Cramer-Rao bound of spatio-temporal linear pre-processing in parameter estimation from sensor array

The knowledge of the Cramer-Rao bound (CRB) for a given estimation problem not only gives a lower bound on the covariance matrix of any unbiased estimator applied to a set of observed data, but it can represent a mean for testing the suitability of given signal structures to provide the necessary information to the parameter estimation. An interesting problem is the evaluation of the CRB in the presence of pre-processing applied to the experimental data. It has been demonstrated that, in general, a linear pre-processing matrix multiplied "on the left-hand side" of the experimental data matrix (left pre-processing) can degrade the performance of any parameter estimator, unless the left pre-processing matrix fulfills proper constraints for the invariance of the pre-processors. This paper is aimed at discussing the performance degradation in parameter estimation due to pre-processing matrices applied "on the right-hand side" of the data matrix (right pre-processing), since this problem seems not to have received the due attention up to this moment, despite this kind of linear pre-processing often occurs in several application contexts in the field of the communications. The paper derives an expression of the deterministic CRB giving a very interesting insight on the structure of the processing matrices, providing the proof that a rank-deficient and/or not orthogonal matrix achieves the same CRB attainable through its full-rank and orthogonalized portion. Furthermore, the conditions upon which the formulation of the CRB for the pre-processed data corresponds to the CRB expression for the unprocessed data are expressed, giving the hypotheses that guarantee any linear pre-processor to be invariant. In order to give practical examples of the meaning and use of the pre-processing approach considered in the paper, some receiving techniques developed for direct sequence code division multiple access systems are briefly investigated and their performance in the context of the direction-of-arrival estimation are discussed on the basis of the CRB of the problem. (C) 2003 Elsevier B.V. All rights reserved.