Wavelet applications in engineering electromagnetics

This resource describes the practical application of wavelets in computational electromagnetics and signal analysis. It shows how to design novel algorithms that solve large electromagnetic field problems using modest computational resources.

Author Notes

Tapan K. Sarkar holds a Ph.D. from Syracuse University.

He is a professor in the department of electrical engineering and computer science at Syracuse University. He is a fellow of the IEEE. He is a co-author of Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling (Artech House, 1998), and is the co-creator of several software packages, including WIPL-D, LINPAR for Windows, Version 2.0, AWAS and MULTIN for Windows (Artech House, 2000, 1999, 1996). He has published extensively.

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List of Figures	p. ix
List of Tables	p. xv
Preface	p. xix
Acknowledgments	p. xxi
Chapter 1 Road Map of the Book	p. 1
1.1 Introduction	p. 1
1.2 Why Use Wavelets?	p. 1
1.3 What Are Wavelets?	p. 2
1.4 What Is the Wavelet Transform?	p. 3
1.5 Use of Wavelets in the Numerical Solution of Electromagnetic Field Problems	p. 4
1.6 Wavelet Methodologies Complement Fourier Techniques	p. 7
1.7 Overview of the Chapters	p. 10
References	p. 11
Chapter 2 Wavelets from an Electrical Engineering Perspective	p. 15
2.1 Introduction	p. 15
2.2 Development of the Discrete Wavelet Methodology from Filter Theory Concepts	p. 16
2.2.1 Preliminaries	p. 16
2.2.2 Development of the Quadrature Mirror Filters	p. 19
2.2.3 Connection Between Filter Theory and the Mathematical Theory of Wavelets	p. 26
2.3 Approximation of a Function by Wavelets	p. 38
2.4 Examples	p. 43
2.5 Conclusion	p. 47
References	p. 47
Appendix 2A Principles of Decimation and Expansion	p. 48
2A.1 Principle of Decimation by a Factor of 2	p. 48
2A.2 Principle of Expansion by a Factor of 2	p. 50
Chapter 3 Application of Wavelets in the Solution of Operator Equations	p. 53
3.1 Introduction	p. 53
3.2 Approximation of a Function by Wavelets	p. 59
3.3 Solution of Operator Equations	p. 62
3.4 Wavelet Basis in the Solution of Integral Equations	p. 64
References	p. 68
Chapter 4 Solving Matrix Equations Using the Wavelet Transform	p. 71
4.1 Introduction	p. 71
4.2 Implementation of the Wavelet-Like Transform Based on the Tensor Product	p. 75
4.3 Numerical Evaluation of the Wavelet-Like Transform	p. 80
4.4 Solution of Large Dense Complex Matrix Equations Using a Wavelet-Like Methodology Based on the Fast Fourier Transform	p. 92
4.4.1 Implementation of the DWT Through the FFT	p. 93
4.4.2 Numerical Implementation of the DWT Through the FFT for a Two-Dimensional System (i.e., a Matrix)	p. 95
4.4.3 Numerical Results	p. 99
4.5 Utilization of Custom FIR Filters	p. 102
4.5.1 Perfect Reconstruction Conditions Using Complex Filters with a Finite Transition Band	p. 104
4.5.2 Application of the Discrete Cosine Transform to a Symmetric Extension of the Data	p. 108
4.5.3 Numerical Results	p. 109
4.5.4 A Note on the Characteristics of the Solution	p. 123
4.6 Conclusion	p. 131
References	p. 132
Chapter 5 Solving the Differential Form of Maxwell's Equations	p. 135
5.1 Introduction	p. 135
5.2 Solution of One-Dimensional Problems Utilizing a Wavelet-Like Basis	p. 136
5.3 Solution of [down triangle, open superscript 2]U + k[superscript 2]U = F for Two-Dimensional Problems Utilizing a Wavelet-Like Basis	p. 142
5.4 Application to Some Waveguide Problems	p. 146
5.5 Conclusion	p. 155
References	p. 155
Chapter 6 Adaptive Multiscale Moment Method	p. 157
6.1 Overview	p. 157
6.2 Introduction of the Multiscaling Methodology	p. 158
6.3 Use of a Multiscale Basis in Solving Integral Equations Via the Moment Method (MM)	p. 165
6.4 Differences Between a Multiscale Basis and a Subdomain Triangular Basis on an Interval [0, L]	p. 171
6.5 Analysis of Electromagnetic Scattering from Materially Coated Strips	p. 174
6.5.1 Integral Equation Relating the Fields to the Excitations	p. 174
6.5.2 Application of the Method of Moments Using a Multiscale Basis	p. 176
6.5.3 Solution of the Integral Equations by the Adaptive Multiscale Moment Method (AMMM)	p. 184
6.5.4 Numerical Results	p. 187
6.6 Extension of the Multiscale Concepts to 2-D Problems	p. 199
6.6.1 Functional Approximation Using a Multiscale Basis	p. 204
6.6.2 A Multiscale Moment Method for Solving Fredholm Integral Equations of the First Kind in Two Dimensions	p. 208
6.6.3 An Adaptive Algorithm Representing a Multiscale Moment Method	p. 211
6.6.4 Application of AMMM for the Solution of Electromagnetic Scattering from Finite-Sized Rectangular Plates	p. 215
6.6.5 Numerical Implementation of the AMMM Methodology for Solving 3-D Problems Using the Triangular Patch Basis Functions	p. 218
6.6.6 Numerical Results	p. 220
6.7 A Two-Dimensional Multiscale Basis on a Triangular Domain and the Geometrical Significance of the Coefficients for the Multiscale Basis	p. 227
6.7.1 Introduction	p. 227
6.7.2 Description of a Multiscale Basis on a Planar Arbitrary Domain	p. 231
6.7.3 A Multiscale Moment Method over an Arbitrary Planar Domain	p. 239
6.7.4 Numerical Results	p. 240
6.8 Discussion	p. 244
6.9 Conclusion	p. 245
References	p. 246
Chapter 7 The Continuous Wavelet Transform and Its Relationship to the Fourier Transform	p. 247
7.1 Introduction	p. 247
7.2 Continuous Transforms	p. 248
7.2.1 Fourier Transform	p. 248
7.2.2 Gabor Transform	p. 251
7.2.3 Continuous Wavelet Transform	p. 254
7.3 Discrete Transforms	p. 257
7.3.1 Discrete Short Time Fourier Transform (DSTFT)	p. 258
7.3.2 Discrete Gabor Transform	p. 259
7.3.3 Discrete Wavelet Transform	p. 266
7.4 Conclusion	p. 271
References	p. 271
Chapter 8 T-Pulse: Windows That Are Strictly Time Limited and Practically Band Limited	p. 273
8.1 Introduction	p. 273
8.2 A Discussion on Various Choices of the Window Function	p. 274
8.3 Development of the T-Pulse	p. 275
8.4 Summary of the Optimization Techniques	p. 278
8.5 Numerical Results	p. 280
8.6 Conclusion	p. 293
References	p. 293
Chapter 9 Optimal Selection of a Signal-Dependent Basis and Denoising	p. 295
9.1 Introduction	p. 295
9.2 Selection of an Optimum Basis	p. 295
9.3 Denoising of Signals Through the Wavelet Transform	p. 301
9.4 Conclusion	p. 303
References	p. 304
Selected Bibliography	p. 305
Books	p. 305
Journal and Conference Papers	p. 306
Wavelet Transform	p. 306
Hybrid Methods	p. 311
Multiresolution Analysis	p. 313
Making Dense Matrices Sparse	p. 315
Scattering	p. 317
Inverse Scattering	p. 324
Target Identification	p. 327
Electromagnetic Compatibility	p. 330
Wireless Communication	p. 330
About the Authors	p. 331
Index	p. 337

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