## A Friendly Guide to Wavelets'The book is indeed what its title promises: A friendly guide to wavelets...In short, Kaiser's book is excellently written and can be considered as one of the best textbooks on this topic presently available...it will enjoy wide distribution among mathematicians and physicists interested in wavelet analysis.' - Int. Math. Nachrichten 'I found this to be an excellent book. It is eminently more readable than the books...which might be considered the principal alternatives for textbooks on wavelets.' - Physics Today 'This volume is probably the most gentle introduction to wavelet theory on the market. As such, it responds to a significant need. The intended audience will profit from the motivation and common-sense explanations in the text. Ultimately, it may lead many readers, who may not otherwise have been able to do so, to go further into wavelet theory, Fourier analysis, and signal processing.' - SIAM Review 'It is well produced and certainly readable...This material should present no difficulty for fourth-year undergraduates...It also will be useful to advanced workers in that it presents a different approach to wavelet theory from the usual one.' - Computing Reviews 'The first half of the book is appropriately named. It is a well-written, nicely organized exposition...a welcome addition to the literature. The second part of the book introduces the concept of electromagnetic wavelets...This theory promises to have many other applications and could well lead to new ways of studying these topics. This book has a number of unique features which...makes it particularly valuable for newcomers to the field.' - Mathematical Reviews 'I wholeheartedly recommend this book for a solid and friendly introduction to wavelets, for anyone who is comfortable with the mathematics required of undergraduate electrical engineers. The book's appeal is that it covers all the fundamental concepts of wavelets in an elegant, straightforward way. It offers truly enjoyable (friendly!) mathematical exposition that is rich in intuitive explanations, as well as clean, direct, and clear in its theoretical developments. I found Kaiser's straightforward end-of-chapter exercises excellent...Kaiser has written an excellent introduction to the fundamental concepts of wavelets. For a book of its length and purpose, I think it should be essentially unbeatable for a long time.' - IEEE Antennas and Propagation Magazine and Proceedings of the IEEE, Nov. 1998 'I loved 'A Friendly Guide to Wavelets'. I advised it to my graduate students.' - Yves Meyer, Universite Paris-Dauphine This volume consists of two parts. Chapters 1-8, Basic Wavelet Analysis, are aimed at graduate students or advanced undergraduates in science, engineering, and mathematics. They are designed for an introductory one-semester course on wavelets and time frequency analysis, and can also be used for self-study or reference by practicing researchers in signal analysis and related areas. The reader is not presumed to have a sophisticated mathematical background; therefore, much of the needed analytical machinery is developed from the beginning. The only prerequisite is a knowledge of matrix theory, Fourier series, and Fourier integral transforms. Notation is introduced which facilitates the formulation of signal analysis in a modern and general mathematical language, and the illustrations should further ease comprehension. Each chapter ends with a set of straightforward exercises designed to drive home the concepts. Chapters 9-11, Physical Wavelets, are at a more advanced level and represent original research. They can be used as a text for a second-semester course or, when combined with Chapters 1 and 3, as a reference for a research seminar. Whereas the wavelets of Part I can be any functions of 'time,' physical wavelets are functions of space-time constrained by differential equations. In Chapter 9, wavelets specifically dedicated to Maxwell's equations are constructed. These wavelets are electromagnetic pulses parameterized by their point and time of emission or absorption, their duration, and the velocity of the emitter or absorber. The duration also acts as a scale parameter. We show that every electromagnetic wave can be composed from such wavelets. This fact is used in Chapter 10 to give a new formulation of electromagnetic imaging, such as radar, accompanied by a geometrical model for scattering based on conformal transformations. In Chapter 11, a similar set of wavelets is developed for acoustics. A relation is established at the fundamental level of differential equations between physical waves and time signals. This gives a one-to-one correspondence between physical wavelets and a particular family of time wavelets. Contents: Preface Suggestions to the Reader Symbols, Conventions, and Transforms Part I: Basic Wavelet Analysis Preliminaries: Background and Notation Windowed Fourier Transforms Continuous Wavelet Transforms Generalized Frames: Key to Analysis and Synthesis Discrete Time-Frequency Analysis and Sampling Discrete Time-Scale Analysis Multiresolution Analysis Daubechies' Orthonormal Wavelet Bases Part II: Physical Wavelets Introduction to Wavelet Electromagnetics Applications to Radar and Scattering Wavelet Acoustics References Index |

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See <http://www.amazon.com/Friendly-Wavelets-Modern-Birkhäuser-Classics/dp/0817681108>.

### Contents

IV | 3 |

V | 12 |

VI | 19 |

VII | 26 |

VIII | 34 |

IX | 41 |

X | 44 |

XII | 50 |

XXXVIII | 124 |

XXXIX | 133 |

XL | 137 |

XLI | 139 |

XLIII | 147 |

XLIV | 160 |

XLV | 169 |

XLVI | 172 |

XIII | 53 |

XIV | 56 |

XV | 58 |

XVI | 60 |

XVIII | 64 |

XIX | 70 |

XX | 73 |

XXI | 77 |

XXII | 78 |

XXIII | 85 |

XXIV | 88 |

XXV | 89 |

XXVI | 91 |

XXVII | 95 |

XXVIII | 96 |

XXIX | 99 |

XXX | 102 |

XXXI | 104 |

XXXII | 110 |

XXXIII | 112 |

XXXIV | 119 |

XXXV | 121 |

XXXVI | 123 |

### Common terms and phrases

adjoint ambiguity function analytic signals arbitrary band-limited called Chapter coefficient function compact support complex components compute conformal transformations construct continuous wavelet transform converges corresponding Daubechies 1992 defined denote discrete electromagnetic wavelets electromagnetic waves example fact factor filter coefficients finite follows Fourier series frame bounds frame vectors frequency domain function f given gives Hence Hilbert space implies inner product integral interpretation interval Lorentz transformations matrix Maxwell's equations measure metric operator mother wavelet multiresolution analysis narrow-band negative-frequency norm notation Note obtained orthogonal orthonormal basis parameterized Plancherel's theorem plane radar reciprocal reflector relation represented reproducing kernel resolution of unity respect rotation sampling satisfy scalar scaling function Section shows signal analysis signal f solution space-time special conformal transformations subset subspace superposition symmetry Theorem time-frequency translations unique unitary values vanishes velocity wavelet analysis wavelet transform wideband zero