Numerical Linear Algebra in Signal Processing Applications
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Editorial Numerical Linear Algebra in Signal Processing Applications Nicola Mastronardi,1 Gene H. Golub,2 Shivkumar Chandrasekaran,3 Marc Moonen,4 Paul Van Dooren,5 and Sabine Van Huffel4 1 Istituto
per le Applicazioni del Calcolo “M. Picone”, sede di Bari, Consiglio Nazionale delle Ricerche Via G. Amendola 122/D, I-70126 Bari, Italy 2 Department of Computer Science, Stanford University, Gates Building 2B, Room 280, Stanford, CA 94305-9025, USA 3 Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA 4 Department of Electrical Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, bus 2446, B-3001 Leuven-Heverlee, Belgium 5 Department of Mathematical Engineering, Catholic University of Louvain, Bˆ atiment Euler (A.202), Avenue Georges Lemaitre 4, B-1348 Leuven-Heverlee, Belgium Received 27 September 2007; Accepted 27 September 2007 Copyright © 2007 Nicola Mastronardi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The cross-fertilization between numerical linear algebra and digital signal processing has been very fruitful in the last decades. In particular, signal processing has been making increasingly sophisticated use of linear algebra on both theoretical and algorithmic fronts. The interaction between them has been growing, leading to many new algorithms. In particular, numerical linear algebra tools, such as eigenvalue and singular value decomposition and their higher-extensions, least squares, total least squares, recursive least squares, regularization, orthogonality and projections, are the kernels of powerful and numerically robust algorithms in many signal processing applications. This special issue contains contributions written by experts of signal processing, computer engineering, and numerical analysis, providing an account of the main results in this interdisciplinary field. Most of the papers are devoted to applications of numerical linear algebra algorithms for solving signal processing problems. Nevertheless, few of them are more theoretically oriented, and describe algorithms for solving linear algebra problems involving structured matrices and tensors, frequently encountered in a variety of signal processing applications. In the paper by H. Reza Bahrami et al., the effect of eigenvalues distribution of spatial correlation matrices on the capacity of frequency-flat and frequency-selective channels is first investigated. Then, a practical scheme, known as linear precoding, is introduced. It can enhance the ergodic capacity of the channel by changing the eigenstructure of the channel, applying a linear transformation. The structures of
precoders using eigenvalue decomposition and linear algebra techniques are derived and their similarities from an algebraic point of view are shown. Numerical methods for finding the maximal symmetric positive definite solution o
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