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THEORY OF STOCHASTIC LOCAL AREA CHANNEL MODELING FOR WIRELESS COMMUNICATIONS by Gregory D. Durgin Final Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPY in Electrical Engineering Theodore S. de Wolf Gary S. Brown Jeffrey H.

Reed Werner Kohler Robert J. Boyle December 2000 Blacksburg, Virginia Keywords: Fading, Mobile Radio Propagation, Wireless Communications Copyright 2000, Gregory D. Durgin THEORY OF STOCHASTIC LOCAL AREA CHANNEL MODELING FOR WIRELESS COMMUNICATIONS Gregory D. Durgin (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.

ii Contents (ABSTRACT) ii 1 Introduction 2 1.1 The Need for Improvement in Channel Modeling Theory .1 Higher and Higher Data Rates .2 Ubiquity of Wireless Devices .4 Faster, Smaller, Cheaper Hardware .6 Multiple-Input, Multiple-Output Systems .2 Key Topics in Small-Scale Channel Modeling .1 Spatial, Temporal, and Frequency Coherence .2 Rigorous Application of Physics to Channel Models .3 Physically-Based Small-Scale Fading Distributions .4 Characterization and Analysis of Angle Spectra .5 Channel Statistics of Rayleigh Fading .6 Spatio-Temporal Peer-to-Peer Measurements .3 How to Read This Dissertation. 9 2 Foundations of Stochastic Channel Modeling 10 2.2 The Baseband Channel .3 Time-Invariant vs. Time-Varying Channels .3 Using the Complete Baseband Channel .1 Spectral Domain Representations .2 General Signal Transmission .3 Static Channel Transmission .4 Mobile Receiver Transmission .4 Stochastic Channel Characterization .3 RMS Power Spectrum Width .4 Channel Duality Principle .A Functions of Three-Dimensional Space .1 Vector Notation for Fourier Transforms .2 Scalar Collapse of Position Vectors .3 Scalar Collapse of Wavevectors. 46 3 The Physics of Small-Scale Fading 49 3.1 Plane Wave Representation .1 Electromagnetic Fields and Received Signals .2 The Maxwellian Basis .3 Homogeneous Plane Waves .4 Inhomogeneous Plane Waves .5 Physics of Homogeneous vs.

Inhomogeneous Plane Waves .2 The Local Area .1 Definition of a Local Area .3 A Wideband Plane Wave .4 The Bandwidth-Distance Threshold .1 Specular Wave Component .2 Non-specular Wave Component .3 Diffuse Wave Component .4 Reduced Wave Grouping .4 The Stochastic Local Area Channel (SLAC) Model .2 Random Phase Models .A Wavevector Criterion for Free Space Plane Waves. 81 4 First-Order Statistics of Small-Scale Channels 82 4.1 Mean Received Power .2 Mean U-SLAC Power .3 Frequency and Spatial Averaging .2 Envelope Probability Density Functions .1 Notes and Concepts .3 Specular Characteristic Function .4 Diffuse, Non-specular Characteristic Function .5 The I-SLAC PDF Generator .3 Closed-Form PDF Solutions .1 The One-Wave PDF .2 The Two-Wave PDF .3 The Three-Wave PDF .4 The Rayleigh PDF .5 The Rician PDF .4 Two-Wave with Incoherent Power (TIP) PDF .3 Rayleigh and Rician Approximations .4 Final Comments on Reduced Wave Groupings .5 TIP PDF Applications .6 Closing Remarks on TIP Fading .A Derivation of Envelope Characteristic Functions .B Derivation for TIP Fading PDF’s .2 Property as a PDF .3 Proper Limiting Behavior .4 Preservation of the Second Moment. 117 5 The Angle Spectrum 118 5.1 Angle Spectrum Concepts .1 Definition of the Angle Spectrum .2 Mapping Angles to Wavenumbers .3 From-the-Horizon Propagation .4 Summary of Angle Spectrum Concepts .2 Fading Rate Variance .1 Definition of a Rate Variance .2 Fundamental Spectral Spread Theorem .3 Multipath Shape Factors .1 Definition of Shape Factors .2 Basic Wavenumber Spread Relationship .3 Comparison to Omnidirectional Propagation .1 Two-Wave Channel Model .2 Sector Channel Model .3 Double Sector Channel Model .4 Rician Channel Model .A Derivation of Shape Factors. 142 6 Rayleigh Fading Channel Statistics 144 6.1 The Level-Crossing Problem .1 Level-Crossing Rate .2 Average Fade Duration .3 Level Crossing in Frequency .4 Level Crossing in Space .2 Envelope Unit Autocovariance .1 Temporal Unit Autocovariance .2 Frequency Unit Autocovariance .3 Spatial Unit Autocovariance .4 Joint Unit Autocovariance .3 Revisiting Classical Spatial Channel Models .1 Classical Spatial Channel Models .2 Channel Model Solutions .A Approximate Spatial Autocovariance Function.

159 7 Spatio-Temporal Measurements 161 7.1 Previous Measurement Campaigns .1 Contribution of this Work .2 Comparison to Other Measurement Campaigns in the Literature .2 Overview of Measurement Campaign .2 Channel Sounding Hardware .3 Automated Antenna Positioning .4 Antenna Specifications .5 Sources of Error in the Experiment .1 Delay Dispersion Results .2 Angle Dispersion Results .3 Joint Angle-Delay Statistics .A Description of Measured Parameters .1 Noncoherent Channel Measurements .3 Time Delay Parameters .4 Angle-of-Arrival Parameters .1 Future Areas of Research .2 Specific Analytical Problems .3 Applications to Wireless Technology. 187 9 Vitae 189 Bibliography 190 viii List of Tables 2.1 Received signal functions used in complex baseband analysis.2 Fourier transform definitions for each channel dependency.3 Channel duality relationships between time, frequency, and space.1 Maximum size of a local area according to the bandwidth-distance threshold for example wireless applications.2 Reduced Wave Grouping Algorithm .1 Summary of envelope PDF’s in different fading environments.2 Exact coefficients for the first five orders of the approximate Two-Wave with Incoherent Power (TIP) fading PDF.3 The TIP PDF contains the Rayleigh, Rician, One-Wave, and Two-Wave PDF’s as special cases.4 Three examples of Two-Wave with Incoherent Power (TIP) fading that may simplify to Rayleigh or Rician PDF’s.1 Summary of dispersion statistics calculated from track measurements.2 Summary of spatial multipath parameters calculated from spatially-averaged azimuthal sweeps of a horn antenna. 176 ix List of Figures 2.1 The many different bandwidths defined for a baseband signal.2 Baseband-Passband transformations in the time domain (inner cycle) and the frequency domain (outer cycle).3 Block diagram of baseband and passband channel models for SISO transmis- sion.4 Spectral diagram of baseband and passband signals and channel.5 Example of a time-varying channel.6 Example of a frequency-varying channel.7 Example of a space-varying channel (one-dimensional cut).8 Example of small-scale and large-scale fading.9 Autocorrelation and power spectrum relationships for time and frequency.10 Autocorrelation and power spectrum relationships for space and frequency.11 Autocorrelation and power spectrum relationships for space, time, and fre- quency.12 Relationship between (x, y, z) and (r, θ, ϕ) coordinates.1 An antenna maps the complex electric field vector, E(˜ r ), to a scalar baseband channel voltage, h̃(r).2 Homogeneous and inhomogeneous plane waves.3 Rules-of-thumb for homogeneous and inhomogeneous plane wave propaga- tion.4 An example linear circuit contains capacitors, inductors, resistors and an AC source.5 A linear circuit solution may be broken into a steady-state and transient solution.6 The size of a local area decreases closer to significant scatterers.7 Illustration of the basic quantities of time-harmonic wave propagation through a scattering environment.8 Examples of different delay spectra using the standard power spectrum, Sh̃ (τ ), and the integrated power spectrum, Fh̃ (τ ).1 Evolution of a Two-Wave PDF and CDF with varying ∆.2 Several CDF’s and PDF’s for Three-Wave local area propagation.3 Evolution of a Rician PDF and CDF as the dominant multipath component
increases.4 TIP PDF and CDF for K = 0 dB.5 TIP PDF and CDF for K = 3 dB.6 TIP PDF and CDF for K = 6 dB.7 TIP PDF and CDF for K = 10 dB.8 A flowchart for determining the simplest way to graph an I-SLAC PDF.1 A local area characterized by an angle spectrum, p(θ, ϕ), will produce differ- ent wavenumber spectra, Sh̃ (k), depending on the orientation in space.2 Multipath power is mapped from the angle spectrum, p(θ), to the wavenum- ber spectrum, Sh̃ (k), as a function of its angle-of-arrival.3 Autocorrelation and spectrum relationships for the space-varying channel.4 Two examples of time-varying, Rayleigh-distributed stochastic processes.5 Two-wave propagation model. Graphs are for the special case of P1 = P2 .6 Multipath sector propagation model.7 Multipath double sector propagation model.8 Rician propagation model.1 Three different multipath-induced mobile-fading scenarios.2 Comparison between Clarke theoretical and approximate envelope autoco- variance functions for Ez -case.3 Comparison between Clarke theoretical and approximate envelope autoco- variance functions for Hx -case.4 Comparison between Clarke theoretical and approximate envelope autoco- variance functions for Hy -case.1 The transmitter-receiver configurations for the 6 local areas measured in- doors.2 In a local area, power delay profiles are measured along two orthogonal linear tracks using an omnidirectional antenna.3 A track measurement is made with an omnidirectional antenna in a campus parking lot.4 In a local area, power delay profiles are measured by spatially averaging angular sweeps with a directional antenna.5 A rotational measurement is made with a directional horn antenna in the lobby of an office building.6 A series of PDP snapshots along a track, measured with an omnidirectional receiver antenna.7 A local area angle-delay spectrum as measured from a set of rotational mea- surements.8 The trend between angle spread and delay spread for indoor and outdoor receivers.

179 xii Acknowledgements Any endeavor as large as a Ph. dissertation requires numerous contributions by people other than the Ph. This dissertation is no exception, as there are a number of organizations and people that must be acknowledged before a single technical detail can be discussed. First, there is a long list of generous sponsors and contributors.

The research in this dissertation was sponsored by a Bradley Fellowship in Electrical and Computer Engineer- ing, the MPRG Industrial Affiliates Program, and ITT Defense & Electronics, Inc. The peer-to-peer measurement campaign could not have been performed without the generous assistance of MPRG students Vikas Kukshya, Jiun Siew, Erica Lau, Christopher Steger, Paulo Cardieri, Jason Aaron, and Bruce Puckett. I also owe an extreme debt of gratitude to everyone on my Ph. Three members in particular – Ted Rappaport, David de Wolf, and Gary Brown – contributed numerous technical criticisms, ideas, and suggestions that found their way into the final dissertation.

I could not have asked for a more supportive (and knowledgeable) Ph. An equal debt of gratitude is owed to the extremely talented students and staff of the Mobile and Portable Radio Research Group of Virginia Tech. What wonderful people I have worked with over the years! There are three students I would like to single out for their help and devoted friendship throughout my dissertation: Neal Patwari, Hao Xu, and Bruce Puckett. Finally, my deepest gratitude and sincerest love is owed to these three people: my mother Lisa, my father Dave, and my Lord and Savior Jesus Christ.

A deep, loving relationship with each has provided all inspiration and direction in life. 1 Chapter 1 Introduction There are few things in nature more unwieldy than the power-limited, space-varying, time- varying, frequency-varying wireless channel. Yet there is great reward for engineers who can overcome these limitations and transmit data through such a harsh environment. The explosive worldwide growth of personal communications services through the 1990’s is a testament to the business opportunities that result from conquering the wireless channel.

However, given the emergence of newer wireless systems that require more and more band- width, the task of conquering the wireless channel is becoming more difficult. This task requires a thorough background in wireless channel modeling.