Báo cáo hóa học: Research Article Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis

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Báo cáo hóa học: " Research Article Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis"Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2007, Article ID 92953, 24 pagesdoi:10.1155/2007/92953Research ArticleSubspace-Based Noise Reduction for Speech Signalsvia Diagonal and Triangular Matrix Decompositions:Survey and Analysis Per Christian Hansen1 and Søren Holdt Jensen2 1 Informatics and Mathematical Modelling, Technical University of Denmark, Building 321, 2800 Lyngby, Denmark 2 Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, 9220 Aalborg, Denmark Received 1 October 2006; Revised 18 February 2007; Accepted 31 March 2007 Recommended by Marc Moonen We survey the definitions and use of rank-revealing matrix decompositions in single-channel noise reduction algorithms for speech signals. Our algorithms are based on the rank-reduction paradigm and, in particular, signal subspace techniques. The focus is on practical working algorithms, using both diagonal (eigenvalue and singular value) decompositions and rank-revealing triangular decompositions (ULV, URV, VSV, ULLV, and ULLIV). In addition, we show how the subspace-based algorithms can be analyzed and compared by means of simple FIR filter interpretations. The algorithms are illustrated with working Matlab code and appli- cations in speech processing. Copyright © 2007 P. C. Hansen and S. H. Jensen. 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.1. INTRODUCTION work and a common notation. In addition to methods based on diagonal (eigenvalue and singular value) decompositions,The signal subspace approach has proved itself useful for we survey the use of rank-revealing triangular decomposi-signal enhancement in speech processing and many other tions. Within this framework, we also discuss alternatives toapplications—see, for example, the recent survey [1]. The the classical least-squares formulation, and we show how sig-area has grown dramatically over the last 20 years, along nals with general (nonwhite) noise are treated by explicit and,with advances in efficient computational algorithms for ma- in particular, implicit prewhitening. Throughout the paper,trix computations [2–4], especially singular value decompo- we provide small working Matlab codes that illustrate the al-sitions and rank-revealing decompositions. gorithms and their practical use. The central idea is to approximate a matrix, derived from We focus on signal enhancement methods which directlythe noisy data, with another matrix of lower rank from which estimate a clean signal from a noisy one (we do not esti-the reconstructed signal is derived. As stated in [5]: “Rank mate parameters in a parameterized signal model). Our pre-reduction is a general principle for finding the right trade-off sentation starts with formulations based on (estimated) co-between model bias and model variance when reconstructing variance matrices, and makes extensive use of eigenvalue de-signals from noisy data.” compositions as well as the ordinary and generalized sin- Throughout the literature of signal processing and ap- gular value decompositions (SVD and GSVD)—the latterplied mathematics, these methods are formulated in terms also referred to as the quotient SVD (QSVD). All these sub-of different notations, such as eigenvalue decompositions, space techniques originate from the seminal 1982 paper [6]Karhunen-Lo` ve transformations, and singular value de- e by Tufts and Kumaresan, who considered noise reductioncompositions. All ...