Formulation and solution of structured total least norm problems for parameter estimation

Abstract

The total least squares (TLS) method is a generalization of the least squares (LS) method for solving overdetermined sets of linear equations Ax/spl ap/b. The TLS method minimizes /spl par/[E|-r]/spl par//sub F/, where r=b-(A+E)x, so that (b-r)/spl isin/Range (A+E), given A/spl isin/C/sup m/spl times/n/, with m/spl ges/n and b/spl isin/C/sup m/spl times/1/. The most common TLS algorithm is based on the singular value decomposition (SVD) of [A/b]. However, the SVD-based methods may not be appropriate when the matrix A has a special structure since they do not preserve the structure. Previously, a new problem formulation known as structured total least norm (STLN), and the algorithm for computing the STLN solution, have been developed. The STLN method preserves the special structure of A or [A/b] and can minimize the error in the discrete L/sub p/ norm, where p=1, 2 or /spl infin/. In this paper, the STLN problem formulation is generalized for computing the solution of STLN problems with multiple right-hand sides AX/spl ap/B. It is shown that these problems can be converted to ordinary STLN problems with one right-hand side. In addition, the method is shown to converge to the optimal solution in certain model reduction problems. Furthermore, the application of the STLN method to various parameter estimation problems is studied in which the computed correction matrix applied to A or [A/B] keeps the same Toeplitz structure as the data matrix A of [A/B], respectively. In particular, the L/sub 2/ norm STLN method is compared with the LS and TLS methods in deconvolution, transfer function modeling, and linear prediction problems.