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Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem

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Abstract

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators.

The fast implementation is compared to two other implementations of algorithms for solving the Toeplitz (Hankel) STLS problem. The comparison is carried out on a recently proposed speech compression scheme. The numerical results confirm the high efficiency of the newly proposed fast implementation: the straightforward implementations have a complexity of O((m+n)3) and O(m3) whereas the proposed implementation has a complexity of O(mn+n2).

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Authors and Affiliations

  1. Department of Electrical Engineering, ESAT-SISTA/COSIC, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94, B-3001, Leuven, Belgium

    Philippe Lemmerling & Sabine Van Huffel

  2. Department of Electrical Engineering, ESAT-SISTA/COSIC, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94, B-3001, Leuven, Belgium

    Nicola Mastronardi

  3. Dipartimento di Matematica, Università della Basilicata, via N. Sauro 85, I-85100, Potenza, Italy

    Nicola Mastronardi

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  1. Philippe Lemmerling
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  2. Nicola Mastronardi
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  3. Sabine Van Huffel
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Lemmerling, P., Mastronardi, N. & Van Huffel, S. Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem. Numerical Algorithms 23, 371–392 (2000). https://doi.org/10.1023/A:1019116520737

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