On optimal dimension reduction for sensor array signal processing

The computational complexity for direction-of-arrival estimation using sensor arrays increases very rapidly with the number of sensors in the array. One way to lower the amount of computations is to employ some kind of reduction of the data dimension. This is usually accomplished by employing linear transformations for mapping full dimension data into a lower dimensional space. Different approaches for selecting these transformations have been proposed. In this paper, a transformation matrix is derived that makes it possible to theoretically attain the full-dimension Cramer-Rao bound also in the reduced space. A bound on the dimension of the reduced data set is given, above which it is always possible to obtain the same accuracy for the estimates of the source localizations, using the lower-dimension data, as that achievable by using the full dimension data. Furthermore, a method is devised for designing the transformation matrix. Numerical examples, using this design method, are presented, where the achievable performance of the (optimal) Weighted Subspace Fitting method with full dimension data is compared to the performance obtained with reduced dimension data. The problem of estimating parameters of sinusoidal signals from noisy data is also addressed by a direct application of the results derived herein.