## A Rapid Introduction to Adaptive FilteringIn this book, the authors provide insights into the basics of adaptive filtering, which are particularly useful for students taking their first steps into this field. They start by studying the problem of minimum mean-square-error filtering, i.e., Wiener filtering. Then, they analyze iterative methods for solving the optimization problem, e.g., the Method of Steepest Descent. By proposing stochastic approximations, several basic adaptive algorithms are derived, including Least Mean Squares (LMS), Normalized Least Mean Squares (NLMS) and Sign-error algorithms. The authors provide a general framework to study the stability and steady-state performance of these algorithms. The affine Projection Algorithm (APA) which provides faster convergence at the expense of computational complexity (although fast implementations can be used) is also presented. In addition, the Least Squares (LS) method and its recursive version (RLS), including fast implementations are discussed. The book closes with the discussion of several topics of interest in the adaptive filtering field. |

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### Contents

1 | |

2 Wiener Filtering | 6 |

3 Iterative Optimization | 19 |

4 Stochastic Gradient Adaptive Algorithms | 32 |

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### Common terms and phrases

ˆdopt A.H. Sayed acoustic echo cancelation adaptive algorithm adaptive filter affine projection algorithm applications approximation assume beamformer Benesty chapter coefficients computational complexity condition number convergence speed cost function defined eigenvalues eigmax EMSE equation error surface estimation error Gamma signal Gaussian given gradient algorithms Hessian matrix IEEE IEEE Trans input regressors input signal input vector input-output pairs inverse Iteration number JMMSE JMSE leads Least Squares linear prediction LMS algorithm LS problem LS solution Mean Square Error method minimum modemax modemin MSD stability NLMS noise v(n obtain optimal filter orthogonal projection output perturbations positive definite positive definite matrix properties pseudoinverse random variables recursion Recursive Least Squares Rey Vega robust samples sensor sequence Signal Process slowest mode speed of convergence statistics steady state error step stochastic gradient transient behavior update Wiener filter Wiener solution wopt zero