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Wavelet Theory and Its Applications : The Springer International Engineering and Computer Science - Randy K. Young

Wavelet Theory and Its Applications

The Springer International Engineering and Computer Science

Hardcover Published: 30th September 1992
ISBN: 9780792392712
Number Of Pages: 223

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This book reviews, extends, and applies wavelet theory, concentrating on the practical applications. Many pictures provide visualizations of wavelet theory and its new extentions, as well as relationships to established concepts. Wavelet theory is integrated with other general theories, including linear systems theory and template matching or matched filtering. These relationships create analogies with related research and connections to practical applications. In addition, by demonstrating the effectiveness of wavelet theory in these general applications, many other specific applications may be improved. Temporal and spatial signals and systems are considered. The properties of the wavelet transform representation are sensitive to the chosen mother wavelet (the kernel of the wavelet transform, analogous to the exponential function in a Fourier transform). These properties are examined and techniques for analyzing these sensitivities are presented.
Wavelet theory is extended with the new mother mapper operator that efficiently maps a wavelet transform with respect to one mother wavelet to a new wavelet transform with respect to a different mother wavelet. The mother mapper efficiently calculates concise wavelet representations that utilize multiple mother wavelets. The mother mapper operator is also employed to efficiently compute 'cross' wavelet transforms or wideband cross ambiguity functions; these 'cross' operators extract the 'commonalities' between two signals or systems to determine the existence or structure of these commonalities.
An original system model, the space-time varying (STV) wavelet operator, is constructed with wavelet theory. As a special case, the STV model can represent linear time-invariant (LTI) systems. LTI systems are represented by the one-dimensional (1D) impulse response. This one-dimensional impulse response is the center slice of the two-dimensional (2D) STV representation. Both the LTI and STV models can also be made to vary with time (leading to 2D and 3D models, respectively). The advantages of the new STV model are expolited to characterize or image an environment.
Physically, the STV wavelet operator creates an output by summing weighted, scaled and translated replicas of the input; these weights are the new system model. This is analogous to the LTI system model in which the output is a weighed sum of translated replicas of the input signal; with the weights being the LTI system model, the impulse response.
Obviously, time scaling is the additional feature of the STV wavelet representation and is also the key to efficient representations of the wideband reflection or scattering process and improved estimation gains. By including the scaling operation as part of the STV system model (that is independent of time), the estimation process for this new system model can account for the linear time variation of the system and thus, have a valid model over a long interval of time. By estimating over a long interval of time, more robust and higher gain and resolution estimates can be formed.

Foreword
Preface
Introduction/Backgroundp. 1
Wavelet Theory Basics - Scaling and Translationp. 4
Appendix 1-A: Time Referencingp. 16
The Wavelet Transformp. 19
Wavelet Transform Definitions and Operatorsp. 19
Wavelet Transform Examplesp. 22
The Resolution of Identityp. 27
Continuous Inverse Wavelet Transform Theoremp. 30
Energy Distribution in the Wavelet Transform Domainp. 34
Discrete Wavelet Transform (Continuous Time Wavelet Series)p. 35
Wideband Matched Filter Interpretation of the Lattice Densityp. 38
Review of the Inverse Discrete Wavelet Transform: Scale-translation Lattice Density and Mother Wavelet Constraintsp. 41
Discrete Wavelet Mathematics - Rigorous Justificationp. 44
Discrete Time Wavelet Seriesp. 50
Multiresolution, Orthogonal, and Biorthogonal Wavelet Transforms (and PR-QMFs)p. 51
Discrete Time Wavelet Series - A Specific Structurep. 52
Multiresolutionp. 55
Orthogonal and Biorthogonal Wavelet Transformsp. 57
An Image Processing Examplep. 60
Appendix 2-A: Nonunique Wavelet Domain Representationsp. 64
Practical Resolution, Gain, and Processing Structuresp. 71
Fourier/Narrowband Gain and Resolution Comparisonsp. 72
Multiple Mother Wavelets - Gain and Resolution Properties Onlyp. 77
Mother Wavelet Properties - Relationships to Established Theoriesp. 79
Signal Analysis - Time-frequency or time-scale Applicationsp. 80
A Physical Interpretation of Scale-translation Resolution: Wideband Ambiguity Function Conditionsp. 81
Wideband Conditionsp. 82
Wideband Signals and the Analytic Signal Modelp. 83
Effective rms Time-bandwidth Productp. 85
Wideband Systems and Signalsp. 87
Active and Passive Sensingp. 92
Ambiguity Functionsp. 95
Reformulation of the WBCAF with Wavelet Transformsp. 101
Properties of the Reformulated WBCAFp. 105
Cross Wavelet Transforms and Signal Commonalitiesp. 110
Appendix 3-A: Wideband/Narrowband Ambiguity Functions: Assumptions, Tradeoffs and Efficienciesp. 113
Appendix 3-B: Narrowband Ambiguity Function Theoryp. 119
Wavelet Theory Extensions and Ambiguity Functionsp. 123
Multiresolution/Orthogonal Wavelets versus Unconstrained Waveletsp. 123
Sampling Gridsp. 126
Unconstrained Wavelet Transforms - Mother Wavelet Freedomp. 127
The Mother Mapper Operatorp. 128
Unconstrained Wavelet Transforms/Mother Mapper Operator Properties and Applicationsp. 133
Mother Mapper Operator Applicationsp. 137
Mother Mapper Operator - Final Considerationsp. 139
Further Research and Applications of the Mother Mapper Operatorp. 140
Linear Systems Modelling with Wavelet Theoryp. 141
Wideband/Nonstationary/Time-varying System Modellingp. 143
The Wideband Signal Reflection Processp. 146
Common Framework of System or Channel Characterizationsp. 151
The STV Wavelet Operator - Space-time-varying System Modelp. 156
STV Wavelet Operator's Energy Distributionp. 159
Estimation of the Wideband System Characterizationp. 160
Properties of the STV Wavelet Operatorp. 160
Examples of the STV Wavelet Operatorp. 164
Wideband Reflection: Comparing the LTI and STV Modelsp. 169
Justification for the STV Operator Instead of Convolution for Signals Represented by Wavelet Transformsp. 174
The STV Wavelet Operator in the Wavelet Transform Domainp. 175
Space-time-varying System Identification Problem with Wideband/Nonstationary Input/Outputsp. 180
Limitations of the STV Wavelet Operator - Time Referencing and Nonlinear Time Variationsp. 181
Bi-wavelet System Representation: Time variation of the Time-varying System Modelp. 184
Wideband Scattering and Environmental Imagingp. 189
Scattering Theoryp. 192
General Scattering Function Backgroundp. 193
Narrowband Scattering Theoryp. 194
Wideband Correlation Receiver and its Outputp. 198
Wideband Point Scatterer Examplep. 201
Wideband Scattering Functions and Resolutionsp. 203
Physical Form of the WBCAFp. 205
Time Delay and Scale Estimationp. 208
WBAAF Moments and Assumptionsp. 209
Wideband Scattering Reviewp. 210
Related Researchp. 211
Referencesp. 215
Subject Indexp. 221
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792392712
ISBN-10: 079239271X
Series: The Springer International Engineering and Computer Science
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 223
Published: 30th September 1992
Publisher: SPRINGER VERLAG GMBH
Country of Publication: US
Dimensions (cm): 23.39 x 15.6  x 1.42
Weight (kg): 0.52

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