From 1990 to 1992, he was with ANT Nachrichtentechnik GmbH, Backnang, Germany, where he was engaged in digital satellite communications. From 1992 to 2001, he was with the Department of Digital Networks at the Technical University Hamburg-Harburg. Since 2001, he has been a full professor of mobile communications with the University of Agder, Grimstad, Norway. He is author of the books “Mobile Radio Channels - Modelling, Analysis, and Simulation” (in German) (Wiesbaden, Germany: Vieweg, 1999) and “Mobile Fading Channels” (John Wiley & Sons, 2002) and “Mobile Radio Channels, 2nd Edition” (John Wiley & Sons, 2011). His current research interests include mobile radio communications, especially multipath fading channel modelling, multiple-input multiple-output (MIMO) systems, channel parameter estimation, and coded-modulation techniques for fading channels.

List of Acronyms xv

List of Symbols xix

1 Introduction 1

1.1 The Evolution of Mobile Radio Systems 1

1.2 Basic Knowledge of Mobile Radio Channels 8

1.3 Structure of this Book 12

2 Random Variables, Stochastic Processes, and Deterministic Signals 17

2.1 Random Variables 17

2.1.1 Basic Definitions of Probability Theory 17

2.1.2 Important Probability Density Functions 24

2.1.3 Functions of Random Variables 35

2.2 Stochastic Processes 37

2.2.1 Stationary Processes 40

2.2.2 Ergodic Processes 42

2.2.3 Level-Crossing Rate and Average Duration of Fades 43

2.2.4 Linear Systems with Stochastic Inputs 45

2.3 Deterministic Signals 48

2.3.1 Deterministic Continuous-Time Signals 48

2.3.2 Deterministic Discrete-Time Signals 50

2.4 Further Reading 52

Appendix 2.A Derivation of Rice’s General Formula for the Level-Crossing Rate 52

3 Rayleigh and Rice Channels 55

3.1 System Theoretical Description of Multipath Channels 56

3.2 Formal Description of Rayleigh and Rice Channels 61

3.3 Elementary Properties of Rayleigh and Rice Channels 62

3.3.1 Autocorrelation Function and Spectrum of the Complex Envelope 62

3.3.2 Autocorrelation Function and Spectrum of the Envelope 65

3.3.3 Autocorrelation Function and Spectrum of the Squared Envelope 67

3.4 Statistical Properties of Rayleigh and Rice Channels 69

3.4.1 Probability Density Function of the Envelope and the Phase 70

3.4.2 Probability Density Function of the Squared Envelope 72

3.4.3 Level-Crossing Rate and Average Duration of Fades 73

3.4.4 The Statistics of the Fading Intervals of Rayleigh Channels 77

3.5 Further Reading 84

Appendix 3.A Derivation of the Jakes Power Spectral Density and the

Corresponding Autocorrelation Function 84

Appendix 3.B Derivation of the Autocorrelation Function of the Envelope 88

Appendix 3.C Derivation of the Autocovariance Spectrum of the Envelope Under

Isotropic Scattering Conditions 90

Appendix 3.D Derivation of the Level-Crossing Rate of Rice Processes with

Different Spectral Shapes of the Underlying Gaussian Random

Processes 91

4 Introduction to Sum-of-Sinusoids Channel Models 95

4.1 Principle of Deterministic Channel Modelling 96

4.2 Elementary Properties of Deterministic Sum-of-Sinusoids Processes 102

4.3 Statistical Properties of Deterministic Sum-of-Sinusoids Processes 107

4.3.1 Probability Density Function of the Envelope and the Phase 108

4.3.2 Level-Crossing Rate and Average Duration of Fades 115

4.3.3 Statistics of the Fading Intervals at Low Signal Levels 120

4.3.4 Stationarity and Ergodicity of Sum-of-Sinusoids Processes 122

4.4 Classes of Sum-of-Sinusoids Processes 123

4.5 Basics of Sum-of-Cisoids Channel Models 126

4.5.1 Elementary Properties of Stochastic Sum-of-Cisoids Processes 127

4.5.2 Probability Density Function of the Envelope and Phase 129

4.6 Criteria for the Performance Evaluation 135

4.7 Further Reading 135

Appendix 4.A Derivation of the Autocorrelation Function of the Squared Envelope

of Complex Deterministic Gaussian Processes 136

Appendix 4.B Derivation of the Exact Solution of the Level-Crossing Rate and the

Average Duration of Fades of Deterministic Rice Processes 137

5 Parametrization of Sum-of-Sinusoids Channel Models 149

5.1 Methods for Computing the Doppler Frequencies and Gains 151

5.1.1 Method of Equal Distances (MED) 151

5.1.2 Mean-Square-Error Method (MSEM) 157

5.1.3 Method of Equal Areas (MEA) 162

5.1.4 Monte Carlo Method (MCM) 170

5.1.5 Jakes Method (JM) 178

5.1.6 Lp-Norm Method (LPNM) 189

5.1.7 Method of Exact Doppler Spread (MEDS) 201

5.1.8 Randomized Method of Exact Doppler Spread (RMEDS) 205

5.1.9 Method of Exact Doppler Spread with Set Partitioning (MEDS-SP) 207

5.2 Methods for Computing the Phases 212

5.3 Fading Intervals of Deterministic Rayleigh Processes 214

5.4 Parametrization of Sum-of-Cisoids Channel Models 222

5.4.1 Problem Description 222

5.4.2 Extended Method of Exact Doppler Spread (EMEDS) 222

5.4.3 Lp-Norm Method (LPNM) 224

5.4.4 Generalized Method of Equal Areas (GMEA) 225

5.4.5 Performance Analysis 228

5.5 Concluding Remarks and Further Reading 234

Appendix 5.A Analysis of the Relative Model Error by Using the Monte Carlo

Method 236

Appendix 5.B Proof of the Convergence of the Sample Mean Autocorrelation

Function by Using the MEDS-SP 238

Appendix 5.C Proof of the Condition for Uncorrelated Inphase and Quadrature

Components of SOC Processes 239

6 Frequency-Nonselective Channel Models 241

6.1 The Extended Suzuki Process of Type I 243

6.1.1 Modelling and Analysis of Short-Term Fading 243

6.1.2 Modelling and Analysis of Long-Term Fading 254

6.1.3 The Stochastic Extended Suzuki Process of Type I 257

6.1.4 The Deterministic Extended Suzuki Process of Type I 262

6.1.5 Applications and Simulation Results 265

6.2 The Extended Suzuki Process of Type II 268

6.2.1 Modelling and Analysis of Short-Term Fading 269

6.2.2 The Stochastic Extended Suzuki Process of Type II 279

6.2.3 The Deterministic Extended Suzuki Process of Type II 283

6.2.4 Applications and Simulation Results 287

6.3 The Generalized Rice Process 290

6.3.1 The Stochastic Generalized Rice Process 291

6.3.2 The Deterministic Generalized Rice Process 295

6.3.3 Applications and Simulation Results 298

6.4 The Modified Loo Model 300

6.4.1 The Stochastic Modified Loo Model 300

6.4.2 The Deterministic Modified Loo Model 311

6.4.3 Applications and Simulation Results 317

6.5 Modelling of Nonstationary Land Mobile Satellite Channels 319

6.5.1 Lutz’s Two-State Channel Model 320

6.5.2 M-State Channel Models 322

6.5.3 Modelling of Nonstationary Real-World LMS Channels 323

7 Frequency-Selective Channel Models 335

7.1 The Ellipse Model of Parsons and Bajwa 336

7.2 System Theoretical Description of Frequency-Selective Channels 338

7.3 Frequency-Selective Stochastic Channel Models 342

7.3.1 Correlation Functions 342

7.3.2 The WSSUS Model According to Bello 344

7.3.3 The COST 207 Channel Models 352

7.3.4 The HIPERLAN/2 Channel Models 358

7.4 Frequency-Selective Sum-of-Sinusoids Channel Models 358

7.4.1 System Functions of Sum-of-Sinusoids Uncorrelated Scattering

(SOSUS) Models 358

7.4.2 Correlation Functions and Power Spectral Densities of

SOSUS Models 364

7.4.3 Delay Power Spectral Density, Doppler Power Spectral Density,

and Characteristic Quantities of SOSUS Models 368

7.4.4 Determination of the Model Parameters of SOSUS Models 372

7.4.5 Simulation Models for the COST 207 Channel Models 376

7.5 Methods for Modelling of Given Power Delay Profiles 378

7.5.1 Problem Description 379

7.5.2 Methods for the Computation of the Discrete Propagation Delays and

the Path Gains 381

7.5.3 Comparison of the Parameter Computation Methods 391

7.5.4 Applications to Measured Power Delay Profiles 393

7.6 Perfect Modelling and Simulation of Measured Wideband Mobile Radio

Channels 396

7.6.1 The Sum-of-Cisoids Uncorrelated Scattering (SOCUS) Model 396

7.6.2 The Principle of Perfect Channel Modelling 403

7.6.3 Application to a Measured Wideband Indoor Channel 404

7.7 Further Reading 406

Appendix 7.A Specification of the L-Path COST 207 Channel Models 409

Appendix 7.B Specification of the L-Path HIPERLAN/2 Channel Models 413

8 MIMO Channel Models 417

8.1 The Generalized Principle of Deterministic Channel Modelling 418

8.2 The One-Ring MIMO Channel Model 421

8.2.1 The Geometrical One-Ring Scattering Model 422

8.2.2 The Reference Model for the One-Ring MIMO Channel Model 423

8.2.3 Simulation Models for the One-Ring MIMO Channel Model 429

8.2.4 Parameter Computation Methods 433

8.2.5 Performance Evaluation 434

8.2.6 Simulation Results 436

8.3 The Two-Ring MIMO Channel Model 438

8.3.1 The Geometrical Two-Ring Scattering Model 439

8.3.2 The Reference Model for the Two-Ring MIMO Channel Model 440

8.3.3 Simulation Models for the Two-Ring MIMO Channel Model 445

8.3.4 Isotropic and Non-Isotropic Scattering Scenarios 449

8.3.5 Parameter Computation Methods 451

8.4 The Elliptical MIMO Channel Model 457

8.4.1 The Geometrical Elliptical Scattering Model 458

8.4.2 The Reference Model for the Elliptical MIMO Channel Model 459

8.4.3 Simulation Models for the Elliptical MIMO Channel Model 463

8.4.4 Model Extensions 466

8.5 Further Reading 469

Appendix 8.A Proof of Ergodicity 472

9 High-Speed Channel Simulators 475

9.1 Discrete-Time Deterministic Processes 476

9.2 Realization of Discrete-Time Deterministic Processes 478

9.2.1 Look-Up Table System 478

9.2.2 Matrix System 481

9.2.3 Shift Register System 483

9.3 Properties of Discrete-Time Deterministic Processes 484

9.3.1 Elementary Properties of Discrete-Time Deterministic Processes 484

9.3.2 Statistical Properties of Discrete-Time Deterministic Processes 491

9.4 Realization Complexity and Simulation Speed 500

9.5 Comparison of the Sum-of-Sinusoids Method with the Filter Method 502

9.6 Further Reading 505

10 Selected Topics in Mobile Radio Channel Modelling 507

10.1 Design of Multiple Uncorrelated Rayleigh Fading Waveforms 507

10.1.1 Problem Description 508

10.1.2 Generalized Method of Exact Doppler Spread (GMEDSq) 511

10.1.3 Related Parameter Computation Methods 516

10.1.4 The Effect of Finite Simulation Time on the Cross-Correlation

Properties 518

10.1.5 Further Reading 520

10.2 Spatial Channel Models for Shadow Fading 521

10.2.1 The Reference Model for Shadow Fading 522

10.2.2 The Simulation Model for Shadow Fading 523

10.2.3 Correlation Models for Shadow Fading 527

10.2.4 Further Reading 535

10.3 Frequency Hopping Mobile Radio Channels 536

10.3.1 The Reference Model for Frequency Hopping Channels 536

10.3.2 The Simulation Model for Frequency Hopping Channels 538

10.3.3 Performance Analysis 540

10.3.4 Simulation Results 544

10.3.5 Further Reading 544

Appendix 10.A Derivation of the Spatial Autocorrelation Function of Lognormal

Processes 545

Appendix 10.B Derivation of the Level-Crossing Rate of Spatial Lognormal

Processes 546

Appendix 10.C Derivation of the Level-Crossing Rate of Sum-of-Sinusoids

Shadowing Simulators 546

Appendix 10.D Application of the Method of Equal Areas (MEA) on the

Gudmundson Correlation Model 548

Appendix 10.E Derivation of the Time-Frequency Cross-Correlation Function of

Frequency Hopping Channels 549

Appendix 10.F Parametrization of Frequency Hopping Channel Simulators 551

References 553

Index 571