Wavelets in Electromagnetics and Device Modeling

GEORGE W. PAN, PhD, is Professor of Electrical Engineering and Director of the Electronic Packaging Lab at Arizona State University. He was previously a professor of electrical engineering and director of the Signal Propagation Lab at the University of Wisconsin-Milwaukee. Professor Pan is a senior member of the IEEE and has served as a technical consultant for Boeing Aerospace Co., Cray Research, Mayo Foundation, and other corporations. He is a contributing author to Modeling and Simulation of High Speed VLSI Interconnects.

Preface xv

1 Notations and Mathematical Preliminaries 1

1.1 Notations and Abbreviations 1

1.2 Mathematical Preliminaries 2

1.2.1 Functions and Integration 2

1.2.2 The Fourier Transform 4

1.2.3 Regularity 4

1.2.4 Linear Spaces 7

1.2.5 Functional Spaces 8

1.2.6 Sobolev Spaces 10

1.2.7 Bases in Hilbert Space H 11

1.2.8 Linear Operators 12

Bibliography 14

2 Intuitive Introduction to Wavelets 15

2.1 Technical History and Background 15

2.1.1 Historical Development 15

2.1.2 When Do Wavelets Work? 16

2.1.3 A Wave Is a Wave but What Is a Wavelet? 17

2.2 What Can Wavelets Do in Electromagnetics and Device Modeling? 18

2.2.1 Potential Benefits of Using Wavelets 18

2.2.2 Limitations and Future Direction of Wavelets 19

2.3 The Haar Wavelets and Multiresolution Analysis 20

2.4 How Do Wavelets Work? 23

Bibliography 28

3 Basic Orthogonal Wavelet Theory 30

3.1 Multiresolution Analysis 30

3.2 Construction of Scalets

3.2.1 Franklin Scalet 32

3.2.2 Battle-Lemarie Scalets 39

3.2.3 Preliminary Properties of Scalets 40

3.3 Wavelet ^ ( r ) 42

3.4 Franklin Wavelet 48

3.5 Properties of Scalets (p(co) 51

3.6 Daubechies Wavelets 56

3.7 Coifman Wavelets (Coiflets) 64

3.8 Constructing Wavelets by Recursion and Iteration 69

3.8.1 Construction of Scalets 69

3.8.2 Construction of Wavelets 74

3.9 Meyer Wavelets 75

3.9.1 Basic Properties of Meyer Wavelets 75

3.9.2 Meyer Wavelet Family 83

3.9.3 Other Examples of Meyer Wavelets 92

3.10 Mallat's Decomposition and Reconstruction 92

3.10.1 Reconstruction 92

3.10.2 Decomposition 93

3.11 Problems 95

3.11.1 Exercise 1 95

3.11.2 Exercise 2 95

3.11.3 Exercise 3 97

3.11.4 Exercise 4 97

Bibliography 98

4 Wavelets in Boundary Integral Equations 100

4.1 Wavelets in Electromagnetics 100

4.2 Linear Operators 102

4.3 Method of Moments (MoM) 103

4.4 Functional Expansion of a Given Function 107

4.5 Operator Expansion: Nonstandard Form 110

4.5.1 Operator Expansion in Haar Wavelets 111

4.5.2 Operator Expansion in General Wavelet Systems 113

4.5.3 Numerical Example 114

4.6 Periodic Wavelets 120

4.6.1 Construction of Periodic Wavelets 120

4.6.2 Properties of Periodic Wavelets 123

4.6.3 Expansion of a Function in Periodic Wavelets 127

4.7 Application of Periodic Wavelets: 2D Scattering 128

4.8 Fast Wavelet Transform (FWT) 133

4.8.1 Discretization of Operation Equations 133

4.8.2 Fast Algorithm 134

4.8.3 Matrix Sparsification Using FWT 135

4.9 Applications of the FWT 140

4.9.1 Formulation 140

4.9.2 Circuit Parameters 141

4.9.3 Integral Equations and Wavelet Expansion 143

4.9.4 Numerical Results 144

4.10 Intervallic Coifman Wavelets 144

4.10.1 Intervallic Scalets 145

4.10.2 Intervallic Wavelets on [0, 1] 154

4.11 Lifting Scheme and Lazy Wavelets 156

4.11.1 Lazy Wavelets 156

4.11.2 Lifting Scheme Algorithm 157

4.11.3 Cascade Algorithm 159

4.12 Green's Scalets and Sampling Series 159

4.12.1 Ordinary Differential Equations (ODEs) 160

4.12.2 Partial Differential Equations (PDEs) 166

4.13 Appendix: Derivation of Intervallic Wavelets on [0, 1] 172

4.14 Problems 185

4.14.1 Exercise 5 185

4.14.2 Exercise 6 185

4.14.3 Exercise 7 185

4.14.4 Exercise 8 186

4.14.5 Project 1 187

Bibliography 187

5 Sampling Biorthogonal Time Domain Method (SBTD) 189

5.1 Basis FDTD Formulation 189

5.2 Stability Analysis for the FDTD 194

5.3 FDTD as Maxwell's Equations with Haar Expansion 198

5.4 FDTD with Battle-Lemarie Wavelets 201

5.5 Positive Sampling and Biorthogonal Testing Functions 205

5.6 Sampling Biorthogonal Time Domain Method 215

5.6.1 SBTD versus MRTD 215

5.6.2 Formulation 215

5.7 Stability Conditions for Wavelet-Based Methods 219

5.7.1 Dispersion Relation and Stability Analysis 219

5.7.2 Stability Analysis for the SBTD 222

5.8 Convergence Analysis and Numerical Dispersion 223

5.8.1 Numerical Dispersion 223

5.8.2 Convergence Analysis 225

5.9 Numerical Examples 228

5.10 Appendix: Operator Form of the MRTD 233

5.11 Problems 236

5.11.1 Exercise 9 236

5.11.2 Exercise 10 237

5.11.3 Project 2 237

Bibliography 238

6 Canonical Multiwavelets 240

6.1 Vector-Matrix Dilation Equation 240

6.2 Time Domain Approach 242

6.3 Construction of Multiscalets 245

6.4 Orthogonal Multiwavelets yjr(t) 255

6.5 Intervallic Multiwavelets xj/(t) 258

6.6 Multiwavelet Expansion 261

6.7 Intervallic Dual Multiwavelets /j/(t) 264

6.8 Working Examples 269

6.9 Multiscalet-Based ID Finite Element Method (FEM) 276

6.10 Multiscalet-Based Edge Element Method 280

6.11 Spurious Modes 285

6.12 Appendix 287

6.13 Problems 296

6.13.1 Exercise 11 296

Bibliography 297

7 Wavelets in Scattering and Radiation 299

7.1 Scattering from a 2D Groove 299

7.1.1 Method of Moments (MoM) Formulation 300

7.1.2 Coiflet-Based MoM 304

7.1.3 Bi-CGSTAB Algorithm 305

7.1.4 Numerical Results 305

7.2 2D and 3D Scattering Using Intervallic Coiflets 309

7.2.1 Intervallic Scalets on [0,1] 309

7.2.2 Expansion in Coifman Intervallic Wavelets 312

7.2.3 Numerical Integration and Error Estimate 313

7.2.4 Fast Construction of Impedance Matrix 317

7.2.5 Conducting Cylinders, TM Case 319

7.2.6 Conducting Cylinders with Thin Magnetic Coating 322

7.2.7 Perfect Electrically Conducting (PEC) Spheroids 324

7.3 Scattering and Radiation of Curved Thin Wires 329

7.3.1 Integral Equation for Curved Thin-Wire Scatterers and Antennae 330

7.3.2 Numerical Examples 331

7.4 Smooth Local Cosine (SLC) Method 340

7.4.1 Construction of Smooth Local Cosine Basis 341

7.4.2 Formulation of 2D Scattering Problems 344

7.4.3 SLC-Based Galerkin Procedure and Numerical Results 347

7.4.4 Application of the SLC to Thin-Wire Scatterers and Antennas 355

7.5 Microstrip Antenna Arrays 357

7.5.1 Impedance Matched Source 358

7.5.2 Far-Zone Fields and Antenna Patterns 360

Bibliography 363

8 Wavelets in Rough Surface Scattering 366

8.1 Scattering of EM Waves from Randomly Rough Surfaces 366

8.2 Generation of Random Surfaces 368

8.2.1 Autocorrelation Method 370

8.2.2 Spectral Domain Method 373

8.3 2D Rough Surface Scattering 376

8.3.1 Moment Method Formulation of 2D Scattering 376

8.3.2 Wavelet-Based Galerkin Method for 2D Scattering 380

8.3.3 Numerical Results of 2D Scattering 381

8.4 3D Rough Surface Scattering 387

8.4.1 Tapered Wave of Incidence 388

8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets 391

8.4.3 Numerical Results of 3D Scattering 394

Bibliography 399

9 Wavelets in Packaging, Interconnects, and EMC 401

9.1 Quasi-static Spatial Formulation 402

9.1.1 What Is Quasi-static? 402

9.1.2 Formulation 403

9.1.3 Orthogonal Wavelets in L2([0, 1]) 406

9.1.4 Boundary Element Method and Wavelet Expansion 408

9.1.5 Numerical Examples 412

9.2 Spatial Domain Layered Green's Functions 415

9.2.1 Formulation 417

9.2.2 Prony's Method 423

9.2.3 Implementation of the Coifman Wavelets 424

9.2.4 Numerical Examples 426

9.3 Skin-Effect Resistance and Total Inductance 429

9.3.1 Formulation 431

9.3.2 Moment Method Solution of Coupled Integral Equations 433

9.3.3 Circuit Parameter Extraction 435

9.3.4 Wavelet Implementation 437

9.3.5 Measurement and Simulation Results 438

9.4 Spectral Domain Green's Function-Based Full-Wave Analysis 440

9.4.1 Basic Formulation 440

9.4.2 Wavelet Expansion and Matrix Equation 444

9.4.3 Evaluation of Sommerfeld-Type Integrals 447

9.4.4 Numerical Results and Sparsity of Impedance Matrix 451

9.4.5 Further Improvements 455

9.5 Full-Wave Edge Element Method for 3D Lossy Structures 455

9.5.1 Formulation of Asymmetric Functionals with Truncation Conditions 456

9.5.2 Edge Element Procedure 460

9.5.3 Excess Capacitance and Inductance 464

9.5.4 Numerical Examples 466

Bibliography 469

10 Wavelets in Nonlinear Semiconductor Devices 474

10.1 Physical Models and Computational Efforts 474

10.2 An Interpolating Subdivision Scheme 476

10.3 The Sparse Point Representation (SPR) 478

10.4 Interpolation Wavelets in the FDM 479

10.4.1 ID Example of the SPR Application 480

10.4.2 2D Example of the SPR Application 481

10.5 The Drift-Diffusion Model 484

10.5.1 Scaling 486

10.5.2 Discretization 487

10.5.3 Transient Solution 489

10.5.4 Grid Adaptation and Interpolating Wavelets 490

10.5.5 Numerical Results 492

10.6 Multiwavelet Based Drift-Diffusion Model 498

10.6.1 Precision and Stability versus Reynolds 499

10.6.2 MWFEM-Based ID Simulation 502

10.7 The Boltzmann Transport Equation (BTE) Model 504

10.7.1 Why BTE? 505

10.7.2 Spherical Harmonic Expansion of the BTE 505

10.7.3 Arbitrary Order Expansion and Galerkin's Procedure 509

10.7.4 The Coupled Boltzmann-Poisson System 515

10.7.5 Numerical Results 517

Bibliography 524

Index 527