## Linear Algebra for Signal ProcessingAdam Bojanczyk, George Cybenko Signal processing applications have burgeoned in the past decade. During the same time, signal processing techniques have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This trend will continue as many new signal processing applications are opening up in consumer products and communications systems. In particular, signal processing has been making increasingly sophisticated use of linear algebra on both theoretical and algorithmic fronts. This volume gives particular emphasis to exposing broader contexts of the signal processing problems so that the impact of algorithms and hardware can be better understood; it brings together the writings of signal processing engineers, computer engineers, and applied linear algebraists in an exchange of problems, theories, and techniques. This volume will be of interest to both applied mathematicians and engineers. |

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### Contents

STRUCTURED MATRICES AND INVERSES | 1 |

STRUCTURED CONDITION NUMBERS FOR LINEAR MATRIX STRUCTURES | 17 |

THE CANONICAL CORRELATIONS OF MATRIX PAIRS AND THEIR NUMERICAL COMPUTATION | 27 |

APPLICATION TO WAVELETS | 51 |

INVERSION OF GENERALIZED CAUCHY MATRICES AND OTHER CLASSES OF STRUCTURED MATRICES | 63 |

WAVELETS FILTER BANKS AND ARBITRARY TILINGS OF THE TIMEFREQUENCY PLANE | 83 |

SYSTOLIC ALGORITHMS FOR ADAPTIVE SIGNAL PROCESSING | 125 |

ADAPTIVE ALGORITHMS FOR BLIND CHANNEL EQUALIZATION | 139 |

SQUAREROOT ALGORITHMS FOR STRUCTURED MATRICES INTERPOLATION AND COMPLETION PROBLEMS | 153 |

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algorithm analysis Applications arbitrary basis functions blind equalization canonical correlations cascade Cauchy matrices coefficients column compute consider construction continuous-time convergence COROLLARY data sequence defined denote diagonal dilation equation discrete discrete-time displacement rank displacement structure Editors eigenvalues entries example factor Figure filter banks frequency given Gohberg Hence HQ(Z IEEE Trans impulse response interpolation problem inverse iteration joint spectral radius Kailath Kalman filter Koltracht LEMMA Linear Algebra linear phase Math Mathematics matrix pair modulated lapped transforms multidimensional nonsingular obtained one-dimensional operator orthogonal orthogonal matrix orthonormal basis perfect reconstruction perturbation Proof QR decomposition QR update recursive recursive least squares result row vectors satisfies scaling function Schur Section SIAM Signal Proc Signal Processing solution solve span{B spectral radius structured condition number structured matrices subspace SVD updating symmetric systolic array Theorem 3.1 Theory tiling time-variant tion Toeplitz matrices triangular Volume wavelet bases wavelet packet wavelet transform zero