Modern Characterization of Electromagnetic Systems and its Associated Metrology

TAPAN K. SARKAR, PhD, is a professor at the Department of Electrical Engineering and Computer Science at Syracuse University, NY, USA. Professor Sarkar has previously published seven books with Wiley.MAGDALENA SALAZAR-PALMA, PhD, is a professor at the Department of Signal Theory and Communications, Carlos III University of Madrid, Leganes, Madrid, Spain. MING DA ZHU, PhD, is an associate professor at the School of Electronic Engineering at Xidian University, Xi’an, Shaanxi, China. HENG CHEN, PhD, is a research assistant at the Department of Electrical Engineering and Computer Science at Syracuse University, NY, USA.

• Preface xiiiAcknowledgments xxiTribute to Tapan K. Sarkar – Magdalena Salazar Palma, Ming Da Zhu, and Heng Chen xxiii1 Mathematical Principles Related to Modern System Analysis 1Summary 11.1 Introduction 11.2 Reduced-Rank Modelling: Bias Versus Variance Tradeoff 31.3 An Introduction to Singular Value Decomposition (SVD) and the Theory of Total Least Squares (TLS) 61.3.1 Singular Value Decomposition 61.3.2 The Theory of Total Least Squares 151.4 Conclusion 19References 202 Matrix Pencil Method (MPM) 21Summary 212.1 Introduction 212.2 Development of the Matrix Pencil Method for Noise Contaminated Data 242.2.1 Procedure for Interpolating or Extrapolating the System Response Using the Matrix Pencil Method 262.2.2 Illustrations Using Numerical Data 262.2.2.1 Example 1 262.2.2.2 Example 2 292.3 Applications of the MPM for Evaluation of the Characteristic Impedance of a Transmission Line 322.4 Application of MPM for the Computation of the S-Parameters Without any A Priori Knowledge of the Characteristic Impedance 372.5 Improving the Resolution of Network Analyzer Measurements Using MPM 442.6 Minimization of Multipath Effects Using MPM in Antenna Measurements Performed in Non-Anechoic Environments 572.6.1 Application of a FFT-Based Method to Process the Data 612.6.2 Application of MPM to Process the Data 642.6.3 Performance of FFT and MPM Applied to Measured Data 672.7 Application of the MPM for a Single Estimate of the SEM-Poles When Utilizing Waveforms from Multiple Look Directions 742.8 Direction of Arrival (DOA) Estimation Along with Their Frequency of Operation Using MPM 812.9 Efficient Computation of the Oscillatory Functional Variation in the Tails of the Sommerfeld Integrals Using MPM 852.10 Identification of Multiple Objects Operating in Free Space Through Their SEM Pole Locations Using MPM 912.11 Other Miscellaneous Applications of MPM 952.12 Conclusion 95Appendix 2A Computer Codes for Implementing MPM 96References 993 The Cauchy Method 107Summary 1073.1 Introduction 1073.2 Procedure for Interpolating or Extrapolating the System Response Using the Cauchy Method 1123.3 Examples to Estimate the System Response Using the Cauchy Method 1123.3.1 Example 1 1123.3.2 Example 2 1163.3.3 Example 3 1183.4 Illustration of Extrapolation by the Cauchy Method 1203.4.1 Extending the Efficiency of the Moment Method Through Extrapolation by the Cauchy Method 1203.4.2 Interpolating Results for Optical Computations 1233.4.3 Application to Filter Analysis 1253.4.4 Broadband Device Characterization Using Few Parameters 1273.5 Effect of Noise Contaminating the Data and Its Impact on the Performance of the Cauchy Method 1303.5.1 Perturbation of Invariant Subspaces 1303.5.2 Perturbation of the Solution of the Cauchy Method Due to Additive Noise 1313.5.3 Numerical Example 1363.6 Generating High Resolution Wideband Response from Sparse and Incomplete Amplitude-Only Data 1383.6.1 Development of the Interpolatory Cauchy Method for Amplitude-Only Data 1393.6.2 Interpolating High Resolution Amplitude Response 1423.7 Generation of the Non-minimum Phase Response from Amplitude-Only Data Using the Cauchy Method 1483.7.1 Generation of the Non-minimum Phase 1493.7.2 Illustration Through Numerical Examples 1513.8 Development of an Adaptive Cauchy Method 1583.8.1 Introduction 1583.8.2 Adaptive Interpolation Algorithm 1593.8.3 Illustration Using Numerical Examples 1603.8.4 Summary 1713.9 Efficient Characterization of a Filter 1723.10 Extraction of Resonant Frequencies of an Object from Frequency Domain Data 1763.11 Conclusion 180Appendix 3A MATLAB Codes for the Cauchy Method 181References 1874 Applications of the Hilbert Transform – A Nonparametric Method for Interpolation/Extrapolation of Data 191Summary 1914.1 Introduction 1924.2 Consequence of Causality and Its Relationship to the Hilbert Transform 1944.3 Properties of the Hilbert Transform 1954.4 Relationship Between the Hilbert and the Fourier Transforms for the Analog and the Discrete Cases 1994.5 Methodology to Extrapolate/Interpolate Data in the Frequency Domain Using a Nonparametric Methodology 2004.6 Interpolating Missing Data 2034.7 Application of the Hilbert Transform for Efficient Computation of the Spectrum for Nonuniformly Spaced Data 2134.7.1 Formulation of the Least Square Method 2174.7.2 Hilbert Transform Relationship 2214.7.3 Magnitude Estimation 2234.8 Conclusion 229References 2295 The Source Reconstruction Method 235Summary 2355.1 Introduction 2365.2 An Overview of the Source Reconstruction Method (SRM) 2385.3 Mathematical Formulation for the Integral Equations 2395.4 Near-Field to Far-Field Transformation Using an Equivalent Magnetic Current Approach 2405.4.1 Description of the Proposed Methodology 2415.4.2 Solution of the Integral Equation for the Magnetic Current 2455.4.3 Numerical Results Utilizing the Magnetic Current 2495.4.4 Summary 2685.5 Near-Field to Near/Far-Field Transformation for Arbitrary Near-Field Geometry Utilizing an Equivalent Electric Current 2765.5.1 Description of the Proposed Methodology 2785.5.2 Numerical Results Using an Equivalent Electric Current 2815.5.3 Summary 2865.6 Evaluating Near-Field Radiation Patterns of Commercial Antennas 2975.6.1 Background 2975.6.2 Formulation of the Problem 3015.6.3 Results for the Near-field To Far-field Transformation 3045.6.3.1 A Base Station Antenna 3045.6.3.2 NF to FF Transformation of a Pyramidal Horn Antenna 3075.6.3.3 Reference Volume of a Base Station Antenna for Human Exposure to EM Fields 3105.6.4 Summary 3115.7 Conclusions 313References 3146 Planar Near-Field to Far-Field Transformation Using a Single Moving Probe and a Fixed Probe Arrays 319Summary 3196.1 Introduction 3206.2 Theory 3226.3 Integral Equation Formulation 3236.4 Formulation of the Matrix Equation 3256.5 Use of an Magnetic Dipole Array as Equivalent Sources 3286.6 Sample Numerical Results 3296.7 Summary 3376.8 Differences between Conventional Modal Expansion and the Equivalent Source Method for Planar Near-Field to Far-Field Transformation 3376.8.1 Introduction 3376.8.2 Modal Expansion Method 3396.8.3 Integral Equation Approach 3416.8.4 Numerical Examples 3446.8.5 Summary 3516.9 A Direct Optimization Approach for Source Reconstruction and NF-FF Transformation Using Amplitude-Only Data 3526.9.1 Background 3526.9.2 Equivalent Current Representation 3546.9.3 Optimization of a Cost Function 3566.9.4 Numerical Simulation 3576.9.5 Results Obtained Utilizing Experimental Data 3586.9.6 Summary 3596.10 Use of Computational Electromagnetics to Enhance the Accuracy and Efficiency of Antenna Pattern Measurements Using an Array of Dipole Probes 3616.10.1 Introduction 3626.10.2 Development of the Proposed Methodology 3636.10.3 Philosophy of the Computational Methodology 3636.10.4 Formulation of the Integral Equations 3656.10.5 Solution of the Integro-Differential Equations 3676.10.6 Sample Numerical Results 3696.10.6.1 Example 1 3696.10.6.2 Example 2 3736.10.6.3 Example 3 3776.10.6.4 Example 4 3796.10.7 Summary 3846.11 A Fast and Efficient Method for Determining the Far Field Patterns Along the Principal Planes Using a Rectangular Probe Array 3846.11.1 Introduction 3856.11.2 Description of the Proposed Methodology 3856.11.3 Sample Numerical Results 3876.11.3.1 Example 1 3876.11.3.2 Example 2 3936.11.3.3 Example 3 3976.11.3.4 Example 4 4016.11.4 Summary 4066.12 The Influence of the Size of Square Dipole Probe Array Measurement on the Accuracy of NF-FF Pattern 4066.12.1 Illustration of the Proposed Methodology Utilizing Sample Numerical Results 4076.12.1.1 Example 1 4076.12.1.2 Example 2 4116.12.1.3 Example 3 4166.12.1.4 Example 4 4196.12.2 Summary 4286.13 Use of a Fixed Probe Array Measuring Amplitude-Only Near-Field Data for Calculating the Far-Field 4286.13.1 Proposed Methodology 4296.13.2 Sample Numerical Results 4306.13.2.1 Example 1 4306.13.2.2 Example 2 4346.13.2.3 Example 3 4376.13.2.4 Example 4 4376.13.3 Summary 4416.14 Probe Correction for Use with Electrically Large Probes 4426.14.1 Development of the Proposed Methodology 4436.14.2 Formulation of the Solution Methodology 4466.14.3 Sample Numerical Results 4476.15 Conclusions 449References 4497 Spherical Near-Field to Far-Field Transformation 453Summary 4537.1 An Analytical Spherical Near-Field to Far-Field Transformation 4537.1.1 Introduction 4537.1.2 An Analytical Spherical Near-Field to Far-Field Transformation 4547.1.3 Numerical Simulations 4647.1.3.1 Synthetic Data 4647.1.3.2 Experimental Data 4657.1.4 Summary 4687.2 Radial Field Retrieval in Spherical Scanning for Current Reconstruction and NF–FF Transformation 4687.2.1 Background 4687.2.2 An Equivalent Current Reconstruction from Spherical Measurement Plane 4707.2.3 The Radial Electric Field Retrieval Algorithm 4727.2.4 Results Obtained Using This Formulation 4737.2.4.1 Simulated Data 4737.2.4.2 Using Measured Data 4757.3 Conclusion 482Appendix 7A A Fortran Based Computer Program for Transforming Spherical Near-Field to Far-Field 483References 4898 Deconvolving Measured Electromagnetic Responses 491Summary 4918.1 Introduction 4918.2 The Conjugate Gradient Method with Fast Fourier Transform for Computational Efficiency 4958.2.1 Theory 4958.2.2 Numerical Results 4988.3 Total Least Squares Approach Utilizing Singular Value Decomposition 5018.3.1 Theory 5018.3.2 Total Least Squares (TLS) 5028.3.3 Numerical Results 5068.4 Conclusion 516References 5169 Performance of Different Functionals for Interpolation/Extrapolation of Near/Far-Field Data 519Summary 5199.1 Background 5209.2 Approximating a Frequency Domain Response by Chebyshev Polynomials 5219.3 The Cauchy Method Based on Gegenbauer Polynomials 5319.3.1 Numerical Results and Discussion 5379.3.1.1 Example of a Horn Antenna 5379.3.1.2 Example of a 2-element Microstrip Patch Array 5399.3.1.3 Example of a Parabolic Antenna 5419.4 Near-Field to Far-Field Transformation of a Zenith-Directed Parabolic Reflector Using the Ordinary Cauchy Method 5439.5 Near-Field to Far-Field Transformation of a Rotated Parabolic Reflector Using the Ordinary Cauchy Method 5529.6 Near-Field to Far-Field Transformation of a Zenith-Directed Parabolic Reflector Using the Matrix Pencil Method 5589.7 Near-Field to Far-Field Transformation of a Rotated Parabolic Reflector Using the Matrix Pencil Method 5649.8 Conclusion 569References 56910 Retrieval of Free Space Radiation Patterns from Measured Data in a Non-Anechoic Environment 573Summary 57310.1 Problem Background 57310.2 Review of Pattern Reconstruction Methodologies 57510.3 Deconvolution Method for Radiation Pattern Reconstruction 57810.3.1 Equations and Derivation 57810.3.2 Steps Required to Implement the Proposed Methodology 58410.3.3 Processing of the Data 58510.3.4 Simulation Examples 58710.3.4.1 Example I: One PEC Plate Serves as a Reflector 58710.3.4.2 Example II: Two PEC Plates Now Serve as Reflectors 59410.3.4.3 Example III: Four Connected PEC Plates Serve as Reflectors 59810.3.4.4 Example IV: Use of a Parabolic Reflector Antenna as the AUT 60410.3.5 Discussions on the Deconvolution Method for Radiation Pattern Reconstruction 60810.4 Effect of Different Types of Probe Antennas 60810.4.1 Numerical Examples 60810.4.1.1 Example I: Use of a Yagi Antenna as the Probe 60810.4.1.2 Example II: Use of a Parabolic Reflector Antenna as the Probe 61210.4.1.3 Example III: Use of a Dipole Antenna as the Probe 61310.5 Effect of Different Antenna Size 61910.6 Effect of Using Different Sizes of PEC Plates 62610.7 Extension of the Deconvolution Method to Three-Dimensional Pattern Reconstruction 63210.7.1 Mathematical Characterization of the Methodology 63210.7.2 Steps Summarizing for the Methodology 63510.7.3 Processing the Data 63610.7.4 Results for Simulation Examples 63810.7.4.1 Example I: Four Wide PEC Plates Serve as Reflectors 64010.7.4.2 Example II: Four PEC Plates and the Ground Serve as Reflectors 64310.7.4.3 Example III: Six Plates Forming an Unclosed Contour Serve as Reflectors 65110.7.4.4 Example IV: Antenna Measurement in a Closed PEC Box 65910.7.4.5 Example V: Six Dielectric Plates Forming a Closed Contour Simulating a Room 66210.8 Conclusion 673Appendix A: Data Mapping Using the Conversion between the Spherical Coordinate System and the Cartesian Coordinate System 675Appendix B: Description of the 2D-FFT during the Data Processing 677References 680Index 683