| Rank reduction is equivalent to the selection of a lower dimension subspace of the space spanned by the columns of the data covariance matrix. The best solutions for rank reduction depend on the subspace selection algorithm. Rank reduction is also a form of data compression prior to processing.; In the past the most widely used approach to rank reduction was based on the principal-components method. This method was optimal in the sense that it minimized the distortion (mean-square error) from that of the original data. It is argued herein that the principal-components method is not always optimal. Two new methods were developed in [1], [2], [3]. It is demonstrated that these two new methods, when applied to obtain a reduced-rank Wiener filter, perform better than the principal-components method.; The first new method of rank reduction is termed the cross-spectral metric which is optimal for basis-vector pruning, when the Karhunen-Loève decomposition of the observed data covariance matrix is used. Here, the optimality is in terms of the minimum mean-square error of the Wiener filter output. The second new method utilizes a decomposition of the data which is based on a sequence of successive orthogonal projections. This latter method of rank reduction is called the multistage reduced-rank Wiener filter, a filter that does not require prior knowledge of the eigenvectors of the observed data covariance matrix.; This thesis is concerned exclusively with the geralization of the rank reduction method and its extension to other applications such as communication systems and image processing. Optimal methods to accomplish rank reduction should depend on the objectives of the problem. By setting an objective function that relates to the performance of the processor, the optimal lower rank solution is the one that optimizes the objective function. However, it is possible that the reduced-rank detection and estimation techniques outperform the full rank method when the statistics are unknown and need to be estimated. This unexpected result is due to the faster convergence rate of the reduced-rank process. The reduced-rank process requires a smaller number of sample support to accurately estimate the parameters of the statistics. |
