## Wavelet Theory and Its ApplicationsThe continuous wavelet transform has deep mathematical roots in the work of Alberto P. Calderon. His seminal paper on complex method of interpolation and intermediate spaces provided the main tool for describing function spaces and their approximation properties. The Calderon identities allow one to give integral representations of many natural operators by using simple pieces of such operators, which are more suited for analysis. These pieces, which are essentially spectral projections, can be chosen in clever ways and have proved to be of tremendous utility in various problems of numerical analysis, multidimensional signal processing, video data compression, and reconstruction of high resolution images and high quality speech. A proliferation of research papers and a couple of books, written in English (there is an earlier book written in French), have emerged on the subject. These books, so far, are written by specialists for specialists, with a heavy mathematical flavor, which is characteristic of the Calderon-Zygmund theory and related research of Duffin-Schaeffer, Daubechies, Grossman, Meyer, Morlet, Chui, and others. Randy Young's monograph is geared more towards practitioners and even non-specialists, who want and, probably, should be cognizant of the exciting proven as well as potential benefits which have either already emerged or are likely to emerge from wavelet theory. |

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### Contents

IntroductionBackground | 1 |

Wavelet Theory Basics Scaling and Translation | 4 |

Time Referencing | 16 |

The Wavelet Transform | 19 |

Wavelet Transform Examples | 22 |

The Resolution of Identity | 27 |

Continuous Inverse Wavelet Transform Theorem | 30 |

Energy Distribution in the Wavelet Transform Domain | 34 |

Sampling Grids | 126 |

Unconstrained Wavelet Transforms Mother Wavelet Freedom | 127 |

The Mother Mapper Operator | 128 |

Unconstrained Wavelet TransformsMother Mapper Operator Properties and Applications | 133 |

Mother Mapper Operator Applications | 137 |

Mother Mapper Operator Final Considerations | 139 |

Further Research and Applications of the Mother Mapper Operator | 140 |

Linear Systems Modelling with Wavelet Theory | 141 |

Discrete Wavelet Transform Continuous Time Wavelet Series | 35 |

Wideband Matched Filter Interpretation of the Lattice Density | 38 |

Scaletranslation Lattice Density and Mother Wavelet Constraints | 41 |

Discrete Wavelet Mathematics Rigorous Justification | 44 |

Discrete Time Wavelet Series | 50 |

Multiresolution Orthogonal and Biorthogonal Wavelet Transforms and PRQMFs | 51 |

Discrete Time Wavelet Series A Specific Structure | 52 |

Multiresolution Wavelet Transforms | 55 |

Orthogonal and Biorthogonal Wavelet Transforms | 57 |

An Image Processing Example | 60 |

Nonunique Wavelet Domain Representations | 64 |

Practical Resolution Gain and Processing Structures | 71 |

FourierNarrowband Gain and Resolution Comparisons | 72 |

Multiple Mother Wavelets Gain and Resolution Properties Only | 77 |

Signal Analysis Timefrequency or timescale Applications | 80 |

Wideband Ambiguity Function Conditions | 81 |

Wideband Conditions | 82 |

Wideband Signals and the Analytic Signal Model | 83 |

Effective rms Timebandwidth Product | 85 |

Wideband Systems and Signals | 87 |

Active and Passive Sensing | 92 |

Ambiguity Functions | 95 |

Reformulation of the WBCAF with Wavelet Transforms | 101 |

Properties of the Reformulated WBCAF | 105 |

Cross Wavelet Transforms and Signal Commonalities | 110 |

WidebandNarrowband Ambiguity Functions Assumptions Tradeoffs and Efficiencies | 113 |

Narrowband Ambiguity Function Theory | 119 |

Wavelet Theory Extensions and Ambiguity Functions | 123 |

WidebandNoastationaryTimevarying System Modelling | 143 |

The Wideband Signal Reflection Process | 146 |

The STV Wavelet Operator Spacetimevarying System Model | 156 |

STV Wavelet Operators Energy Distribution | 159 |

Estimation of the Wideband System Characterization | 160 |

Examples of the STV Wavelet Operator | 164 |

Comparing the LTI and STV Models | 169 |

Justification for the STV Operator Instead of Convolution for Signals Represented by Wavelet Transforms | 174 |

The STV Wavelet Operator in the Wavelet Transform Domain | 175 |

Spacetimevarying System Identification Problem with WidehandNonstationary InputOutputs | 180 |

Limitations of the STV Wavelet Operator Time Referencing and Nonlinear Time Variations | 181 |

Time variation of the Timevarying System Model | 184 |

Wideband Scattering and Environmental Imaging | 189 |

Scattering Theory | 192 |

General Scattering Function Background | 193 |

Narrowband Scattering Theory | 194 |

Wideband Correlation Receiver and its Output | 198 |

Wideband Point Scatterer Example | 201 |

Wideband Scattering Functions and Resolutions | 203 |

Physical Form of the WBCAF | 205 |

Time Delay and Scale Estimation | 208 |

WBAAF Moments and Assumptions | 209 |

Wideband Scattering Review | 210 |

Related Research | 211 |

215 | |

221 | |

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### Common terms and phrases

achieved allows ambiguity function analysis applications approximation assumed assumptions bandwidth becomes biorthogonal wavelet Chapter characteristics condition considered constant constraints continuous correlation create cross defined delay depends derived desirable detailed discrete discussed distribution duration efficient elements employed energy environment equation estimate example exist Figure filter formulated Fourier frame frequency gain grid heart identity impulse response input signal interpretation interval invalid inverse later lattice limited linear maps matched mathematics mother wavelet multiple multiresolution narrowband Note original orthogonal output parameter particular performance plane position presented problem processing processing interval processor properties range received signal reconstruction reference reflection represent representation resolution resolution properties respect sample scale and translation scattering shift shown space specific structure STV wavelet operator surface system model term time-varying transmitted signal valid values variation wavelet domain wavelet theory wavelet transform WBCAF wideband system characterization

### Popular passages

Page 218 - Papoulis, A. , Probability, Random Variables, and Stochastic Processes, McGraw-Hill Book Co., New York, New York, 1965.