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 xiii

Acknowledgments xxi

Tribute to Tapan K. Sarkar – Magdalena Salazar Palma, Ming Da Zhu, and Heng Chen xxiii

1 Mathematical Principles Related to Modern System Analysis 1

Summary 1

1.1 Introduction 1

1.2 Reduced-Rank Modelling: Bias Versus Variance Tradeoff 3

1.3 An Introduction to Singular Value Decomposition (SVD) and the Theory of Total Least Squares (TLS) 6

1.3.1 Singular Value Decomposition 6

1.3.2 The Theory of Total Least Squares 15

1.4 Conclusion 19

References 20

2 Matrix Pencil Method (MPM) 21

Summary 21

2.1 Introduction 21

2.2 Development of the Matrix Pencil Method for Noise Contaminated Data 24

2.2.1 Procedure for Interpolating or Extrapolating the System Response Using the Matrix Pencil Method 26

2.2.2 Illustrations Using Numerical Data 26

2.2.2.1 Example 1 26

2.2.2.2 Example 2 29

2.3 Applications of the MPM for Evaluation of the Characteristic Impedance of a Transmission Line 32

2.4 Application of MPM for the Computation of the S-Parameters Without any A Priori Knowledge of the Characteristic Impedance 37

2.5 Improving the Resolution of Network Analyzer Measurements Using MPM 44

2.6 Minimization of Multipath Effects Using MPM in Antenna Measurements Performed in Non-Anechoic Environments 57

2.6.1 Application of a FFT-Based Method to Process the Data 61

2.6.2 Application of MPM to Process the Data 64

2.6.3 Performance of FFT and MPM Applied to Measured Data 67

2.7 Application of the MPM for a Single Estimate of the SEM-Poles When Utilizing Waveforms from Multiple Look Directions 74

2.8 Direction of Arrival (DOA) Estimation Along with Their Frequency of Operation Using MPM 81

2.9 Efficient Computation of the Oscillatory Functional Variation in the Tails of the Sommerfeld Integrals Using MPM 85

2.10 Identification of Multiple Objects Operating in Free Space Through Their SEM Pole Locations Using MPM 91

2.11 Other Miscellaneous Applications of MPM 95

2.12 Conclusion 95

Appendix 2A Computer Codes for Implementing MPM 96

References 99

3 The Cauchy Method 107

Summary 107

3.1 Introduction 107

3.2 Procedure for Interpolating or Extrapolating the System Response Using the Cauchy Method 112

3.3 Examples to Estimate the System Response Using the Cauchy Method 112

3.3.1 Example 1 112

3.3.2 Example 2 116

3.3.3 Example 3 118

3.4 Illustration of Extrapolation by the Cauchy Method 120

3.4.1 Extending the Efficiency of the Moment Method Through Extrapolation by the Cauchy Method 120

3.4.2 Interpolating Results for Optical Computations 123

3.4.3 Application to Filter Analysis 125

3.4.4 Broadband Device Characterization Using Few Parameters 127

3.5 Effect of Noise Contaminating the Data and Its Impact on the Performance of the Cauchy Method 130

3.5.1 Perturbation of Invariant Subspaces 130

3.5.2 Perturbation of the Solution of the Cauchy Method Due to Additive Noise 131

3.5.3 Numerical Example 136

3.6 Generating High Resolution Wideband Response from Sparse and Incomplete Amplitude-Only Data 138

3.6.1 Development of the Interpolatory Cauchy Method for Amplitude-Only Data 139

3.6.2 Interpolating High Resolution Amplitude Response 142

3.7 Generation of the Non-minimum Phase Response from Amplitude-Only Data Using the Cauchy Method 148

3.7.1 Generation of the Non-minimum Phase 149

3.7.2 Illustration Through Numerical Examples 151

3.8 Development of an Adaptive Cauchy Method 158

3.8.1 Introduction 158

3.8.2 Adaptive Interpolation Algorithm 159

3.8.3 Illustration Using Numerical Examples 160

3.8.4 Summary 171

3.9 Efficient Characterization of a Filter 172

3.10 Extraction of Resonant Frequencies of an Object from Frequency Domain Data 176

3.11 Conclusion 180

Appendix 3A MATLAB Codes for the Cauchy Method 181

References 187

4 Applications of the Hilbert Transform – A Nonparametric Method for Interpolation/Extrapolation of Data 191

Summary 191

4.1 Introduction 192

4.2 Consequence of Causality and Its Relationship to the Hilbert Transform 194

4.3 Properties of the Hilbert Transform 195

4.4 Relationship Between the Hilbert and the Fourier Transforms for the Analog and the Discrete Cases 199

4.5 Methodology to Extrapolate/Interpolate Data in the Frequency Domain Using a Nonparametric Methodology 200

4.6 Interpolating Missing Data 203

4.7 Application of the Hilbert Transform for Efficient Computation of the Spectrum for Nonuniformly Spaced Data 213

4.7.1 Formulation of the Least Square Method 217

4.7.2 Hilbert Transform Relationship 221

4.7.3 Magnitude Estimation 223

4.8 Conclusion 229

References 229

5 The Source Reconstruction Method 235

Summary 235

5.1 Introduction 236

5.2 An Overview of the Source Reconstruction Method (SRM) 238

5.3 Mathematical Formulation for the Integral Equations 239

5.4 Near-Field to Far-Field Transformation Using an Equivalent Magnetic Current Approach 240

5.4.1 Description of the Proposed Methodology 241

5.4.2 Solution of the Integral Equation for the Magnetic Current 245

5.4.3 Numerical Results Utilizing the Magnetic Current 249

5.4.4 Summary 268

5.5 Near-Field to Near/Far-Field Transformation for Arbitrary Near-Field Geometry Utilizing an Equivalent Electric Current 276

5.5.1 Description of the Proposed Methodology 278

5.5.2 Numerical Results Using an Equivalent Electric Current 281

5.5.3 Summary 286

5.6 Evaluating Near-Field Radiation Patterns of Commercial Antennas 297

5.6.1 Background 297

5.6.2 Formulation of the Problem 301

5.6.3 Results for the Near-field To Far-field Transformation 304

5.6.3.1 A Base Station Antenna 304

5.6.3.2 NF to FF Transformation of a Pyramidal Horn Antenna 307

5.6.3.3 Reference Volume of a Base Station Antenna for Human Exposure to EM Fields 310

5.6.4 Summary 311

5.7 Conclusions 313

References 314

6 Planar Near-Field to Far-Field Transformation Using a Single Moving Probe and a Fixed Probe Arrays 319

Summary 319

6.1 Introduction 320

6.2 Theory 322

6.3 Integral Equation Formulation 323

6.4 Formulation of the Matrix Equation 325

6.5 Use of an Magnetic Dipole Array as Equivalent Sources 328

6.6 Sample Numerical Results 329

6.7 Summary 337

6.8 Differences between Conventional Modal Expansion and the Equivalent Source Method for Planar Near-Field to Far-Field Transformation 337

6.8.1 Introduction 337

6.8.2 Modal Expansion Method 339

6.8.3 Integral Equation Approach 341

6.8.4 Numerical Examples 344

6.8.5 Summary 351

6.9 A Direct Optimization Approach for Source Reconstruction and NF-FF Transformation Using Amplitude-Only Data 352

6.9.1 Background 352

6.9.2 Equivalent Current Representation 354

6.9.3 Optimization of a Cost Function 356

6.9.4 Numerical Simulation 357

6.9.5 Results Obtained Utilizing Experimental Data 358

6.9.6 Summary 359

6.10 Use of Computational Electromagnetics to Enhance the Accuracy and Efficiency of Antenna Pattern Measurements Using an Array of Dipole Probes 361

6.10.1 Introduction 362

6.10.2 Development of the Proposed Methodology 363

6.10.3 Philosophy of the Computational Methodology 363

6.10.4 Formulation of the Integral Equations 365

6.10.5 Solution of the Integro-Differential Equations 367

6.10.6 Sample Numerical Results 369

6.10.6.1 Example 1 369

6.10.6.2 Example 2 373

6.10.6.3 Example 3 377

6.10.6.4 Example 4 379

6.10.7 Summary 384

6.11 A Fast and Efficient Method for Determining the Far Field Patterns Along the Principal Planes Using a Rectangular Probe Array 384

6.11.1 Introduction 385

6.11.2 Description of the Proposed Methodology 385

6.11.3 Sample Numerical Results 387

6.11.3.1 Example 1 387

6.11.3.2 Example 2 393

6.11.3.3 Example 3 397

6.11.3.4 Example 4 401

6.11.4 Summary 406

6.12 The Influence of the Size of Square Dipole Probe Array Measurement on the Accuracy of NF-FF Pattern 406

6.12.1 Illustration of the Proposed Methodology Utilizing Sample Numerical Results 407

6.12.1.1 Example 1 407

6.12.1.2 Example 2 411

6.12.1.3 Example 3 416

6.12.1.4 Example 4 419

6.12.2 Summary 428

6.13 Use of a Fixed Probe Array Measuring Amplitude-Only Near-Field Data for Calculating the Far-Field 428

6.13.1 Proposed Methodology 429

6.13.2 Sample Numerical Results 430

6.13.2.1 Example 1 430

6.13.2.2 Example 2 434

6.13.2.3 Example 3 437

6.13.2.4 Example 4 437

6.13.3 Summary 441

6.14 Probe Correction for Use with Electrically Large Probes 442

6.14.1 Development of the Proposed Methodology 443

6.14.2 Formulation of the Solution Methodology 446

6.14.3 Sample Numerical Results 447

6.15 Conclusions 449

References 449

7 Spherical Near-Field to Far-Field Transformation 453

Summary 453

7.1 An Analytical Spherical Near-Field to Far-Field Transformation 453

7.1.1 Introduction 453

7.1.2 An Analytical Spherical Near-Field to Far-Field Transformation 454

7.1.3 Numerical Simulations 464

7.1.3.1 Synthetic Data 464

7.1.3.2 Experimental Data 465

7.1.4 Summary 468

7.2 Radial Field Retrieval in Spherical Scanning for Current Reconstruction and NF–FF Transformation 468

7.2.1 Background 468

7.2.2 An Equivalent Current Reconstruction from Spherical Measurement Plane 470

7.2.3 The Radial Electric Field Retrieval Algorithm 472

7.2.4 Results Obtained Using This Formulation 473

7.2.4.1 Simulated Data 473

7.2.4.2 Using Measured Data 475

7.3 Conclusion 482

Appendix 7A A Fortran Based Computer Program for Transforming Spherical Near-Field to Far-Field 483

References 489

8 Deconvolving Measured Electromagnetic Responses 491

Summary 491

8.1 Introduction 491

8.2 The Conjugate Gradient Method with Fast Fourier Transform for Computational Efficiency 495

8.2.1 Theory 495

8.2.2 Numerical Results 498

8.3 Total Least Squares Approach Utilizing Singular Value Decomposition 501

8.3.1 Theory 501

8.3.2 Total Least Squares (TLS) 502

8.3.3 Numerical Results 506

8.4 Conclusion 516

References 516

9 Performance of Different Functionals for Interpolation/Extrapolation of Near/Far-Field Data 519

Summary 519

9.1 Background 520

9.2 Approximating a Frequency Domain Response by Chebyshev Polynomials 521

9.3 The Cauchy Method Based on Gegenbauer Polynomials 531

9.3.1 Numerical Results and Discussion 537

9.3.1.1 Example of a Horn Antenna 537

9.3.1.2 Example of a 2-element Microstrip Patch Array 539

9.3.1.3 Example of a Parabolic Antenna 541

9.4 Near-Field to Far-Field Transformation of a Zenith-Directed Parabolic Reflector Using the Ordinary Cauchy Method 543

9.5 Near-Field to Far-Field Transformation of a Rotated Parabolic Reflector Using the Ordinary Cauchy Method 552

9.6 Near-Field to Far-Field Transformation of a Zenith-Directed Parabolic Reflector Using the Matrix Pencil Method 558

9.7 Near-Field to Far-Field Transformation of a Rotated Parabolic Reflector Using the Matrix Pencil Method 564

9.8 Conclusion 569

References 569

10 Retrieval of Free Space Radiation Patterns from Measured Data in a Non-Anechoic Environment 573

Summary 573

10.1 Problem Background 573

10.2 Review of Pattern Reconstruction Methodologies 575

10.3 Deconvolution Method for Radiation Pattern Reconstruction 578

10.3.1 Equations and Derivation 578

10.3.2 Steps Required to Implement the Proposed Methodology 584

10.3.3 Processing of the Data 585

10.3.4 Simulation Examples 587

10.3.4.1 Example I: One PEC Plate Serves as a Reflector 587

10.3.4.2 Example II: Two PEC Plates Now Serve as Reflectors 594

10.3.4.3 Example III: Four Connected PEC Plates Serve as Reflectors 598

10.3.4.4 Example IV: Use of a Parabolic Reflector Antenna as the AUT 604

10.3.5 Discussions on the Deconvolution Method for Radiation Pattern Reconstruction 608

10.4 Effect of Different Types of Probe Antennas 608

10.4.1 Numerical Examples 608

10.4.1.1 Example I: Use of a Yagi Antenna as the Probe 608

10.4.1.2 Example II: Use of a Parabolic Reflector Antenna as the Probe 612

10.4.1.3 Example III: Use of a Dipole Antenna as the Probe 613

10.5 Effect of Different Antenna Size 619

10.6 Effect of Using Different Sizes of PEC Plates 626

10.7 Extension of the Deconvolution Method to Three-Dimensional Pattern Reconstruction 632

10.7.1 Mathematical Characterization of the Methodology 632

10.7.2 Steps Summarizing for the Methodology 635

10.7.3 Processing the Data 636

10.7.4 Results for Simulation Examples 638

10.7.4.1 Example I: Four Wide PEC Plates Serve as Reflectors 640

10.7.4.2 Example II: Four PEC Plates and the Ground Serve as Reflectors 643

10.7.4.3 Example III: Six Plates Forming an Unclosed Contour Serve as Reflectors 651

10.7.4.4 Example IV: Antenna Measurement in a Closed PEC Box 659

10.7.4.5 Example V: Six Dielectric Plates Forming a Closed Contour Simulating a Room 662

10.8 Conclusion 673

Appendix A: Data Mapping Using the Conversion between the Spherical Coordinate System and the Cartesian Coordinate System 675

Appendix B: Description of the 2D-FFT during the Data Processing 677

References 680

Index 683

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