. Finally, an application is studied, considering the potential use of sensor arrays in mobile communication systems.