[FONT=Times New Roman](ABSTRACT)

This report was written to satisfy the final dissertation requirements toward a doctoral

degree in electrical engineering. The dissertation outlines work accomplished in the pursuit

of this degree. This report is also designed to be a general introduction to the concepts and

techniques of small-scale radio channel modeling. At the present time, there does not exist

a comprehensive introduction and overview of basic concepts in this field. Furthermore, as

the wireless industry continues to mature and develop technology, the need is now greater

than ever for more sophisticated channel modeling research.

Each chapter of this preliminary report is, in itself, a stand-alone topic in channel

modeling theory. Culled from original reports and journal papers, each chapter makes a

unique contribution to the field of channel modeling. Original contributions in this report

include

joint characterization of time-varying, space-varying, and frequency-varying channels

under the rubric of duality

rules and definitions for constructing channel models that solve Maxwell’s equations

overview of probability density functions that describe random small-scale fading

techniques for modeling a small-scale radio channel using an angle spectrum

overview of techniques for describing fading statistics in wireless channels

results from a wideband spatio-temporal measurement campaign

Together, the chapters provide a cohesive overview of basic principles. The discussion of

the wideband spatio-temporal measurement campaign at 1920 MHz makes an excellent case

study in applied channel modeling and ties together much of the theory developed in this

dissertation.

Contents

(ABSTRACT) ii

1 Introduction 2

1.1 The Need for Improvement in ChannelModeling Theory . . . . . . . . . . . 3

1.1.1 Higher and Higher Data Rates . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Ubiquity ofWireless Devices . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Smart Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 Faster, Smaller, Cheaper Hardware . . . . . . . . . . . . . . . . . . . 4

1.1.5 Frequency Congestion . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.6 Multiple-Input, Multiple-Output Systems . . . . . . . . . . . . . . . 5

1.2 Key Topics in Small-Scale ChannelModeling . . . . . . . . . . . . . . . . . 6

1.2.1 Spatial, Temporal, and Frequency Coherence . . . . . . . . . . . . . 6

1.2.2 Rigorous Application of Physics to ChannelModels . . . . . . . . . 6

1.2.3 Physically-Based Small-Scale Fading Distributions . . . . . . . . . . 7

1.2.4 Characterization and Analysis of Angle Spectra . . . . . . . . . . . . 7

1.2.5 Channel Statistics of Rayleigh Fading . . . . . . . . . . . . . . . . . 7

1.2.6 Spatio-Temporal Peer-to-Peer Measurements . . . . . . . . . . . . . 8

1.3 How to Read This Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Foundations of Stochastic Channel Modeling 10

2.1 Baseband Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 SignalModulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 The Baseband Channel . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Time-Invariant vs. Time-Varying Channels . . . . . . . . . . . . . . 15

2.1.4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Channel Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Coherence vs. Selectivity . . . . . . . . . . . . . . . . . . . . . . . . 20

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2.2.2 Temporal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.3 Frequency Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.4 Spatial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Using the Complete Baseband Channel . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Spectral Domain Representations . . . . . . . . . . . . . . . . . . . . 26

2.3.2 General Signal Transmission . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3 Static Channel Transmission . . . . . . . . . . . . . . . . . . . . . . 28

2.3.4 Mobile Receiver Transmission . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Stochastic Channel Characterization . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Autocorrelation Relationships . . . . . . . . . . . . . . . . . . . . . 30

2.4.2 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 RMS Power SpectrumWidth . . . . . . . . . . . . . . . . . . . . . . 36

2.4.4 Channel Duality Principle . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.A Functions of Three-Dimensional Space . . . . . . . . . . . . . . . . . . . . . 44

2.A.1 Vector Notation for Fourier Transforms . . . . . . . . . . . . . . . . 44

2.A.2 Scalar Collapse of Position Vectors . . . . . . . . . . . . . . . . . . 45

2.A.3 Scalar Collapse ofWavevectors . . . . . . . . . . . . . . . . . . . . . 46

3 The Physics of Small-Scale Fading 49

3.1 PlaneWave Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 Electromagnetic Fields and Received Signals . . . . . . . . . . . . . 50

3.1.2 The Maxwellian Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.3 Homogeneous PlaneWaves . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.4 Inhomogeneous PlaneWaves . . . . . . . . . . . . . . . . . . . . . . 53

3.1.5 Physics of Homogeneous vs. Inhomogeneous PlaneWaves . . . . . . 55

3.2 The Local Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.1 Definition of a Local Area . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 Scatterer Proximity . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.3 AWideband PlaneWave . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.4 The Bandwidth-Distance Threshold . . . . . . . . . . . . . . . . . . 64

3.3 Wave Groupings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.1 SpecularWave Component . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.2 Non-specularWave Component . . . . . . . . . . . . . . . . . . . . . 67

3.3.3 DiffuseWave Component . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.4 ReducedWave Grouping . . . . . . . . . . . . . . . . . . . . . . . . 68

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3.4 The Stochastic Local Area Channel (SLAC)Model . . . . . . . . . . . . . 70

3.4.1 StochasticModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.2 RandomPhaseModels . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4.4 Autocorrelation Functions . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4.5 Heterogeneous Scattering . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.6 Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.A Wavevector Criterion for Free Space PlaneWaves . . . . . . . . . . . . . . 81

4 First-Order Statistics of Small-Scale Channels 82

4.1 Mean Received Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.2 Mean U-SLAC Power . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1.3 Frequency and Spatial Averaging . . . . . . . . . . . . . . . . . . . . 85

4.1.4 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Envelope Probability Density Functions . . . . . . . . . . . . . . . . . . . . 89

4.2.1 Notes and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.2 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Specular Characteristic Function . . . . . . . . . . . . . . . . . . . . 90

4.2.4 Diffuse, Non-specular Characteristic Function . . . . . . . . . . . . . 91

4.2.5 The I-SLAC PDF Generator . . . . . . . . . . . . . . . . . . . . . . 92

4.3 Closed-Form PDF Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.1 The O ne-Wave PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.2 The Two-Wave PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.3 The Three-Wave PDF . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.4 The Rayleigh PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.3.5 The Rician PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Two-Wave with Incoherent Power (TIP) PDF . . . . . . . . . . . . . . . . 100

4.4.1 Approximate Representation . . . . . . . . . . . . . . . . . . . . . . 100

4.4.2 Graphical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.4.3 Rayleigh and Rician Approximations . . . . . . . . . . . . . . . . . . 102

4.4.4 Final Comments on ReducedWave Groupings . . . . . . . . . . . . 108

4.4.5 TIP PDF Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4.6 Closing Remarks on TIP Fading . . . . . . . . . . . . . . . . . . . . 110

4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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4.A Derivation of Envelope Characteristic Functions . . . . . . . . . . . . . . . . 113

4.B Derivation for TIP Fading PDF’s . . . . . . . . . . . . . . . . . . . . . . . . 115

4.B.1 Approximate Representation . . . . . . . . . . . . . . . . . . . . . . 115

4.B.2 Property as a PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.B.3 Proper Limiting Behavior . . . . . . . . . . . . . . . . . . . . . . . . 117

4.B.4 Preservation of the SecondMoment . . . . . . . . . . . . . . . . . . 117

5 The Angle Spectrum 118

5.1 Angle SpectrumConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.1.1 Definition of the Angle Spectrum . . . . . . . . . . . . . . . . . . . . 119

5.1.2 Mapping Angles toWavenumbers . . . . . . . . . . . . . . . . . . . . 120

5.1.3 From-the-Horizon Propagation . . . . . . . . . . . . . . . . . . . . . 121

5.1.4 Summary of Angle SpectrumConcepts . . . . . . . . . . . . . . . . . 124

5.2 Fading Rate Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.1 Definition of a Rate Variance . . . . . . . . . . . . . . . . . . . . . . 127

5.2.2 Fundamental Spectral Spread Theorem . . . . . . . . . . . . . . . . 129

5.3 Multipath Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 Definition of Shape Factors . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.2 BasicWavenumber Spread Relationship . . . . . . . . . . . . . . . . 131

5.3.3 Comparison to Omnidirectional Propagation . . . . . . . . . . . . . 132

5.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.1 Two-Wave ChannelModel . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.2 Sector ChannelModel . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4.3 Double Sector ChannelModel . . . . . . . . . . . . . . . . . . . . . . 136

5.4.4 Rician ChannelModel . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.A Derivation of Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6 Rayleigh Fading Channel Statistics 144

6.1 The Level-Crossing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.1.1 Level-Crossing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

6.1.2 Average Fade Duration . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.1.3 Level Crossing in Frequency . . . . . . . . . . . . . . . . . . . . . . . 146

6.1.4 Level Crossing in Space . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2 Envelope Unit Autocovariance . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2.1 Temporal Unit Autocovariance . . . . . . . . . . . . . . . . . . . . . 148

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6.2.2 Frequency Unit Autocovariance . . . . . . . . . . . . . . . . . . . . . 149

6.2.3 Spatial Unit Autocovariance . . . . . . . . . . . . . . . . . . . . . . . 150

6.2.4 Joint Unit Autocovariance . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3 Revisiting Classical Spatial ChannelModels . . . . . . . . . . . . . . . . . 153

6.3.1 Classical Spatial ChannelModels . . . . . . . . . . . . . . . . . . . . 153

6.3.2 ChannelModel Solutions . . . . . . . . . . . . . . . . . . . . . . . . 154

6.3.3 Additional Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.A Approximate Spatial Autocovariance Function . . . . . . . . . . . . . . . . 159

7 Spatio-Temporal Measurements 161

7.1 PreviousMeasurement Campaigns . . . . . . . . . . . . . . . . . . . . . . . 162

7.1.1 Contribution of thisWork . . . . . . . . . . . . . . . . . . . . . . . . 162

7.1.2 Comparison to Other Measurement Campaigns in the Literature . . 162

7.2 O verview ofMeasurement Campaign . . . . . . . . . . . . . . . . . . . . . 164

7.2.1 Measured Locations . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.2.2 Channel Sounding Hardware . . . . . . . . . . . . . . . . . . . . . . 164

7.2.3 Automated Antenna Positioning . . . . . . . . . . . . . . . . . . . . 166

7.2.4 Antenna Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.2.5 Sources of Error in the Experiment . . . . . . . . . . . . . . . . . . . 169

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3.1 Delay Dispersion Results . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3.2 Angle Dispersion Results . . . . . . . . . . . . . . . . . . . . . . . . 175

7.3.3 Joint Angle-Delay Statistics . . . . . . . . . . . . . . . . . . . . . . 178

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.A Description ofMeasured Parameters . . . . . . . . . . . . . . . . . . . . . . 181

7.A.1 Noncoherent ChannelMeasurements . . . . . . . . . . . . . . . . . . 181

7.A.2 Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.A.3 Time Delay Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 182

7.A.4 Angle-of-Arrival Parameters . . . . . . . . . . . . . . . . . . . . . . . 184

8 Conclusions 185

8.1 Future Areas of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.1.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.1.2 Specific Analytical Problems . . . . . . . . . . . . . . . . . . . . . . 186

8.1.3 Applications toWireless Technology . . . . . . . . . . . . . . . . . . 187

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8.1.4 Measurement Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8.1.5 Computer Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9 Vitae 189

Bibliography 190