Wavelets in electromagnetics and device modeling

* The first book on the subject.
* Written by an acknowledged expert in the field.
* The techniques discussed have important applications to wireless engineering.

Author Notes

George W. Pan is Professor of Electrical Engineering and Director of the Electronic Packaging Lab at Arizona State University.

Preface	p. xv
1 Notations and Mathematical Preliminaries	p. 1
1.1 Notations and Abbreviations	p. 1
1.2 Mathematical Preliminaries	p. 2
1.2.1 Functions and Integration	p. 2
1.2.2 The Fourier Transform	p. 4
1.2.3 Regularity	p. 4
1.2.4 Linear Spaces	p. 7
1.2.5 Functional Spaces	p. 8
1.2.6 Sobolev Spaces	p. 10
1.2.7 Bases in Hilbert Space H	p. 11
1.2.8 Linear Operators	p. 12
Bibliography	p. 14
2 Intuitive Introduction to Wavelets	p. 15
2.1 Technical History and Background	p. 15
2.1.1 Historical Development	p. 15
2.1.2 When Do Wavelets Work?	p. 16
2.1.3 A Wave Is a Wave but What Is a Wavelet?	p. 17
2.2 What Can Wavelets Do in Electromagnetics and Device Modeling?	p. 18
2.2.1 Potential Benefits of Using Wavelets	p. 18
2.2.2 Limitations and Future Direction of Wavelets	p. 19
2.3 The Haar Wavelets and Multiresolution Analysis	p. 20
2.4 How Do Wavelets Work?	p. 23
Bibliography	p. 28
3 Basic Orthogonal Wavelet Theory	p. 30
3.1 Multiresolution Analysis	p. 30
3.2 Construction of Scalets [phi]([tau])	p. 32
3.2.1 Franklin Scalet	p. 32
3.2.2 Battle-Lemarie Scalets	p. 39
3.2.3 Preliminary Properties of Scalets	p. 40
3.3 Wavelet [psi]([tau])	p. 42
3.4 Franklin Wavelet	p. 48
3.5 Properties of Scalets [phi]([omega])	p. 51
3.6 Daubechies Wavelets	p. 56
3.7 Coifman Wavelets (Coiflets)	p. 64
3.8 Constructing Wavelets by Recursion and Iteration	p. 69
3.8.1 Construction of Scalets	p. 69
3.8.2 Construction of Wavelets	p. 74
3.9 Meyer Wavelets	p. 75
3.9.1 Basic Properties of Meyer Wavelets	p. 75
3.9.2 Meyer Wavelet Family	p. 83
3.9.3 Other Examples of Meyer Wavelets	p. 92
3.10 Mallat's Decomposition and Reconstruction	p. 92
3.10.1 Reconstruction	p. 92
3.10.2 Decomposition	p. 93
3.11 Problems	p. 95
3.11.1 Exercise 1	p. 95
3.11.2 Exercise 2	p. 95
3.11.3 Exercise 3	p. 97
3.11.4 Exercise 4	p. 97
Bibliography	p. 98
4 Wavelets in Boundary Integral Equations	p. 100
4.1 Wavelets in Electromagnetics	p. 100
4.2 Linear Operators	p. 102
4.3 Method of Moments (MoM)	p. 103
4.4 Functional Expansion of a Given Function	p. 107
4.5 Operator Expansion: Nonstandard Form	p. 110
4.5.1 Operator Expansion in Haar Wavelets	p. 111
4.5.2 Operator Expansion in General Wavelet Systems	p. 113
4.5.3 Numerical Example	p. 114
4.6 Periodic Wavelets	p. 120
4.6.1 Construction of Periodic Wavelets	p. 120
4.6.2 Properties of Periodic Wavelets	p. 123
4.6.3 Expansion of a Function in Periodic Wavelets	p. 127
4.7 Application of Periodic Wavelets: 2D Scattering	p. 128
4.8 Fast Wavelet Transform (FWT)	p. 133
4.8.1 Discretization of Operation Equations	p. 133
4.8.2 Fast Algorithm	p. 134
4.8.3 Matrix Sparsification Using FWT	p. 135
4.9 Applications of the FWT	p. 140
4.9.1 Formulation	p. 140
4.9.2 Circuit Parameters	p. 141
4.9.3 Integral Equations and Wavelet Expansion	p. 143
4.9.4 Numerical Results	p. 144
4.10 Intervallic Coifman Wavelets	p. 144
4.10.1 Intervallic Scalets	p. 145
4.10.2 Intervallic Wavelets on [0, 1]	p. 154
4.11 Lifting Scheme and Lazy Wavelets	p. 156
4.11.1 Lazy Wavelets	p. 156
4.11.2 Lifting Scheme Algorithm	p. 157
4.11.3 Cascade Algorithm	p. 159
4.12 Green's Scalets and Sampling Series	p. 159
4.12.1 Ordinary Differential Equations (ODEs)	p. 160
4.12.2 Partial Differential Equations (PDEs)	p. 166
4.13 Appendix: Derivation of Intervallic Wavelets on [0, 1]	p. 172
4.14 Problems	p. 185
4.14.1 Exercise 5	p. 185
4.14.2 Exercise 6	p. 185
4.14.3 Exercise 7	p. 185
4.14.4 Exercise 8	p. 186
4.14.5 Project 1	p. 187
Bibliography	p. 187
5 Sampling Biorthogonal Time Domain Method (SBTD)	p. 189
5.1 Basis FDTD Formulation	p. 189
5.2 Stability Analysis for the FDTD	p. 194
5.3 FDTD as Maxwell's Equations with Haar Expansion	p. 198
5.4 FDTD with Battle-Lemarie Wavelets	p. 201
5.5 Positive Sampling and Biorthogonal Testing Functions	p. 205
5.6 Sampling Biorthogonal Time Domain Method	p. 215
5.6.1 SBTD versus MRTD	p. 215
5.6.2 Formulation	p. 215
5.7 Stability Conditions for Wavelet-Based Methods	p. 219
5.7.1 Dispersion Relation and Stability Analysis	p. 219
5.7.2 Stability Analysis for the SBTD	p. 222
5.8 Convergence Analysis and Numerical Dispersion	p. 223
5.8.1 Numerical Dispersion	p. 223
5.8.2 Convergence Analysis	p. 225
5.9 Numerical Examples	p. 228
5.10 Appendix: Operator Form of the MRTD	p. 233
5.11 Problems	p. 236
5.11.1 Exercise 9	p. 236
5.11.2 Exercise 10	p. 237
5.11.3 Project 2	p. 237
Bibliography	p. 238
6 Canonical Multiwavelets	p. 240
6.1 Vector-Matrix Dilation Equation	p. 240
6.2 Time Domain Approach	p. 242
6.3 Construction of Multiscalets	p. 245
6.4 Orthogonal Multiwavelets [psi](t)	p. 255
6.5 Intervallic Multiwavelets [psi](t)	p. 258
6.6 Multiwavelet Expansion	p. 261
6.7 Intervallic Dual Multiwavelets [psi](t)	p. 264
6.8 Working Examples	p. 269
6.9 Multiscalet-Based 1D Finite Element Method (FEM)	p. 276
6.10 Multiscalet-Based Edge Element Method	p. 280
6.11 Spurious Modes	p. 285
6.12 Appendix	p. 287
6.13 Problems	p. 296
6.13.1 Exercise 11	p. 296
Bibliography	p. 297
7 Wavelets in Scattering and Radiation	p. 299
7.1 Scattering from a 2D Groove	p. 299
7.1.1 Method of Moments (MoM) Formulation	p. 300
7.1.2 Coiflet-Based MoM	p. 304
7.1.3 Bi-CGSTAB Algorithm	p. 305
7.1.4 Numerical Results	p. 305
7.2 2D and 3D Scattering Using Intervallic Coiflets	p. 309
7.2.1 Intervallic Scalets on [0, 1]	p. 309
7.2.2 Expansion in Coifman Intervallic Wavelets	p. 312
7.2.3 Numerical Integration and Error Estimate	p. 313
7.2.4 Fast Construction of Impedance Matrix	p. 317
7.2.5 Conducting Cylinders, TM Case	p. 319
7.2.6 Conducting Cylinders with Thin Magnetic Coating	p. 322
7.2.7 Perfect Electrically Conducting (PEC) Spheroids	p. 324
7.3 Scattering and Radiation of Curved Thin Wires	p. 329
7.3.1 Integral Equation for Curved Thin-Wire Scatterers and Antennae	p. 330
7.3.2 Numerical Examples	p. 331
7.4 Smooth Local Cosine (SLC) Method	p. 340
7.4.1 Construction of Smooth Local Cosine Basis	p. 341
7.4.2 Formulation of 2D Scattering Problems	p. 344
7.4.3 SLC-Based Galerkin Procedure and Numerical Results	p. 347
7.4.4 Application of the SLC to Thin-Wire Scatterers and Antennas	p. 355
7.5 Microstrip Antenna Arrays	p. 357
7.5.1 Impedance Matched Source	p. 358
7.5.2 Far-Zone Fields and Antenna Patterns	p. 360
Bibliography	p. 363
8 Wavelets in Rough Surface Scattering	p. 366
8.1 Scattering of EM Waves from Randomly Rough Surfaces	p. 366
8.2 Generation of Random Surfaces	p. 368
8.2.1 Autocorrelation Method	p. 370
8.2.2 Spectral Domain Method	p. 373
8.3 2D Rough Surface Scattering	p. 376
8.3.1 Moment Method Formulation of 2D Scattering	p. 376
8.3.2 Wavelet-Based Galerkin Method for 2D Scattering	p. 380
8.3.3 Numerical Results of 2D Scattering	p. 381
8.4 3D Rough Surface Scattering	p. 387
8.4.1 Tapered Wave of Incidence	p. 388
8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets	p. 391
8.4.3 Numerical Results of 3D Scattering	p. 394
Bibliography	p. 399
9 Wavelets in Packaging, Interconnects, and EMC	p. 401
9.1 Quasi-static Spatial Formulation	p. 402
9.1.1 What Is Quasi-static?	p. 402
9.1.2 Formulation	p. 403
9.1.3 Orthogonal Wavelets in L[superscript 2]([0,1])	p. 406
9.1.4 Boundary Element Method and Wavelet Expansion	p. 408
9.1.5 Numerical Examples	p. 412
9.2 Spatial Domain Layered Green's Functions	p. 415
9.2.1 Formulation	p. 417
9.2.2 Prony's Method	p. 423
9.2.3 Implementation of the Coifman Wavelets	p. 424
9.2.4 Numerical Examples	p. 426
9.3 Skin-Effect Resistance and Total Inductance	p. 429
9.3.1 Formulation	p. 431
9.3.2 Moment Method Solution of Coupled Integral Equations	p. 433
9.3.3 Circuit Parameter Extraction	p. 435
9.3.4 Wavelet Implementation	p. 437
9.3.5 Measurement and Simulation Results	p. 438
9.4 Spectral Domain Green's Function-Based Full-Wave Analysis	p. 440
9.4.1 Basic Formulation	p. 440
9.4.2 Wavelet Expansion and Matrix Equation	p. 444
9.4.3 Evaluation of Sommerfeld-Type Integrals	p. 447
9.4.4 Numerical Results and Sparsity of Impedance Matrix	p. 451
9.4.5 Further Improvements	p. 455
9.5 Full-Wave Edge Element Method for 3D Lossy Structures	p. 455
9.5.1 Formulation of Asymmetric Functionals with Truncation Conditions	p. 456
9.5.2 Edge Element Procedure	p. 460
9.5.3 Excess Capacitance and Inductance	p. 464
9.5.4 Numerical Examples	p. 466
Bibliography	p. 469
10 Wavelets in Nonlinear Semiconductor Devices	p. 474
10.1 Physical Models and Computational Efforts	p. 474
10.2 An Interpolating Subdivision Scheme	p. 476
10.3 The Sparse Point Representation (SPR)	p. 478
10.4 Interpolation Wavelets in the FDM	p. 479
10.4.1 1D Example of the SPR Application	p. 480
10.4.2 2D Example of the SPR Application	p. 481
10.5 The Drift-Diffusion Model	p. 484
10.5.1 Scaling	p. 486
10.5.2 Discretization	p. 487
10.5.3 Transient Solution	p. 489
10.5.4 Grid Adaptation and Interpolating Wavelets	p. 490
10.5.5 Numerical Results	p. 492
10.6 Multiwavelet Based Drift-Diffusion Model	p. 498
10.6.1 Precision and Stability versus Reynolds	p. 499
10.6.2 MWFEM-Based 1D Simulation	p. 502
10.7 The Boltzmann Transport Equation (BTE) Model	p. 504
10.7.1 Why BTE?	p. 505
10.7.2 Spherical Harmonic Expansion of the BTE	p. 505
10.7.3 Arbitrary Order Expansion and Galerkin's Procedure	p. 509
10.7.4 The Coupled Boltzmann-Poisson System	p. 515
10.7.5 Numerical Results	p. 517
Bibliography	p. 524
Index	p. 527

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