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 xv1 Notations and Mathematical Preliminaries 11.1 Notations and Abbreviations 11.2 Mathematical Preliminaries 21.2.1 Functions and Integration 21.2.2 The Fourier Transform 41.2.3 Regularity 41.2.4 Linear Spaces 71.2.5 Functional Spaces 81.2.6 Sobolev Spaces 101.2.7 Bases in Hilbert Space H 111.2.8 Linear Operators 12Bibliography 142 Intuitive Introduction to Wavelets 152.1 Technical History and Background 152.1.1 Historical Development 152.1.2 When Do Wavelets Work? 162.1.3 A Wave Is a Wave but What Is a Wavelet? 172.2 What Can Wavelets Do in Electromagnetics and Device Modeling? 182.2.1 Potential Benefits of Using Wavelets 182.2.2 Limitations and Future Direction of Wavelets 192.3 The Haar Wavelets and Multiresolution Analysis 202.4 How Do Wavelets Work? 23Bibliography 283 Basic Orthogonal Wavelet Theory 303.1 Multiresolution Analysis 303.2 Construction of Scalets 3.2.1 Franklin Scalet 323.2.2 Battle-Lemarie Scalets 393.2.3 Preliminary Properties of Scalets 403.3 Wavelet ^ ( r ) 423.4 Franklin Wavelet 483.5 Properties of Scalets (p(co) 513.6 Daubechies Wavelets 563.7 Coifman Wavelets (Coiflets) 643.8 Constructing Wavelets by Recursion and Iteration 693.8.1 Construction of Scalets 693.8.2 Construction of Wavelets 743.9 Meyer Wavelets 753.9.1 Basic Properties of Meyer Wavelets 753.9.2 Meyer Wavelet Family 833.9.3 Other Examples of Meyer Wavelets 923.10 Mallat's Decomposition and Reconstruction 923.10.1 Reconstruction 923.10.2 Decomposition 933.11 Problems 953.11.1 Exercise 1 953.11.2 Exercise 2 953.11.3 Exercise 3 973.11.4 Exercise 4 97Bibliography 984 Wavelets in Boundary Integral Equations 1004.1 Wavelets in Electromagnetics 1004.2 Linear Operators 1024.3 Method of Moments (MoM) 1034.4 Functional Expansion of a Given Function 1074.5 Operator Expansion: Nonstandard Form 1104.5.1 Operator Expansion in Haar Wavelets 1114.5.2 Operator Expansion in General Wavelet Systems 1134.5.3 Numerical Example 1144.6 Periodic Wavelets 1204.6.1 Construction of Periodic Wavelets 1204.6.2 Properties of Periodic Wavelets 1234.6.3 Expansion of a Function in Periodic Wavelets 1274.7 Application of Periodic Wavelets: 2D Scattering 1284.8 Fast Wavelet Transform (FWT) 1334.8.1 Discretization of Operation Equations 1334.8.2 Fast Algorithm 1344.8.3 Matrix Sparsification Using FWT 1354.9 Applications of the FWT 1404.9.1 Formulation 1404.9.2 Circuit Parameters 1414.9.3 Integral Equations and Wavelet Expansion 1434.9.4 Numerical Results 1444.10 Intervallic Coifman Wavelets 1444.10.1 Intervallic Scalets 1454.10.2 Intervallic Wavelets on [0, 1] 1544.11 Lifting Scheme and Lazy Wavelets 1564.11.1 Lazy Wavelets 1564.11.2 Lifting Scheme Algorithm 1574.11.3 Cascade Algorithm 1594.12 Green's Scalets and Sampling Series 1594.12.1 Ordinary Differential Equations (ODEs) 1604.12.2 Partial Differential Equations (PDEs) 1664.13 Appendix: Derivation of Intervallic Wavelets on [0, 1] 1724.14 Problems 1854.14.1 Exercise 5 1854.14.2 Exercise 6 1854.14.3 Exercise 7 1854.14.4 Exercise 8 1864.14.5 Project 1 187Bibliography 1875 Sampling Biorthogonal Time Domain Method (SBTD) 1895.1 Basis FDTD Formulation 1895.2 Stability Analysis for the FDTD 1945.3 FDTD as Maxwell's Equations with Haar Expansion 1985.4 FDTD with Battle-Lemarie Wavelets 2015.5 Positive Sampling and Biorthogonal Testing Functions 2055.6 Sampling Biorthogonal Time Domain Method 2155.6.1 SBTD versus MRTD 2155.6.2 Formulation 2155.7 Stability Conditions for Wavelet-Based Methods 2195.7.1 Dispersion Relation and Stability Analysis 2195.7.2 Stability Analysis for the SBTD 2225.8 Convergence Analysis and Numerical Dispersion 2235.8.1 Numerical Dispersion 2235.8.2 Convergence Analysis 2255.9 Numerical Examples 2285.10 Appendix: Operator Form of the MRTD 2335.11 Problems 2365.11.1 Exercise 9 2365.11.2 Exercise 10 2375.11.3 Project 2 237Bibliography 2386 Canonical Multiwavelets 2406.1 Vector-Matrix Dilation Equation 2406.2 Time Domain Approach 2426.3 Construction of Multiscalets 2456.4 Orthogonal Multiwavelets yjr(t) 2556.5 Intervallic Multiwavelets xj/(t) 2586.6 Multiwavelet Expansion 2616.7 Intervallic Dual Multiwavelets \j/(t) 2646.8 Working Examples 2696.9 Multiscalet-Based ID Finite Element Method (FEM) 2766.10 Multiscalet-Based Edge Element Method 2806.11 Spurious Modes 2856.12 Appendix 2876.13 Problems 2966.13.1 Exercise 11 296Bibliography 2977 Wavelets in Scattering and Radiation 2997.1 Scattering from a 2D Groove 2997.1.1 Method of Moments (MoM) Formulation 3007.1.2 Coiflet-Based MoM 3047.1.3 Bi-CGSTAB Algorithm 3057.1.4 Numerical Results 3057.2 2D and 3D Scattering Using Intervallic Coiflets 3097.2.1 Intervallic Scalets on [0,1] 3097.2.2 Expansion in Coifman Intervallic Wavelets 3127.2.3 Numerical Integration and Error Estimate 3137.2.4 Fast Construction of Impedance Matrix 3177.2.5 Conducting Cylinders, TM Case 3197.2.6 Conducting Cylinders with Thin Magnetic Coating 3227.2.7 Perfect Electrically Conducting (PEC) Spheroids 3247.3 Scattering and Radiation of Curved Thin Wires 3297.3.1 Integral Equation for Curved Thin-Wire Scatterers and Antennae 3307.3.2 Numerical Examples 3317.4 Smooth Local Cosine (SLC) Method 3407.4.1 Construction of Smooth Local Cosine Basis 3417.4.2 Formulation of 2D Scattering Problems 3447.4.3 SLC-Based Galerkin Procedure and Numerical Results 3477.4.4 Application of the SLC to Thin-Wire Scatterers and Antennas 3557.5 Microstrip Antenna Arrays 3577.5.1 Impedance Matched Source 3587.5.2 Far-Zone Fields and Antenna Patterns 360Bibliography 3638 Wavelets in Rough Surface Scattering 3668.1 Scattering of EM Waves from Randomly Rough Surfaces 3668.2 Generation of Random Surfaces 3688.2.1 Autocorrelation Method 3708.2.2 Spectral Domain Method 3738.3 2D Rough Surface Scattering 3768.3.1 Moment Method Formulation of 2D Scattering 3768.3.2 Wavelet-Based Galerkin Method for 2D Scattering 3808.3.3 Numerical Results of 2D Scattering 3818.4 3D Rough Surface Scattering 3878.4.1 Tapered Wave of Incidence 3888.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets 3918.4.3 Numerical Results of 3D Scattering 394Bibliography 3999 Wavelets in Packaging, Interconnects, and EMC 4019.1 Quasi-static Spatial Formulation 4029.1.1 What Is Quasi-static? 4029.1.2 Formulation 4039.1.3 Orthogonal Wavelets in L2([0, 1]) 4069.1.4 Boundary Element Method and Wavelet Expansion 4089.1.5 Numerical Examples 4129.2 Spatial Domain Layered Green's Functions 4159.2.1 Formulation 4179.2.2 Prony's Method 4239.2.3 Implementation of the Coifman Wavelets 4249.2.4 Numerical Examples 4269.3 Skin-Effect Resistance and Total Inductance 4299.3.1 Formulation 4319.3.2 Moment Method Solution of Coupled Integral Equations 4339.3.3 Circuit Parameter Extraction 4359.3.4 Wavelet Implementation 4379.3.5 Measurement and Simulation Results 4389.4 Spectral Domain Green's Function-Based Full-Wave Analysis 4409.4.1 Basic Formulation 4409.4.2 Wavelet Expansion and Matrix Equation 4449.4.3 Evaluation of Sommerfeld-Type Integrals 4479.4.4 Numerical Results and Sparsity of Impedance Matrix 4519.4.5 Further Improvements 4559.5 Full-Wave Edge Element Method for 3D Lossy Structures 4559.5.1 Formulation of Asymmetric Functionals with Truncation Conditions 4569.5.2 Edge Element Procedure 4609.5.3 Excess Capacitance and Inductance 4649.5.4 Numerical Examples 466Bibliography 46910 Wavelets in Nonlinear Semiconductor Devices 47410.1 Physical Models and Computational Efforts 47410.2 An Interpolating Subdivision Scheme 47610.3 The Sparse Point Representation (SPR) 47810.4 Interpolation Wavelets in the FDM 47910.4.1 ID Example of the SPR Application 48010.4.2 2D Example of the SPR Application 48110.5 The Drift-Diffusion Model 48410.5.1 Scaling 48610.5.2 Discretization 48710.5.3 Transient Solution 48910.5.4 Grid Adaptation and Interpolating Wavelets 49010.5.5 Numerical Results 49210.6 Multiwavelet Based Drift-Diffusion Model 49810.6.1 Precision and Stability versus Reynolds 49910.6.2 MWFEM-Based ID Simulation 50210.7 The Boltzmann Transport Equation (BTE) Model 50410.7.1 Why BTE? 50510.7.2 Spherical Harmonic Expansion of the BTE 50510.7.3 Arbitrary Order Expansion and Galerkin's Procedure 50910.7.4 The Coupled Boltzmann-Poisson System 51510.7.5 Numerical Results 517Bibliography 524Index 527