Subspace Tracking for Signal Processing [chapter]

Jean Pierre Delmas
2010 Adaptive Signal Processing  
Research in subspace and component-based techniques were originated in Statistics in the middle of the last century through the problem of linear feature extraction solved by the Karhunen-Loève Transform (KLT). Then, it application to signal processing was initiated three decades ago, and has met considerable progress. Thorough studies have shown that the estimation and detection tasks in many signal processing and communications applications such as data compression, data filtering, parameter
more » ... stimation, pattern recognition, and neural analysis can be significantly improved by using the subspace and component-based methodology. Over the past few years new potential applications have emerged, and subspace and component methods have been adopted in several diverse new fields such as smart antennas, sensor arrays, multiuser detection, time delay estimation, image segmentation, speech enhancement, learning systems, magnetic resonance spectroscopy, and radar systems, to mention only a few examples. The interest in subspace and component-based methods stems from the fact that they consist in splitting the observations into a set of desired and a set of disturbing components. They not only provide new insight into many such problems, but they also offer a good tradeoff between achieved performance and computational complexity. In most cases they can be considered to be low-cost alternatives to computationally intensive maximum-likelihood approaches. In general, subspace and component-based methods are obtained by using batch methods, such as the eigenvalue decomposition (EVD) of the sample covariance matrix or the singular value decomposition (SVD) of the data matrix. However, these two approaches are not suitable for adaptive applications for tracking nonstationary signal parameters, where the required repetitive estimation of the subspace or the eigenvectors can be a real computational burden because their iterative implementation needs O(n 3 ) operations at each update, where n is the dimension of the vector-valued data sequence. Before proceeding with a brief literature review of the main contributions of adaptive estimation of subspace or eigenvectors, let us first classify these algorithms with respect to their computational complexity. If r denotes the rank of the principal or dominant) or minor subspace we would like to estimate, since usually r ≪ n, it is classic to refer to the following classification. Algorithms requiring O(n 2 r) or O(n 2 ) operations by update are classified as high complexity;
doi:10.1002/9780470575758.ch4 fatcat:pwbi2mtzora4dpouyscxf6hite