## Nonlinear System Identification by Haar WaveletsIn order to precisely model real-life systems or man-made devices, both nonlinear and dynamic properties need to be taken into account. The generic, black-box model based on Volterra and Wiener series is capable of representing fairly complicated nonlinear and dynamic interactions, however, the resulting identification algorithms are impractical, mainly due to their computational complexity. One of the alternatives offering fast identification algorithms is the block-oriented approach, in which systems of relatively simple structures are considered. The book provides nonparametric identification algorithms designed for such systems together with the description of their asymptotic and computational properties. |

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### Contents

Chapter 1 Introduction
| 1 |

Chapter 2 Hammerstein Systems
| 5 |

Chapter 3 Identification Goal
| 12 |

Chapter 4 Haar Orthogonal Bases
| 17 |

Chapter 5 Identification Algorithms
| 43 |

Chapter 6 Computational Algorithms
| 76 |

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### Common terms and phrases

ˆKn ˇ ˇ ˇ ˇmn 2IKN algorithm converges analysis phase approximation error asymptotic binary rational bounded Chap classic Haar coefficients Oˇmn compactly supported computed cones of influence convergence conditions convergence rate discontinuous distribution algorithm dyadic empirical coefficients O˛Kn empirical cones empirical scaling function empirical wavelet coefficients estimate external noise EZW scheme f.uk yk/g factor fast wavelet transform function f growing number Haar functions Haar wavelet Hammerstein system identification algorithms identified nonlinearity implementation input probability density input signal input–output Lemma linear algorithms Lipschitz continuous Lipschitz functions log2 logN MISE error sense MISEOMK D O multiscale nonlinear algorithms nonparametric number of measurements NX kD1 yk O˛Mn order statistics orthogonal series piecewise-constant piecewise-Lipschitz nonlinearities probability density function properties QOS algorithm random scale selection rule scaling function coefficients Sect Sliwi´nski synthesis phase system identification system nonlinearity Theorem unbalanced Haar variance x/dx xk x k1