## Linear Estimation and Detection in Krylov SubspacesOne major area in the theory of statistical signal processing is reduced-rank - timation where optimal linear estimators are approximated in low-dimensional subspaces, e.g., in order to reduce the noise in overmodeled problems, - hance the performance in case of estimated statistics, and/or save compu- tional complexity in the design of the estimator which requires the solution of linear equation systems. This book provides a comprehensive overview over reduced-rank ?lters where the main emphasis is put on matrix-valued ?lters whose design requires the solution of linear systems with multiple right-hand sides. In particular, the multistage matrix Wiener ?lter, i.e., a reduced-rank Wiener ?lter based on the multistage decomposition, is derived in its most general form. In numerical mathematics, iterative block Krylov methods are very po- lar techniques for solving systems of linear equations with multiple right-hand sides, especially if the systems are large and sparse. Besides presenting a - tailed overview of the most important block Krylov methods in Chapter 3, which may also serve as an introduction to the topic, their connection to the multistage matrix Wiener ?lter is revealed in this book. Especially, the reader will learn the restrictions of the multistage matrix Wiener ?lter which are necessary in order to end up in a block Krylov method. This relationship is of great theoretical importance because it connects two di?erent ?elds of mathematics, viz., statistical signal processing and numerical linear algebra. |

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### Contents

Introduction | 1 |

Part I Theory Linear Estimation in Krylov Subspaces | 11 |

Part II Application Iterative Multiuser Detection | 110 |

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### Common terms and phrases

applied approximation auto-covariance matrix BCG algorithm block Arnoldi block Krylov methods blocking matrices CDMA Chebyshev polynomial CN×M coded data bits computational complexity computationally decoder decomposition defined denotes diagonal DS-CDMA eigen decomposition eigensubspace eigenvalues Equa equalizer-demapper error estimate EXIT characteristic extrinsic information Figure full-rank Gray code Hermitian and positive Hessenberg matrix IEEE IEEE Transactions interleaver inverse J. S. Goldstein Krylov subspace Lanczos algorithm linear equations lower triangular lower triangular matrix Matrix Wiener Filter MMSE MSWF multiuser system Note number of FLOPs observation vector optimal orthogonal orthonormal output performance polynomial positive definite power delay profile prefilter matrix priori information priori LLRs Proakis Proposition QPSK random channel random variable rank reduced-rank MSMWF right-hand side Section Signal Processing single-user system solution solving Subsection system matrix system of linear time-variant tion Turbo iteration users vector random sequence Wiener–Hopf equation zero-mean Zoltowski