iples
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`Page 1 of 172
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`

`
`Wireless
`
`Communications
`
`Principles and Practice
`
`Theodore S. Rappaport
`
`For book and bookstore informarion
`
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`Page 2 of 172
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`

`
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`© 1996 by Prentice Hall PTR
`Prentice-Hall, Inc.
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`Printed in the United States of America
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`ISBN O-13—375536-3
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`

`
`The LORD has blessed me i
`
`with a wonderfulfamily
`to whom this book is dedicated
`
`To my wife Brenda Marie,
`and. to our children
`
`Matthew, Natalie, and Jennifer
`
`Page 4 of 172
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`Page 4 of 172
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`

`
`Contents
`
`Preface
`
`1 Introduction to Wireless Communication Systems
`1.1 Evolution or Mobile Radio Communications
`1.2 Mobile Radiotelephone in the U.S.
`
`1.3 Mobile Radio Systems Around the World
`
`1.4 Examples of Mobile Radio Systems
`
`1.4.1 Paging Systems
`
`1.4.2 Cordless Telephone Systems
`1.4.3 Cellular Telephone Systems
`
`1.4.4 Comparison of Common Mobile Radio Systems
`
`1.5 Trends in Cellular Radio and Personal Communications
`
`1.6 Problems
`
`2 The Cellular Concept — System Design Fundamentals
`2.1 Introduction
`
`2.2 Frequency Reuse
`
`2.3 Channel Assignment Strategies
`
`2.4 Handoff Strategies
`
`2.4.1 Prioritizing Handoffs
`
`2.4.2 Practical Handoff Considerations
`
`2.5 Interference and System Capacity
`
`2.5.1 Co-channel Interference and System Capacity
`
`2.5.2 Adjacent Channel Interference
`
`2.5.3 Power Control for Reducing Interference
`
`2.6 Trunking and Grade of Service
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`vi
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`Contents
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`2.7 Improving Capacity in Cellular Systems
`
`2.7.1 Cell Splitting
`
`2.7.2 Sectoring
`
`2.7.3 A Novel Microcell Zone Concept
`
`2.8 Summary
`2
`alems
`
`3 l
`
`e Radio Propagation: Large-Scale Path Loss
`
`3.1 muoducttion to Radio Wave Propagation
`
`3.2 Free Space Propagation Model
`
`3.3 Relating Power to Electric Field
`
`3.4 The Three Basic Propagation Mechanisms
`3 .5‘ Reflection
`
`3.5.1 Reflection from Dielectrics
`
`3.5.2 Brewster Angle
`3.5.3 Reflection from Perfect Conductors
`
`3.6 Ground Reflection. (2-ray) Model
`3.7 Diffraction
`
`3.7.1 Fresnel Zone Geometry —
`
`3.7.2 Knife-edge Diffraction Model
`3.7.3 Multiple Knife-edge Diffraction
`
`3.8 Scattering
`3.8.1 Radar Cross Section Model
`
`3.9 ‘Practical Link Budget Design using Path Loss Models
`
`3.9.1 Log-distance Path Loss Model
`3.9.2 Log-normal Shadowing V
`3.9.3 Determination of Percentage of Coverage Area»
`3.10 Outdoor Propagation Models
`
`3.10.1 Longley-Rice Model
`3.10.2 Durkin’s Model — A Case Study‘
`3.10.3 Okumura Model
`
`3.10.4 Hata Model
`
`3.10.5 PCS Extension to Hata Model
`
`3.10.6 Walfisch and Bertoni Model
`
`3.10.7 Wideband PCS Microcell Model
`
`3.11 Indoor Propagation Models
`3.11.1 Partition Losses (same floor)»
`3.11.2 Partition Losses between Floors
`
`3.11.3 Log-distance Path Loss Model
`
`3.11.4»Ericsson Multiple Breakpoint Model
`
`3.11.5 Attenuation Factor Model
`
`b
`
`3.12 Signal Penetration into Buildings
`
`3.13 Ray Tracing and Site Specific Modeling
`3 .14 Problems
`
`.
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`54
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`57
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`61
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`63
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`Contents
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`vii
`
`4 Mobile Radio Propagation: Small-Scale Fading and Multipath 139
`139
`4.1 Small-Scale Multipath Propagation
`.
`4.1.1 Factors Influencing Small-Scale Fadin
`4.1.2 Doppler Shift
`4.2 Impulse Response Model of a Multipath Channel
`4.2.1 Relationship Between Bandwidth and Received Power
`4.3 Small-S cale Multipath Measurements
`p
`4.3.1 Direct RF Pulse System
`4.3.2 Spread Spectrum Sliding Correlator Channel Sounding
`4.3.3 Frequency Domain Channel Sounding
`4.4 Parameters of Mobile Multipath Channels
`4.4.1 Time Dispersion Parameters
`4.4.2 Coherence Bandwidth
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`140
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`141
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`143
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`147
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`153
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`154
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`155
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`158
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`159
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`160
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`163
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`4.4.3 Doppler Spread and Coherence Time
`4.5 Types of Small-Scale Fading
`4.5.1 Fading Effects Due to Multipath Time Delay Spread
`4.5.2 Fading Effects Due to Doppler Spread
`4.6 Rayleigh and Ricean Distributions
`4.6.1 Rayleigh Fading Distribution
`4.6.2 Ricean Fading Distribution
`4.7 Statistical Models for Multipath Fading Channels
`4.7.1 Clarl-:e’s Model for Flat Fading
`4.7.2 Simulation of Clarke and Gans Fading Model
`4.7.3 Level Crossing and Fading Statistics
`.
`4.7.4 Two-ray Rayleigh Fading Model
`4.7.5 Saleh and Valenzuela Indoor Statistical Model
`
`7
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`165
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`167
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`168
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`170
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`172
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`172
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`174
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`176
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`177
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`181
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`185
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`188
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`188
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`4.7.6 SIRCIIVI and SMRCIM Indoor and Outdoor Statistical Models
`
`189 I
`
`4.8 Problems
`
`. 5 Modulation Techniques for Mobile Radio
`5.1 Frequency Modulation vs. Amplitude Modulation
`5.2 Amplitude Modulation
`5.2.1 Single Sideband AM
`5.2.2 Pilot Tone SSB
`
`5.2.3 Demodulation of AM signals
`5.3 Angle Modulation
`5.3.1 Spectra and Bandwidth of FM Signals
`5.3.2 FM Modulation Methods
`
`5.3.3 FM Detection Techniques
`5.3.4 Tradeoff Between SNR and Bandwidth in an FM Signal.
`
`5.4 Digital Modulation — an Overview .
`5.4.1 Factors That Influence the Choice of Digital Modulation
`5.4.2 Bandwidth and Power Spectral Density of Digital Signals
`5.4.3 Line Coding
`5.5 Pulse Shaping Techniques
`
`192
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`197
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`198
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`199
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`I 202
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`203
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`206
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`206
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`

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`
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`viii
`
`Contents
`
`5.5.2 Raised Cosine Rolloff Filter
`
`Z
`
`5.5.3 Gaussian Pulse—shaping Filter
`5.6 Geometric Representation of Modulation Signals
`5.7 Linear Modulation Techniques
`5.7.1 Binary Phase Shift Keying (BPSK)
`5.7.2 Differential Phase Shift Keying (DPSK)
`5.7.3 Quadrature Phase Shift Keying (QPSK)
`5.7.4 QPSK Transmission and Detection Techniques
`5.7.5 Offset QPSK
`5.7.6 TC/4 QPSK
`5.7.7 1t/4 QPSK Transmission Techniques
`5.7.8 1t/4 QPSK Detection Techniques
`5.8 Constant Envelope Modulation
`5.8.1 Binary Frequency Shift Keying
`5.8.2 Minimum Shift Keying (MSK)
`5.8.3 Gaussian Minimum Shift Keying (GMSK)
`5.9 Combined Linear and Constant Envelope Modulation Techniques
`5.9.1 M-ary Phase Shift Keying (MPSK)
`5.9.2 M-ary Quadrature Amplitude Modulation (QAM)
`5.9.3 M-ary Frequency Shift Keying (MFSK)
`5.10 Spread Spectrum Modulation Techniques
`5.10.1 Pseudo-noise (PN) Sequences
`5.10.2 Direct Sequence Spread Spectrum (DS-SS)
`5.10.3 Frequency Hopped Spread Spectrum (FH-SS)
`5.10.4 Performance of Direct Sequence Spread Spectrum
`5.10.5 Performance of Frequency Hopping Spread Spectrum
`5.11 Modulation Performance in Fading and Multipath Channels
`5.11.1 Performance of Digital Modulation in Slow, Flat Fading Channels
`5.11.2 Digital Modulation in Frequency Selective Mobile Channels
`5.11.3 Performance of TE/4 DQPSK in Fading and Interference
`5.12 Problems
`
`6 Equalization, Diversity, and Channel Coding
`6.1Introduction
`
`A 6.2 Fundamentals of Equalization
`6.3 A Generic Adaptive Equalizer
`6.4 Equalizers in a Communications Receiver
`6.5 Survey of Equalization Techniques
`6.6 Linear Equalizers
`-
`6.7 Nonlinear Equalization
`6.7.1 Decision Feedback Equalization (DFE)
`6.7.2 Maximum Likelihood Sequence Estimation (MLSE) Equalizer
`6.8 Algorithms for Adaptive Equalization
`‘
`6.8.1 Zero Forcing Algorithm
`6.8.2 Least Mean Square Algorithm
`6.8.3 Recursive Least Squares Algorithm
`6.8.4 Summary of Algorithms
`
`229
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`233
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`243
`246
`247
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`249
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`Contents
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`ix
`
`‘ 6.9 Fractionally Spaced Equalizers
`6.10 Diversity Techniques
`6.10.1 Derivation of Selection Diversity Improvement
`6.10.2 Derivation of Maximal Ratio Combining Improvement
`6.10.3 Practical Space Diversity Considerations
`6.10.4 Polarization Diversity
`V
`6.10.5 Frequency Diversity
`6.10.6 Time Diversity
`6.11 RAKE Receiver
`
`6.12 Interleaving
`6.13 Fundamentals of Channel Coding
`6.14 Block Codes
`
`6.14.1 Examples of Block Codes
`6.14.2 Case Study of Reed-Solomon Codes
`6.15 Convolutional Codes
`
`6.15.1 Decoding of Convolutional Codes
`6.16 Coding Gain
`6.17 Trellis Coded Modulation
`
`6.18 Problems
`
`I
`
`7 Speech Coding
`7.1 Introduction
`
`7.2 Characteristics of Speech Signals
`7.3 Quantization Techniques
`7.3.1 Uniform Quantization
`7.3.2 Nonuniform Quantization
`7.3.3 Adaptive Quantization
`7.3.4 Vector Quantization
`7.4 Adaptive Differential Pulse Code Modulation
`7 .5 Frequency Domain Coding of Speech
`7.5.1 Sub-band Coding
`7.5.2 Adaptive Transform Coding
`7.6 Vocoders
`
`7.6.1 Channel Vocoders
`
`7.6.2 Formant Vocoders
`
`7.6.3 Cepstrum Vocoders
`7.6.4 Voice-Excited Vocoder
`
`7.7 Linear Predictive Coders
`
`7.7.1 LPC Vocoders
`
`7.7.2 Multi-pulse Excited LPC
`7.7.3 Code—Excited LPC
`
`7.7.4 Residual Excited LPC
`
`’
`
`.
`
`‘
`
`.
`
`7.8 Choosing Speech Codecs for Mobile Communications
`7.9 The GSM Codec
`
`7.10 The USDC Codec
`
`7.11 Perfonnance Evaluation of Speech Coders
`
`323
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`

`
`
`
`Contents
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`463
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`463
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`465
`
`Multiple Access Techniques for Wireless Communications
`8.1 Introduction
`
`X 8
`
`8.1.1 Introduction to Multiple Access
`8.2 Frequency Division Multiple Access (FDMA)
`98.3 Time Division Multiple Access (TDMA)
`8.4 Spread Spectrum Multiple Access
`8.4.1 Frequency Hopped Multiple Access (FHMA)
`8.4.2 Code Division Multiple Access (CDMA)
`8.4.3 Hybrid Spread Spectrum Techniques
`8.5 Space Division Multiple Access (SDMA)
`8.6 Packet Radio
`
`_ 8.6.1 Packet Radio Protocols
`8.6.2 Carrier Sense Multiple Access (CSMA) Protocols
`8.6.3 Reservation Protocols
`
`8.6.4 Capture Effect in Packet Radio
`8.7 Capacity of Cellular Systems
`8.7.1 Capacity of Cellular CDMA
`8.7.2 Capacity of CDMA with Multiple Cells
`8.7.3 Capacity of Space Division Multiple Access
`8.8 Problems
`
`9 Wireless Networking
`9.1 Introduction to Wireless Networks
`
`9.2 Differences Between Wireless and Fixed Telephone Networks
`9.2.1 The Public Switched Telephone Network (PSTN)
`9.2.2 Limitations in Wireless Networking
`9.2.3 Merging Wireless Networks and the PSTN
`9.3 Development of Wireless Networks
`9.3.1 First Generation Wireless Networks
`
`9.3.2 Second Generation Wireless Networks
`
`9.3.3 Third Generation Wireless Networks
`
`9.4 Fixed Network Transmission Hierarchy
`9.5 Traffic Routing in Wireless Networks
`
`9.5.1 Circuit Switching
`9.5.2 Packet Switching
`9.5.3 The X.25 Protocol
`
`9.6 Wireless Data Services 9
`
`9.6.1 Cellular Digital Packet Data (CDPD)
`
`9.6.2 Advanced Radio Data Information Systems (ARDIS)
`9.6.3 RAM Mobile Data (RMD)
`
`9.7 Common Channel Signaling (CCS)
`A 9.7.1 The Distributed Central Switching Office for CCS
`9.8 Integrated Services Digital Network (ISDN)
`9.8.1 Broadband ISDN and ATM
`
`9.9 Signaling System No.7 (SS7)
`9.9.1 Network Services Part (NSP) of SS7
`
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`Contents
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`xi
`
`9.9.2 The SS7 User Part
`
`9.9.3 Signaling Traffic in SS7
`9.9.4 SS7 Services
`
`9.9.5 Performance of SS7
`
`9.10 An example of SS7 —— Global Cellular Network Interoperability
`
`9.11 Personal Communication Services/Networks (PCS/PCN)
`
`9.11.1 Packet vs. Circuit switching for PCN
`9.11.2 Cellular Packet-Switched Architecture
`9.12 Protocols for Network Access
`
`9
`
`9.12.1 Packet Reservation Multiple Access (PRMA)
`9.13 Network Databases
`’
`
`9.13.1 Distributed Database for Mobility Management
`9.14 Universal Mobile Telecommunication System (UMTS)
`9.15 Summary
`‘
`10 Wireless Systems and Standards
`10.1 AMPS and ETACS
`S
`10.1.1 AMPS and ETACS System Overview
`10.1.2 Call Handling in AMPS and ETACS
`10.1.3 AMPS and ETACS Air Interface
`
`10.1.4 N-AMPS
`
`10.2 United States Digital Cellular (IS-54)
`1
`10.2.1 USDC Radio Interface
`
`10.2.2 United States Digital Cellular Derivatives (IS-94 and IS-136)
`10.3 Global System for Mobile (GSM)
`10.3.1 GSM Services and Features
`10.3.2 GSM System Architecture
`10.3.3 GSM Radio Subsystem
`10.3.4 GSM Channel Types
`10.3.5 Example of a GSM Call
`10.3.6 Frame Structure for GSM
`
`1
`
`10.3.7 Signal Processing in GSM
`10.4 CDMA Digital Cellular Standard (IS-95)
`10.4.1 Frequency and Channel Specifications
`10.4.2 Forward CDMA Channel
`
`10.4.3 Reverse CDMA Channel
`
`10.4.4 IS-95 with 14.4 kbps Speech Coder [ANS95]
`10.5 CT2 Standard For Cordless Telephones
`10.5.1 CT2 Services and Features
`
`10.5.2 The CT2 Standard
`
`10.6 Digital European Cordless Telephone (DECT)
`10.6.1 Features and Characteristics
`
`10.6.2 DECT Architecture
`
`10.6.3 DECT Functional Concent
`
`466
`
`467
`468
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`469
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`469
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`472
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`472
`473
`477
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`478
`479
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`479
`480
`481
`483
`483
`484
`485
`487
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`491
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`491
`493,; ..
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`500
`500
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`520
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`
`
`
`xii
`
`10.7 PACS — Personal Access Communication Systems
`10.7.1 PACS System Architecture
`10.7.2 PACS Radio Interface
`
`10.8 Pacific Digital Cellular (PDC)
`10.9 Personal Handyphone System (PHS)
`10.10 U.S. PCS and ISM Bands
`
`10.11 U.S. Wireless Cable Television
`
`10.12 Summary of Standards Throughout the World
`10.13 Problems
`
`Contents
`
`539
`540
`541
`
`543
`544
`544
`
`547
`
`548
`551
`
`APPENDICES
`
`.
`
`555
`A Trunking Theory
`556
`A.1 Erlang B
`556
`A.1.1 Derivation of Erlang B
`561
`A2 Erlang C
`561
`A.2.1 Derivation of Erlang C
`565
`B Noise Figure Calculations for Link Budgets
`569
`C Gaussian Approximations for Spread Spectrum CDMA
`577
`C.1 The Gaussian Approximation .
`582
`C2 The Improved Gaussian Approximation (IGA)
`C.3 A Simplified Expression for the Improved Gaussian Approximation (SEIGA) 585
`D Q, erf & erfc Functions 7
`593
`D.1 The Q-Function
`593
`D.2 The erf and erfc functions
`595
`
`E Mathematical Tables
`
`F Abbreviations and Acronyms
`G References
`
`Index
`
`599
`
`607
`617
`
`635
`
`Page 12 of 172
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`Page 12 of 172
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`

`
`>!<
`
`* * *
`
`>!<
`
`>!<
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`Page 13 of 172
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`Page 13 of 172
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`

`
`
`
`CHAPTER
`
`4
`
`Mobile Radio Propagation:
`Small-Scale Fading and
`Multipath
`
`Small’-scale fading, or simply fading, is used
`
`to describe the rapid fluctuation of the amplitude of a radio signal over a short
`period of time or travel distance, so that large-scale path loss effects may be
`ignored. Fading is caused by interference between two or more versions of the
`transmitted signal which arrive at the receiver at slightly different times. These
`Waves, called multipath waves, combine at the receiver antenna to give a result-
`ant signal which can vary widely in amplitude and phase, depending on the dis-
`tribution of the intensity and relative propagation time of the waves and the
`bandwidth of the transmitted signal.
`
`4.1 Small-Scale Multipath Propagation
`
`Multipath in the radio channel creates small-scale fading effects. The three
`
`most important effects are:
`
`° Rapid changes in signal strength over a small travel distance or time inter-
`val
`
`0 Random frequency modulation due to varying Doppler shifts on different
`multipath signals
`Time dispersion (echoes) caused by multipath propagation delays.
`
`°
`
`In built-up urban areas, fading occurs because the height of the mobile
`antennas are well below the height of surrounding structures, so there is no sin-
`gle line-of-sight path to the base station. Even when a line-of-sight exists, multi-
`path still occurs due to reflections from the ground and surrounding structures.
`The incoming radio waves arrive from different directions with different propa-
`
`gation delays. The signal received by the mobile at any point in space may con-
`sist of a large number of plane waves having randomly distributed amplitudes,
`
`139
`
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`

`
`140
`
`Ch. 4 - Mobile Radio Propagation: Sma|l—Scale Fading and Multipath
`
`phases, and angles of arrival. These multipath components, combine vectorially
`at the receiver antenna, and can cause the signal received by the mobile to dis-
`tort or fade. Even when a mobile receiver is stationary, the received signal may
`
`fade due to movement of surrounding objects in the radio channel.
`
`If objects in the radio channel are static, and motion is considered to be only
`
`due to that of the mobile, then fading is purely a spatial phenomenon. The spa-
`
`tial variations of the resulting signal are seen as temporal variations by the
`
`receiver as it moves through the multipath field. Due to the constructive and
`destructive effects of multipath waves summing at various points in space, a
`receiver moving at high speed can pass through several fades in a small period of
`
`time. In a more serious case, a receiver may stop at a particular location at which
`the received signal is in a deep fade. Maintaining good communications can then
`become very difficult, although passing vehicles or people walking in the vicinity
`of the mobile can often disturb the field pattern, thereby diminishing the likeli-
`hood of the received signal remaining in a deep null for a long period of time.
`Antenna space diversity can prevent deep fading nulls, as shown in Chapter 6.
`Figure 3.1 shows typical rapid variations in the received signal level due to
`
`small—scale fading as a receiver is moved over a distance of a few meters.
`
`Due to the relative motion between the mobile and the base station, each
`
`multipath wave experiences an apparent shift in frequency. The shift in received
`signal frequency due to motion is called the Doppler shift, and is directly propor-
`tional to the velocity and direction of motion of the mobile with respect to the
`direction of arrival of the received multipath wave.
`
`4.1.1 Factors Influencing Small-Scale Fading
`
`Many physical factors in the radio propagation channel influence small-
`scale fading. These include the following:
`
`0 Multipath propagation —— The presence of reflecting objects and scatterers
`’
`in the channel creates a constantly changing environment that dissipates
`the signal energy in amplitude, phase, and time. These effects result in mul-
`tiple versions of the transmitted signal that arrive at the receiving antenna,
`displaced with respect to one another in time and spatial orientation. The
`random phase and amplitudes of the different multipath components cause
`fluctuations in signal strength, thereby inducing small—scale fading, signal
`distortion, or both. Multipath propagation often lengthens the time required
`for the baseband portion of the signal to reach the receiver which can cause
`signal smearing due to intersymbol interference.
`
`0 Speed of the mobile —— The relative motion between the base station and
`
`the mobile results in random frequency modulation due to different Doppler
`shifts on each of the multipath components. Doppler shift will be positive or
`negative depending on whether the mobile receiver is moving toward or
`away from the base station.
`
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`

`
`Small—Sca|e Multipath Propagation
`
`141
`
`0 Speed of surrounding objects —— If objects in the radio channel are in
`
`motion, they induce a time varying Doppler shift on multipath components.
`If the surrounding objects move at a greater rate than the mobile, then this
`effect dominates the small-scale fading. Otherwise, motion of surrounding
`
`objects may be ignored, and only the speed of the mobile need be considered.
`
`0 The transmission bandwidth of the signal — If the transmitted radio
`
`signal bandwidth is greater than the “bandwidth” of the multipath channel,
`
`the received signal will be distorted, but the received signal strength will not
`
`fade much over a local area (i.e., the small-scale signal fading will not be sig-
`nificant). As will be shown, the bandwidth of the channel can be quantified
`by the coherence bandwidth which is related to the specific multipath struc-
`ture of the channel. The coherence bandwidth is a measure of themaximum
`
`frequency difference for which signals are still strongly correlated in ampli-
`tude. If the transmitted signal has a narrow bandwidth as compared to the
`channel, the amplitude of the signal will change rapidly, but the signal will
`not be distorted in time. Thus, the statistics of small-scale signal strength
`and the likelihood of signal smearing appearing over small-scale distances
`are very much related to the specific amplitudes and delays of the multipath
`channel, as well as the bandwidth of the transmitted signal.
`
`4.1.2 Doppler Shift
`
`Consider a mobile moving at a constant velocity v, along a path segment
`having length d between points X and Y, while it receives signals from a remote
`source S, as illustrated in Figure 4.1. The difference in path lengths traveled by
`the wave from source S to the mobile at points X and Y is A = d cost) = vAt cost),
`where At
`is the time required for the mobile to travel from X to Y, and 9 is
`assumed to be the same at points X and Y since the source is assumed to be very
`far away. The phase change in the received signal due to the difference in path
`lengths is therefore
`
`Ad)=
`
`27rAl : 27cvAt
`X
`X
`
`cos6
`
`(4.1)
`
`and hence the apparent change in frequency, or Doppler shift, is given by fd ,
`where
`
`. cose
`
`(4.2)
`
`Equation (4.2) relates the Doppler shift to the mobile velocity and the spa-
`tial angle between the direction of motion of the mobile and the direction of
`arrival of the Wave. It can be seen from equation (4.2) that if the mobile is mov-
`ing toward the direction of arrival of the wave, the Doppler shift is positive (i.e.,
`the apparent received frequency is increased), and if the mobile is moving away
`from the direction of arrival of the wave, the Doppler shift is negative (i.e. the
`
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`

`
`142
`
`Ch. 4 - Mobile Radio Propagation: Small-Scale Fading and Multipath
`
`apparent received frequency is decreased). As shown in section 4.7.1, multipath
`components from a CW signal, which arrive from different directions contribute
`
`to Doppler spreading of the received signal, thus increasing the signal band-
`width.
`
`
`
`Figure 4.1
`Illustration of Doppler effect.
`
`
`
`Example 4.1
`
`Consider a transmitter which radiates a sinusoidal carrier frequency of 1850
`MHz. For a vehicle moving 60 mph, compute the received carrier frequency if
`the mobile is moving (a) directly towards the transmitter, (b) directly away
`from the transmitter, (c) in a direction which is perpendicular to the direction
`of arrival of the transmitted signal.
`
`Solution to Example 4.1
`Given:
`
`Carrier frequency fc = 1850 MHz
`
`Therefore, wavelength
`
`= c/fc =
`
`3 x 108
`
`1850 x 106
`
`= 0.162m
`
`Vehicle speed v = 60 mph = 26.82 rn/s
`
`(a) The vehicle is moving directly towards the transmitter.
`The Doppler shift in this case is positive and the received frequency is given
`by equation (4.2)
`‘
`f= fc+fd =1850><10 +(2%%
`.(b) The vehicle is moving directly away from the transmitter.
`The Doppler shift in this case is negative and hence the received frequency
`is given by
`
`6
`
`.
`
`= 1850.000l 6 MHZ
`
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`
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`

`
`Impulse Response Model of a Multipath Channel
`
`143
`
`f: fc—fd = l850><106—2—6§g = 1849.999834 MHZ
`0.162
`
`(c) The vehicle is moving perpendicular to the angle of arrival of the transmit-
`ted signal.
`In this case, 9 = 90°, cost) = 0, and there is no Doppler shift.
`The received signal frequency is the same as the transmitted frequency of
`1850 MHz.
`
`4.2
`
`Impulse Response Model of a Multipath Channel .
`
`The small-scale variations of a mobile radio signal can be directly related to
`the impulse response of the mobile radio channel. The impulse response is a
`wideband channel characterization and contains all information necessary to
`simulate or analyze any type of radio transmission through the channel. This
`stems from the fact that a mobile radio channel may be modeled as a linear filter
`with a time varying impulse response, where the time variation is due to
`
`receiver motion in space. The filtering nature of the channel is caused by the
`summation of amplitudes and delays of the multiple arriving waves at any
`instant of time. The impulse response is a useful characterization of the channel,
`since it may be fused to predict and compare the performance of many different
`mobile communication systems and transmission bandwidths for a particular
`mobile channel condition.
`
`To show that a mobile radio channel may be modeled as a linear filter with
`a time varying impulse response, consider the case where time variation is due
`strictly to receiver motion in space. This is shown in Figure 4.2.
`
`
`
`'
`
`d
`
`9
`
`Figure 4.2
`The mobile radio channel as a function of time and space.
`
`spatial position
`‘
`
`In Figure 4.2, the receiver moves along the ground at some constant veloc-
`ity v. For a fixed position d, the channel between the transmitter and the receiver
`can be modeled as a linear time invariant system. However, due to the different
`multipath waves which have propagation delays which vary over different spa-
`tial locations of the receiver, the impulse response of the linear time invariant
`channel should be a function of the position of the receiver. That is, the channel
`impulse response can be expressed as h(d,t). Let x(t) represent the transmitted
`signal, then the received signal y(d,t) at position d can be expressed as a convo-
`lution of x (t) with h(d,t).
`
`Page 18 of 1,72
`
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`

`
`144
`
`Ch. 4 - Mobile Radio Propagation: Small-Scale Fading and Multipath
`
`y(d,t) = x(t) ®h(d,t) = I00 x(t)h(d,t—t)dt
`
`(4.3)
`
`For a causal system, h (d, t) = O for t<0, thus equation (4.3) reduces to
`t
`
`y(d, 1:) = J‘x(r)h(d,t—t)dt
`
`(4.4)
`
`Since the receiver moves along the ground at a constant velocity v, the posi-
`tion of the receiver can by expressed as
`
`Substituting (4.5) in (4.4), We obtain
`t
`
`d = vt
`
`y(vt,t) = J‘.7C(’C)h(vt,t-—'C)d‘C
`
`(4.5)
`
`(4.6)
`
`Since v is a constant, y (vt, t)
`can be expressed as
`
`t
`
`is just a function of 1.‘. Therefore, equation (4.6)
`
`y(t) =- Jx(x)h(ut.t—c)d: = x(t) ®h(’vt,t) = x(t) ®h(d, t)
`
`(4.7)
`
`From equation (4.7) it is clear that the mobile radio channel can be modeled as a
`
`linear time varying channel, where the channel changes with time and distance.
`Since u may be assumed constant over a short time (or distance) interval,
`we may let x (t)
`represent the transmitted bandpass waveform, y (t)
`the
`received waveform, and h (t», 1:)
`the impulse response of the time varying multi-
`path radio channel. The impulse response h (t, '5) completely characterizes the
`channel and is a function of both t and 1:. The variable t represents the time
`variations due to motion, whereas ‘I.’ represents the channel multipath delay for
`a fixed value of t . One may think of 1: as being a Vernier adjustment of time. The
`received signal y (t) can be expressed as a convolution of the transmitted signal
`x (t) with the channel impulse response (see Figure 4.3a).
`
`y(t) = [00 x(t)h(t,1:)dt = x(t) ®h(t,1:)
`
`—-(X)
`
`i
`
`(4.8)
`
`If the multipath channel is assumed to be a bandlimited bandpass channel,
`which is reasonable, then h(t,t) may be equivalently described by a complex
`baseband impulse response h b (t, T) with the input and output being the com-
`plex envelope representations of the transmitted and received signals, respec-
`tively (see Figure 4.3b). That is,
`
`gm) = %c(t) ®%hb(t,t)
`
`(4.9)
`
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`

`
`Impulse Response Model of a Multipath Channel
`
`145
`
`W)
`
`h(t,1;) = Re{h,,(:,«:>e’“’”‘} }}-..-.-_.y(t)
`
`1 7§">"l
`
`!
`
`t
`ya) =Re{r<t>ej‘””}
`
`ya) = xc) ®h<t>
`
`c(t
`
`éhg, (17, T)
`
`Tft)
`gm) = 50 (t)' 69 éhb (t)
`
`Figure 4.3
`(a) Bandpass channel impulse response model.
`(b) Baseband equivalent channel impulse response model.
`
`where c (t) and r (t) are related to x (t) and y(t), respectively, through [Cou93]
`
`x (t) = Re {c (t) exp (j2TCfct)}
`
`y (t) = Re {r (t) exp (j27Tfct)}
`
`(4.10)
`
`(4.11)
`
`The factors of 1/2 in equation (4.9) are due to the properties of the complex
`
`envelope, in order to represent the passband radio system at baseband. The low-
`pass characterization removes the high frequency Variations caused by the car-
`rier, making the signal analytically easier to handle. It is shown by Couch
`[Cou93] that the average power of a bandpass signal x2 (t)
`is equal to %!c (t) I2 ,
`where the overbar denotes ensemble average for a stochastic signal, or time
`average for a deterministic or ergodic stochastic signal.
`It is useful to discretize the multipath delay axis 1: of the impulse response
`into equal time delay segments called excess delay bins, where each bin has time
`a delay width equal to ‘Ci + 1 ~ri , where ‘[0 is equal to 0, and represents the first
`arriving signal at the receiver. Letting i = 0, it is seen that ‘Cl — 1:0 is equal to the
`time delay bin width given by At. For convention,
`To = 0 ,
`‘E1 = A1, and
`‘Ci = iA1: , for i = O to N —1 , where N represents the total number of possible
`equally-spaced multipath components, including the first arriving component.
`Any number of multipath signals received within the i th bin are represented by
`a single resolvable multipath component having delay Ti. This technique of
`quantizing the delay bins determines the time delay resolution of the channel
`model, and the useful frequency span of the model can be shown to be 1/ (2Ar) .
`That is, the model may be used to analyze transmitted signals having band-
`widths which are less than 1/ (ZAI) . Note that To = 0 is the excess time delay
`
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`
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`

`
`146
`
`Ch. 4 - Mobile Radio Propagation: Small-Scale Fading and Multipath
`
`of the first arriving multipath component, and neglects the propagation delay
`between the transmitter and receiver. Excess delay is the relative delay of the
`
`ith multipath component as compared to the first arriving component and is
`given by ‘Ci
`. The maximum excess delay of the channel is given by NAT .
`Since the received signal in a multipath channel consists of a series of
`
`attenuated, time-delayed, phase shifted replicas of the transmitted signal, the
`
`baseband impulse response of a multipath channel can be expressed as
`
`N—1
`
`‘
`
`hb (t, ‘c) = Za,.(t, 17) exp [j21tfcr,. (t) +¢(t, 'c)]5(1:—t,- (t))
`i’=0
`
`(4.12)
`
`where a,- (t, t) and Ti (t) are the real amplitudes and excess delays, respectively,
`of ith multipath component at time t. The phase term 21tfcr,.(t) + (1),. (t, T) in
`(4.12) represents the phase shift due to free space propagation of the ith multi-
`
`path component, plus any additional phase shifts which are encountered in the
`
`channel._ In general, the phase term is simply represented by a single variable
`9,. (t, T) which lumps together all the mechanisms for phase shifts of a single
`multipath component within the ith excess delay bin. Note that some excess
`
`delay bins may have no multipath at some time t and delay Ti , since ai (t, 1:)
`may be zero. In equation (4.12), N is the total possible number of multipath com-
`
`ponents (bins), -and 8 (-)
`is the unit impulse function which determines the spe-
`cific multipath bins that have components at time t and excess delays Ti .‘ Figure
`4.4 illustrates an example of different snapshots of h b (t, t) , where t varies into
`the page, and the time delay bins are quantized to Widths of AT .
`
`
`
`T0
`
`T1
`
`T2
`
`T3
`
`T4
`
`’”—+—*“—’ mo)
`‘
`TN-2
`TN-1
`
`Figure 4.4
`An example of the time varying discrete-time impulse response model for a multipath radio channel.
`
`Page 21 of 172
`
`x—i——4————> my
`
`I——‘%—:> my
`
`F“t“‘j—> rm)
`
`Page 21 of 172
`
`

`
`Impulse Response Model of a Multipath Channel
`
`147
`
`If the channel impulse response is assumed to be time invariant, or is at
`
`least wide sense stationary over a small-scale time or distance interval, then