Resource Allocation for Multicarrier Communications

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Resource Allocation for Multicarrier Communications

Didem Kivanc Tureli

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

University of Washington

2005

Program Authorized to Offer Degree: Electrical Engineering

University of Washington Graduate School

This is to certify that I have examined this copy of a doctoral dissertation by Didem Kivanc Tureli and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.

Chair of the Supervisory Committee:

Hui Liu

Reading Committee:

Hui Liu James Ritcey James K. Peckol

Date:

In presenting this dissertation in partial fulfillment of the requirements for the doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U.S. Copyright Law. Requests for copying or reproduction of this dissertation may be referred to Proquest Information and Learning, 300 North Zeeb Road, Ann Arbor, MI 48106-1346, 1-800-521-0600, to whom the author has granted “the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform.”

Signature

Date

University of Washington Abstract

Resource Allocation for Multicarrier Communications Didem Kivanc Tureli Chair of the Supervisory Committee: Professor Hui Liu Department of Electrical Engineering

Multicarrier transmission schemes have been widely adopted in wireless communication systems. Multiple users can share a multicarrier channel using multiple access mechanisms such as OFDMA or MC-CDMA. This dissertation studies some unique features of multicarrier transmission and how these effect the performance of a multiple access system. Analytical expressions are derived for the effect of frequency offset error on MC-CDMA. Algorithms are derived for resource allocation in static and dynamic OFDMA systems. Signal degradation due to frequency offset is one of the main problems in multicarrier communication. This dissertation presents analytical expressions for the expected drop in signal to interference and noise ratio (SINR) experienced in an uplink MC-CDMA system. In an orthogonal frequency division multiple access (OFDMA) system users are assigned different carriers instead of sharing them. Optimal algorithms for frequency allocation are found to be computationally demanding. In this dissertation a new class of algorithms is proposed which achieves the users’ QoS objectives while keeping transmission power low compared to existing algorithms and at lower computational cost. In a dynamic system, users can enter and exit the cell at any time. This dissertation presents an analysis of the issue of “fairness” in an OFDMA system, and introduces two online nonpreemptive scheduling algorithms.

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1: Introduction . . . . . 1.1 Notation . . . . . . . . . . . 1.2 Dissertation Overview . . . 1.3 Summary of Contributions .

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Chapter 2: The Effect of Frequency Offset on 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 System Model . . . . . . . . . . . . . . . 2.3 The Effect of Frequency Offset . . . . . 2.4 Simulation Results and Discussion . . . 2.5 Conclusions . . . . . . . . . . . . . . . . Chapter 3: 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

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1 3 6 7

MC-CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 10 15 22 35 41

Frequency Division Multiple Ac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49 49 51 59 67 75 79 88 102

Resource Allocation for Orthogonal cess (OFDMA) . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . System Model and Problem Formulation . Some Previous Approaches . . . . . . . . The Sensible Greedy Approach . . . . . . Algorithmic Complexity . . . . . . . . . . Simulation Setup . . . . . . . . . . . . . . Simulation Results . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . .

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Online Scheduling for Othogonal Frequency Division Multiple Access (OFDMA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 i

4.2 4.3 4.4 4.5 4.6

System Model and Dynamic Scheduling Problem Formulation The Online Greedy Scheduling Algorithms . . . . . . . . . . . Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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108 115 118 121 124

Chapter 5: Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Appendix A: Summary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.1 General Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.2 Symbols Relating to MC-CDMA and OFDMA . . . . . . . . . . . . . . . . . 145 Appendix B: Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 149 B.1 Average SINR for known channels . . . . . . . . . . . . . . . . . . . . . . . . 150 B.2 Asymptotic SINR for random channels . . . . . . . . . . . . . . . . . . . . . . 163 Appendix C: Appendix for Chapter 3 . . . . . . . C.1 The Modified Lagrange Algorithm . . . . . C.2 The Bandwidth Assignment Based on SINR C.3 The Optimal Rate Craving Algorithm . . .

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189 189 198 203

Appendix D: Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 217 D.1 OFDMA and Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

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LIST OF FIGURES

Figure Number

Page

1.1

Cellular communication system. . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1 2.2 2.3

Combined baseband model of transmitter and channel for user k. . . . . . . Baseband receiver model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multicarrier frequency spectra. (a) 8 subcarriers, N = 8, M = 8, α = 1, (b) 4 subcarriers, N = 4, M = 4, α = 1, (c) 8 subcarriers, N = 8, M = 4, α = 2. Plot of Ψn−m for σF T = 0.2, and N = 128. . . . . . . . . . . . . . . . . . . . SINR from Theorem 2.2 and simulation results. LMMSE receiver, M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER from Theorem 2.2 and simulation results. LMMSE receiver, M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SINR from Theorem 2.2 and simulation results. LMMSE receiver, M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER from Theorem 2.2 and simulation results. LMMSE receiver, M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SINR from Theorem 2.2 and simulation results. MF receiver, M = 128, α = 1, K = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER from Theorem 2.2 and simulation results. MF receiver, M = 128, α = 1, K = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SINR from Theorem 2.2 and simulation results. MF receiver, M = 64, α = 2, K = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BER from Theorem 2.2 and simulation results. MF receiver, M = 64, α = 2, K = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation in SINR at the output of the LMMSE receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation in SINR at the output of the LMMSE receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . Degradation in SINR at the output of the LMMSE receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . .

16 16

2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13

2.14

2.15

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17 27 37 37 38 38 39 39 40 40

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2.16 Degradation in SINR at the output of the LMMSE receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.17 Degradation in SINR at the output of the LMMSE receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . 44 2.18 Degradation in SINR at the output of the LMMSE receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . 44 2.19 Degradation in SINR at the output of the MF receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

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2.20 Degradation in SINR at the output of the MF receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.21 Degradation in SINR at the output of the MF receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.22 Degradation in SINR at the output of the MF receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . 46 2.23 Degradation in SINR at the output of the MF receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.24 Degradation in SINR at the output of the MF receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1

System model for OFDMA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2

Continuous approximation to the rate function. . . . . . . . . . . . . . . . . . 56

3.3

Example 1: Carrier allocation by the RCG algorithm. . . . . . . . . . . . . . 74

3.4

Example 2: Carrier allocation by the ACG algorithm. . . . . . . . . . . . . . 76

3.5

Channel simulator.

3.6

A single circular cell, users are distributed in a 2D Gaussian distribution centered around (0, 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.7

The Saleh-Valenzuela multipath channel model. . . . . . . . . . . . . . . . . . 84

3.8

Sample channel profiles.

3.9

Adaptive modulation and power loading, outage probability vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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3.10 Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 iv

3.11 Adaptive modulation without power loading for Channel 1, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . 90 3.12 Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.13 Adaptive modulation without power loading for Channel 2, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . 91 3.14 Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.15 Adaptive modulation without power loading for Channel 1, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . 92 3.16 Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.17 Adaptive modulation without power loading for Channel 2, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . 93 3.18 Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.19 Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.20 Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.21 Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.22 Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.23 Adaptive modulation and power loading for Channel 1, N = 20 carriers, transmission power per bit vs. number of users. . . . . . . . . . . . . . . . . 98 3.24 Percentage increase in transmission power due to not using power loading on Channel 1 vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.25 Percentage increase in transmission power due to not using power loading on Channel 1 vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . 100 3.26 Percentage increase in transmission power due to not using power loading on Channel 2 vs. number of users. . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.27 Percentage increase in transmission power due to not using power loading on Channel 2 vs. number of carriers. . . . . . . . . . . . . . . . . . . . . . . . . 101 3.28 Adaptive modulation and power loading, average CPU time required vs. number of users (128 subcarriers). . . . . . . . . . . . . . . . . . . . . . . . . 102 3.29 Adaptive modulation and power loading, average CPU time required vs. number of subcarriers (20 users). . . . . . . . . . . . . . . . . . . . . . . . . . 103 v

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Simple Scheduling Algorithm flowchart. . . . . . . . . . . . . . . . . . . . . . Effect of Mmax , time vs. outage probability. . . . . . . . . . . . . . . . . . . Effect of Mmax , time vs. transmission power per bit. . . . . . . . . . . . . . . M-BABS and rate proportional bandwidth allocation, time vs. outage probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability of outage for scheduling algorithms. . . . . . . . . . . . . . . . . . Transmission power per bit for scheduling algorithms. . . . . . . . . . . . . . The number of carrier allocations and reallocations in scheduling algorithms. CPU cycles per frame for the scheduling algorithms. . . . . . . . . . . . . . .

Path p = (k1 , n1 , k2 , n2 , ..., kP −1 , nP ). . . . Graph G(AM +1 ) and circuit c. . . . . . . . Graph G(AM ), path p, circuit c0 , path p0 . Graph G(AM ), path p, circuit c0 , and sets {d01 , d02 }, P 0 = {p1 , p2 }. . . . . . . . . . . . C.5 Partition of graph G(AM ), and path p. . . C.6 Graph G(AM ), path p, circuit c0 , path p0 . C.7 Graph G(AM ), path p, circuit c0 . . . . . . C.1 C.2 C.3 C.4

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. . . . . . . . . . . . . . . . . . . . . . . . . . . of paths: B 0 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . {b01 , b02 }, D0 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

118 121 122 123 124 125 126 126

. 204 . 207 . 208 . . . .

210 211 211 213

LIST OF TABLES

Table Number 3.1 3.2 3.3 3.4 3.5

Page

Transfer function coefficients for chosen RCPC codes. . . . . . . . . . . . . Modulation constellations and coding rates available for transmission, with 128 carriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of lookup table size (Ntable ) on probability of outage for the MLR algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Order of complexity of the algorithms. . . . . . . . . . . . . . . . . . . . . . System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 55 . 56 . 63 . 80 . 86

C.1 Variables used in LR and MLR Algorithms . . . . . . . . . . . . . . . . . . . 191

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LIST OF ALGORITHMS

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3.1

Branch and Bound Algorithm

3.2

BABS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3

RCG Algorithm

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3.4

ACG Algorithm

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4.1

Modified BABS (M-BABS) Algorithm.

4.2

Simple Scheduling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.3

Adaptive Scheduling algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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C.1 Lagrangian Relaxation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.2 Modified Lagrangian Relaxation Algorithm . . . . . . . . . . . . . . . . . . . 195 C.3 BABS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 C.4 RCO Graph Construction (G CONS) Algorithm . . . . . . . . . . . . . . . . 205 C.5 Rate Craving Optimal (RCO) Algorithm

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GLOSSARY Mathematical Abbreviations i.i.d.:

independent, identically distributed.

pdf:

probability density function.

cdf:

cumulative distribution function.

Acronyms Used Asymmetric Digital Subscriber Lines.

ADSL:

AWGN:

BER:

Bit Error Rate.

BRMP:

BS:

Additive White Gaussian Noise.

Bit Rate Maximization Problem.

Base Station.

BTRC:

“Better Than” Raised Cosine (Pulse Shaping Filter).

CCI:

Co-Channel Interference.

CCR:

Coherent Combining Receiver.

CDMA:

CFO:

Code Division Multiple Access.

Carrier Frequency Offset. ix

CP:

Cyclic Prefix, or guard interval for multicarrier transmission.

DFT:

Discrete Fourier Transform.

DQPSK:

DMT:

1/4-Shift Differentially Encoded PSK.

Discrete Multitone.

DS-CDMA:

Direct Sequence Code Division Multiple Access.

ECC:

Error Control Coding.

EGC:

Equal Gain Combining (Receiver).

FDM:

Frequency Division Multiplexing.

FDMA:

Frequency Division Multiplexing Multiple Access.

FFT:

Fast Fourier Transform.

FMT:

Filtered Multitone.

GI:

Guard Interval, or cyclic prefix for multicarrier transmission.

GSM:

Global System of Mobile Communication.

H.263:

ITU Standard on Video Coding for Low Bit Rate Communication.

HIPERLAN:

ICI:

IDFT:

HIgh PErformance Radio LAN: the ETSI standard for Local Area Networks.

Inter-Carrier Interference. Inverse Discrete Fourier Transform. x

IEEE 802.11:

IEEE Working Group for Wireless Local Area Network Standards.

IEEE 802.15:

IEEE Working Group for Wireless Personal Area Network Standards.

IEEE 802.16:

IEEE Working Group for Broadband Wireless Access Standards.

ISI:

Inter-Symbol Interference.

ITU:

International Telecommunications Union.

kbps:

Kilo Bits Per Second.

MAC:

Medium Access Control.

MCM:

Multi-Carrier Modulation.

MC-CDMA:

Multi-Carrier Code Division Multiple Access.

MC-DS-CDMA:

MC-FDMA:

MF:

Multi-Carrier Direct Sequence Code Division Multiple Access.

Multicarrier Frequency Division Multiple Access.

Matched Filter (Receiver).

MIMO:

Multiple Input, Multiple Output. Margin Maximization Problem.

MMP:

MMSE:

Minimum Mean Square Error (Receiver).

MPFB:

Multitone with Polyphase Filterbank.

MRC:

Maximum Ratio Combining (Receiver). xi

MU:

Mobile Unit. Multi-User Detection.

MUD:

Linear Minimum Mean Square Error (Receiver).

LMMSE:

OFDM:

Orthogonal Frequency Division Multiplexing.

OFDMA:

Orthogonal Frequency Division Multiplexing Multiple Access (also known as

OFDM-FDMA, Multiuser OFDM). PCC-OFDM:

Polynomial Cancellation Coding OFDM.

PCC-CDMA:

Polynomial Cancellation Coding Multicarrier CDMA.

PCS:

Personal Communication Systems. Quadrature Amplitude Modulation.

QAM:

QOS:

Quality of Service.

RC:

Raised Cosine (Filter). Rate Compatible Punctured Convolutional (Code).

RCPC:

SINR:

Signal to Interference and Noise Ratio.

SISO:

Single Input, Single Output.

SNR:

Signal to Noise Ratio.

TDMA:

VA:

Time Division Multiple Access.

Viterbi Algorithm. xii

VAD:

Voice Activity Detection.

WGN:

White Gaussian Noise.

WLAN:

Wireless Local Area Network.

WSSUS:

Wide Sense Stationary Uncorrelated Scattering (Wireless Channel).

xiii

ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Hui Liu for his invaluable guidance and support. He has been an excellent academic advisor and role model over the years. Without him this dissertation would not have been possible. My thanks are also extended to the members of my supervisory committee: Prof. James Ritcey, Prof. Sumit Roy, Prof. James K. Peckol, and Prof. Minqin Zhang for their valuable comments and efforts. I would like to thank my friends, especially Hujun Yin, Kemin Li, Guanbin Xing, Manyuan Shen, Jing Ye and Ying Li for their help, collaboration and camaraderie. Above all I would like to thank my husband Ufuk Tureli, and my parents Mujde and Ulku Kivanc for their generous help and unwavering support.

xiv

1

Chapter 1 INTRODUCTION

In recent years, wireless communications has grown to permeate all facets of life. From cellular access to wireless data networks, there is a demand for clear, fast transmission of multimedia information. Wireless communications systems deal with limited resources. Wireless channels are inherently noisy compared to fiber or wire channels, because the signal does not follow a direct path to the receiver, and has to go through and around trees, buildings, people, and other obstacles. The channel is frequency-selective and time-varying due to changes in the environment. There are three main resources that can be utilized to overcome the environment: time, frequency and transmission power. The transmission power available to mobile devices is often limited, since they rely on batteries and cause interference for other users. When error control coding is used to detect and correct the error introduced by the channel, the user may either keep the symbol transmission rate constant and use more time, or increase the symbol transmission rate, which according to the Shannon-Hartley theorem [25] increases the signal bandwidth. In a multiuser transmission system, both these resources – time and frequency – are scarce, and must be carefully allocated between users. A broadband communication channel is one in which the total bandwidth available for transmission is not negligibly small in relation to the carrier frequency. In such systems, the channel response is frequency selective. Traditional single carrier systems use error protection and equalization mechanisms to compensate for the effect of the channel. An alternative to single carrier communication is to divide the total available spectrum into many smaller channels. This process is referred to as multicarrier modulation (MCM). Just as in radio or television broadcasting the total bandwidth is divided into consecutive channels

2

each of which is allocated to a different broadcaster, in multicarrier communications the available bandwidth is divided into subchannels. Since the length of each symbol in MCM is larger, the multicarrier signal is inherently more resilient to the inter-symbol interference (ISI) introduced by the channel. This resilience comes at a price. The many carriers have much smaller bandwidth, and a small frequency synchronization error can have significant effect on signal quality. This is particularly a problem in uplink communication, where the receiver can only be synchronized to the frequency offset of a single user while many other users may be transmitting simultaneously and causing interference, each with a different carrier frequency offset error. Orthogonal frequency division multiplexing (OFDM) is a form of multicarrier modulation in which the carrier frequencies are spaced at intervals of 1/Ts , where Ts is the symbol duration. This is the most efficient use of bandwidth in a multicarrier system without introducing Inter-Carrier Interference (ICI). All three chapters of this dissertation consider multicarrier communication systems with orthogonal waveforms (i.e. OFDM), but with different multiple access strategies. In general, there are three ways to divide resources in a communication system: Time Division Multiple Access (TDMA), Frequency Division Multiple Access (FDMA) or Code Division Multiple Access (CDMA). In TDMA, the users are allowed to access any or all of the bandwidth available, at different times. In FDMA, the users can only use a fraction of the available frequency spectrum, but for extended periods of time. In CDMA, the users transmit simulateneously in frequency and time, but must use a special code while transmitting. The two medium access strategies studied in this dissertation are MC-CDMA and OFDMA. Multicarrier CDMA (MC-CDMA) is one way of using Code Division Multiple Access across a multicarrier channel [134]. OFDMA is Orthogonal Frequency Division Multiplexing with Frequency Division Multiple Access, where each user gets one or more carriers to themselves [29]. This dissertation studies the problem of resource allocation in a multiuser multicarrier communication system. The first part of the dissertation looks at the effect of a frequency offset error on the uplink of MC-CDMA, and quantifies the effect of this offset on the signal to interference and noise ratio (SINR). The second part of the dissertation studies

3

resource allocation algorithms for OFDMA systems. A new class of suboptimal algorithms are developed for resource allocation among multiple users. In the final section, these algorithms are modified to work in a time varying multiuser environment. Base Station

K Users N Subcarriers

Figure 1.1: Cellular communication system.

1.1

Notation

Throughout this dissertation, boldface lower case fonts are used for vectors, x = [x1 , x2 , ..., xN ]. Element i of vector x is denoted xk or x(k) and [xk ] denotes the vector with kth element xk . Boldface upper case fonts are used for matrices. Row n of matrix A is denoted an , and element (n, m) is an,m or an (m)  a1,1    a2,1 A=  ..  . 

a1,2

···

a1,M

a2,2 .. .

··· .. .

a2,M .. .

aN,1 aN,2 · · ·

aN,M

       =    

 aT1  ..  .   T aN

[an,m ] denotes the matrix with elements an,m in the nth row and mth column. diag [d0 , d1 , · · · , dN ] represents a diagonal matrix with diagonal elements d0 , ..., dN . sˆk represents the estimate of the value sk .

4

Some of the symbols used in this dissertation are summarized below. Appendix A contains a more complete list. I

Identity matrix.

1

Vector of 1’s, 1 = [1, 1, ..., 1]T .

AH

Hermitian of matrix A.

tr(A) ||A||22

Trace of matrix A, the sum of its diagonal elements.  The L2 -norm of matrix A, ||A||2 = tr AAH .

A B

Hadamard (element-by-element) product of matrices.

N

The set of natural numbers, N = {1, 2, ...}.

R

The set of real numbers.

E f {X} Expectation of random variable X with respect to f . dxe

The smallest integer greater than x (ceiling function).

IA

The indicator function, where A is a boolean expression:   1 if A is true, IA =  0 otherwise.

[x]+

The function:

  x + [x] =  0

if x > 0, otherwise,

The following are the symbols used that are related to OFDMA or MC-CDMA: K

Number of users.

N

Number of carriers.

M

Number of carriers used for transmission in MC-CDMA modulation.

α

Bandwidth scaling ratio defined (p. 17) as the ratio of intercarrier spacing to subcarrier bandwidth α = N/M .

β

Ratio of number of users to number of carriers or degree of freedom for multicarrier CDMA, β = K/N .

5

H WM

Modulation matrix for OFDM.

δFk

Frequency offset error for user k.

SINRj

Signal to interference and noise ratio for user j at the output of a multiuser detector.

SINR∞ detector,j

Asymptotic SINR for user j at the output of a multiuser detector of type “detector” in the limit as the number of users and the number of carriers go to infinity while the ratios α and β remain constant.

∆ SINRj

Degradation in SINR due to frequency offset: ∆ SINRj (dB) = SINRj (dB) − SINRj |δF =0 (dB)

∆ SINR∞ detector,j Asymptotic degradation in SINR due to frequency offset: ∞ ∞ ∆ SINR∞ detector,j (dB) = SINRdetector,j (dB) − SINRdetector,j δF =0 (dB)

Rmax

Maximum transmission rate per carrier, determined by coding and modulation schemes available.

Pe

Maximum tolerable probability of error.

k Rmin

Transmission rate requested by user k.

k Pmax

Maximum attainable transmission power for user k.

hk (n)

Channel gain for user k on carrier n.

rk (n)

Transmission rate for user k on carrier n.

f (·)

Rate function, relating the transmission power to the transmission rate, minimum acceptable probability of error and channel gain: f (rk (n)) = pk (n) |hk (n)|2 for user k on carrier n for a given Pe .

mk

Number of carriers allocated to user k in an FDMA algorithm.

ρk (n)

Proportion of resources (e.g. time) allocated to user k on carrier n.

6

1.2

Dissertation Overview

Chapter 2 provides an analysis of the effect of frequency offset on uplink Multicarrier CDMA (MC-CDMA) systems. The first section discusses other work in estimating the effects of frequency and timing offset. The second section introduces an analytical derivation of the effect of frequency offset on the SINR at the output of an MC-CDMA system. Next, the asymptotic degradation in SINR due to frequency offset is derived for the MC-CDMA system. Simulations verify the analytical formulations, and the results are interpreted in terms of their effect on the quality of service that users will receive. When comparing the results to previous work by Steendam [122], it is found that uplink OFDMA communication is less sensitive to carrier frequency offset error than uplink MC-CDMA with no multi-user interference. In a multiuser OFDM system, the users’ responses must be coordinated so that they can access the channel simultaneously. In an uplink scenario where many users are trying to access a single base station, centralized allocation allows the most efficient use of resources. In Chapter 3, the problem of minimizing transmission power while allowing users to transmit at some minimum acceptable transmission rate with fixed probability of bit error is formulated mathematically. The Sensible Greedy algorithms – Bandwidth Assignment Based on SINR (BABS), Amplitude Craving Greedy (ACG), Rate Craving Greedy (RCG) and Rate Craving Optimal (RCO) algorithms – are introduced, and compared to previous methods for subcarrier and power allocation in a multiuser OFDMA system. In real communication systems, wireless channels vary slowly in time, and users will enter and exit a cell. This presents a problem for a centralized resource distribution system, to retain good quality of service (QoS) all resources must be reallocated and information about the new system-wide allocation must be conveyed to all mobile units. In this case, a simplified online scheduling algorithm may be used, which reduces messaging overhead by adapting slowly to channel variation and passing on some of the resource allocation decisions to mobile units. Chapter 4 describes two online scheduling algorithms for resource allocation which are based on the Sensible Greedy algorithms, and compares these to centralized

7

allocation, as well as random allocation using simulations of a time varying channel with time varying traffic. Chapter 5 summarizes the contributions of the dissertation and presents ideas for further work. Appendix A summarizes the mathematical notation used in this dissertation. Appendix B contains the proofs of theorems introduced in Chapter 2. Appendix C contains proofs of theorems introduced in Chapter 3, and an overview of the modified Lagrangian relaxation (MLR) algorithm. Appendix D contains proofs of theorems introduced in Chapter 4. 1.3

Summary of Contributions

This dissertation presents performance analysis of and resource allocation algorithms for a multicarrier wireless system. The main contributions of this dissertation are: 1. The effect of frequency offset on an MC-CDMA system (Chapter 2, [59]) Analytical expressions have been derived for the Signal to Interference and Noise Ratio (SINR) as a function of frequency offset at the output of a Matched Filter receiver (MF) and a Linear Minimum Mean Square Error receiver (LMMSE) in a multicarrier CDMA system. • For known channel coefficients, Theorems 2.2-2.4 show the effect of frequency offset on the SINR at the output of MF and LMMSE receivers. • For unknown channel coefficients with known statistical distribution, Theorems 2.5-2.6 describe the asymptotic effect of frequency offset on the SINR at the output of MF and LMMSE receivers as the number of carriers and number of users go to infinity while the ratio of carriers to users stays constant. 2. Resource allocation in a static OFDMA system (Chapter 3, [58][60]) Centralized, static algorithms are studied for the efficient allocation of carriers among multiple users in an orthogonal frequency division multiple access (OFDMA) system. A suite of Sensible Greedy algorithms are introduced to address this problem.

8

• A modified Lagrangian Relaxation algorithm is introduced for resource allocation in an OFDMA system, with lower probability of outage and faster execution time for discrete constellations. • An optimal branch and bound algorithm is introduced for resource allocation in an OFDMA system, which minimizes the total transmission power with given probability of error and minimum transmission rates for all users. • A suite of Sensible Greedy algorithms are introduced for resource allocation in OFDMA systems which yield close to minimum transmission power with given probability of error and minimum transmission rates for all users. The algorithms are compared through simulations and analytical complexity analysis. 3. Dynamic resource allocation in an OFDMA system (Chapter 4) Online nonpreemptive resource allocation and scheduling algorithms are derived for an OFDMA based system with a time varying frequency selective channel and dynamic variation of traffic demand in time. • Upper bounds are derived on the outcome fairness and effort fairness of a resource allocation scheme in an OFDMA sytem based on system parameters such as frequency spacing, time granularity and available transmission rates. These bounds are used to derive a formulation of the resource allocation problem which takes into account the fairness of the allocation. • Two online scheduling algorithms are introduced: the Simple Scheduling Algorithm, and the Adaptive Scheduling Algorithm for scheduling and resource allocation. The algorithms are compared using simulations.

9

Chapter 2 THE EFFECT OF FREQUENCY OFFSET ON MC-CDMA

One of the biggest problems in multicarrier communication is the effect of frequency offset error. Frequency offset error decreases the received useful signal power and introduces interference between carriers, thereby increasing the bit error rate (BER) at the receiver. It is important to analyze the effect of such frequency offset error to determine the effect of system design parameters such as the number of carriers in the system, the precision of the oscillator being used, the minimum transmission power necessary to overcome given offset error and the number of users which can be supported by the system. Channel impulse variation during an OFDM symbol [99], clock frequency offset [126] and carrier frequency offset [127] cause frequency offset error. This chapter focuses on the effect of carrier frequency offset (CFO) on the capacity of an uplink multi-carrier CDMA (MCCDMA) system. Tight filtering of individual subcarriers is discussed as one way to mitigate the effect of frequency offset error. Expressions are derived for the Signal to Interference and Noise Ratio (SINR) at the output of a linear receiver for known channels. The chapter then focuses on systems in which the individual subcarrier gains are independent identically distributed (i.i.d.) random variables. It is shown that for such a system, in the limit as the number of users and the number of carriers go to infinity, the SINR at the receiver converges to a function of system load, channel statistics and frequency offset statistics. The BER of the received signal is predicted from the SINR by assuming that the intercarrier interference (ICI) introduced by the frequency offset error can be modeled as additional Gaussian noise. Simulations are used to verify this model. The chapter is organized as follows. Section 1 introduces the topic of frequency offset in multicarrier systems, and summarizes some previous work in this area. Section 2 describes a baseband model for the MC-CDMA system that is considered. Section 3 presents an analysis of the effect of frequency offset in an MC-CDMA system, for known channel

10

coefficients and the asymptotic result for systems with a large number of unknown but independent and identically distributed carrier gains. Numerical results are discussed in Section 4. Conclusions are summarized in Section 5. 2.1

Introduction

Multicarrier modulation provides an elegant solution to the problem of intersymbol interference induced by multipath channels. Large symbol lengths ensure that signals arriving from different paths will combine coherently without additional processing. A multiple access multicarrier scheme should retain this effect of frequency diversity for each user, while minimizing Multiple Access Interference (MAI). Several schemes have been proposed which use CDMA to do just this [42]: Multitone CDMA (MT-CDMA)[133], multicarrier direct sequence CDMA (MC-DS-CDMA) [27][63] [118][119], and multicarrier CDMA (MC-CDMA) [140][32][18]. In all these schemes, users are allowed to transmit on every available carrier, thus obtaining the maximum spectral utilization. Each user is assigned a CDMA code, which is used to differentiate between signals belonging to different users at the receiver. In the first two systems the users’ symbols are spread in time, the spreading occurs after multicarrier modulation. In MC-CDMA, the users’ symbols are spread prior to multicarrier modulation, each chip is transmitted on a different subcarrier. MC-CDMA could be particularly useful in the uplink of a mobile system, since it retains many of the benefits of CDMA such as soft handoff and high spectral efficiency. An uplink channel generally makes the receiver’s job more difficult since each user experiences different channel gains and frequency offset. It is important to analyze how these factors influence the efficacy of the transmission scheme. For single user OFDM with no channel effects, Pollet et.al. [86][85] quantified the degradation in SINR resulting from frequency offset, in the limit as the number of carriers goes to infinity. Moose [78] has also analyzed the effect of frequency offset in OFDM, and established a lower bound on the SINR. This lower bound is related to the lower bound which is derived in Appendix B. Steendam et.al. [115] have worked extensively on generalizing Pollet and Moeneclaey’s re-

11

sults to multiuser systems, including OFDMA [122], MC-CDMA [117], and MC-DS-CDMA [121]. Steendam studies various scenarios, such as the effect of carrier frequency offset, carrier phase jitter [116], synchronization [117], and timing jitter [117] in both uplink and downlink communications [120]. Their analysis of frequency offset in uplink MC-CDMA and MC-DS-CDMA systems, however, assumes a simplified channel model, in which all the carriers of any given user experience the same carrier gain, that is, flat frequency fading across all carriers. An upper bound is derived for the effect of frequency offset in an uplink system, by assuming that all interfering users will have maximum frequency offset [120]. Jiho Jang and Kwang Bok Lee [51][52] have analyzed the degradation SINR in response to fixed frequency offset at the output of a downlink MC-CDMA system transmitting over a Rayleigh fading channel. Their work compares the robustness of Equal Gain Combining (EGC) and Maximum Ratio Combining (MRC) receivers to frequency offset error. The results confirm previous findings, that MRC receivers result in higher BER at the receiver than EGC receivers, since they preserve the orthogonality of the MC-CDMA system better than MRC receivers even in the presence of frequency offset. However since the SINR at the output of the MRC receiver is low to begin with, from the definition of SINR degradation that is chosen its degradation must also be less. Taeyoung Kim et.al. [57] have studied fixed carrier frequency offset error in an MC-CDMA system in correlated fading channels, and conclude that the effect of correlated channels on the SINR is more pronounced than the effect of frequency offset error. KyunByoung Ko et.al. [62] have studied carrier frequency offset error in an asynchronous MC-CDMA system. Tomba and Krzymien [125][127] have studied the effect of downlink phase noise and frequency offset error on the BER at the output of an MC-CDMA system, assuming random phase noise and frequency offset error with known probability density functions. Tomba and Krzymien used a semi-analytical approach, by using a Markov process model for the frequency and phase noise introduced. The inter-carrier interference at the output of a decorrelating linear receiver was calculated using this approach. This interference was assumed to be Gaussian, an assumption verified using the Kolmogorov-Smirnov test, and the BER was calculated using standard formulas. This approach has been used again by Tomba and Krzymien [126] to study chip timing jitter in an MC-CDMA system, and by Robertson

12

and Kaiser [99] to study Doppler frequency offset in OFDM and OFDMA systems. Mashury [76] has studied carrier frequency offset error similarly, for an uplink MC-CDMA system using a linear MMSE receiver, and a decorrelating receiver. Sathananthan and Tellambura [103][104] have applied the method of Markov integration to an OFDM system, to calculate the BER directly, instead of finding the SINR then calculating the BER by assuming that the intercarrier interference is Gaussian. Their results show a more accurate prediction of the effect of frequency offset error. Some researchers have also considered the role of pulse shaping and subcarrier filtering on the effect of frequency offset error and phase jitter. Benvenuto et.al. [8] have compared Discrete Multitone (DMT) to Filtered Multitone (FMT), Wongjong Rhee et.al. [98] have compared MC-CDMA to and Multitone with Polyphase Filterbank (MPFB), and Steendam et.al. [122] have compared OFDMA to FDMA. There appears to be considerable improvement in resilience to phase jitter and to frequency offset error when comparing the OFDM-based systems (DMT, MC-CDMA, OFDMA) with overlapping subcarrier spectra, and the FDM-based systems (FMT, MPFB, FDMA) which have non-overlapping subcarrier spectra. Most recently, Peng Tan and Beaulieu [124] have proposed a “better than” raised cosine (BTRC) pulse for reducing intercarrier interference in an OFDM system. The pulse shaping described in their paper differs from traditional pulse shaping for communications in that the raised cosine pulse is reproduced in the frequency domain by applying the frequency domain function for that filter to the time domain signal being transmitted. Thus it is similar to the MPFB scheme described by Wongjong Rhee et.al. [98]. In this case also, the main source of improved resistance to frequency offset is that subcarriers in FDM-based systems are spaced much further apart in frequency than subcarriers in OFDM-based systems, and as demonstrated by Peng Tan and Beaulieu [124], the main lobes of the spectra of each individual carrier are much narrower. Since the main component of inter-carrier interference is the input from the adjacent subcarriers, a narrower spectrum reduces interference considerably. In their research on pulse shaping and modified constellations for OFDM systems, Remvik et.al. [96] have shown that improved modulation scheme and optimized pulse shaping combined have a modest effect on the resilience of an OFDM system to frequency and

13

phase jitter. Thus for systems utilizing pulse shaping filters of the same order, the improved resilience to frequency offset error comes at the cost of bandwidth efficiency. A higher order filter allows a narrower main lobe, which can be used to improve resistance to inter carrier interference or to increase the total transmission rate for the system, but not both. The FDM-based schemes are generally half as bandwidth efficient as the OFDM-based schemes. Polynomial Cancellation Coding for OFDM (PCC-OFDM) [3] and MC-CDMA (PCC-CDMA) [4] also work on the same trade-off, between bandwidth utilization and resilience to frequency offset error and phase jitter. When the total data transmission per bandwidth is fixed, the effect of carrier frequency offset error is very close for PCC-CDMA and for MC-CDMA. PCC-CDMA is advantageous for fading channels, however, since MCCDMA performance degrades rapidly when the channel coherence time is longer than the guard interval (GI) for the OFDM symbol, whereas PCC-CDMA shows a much more graceful degradation [4]. In the FMT, MPFB, FDMA and PCC-CDMA schemes describe above, the frequency spacing is fixed at exactly twice the spacing between the carriers of the underlying OFDM scheme, to take advantage of the orthogonality between carriers. This results in a multicarrier system in which every other carrier is used for data transmission, and the remaining carriers are unused “virtual carriers.” For Peng Tan and Beaulieu [124]’s scheme, the ICI is found for an excess bandwidth of α = 1 for the raised cosine filter, in other words at approximately 1/2 bandwidth utilization. Although the BTRC pulse has not been analyzed, since it shows a narrower main lobe than the RC pulse, it can be inferred that BTRC with α = 1 is also not bandwidth efficient. Other schemes to combat ICI using pulse shaping generally require knowledge about the channel state at the receiver [95], and are outside the scope of this chapter. In this chapter, the effect of carrier frequency offset on the SINR at the output of a linear Multi-User Detector (MUD) is studied using asymptotic analysis. Random matrices have been studied for many years in the context of theoretical physics and mathematics [114]. Tse and Hanly [130] have used free probability theory [10][128] and random matrix theory [77] to analyze CDMA systems with linear MMSE, MF and decorrelating receivers. Asymptotic analysis has also been used to study multiple-input multiple-output systems [79], and recently Peacock et.al. have studied MC-CDMA systems [83]. In asymptotic anal-

14

ysis, the SINR at the output of a linear system is evaluated as the number of transmitting channels (code chips, antennas, or frequency carriers) and the number of users both go to infinity, while the ratio of users to channels stays constant. For the above systems, it is shown that when the number of interferers is very large, the effect of each interferer can be “decoupled,” and can be evaluated separately. Mantravadi and Veeravalli [75] have used this method to show that asynchronous CDMA and synchronous CDMA have the same asymptotic capacity. This chapter introduces analytical expressions for the degradation in SINR caused by inter-carrier and inter-user interference resulting from a random carrier frequency offset with known probability distribution. The system being studied is an uplink MC-CDMA system. The analysis is based on Pollet’s approach [85], and does not rely on MonteCarlo integration to find the effect of frequency offset [127]. A factor α is introduced which regulates the spacing between carriers, allowing these results to be generalized to an FMT/MPFB/FDMA system without spectral overlap. Section 2.3.1, presents an analytical expression for the average SINR at the output of a linear receiver with known channel coefficients, and random frequency offset. Two linear receivers are studied:

• the multi-user linear minimum mean square error (LMMSE) receiver, and

• the conventional matched filter (MF) receiver.

Further analysis in Section 2.3.2, shows that in the limit as the number of the number of subcarriers (N ), the number of data subcarriers (M ), and the number of users (K) go to infinity while the ratios of these terms, α = N/M and β = K/M remain constant, the SINR at the output of a MF receiver and the SINR at the output of a LMMSE receiver both approach functions of the probabilistic distribution of the channel coefficients and the carrier frequency offset experienced by the users. These results are a generalization of the result presented by Peacock et.al. [83] to a system with users experiencing i.i.d. random frequency offsets.

15

2.2

System Model

This chapter explores a multi-carrier CDMA system. In MC-CDMA systems, all mobile stations transmit simultaneously on all carriers marked for transmission, using different transmission codes. The transmission rate per mobile station is one symbol per time interval, this symbol is multiplied by a different code before transmission on every carrier. At the receiver, the users’ signals are recovered using a linear multiuser detector. Fig. 2.1 illustrates the baseband model of the combined transmitter and channel response for a single user. Fig. 2.2 illustrates the receiver at the base station. In this chapter, the receiver is assumed to have perfect channel state information, that is, all carrier gains of all mobile stations are known at the receiver. The transmitter is assumed to have no channel state information. The channel and the frequency offset of the users are assumed to be constant for the duration of the analysis. The channel is modeled as a wide sense stationary uncorrelated scattering (WSSUS) Raleigh fading channel, where the guard interval (GI) is larger than the maximum channel delay spread. Each subcarrier signal is assumed to undergo flat fading, where the channel gain is an independent identically distributed random variable. It is assumed that the users are perfectly synchronized in time, and there is no interference from adjacent cells to the system, or such interference can be modeled as Gaussian noise. Suppose that there are M carrier frequencies in an MC-CDMA system, and the bandwidth of a subcarrier is 2π/N . Fig. 2.3 shows the frequency-domain spectra of three multicarrier systems. The first and second spectra show systems that are “tightly packed” in frequency, as in a standard OFDM system. The third system has subcarriers which are spaced further apart, similar to the pulse shaping FDM schemes proposed in literature [8][98][122][124]. Consider a system with M carrier frequencies and subcarrier bandwidth 2π/N . If the inter-subcarrier spacing 2π/M is an integer multiple of the subcarrier bandwidth 2π/N , then the subcarriers retain their orthogonality, eliminating ICI in a perfectly synchronized system. The simplest implementation of this system, is using a rectangular waveform, in which case the polyphase filterbank implementation is simply using N -ary IFFT and FFT receivers at the transmitter and receiver respectively, that is, by implement-

16

Channel 1

α

sk,l ck,1

α

S/P

α

S/P α

hk,1 ej2πδFk,l αT /N

P/S, add GI

hk,0

IDFT

ck,0

nk,lN +n xk,lN +n

α α ck,M −1

S/P

hk,M −1 ej2πδFk,l T α(M −1)/N Transmitter

Figure 2.1: Combined baseband model of transmitter and channel for user k.

P/S rl

DFT

rlN +n

S/P, remove GI

P/S

α α

α

P/S

α

Multi-User Detector (MUD)

α

α Figure 2.2: Baseband receiver model.

C, H, δF, σz2

Channel Estimator

ˆ sl

17

Figure 2.3: Multicarrier frequency spectra. (a) 8 subcarriers, N = 8, M = 8, α = 1, (b) 4 subcarriers, N = 4, M = 4, α = 1, (c) 8 subcarriers, N = 8, M = 4, α = 2.

ing the N -ary transmission scheme in (a) and assigning zeros to the unused carriers. From this perspective, the transmission scheme in (c) can be viewed as a variation on (b), where the subcarriers are bandpass filtered prior to transmission or a variation on (a) where every other carrier is used for transmission. As discussed in the introduction, optimized pulse shape filtering can introduce some additional improvement in the system’s resistance to frequency offset, however the most important effect of pulse shaping is to reduce the spectrum of each carrier [96], and this effect will be analyzed in this chapter. In this dissertation, the pulse shaping waveform used in transmission and reception is assumed to be rectangular. To quantify the subcarrier bandwidth, bandwidth scaling ratio is defined as follows: Definition 2.1. Let M be the number of subcarriers in a multicarrier transmission system, and let the subcarrier bandwidth, defined as the width of the main lobe of the pulse shaping filter for transmission be 2π/N . Then α = N/M is defined as the bandwidth scaling ratio. In this dissertation, α is assumed to be integer. The frequency offset experienced by user k during symbol interval l is modeled as a random variable δFk,l with probability density function (pdf) Gk (δFk,l ). The effect of the

18

frequency offset on carrrier n is multiplication by a factor ej2πnδFk,l T /N . In the analysis, it is assumed that all interfering users have the same frequency offset distribution, G(δFk,l ). Let K be the number of users in the system. The CDMA code assigned to the kth user is ck = [ck,0 , ..., ck,M −1 ]T , and the channel attenuation vector experienced on the M data channels is hk = [hk,0 , ..., hk,M −1 ]T . Thus, the contribution of user k to the signal at the receiver on subcarrier m is its data symbol multiplied by a factor ak,m = ck,m hk,m . The effective channel spreading vector for user k is defined as ak = [ak,0 , ak,1 , · · · , ak,M −1 ]T . The matrix A = C H = [a1 a2 · · · aK ] is defined as effective channel spreading matrix. For the channel model considered in this chapter, the channel gains hk,n are Gaussian random variables. Assuming that the code chips ck,n are equal to +1 or −1 with equal probability, the elements of the effective channel spreading matrix ak,n are also independent Gaussian random variables. So for this analysis, the CDMA code ck is not relevant. This conclusion will not hold when subcarrier fading gains are correlated. Let sk,l be the complex data symbol transmitted by user k ∈ {1, 2, ..., K} in the symbol interval t ∈ [(l − 1)T, lT ]. The data symbols are assumed to be independent identically n o distributed, with variance E |sk,l |2 = 1. Under perfect timing synchronization, the vector of N samples received during time interval l, rl = [r0,l , ..., rN −1,l ]T , is given by:

rl =

K X

D(δFk,l )Mak sk,l + zl ,

(2.1)

k=1

where M is the modulation matrix, ak is the effective channel spreading vector for user k, zl = [z0,l , ..., zN −1,l ]T is white Gaussian noise, and   D(δFk,l ) = diag 1, ej2πδFk,l T /N , · · · , ej2π(N −1)δFk,l T /N

(2.2)

is the frequency offset matrix of the kth user. The form of the modulation matrix M depends on the number of subcarriers that are

19

used, and the spacing between subcarriers. Let WM be the M -ary DFT matrix:   1 1 ··· 1       1 ej2π/M ··· ej2π(M −1)/M .  WM =  .  .. .. ..   .. . . .   j2π(M −1)/M j2π(M −1)(M −1)/M 1 e ··· e Matrix M can be built by stacking α M-ary IDFT matrices vertically:   H  WM     1  ..   M = √  .  α matrices   α   H WM The demodulator for M is its Hermitian, M−1 = MH . In order to recover the desired signal from user j, sj,l , the linear multi-user detector (MUD) at the receiver will combine information from all carriers using a weighting vector bj . The receiver’s estimate of the symbol transmitted by user j in time interval l is: ! K X sˆj,l = bH MH D(δFk,l )Mak sk,l + zl . (2.3) j k=1

Equation (2.3) can be rewritten to separate the signal of interest from other interference, H sˆj,l = bH j M D(δFj,l )Maj sj,l +

K X

H H bH j M D(δFk,l )Mak sk,l + bj zl .

(2.4)

k=1 k6=j

The first term in Equation (2.4) is the signal of interest, the second term represents interference from other users, and the last term is Gaussian noise. The SINR at the output of the receiver is defined as:  2  H H AVGl bj M D(δFj,l )Maj sj,l ( SINRj = 2 ) , PK H H AVGl k=1 bH j M D(δFk,l )Mak sk,l + bj zl

(2.5)

k6=j

n o where AVGl {·} denotes the average over time. Assuming that E |sk |2 = 1, and that the frequency offsets of the users are independent identically distributed random variables, this expression reduces to: SINRj =

σz2



Ej  bH b j j + V j − Ej

(2.6)

20

where Ej Vj

= =

n 2 o H E δFj bH M D(δF )Ma j j j K X

n 2 o H E δFk bH j M D(δFk )Mak

(2.7) (2.8)

k=1

Self interference occurs through Inter-Channel Interference (ICI) between different frequencies modulated with different CDMA chips. Since the same symbol is transmitted on each channel, it results in a scaling of the signal of interest. Multiple access interference can be caused by interference from signals transmitted by different users on the same carrier, as well as by ICI. The asymptotic SINR at the output of a given multi-user detector is defined as: SINR∞ detector,j =

lim

N,M,K→∞ K/M =β,N/M =α

SINRj .

(2.9)

Previous work by Tomba and Krzymien [127] has shown that the effect of ICI introduced by carrier frequency offset is similar to the effect of white Gaussian noise with the same mean and variance as the ICI. Assuming that the sum of the additive white Gaussian noise and the total interference is a Gaussian random variable [76], the BER at the output of the DQPSK receiver will be [89]:    1 2 1 2 Pe = Q1 (a, b) − I0 (ab) exp − a + b 2 2

(2.10)

where v u u Eb a = t2 N0 v u u Eb b = t2 N0

r ! 1 1− 2 r ! 1 1+ 2

I0 (ab) is the modified Bessel function of the first kind, and Q1 (a, b) is the Marcum Q-function of order 1 [23], Z Q1 (a, b) = b



x2 + a2 x exp − 2 

 I0 (ax)dx.

This assumption is verified by simulations in Section 4 of this chapter.

21

When there is no frequency offset, δFk = 0 for k = 1, 2, · · · , K, the signal at the output of the transmitter is: sˆj,l =

bH j aj sj,l

+

K X

H bH j ak sk,l + bj zl .

(2.11)

k=1 k6=j

and the SINR of the signal at the output of the MUD is:

SINRj |δF =0

H 2 bj aj = 2   P K H b a b + σz2 bH k=1 j k j j

(2.12)

k6=j

Defining the vector of symbols transmitted as sl = [s1,l , · · · , sK,l ]T and the vector of symbol estimates at the receiver as ˆsl = [ˆ s1,l , · · · , sˆK,l ]T , in the absence of frequency offset the two are related by: ˆsl = BH Asl + BH zl .

(2.13)

In this chapter, the effect of carrier frequency is measured in terms of the degradation in SINR caused by frequency offset, defined as: ∆ SINRj

SINRj SINRj |δF =0 = SINRj (dB) − SINRj |δF =0 (dB)

=

(2.14) (2.15)

and the asymptotic degradation in SINR caused by frequency offset, defined as: ∆ SINR∞ detector,j

=

SINR∞ detector,j SINR∞ detector,j

(2.16)

δF =0

∞ = SINR∞ detector,j (dB) − SINRdetector,j δF =0 (dB)

(2.17)

where SINR∞ detector,j δF =0 is the asymptotic SINR at the output of the receiver when the frequency offset is equal to zero. For a given frequency offset, ∆ SINRj and ∆ SINR∞ detector,j predict how much more transmission power will be required to achieve the same bit error rate at the receiver. This chapter considers two linear detectors, the MF receiver, and the LMMSE multiuser receiver. In both cases, the channel coefficients are assumed to be known at the transmitter. The choice of weighting vectors bk depends on the type of detector that is implemented.

22

• The Matched Filter (MF) Receiver : The detection matrix for the MF weights each channel by the channel coefficients for the user being received, to realize the optimal single user detector. The combining vector for each user, bk , minimizes the mean square error |ˆ sk,l − sk,l |2 in the presence of unknown, uncorrelated noise. The detector matrix for the MF receiver is BH = AH . The MF receiver allows for reasonable performance when the number of interfering users is small, at lower complexity than the MMSE receiver. It is also known as the Maximum Ratio Combining (MRC) receiver in the context of MC-CDMA [52]. • The Linear MMSE receiver : For the LMMSE receiver, the matrix B is chosen so as P to minimize K sk,l − sk,l |2 . Then B satisfies the orthogonality principle: k=1 |ˆ n o E sl (sˆl − sl ) (Asl + zl )H = 0. and can be found to be equal to: BH = AH AAH + σz2 I

−1

,

(2.18)

where σz2 is the variance of white Gaussian noise, and α = N/M is the bandwidth scaling ratio. The next section presents an analysis of the effect of frequency offset on the SINR at the output of the channel. Expressions are derived for both SINRj and SINR∞ detector,j . The channel is assumed to be constant for the analysis. For the MF and LMMSE receiver, the asymptotic SINR at the output of the receiver is found to depend only on the type of receiver, the system parameters (N , K, α) and the distribution of channel parameters, not on the individual realization of the channel. 2.3

The Effect of Frequency Offset

This section presents two analyses of the effect of frequency offset error on the SINR at the output of the receiver. The first analysis yields an analytical expression which interprets the effects of the channel coefficients, and random frequency offset error. The second analysis is an extension of work by Tse and Hanly [129] on the effective interference of each user in a

23

multiuser system with linear multiuser receivers. Tse and Hanly show that as the number of users and the number of carriers in a multi-dimensional system such as MC-CDMA goes to infinity, with the ratio of users to carriers remaining constant, the effect of every interferer can be shown to converge to a fixed level of effective interference. The second analysis shows the effect of frequency offset on the fixed interference introduced by a user in a MCCDMA system. The resulting expressions for the degradation in SINR are important in two respects. First, when the inter-user interference is modeled as Gaussian, the BER depends on the SINR through well-known formulas [85]. Second, the SINR can be related to the capacity of the channel. In an MC-CDMA channel, as in a CDMA channel, any vertex of the capacity region can be achieved using successive interference decoding using a particular decoding order, and a LMMSE receiver, with the information-theoretic rate at the output of the receiver given by: 1 log(1 + SINRk ) 2 where SINRk is the SINR at the output of the LMMSE receiver for user k. Throughout this section, it is assumed that the users’ data symbols sk,l , the noise terms zn,l , and the frequency offset δFk,l for all symbols l are independent and identically distributed. For convenience, in the following, the subscript l will be dropped.

2.3.1

Effect of Frequency Offset for Linear Receivers with Channel State Information

Given the system described in Equation (2.4), the effect of frequency offset on the SINR at the output of a receiver is analyzed for a linear multiuser receiver. It is assumed that the channel coefficients are known perfectly at the receiver, but the frequency offset is not known or compensated for.

Theorem 2.2. Consider an MC-CDMA system where user j’s signal is the signal of interest. If the frequency offsets of all users, δFk for k = 1, ..., K are independent, and user j’s frequency offset has a probability density function Gj (δFj ) and all interfering users k = 1, ..., K, k 6= j are identically distributed with probability density function G(δFk ) then

24

the SINR for user j at the output of the receiver is: SINRj =

Ej   Vj − Ej + σz2 tr bH j bj

(2.19)

where Ej represents the desired signal power (first term in Equation (2.4)):     H H ∗ ¯ j Maj aH Ej = 1H Ψ Mbj bH 1, j M j M

(2.20)

and Vj represents total signal and interference power:     H ∗ ¯ MAAH MH Mbj bH M 1. Vj = 1H Ψ j

(2.21)

¯ j and Ψ ¯ represent the effect of frequency offset on self interference and The matrices Ψ multiple access interference power. ¯j Ψ

 E δFj p(δFj )p(δFj )H ,  ¯ = E δF p(δFk )p(δFk )H Ψ for k 6= j, k h iT p(δF ) = 1, ej2πδF T /N , · · · , ej2πδF T (N −1)/N . =

(2.22) (2.23) (2.24)

¯ j is given by: Element (n, m) of matrix Ψ ¯j Ψ

 n,m

  n o (n − m) j2π n−m δF T j N = E e = ΦGj 2π N

(2.25)

¯ is given by: and element (n, m) of matrix Ψ ¯ n,m Ψ

  n o (n − m) j2π n−m δF T k N = E e = ΦG 2π N

(2.26)

where ΦGj (·) and ΦG (·) are the characteristic functions of the distribution functions Gj and G of normalized frequency offsets of the user of interest and the interfering users respectively. The proof of this Theorem can be found in Appendix B. The expression in Equation (2.19) is significant because it gives insight into how the frequency offset induces error. MC-CDMA is generally described as CDMA in frequency. In this case it can also be viewed as a CDMA scheme spread across time, using the code Mak , and despread at the receiver using the filter Mbk . Ej is the total power of terms that are spread and despread using matching sequences. Vj is the sum of Ej , and the inter-user

25

interference. When there is no frequency offset, the matrix Ψ is equal to the matrix of all 1’s. The complex scaling of the non-diagonal terms of Ψ introduces excess interference, which causes inter-channel interference (ICI), caused by mismatch between spreading and despreading chips due to the frequency offset. ¯ j and Ψ ¯ depends on the distribution of the frequency offsets The form of the matrices Ψ of the user of interest δFj and the interfering users δFk for k 6= j. Both matrices depend on the characteristic function of the distribution, which is the Fourier transform of the probability density function. For known distributions they can be derived easily, since the elements correspond to the characteristic functions of the frequency offset distribution functions, ΦGj (t) and ΦG (t) evaluated at t = 2π(n − m)/N . For instance: • The frequency offsets, δFk for users k = 1, ..., K, are uniformly distributed in the interval [−F, F ],      (n − m) (n − m) E δF exp j2π δF T = sinc 2π FT N N   (n − m) √ = sinc 2π 3σF T . N

(2.27)

• The frequency offsets, δFk for users k = 1, ..., K, are Gaussian with zero mean and variance σF ,     2 ! 1 (n − m) (n − m) δF T = exp − 2π σF T . E δF exp j2π N 2 N

(2.28)

More generally, the following result is proven in Appendix B: Theorem 2.3. Assume that the probability distribution of the normalized frequency offset, G(δF T ), has zero mean and is symmetric about δF T = 0. Assume also that δF lies in the ¯ is bounded interval [−1/(2T ), 1/(2T )] with probability 1. Then element (n, m) of matrix Ψ from below by:

where σF

  (n − m) ¯ σF T (2.29) Ψn,m ≥ cos 2π N r n o n o 2 = E (δF T ) T . As the fourth moment of δF T , E (δF T )4 approaches

(σF T )4 , the distribution of δF T approaches a binomial distribution with δF = ±σF , and the above bound becomes arbitrarily close.

26

¯ and Ψ ¯ j . The assumption that the The above theorem applies to both matrices Ψ frequency offset lies in the interval [−1/(2T ), 1/(2T )] with probability 1 is reasonable, since if it lies outside this interval, the signal from the neighboring carrier is stronger than that from the carrier of interest, and the signal of interest can not be reliably retrieved. The bound on Ψn,m is of interest because as the frequency offset goes to zero, the entries Ψn,m will go to 1. It is then reasonable to assume that a lower bound on Ψn,m gives an upper bound on the resulting degradation in SINR. Theorem 2.3 can be used to justify approximating the entries of the carrier offset matrix by Ψn,m ≈ cos(2π(n − m)σF T /N ), where σF is the standard deviation of the distribution for δF much less than the inter-carrier spacing ∆F . Figure 2.4 compares the values of Ψn,m as n − m goes from −N to N for the uniform distribution, the Gaussian distribution, and using the approximation in Equation (2.29). The plot corresponds to the elements across the anti-diagonal of the matrix Ψ, when σF T = 0.2. Although this is a large offset, the plots are still close. The next theorem describes the effect of the bandwidth scaling ratio on the degradation in SINR caused by frequency offset. Theorem 2.4. For the system described in Theorem 2.2, with integer bandwidth scaling ratio α > 1, the SINR at the output of the receiver is: SINRj =

Ej   Vj − Ej + σz2 tr bH b j j

(2.30)

where Ej represents the desired signal power:   ∗  H H H ¯ j WM Ej = 1H Ψ aj aH 1, j WM WM bj bj WM

(2.31)

and Vj represents total signal and interference power:   ∗  H H ¯ WM Vj = 1H Ψ 1 AAH WM WM bj bH j WM

(2.32)

where ¯j Ψ



 n,m

=

 ¯ Ψ = n,m



  (n − m) δFj T , E δFj ∆α (δFj T ) exp j2π M α     (n − m) δFk T E δFk ∆α (δFk T ) exp j2π for k 6= j, M α

(2.33) (2.34)

27

1

0.9

0.8

0.7

0.6

0.5

´ ` exp − (2πσF T n/N )2 /2 ` √ ´ sinc 2π 3 σF T n/N cos (2πσF T n/N )

0.4

0.3

−100

−50

0

50

100

Carrier Index n = k − l

Figure 2.4: Plot of Ψn−m , the elements across the antidiagonal of matrix Ψ for σF T = 0.2, and N = 128:   Ψ0 Ψ1 · · · ΨN −2 ΨN −1  Ψ−1 Ψ0 · · · ΨN −3 ΨN −2     ..  . . . . . . Ψ= . . . . .    Ψ−(N −2) Ψ−(N −3) · · · Ψ0  Ψ1 Ψ−(N −1) Ψ−(N −2) · · · Ψ−1 Ψ0

28

and  ∆α (δF T ) = 2.3.2

sin (πδF T ) α sin (πδF T /α)

2 .

(2.35)

Asymptotic Analysis of the Effect of Frequency Offset for Matched Filter and Linear MMSE Receivers

This section analyzes the performance of two linear multiuser receivers in the presence of frequency offset. In their work on linear multiuser receivers, Tse and Hanly [129] introduced the concept of effective interference. They showed that given certain conditions on carrier gains, as the number of users and the number of carriers become arbitrarily large, while the ratio of users to carriers - the number of users per degree of freedom - remains fixed, the effect of a single user on the output SINR converges to a fixed ratio. This ratio depends on the distribution of users’ transmit powers, and on the receiver, but not on the particular realization of the channel. The effect of frequency offset on the asymptotic performance of the LMMSE receiver and the conventional MF receiver is studied in this section. It is assumed, as previously, that the frequency offsets of the users’ are independent of channel  gains. It is also assumed that the channel gains are random variables, with E |sk |2 = 1,    | E ak,m |2 = 1/M , E |sk |4 < ∞ and E |ak,m |4 < ∞ for all users k and all carriers m. The Matched Filter (MF) Receiver The conventional matched filter receiver aims to maximize a user’s signal in Gaussian noise. It does not take into account the structure of interference from other users. In this sense, it is more like a single user receiver than a multi-user receiver. In the context of MC-CDMA, a matched filter receiver is also known as a Coherent Combining Receiver. The receiver H matrix which represents the MF Receiver is BH M F = A . The symbol at the output of a

MF receiver is: H sˆj,l = aH j M D(δFj,l )Maj sj,l +

K X

H H aH j M D(δFk,l )Mak sk,l + aj zl .

(2.36)

k=1 k6=j

Using assumptions about the users’ spreading codes and average transmission power, the expectation of the SINR over the set of all possible signatures for user j can be calculated

29

for the case when both the number of carriers and the number of users become arbitrarily large [129]. Theorem 2.5. Let the channel signatures ak = [ak,0 , ak,1 , ..., ak,M −1 ]T for users k = 1, ..., K, be random vectors whose entries are independent random variables. The elements of vector ak are identically distributed with variance σk :   ∗  σj2 /M if k = j and m = n, E aj,m ak,n =  0 otherwise. Let δFj be the carrier frequency offset error and σj2 be the total transmit power per carrier for the user of interest, user j. Assume that the transmission powers σk2 and the carrier frequency offsets errors δFk for all interfering users k = 1, ..., j − 1, j + 1, ..., K are independent random variables, and that they are drawn from the same probability distribution. As the number of users K and the number of subcarriers M go to infinity, while the degree of freedom of the system β = K/M and bandwidth scaling ratio α = N/M remain constant, the SINR at the output of the Matched Filter converges almost surely to: SINR∞ MF,j

=

σj2 sinc2 (δFj T )  2   2 sin(πδFk T ) 2 σz + β E k6=j σk E k6=j α sin(πδFk T /α)

=

σj2 sinc2 (δFj T )   Pα−1 σz2 + αβ E k6=j σk2 p=−(α−1) 1 −

|p| α



ΦG 2π αp

(2.37)



(2.38)

where ΦG (·) is the characteristic function of the distribution G of relative frequency offsets δFk T of the interfering users. When the carrier frequency offset error of user j is also a ran dom variable, the numerator of the above expression can be replaced by E δFj sinc2 (δFj T ) . When α = 1, the SINR at the output of the matched filter depends only on the frequency offset of the user whose signal is being demodulated. This is because the matched filter does not try to cancel out other users’ signals. The SINR degradation caused by other users’ frequency offsets is negligible compared to the interference which is not canceled by the matched filter. Increasing α increases the frequency separation between the components of the signal. 2  sin(πδFk T ) For relative frequency offset δFk T less than half a carrier spacing, α sin(πδF is less k T /α)

30

than 1, and serves to reduce the interference shown in the denominator of the SINR in Equation (2.37). The Linear Minimum Mean Square Error (LMMSE) Receiver The linear MMSE receiver is the linear receiver which minimizes the mean square error between the transmitted symbols and the symbols at the output of the receiver. While it introduces additional complexity in calculating weighting coefficients, the performance of the system is significantly improved. The receiver matrix for the linear MMSE receiver is H BH σz2 I + AAH M M SE = A

The receiver vector for user j is bj = σz2 I + AAH

−1

−1

(2.39)

aj , and the symbol at the output

of a MMSE receiver is: 2 H sˆj,l = aH j σz I + AA

+

K X

−1

MH D(δFj,l )Maj sj,l

2 H aH j σz I + AA

−1

2 H MH D(δFk,l )Mak sk,l + aH j σz I + AA

−1

MH zl .

k=1 k6=j

The first term is the useful signal power, the second term is the power of interference from other users, and the last term results from AWGN from the channel. When there is no frequency or timing offset, the SINR at the output of a LMMSE receiver is given by [129]: SINRLMMSE,j = aj σz2 I + Aj AH j

−1

aj

(2.40)

where Aj =

h

a1 a 2 · · ·

aj−1 aj+1 · · ·

aK

i

(2.41)

For convenience, the matrix Xj is defined as: 2 Xj = Aj AH j + σz I

(2.42)

will be used in the remainder of this chapter as well as in Appendix B. Since the different users’ spreading gains are independent of each other, matrix Xj is not dependent on the

31

spreading gains of user j. The receiver vector bj can be written in terms of matrix Xj as  −1 bj = Xj + aj aH aj . j While the expression for the output SINR in a system with frequency offset cannot be simplified to this form, it can be expressed in terms of a product of vector aj and Xj . Theorem 2.6. Let the channel signatures ak = [ak,0 , ak,1 , ..., ak,M −1 ]T for users k = 1, ..., K, be random vectors whose entries are independent random variables. The elements of vector ak are identically distributed with total transmission power σk2 : E a∗j,m ak,n 



  σ 2 /M j =  0

if k = j and m = n, otherwise.

n o σk2 < ∞ and E |ak,n |4 < ∞ for all users k. Let δFj be the carrier frequency offset error and σj2 be the total transmit power per carrier for the user of interest, user j. Assume that the transmission powers σk2 and the carrier frequency offsets errors δFk for all interfering users k = 1, ..., j − 1, j + 1, ..., K are independent random variables, and that they are drawn from the same probability distribution. As the number of users K and the number of subcarriers M go to infinity, while the degree of freedom of the system β = K/M and bandwidth scaling ratio α = N/M remain constant, the SINR at the output of a linear MMSE receiver converges to: SINR∞ LMMSE,j =

∆α,k Ej Nj

(2.43)

where

Cp2 = Cn2 = C1,sum = C2,sum =

Ej

 2 = C1,sum sinc2 (πδFj T /α) + Cp2 1 − sinc2 (πδFj T /α)

(2.44)

Nj

= C1,sum S2 + Cn2 (S1 − S2 ) + σz2 C2,sum (1 − S1 )

(2.45)

1 M C2,sum



+

Aj,4 2

(σj2 /M )

2

 2

2 C1,sum

β E k6=j σk + σz C2,sum  −1  σj2 H + σ2I tr A A j j z M  −2  2 σj H 2 Aj Aj + σz I M tr

E k6=j {∆α,k (δFk )}  = E k6=j ∆α,k (δFk ) sinc2 (πδFk T /α)  2 sin(πδFk T ) = α sin(πδFk T /α) n o  2 = E |aj,n |4 − σj2 /M

S1 = S2 ∆α,k Aj,4

32

The proof is given in Appendix B. The above expression for the asymptotic SINR at the output of a LMMSE receiver still seems to depend on the channel coefficients. However in the limit as M and K go to infinity while α and β remain constant, it can be shown using random matrix theory [77] that this dependence disappears. The matrix Xj includes the sum of the covariance matrices of K −1 spreading sequences. Since the columns of matrix Aj , the spreading signatures of the users, are independent identically distributed complex Gaussian random vectors, the matrix Aj AH j follows the Wishart distribution with K − 1 degrees of freedom [73]. In the limit as N goes to infinity, Aj AH j becomes a Wishart random matrix [5]. Since the trace of a matrix is the sum of its eigenvalues, in the limit as N goes to infinity Z ∞  1 1 2 −1 tr Aj AH + σ I = dG(λ) (2.46) lim j z M →∞ M λ + σz2 0 where G(λ) is the limiting empirical distribution of the eigenvalues of matrix Aj AH j , defined below: Definition 2.7 (Empirical distribution function [5][39, p.387][83]). The empirical distribution function of the eigenvalues of matrix Aj AH j with eigenvalues λ1 ≤ · · · ≤ λM is defined as: GM A (x) =

1 #{i : λi ≤ x} M

where #S denotes the cardinality of the set S. Tse and Hanly [130] have used free probability theory [10][128] and random matrix theory [77] to find recursive equations to evaluate Equations (2.46). For the case of equal transmission powers for all users considered in this dissertation there is an analytical solution:  1 2 −1 tr Aj AH j + σz I M →∞ M s   2 1 1−β 1 4 1−β = −1 + − 1 + 2. 2 2 2 2σz 2 2σz σz

lim C1,sum =

M →∞

lim

Similarly, the term C2,sum can be written as [68][69]: Z ∞  1 1 H 2 −2 lim C2,sum = lim tr Aj Aj + σz I = dG(λ). M →∞ M →∞ M (λ + σz2 )2 0

(2.47) (2.48)

(2.49)

33

An analytical solution to the above equation is difficult to find, an empirical solution was proposed by Li et.al. [69]. In this dissertation, C2,sum was found empirically by averaging over a large number of random instances of matrix Aj for fixed N , K and σz2 . These properties remove the dependence of the asymptotic SINR in Theorem 2.6 on the instance of the channel under consideration, yielding a general expression for the asymptotic SINR at the output of a LMMSE receiver. Effect of carrier bandwidth on MC-CDMA with LMMSE receiver This section will investigate the effect of carrier bandwidth on frequency offset error. The systems compared will both have M carriers, the first has subcarrier spacing α = 1, N = M , while the second has α = 2, N = 2M . For the case when α = 1, σj2 = 1, the results of Theorem 2.6 reduces to the following: SINR∞ LMMSE,j =

Ej Nj

where Ej

2 = C1,sum sinc2 (πδFj T )  n    o 4   E |aj,n |  2  1 2 +  C2,sum +  n o2 − 1 C1,sum  1 − sinc (πδFj T ) (2.50) M E |aj,n |2

Nj

= C1,sum sinc2 (πδFk T )     + β σ¯k2 + σz2 C2,sum 1 − sinc2 (πδFk T )    1 H 2 −1 = tr Aj Aj + σz I M    1 2 −2 = tr Aj AH + σ I j z M

C1,sum C2,sum

(2.51) (2.52) (2.53)

where the bar denotes expectation over interfering users. As expected, when the frequency offset is zero, the sinc2 (·) terms will equal 1 and SINR∞ LMMSE,j = C1,sum =

   1 2 −1 tr Aj AH + σ I . j z M

The first term of Ej is generally much larger than the second, and represents the reduction in the signal of interest due to frequency offset. The second term represents a

34

slight increase in the signal of interest which is picked up indirectly due to sidelobes and inter-carrier correlation. The first term of Nj represents interference from users which are transmitting on the same carrier. This term does not appear in OFDM or OFDMA systems in which any signal on the correctly synchronized carrier must come from the user of interest. The second term of Nj represents inter-user interference which becomes worse due to mismatch between the receiver and the channel resulting from carrier frequency offset. Next consider α = 2 and N = 2M . If the users have identical constant frequency offset, the ∆α,k terms will cancel out. The net effect of reducing carrier bandwidth by α is to reduce the frequency offset by the same factor. However this does not correspond to an α-fold increase in SINR, since SINR does not degrade linearly with frequency offset. In fact, beyond a factor of α = 2, which eliminates the effect of adjacent main lobes, further increasing α will not improve SINR significantly.

OFDMA vs. MC-CDMA with LMMSE receiver

In an OFDMA system if channel coefficients are known they can be compensated for perfectly. The system does not suffer from any multi-user interference when there are no synchronization errors. Assume that for K users in an M carrier system, each user will transmit on average β = K/M of its total transmission power on every carrier it is assigned. If there is frequency offset error, it causes interference similar to that in a single user OFDM system [115]:

SINROF DM A =

sinc2 (δFj T )  βσk2 1 − sinc2 (δFj ) + σz2

(2.54)

Consider an MC-CDMA system where there is no transmission channel, α = 1, and the spreading code is a binary random variable. As the number of carriers go to infinity, they will   2I become orthogonal and inter-user interference will go to zero. The matrix Aj AH + σ z j

35

becomes diagonal and   C1,sum = 1/ βσk2 + σz2 ,  2 C1,sum = 1/ βσk2 + σz2 Cp,2 = 0

(2.55) (2.56) (2.57)





Cn,2 = 1/ βσk2 + σz2 .

(2.58)

Substituting these variables into Equation yields an asymptotic SINR of SINRM C−CDM A =

sinc2 (δFj T ) βσk2 + σz2

(2.59)

Although the codes become orthogonal in the limit as the number of carriers go to infinity, the receiver used is a LMMSE receiver, and as a result the expected SINR starts   at 1/ βσk2 + σz2 for no frequency offset. The denominator in Equation (2.54) decreases slightly with an increase in transmission power, as in single user OFDM. This does not happen with MC-CDMA since the power spectrum of the K − 1 interfering users overlaps with the power spectrum of the user of interest. This analysis shows that in uplink communications where channel coefficients are perfectly known and compensated for, OFDMA is slightly more sensitive to frequency offset error than MC-CDMA. However, MC-CDMA suffers from higher levels of inter-user interference when there is no frequency offset, and OFDMA approaches this base level of inter-user interference.

2.4

Simulation Results and Discussion

This section describes simulations of the MC-CDMA system described in Section 2.2. In Figures 2.5-2.12, the BER and SINR at the output of the receiver averaged over the transmitted symbols is shown to converge to the expression in Theorem 2.4 for both LMMSE and MF receivers. Next, in Figures 2.13-2.24, the SINR for a given user in a given MC-CDMA system is plotted against the predictions from Theorems 2.5 and 2.6. The results show that these theorems can be used to give a good prediction of the effect of frequency offset for many, but not all users.

36

2.4.1

Outline of the Simulated System

Two systems are studied: a tightly packed, OFDM-like system (α = 1, Fig. 2.3(a)), and a system with no spectral overlap (α = 2, Fig. 2.3(c)) [98]. When the bandwidth scaling ratio is an integer, the resulting tones are orthogonal. To achieve a fair comparison, both systems are given the same bandwidth, and the same number of carriers. The former system suffers from frequency offset, while the latter is affected more by inter user interference introduced by the channel. The total power of each user is set to 1. Differential Quadrature Phase Shift Keying (DQPSK) modulation is used in both systems. The channel gains on each carrier are assumed to be independent identically distributed circular complex Gaussian random variables.

2.4.2

Time Average SINR (Theorem 2.4)

This section examines the accuracy of the result derived in section 2.3.1 for MC-CDMA with bandwidth scaling ratio α = 1 (128 carriers) and α = 2 (64 carriers). The total transmission bandwidth and the pulse shaping filters are the same for both systems, which means that the transmission rate of the first system is twice that of the second system. Figures 2.5-2.12 show the simulated and predicted SINR and BER at the output of a LMMSE and a MF receiver, in the presence of frequency offset. The SINR is predicted from Theorem 2.4. The BER at the output of the receiver is predicted using the SINR from Theorem 2.4 and Equation (2.10). Figures 2.5-2.8 show the result for MC-CDMA with a LMMSE receiver with 40 users, while Figures 2.9-2.12 compare the predicted values against simulations for a MF receiver with 10 users. Predicted values are plotted marked with “×” and simulation values are plotted as solid lines. The predicted SINR matches the actual SINR very well in all cases. The predicted BER matches the actual BER quite well for the LMMSE receiver. For the MF receiver however, when SINR is high the BERs do not match well, since the interference no longer resembles white Gaussian noise. This effect is not as pronounced for the LMMSE receiver because in the limit as the spreading code length goes to infinity, the distribution of the multiple access interference at the output of a LMMSE receiver approaches the Gaussian distribution [143][41].

37

14 SNR = 18dB

Signal to Interference and Noise Ratio (SINR)

12 SNR = 15dB 10 SNR = 12dB 8 SNR = 9dB 6

4

5

1 × 10 bits transmitted K = 10, N = 128, M = 128 Deterministic (known) frequency offset

2

−2

−1

10

10 Relative Frequency Offset (δF T)

Figure 2.5: SINR from Theorem 2.2 and simulation results. LMMSE receiver, M = 128, α = 1, K = 40.

1 × 105 bits transmitted K = 10, N = 128, M = 128 Deterministic (known) frequency offset

−1

10

Bit Error Rate (BER)

SNR = 9dB

−2

10

SNR = 12dB

SNR = 15dB

−3

10

SNR = 18dB

−4

10

−2

10

−1

10 Relative Frequency Offset (δF T)

Figure 2.6: BER from Theorem 2.2 and simulation results. LMMSE receiver, M = 128, α = 1, K = 40.

38

7

SNR = 18dB

Signal to Interference and Noise Ratio (SINR)

6.5

SNR = 15dB

6 SNR = 12dB 5.5 5 SNR = 9dB

4.5 4 3.5 3

1 × 105 bits transmitted K = 40, N = 128, M = 64 Deterministic (known) frequency offset

2.5 2 −2

−1

10

10

Relative Frequency Offset ( δF T )

Figure 2.7: SINR from Theorem 2.2 and simulation results. LMMSE receiver, M = 64, α = 2, K = 40.

Bit Error Rate (BER)

1 × 105 bits transmitted K = 40, N = 128, M = 64 Deterministic (known) freq. offset

SNR = 9dB −1

10

SNR = 12dB

SNR = 15dB

SNR = 18dB −2

10

−1

10

Relative Frequency Offset ( δF T )

Figure 2.8: BER from Theorem 2.2 and simulation results. LMMSE receiver, M = 64, α = 2, K = 40.

39

14 SNR = 18dB 12 Signal to Interference and Noise Ratio (SINR)

SNR = 15dB

10

SNR = 12dB

8 SNR = 9dB

6

4

5

1 × 10 bits transmitted K = 10, N = 128, M = 128 Deterministic (known) frequency offset

2

−2

−1

10

10 Relative Frequency Offset (δF T)

Figure 2.9: SINR from Theorem 2.2 and simulation results. MF receiver, M = 128, α = 1, K = 10.

1 × 105 bits transmitted K = 10, N = 128, M = 128 Deterministic (known) frequency offset

−1

10

Bit Error Rate (BER)

SNR = 9dB −2

SNR = 12dB

10

SNR = 15dB −3

10

SNR = 18dB

−4

10

−2

10

−1

10 Relative Frequency Offset (δF T)

Figure 2.10: BER from Theorem 2.2 and simulation results. MF receiver, M = 128, α = 1, K = 10.

40

8.5 SNR = 18dB

Signal to Interference and Noise Ratio (SINR)

8 SNR = 15dB 7.5

7

SNR = 12dB

6.5

6 SNR = 9dB 5.5

5

4.5

1 × 105 bits transmitted K = 10, N = 128, M = 64 Deterministic (known) frequency offset

4 −2

−1

10

10 Relative Frequency Offset (δF T)

Figure 2.11: SINR from Theorem 2.2 and simulation results. MF receiver, M = 64, α = 2, K = 10.

1 × 105 bits transmitted K = 10, N = 128, M = 64 Deterministic (known) frequency offset −1

10

Bit Error Rate (BER)

SNR = 9dB

SNR = 12dB

SNR = 15dB

SNR = 18dB

−2

10

−1

10 Relative Frequency Offset (δF T)

Figure 2.12: BER from Theorem 2.2 and simulation results. MF receiver, M = 64, α = 2, K = 10.

41

2.4.3

Asymptotic SINR (Theorem 2.5, Theorem 2.6)

Since it has been established that Theorem 2.4 can be used to represent the average SINR at the output of a linear receiver for a given user, this expression will be used to evaluate the results of Theorem 2.5 for the MF receiver, and Theorem 2.6 for the LMMSE receiver. These theorems show the asymptotic SINR at the output of a MC-CDMA system as the number of carriers and the number of users go to infinity, while the ratio between the two remains constant. Figures 2.13-2.18 show the asymptotic degradation in SINR with frequency offset for the LMMSE receiver, and the actual degradation for 100 different users in MC-CDMA systems with random channel coefficients. There are 128 carriers and 40 users in the system. In Figures 2.13-2.15, α = 1, and every subcarrier is used for data transmission, while in Figures 2.14-2.18, α = 2, and every other subcarrier is used. In all cases, the SINR is centered around the asymptotic value. The variance around the mean is greatest for the Gaussian distributed frequency offset, and lowest when the frequency offset is constant for all users, as can be inferred from Theorem 2.3. The variance is larger in Figures 2.14-2.18, and there is one user with a degradation much less than the mean. As N increases, the probability of such an outlier goes to zero. Figures 2.19-2.24 show the asymptotic degradation in SINR for a MF receiver, and the actual degradation for 100 users. This system also has 128 carriers but the number of users is reduced to 10, which reduces the inter-carrier interference to more reasonable levels. Although the average SINR at the output of the MF receiver is less affected by frequency offset, the effect varies more for individual users. This is partly because the number of users is much fewer, and partly because the SINR at the output of the MF receiver is more dependent on the effective spreading code than that of the LMMSE receiver. 2.5

Conclusions

This chapter presented an analytical expression for the SINR at the output of the detector of an uplink communication system. Simulations have shown that the effect of frequency offset for an uplink system depends more on the standard deviation of the frequency offset

42

predicted limit, N −> ∞ single user, N = 128

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 128 Deterministic (known) frequency offset

−3

10

−2

−1

10

10 Relative Frequency Offset ( δF T )

Figure 2.13: Degradation in SINR at the output of the LMMSE receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

predicted limit, N −> ∞ single user, N = 128

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 128 Uniform distributed frequency offset , averaged over 20 samples

−3

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.14: Degradation in SINR at the output of the LMMSE receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

43

predicted limit, N −> ∞ single user, N = 128

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 128 Gaussian distributed frequency offset , averaged over 20 samples

−3

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.15: Degradation in SINR at the output of the LMMSE receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 64 Deterministic (known) frequency offset −3

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.16: Degradation in SINR at the output of the LMMSE receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

44

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 64 Uniform distributed frequency offset , averaged over 20 samples −3

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.17: Degradation in SINR at the output of the LMMSE receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64

1

Degradation in Signal to Interference and Noise Ratio (SNR−SNR0)

10

0

10

−1

10

−2

10

K = 40, N = 128, M = 64 Gaussian distributed frequency offset , averaged over 20 samples −3

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.18: Degradation in SINR at the output of the LMMSE receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

45

predicted limit, N −> ∞ single user, N = 128 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 128 Deterministic (known) frequency offset −4

10

−2

−1

10

10 Relative Frequency Offset ( δF T )

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

Figure 2.19: Degradation in SINR at the output of the MF receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

predicted limit, N −> ∞ single user, N = 128 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 128 Uniform distributed frequency offset , averaged over 20 samples −4

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.20: Degradation in SINR at the output of the MF receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

46

predicted limit, N −> ∞ single user, N = 128 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 128 Gaussian distributed frequency offset , averaged over 20 samples −4

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

Figure 2.21: Degradation in SINR at the output of the MF receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 128, α = 1, K = 40.

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 64 Deterministic (known) frequency offset −4

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.22: Degradation in SINR at the output of the MF receiver, fixed frequency offset for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

47

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 64 Uniform distributed frequency offset , averaged over 20 samples −4

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Degradation in Signal to Interference and Noise Ratio (SNR − SNR0)

Figure 2.23: Degradation in SINR at the output of the MF receiver, uniformly distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

predicted limit, N/M = 2 and N,M −> ∞ single user, N = 128, M = 64 0

10

−1

10

−2

10

−3

10

K = 10, N = 128, M = 64 Uniform distributed frequency offset , averaged over 20 samples −4

10

−2

10

−1

10 Relative Frequency Offset ( δF T )

Figure 2.24: Degradation in SINR at the output of the MF receiver, Gaussian distributed frequency offset, different for all users, from Theorem 2.2 and Theorem 2.6. M = 64, α = 2, K = 40.

48

than on the particular distribution. Asymptotic expressions have been derived for the SINR at the output of a LMMSE and a MF receiver. Simulations show that while the average SINR approaches this asymptotic value, the SINR for a given user can vary greatly from the mean. The actual spreading gain for a user has more effect on the SINR at the output of the MF receiver than that of the LMMSE receiver. The channel model used in simulations and the derivation of Theorems 2.5 and 2.6 is a simplified model, recent work has shown that similar results can be derived for an MCCDMA system with random spreading codes in a multipath channel [69] [83]. The analysis by Li et.al. [69] may be used to generalize the above results to a channel which is correlated in frequency. The results of this chapter show that while frequency offset error does affect the SINR and BER at the output of a linear receiver, in the downlink scenario where different users see different channels, the inter-carrier interference within the users’ own signal is just as bad as inter-user interference between different users. Even when the users see different frequency offsets, when the LMMSE receiver is used, the output depends only on the mean square offset. This result seems to favor the use of an OFDMA system, where instead of transmitting across all carriers using a fixed code, users transmit on a few chosen carriers. OFDMA allows for multiple-access interference between users to be minimized, since no two users transmit on the same carrier. The preceding analysis shows that the effect of frequency offset on an OFDMA system will not be significantly worse than the effect on an MC-CDMA system, even though the user may not use carriers which are consecutive in frequency, and may receive more interference from other users than from its own signal. The next chapter will describe an OFDMA system, and develop a group of algorithms for the allocation of subcarriers among users in an OFDMA system.

49

Chapter 3 RESOURCE ALLOCATION FOR ORTHOGONAL FREQUENCY DIVISION MULTIPLE ACCESS (OFDMA)

As mobile communication becomes more common, there is a need for power-saving algorithms which extend battery life for devices while retaining a guaranteed quality of service. Multiuser power loading and resource allocation strategies allow available resources to be used more efficiently. The previous chapter described the effect of frequency offset in an MC-CDMA system. In this chapter, multiuser allocation is explored in the context of an OFDM based frequency division multiple access (OFDMA) system. An OFDMA system is defined as one in which each user is assigned a subset of the subcarriers for use, and each carrier is assigned exclusively to one user. The aim of this chapter is to introduce the problem of bandwidth and power allocation among users in an OFDMA system to minimize the total transmission power while maintaining a minimum quality of service. This chapter begins with an introduction, and an overview of prior work. In the next section, the system model used is described, and the problem is formulated. Section 3 describes some of the previous work in this area. In Section 4, a new class of algorithms, described as the Sensible Greedy algorithms are introduced. These algorithms are compared to those described in Section 5 analytically, in terms of computational complexity. In Section 6, the simulation model is introduced, and in Section 7 the algorithms are compared through simulations. The final section concludes the chapter. 3.1

Introduction

One of the biggest advantages of OFDM systems is the ability to allocate power and rate optimally across frequency, using “waterfilling” over the inverse of the channel spectrum. Computationally efficient algorithms exist to perform discrete waterfilling for single user communication [19]. Wahlqvist et.al. [136] was one of the first to show that dynamic resource allocation can

50

improve quality of service. Yu and Cioffi [141] calculate the capacity region for an OFDMA system with two users. Jang and Lee [53] show that a multiuser waterfilling algorithm which maximizes the transmission rate in an OFDMA system will assign each carrier to the user with highest gain on it. Similar results have been found for TDMA systems by Knopp and Humblet [61], and for the power allocation which maximizes the capacity of a general multichannel, multiuser system by Tse and Hanly [129]. However this algorithm does not support minimum rate requirements for individual users. The first algorithm for resource allocation was introduced by Rohling and Grunheid [100]. They present a simple heuristic greedy algorithm, and show that it performs better than simple banded OFDMA. Several problems have been studied, including maximizing the number of users with equal rate requirements [1], maximizing the minimum transmission rate for users with limited power [97]. Hoo et.al. have studied high speed cable systems, where all users see the same gain on all the carriers. The capacity of the system was calculated [44], and both optimal and suboptimal algorithms have been developed [45]. Efforts to exploit the full extent of centralized resource allocation on the wireless uplink channel prove to be computationally hazardous. Smaller, simplified system can be solved using integer programming [56], and a branch and bound algorithm has been proposed in this chapter which finds the best allocation of subcarriers among users much more efficiently than the use of a standard integer programming suite [74]. Unfortunately, this algorithm is still computationally prohibitive. An innovative technique, introduced by Wong et.al. [137] applies Lagrangian relaxation (LR) to this problem. In Lagrange relaxation, the Lagrange method of optimization is used on an integer parameter, which is “relaxed” to take on non-integer values. In this case the subcarrier assignment function ρk (n), which yields 1 when a user k is assigned subcarrier n and 0 otherwise, is allowed to take on any value between 0 and 1. Despite the significant gain over fixed assignment strategies, the algorithm is computationally intensive and is difficult to implement. A modified version of the algorithm proposed by Wong et.al. is described in this chapter, and shown to converge to values which are close to optimal. Although the modified Lagrangian Relaxation algorithm is much faster than the branch and bound algorithm, it is still prohibitively expensive for implementation.

51

More recent work focuses on reducing computational load of resource allocation by starting with a simple solution and iteratively updating to satisfy constraints. The simplest of these by Saulnier and Seyedi [109] introduces a distributed algorithm where users drop their worst carrier and pick up a random new carrier in each time period. Pietrzyk and Janssen [84] reduce the computational load by initializing with simple algorithms, and implementing the Lagrangian relaxation algorithm to iteratively update the allocation. Munz et.al. [80] introduce a multiuser waterfilling algorithm which estimates the Lagrange “cost” variables. Ergen et.al. [31] introduce an adaptive set of algorithms that consist of initialization and update steps to reallocate carriers among users. The main result of this chapter is a class of algorithms which have been proposed for power allocation and subcarrier assignment. These algorithms achieve comparable performance to the LR algorithm but do not require intensive computation. A single cell with one base station and many mobile stations is considered. The algorithms assume perfect information about the channel state due to multipath fading as well as path loss and shadowing effects, and the presence of a medium access protocol to convey information about channel state and subcarrier allocation between the base station and the mobile stations [21]. 3.2

System Model and Problem Formulation

This section introduces the system model that is used, and formally introduces the problem being addressed in this chapter. The criteria used for quality of service are introduced, and the power loading method is formulated. The system is shown in Figure 3.1.

3.2.1

Quality of Service, Bit Error Rates and the Rate-Power Function

Quality of Service (QoS) is a concept that tries to quantify the overall satisfaction of a user with the experience of using a communication system. In this chapter, the following factors are used to define the quality of service experienced by a user:

• The Transmission Bandwidth, which affects how quickly a file downloads or how much resolution a video transmission can show,

52

• The Bit Error Rate (BER), which affects whether the file downloaded has errors in it, or whether the video can be decoded at the receiver.

• The Availability of Service, whether a user who requests bandwidth can actually be allocated it, or whether they have to wait or give up. k , while the The minimum transmission bandwidth required by user k is represented as Rmin

maximum BER the user is able to tolerate is represented as Pe . The availability of service in k the system, Pout , is evaluated by measuring the probability that a user will not receive Rmin k bandwidth in simulations. Since Rmin is the minimum rate requirement for user k, users

who are not allocated this bandwidth are dropped from the system and do not transmit. The maximum bit error rate and minimum transmission rate achievable in the system depend not only on the available bandwidth but also on the transmission power. When bandwidth is available, the BER, transmission rate and transmission power on a single carrier are related by the rate-power function: P = f (R, Pe ). The rate-power function f (·, ·) describes the mathematical relationship between the minimum BER that can be tolerated, Pe , the transmission rate, and the minimum power requirement. The rate function is determined by the available coding and modulation schemes. In general, it is a discontinuous function, since the number of coding and modulation schemes applicable are finite. Figure 3.2 shows the function used in simulations and the continuous function approximation used for some of the algorithms. A subcarrier can transmit at most Rmax bits per unit time. The remainder of this section describes error control coding and modulation methods that are used in this chapter, and explains the rationale behind the continuous rate-power function used in algorithms. Table 3.2 summarizes the available transmission rates.

Convolutional Encoding The error control code used in this system is a Rate Compatible Punctured Convolutional (RCPC) code [105, 35]. A punctured convolutional code is a convolutional code where every pth bit of the coded sequence is dropped, or not transmitted. Rate Compatible Punctured Convolutional (RCPC) codes are sets of codes that are derived from the same convolutional

DATA OUT

DATA IN

Carrier n

Viterbi Decoder

Rate 1/2 Convolutional Encoder

Single User Power, Rate Allocator

QAM Demodulator

Modulation Rate Selector

QAM Modulator

Carrier n Channel State Estimator

Channel Estimation Data from N carriers

Deinterleaver

Figure 3.1: System model for OFDMA.

Carrier Allocation from Base Station

Inserter

Code Rate Selector

Deleter

Interleaver

OFDM Demodulator

Channel

OFDM Modulator

Modulated Signal from N carriers

53

54

encoder, but with different puncturing matrices. The process of puncturing provides an easy way for the receiver to change the coding rate without modifying the encoder or decoder. Good patterns for perforation are mostly found by searching over all sets of puncturing patterns for some criterion. RCPC codes have been used as part of the standard for the IEEE 802.11a Local Area Network (LAN) system [48]. In this dissertation, a convolutional encoder with generator polynomials g0 = 1338 and g1 = 1718 , is used [139, 48]. The code has rate R = 1/2, constraint length K = 7. It is punctured with period p = 2 to yield a rate R = 2/3 code with free distance df ree = 6, and with puncturing period p = 3 to yield a rate R = 2/3 code with free distance df ree = 5. The sum of bit errors (the information error weight) for error events of distance j (i.e. jbit errors prior to decoding) for a given code are known as the distance spectra of that code, and are represented as cj . For the codes used in this dissertation, the distance spectra cj , and the number of error paths of distance j, aj , are given in Table 3.1. These coefficients are important in finding the probability of error after decoding for a given probability of error after demodulation of the received signal [89, p.487]. An upper bound on the probability of error in Gaussian channels, or assuming infinite interleaving, can be calculated using the transfer function of the code:

∞ 1 X cj Pj PB ≤ l j=df ree

where Pj is the probability that an incorrect path with a Hamming weight j is selected in the Viterbi decoding process, and depends on the modulation scheme used as well as the channel fading process. In this dissertation, the BER at the output of the decoder was approximated by: 1 PB ≈ cdf ree P2 . l More sophisticated models can be found using simulations [35] and theoretical analysis [6].

Modulation The process of digital modulation involves mapping bits to a constellation of symbols. The digital symbols are then modulated to the carrier frequency and transmitted across the

55

Table 3.1: Transfer function coefficients for chosen RCPC codes. Rate

Spectral Coefficients (ad , d = df ree , df ree + 1, ..., df ree + 9) [cd , d = df ree , df ree + 1, ..., df ree + 9]

1/2

(11, 0, 38, 0, 193, 0, 1331, 0, 7275, 0, 40406, 0, 234969, 0, 1337714, 0, 7594819, 0, 43375588, 0) [36, 0, 211, 0, 1404, 0, 11633, 0, 77433, 0, 502690, 0, 3322763, 0, 21292910, 0, 134365911, 0, 843425871, 0]

2/3

(1, 16, 48, 158, 642, 2435, 9174, 34701, 131533, 499312) [3, 70, 285, 1276, 6160, 27128, 117019, 498835, 2103480, 8781268]

3/4

(8, 31, 160, 892, 4512, 23297, 120976, 624304, 3229885, 16721329) [42, 201, 1492, 10469, 62935, 379546, 2252394, 13064540, 13064540, 75080308, 427474864]

channel. In this dissertation, the modulation methods considered are BPSK, QPSK, 16QAM, and 64-QAM. It is assumed, as in the previous section, that the total noise and interference term is normally distributed. The formulas for the BER at the output of a QAM system in white Gaussian noise are well known [89].

Rate-Power Function Table 3.2 lists the modulation schemes and coding rates available. Users are only allowed to transmit at these given bit rates. The discontinuous nature of the rate-power function makes the problem of subcarrier allocation even more difficult. While it is possible to work with the staircase rate-power function directly [56], many algorithms, including the sensible greedy algorithms, simplify the problem by assuming that users can transmit at any rate between the minimum and maximum transmission rates listed in the table. The Lagrange Relaxation and BABS algorithms further require that the continuous rate-power function be continuous, convex and differentiable. For these algorithms, a continuous ratepower function has been defined using cubic regression. The rate-power function used in

56

Table 3.2: Modulation constellations and coding rates available for transmission, with 128 carriers. Information

Information bits

Modulation

Coding rate

bits per carrier

per OFDM symbol

BPSK

1/2

0.5

64

BPSK

2/3

0.67

86

BPSK

3/4

0.75

96

QPSK

1/2

1.0

128

QPSK

2/3

1.33

156

QPSK

3/4

1.50

192

16-QAM

1/2

2.0

256

16-QAM

2/3

2.67

341

16-QAM

3/4

3.0

384

64-QAM

1/2

3.0

384

64-QAM

2/3

4.0

512

64-QAM

3/4

4.5

576

90

64 64

80

(R

=3

(R

=2

70

/4 )

/3

16 60

SNR

50

16

40

M

−Q AM

30

QP 20

SK

QP

BP SK

SK

/2

/2

0 0

0.5

/4 )

(R =

2/

(R

3)

=2 /3 )

=1

=1

(R =3

)

=1 /2 )

(R

(R

(R

−Q AM

−Q A

16

10

−Q AM

−Q AM

)

) 1

1.5 2 2.5 3 3.5 Transmission rate per subcarrier

4

4.5

Figure 3.2: Continuous approximation to the rate function.

57

this dissertation, for the modulation and coding mechanisms in Table 3.2 and a minimum admissible Bit Error Rate (BER) of 1 × 10−4 is shown in Figure 3.2, together with the continuous rate-power function. When Phase Shift Keying (PSK) modulation is used, it is customary to use the approximation [14]: p(n)|H(n)|2 r(n) = log 1 + 2Γn σn 

 (3.1)

where Γn is known as the SNR gap. It is defined as the gap between the ideal channel capacity and the bandwidth efficiency of a real modulation scheme:    1 SERn 2 Q−1 . Γn = 3 4

(3.2)

An expression for the SNR gap for QAM signals has also been developed [2], however it is less accurate. In this dissertation it is assumed that both coding rate and modulation scheme can be different on different carriers. This makes finding an analytical expression for a continuous approximation difficult. Instead, a best-fit polynomial model was used as a continuous ratepower function. Initially, the goal was to find a 3rd degree polynomial which would give the best fit to the discrete rate-power function, under the condition that the polynomial function is convex and increasing, and f (0, Pe ) = 0. It was found that for the modulation and coding schemes used, and minimum error rates between 1 × 10−4 and 1 × 10−6 , best fit polynomials had negative coefficients for both the linear and quadratic terms. This is inconsistent the requirement that the function is increasing and convex for all r. So a polynomial of the form f (R, Pe ) = A(Pe ) r3

(3.3)

was chosen to be the continuous rate-power function, where A(Pe ) is found using least squares regression with the discrete set of rate-power mappings as the data. Another advantage of this approximation is that it provides a convenient way to represent users with different BER requirements using the same rate-power function, using the property that s f (r, Pe,1 ) = f

3

A(Pe,2 ) r, Pe,2 A(Pe,1 )

! .

58

For instance if most of the users in the system require a minimum probability of error of Pe = 1 × 10−3 and f (r, 1 × 10−3 ) = A(1 × 10−3 ) r3 , while user k requires a minimum probability of error Pe = 1 × 10−6 and f (r, 1 × 10−6 ) = A(1 × 10−6 ) r3 , then user k can use p the rate-power function for Pe = 1 × 10−3 but request 3 A(1 × 10−6 )/A(1 × 10−3 ) times its minimum rate requirement. Algorithms which use the minimum rate-power function do not need to be modified to allow for the new user’s more stringent error requirement.

3.2.2

Problem Formulation

The system under consideration is an OFDM system with frequency division multiple access (FDMA). Perfect channel state information is assumed at both the receiver and the transmitter, i.e. the channel gain on each subcarrier due to path loss, shadowing, and multipath fading is assumed to be known. Channel parameters are assumed to be perfectly estimated by a channel estimation algorithm for uplink OFDMA [66][90][110]. The design and performance of this algorithm is outside the scope of this dissertation. The system does not employ spreading in either time or frequency, and each subcarrier can only be used by one user at any given time. Subcarrier allocation is performed at the base station and the users are notified of the carriers chosen for them. After the allocation, each user performs power allocation and bit loading across the subcarriers allocated to it to find the transmission power. Consider a system with K users, and N subcarriers. Each user k must transmit at least k Rmin bits per unit time. Let Hk (n) be the channel gain, pk (n) the transmission power and

rk (n) the transmission rate for user k on subcarrier n. The quantities are related by a function f (rk (n)) = pk (n)|Hk (n)|2 as shown in Figure 3.2. The objective is to find a subcarrier allocation which allows each user to satisfy its rate

59

requirements while using minimum power: min

PT =

N X K X

pk (n)

(3.4)

n=1 k=1 N X

s.t.

n=1 K X

k rk (n) ≥ Rmin ,

k = 1, ..., K,

(3.5)

δ [rk (n)] ≤ 1

n = 1, ..., N.

(3.6)

k=1

where δ [.] is the Kronecker delta function and pk (n) =

3.3

1 f −1 (rk (n)). |Hk (n)|2

(3.7)

Some Previous Approaches

This section describes several algorithms derived from previous work on resource allocation for OFDMA. The first algorithm is fixed assignment, which does not take into account channel information. Next, Rohling and Grunheid’s the SSFA algorithm [100] is described. The Lagrange Relaxation Algorithm was introduced by Wong et.al. [137], some modifications were introduced to this algorithm in this dissertation, and the resulting algorithm is called the Modified Lagrange Relaxation (MLR) algorithm. The MLR algorithm reduces computational complexity and ensures that rate requirements are met in a system with a discrete rate-power function. Finally, using the dynamic programming approach introduced by Inhyoung Kim et.al. [56], a branch and bound algorithm is designed for power and carrier allocation.

3.3.1

Fixed assignment methods

The simplest approach to subcarrier assignment is to ignore channel information, and allocate carriers to users proportional to their rate requirements. Subcarriers can be allocated in consecutive chunks (bands) or interleaved to improve frequency diversity. While this method is the easiest to implement and requires minimum communication between the mobile station and the base station, it also leads to high minimum power requirements [100].

60

3.3.2

Single Step Frequency Allocation (SSFA) Algorithm

The centralized frequency allocation algorithm proposed by Rohling and Grunheid [100] will be referred to as the Single Step Frequency Allocation (SSFA) algorithm hereafter. k . The base station In this algorithm, the user requests Nk carriers, proportional to Rmin

first makes a list Vk of favorite subcarriers for each user k. In each stage, a subcarrier is allocated to the user with the lowest ratio of allocated to requested carriers, going down the favorites list for the user. For each user k, there is also a list Ak of the nk previously allocated carriers and the Nk − nk carriers that could still potentially be allocated. When a user requests a carrier that is already allocated, the carrier is given to the user with the highest accumulated relative power loss. The power lost by user k from not getting a subcarrier Vk (i) is defined as L(Vk (i), Vk (j)) = |Hk (Vk (i))|2 − |Hk (Vk (j))|2 where Vk (j) is the carrier user k will get instead of Vk (i). The accumulated power loss hkacc is the sum of L(Vk (i), Vk (j)) for all carriers lost by user k until that stage, and the accumulated relative power loss is defined as |Hk (Vk (i))|2 − |Hk (Vk (j))|2 + hkacc P = Pnk 2 n=1 |Hk (Ak (n))| k

(3.8)

A similar algorithm was introduced Rhee and Cioffi [97], to maximize the minimum transmission rate per user subject to constraints on maximum transmission power. Their algorithm uses a slightly different cost function, but does not allow carriers to be reassigned to another user once they have been allocated. This improves the algorithm in terms of computational complexity, but the average transmission power is slightly higher than the SSFA algorithm.

3.3.3

The Modified Lagrange Relaxation (MLR) Algorithm

The mixed integer optimization problem in (3.4) can be solved using the Lagrange Relaxation algorithm [137]. The algorithm approaches the solution by slowly increasing the power level for each user. Each user is given a power coefficient λk , which determines their transmit power. This is not a cap on the total power allocated to the user, but can be interpreted as the “water level” in single user waterfilling. λk has the dual role of regulating both the

61

subcarrier allocation and the total transmission power for each user. The algorithm iterates, by incrementing λk by ∆λ for the user who needs the rate increase the most, reassigning channels and finding the new rates [137]. Although this algorithm is iterative, it converges to a good solution. However, due to the nonlinear nature of the integer problem, the LR algorithm requires a large number of iterations to converge. When the algorithm is terminated after a fixed number of iterations, even when the number of iterations is large, the resulting solution is not close to the final result since the convergence to the optimal result is not smooth. The accuracy of the result and the speed of convergence can be controlled by varying the increment ∆λ. For small ∆λ the convergence is slow but the result is accurate, for larger ∆λ there is a higher probability of outage and the optimal solution cannot be found. The computational complexity is inversely proportional to the accuracy of the final solution. In the worst case, it can take up to K × λmax /∆λ iterations to converge. Several modifications were made to this algorithm which allow faster convergence and significantly reduce the outage probability for a given set of users: • In the new implementation, the minimum number of carriers required by each user are calculated at the beginning of the program. If there are not enough carriers to support all the users, the users who require the most bandwidth will be dropped to reduce the number of iterations of the algorithm. When users can not be accommodated, they will continue raising the water level until they top out, this will result in at least λmax /∆λ extra iterations. • All function calls are implemented through table lookup. Since calculating λ, the “water level” in each iteration requires taking the cube root (for the rate-power function considered in this dissertation) or logarithm (for the rate-power function in the original article by Wong et.al. [137]), this speeds the program up considerably. In addition, it allows the programmer to control the trade-off between faster program execution and more precise output. • In every iteration of the algorithm, the chosen user is made to release all “shared”

62

carriers in which it is the second user. This reduces the number of shared carriers in the relaxed solution, and thus the probability of outage. • The original algorithm has two modes. In the first mode, users are initially allowed to share carriers, to determine a solution to the relaxed problem. When the relaxed solution is mapped to a frequency allocation without shared carriers, the allocation is made to the user with the highest stake in the carrier (largest ρ). In the second mode, users are not allowed to share carriers at all. It turns out, however, that both algorithms can result in a rather high probability of outage. In the first case, this occurs because users who share all their carriers may be left with too few to satisfy their rate requirements when the integer constraints are invoked. In the second case, the step size ∆λ may sometimes be too large, and the algorithm may step over the optimal solution. In the new implementation, the assignment does not allow users to share carriers during the iteration phase, but does keep track of an alternative user for the carrier. After the algorithm converges to a solution, the algorithm reassigns subcarriers to ensure that all users have enough subcarriers to satisfy their minimum rate requirements. This step takes very little time, compared to the total execution time of the algorithm. With the modifications above, the MLR algorithm performs as well as the algorithms proposed below in terms of outage probability and around the same or slightly better in terms of total power consumption in scenarios of interest. This improvement comes at a price, however. Consider the case when the users see the same channel profile and request the same minimum rate. In this case, the best carrier allocation assigns the carrier with the highest gain to one user, the carrier with the second highest gain to the next user, and so on. But since channel sharing is now limited, the MLR algorithm will yield an allocation where the carriers are ranked according to gain, and each user receives a consecutive group of carriers. For the system in this dissertation, carrier gains are assumed to be independent. When the number of carrier is large, the probability

63

Table 3.3: Effect of lookup table size (Ntable ) on probability of outage for the MLR algorithm.

Pout for LR algorithm

Pout for MLR algorithm

200

0.090844

0.0085138

500

0.037324

0.0013671

1000

0.018830

0.00034194

2000

0.0094574

8.5496 × 10−5

5000

0.0037932

1.3680 × 10−5

Ntable

of having a number of correlated carriers is small. Nonetheless, when the number of carriers and the number of users is small, a small group of correlated carriers can cause a significant deviation from the best allocation, as shown in Figure 3.23. The computational complexity of the modified LR algorithm is still very high. It is shown in Section 3.5.2 of this chapter that the algorithm is O(KN Ntable ), where Ntable is the number of elements in the lookup table used to calculate the function Hf (·) defined in Table C.1. Table 3.3 shows the effect of table size on the probability of outage at the output of the original LR and the modified LR algorithms for a system with N = 128 carriers and K = 20 users. The table shows that the modifications made to the Lagrangian relaxation algorithm result in an order of magnitude reduction in outage probability. The step size ∆λ is determined from Ntable by using the maximum transmission rate per carrier (4.5 bits per carrier) and the minimum transmission rate per carrier (0.5 bits per carrier) to find the minimum and maximum feasible values for Hf (·). A value of Ntable = 500 was chosen for simulations, as this seems to give the best trade-off between outage probability and computational complexity.

3.3.4

Integer Programming and the Branch and Bound Algorithm

The integer programming formulation proposed by Inhyoung Kim et.al. [56] searches all possible assignments to find the global optimum frequency and power assignment. With a simplified rate-power function, they present results for a system as large as 20 carriers

64

Algorithm 3.1: Branch and Bound Algorithm 1:

Initialize M AX COST to cost incurred in BABS-RCO algorithm, initialize BestAlloc to the allocation.

2:

Form the array f avorites, where f avorites(k, n) is the index of the nth highest gain

(favorite) channel of user k. l k m R 3: Nmin (k) ← R min l max m Rk 4: Nmax (k) ← Rmin min 5:

L0 .assignment ← {X, X, · · · , X}

6:

L0 .Nempty ← N

7:

L0 .nextnode ← f avorites(1, 1)

8:

LEAF N ODES ← L0

9:

for L ∈ LEAF N ODES do

10:

if min cost(L) > M AX COST then LEAF N ODES ← LEAF N ODES − L

11: 12:

else if L.Nempty = 0 then

13:

M AX COST ← min cost(L),

14:

BestAlloc ← L

15:

else

16:

LEAF N ODES ← LEAF N ODES − L

17:

for k = 1 : K do,

18:

Lchild .assignment ← L.assignment

19:

Lchild .assignment(L.nextnode) ← k

20:

Lchild .Nempty ← L.Nempty − 1

21:

Lchild .nextnode ← f avorites((k + 1) mod K, 1)

22:

LEAF N ODES ← LEAF N ODES ∪ Lchild end for

23: 24: 25:

end if end for

65

and 5 users. Simulation experiments using the setup described in this dissertation, and the GNU Linear Programming Kit (GLPK) [74] did not yield results in a reasonable length of time (over a month) for a system with only 16 carriers and 4 users. Smaller problems could be solved, but did not yield data that could help in a meaningful comparison of the different methods, for very small problems all methods work well. The execution time varied greatly depending on the particular channels and rate distributions which were chosen. This variance in the behavior of search methods for integer programming problems is well documented in the literature, and is a subject of research in the field of artificial intelligence [43]. While the integer programming formulation presented by Inhyoung Kim et.al. [56] is feasible for some systems, it does not work for the system described in this dissertation. One algorithm to solve a mixed integer programming problem like carrier assignment is the branch and bound algorithm [22]. The branch and bound algorithm works by dividing the set of all possible solutions into P subsets. The subsets are then evaluated to find the minimum and maximum cost of a solution which belongs in that subset. If the minimum cost for a subset is greater than the maximum cost for another subset, then the global minimum cost solution does not belong to that subset, and that branch of the solution tree is “bound” or removed from consideration. When all subsets that can be bound have been removed, one of the remaining branches is “branched” into smaller subsets. This continues until there is only one set, containing only the minimum cost solution remaining. The critical part of designing a branch and bound algorithm for a particular problem is to find good strategies for branching and for bounding. Algorithm 3.1 describes the branch and bound algorithm that has been implemented in this dissertation. The variable M AX COST keeps track of the best solution to the carrier assignment problem (the maximum value for the minimum cost). Nmin (k) is the minimum number of carriers that needs to be allocated to user k to satisfy minimum rate requirements, while Nmax (k) is the maximum number of carriers it can use. The variable M AX COST is initialized using the BABS-RCO algorithm, which allows the bounding of the search tree to begin early in the algorithm. Next, the carriers are ranked according to channel gain for each user. The array of rankings is called f avorites, element f avorites(k, 1) is the carrier user k sees the highest gain on.

66

Every node in the branch and bound tree represents partial allocation. At the top level, none of the carriers have been allocated to users. All nodes at level p represent partial allocations where p carriers have been allocated, N − p carriers are still vacant. A node is “branched” into up to K new nodes by considering the allocation of one additional carrier to one of K possible users. After branching, the minimum possible cost at any child node is calculated by summing the minimum cost for all K users. The minimum cost for user k is found by using all Nk carriers that have already been allocated to user k, and continuing to add the most f avorite carriers that have not been allocated until the number of carriers reaches Nmax (k) or there are no more unallocated carriers left, and using single user waterfilling over this set. The carrier from which a node will branch can be different in every node, and is set when the node is created. The next branching carrier is initialized to the most “favorite,” unallocated carrier for some user k such that the number of carriers allocated user k is less than its maximum Nmax (k). This branching strategy is aimed to avoid a specific problem that occurs due to the discrete constellation space, whereby a branch containing the optimal solution is bounded because a user has been allocated a very low gain carrier which would otherwise remain unallocated. Then the minimum cost calculated for this node will be greater than the actual minimum cost, since the user given the node is stuck transmitting on it, instead of on a higher gain carrier which has not yet been allocated. With the branching strategy described above, when such a node is bounded, there is always a better solution in a competing branch in which the low gain carrier is allocated to the user whose f avorite list determined the choice to branch on this carrier. The order in which the search is conducted – depth first or breadth first – is determined by the order in which the nodes are processed in the for loop on Line 9 of Algorithm 3.1. In general depth first search works better, since breadth first search uses too much memory. It is also advantageous to begin searching the tree by going down the branches to the initial solution calculated using BABS-RCO or some other very good estimate of the best allocation. The optimal solution generally lies in the neighborhood of the estimate, and beginning the search there reduces M AX COST , leading to faster pruning of the search tree.

67

3.4

The Sensible Greedy Approach

The problem posed by Equation (3.4) is computationally intractable, and as described above, a direct approach to solving it does not yield a good algorithm. This section examines two algorithms which use information about users’ channel and rate requirements to find a close approximation to the solution. Intuitively, the problem is separated into two stages:

1. Resource Allocation: Decide the number of subcarriers each user gets, its bandwidth, based on rate requirements and the users’ average channel gain.

2. Subcarrier Allocation: Use the result of the resource allocation stage and channel information to allocate the subcarriers to the users.

By solving these subproblems separately a good, but not necessarily optimal, solution is found which guarantees a certain level of service for each user.

3.4.1

Resource Allocation Algorithm

In a wireless environment, some users will see a lower overall SNR than other users. These users tend to require the most power. Studying the subcarrier allocations from the LR algorithm shows that once users have enough subcarriers to satisfy their minimum rate requirements, giving more subcarriers to users with lower average SNR helps to reduce the total transmission power. This section describes the Bandwidth Assignment Based on SNR (BABS) algorithm which uses the average SNR of a user to decide the number of subcarriers assigned to it. Consider the problem described in Section 3.2.2, but assume that each user k experiP  N 2 /N on every subcarrier, and the maximum ences a channel gain of Hk = |H (n)| k n=1 transmission rate per carrier is Rmax . Let user k be allocated mk subcarriers.

68

This section studies the following problem: K X mk

min

k=1 K X

s.t.

Hk

k f (Rmin /mk )

(3.9)

mk = N

(3.10)

k=1

 mk ∈

  k Rmin , ..., N Rmax

1≤k≤K

(3.11)

The objective is to find a set of K feasible bandwidth allocations, defined below: Definition 3.1. Let the bandwidth allocation, [mk ], be the set of integers {mk : 1 ≤ k ≤ K}, where mk represents the number of carriers assigned to user k. Let F denote the set of feasible bandwidth allocations, i.e. [mk ] ∈ F if and only if [mk ] satisfies Equations (3.10)-(3.11). When the gain on every subcarrier is the same, the optimal rate-power allocation k /m bits on each subcarrier, resulting in total transmission power is to transmit Rmin k k /m )/H . Define the transmission power when m carriers are allocated to user mk f (Rmin k k k

k: mk f Gk (mk ) = Hk



k Rmin mk

 (3.12)

Whenever a new carrier is allocated to user k, the power received at the base station from user k is reduced by by Fk (mk ) = Hk Gk (mk + 1) − Hk Gk (mk ).

(3.13)

To find the optimal distribution of subcarriers among users given the flat channel assumption, a greedy descent algorithm is proposed, similar to discrete waterfilling. The Bandwidth Assignment Based on SNR (BABS) algorithm is depicted in Algorithm 3.2. In Appendix C Section C.2, this algorithm is shown to converge to the distribution of subcarriers among users that solves (3.9), and thus (3.4) for the special case when the subcarriers for a given user all experience identical fading gains |Hk (n)|2 = Hk ∀k, n given the following assumptions:

69

Algorithm 3.2: BABS Algorithm 1:

mk ←

2:

while

k Rmin Rmax , PK k=1 mk

m

l

k = 1, .., K. > N do

3:

k ∗ ← arg min mk ,

4:

m

1≤k≤K

k∗

← 0,

end while PK 6: while k=1 mk < N , do  k  Rmin 7: Gk ← mHk +1 f (mk +1) − k 5:

8:

l ← arg min Gk ,

9:

ml ← ml + 1,

mk Hk f



k Rmin mk



,

k = 1, .., K

1≤k≤K

10:

end while

Assumption 3.2. Assume that the set of feasible bandwidth allocations, F, for the optimization problem described in Equations (3.9)-(3.11) is not empty, i.e. there exists at least one distribution of carriers [lk ] that satisfies Equations (3.10)-(3.11). Assumption 3.3. Assume that for all users k, the function Fk (mk ) defined in Equation (3.13): • Negative definite Fk (mk ) < 0

∀mk ∈ N

(3.14)

• A monotonically increasing function of mk , Fk (mk + 1) > Fk (mk )

∀mk ∈ N

(3.15)

The last assumption is always true for flat fading channels when the rate-power function f (·) is strictly convex and uniformly increasing, like the rate-power function in Figure 3.2. Given these assumptions, the following theorem is proven in AppendixC Section C.2: Theorem 3.4. If Assumptions 3.2 and 3.3 hold, then the Bandwidth Assignment Based on SNR (BABS) Algorithm, described in Algorithm 3.2 solves the problem stated in Equations (3.9) - (3.11).

70

3.4.2

Subcarrier Assignment Algorithms

Once the number of subcarriers is determined, the next step is to assign specific subcarriers to users. As shown in Appendix C Section C.3, the problem is still difficult to solve since different users see different channel gains on the same subcarrier. In this section, two suboptimal algorithms are proposed to allocate subcarriers to users. The RCG algorithm begins with an estimate of the users’ transmission rate on each carrier and aims to maximize the total transmission rate. The ACG algorithm is a modification of RCG which achieves comparable performance at reduced computational complexity. The Rate Craving Optimal (RCO) and Rate Craving Greedy (RCG) Algorithms Since the number of subcarriers assigned to user k by the BABS algorithm, mk must be l k m Rmin greater than Rmax , the condition that the total transmission power for user k over all k carriers be greater than or equal to the minimum transmission rate requested, Rmin can be

satisfied whenever user k is assigned at least mk carriers, and the transmission power pk (n) on each carrier is large enough. For any allocation of subcarriers among users, the total transmission power is minimized when each user’s transmission power is minimized. A given user k can minimize its transmission power by using single user “waterfilling” allocation over all the carriers it is assigned. If the “water level”, λk , for user k were known, then the transmission rate for  user k on carrier n would also be known, rk∗ (n) = f 0−1 λk |Hk (n)|2 . The BABS algorithm, in addition to determining the number of carriers to be assigned to user k, also introduces an estimate of the “water level” for each user, based on a flat fading system. Using this estimate, the problem can be reformulated by: • Replacing the condition

PN

n=1 rk (n)

k ≥ Rmin by the conditions

P

δ [rk (n)] = mk , and

+

rk (n) = [rk∗ (n)] whenever subcarrier n is allocated to user k.     P PK • Replacing the objective function min K p (n) by max r (n) . k k k=1 k=1 This new problem is a version of the well known combinatorial set partitioning problem, with a linear objective function, it is less computationally expensive to solve than the original

71

problem in Equation (3.4). The remainder of this section will introduce some some terms used to describe the optimal carrier allocation among users. The new problem is defined in Equation (3.20)-(3.21), and an algorithm for solving (3.20) is introduced. This optimal algorithm is used to derive a heuristic algorithm, called the Rate Craving Greedy (RCG) algorithm. Definition 3.5. A partition P of a set A is a set of subsets with the following properties: ∀Ai , Aj ∈ P, Ai ∩ Aj = {} ∪Ai ∈P Ai = A

(subsets are disjoint),

(3.16)

(subsets cover the original).

(3.17)

The number of elements in the set A (cardinality) is denoted #A. If A is the set of carriers, a frequency allocation can be described as a partition of the set of all carriers: Definition 3.6. Define carrier allocation, A = {Ak : 1 ≤ k ≤ K}, as a partition of subcarriers {1, ..., N } into K sets A1 , ..., AK (an allocation of N subcarriers to K users). Let P ([mk ]) denote the set of feasible carrier allocations for bandwidth allocation [mk ], defined as the set of all possible carrier allocations A in a system with N subcarriers and K users such that: #Ak = mk

1 ≤ k ≤ K.

(3.18)

Define the total transmission rate of carrier allocation A as: Rate(A) = max

λ1 ,...,λK

K X X

f 0−1 λk |Hk (n)|2



(3.19)

k=1 n∈Ak

Let fA (·) : {1, .., N } → {1, .., K} denote the function that, given the carrier index, returns the index of the user that this carrier has been allocated to in carrier allocation A. The problem that is solved in this section is: max A

s.t.

Rate(A) #Ak = mk

(3.20) 1 ≤ k ≤ K.

(3.21)

72

The coefficients λk that maximize transmission rate are not known in advance, however they can be estimated based on the average SNR. For the model used in simulations, f (r) = 0.6 r3 , the water level λk is estimated to be λ∗k

=

k Rmin /

N X

!2 |Hk (m)|

,

m=1

and the corresponding estimate of the transmission rate for user k on carrier n is: |Hk (n)| k rk (n) = PN Rmin . |H (m)| k m=1

(3.22)

Problem (3.20) can be solved by using the Rate Craving Optimal (RCO) Algorithm, listed in Appendix C Section C.3. A summary of the algorithm is given below: • Allocate each carrier n to the user with maximum transmission rate rk (n). • While there exists some user k such that #Ak > mk , remove a subcarrier from user k, and add a subcarrier to a user l such that #Al < ml using a sequence of reallocations: give carrier n from user k to user k1 , give carrier n1 from user k1 to user k2 , ..., give carrier np from user kp to user l. In Appendix C Section C.3, the RCO algorithm is proven to solve the optimization problem in (3.20). Unfortunately, this algorithm is computationally intensive, since it involves finding the shortest path through a K node graph and re-calculating graph weights in each iteration. A family of suboptimal Rate Craving (RC) algorithms can be derived from the RCO algorithm, where algorithm RC-p searches for only p-stage re-allocations, with p < K. In particular for p = 1 (nearest neighbors search), the algorithm can be greatly simplified. The simplified version of the algorithm is called the Rate Craving Greedy (RCG) algorithm and is outlined in Algorithm 3.3. Example 3.1. Figure 3.3 demonstrates an example run of the RCG algorithm for N = 8 subcarriers and K = 4 users which require 2 subcarriers each. Columns represent different users and rows are different subcarriers. The number in row n column k represents an estimate of the rate at which user k would transmit on n if it were allocated that carrier,

73

Algorithm 3.3: RCG Algorithm Ensure: mk is the number of subcarriers allocated to each user, rk (n) = f 0−1 (λ∗k |Hk (n)|2 ) is the estimated transmission rate of user k on subcarrier n, Ak ← {} for k = 1, .., K. 1: 2:

for each subcarrier n = 1:N, do k ∗ ← arg max rk (n) 1≤k≤K

3:

Ak∗ ← Ak∗ ∪ {n}.

4:

end for

5:

for all users k such that #Ak > mk , do

6:

while #Ak > mk do

7:

l∗ ← arg

8:

n∗ ← arg min −rk (n) + rl∗ (n)

9:

Ak ← Ak /{n∗ }, Al∗ ← Al∗ ∪ {n∗ }.

10: 11:

min

min −rk (n) + rl (n)

{l:#Al mk . The best user that each subcarrier can be assigned to is found. Then the subcarriers are sorted according to cost and reassigned from the largest one down until all users have exactly mk subcarriers. In practice, number of subcarriers to be reassigned is usually much less than N. ACG algorithm: The ACG algorithm also iterates N times, with 2K comparisons in each step, once to make check if user k has finished carrier assignment, and the second to find the maximum carrier gain. It is Θ(KN ). 3.6

Simulation Setup

In simulations Bandwidth Assignment Based on SNR with Amplitude Craving Greedy subcarrier assignment (BABS-ACG), BABS with Rate Craving Greedy subcarrier assignment (BABS-RCG), and BABS-RC-2 described in Section 3.4 are compared to the Single Step

80

Table 3.4: Order of complexity of the algorithms. Method

Order of operations

MLR

Ω(KN ), O(KN Ntable )

SSFA

Ω(KN ), O(KN log N )

BABS

Θ(KN )

ACG

Θ(KN )

RCG

Ω(KN ), O(KN + N log N )

Frequency Allocation (SSFA) and the Modified Lagrange Relaxation (MLR) and branch and bound algorithms described in Section 3.3. The BABS-RC-2 algorithm is a slightly modified version of the RCG algorithm, which searches all two step re-allocation procedures as well as the one step reallocations searched by RCG. Since RC-2 is closer to the full search required to solve (3.20), it results in a better subcarrier distribution. The RCO algorithm is not implemented, however a BABS-Linear Programming (BABS-LP) algorithm was implemented which solved (3.20) using linear programming. The proof in Appendix C Section C.3 shows that the output of this algorithm will be the same as that of the BABS-RCO algorithm. The modifications to the Lagrange Relaxation algorithm described by Wong et.al. [137] that produced the Modified LR algorithm are described in detail in Appendix C Section C.1. All algorithms are followed by running the single user power allocation algorithm for each user. The system under consideration has parameters given in Table 3.5. The channel model and traffic source models used in simulations are described in more detail below.

3.6.1

Channel Model

In a wireless communication system, data is modulated onto a carrier frequency and transmitted across the air and is modulated by a wireless channel. The signal is impaired in several ways during its travels. The channel simulator used in this dissertation is illustrated in Figure 3.5, and described in detail below.

81

β1 ?  -

e−τ1

-

e−τL

- @ @ @  @   R @    6  - @ @  n(t) + Z(t) 6

 - @ @  6

α

βL Slow Fading

Channel Noise

Multipath Fading

and Interference

Figure 3.5: Channel simulator.

Propagation Loss Path loss from propagation is the change in signal strength that results from the mobile units’ distance from the base station. This form of attenuation depends on both the distance between the base station and the mobile, and the properties of the propagation channel, such as whether it is in space, in the atmosphere, through foliage, through a city, etc. The simplest model for this form of loss is when the signal is transmitted across free space. In this case, the power at the receiver, αR (d), at a distance d from the transmitter is related to the transmitted power, pT , by: pR (d) =

λ2 G T G R pT (4πd)2

(3.27)

where GR is the aperture of the antenna, λ is the wavelength of the carrier signal and GT is the antenna gain at the transmitter. Since several of these terms are difficult to calculate, the antenna gain can be measured for a known transmission power, at some location d0 in the far field of the antenna:  pR (d) = pR (d0 )

d0 d

2 d ≥ d0 ≥ df

where df is the inner limit of the far field for the antenna.

(3.28)

82

y

R x (0,0)

Figure 3.6: A single circular cell, users are distributed in a 2D Gaussian distribution centered around (0, 0).

In the more general case, the signal does not propagate through free space but through an environment which contains air, rain, foliage, and buildings. When the environment has multipath reflection from the ground and other sources, as well as various materials – such as the atmosphere, rain, etc. – to travel through, the effect of distance on the received power becomes:  pR (d) = pR (d0 )

d0 d

η d ≥ d0 ≥ df

(3.29)

where the path loss exponent η is typically between 2 and 6. In this dissertation, the propagation loss on the channel is modeled as: α(d) = pR (d)/pT =

A d2.5

(3.30)

This reflects an open area or suburban setting. A is chosen such that 75% of users get 10dB or better. The users are assumed to lie in a circular symmetric distribution around the base station, centered at (0, 0) with radius R, as illustrated in Figure 3.6. Their location within the cell has a truncated Gaussian distribution with variance R/2: f (x, y) =

1 (x2 +y2 )/2σ2 e 2πσ 2

(3.31)

For this application, σ = 1 and it is assumed that σ ≈ R/6. The probability that a user is

83

within a radius d of the base station is Z dZ π 1 2 2 2 2 m= er /2σ rdrdθ = 1 − e−d /2σ 2 2πσ 0 −π

(3.32)

To ensure that 75% of users will are within a radius r of the base station, where r = p p 2 ln(1 − 0.75) = ln(16). To ensure that 75% of users receive less than 10dB attenuation due to the channel, pR (d) A =1= pT (ln(16))2.5/2

(3.33)

Thus A = (ln(16))1.25 was chosen. Multipath Fading Smaller scale variations in the wireless channel result from multipath fading. As the signal is sent across the wireless channel, it may bounce off different objects in the environment. Rays which arrive at the receiver at different times cause the channel to appear as a filter. In the time domain, such a wireless channel can be modeled as the sum of multipath components: h(t) =

∞ X

βl ejφl δ(t − τl )

(3.34)

l=0

where τl is the timing offset between different multipath components. Inter-arrival times are exponentially distributed. φl is the phase offset for this component, it is uniformly in the interval [0, 2π). βl is the path gain, Rayleigh distributed with mean square value βl2 = β 2 (0)e−τl /γ . The channel model used in this dissertation is based on the Saleh-Valenzuela double exponential multipath fading model [102], which builds on the simple multipath model. Looking at experimental data gathered in various frequency bands, the authors note that multipath rays come in batches, which are also exponentially distributed. The double exponential model used is illustrated in Figure 3.7. The channel identity function is: h(t) =

∞ X ∞ X

βk,l ejφk,l δ(t − Tl − τk,l )

(3.35)

t=0 k=0

where φk,l is uniformly distributed on [0, 2π), βk,l is Rayleigh distributed with mean square 2 = β 2 (0, 0)e−Tl /T e−τk,l /γ , and the cluster and ray arrival times, T and τ value βk,l l k,l are

84

τ

Multipath blocks

time

Figure 3.7: The Saleh-Valenzuela multipath channel model.

distributed with exponential inter-arrival times: p(Tl |Tl−1 ) = Λe−Λ(Tl −Tl−1 ) p(τkl |τ(k−1)l ) = λe−λ(τkl −τ(k−1)l ) These particular models were chosen firstly, because they are shown to fit experimental data, and secondly because they are easily extended to the case of multiple antenna systems. The channel model used is based on the Saleh-Valenzuela multipath fading model [102]. A double exponential channel model is used, with independent fading based on distance from the base station. h(t) =

∞ X ∞ X

βk,l ejφk,l δ(t − Tl − τk,l )

(3.36)

t=0 k=0

where φk,l is uniformly distributed on [0, 2π), βk,l is Rayleigh distributed with mean square 2 = β 2 (0, 0)e−Tl /T e−τk,l /γ , and the cluster and ray arrival times, T and τ value βk,l l k,l are

distributed with exponential inter-arrival times: p(Tl |Tl−1 ) = Λe−Λ(Tl −Tl−1 ) p(τkl |τ(k−1)l ) = λe−λ(τkl −τ(k−1)l ) Two sets of channel parameters are studied: • Channel 1 is highly uncorrelated with only a single cluster at Tl = 0 and ray arrival parameters γ = 3.75 × 10−6 , λ = 3.0 × 10−6 ,

85

2.5

Channel 1

2 1.5 1 0.5 0

0

20

40

60 Subcarrier No.

80

100

120

0

20

40

60 Subcarrier No.

80

100

120

6

Channel 2

5 4 3 2 1 0

Figure 3.8: Sample channel profiles.

• Channel 2 is a highly correlated channel with parameters Γ = 336 × 10−9 , 1/Λ = 168 × 10−9 , γ = 286 × 10−9 , 1/λ = 51 × 10−9 .

The channel profile is then normalized to model slow fading for users who are arranged in a two dimensional Gaussian distribution around the base station. The total power received P −2.5 2 by the base station from user k is given by Pk = N , where dk is the n=1 |Hk (n)| = P0 dk ratio distance between the user and the base station to the cell radius. Choosing P0 = 3.578 allows a SNR that is greater than or equal to 10dB for 75% of all users. Two example channel profiles are shown in Figure 3.8. The coding and modulation schemes used are shown in Table 3.5. Adaptive modulation is used both with and without power loading of subcarriers in simulations. While power based loading increases the throughput, it is computationally costly and may not be used in a practical system [101][55][26].

86

Table 3.5: System Parameters

3.6.2

Bandwidth

4 MHz

No. of subcarriers

512

Rate per subcarrier

7.81 KSymbols/sec

Modulation

BPSK, QPSK, 16-, 64-QAM

Convolutional codes

Rate 1/2, 2/3, 3/4

Traffic Model

Three classes of users are considered: data, voice and video. It is assumed that 10% of the users will transmitting video, 40% will be transmitting voice and the remaining 50% of the users will be transmitting data. Video and voice traffic are given a constant transmission rate of 64kbps and 16kbps respectively, whereas data traffic is assumed to be exponentially distributed, with a mean of 30kbps. In the next chapter these models are modified to take into account the bursty nature of the traffic. This section describes some of the functions that are implemented in the transmitter side of a communication system. At the transmitter, voice, video or data traffic generated by the user is encoded using some standard source encoding mechanism that distills the information from the source into a sequence of symbols. This process is known as source coding. Next, the symbols are encoded to protect against error in the wireless channel. Finally, the sequence of bits generated by the Error Control Coding (ECC) encoder are modulated, that is, mapped onto a known set of constellation points. The modulated signal is then transmitted at some fixed carrier frequency fc across the wireless channel.

Voice Traffic Conventional cellular communications systems generally transmit voice information that is encoded using Vector Sum Excited Linear Predictive (VSLEP). Given typical conversation, VSLEP generates data at the rate of 7.95kbps. GSM uses Regular Pulse Excited - Linear Predictive Coder (RPE-LPC) with a Long Term Predictor loop. This method encodes 20ms

87

of speech in 260 bits, requiring a transmission rate of 13kbps [13][108]. In this dissertation, a transmission rate of 16kbps is assumed for voice traffic.

Video Traffic Although wireless video transmission is not yet standardized, the new range of camera equipped PDAs seem to point to the possibility. The most common video coding standards for wireless communications are ITU H.263 and MPEG-4 [24][54][34]. MPEG-4 supports both Constant Bit Rate (CBR) and Variable Bit Rate (VBR) transmission, at rates ranging from below 64kbps to 4Mbps. In general, rates between 22kbps and 64kbps are considered to be useful for wireless transmission of video [34].

Data Traffic Data traffic is modeled as a exponentially distributed random variable with a mean of 30kbps. The minimum probability of error is set to 1 × 10−4 as for voice and video traffic, however the same system can be used to model data transfer with lower probability of error and transmission rate as described in Section 3.2.1.

3.6.3

Loading Algorithms for Single User OFDM

Numerous algorithms have been developed to calculate how best to distribute transmission power, and consequently the transmission rate, among the carriers. One of the earliest of these is the Hughes-Hartogs Algorithm. This algorithm assigns bits one at a time to carriers, choosing the carrier that requires the least incremental power [11]. The aim is to get minimum bit error rate, however it is computationally expensive, O(Rtot × N ), where Rtot is the total number of bits transmitted, N is the number of subcarriers. Despite giving excellent results, the Hughes-Hartogs algorithm may be too computationally expensive. Chow, Cioffi and Bingham have proposed an algorithm that uses the SNR gap Γ to meet minimum transmission rates. Γ is the gap between channel capacity and the bandwidth efficiency of real modulation schemes, for more information see Section 3.2.1 of this dissertation. This algorithm gives similar performance to Hughes-Hartogs, at

88

lower cost. O(Niter × 2N − N ). Another well known algorithm by Fischer and Huber changes the problem, to maximizing SNR with both power and rate constraints: max s. t.

SNR0 = N X n=1 N X

Vi2 Ni /2

(3.37)

r(n) = Rmin

(3.38)

p(n) = Pmax

(3.39)

n=1

Solving this problem using Lagrange multipliers leads to the assignment of higher rates to channels with less noise, transmit power is allocated to ensure equal error probability. The algorithm has lower complexity than Chow et. al. and gives better SNR output. Because of the ease of implementation, the Hughes-Hartogs algorithm was used in the simulations in this chapter. 3.7

Simulation Results

The algorithms are compared based on three criteria: the probability that a user k is k able to transmit less than Rmin per unit time, the total transmission power PT and the

computational complexity used in CPU cycles. A total of 1 × 104 frames are simulated, each with a different channel response and user profile. The outage probability is about the same for all channels and subcarrier loading strategies used. The probability for Channel 1, with power loading is shown in figure 3.9. All algorithms were programmed to begin by checking whether the minimum rate requirements can be satisfied with the given number of users, if they cannot, the user with the largest rate request is dropped from the system. By isolating resource assignment from subcarrier assignment, the system allow the system designer to decide whether to drop users or reduce rates, and who to drop based on fairness and other requirements. The modifications that have been made to the Lagrange Relaxation algorithm prevent the algorithm from simply dropping users with bad channels, allowing the algorithms to be compared (Fig. 3.10-3.17). Figures 3.10-3.13 show that when the number of carriers is

89

0.3 SSFA with waterfilling BABS with waterfilling MLR with waterfilling 0.25

Outage Probability

0.2

0.15

0.1

0.05

0

10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.9: Adaptive modulation and power loading, outage probability vs. number of users.

fixed at 128, initially as traffic increases the system becomes more crowded. Users do not have as much freedom in choosing carriers, the constellation size goes up and the coding rate is reduced. This leads to the increase in the transmission power requirement. After 20-30 users join the system, the utilization peaks and the probability of outage begins to increase sharply. This allows the system to drop users who need a lot of bandwidth, and the transmission power required per unit bit transmitted decreases. Figures 3.14-3.17 show the same phenomenon in reverse. As the number of carriers increases, the probability of outage decreases, causing the price of transmission to go up because the system is accommodating difficult users. When the number of carriers goes above 80, the outage probability becomes much smaller, the new subcarriers increase diversity and reduce transmission power. In all cases the MLR algorithm requires the least transmission power per bit, followed closely by BABS-RC-2. The BABS-RCG algorithm finds good allocations for channels with low subcarrier correlation, but not for Channel 2, where the nearest neighbors search used by the RCG algorithm is not thorough enough. By also searching for two stage reallocations,

90

2

Power per Bit Transmitted (PT/Rtot )

10

SSFA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading MLR with power loading

1

10

10

20

30

40

50

60

70

80

90

100

Number of Users

Figure 3.10: Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users.

2

T

Power per Bit Transmitted (P /R

tot

)

10

SSFA w/o power loading BABS+ACG w/o power loading BABS+RCG w/o power loading BABS+RC−2 w/o power loading MLR w/o power loading

1

10

10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.11: Adaptive modulation without power loading for Channel 1, transmission power per bit vs. number of users.

91

2

Power per Bit Transmitted (PT/Rtot )

10

SSFA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading MLR with power loading 1

10

10

20

30

40

50

60

70

80

90

100

Number of Users

Figure 3.12: Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of users.

2

T

Power per Bit Transmitted (P /R

tot

)

10

SSFA w/o power loading BABS+ACG w/o power loading BABS+RCG w/o power loading BABS+RC−2 w/o power loading MLR w/o power loading

1

10

10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.13: Adaptive modulation without power loading for Channel 2, transmission power per bit vs. number of users.

92

SSFA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading MLR with power loading 2

T

Power per Bit Transmitted (P /R

tot

)

10

1

10

20

40

60

80

100 120 Number of Subcarriers

140

160

180

200

Figure 3.14: Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of carriers.

SSFA w/o power loading BABS+ACG w/o power loading BABS+RCG w/o power loading BABS+RC−2 w/o power loading MLR w/o power loading 2

T

Power per Bit Transmitted (P /R

tot

)

10

1

10

20

40

60

80

100 120 Number of Subcarriers

140

160

180

200

Figure 3.15: Adaptive modulation without power loading for Channel 1, transmission power per bit vs. number of carriers.

93

SSFA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading MLR with power loading 2

T

Power per Bit Transmitted (P /R

tot

)

10

1

10

20

40

60

80

100 120 Number of Subcarriers

140

160

180

200

Figure 3.16: Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of carriers.

SSFA w/o power loading BABS+ACG w/o power loading BABS+RCG w/o power loading BABS+RC−2 w/o power loading MLR w/o power loading 2

T

Power per Bit Transmitted (P /R

tot

)

10

1

10

20

40

60

80

100 120 Number of Subcarriers

140

160

180

200

Figure 3.17: Adaptive modulation without power loading for Channel 2, transmission power per bit vs. number of carriers.

94

2

T

Power per Bit Transmitted (P / R

tot

)

10

BFDMA with power loading BABS+BFDMA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading BABS+LP with power loading

1

10

10

20

30

40

50

60

70

80

90

100

Number of Users

Figure 3.18: Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users.

the BABS-RC-2 algorithm still finds a subcarrier allocation with very low power. Both BABS-ACG and BABS-RCG do considerably better than SSFA.

3.7.1

Fixed Carrier Allocation

The previous simulations have compared the transmission power for different adaptive carrier allocation schemes. Figures 3.18-3.21 compare adaptive carrier allocation to fixed carrier allocation, which does not take into any information about the channel, and BABS-FDMA, in which the average SNR at the output of the channel is used to decide the number of carriers a user will be assigned but the assignment of carriers to users is fixed. The BABS-FDMA scheme provides some improvement over fixed FDMA when the system is not fully loaded and probability of outage is relatively low (few users or any carriers), and when the channel gains are correlated, as in the profile of Channel 2. The assignment of carriers to users plays a major role in the reduction of transmission power that results from an adaptive carrier allocation algorithm.

95

2

T

Power per Bit Transmitted (P / R

tot

)

10

BFDMA with power loading BABS+BFDMA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading BABS+LP with power loading

1

10

20

40

60

80

100

120

140

160

180

200

Number of Subcarriers

Figure 3.19: Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of carriers.

2

T

Power per Bit Transmitted (P / R

tot

)

10

BFDMA with power loading BABS+BFDMA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading BABS+LP with power loading

1

10

10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.20: Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of users.

96

2

T

Power per Bit Transmitted (P / R

tot

)

10

BFDMA with power loading BABS+BFDMA with power loading BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading BABS+LP with power loading

1

10

20

40

60

80

100

120

140

160

180

200

Number of Subcarriers

Figure 3.21: Adaptive modulation and power loading for Channel 2, transmission power per bit vs. number of carriers.

3.7.2

The Sensible Greedy Algorithms and Rate Craving Optimal (RCO) Algorithm

Figures 3.18-3.21 also show plots of an algorithm labeled as BABS-LP. The BABS-LP algorithm consists of the BABS algorithm, followed by a linear programming formulation of (3.20) implemented using the GNU Linear Programming Kit (GLPK) [74]. The RCO algorithm was introduced mainly to develop the RCG algorithm, which is a heuristic approach to solving (3.20). In Appendix C Section C.3 it is shown that the BABS-RCO algorithm will solve (3.20), a linear programming algorithm will solve it much more efficiently. Since both algorithms find the optimal solution to (3.20), the output of the BABS-LP algorithm will be the same as that of the BABS-RCO algorithm if it were implemented. Figure 3.22 gives a closer look at the BABS-ACG, BABS-RCG, BABS-RC-2 and BABS-LP algorithms for Channel 1. The plot for BABS-LP is difficult to see, since it falls almost on the same line as BABS-RC-2. The simulations show that BABS-RC-2 is a very close approximation to BABS-RCO for the system under consideration.

97

70 60

Power per Bit Transmitted (PT / Rtot )

50

40

30

20

BABS+ACG with power loading BABS+RCG with power loading BABS+RC−2 with power loading BABS+LP with power loading 10

10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.22: Adaptive modulation and power loading for Channel 1, transmission power per bit vs. number of users.

3.7.3

Modified Lagrangian Relaxation Algorithm

While the MLR algorithm achieves a better carrier allocation for an OFDMA system than the Sensible Greedy algorithms, due to the factors outlined in Section 3.3.3 it may not find the optimum allocation. Figure 3.23 shows the transmission power per bit for the subcarrier allocation produced by the algorithms BABS-ACG, BABS-RC-2, MLR and the branch and bound algorithm. The branch and bound algorithm searches all possible allocations, and yields the lowest transmission power per bit. Since the number of carriers is relatively small (N = 20) the BABS-RC-2 and BABS-ACG algorithms perform well relative to the Modified Lagrangian Relaxation algorithm. Since the MLR algorithm yields better allocations when the number of carriers is large, for 20 carriers its performance is not very good. Unfortunately the branch and bound algorithm is too computationally intensive for larger systems to be simulated.

98

BABS+ACG with power loading BABS+RC−2 with power loading SSFA with power loading MLR with power loading Branch and Bound Algorithm (Full search)

Power per Bit Transmitted (PT/Rtot )

40

30

20

10

2

2.5

3

3.5

4

4.5

5

Number of Users

Figure 3.23: Adaptive modulation and power loading for Channel 1, N = 20 carriers, transmission power per bit vs. number of users.

3.7.4

Bit loading vs. Power loading

In simulations, two forms of single user channel loading were used, bit loading only, and power loading with bit loading. When only bit loading is used, the user transmits with the same power on every subcarrier it is allocated. However the number of bits transmitted on that carrier, as determined by the constellation size and the coding rate is varied. In this implementation, users are allowed to transmit 0 bits on any carrier, which sets the transmission power on that carrier to zero. This is necessary in general for simulating Rayleigh fading channels, where channel gains can be very small. In bit loading with power loading, both the power transmitted per subcarrier and the number of bits transmitted per subcarrier are allowed to differ. Bit loading without power loading is much simpler to implement in terms of computational complexity, however power loading maximizes the capacity of the channel, and allows the same transmission rate to occur with less transmission power.

99

Percentage Change in Transmission Power Due to Power Allocation (∆P /P ) T T

120 BFDMA BABS+BFDMA SSFA BABS+ACG BABS+RCG BABS+RC−2 MLR

100

80

60

40

20

0 10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.24: Percentage increase in transmission power due to not using power loading on Channel 1 vs. number of users.

Figures 3.24-3.27 show plots of the percentage increase in transmission power per bit due to switching from bit loading with power loading to bit loading only. The change is largest for the BABS algorithm with no subcarrier selection algorithm (BABS-BFDMA) suggesting that the algorithm should be modified when being used in such a system. In general algorithms such as MLR and BABS-RC-2 which yield the lowest transmission power per bit in both channel and loading scenarios show the least degradation from using only bit loading. For systems which are under heavy traffic (more users, fewer carriers), the users have fewer carriers each, and power loading is not as important as for lightly loaded systems, where many carriers are used per user.

3.7.5

Simulation Analysis of Computational Complexity

Finally, the algorithms can be compared using simulations on a real computer. While the CPU time used may be different for every set of values, by running the algorithms for a large ensemble of channels and traffic requirements, an average CPU time can be found.

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Percentage Change in Transmission Power Due to Power Allocation ( ∆ PT / PT )

120

100

BFDMA BABS+BFDMA SSFA BABS+ACG BABS+RCG BABS+RC−2 MLR

80

60

40

20

0 10

20

30

40

50

60

70

80

90

100

Number of Subcarriers

Figure 3.25: Percentage increase in transmission power due to not using power loading on Channel 1 vs. number of carriers.

Percentage Change in Transmission Power Due to Power Allocation ( ∆ PT / PT )

120 BFDMA BABS+BFDMA SSFA BABS+ACG BABS+RCG BABS+RC−2 MLR

100

80

60

40

20

0 10

20

30

40

50 60 Number of Users

70

80

90

100

Figure 3.26: Percentage increase in transmission power due to not using power loading on Channel 2 vs. number of users.

101

Percentage Change in Transmission Power Due to Power Allocation ( ∆ PT / PT )

120

100

BFDMA BABS+BFDMA SSFA BABS+ACG BABS+RCG BABS+RC−2 MLR

80

60

40

20

0 10

20

30

40

50

60

70

80

90

100

Number of Subcarriers

Figure 3.27: Percentage increase in transmission power due to not using power loading on Channel 2 vs. number of carriers.

The SPEC int and SPEC fp numbers are used in this dissertation to aid in assessing the computing environment on which the algorithms were implemented. The Standard Performance Evaluation Corporation (SPEC) was founded in 1988 to provide a better foundation for comparing different workstations, using a standardized suite of source code based on existing applications that are ported to different machines by the members of the organization. SPEC benchmark results are available from the webpage at http://www.specbench.org [113]. Figures 3.28-3.29 compare the computational complexity of the algorithms. These plots aid the comparison in two respects. First, the number of iterations required by the Modified Lagrange Relaxation algorithm is difficult to predict, and thus a theoretical comparison is not possible. Second, while the upper bound for the number of iterations the RCG algorithm requires is O(KN + N log N ), in practice the number of subcarrier reassignments required in step 2 is not very large, and the algorithm executes much more quickly than could be expected. The two methods were coded in C, the platform for implementation was an AMD

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8

10

7

CPU time

10

6

10

MLR BABS+RC−2 SSFA BABS+RCG BABS+ACG

5

10

5

10

15

20

25 30 Number of Users

35

40

45

50

Figure 3.28: Adaptive modulation and power loading, average CPU time required vs. number of users (128 subcarriers).

Duron based PC running Linux with 42.9 SPECint 95 and 29.4 SPECfp 95 ratings. The algorithms were tested on the same 300 channel conditions and rate requirements. Plots of the required CPU time show that both greedy algorithms perform an order of magnitude faster than the iterative MLR algorithm, but BABS-ACG performs about twice as fast as BABS-RCG, demonstrating that the worst case performance is a pessimistic lower bound for the RCG algorithm. The SSFA algorithm is about twice as slow as the others, and has about the same computational complexity as BABS-RC-2. 3.8

Conclusion

Fast and efficient medium access algorithms allow mobile networks to adapt quickly to changes in the environment. In this chapter, a computationally efficient class of algorithms for allocating subcarriers and power among users in a multicarrier system has been described. Dividing the problem into two stages enabled the design of algorithms with low computational complexity, which operate well under realistic channel and data traffic as-

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7

CPU time

10

6

10

MLR BABS+RC−2 SSFA BABS+RCG BABS+ACG

5

10

20

40

60 80 Number of Subcarriers

100

120

Figure 3.29: Adaptive modulation and power loading, average CPU time required vs. number of subcarriers (20 users).

sumptions. This approach allows efficient use of system resources in terms of transmission power, bandwidth efficiency and computation time. Simulation show that the algorithms yield low outage probability, and low power requirements at reasonable complexity, showing that a good resource allocation strategy can be achieved by efficient algorithms in a practical system. The feasibility of the algorithms will depend on factors such as how quickly the channel varies, the accuracy and overhead of the channel estimation algorithm, and the latency in the medium access protocol. The interaction of power and rate control protocols with such factors is a course for further study.

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Chapter 4 ONLINE SCHEDULING FOR OTHOGONAL FREQUENCY DIVISION MULTIPLE ACCESS (OFDMA)

The previous chapter introduced a family of resource allocation algorithms for an OFDMA system. The system considered snapshots in time of the resource requirements and channel coefficients of a number of users accessing the system. In a real system, users will enter and exit the system continuously, and channel coefficients will change due to mobility and channel fading. Scheduling for broadband wireless systems is a topic which has gained interest recently. This chapter derives limits on the outcome fairnes and effort fairness of an OFDMA system. The first theorem of this chapter introduces an upper bound on outome fairness based on system parameters such as frequency spacing, time granularity and available transmission rates. The second theorem relates the effort fairness of a system to the outcome fairness, a corollary results in an upper bound on effort fairness of a system, which depends the minimum and maximum possible transmission rates per carrier. Based on the expression for effort fairness, the resource allocation problem is modified for a time varying OFDMA system. Two online scheduling and resource allocation algorithms are introduced for the time varying OFDMA system. The algorithms provide support for multiple classes of traffic in the same cell to efficiently use the wireless spectrum without excessive messaging overhead or complicated computational algorithms. The results show that at some cost to the total transmission power, a scheme can be found which does not require frequent reassignment of carriers among users or complex reassignment algorithms. This chapter is organized as follows. Section 1 introduces the topic of scheduling for OFDMA systems, and presents some previous work in this area. Section 2 defines what is meant by the terms online, nonpreemptive and fair. Two theorems are derived for effort and outcome fairness in an OFDMA system. The modified resource allocation problem is defined. Section 3 introduces the online scheduling algorithms derived from the Sensible

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Greedy Algorithms, which are used to tackle the modified resource allocation problem. Section 4 contains details of the simulation model used. Section 5 contains an analysis of simulation results and Section 6 concludes the chapter. 4.1

Introduction

Scheduling for OFDMA systems was first studied for practical systems such as DECT and ACIS [38][20][21]. These algorithms are designed to be simple and distributed, but are not designed for a very heavily loaded system. The scheduling and resource allocation algorithms described in this chapter are based on concepts from weighted fair queueing and energy aware scheduling algorithms.

4.1.1

Weighted Fair Queueing

Fair scheduling was first introduced in the context of wired networks. It generally refers to a condition on the bandwidth allocation among users accessing a common channel which must be met by a scheduling algorithm. A common fairness criterion is weighted fairness [81], defined as: Sk (τ, t) Φk ≥ Sl (τ, t) Φl

∀k, l = 1, · · · , K

(4.1)

where Sk (τ, t) is the ammount of traffic belonging to user k that is served in an interval (τ, t), and Φk is the weight assigned to user k by the system. Fairness can be defined over different time scales. Long term fairness refers to the requirement that the ratio of Sk (τ, t) to Sl (τ, t) should approach Φk /Φl as t − τ goes to infinity, whereas short term fairness imposes constraints on the ratio for finite values of t. The most common algorithms used for scheduling in wireless networks are variants of the Weighted Fair Queueing algorithm [15][9]. Weighted Fair Queueing (WFQ) was first proposed independently by Parekh and Gallagher [81] and Demers et.al. [28] for wireline networks. Variants of the algorithm have been developed for wireline networks [142] and for wireless networks [9][15]. Until recently, most wireless WFQ algorithms focused on systems with a single transmission channel, and a two state channel model was assumed, where the channel state is either “good” and transmission is successful, or it is “bad” and transmission

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will be unsuccessful. The channel state of all users are assumed to be known at the time of scheduling. In recent work, WFQ algorithms have been designed for more realistic channel models, where the transmission rate and channel SNR are related though the maximum tolerable probability of error per bit [70][65][16]. The definition of weighted fairness in Equation (4.1) is relevant to wired and two state wireless channels, but is not sufficient for wireless channels with multiple states. When there are multiple transmission states, it is not clear whether the definition of “traffic served”, Sk (τ, t) should refer to the opportunity to transmit that is offered to a user, or the actual volume of traffic that the user is able transmit. Defining fairness based on actual throughput can mean that a user with a chronically bad channel may take on a large portion of the system resources, reducing overall system throughput. Following Cao [16], this dissertation defines two kinds of fairness: • Effort fairness: S¯k (τ, t) is defined as the total bandwidth offered to user k. • Outcome fairness: Sk (τ, t) is defined as the total transmission rate of user k. Cao’s scheduling algorithm aims to provide short term outcome fairness for delay sensitive flows, and long term outcome fairness for best effort flows. On the other hand, Liu et.al. [70] and Kulkarni and Rosenberg [64][65] introduce scheduling algorithms to allow effort fairness for all flows, while using “opportunistic scheduling” to take advantage of channel conditions to maximize the total throughput of all users. Condition (3.5) in the OFDMA resource allocation problem addressed in the previous chapter of this dissertation can be restated as a weighted outcome fairness criterion by P PN noting that Sk (τ, t) = M m=1 n=1 rk (m, n): Rk Sk (τ, t) = min l Sl (τ, t) Rmin

∀k, l = 1, · · · , K

(4.2)

Then the problem in Equation (3.4) imposes short term outcome fairness constraints for all time. The dichotomy between outcome fairness and effort fairness stems from having variable transmission rates. As shown in the next section, if the transmission rate is constant for

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all users at all times, there is no upper bound on the transmission power, and interference between users is not an issue, then transmission power can be chosen for any user to ensure that the signal to noise ratio at the receiver is arbitrarily large. Although there has been work on multichannel scheduling [64][71], there still remains a disconnect between the literature of resource allocation for OFDMA, outlined in the previous chapter, and that of multichannel scheduling. While some resource allocation algorithms have been proposed which adapt to channel conditions over time [31][40], for the most part these algorithms use time as an additional dimension to relax the very strict fairness condition imposed by Equation (3.5), and do not take into account the additional constraints introduced by time varying traffic and channel conditions. Li and Liu [67] study dynamic resource allocation for a multichannel system, where packet delay is constrained by the finite buffer size at mobile stations. This chapter describes an online resource allocation and scheduling algorithm for OFDMA. The algorithm is based on the resource allocation algorithm of the previous section, but aims to satisfy both outcome and effort fairness criteria, while minimizing the total transmission power of all users in the system. The algorithm is “online” in the sense that when a transmission request arrives at the base station, the channel conditions for the requesting user at the time of transmission are revealed to the base station. Decisions about resource allocation and scheduling must be made without knowledge of future traffic requirements.

4.1.2

Energy Aware Scheduling

Energy Efficient Fair Scheduling has been studied by Raghunathan, Schurgers, Srivastava et.al. [106][107][91] for single user scheduling and Zhang et.al. [144]. Schurgers et.al. study the application of Dynamic Voltage Scaling (DVS) algorithms, which minimize processing power in real time operating systems to scheduling for wireless communications. Zhang et.al. also study the trade-off between transmission duration and transmission power for IEEE 802.11 LANs. Both groups study power saving modes for actual devices, which have power consumption behaviour that is slightly more complicated than that depicted in the previous chapter. Another group of papers by Rajan, Sabharwal and Aazhang [94][93][92], and Uysal-Biyikoglu, Prabhakar, Aazhang et.al. [88][131][132] focuses on more abstract

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models, and explores the tradeoff between transmission delay and transmission power. Zhang et.al. [144] point out that a mobile station which ceases transmission will continue to consume a significant ammount of energy until it enters sleep mode. However the device must wait some time after finishing transmission before it can enter reduced power mode. For instance, the AT&T WaveLAN interface card waits for 80µs to enter sleep mode, and needs 250µs to enter transmission mode again [144]. This means that once a user begins to transmit, it is preferable not to stop transmission. Keeping the time between periodic transmissions for delay sensitive traffic as long as possible allows time for the transmitter to enter sleep mode. In addition, nonpreemptive scheduling reduces the scheduling overhead for the system. Raghunathan et.al. use a similar weighted round robin scheduling mechanism, but in addition to maximizing sleep time for transceivers, minimize the transmission rate to reduce power consumption [91]. Uysal-Biyikoglu et.al. [131][132] study single user scheduling. Their method is called “Lazy Scheduling,” and exploits the tradeoff between the maximum transmission delay allowed and maximum transmission power. Optimal offline algorithms and online algorithms are developed which use the length of the buffer at the mobile stations to set up a schedule which minimizes the total transmission energy while meeting transmission deadlines. Prabhakar et.al. [88] show how this algorithm can be generalized to multiple users, packets with variable deadlines and time varying channels. Rajan, Sabharwal and Aazhang [94][93][92] use the Value Iteration Algorithm to solve the same problem, but focus on time-varying channels. 4.2

System Model and Dynamic Scheduling Problem Formulation

This chapter studies online scheduling and resource allocation for an OFDMA system with no preemption and perfect channel state information. A mobile station entering the system transmits to the base station the minimum transmission rate and maximum probability of error it can be assigned, and information on channel conditions. If there are enough resources to allow the mobile to transmit, the mobile station is admitted into the network, and assigned a number of carriers to transmit on. Once assigned carriers, the mobile station retains its allocation until it transmits all packets in its immediate queue. After transmission

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is completed, the carriers are released and the user exits the system. The BS is not notified of the duration of the transmission before carrier allocation, nor does it have any knowledge of future channel conditions or traffic arrival.

4.2.1

Online Scheduling and Preemption

Scheduling for OFDMA systems is a topic of recent interest, since wireless local area network products based on the IEEE 802.11A[48] and IEEE 802.16 [49][50] standards have attained commercial success. The main difference between the two systems is that the IEEE 802.11 protocol was designed for smaller networks, and is decentralized, whereas the IEEE 802.16 protocol was designed for wide area networks, and it is connection oriented. The IEEE 802.16 protocol has two modes: Grant per Connection (GPC) mode in which every flow which a given mobile station initiates is allocated resources separately by the base station, and the Grant per Subscriber Station (GPSS) mode in which a mobile requests a block of bandwidth to satisfy the requirements of all connections originating from the mobile station. The GPSS mode is preferable to reduce overhead in the allocation protocol and to reduce the computational complexity of scheduling. This chapter focuses on a connection oriented system. The results of the previous chapter, as well as prior work on energy efficient scheduling summarized above show that allocating more carriers to a connection for longer durations of time will reduce power requirements. However tying up resources in this way will delay the access of new users to the system. Preemption is the process of interrupting an active connection in order to allocate resources to another connection. In a system with multiple resources, such as a multichannel communication system, various forms of preemption are possible [33]. A system may be nonpreemptive, once channels are allocated they are allocated for the duration of a call. In migratable or local preemption, communication on channels may be preempted and resumed individually. Local and migratable preemption do not necessarily cause communication outage. In gang scheduling, all active channels assigned to a mobile station are suspended and resumed simultaneously. The main drawback of preemption is the communication overhead needed to coordinate

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resource reallocation. As discussed in Section 4.1.2, switching a transmitter off then on again to allow preemption also requires additional power. Since transmission power and communication overhead are important concerns for the system of interest, both algorithms studied in this chapter are completely nonpreemptive. In the Simple Scheduling Algorithm, carrier allocation remains constant for the duration of a flow. The Adaptive Scheduling Algorithm allows periodic reallocation of carriers but only to add carriers, no carrier is released by a mobile station before it completes transmission. This policy allows the system to adapt to channel conditions, but requires more overhead. Another characteristic of the algorithms introduced in this chapter is that they are online [111]. There are several types of online algorithms, which assume various degrees of knowledge about the system. In this chapter the resource allocation and scheduling algorithms use information about overall traffic and channel statistics, but have no specific knowledge about future traffic and channel patterns. The algorithm is also nonclairvoyant [111], since the duration of a call is not known until it leaves the system. Offline algorithms can do a much better job of resource allocation, but they introduce a scheduling delay (or latency) which may not be practical for a wireless system.

4.2.2

Weighted Fairness and OFDMA Resource Allocation

Since the system under consideration is nonpreemptive and nonclairvoyant, resources must be carefully allocated to ensure fairness between users who arrive when the system is empty and those who arrive when the system is full, as well as users who see different degrees of noise and interference on the transmission channels. In the previous section, outcome fairness and effort fairness were defined for a multi-user communication system. In this section, these terms are defined more specifically for an OFDMA system: • Effort fairness: The degree of effort fairness of a carrier allocation in an OFDMA system can be represented by vM Nk vM Nl − l Rk Rmin min where Nk is the number of carriers assigned to user k in this allocation.

(4.3)

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• Outcome fairness: The degree of outcome fairness of a carrier allocation in an OFDMA system can be represented by Rk R l Rk − Rl min min

(4.4)

where Rk is the actual transmission rate for user k. In the problem studied in Chapter 3, any feasible allocation which minimizes the total transmit power will set transmission rates as close as possible to minimum requirements. If the transmission rate can be set to any real value, then the power and rate allocation for user k will satisfy: M X N X

min

m=1 n=1 M X N X

s.t.

f (rk (m, n)) k rk (m, n) = Rmin .

m=1 n=1

When all transmission rates rk (m, n) which solve the above problem are achievable, the allocation is perfectly outcome fair, but there are no bounds on the effort fairness of the system. Just as the large packet size causes a deviation from perfect fairness for weighted fair queueing algorithms in wireline systems [81], when the set of available transmission rates is finite, there is a limit on the outcome fairness achievable in an OFDMA system. This bound is given by Theorem 4.1 below. Theorem 4.1. Consider an OFDMA system with N carriers, M time slots, and K users. k . Let r (m, n) represent the transEach user k has minimum transmission requirement Rmin k

mission rate for user k on carrier n in time slot m. Let T be the duration of a single time slot, let {0, Y1 , · · · , YQ } be the set of modulation and coding levels per carrier, and V = {v0 , v1 , · · · , vQ } where v0 = 0 and vk = Yk /(M N T ) be the set of available transmission rates per carrier. Let f (vq ) represent minimum SNR requirement for any user to transmit at rate vq on some carrier. Define ∆ = Rk =

max (Yq − Yq−1 )

1≤q≤Q

N X M X n=1 m=1

rk (m, n).

(4.5) (4.6)

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Then any resource allocation which satisfies the conditions k Rk ≥ Rmin , K X

δ [rk (m, n)] ≤ 1

k = 1, ..., K n = 1, ..., N,

(4.7) m = 1, ..., M,

(4.8)

k=1

will satisfy the following fairness condition: Rk ∆ 1 Rl . ≤ − Rk l k l M N T Rmin min Rmin , Rmin min

(4.9)

The proof of this theorem is given in Appendix D. The limiting term in Equation (4.9) shows the effect of frequency granularity, the time horizon and the granularity of the modulation scheme on the achievable fairness. As the granularity, defined here as the maximum difference between two consecutive transmission rates, increases, ∆ will increase, and the fairness bound will become less strict. As the number of frequencies increase, the total transmission rate per carrier decreases, and the outcome fairness improves. Finally, the fairness depends on the time horizon, that is, on what time scale fairness is being considered. In the limit as the time slots M goes to infinity, perfect fairness can be achieved. The finite granularity of the modulation and coding scheme also limits effort fairness. Since there is a maximum and minimum number of bits which can be transmitted per carrier, there are upper and lower bounds on the number of carriers a user can transmit on. The relationship between outcome fairness and effort fairness in an OFDMA system with a finite number of transmission rates is given by Theorem 4.2. Theorem 4.2. Consider an OFDMA system with N carriers, M time slots, and K users. k . Let r (m, n) represent the transEach user k has minimum transmission requirement Rmin k

mission rate for user k on carrier n in time slot m. Let T be the duration of a single time slot, let {0, Y1 , · · · , YQ } be the set of modulation and coding levels per carrier, and V = {v0 , v1 , · · · , vQ } where v0 = 0 and vk = Yk /(M N T ) be the corresponding set of available transmission rates per carrier. Let f (vq ) represent minimum SNR requirement for any user to transmit at rate vq on some carrier. If Rk Rl Rk − Rl ≤ , min min

(4.10)

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then     vM Nk vM 1 Rk vM Nl Rl −1 − l ≤ + l Rk k v1 2 Rmin Rmin Rmin min   vM  vM  +1 + + k , Rl 2 v1 min Rmin min

(4.11)

The proof of this theorem is also in Appendix D. Equation (4.11) shows the effect of various parameters on the effort fairness of the OFDMA system. The first term shows the effect of having more than one transmission rate per carrier, if v1 = vM , the modulation and coding schemes would be fixed and this term would be 0. The second term shows the effect of the discrepancy in effort fairness on outcome fairness. In the proof in Appendix D it is shown that the last term is a result of the granularity of the transmission rates. 4.2.3

The Modified Resource Allocation Problem

Based on the two theorems above, it is possible to derive an upper bound on effort fairness for an OFDMA system: Corollary 4.3. If the conditions of Theorem 4.1 and 4.2 are both met, then     vM Nk  v vM v N v M M M l  −1 + 3 −1 + − l ≤ Rk k , Rl v1 2 v1 Rmin min Rmin min min

(4.12)

where =

∆ 1 . k l M N T min Rmin , Rmin

(4.13)

Whenever the transmission rate is much greater than vM = YM /(M N T ), and  is much less than zero, the largest term on the right hand side of Equation (4.12) is the first term. The effort fairness of the system can be improved by decreasing YM or increasing Y1 . Theoretically, decreasing YM decreases the total throughput of the system, while increasing Y1 reduces the number of transmission rates available to reduce the total transmission power. In practice, it is assumed that individual mobile stations will determine their own transmission powers and transmission rates. In this case, decreasing YM will still reduce the number of users who can be served by the system, while increasing Y1 reduces the maximum

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number of carriers per user. This means that more users can be accomodated, while the decision of transmission rates per carrier are left up to the mobiles. The Bandwidth Assignment Based on SNR (BABS) algorithm from Section 3.4.1 is used to allocate enough bandwidth to users to satisfy minimum rate requirements. In this chapter, the BABS algorithm is modified to limit the maximum number of carriers that are allocated to a user according to M X N X

δ [rk (m, n)] ≤

m=1 n=1

k Rmin vmin

(4.14)

k for all users k and all time slots m. Rmin is the minimum transmission rate requested by

user k, and vmin is the minimum allowed modulation rate. vmin is determined based on the ammount of traffic in the system. With the addition of this condition, the scheduling problem becomes: min

M X N X K X f (rk (m, n))

PT =

s.t.

M X N X

k rk (m, n) ≥ Rmin ,

m=1 n=1 N X



δ [rk (m, n)] ≤

n=1 K X

(4.15)

|Hk (n)|2

m=1 n=1 k=1

 k Rmin , vmin

δ [rk (m, n)] ≤ 1,

1 ≤ k ≤ K,

1 ≤ k ≤ K,

1 ≤ m ≤ M,

(4.16)

1 ≤ m ≤ M,

1 ≤ n ≤ N,

(4.17)

(4.18)

k=1

where δ [.] is the Kronecker delta function. Ideally, vmin is chosen to solve min s.t.

vmin

(4.19)

vmin ≥ v1  k   Rmin Pr < Navail < Ptarget Rmax

(4.20) (4.21)

where Navail is the number of carriers available for assignment, and Ptarget is the targeted   k ) = Rk /v probability of user blocking. Let Mmax (Rmin min min be the maximum number of carriers that a user can transmit on when the minimum transmission rate is set to vmin . As

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vmin is decreased, Mmax decreases for all users, and the number of available carriers Navail at any time will increase. In this chapter vmin was chosen based on simulation results. 4.3

The Online Greedy Scheduling Algorithms

This section introduces the scheduling and resource allocation algorithms which were developed for the OFDMA systems, the Simple Scheduling Algorithm and the Adaptive Scheduling Algorithm. Both algorithms are based on the sensible greedy principle described in the previous chapter. It is assumed that only one new user will arrive to a cell per frame. In a multicellular system, the cells may employ a staggered frame MAC protocol [21], where a frame is divided into several subframes, and each cell performs subcarrier allocation during one of the subframes. This allows users in one cell to sense interference from users in the other cell who are using the same frequencies. The first stage of the scheduling algorithm decides the number of carriers that will be allocated to a user, while the second stage decides the allocation of carriers. Both algorithms use a modified version of the Bandwidth Assignment Based on SNR (BABS) algorithm from Section 3.4.1 of this dissertation. The Modified BABS (M-BABS) algorithm is described in Algorithm 4.1. It differs from the original algorithm in two respects. First, the number of carriers that a user who is already in the system is allocated cannot be decreased until that user exits the system. Second, the maximum number of carriers that a user is allowed to have is bounded   k ) = Rk /v from above by Mmax (Rmin min min . The parameter vmin is determined based on simulation results. Specifically for the traffic model described in Section 4.4, the simulations in Section 4.5 show that vmin = 2.0 yields good results. 4.3.1

Simple Scheduling Algorithm

The Simple Scheduling algorithm consists of the Modified BABS algorithm, followed by the Least Interference Algorithm [17]. The Least Interference Algorithm chooses the carrier with highest SINR for the user to transmit on. In this dissertation instead of being restricted to a single channel, users are allowed to reserve and transmit on more than one channel at a

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Algorithm 4.1: Modified BABS (M-BABS) Algorithm. Ensure: Mobile Unit l (MU l) enters active state. K is set of users in active state, who may be assigned more carriers. l ∈ K. l l m R 1: ml ← R min . max PK 2: if k=1 mk >= N then ml ← 0.

3: 4: 5:

else while

PK

< N do  k  Rmin Gk ← mHk +1 f (mk +1) − k

6:

k=1 mk

7:

k ∗ ← arg min Gk .

8:

mk∗ ← mk∗ + 1.

mk Hk f



k Rmin mk



,

k ∈ K.

k∈K

9:

end while

11:

for all users k ∈ K do  k ) . mk ← min mk , Mmax (Rmin

12:

end for

10:

13:

end if

time. The number of carriers allocated to a user is determined by the M-BABS algorithm. The difference between this algorithm and the ACG algorithm of the previous chapter is that in this algorithm is online, meaning that a user’s transmission requirements are revealed to the base station only when the user requests access. This means that the number of carriers that are used must be limited, so that incoming calls will not be blocked. By contrast the ACG algorithm is an offline algorithm, and all present and future information about mobile stations are assumed to be known at the time the scheduling algorithm is executed. The algorithm is described in Algorithm 4.2 and Figure 4.1. Second, the algorithm is partially distributed, that is, the decision of which carriers the user will transmit on is left to the user rather than the base station. This distributed implementation requires that there be a very low probability of two users simultaneously beginning transmition on the same carrier. Alternatively, a multiple access protocol may

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Algorithm 4.2: Simple Scheduling Algorithm Ensure: Mobile Unit k (MU k) enters active state. mk is the number of subcarriers allocated to MU k, Ak ← {}. 1:

for each subcarrier n = 1:N, do MU k polls carrier to obtain hk (n).

2: 3:

end for

4:

MU k : Hk ←

5:

MU k transmits Hk to BS.

6:

BS : K ← {k}.

7:

BS : Run M-BABS Algorithm (Algorithm 4.1) and obtain mk , number of carriers as-

1 N

PN −1 n=0

|hk (n)|2 .

signed to MU k. 8:

BS transmits mk to MU k.

9:

while #Ak > mk do

10:

MU k : n∗ ← arg max1≤n≤N |Hk (n)|2

11:

MU k : Ak ← Ak ∪ {n∗ }.

12:

end while

13:

MU k transmits Ak to BS.

be used to allow the base station to coordinate requests from the mobiles. Mobile units will contact the base station after selecting their carriers, and can begin transmission after receiving approval. It is assumed that users have full knowledge of who is transmitting on each channel. Either the multiple access protocol ensures that this information is available to all mobiles at all times, or the carriers can be gain can be “sensed” as in the IEEE 802.11 protocol, determining the busy carriers based on SNR.

4.3.2

Adaptive Scheduling Algorithm

The Simplified Scheduling algorithm allocates carriers to a user when that user begins transmission, but does not allow any reallocation of carriers once a user begins transmitting. The

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Mobile Unit

(1) Poll all carriers, find average SINR (4) Choose mk carriers with highest gain.

Base Station

(1) Hk : average SINR. (2) Run M-BABS. (3) mk : no. carriers. (5) Ak : list of carriers.

(6) Record used carriers.

Figure 4.1: Simple Scheduling Algorithm flowchart.

Adaptive Scheduling Algorithm introduced in this section builds on the Simple Scheduling Algorithm by allowing new carriers to be allocated to a user who is already transmitting. Since a carrier assigned to a user can’t be unassigned, the algorithm is nonpreemptive. The algorithm remains centralized in allocating the number of carriers. The Adaptive Scheduling algorithm is described in Algorithm 4.3. It differs from the Simple Scheduling Algorithm of the previous section in that the set of users to be updated using the M-BABS algorithm includes all users currently transmitting, rather than only the incoming user. The Adaptive Scheduling Algorithm requires that users keep track of channel conditions while they are transmitting and that this information is regularly relayed to the base station, so that the M-BABS algorithm can yield a good allocation scheme. At the end of the algorithm any users who are allocated additional bandwidth must be notified of this. It may be necessary to stagger the notification to prevent two users from choosing the same carrier and causing interference in the system [21].

4.4

Simulation Setup

The simulation model used in this chapter is based on the model in Chapter 3. Unlike the static model in the previous chapter, both traffic and channel coefficients are assumed to change in time. This section describes how the system changes.

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Algorithm 4.3: Adaptive Scheduling algorithm Ensure: Mobile Unit k (MU k) enters active state. mk is the number of subcarriers allocated to MU k, Ak ← {}. 1:

for all subcarriers n = 1:N, do MU k polls carrier to obtain hk (n).

2: 3:

end for

4:

MU k : Hk ←

5:

MU k transmits Hk to BS.

6:

BS : K ← {1 ≤ k ≤ K : k is talking}.

7:

BS : Run M-BABS Algorithm (Algorithm 4.1) and obtain ml , number of carriers as-

1 N

PN −1 n=0

|hk (n)|2 .

signed to all MU l ∈ K. 8:

for all users l ∈ K do BS transmits ml to MU l.

9:

while #Al > ml do

10: 11:

MU l : n∗ ← arg max1≤n≤N |Hl (n)|2

12:

MU l : Al ← Al ∪ {n∗ }.

13:

end while

14:

MU l transmits Al to BS.

15:

end for

4.4.1

Time Varying Channel

In the slow fading coefficients are not assumed to change in time. The multipath components, however, are modeled as time varying [123]. Despite some reservations expressed in the literature [87], this model is chosen because of ease of implementation, and because it can easily be extended to a MIMO system. We start with the standard multipath channel model: h(t) =

∞ X

βl ejφl δ(t − τl )

(4.22)

l=0

In this model, the fast fading coefficients are assumed to be Rayleigh distributed random

120

processes. Each path l is independently generated from a complex Gaussian process, a first order filter was implemented, so that the path coefficient for time t is βlt = 0.8βlt−1 + 0.2Blt where B is also a complex Gaussian random variable. For this application, it is assumed that mobile units move slowly compared to the rate of traffic arrival and departure, thus the slow fading coefficients were assumed to be constant for the duration of a reservation. 4.4.2

Dynamic Traffic Model

In this dissertation, all traffic is assumed to be distributed according to the classical two state Markov Modulated process (on-off model) [12], in which sources alternate between an active state (“talkspurt”) during which they generate a constant stream of packets, and a silent state (“principle gap”) during which they are silent. The duration of a talkspurt (principle gap) is assumed to be exponentially distributed with a mean of tON = tOF F = 1 second [12]. The channel for a user changes slowly during a talkspurt. The channels are modeled as being completely independent between different talkspurts of the same user. Throughout this dissertation, system users are divided into three categories: voice, video and data. In simulations, 10% of users are video users, 40% are voice and 50% are data. When a user enters active mode, it is assigned randomly to transmit voice, video or data. • Voice and Video Traffic: Voice terminals generate a steady flow of 16kbps during the ON state, whereas video terminals generate a flow of 64kbps. • Data Traffic:

Data traffic is assumed fit the exponential arrival, exponential mes-

sage length model, for the purposes of this study, to reduce latency, data traffic which requires high bandwidth is transmitted at a rate proportional to the bandwidth request. The transmission time is kept constant. The mean packet size is assumed to be 180kB, which requires transmission rates which are exponentially distributed with mean 180kbps, and average duration 1 second. All other system parameters are the same as those described in Chapter 3.

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4.5

Simulation Results

In simulations, K = 60 users were put into the system, with N = 128 subcarriers. The algorithms are compared based on the probability that a user could not be allocated enough resources to satisfy minimum rate requirements (outage probability), the average transmission power per bit, the number of subcarrier allocations per unit time, and the computational complexity measured in CPU cycles. 4.5.1

The M-BABS Algorithm

0

10

Mmax = 0.5 Mmax = 1.0

−1

10

Outage Probability

Mmax = 1.5 −2

10

M

max

= 1.8

Mmax = 2.0

−3

Mmax = 2.2

10

−4

10

−5

10

0

50

100

150

200

250 Time Index

300

350

400

450

500

Figure 4.2: Effect of Mmax , time vs. outage probability.

Simulation results were used to choose Mmax . Keeping the average load per carrier is kept above some threshold. will reserve carriers for incoming users without significantly degrading performance (total transmission power) for the times when there are few users in the system. Figures 4.2-4.3 illustrate the resulting outage probability and total transmission power when choosing Mmax is varied. The average transmission power per bit is reduced significantly as Mmax is increased until 1.8. The lowest average transmission power per

122

30

Mmax = 0.5

Power per Bit Transmitted (PT / Rtot )

25

Mmax = 1.0 20

Mmax = 1.5 15

Mmax = 1.8

10

Mmax = 2.0 Mmax = 2.2

5

0

50

100

150

200

250

300

350

400

450

500

Time Index

Figure 4.3: Effect of Mmax , time vs. transmission power per bit.

bit is obtained at Mmax = 2.0. At this point the advantage of having free carriers for an incoming user to choose from is outweighed by the cost of using a higher modulation rate. In the following simulations, Mmax = 2.0. Figure 4.4 compares bandwidth allocation using M-BABS to rate proportional bandwidth allocation. A carrier allocation scheme is not implemented, and in both algorithms the mobile unit chooses carriers randomly. As in the Simple Scheduling Algorithm, users keep their assigned carriers and do not receive any additional carriers until the end of their talkspurt. The results show that the M-BABS algorithm alone causes significant reduction in the probability of outage.

4.5.2

The Online Greedy Scheduling Algorithms

Figures 4.5-4.8 compare four algorithms. The first is the BABS-ACG algorithm from the previous chapter, re-run during every time slot. The second is a random FDMA algorithm, where the number of carriers a user is allocated is chosen using the M-BABS algorithm, but

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0.06

Rate Proportional Bandwidth + Random Carrier Allocation

Probability of Outage

0.04

0.02

M−BABS + Random Carrier Allocation

0

0

50

100

150

200

250 Time Index

300

350

400

450

500

Figure 4.4: M-BABS and rate proportional bandwidth allocation, time vs. outage probability.

the subcarrier allocation is random. The users are not required to change channels until they finish transmission. The reason that the MBABS algorithm was used in the FDMA example was that as shown in Figure 4.4, if the bandwidth distribution is based only on the transmission rate requirements the outage rate is unacceptably high and does not allow a fair comparison between the methods. The results show that the Simple Scheduling algorithm performs considerably better than random carrier allocation in terms of outage probability and the average power requirements. The Adaptive Scheduling Algorithm retains a slight advantage in terms of power requirements over the Simple Scheduling algorithm, despite having very low probability of outage, on the order of that of the BABS-ACG algorithm. This result depends on the rate at which the wireless channel changes over time, however. For a channel which is not highly correlated in time, the BABS-ACG algorithm may perform considerably better. Figure 4.7 shows that although the BABS-ACG algorithm, as expected, achieves the lowest transmission power and outage probability, it requires a large number of reallocations of

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carriers. Surprisingly, the Adaptive Scheduling Algorithm seems to require fewer carrier reallocations than the simple scheduling algorithm. This is due to the higher outage probability of the Simple Scheduling Algorithm, since a user may be dropped and reaccepted to the system several times, and must be reallocated carriers in each try. The random FDMA algorithm is not shown, but has about the same number of carrier reallocations as the Simple Scheduling Algorithm. Figure 4.8 compares the combined computational load of scheduling on both the base station and the mobile units. The simulations are carried out on an AMD Duron based PC running Linux with 42.9 SPECint 95 and 29.4 SPECfp 95 ratings. In this implementation, the Adaptive Scheduling Algorithm did not require significant extra processing load over the Simple Scheduling Algorithm.

M−BABS + Random Carrier Allocation

−2

10

Simple Scheduling Algorithm

−3

Probability of Outage

10

Adaptive Scheduling Algorithm −4

10

−5

10

BABS + ACG

−6

10

0

50

100

150

200

250

300

350

400

450

Time Index

Figure 4.5: Probability of outage for scheduling algorithms.

500

125

2

10 80

60

T

Power per Bit Transmitted (P / R

tot

)

M−BABS + Random Carrier Allocation 40

20 Simple Scheduling Algorithm Adaptive Scheduling Algorithm

1

10

BABS + ACG 8 6 0

50

100

150

200

250 Time Index

300

350

400

450

500

Figure 4.6: Transmission power per bit for scheduling algorithms.

4.6

Conclusion

For a time varying system where reducing latency is important, online scheduling and resource allocation need to be performed together. In this chapter the problem of resource allocation for OFDMA is analyzed in the context of fairness, and a modified resource allocation and scheduling problem is proposed. Two simple resource allocation and scheduling algorithms are introduced which extend the the Sensible Greedy algorithms described in the previous chapter for use in time varying channels and traffic. It is shown that adapting the bandwidth allocated to the channel conditions for a user results in considerable improvement in outage probability and transmission power. The scheduling algorithms proposed are simple online nonpreemptive partially distributed algorithms. The system is allowed to adjust to channel conditions by allocating more resources to users whose channel conditions may have worsened. This approach allows efficient use of system resources with lower overhead due to channel reallocation and lower computational complexity. Simulations show that the algorithms yield low outage probability, and low power requirements at reasonable

126

0.4

No. Carrier Allocations and Reallocations.

0.35

0.3

0.25

0.2

BABS ACG

0.15

Simple Scheduling Algorithm 0.1 Adaptive Scheduling Algorithm 0.05

0

0

50

100

150

200

250 Time Index

300

350

400

450

500

Figure 4.7: The number of carrier allocations and reallocations in scheduling algorithms.

complexity and overhead. The convergence and performance of these algorithms will depend on how quickly channel conditions and traffic conditions change, and this is a course for futher study.

127

6

4

x 10

3.5

3

CPU Time

2.5

2 Adaptive Scheduling Algorithm Simple Scheduling Algorithm

1.5

1

0.5

0

0

50

100

150

200

250 Time Index

300

350

400

450

500

Figure 4.8: CPU cycles per frame for the scheduling algorithms.

128

Chapter 5 SUMMARY AND CONCLUSIONS

This dissertation has explored analytical performance estimation and resource allocation for multicarrier communications. The effect of frequency offset on MC-CDMA systems was formulated. Analytical formulas were derived to show the degradation in SINR for known channels, and the asymptotic limit of such degradation for unknown channels as the number of carriers and the number of users go to infinity. Static resource allocation algorithms were derived for OFDMA systems. Previous algorithms were studied, a modified Lagrangian Relaxation algorithm was derived which was shown to work better than the original algorithm proposed by Wong et.al. [137], and an optimal branch and bound algorithm was derived for carrier and power allocation. A family of heuristic algorithms were derived which were shown to yield good allocation strategies at low complexity. Finally, dynamic online scheduling algorithms were derived for OFDMA systems with known carrier coefficients with time-varying traffic. 5.1

Summary

Chapter 1 of this dissertation provided an introduction to the topic of multicarrier communications and introduced the notation that has been used in the remainder of the dissertation. In Chapter 2, an analytical expression was derived for the error incurred through frequency offset on the uplink channel of MC-CDMA. One of the most important drawbacks of multicarrier schemes is their sensitivity to frequency offset. The framework developed considered variable spacing between carrier frequencies. Each user was assumed to experience a different frequency offset. LMMSE and MF receivers were considered. The analysis was verified through simulations. Chapter 3 studied the problem of subcarrier and power allocation for uplink communication to multiple users in an OFDM based wireless system. The problem of minimizing

129

total power consumption with constraints on BER and transmission rate for users requiring different classes of service was formulated and simple algorithms with good performance were derived. The problem of joint allocation was divided into two steps. In the first step, the number of subcarriers that each user will get was determined based on the users’ average SNR. The algorithm was shown to find the distribution of subcarriers that minimizes the total power required when every user experiences a flat fading channel. The second stage of the algorithm finds the best assignment of subcarriers to users. Two different approaches were presented, the Rate Craving Greedy (RCG) algorithm and the Amplitude Craving Greedy (ACG) algorithm. Numerical results demonstrated that the proposed low complexity algorithms offer comparable performance with an existing iterative algorithm Chapter 4 studied the problem of finding an optimal subcarrier and power allocation strategy for uplink communication to multiple users in a time varying OFDM based wireless system. Bounds were derived for the rate of effort and outcome fairness achievable in an OFDMA scheduling system. The problem of minimizing total power consumption with constraints on BER and transmission rate for users requiring different classes of service was studied for a time varying system. The problem of resource allocation was reformulated to take into account the fairness of the allocation. Two new resource allocation and scheduling algorithms were introduced, a simple scheduling algorithm and an adaptive scheduling algorithm. These algorithms reduced the overhead required for carrier and power allocation compared to previous approaches while retaining acceptable level of service at affordable power levels. The algorithms were studied through simulation. 5.2

Future Work

Future work is divided into three areas: performance estimation for multicarrier communications, resource allocation and scheduling for OFDMA. In Chapter 2, the analytic effect of frequency offset on an MC-CDMA system was investigated. An asymptotic expression was derived for the SINR at the output of a MF and a LMMSE receiver as the number of carriers and the number of users go to infinity, while the ratio of users to carriers remain constant. The analysis assumed that channel coefficients were unknown, but independent and identically distributed with a known probability

130

distribution. Theorem 2.6 shows that for a LMMSE receiver, the SINR depends on the properties of the matrix   2 ˜ jA ˜H Xj = A j + σz I . It remains to be seen whether such a relationship will hold for correlated carriers. In previous work, Peacock, Collings and Honig [83] note that when carrier coefficients are correlated, the asymptotic SINR at the output of an MC-CDMA system without carrier offset will depend on the eigenvalues of matrix Xj for the particular distribution of carriers. However, in earlier work Tse [128] notes that for random matrices, it is “freeness” and not the independence of the random vectors that form the spreading codes for the receiver that result in the expressions that have been derived for MIMO systems. Li, Tulino and Verdu [69] derive a receiver for MC-CDMA systems based on the asymptotic properties of correlated channels. Based on these results, it remains to be seen whether these techniques can be used to generalize Theorem 2.6 to correlated channels. Another topic of interest is to generalize the results to systems which do not have perfect power control, where users have different average transmission power, and to systems with timing offset in addition to frequency offset between users. Finally, with the growing use of wideband multicarrier transmission systems, frequency offset due to timing offset error and Doppler spread have become increasingly important, and further work will focus on frequency offset which is not constant for all carriers. In Chapter 3, resource allocation algorithms were derived for OFDMA systems. One problem in this research has been the computational complexity of optimal algorithms. While a branch and bound algorithm has been proposed for resource allocation, the performance of this algorithm and related algorithms proposed earlier by Inhyoung Kim et.al. [56] is highly dependent on the distribution of carrier gains, rate requirements, available modulation levels, and the rate-power function. Hogg et.al. [43] give an introduction to the importance of phase transitions in the study of search algorithms. The worst case complexity of heuristic search methods such as branch and bound algorithms show a two-state behavior. Either the search algorithm terminates very quickly or it takes a very long time. It may be of interest to investigate this field to see what conditions produce algorithms that

131

terminate quickly. Chapter 4 addresses the problem of scheduling for OFDMA systems. There are several avenues for further research in this field. Low power scheduling is a topic of interest in sensor networks and other ad hoc networks. When transmission power is limited, it may be of interest to limit transmission to other nodes and collaborate in forwarding packets. If a joint scheduling approach is used, then the problem becomes much more complicated. Even when a distributed algorithm is employed, as proposed by Somani and Zhou [112], the routing topology used is also important. Pattara-Atikom, Krishnamurthy and Banerjee [82] present a review of distributed allocation algorithms for 802.11 systems, research in this area is ongoing. More generally, distributed resource allocation and scheduling algorithms involve little or no information being transmitted to the base station, whereas centralized resource allocation and scheduling algorithms require all channel and rate information to be perfectly known. The information transmitted between the mobile stations and the base station is known as protocol information [36]. The relationship between protocol information, system model and attainable rate, delay, power and bandwidth requirements, first investigated by Gallager [36] remains a topic of interest [72][7][93][138]. The extension to multi-antenna systems is also of interest. This dissertation has only explored single antenna systems. Recently Viswanath, Tse and Laroia [135] have introduced the concept of opportunistic scheduling, whereby the channel is actively randomized to improve the performance of the multiuser system. Further work in scheduling may incorporate opportunistic beamforming or space-time coding.

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144

Appendix A SUMMARY OF NOTATION A.1

General Terms

Boldface lower case fonts are used for vectors, x = [x0 , x1 , ..., xN −1 ]T . Element k of vector x is denoted xk or x(k). [xk ] denotes the vector with elements xk . Boldface upper case fonts are used for matrices,     A=   

a1,1

a1,2

···

a1,M

a2,1 .. .

a2,2 .. .

··· .. .

a2,M .. .

aN,1 aN,2 · · ·

aN,M

       =    

 aT1  ..  . .  T aN

Column n of matrix A is denoted an , and element (m, n) is Am,n or an (m). [am,n ] denotes the matrix with elements am,n . diag [d0 , d1 , · · · , dN −1 ] represents a diagonal matrix with non-zero elements d0 , ..., dN −1 . Estimates of a variable are denoted by the hat symbol, sˆk represents the estimate of the value sk . I

Identity matrix.

IN ×N

Identity matrix of size N .

1

Vector of 1’s, 1 = [1, 1, ..., 1]T .

1N

Vector of 1’s, of length N .

A∗

Complex conjugate of matrix A.

AT

Transpose of matrix A.

AH

Hermitian of matrix A.

1H A1

Sum of all elements of matrix A.

tr(A)

Trace of matrix A.

145

||A||22

 L2 -norm of matrix A, ||A||2 = tr AAH .

A B

Hadamard (element-by-element) product of matrices; If C = A B then Ck,n = Ak,n Bk,n .

A⊗B

Tensor product of matrices, 

A B ···  1,1  . .. A ⊗ B =  .. .  AM,1 B · · ·

 A1,N B  ..  . .  AM,N B

N

Set of natural numbers, N = {1, 2, ...}.

R

Set of real numbers.

C

Field of complex numbers.

E f {X}

Expectation of random variable X with respect to f .

Pr[A]

Probability of event A

Pr[A|B]

Probability of event A given B

dxe

Smallest integer greater than x (ceiling function).

bxc

Greatest integer less than x (floor function).

IA

Indicator function, where A is a boolean expression:   1 if A is true, IA =  0 otherwise.

#S

Cardinality of (number of elements in) set S.

[x]+

The function: [x]+ = x I{x>0}

log

logarithm base 10

ln

natural logarithm

f :A→B

denotes that function f maps elements in set A to elements in set B

f 0 (·)

Derivative of f (·).

146

f −1 (·)

Inverse function of function f (·).

O(f )

big-oh notation

Ω(f )

big-omega notation

Θ(f )

big-theta notation

A.2

Symbols Relating to MC-CDMA and OFDMA

K

Number of users.

N

Number of carriers.

M

Number of carriers used for transmission in MC-CDMA modulation.

α

Bandwidth scaling ratio defined (p. 17) as the ratio of intercarrier spacing to subcarrier bandwidth α = N/M .

β

Ratio of number of users to number of carriers or degree of freedom for Multicarrier CDMA, β = K/N .

Ts

Duration of useful portion of an OFDM symbol.

Tg

Duration of cyclic prefix.

Ts + Tg

Duration of one OFDM symbol.

∆f

Inter-carrier frequency spacing.

fn

Baseband frequency of carrier n, fn = n∆f .

sk (l)

Symbol transmitted by user k in frame l.

xn (l)

Symbol received on carrier n in frame l.

zn (l)

Instance of white Gaussian noise received on carrier n in frame l.

W

DFT matrix, W = [wn,m ] where wn,m = e−j2πnm/N .

H WM

M-ary DFT matrix.

M

Modulation matrix for an OFDM system with M carriers bandwidth i h √1 ej2πnm/M . scaling ratio α and with elements MH = n,m α

C

Matrix of CDMA codes for users in MC-CDMA system.

ck

CDMA code for user k.

147

H

Matrix of channel gains.

hk

Vector of channel gains for user k.

hk (n)

Channel gain for user k on carrier n.

A

Matrix of composite effect of CDMA codes and channel gains for users in MC-CDMA system, A = C H.

Aj

Matrix of composite effect of CDMA codes and channel gains for all users except user j in an MC-CDMA system, matrix A with column j removed.

σz

 Covariance of signal z, σz2 = E |z|2 .

X

Covariance of received signal in an MC-CDMA system, X = AAH +σz2 I.

Xj

Covariance of interference and noise for user j in an MC-CDMA system, 2 Xj = Aj AH j + σz I.

δFk

Frequency offset error for user k.

SINRj

Signal to interference and noise ratio for user j at the output of a multiuser detectorr.

SINR∞ detector,j

Asymptotic SINR for user j at the output of a multiuser detector of type “detector” in the limit as the number of users and the number of carriers go to infinity while the ratios α and β remain constant.

∆ SINRj

Degradation in SINR due to frequency offset: ∆ SINRj

∆ SINR∞ detector,j

SINRj SINRj |δF =0 = SINRj (dB) − SINRj |δF =0 (dB) =

Asymptotic degradation in SINR due to frequency offset: ∆ SINR∞ detector,j

=

SINR∞ detector,j ∞ SINRdetector,j

δF =0 ∞ = SINR∞ ( detector,j dB) − SINRdetector,j δF =0 (dB)

148

G(·)

Probability density function (pdf) G of relative frequency offsets δFk T .

ΦG (·)

Characteristic distribution function (cdf) of the distribution G of relative frequency offsets δFk T .

σF

Covariance of frequency offset, averaged over the frequency offset dis tribution of received signals, σn2 = E t |δF |2 .

D(δF )

Frequency offset matrix for user k:  D(δF ) = diag 1, ej2πδFk T /N , · · · , ej2π(N −1)δFk T /N .

p(δF )

Frequency offset vector for user k:  T p(δFk ) = 1, ej2πδFk T /N , · · · , ej2(N −1)πδFk T /N .

Ψ

Matrix, expected effect of frequency offsets, Ψ = [Ψn,m ] Ψn,m = E δF {cos (2π(n − m)δF T /N )}.

Ek

Total recieved signal power for user k after multi-user receiver.

Vk

Total power of multiple-access interference for user k after multi-user receiver.

Rmax

Maximum transmission rate per carrier, determined by coding and modulation schemes available.

Pe

Maximum tolerable probability of error.

k Rmin

Transmission rate requested by user k.

k Pmax

Maximum available transmission power for user k.

Rk

Total transmission rate for user k.

Pk

Total transmission power for user k.

rk (n)

Transmission rate for user k on carrier n.

pk (n)

Transmission power for user k on carrier n.

f (·)

Rate function, relating the maximum available transmission power to the transmission rate, minimum acceptable probability of error and channel gain: f (rk (n)) = pk (n) |hk (n)|2 for user k on carrier n for a given Pe .

149

kmin (n)

For FDMA systems, user transmitting on carrier n

mk

Number of carriers allocated to user k in an FDMA algorithm.

ρk (n)



For FDMA systems, without relaxation, ρk (n) = 1 if user k transmits on carrier n, ρk (n) = 0 otherwise.



For FDMA-TDMA or “relaxed” FDMA solutions, fraction of resources (e.g. time) allocated to user k on carrier n.

Hf (x)

Function used by the Lagrangian Relaxation algorithm, to calculate the “shadow pricing coefficient”    Hf (x) = f f 0−1 (x) − xf 0−1 (x) ,

λk

(A.1)

Shadow power prices, or “water level” used to rank users in the Lagrangian Relaxation algorithm.

F

Set of feasible subcarrier allocations among users for ACG or RCO algorithms.

150

Appendix B APPENDIX FOR CHAPTER 2

In Chapter 2, expressions are derived which describe the effect of random frequency offset in an MC-CDMA system. In the system of interest, there are K users and M carrier frequencies. The effective channel spreading vector for user k is ak = [ak,0 ak,1 · · · ak,M −1 ]T . The modulation matrix is    1   √ M= α  

1

1

···

1



1 .. .

ej2πα/N .. .

··· .. .

ej2π(M −1)α/N .. .

···

ej2π(N −1)(M −1)α/N

      

1 ej2π(N −1)α/N

(B.1)

and the demodulation matrix is MH . After spreading, modulation and demodulation, the received signal is filtered using a linear multiuser receiver. The received signal from user j is given below, K X

H sˆj,l = bH j M D(δFj,l )Maj sj,l +

H H bH j M D(δFk,l )Mak sk,l + bj z.

(B.2)

k=1,k6=j

and the SINR of this signal is: SINRj =

σz2



Ej  bH b + V j − Ej j j

(B.3)

where Ej

2 H = bH j M D(δFj )Maj

Vj

=

K X H H bj M D(δFk )Mak 2

(B.4) (B.5)

k=1

In this expression, Ej represent the signal power for user j (term 1 in (B.2)) and Vj represent the total signal and interference power (term 1 + term 2).

151

The following terms are used throughout the proofs in this chapter: H ˜ aj = WM aj

(B.6)

H ˜ j = WM b bj H H 2 Vj,k = bj M D (δFk ) Mak  2 sin (πδFk T ) ∆α,k = α sin (πδFk T /α) θk = πδFk T   DN (δF ) = diag 1, ej2πδF T /N , · · · , ej2πδF T (N −1)/N h iT pN (δF ) = 1, ej2πδF T /N , · · · , ej2πδF T (N −1)/N

B.1

(B.7) (B.8) (B.9) (B.10) (B.11) (B.12)

Average SINR for known channels

This section contains the proofs of Theorems 2.2 – 2.4. These theorems describe the SINR at the output of a linear receiver in a channel with known coefficients. B.1.1

Lemmas:

There are several basic properties of matrices which are used in the derivations in this chapter. Those that were found in literature are listed with references, those not found directly in the literature are included with proofs. Lemma B.1. Let A ∈ C N ×N and D ∈ C N ×N be a diagonal matrix with nonzero elements D = diag [D0 , D1 , · · · , DN −1 ], and p = [D0 , D1 , · · · , DN −1 ]T . Then  DADH = ppH A. Proof of Lemma B.1: 1. Let Q = DADH . Then element (i, j) of matrix Q is: Qi,j

=

L1 X

Di,l1

l1 =0

=

L2 X

l2 =0 ∗ Di,i Ai,j Dj,j

= Di Dj∗ Ai,j

∗ Al1 ,l2 Dj,l 2

(B.13)

152

2. Similarly, let R = ppH A. Then, element (i, j) of matrix R is: Ri,j

= Di Dj∗ Ai,j = Di Dj∗ Ai,j = Qi,j .

Lemma B.2 ([46, p.306]). Let A, B ∈ C N ×N . Then  tr (AB) = 1H A BT 1. Lemma

B.3. Let x, y



CN ,

D



C N ×N

be a diagonal matrix D

=

T

diag [D0 , D1 , · · · , DN −1 ], let p ∈ C N be p = [D0 , D1 , · · · , DN −1 ] and 1 = [1 · · · 1] ∈ C N  H    x Dy 2 = 1H xxH ∗ ppH yyH 1 (B.14) Proof of Lemma B.3 Using Lemma B.1 and Lemma B.2, H x Dy 2 = xH DyyH Dx   = xH yyH ppH x    = tr xH yyH ppH x     = tr yyH ppH xxH    T  = 1H yyH ppH xxH 1   ∗   = 1H xxH ppH yyH 1

Lemma B.4. N −1 X n=0

  n (N − 1) sin (πδF T ) exp j2πδF T = exp jπδF T . N N sin (πδF T /N ) 

(B.15)

Proof of Lemma B.4   N −1  X e(j2πδF T ) − 1 (N − 1) sin (πδF T ) n = (j2πδF T /N ) = exp jπδF T . exp j2πδF T N N sin (πδF T /N ) e −1 n=0

153

Lemma B.5 (Jensen’s Inequality [39, p.161]). A function u : R → R is called convex if for all real a there exists λ, depending on a such that u(x) ≥ u(a) + λ(x − a) for all x. If u is convex and X is a random variable with finite mean, then E {u(X)} ≥ u ( E {X}). Lemma B.6 ([46, p.244]). Let A, C ∈ C M ×M , and B, D ∈ C N ×N . Then (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD). Lemma B.7. Let A ∈ C M ×M , and B ∈ C N ×N . Then (A ⊗ B) (C ⊗ D) = (A C) ⊗ (B D) Proof of Lemma B.7: (A ⊗ B) (C ⊗ D)     A1,1 B · · · A1,M B C1,1 D · · · C1,M D     .. ..  .   ..  .. .. =  ..    . . . . .     AM,1 B · · · AM,M B CM,1 D · · · CM,M D   (A1,1 C1,1 ) (B D) · · · (A1,M C1,M ) (B D)   .  .  .. =  ..  . ..   (AM,1 CM,1 ) (B D) · · · (AM,M CM,M ) (B D) = (A C) ⊗ (B D)

Lemma B.8. Let A ∈ C M ×M , and B ∈ C N ×N . Then H 1H M ×N (A ⊗ B) 1M ×N = 1M A 1M



 1H N B 1N .

Proof of Lemma B.8: Since 1M ×N = 1N ⊗ 1M , using Lemma B.6 1H M ×N (A ⊗ B) 1M ×N

=

H 1H M ⊗ 1M

=

1H M

=

1H M



(A ⊗ B) (1M ⊗ 1N )   A 1M ⊗ 1H N B 1N   A 1M 1H N B 1N

154

Lemma B.9. If Vj,k is defined as 2 H Vj,k = bH j M DN (δFk ) Mak

(B.16)

where M is a modulation matrix for an MC-CDMA system with M carriers defined in Equation (B.1) and DN (δFk ) is the diagonal frequency offset matrix defined in Equation (B.11) then  ∗    ˜ ˜ H ΨM (δFk /α) 1. Vj,k = ∆α,k (δFk T ) 1H ˜ ak ˜ aH k bj bj 2 ˜H = ∆α,k (δFk T ) b D (δF /α)˜ a M k k j

(B.17) (B.18)

˜ j , ∆α,k , DN (δFk /α), pN are where ΨN (δFk ) = pN (δFk )pN (δFk )H , and the terms ˜ ak , b defined in Equations (B.6)-(B.12). Proof of Lemma B.9: 1. Define Vj,k as shown in (B.16). Using Lemma B.3, Vj,k = 1H



H Mbj bH j M

∗

  H Maj aH pN pH 1 j M N

(B.19)

 T where pN (δFk ) = 1, ej2πδFk T /N , · · · , ej2(N −1)πδFk T /N , is composed of the elements of the diagonal frequency offset matrix DN (δFk ). H a and b ˜ j = WH bj , 2. Since ˜ aj = WM j M



H αMaj aH j M

 ˜ aj  h i  .  H =  ..  ˜ aj · · · ˜ aH = j   ˜ aj

˜ a˜ aH  j j  ..  .  ˜ aj ˜ aH j

··· .. . ···

 ˜ aj ˜ aH j  ..  .   ˜ aj ˜ aH j

|

{z

}



    

α rows

   

α columns This can be rewritten as: H Maj aH j M =

where ⊗ denotes the tensor product.

 1 ˜ aj ˜ aH ⊗ 1α×α j α

(B.20)

155

H 3. Using the same expansion for Mbj bH j M ,

Vj,k =

∗  i    1 h H H ˜ ˜ ˜ a ˜ a ⊗ 1 b b ⊗ 1 Ψ (δF ) j α×α j α×α N k j j α2

(B.21)

where ΨN (δFk ) is defined in the lemma. 4. Using Equation (B.12) h

pN (δF ) =

1, ej2πδF T /N , · · · , ej2(N −1)πδF T /N

iT

= pα (δF ) ⊗ pM (δF/α)

5. Using Lemma B.6, ΨN (δF ) = (pα (δF ) ⊗ pM (δF/α))H (pα (δF ) ⊗ pM (δF/α))   = pα (δF )pα (δF )H ⊗ pM (δF/α)pM (δF/α)H = Ψα (δF ) ⊗ ΨM (δF/α) 6. Using Equation (B.21), and Lemma B.7 Vj,k =

=

 ∗    1 Hh H ˜j b ˜H 1 ˜ a ˜ a ⊗ 1 b ⊗ 1 j α×α α×α j j α2 ((ΨM (δF/α)) ⊗ (Ψα (δF )))] 1 h ∗  i   1 H H H ˜ ˜ 1 ˜ a ˜ a b b Ψ (δF/α) ⊗ (Ψ (δF )) 1 j j j j α M α2

7. Using Lemma B.8, Vj,k =

1 α2

∗  i   H ˜ ˜ ˜ aj ˜ aH b b Ψ (δF/α) 1 × j M j j  H  1 (Ψα (δF )) 1

h

1H



(B.22)

8. Using Lemma B.4 H

1 (Ψα (δFk )) 1 =

α−1 X α−1 X n=0 m=0



(n − m) exp j2πδFk T α



 =

sin (πδFk T ) sin (πδk F T /α)

By the definition of ∆α,k (δF T ) in Equation (B.9)  ∆α (δFk T ) =

sin (πδFk T ) α sin (πδFk T /α)

2 =

1 H 1 (Ψα (δFk )) 1 α2

Substituting ∆α (δFk T ) in Equation (B.22) yields Equation (B.17).

2 .

156

9. Using Lemma B.3,  ∗    ˜j b ˜H Vj,k = ∆α,k (δFk T ) 1H ˜ ak ˜ aH b Ψ (δF /α) 1 M k j k 2 ˜H = ∆α,k (δFk T ) b ak j DM (δFk /α)˜

B.1.2

Proof of Theorem 2.2:

Assuming that the users’ information symbols {sk , 1 ≤ k ≤ K} are independent identically distributed with variance σs2 = 1, the noise terms zn are independent identically distributed white Gaussian noise with variance σz2 , and the frequency offset δFk for users k = 1, ..., K are all independent and identically distributed, the effect of frequency offset on the signal to noise ratio at the output of the receiver are described in the following theorem, Theorem 2.2 repeated from Chapter 2, p.23: Theorem. 2.2. Consider an MC-CDMA system where user j’s signal is the signal of interest. If the frequency offsets of all users, δFk for k = 1, ..., K are independent, and user j’s frequency offset has a cumulative density function ΦGj (δFj ) and all interfering users k = 1, ..., K, k 6= j are identically distributed with cumulative density function ΦG (δF ) then the SINR for user j at the output of the receiver is: SINRj =

Ej   Vj − Ej + σz2 tr bH j bj

(B.23)

where Ej represents the desired signal power (first term in Equation (B.23)):     H H H ∗ ¯ j Maj aH Ej = 1H Ψ M Mb b M 1, j j j

(B.24)

and Vj represents total signal and interference power:     H ∗ ¯ MAAH MH Mbj bH Vj = 1H Ψ M 1. j

(B.25)

157

¯ j and Ψ ¯ represent the effect of frequency offset on self interference and The matrices Ψ multiple access interference power. ¯j Ψ

 E δFj p(δFj )p(δFj )H .  ¯ = E δF p(δFk )p(δFk )H Ψ for k 6= j. k h iT p(δF ) = 1, ej2πδF T /N , · · · , ej2πδF T (N −1)/N . =

(B.26) (B.27) (B.28)

¯ j is given by: Element (n, m) of matrix Ψ ¯j Ψ

 n,m

  o n (n − m) δF T j2π n−m j N = ΦGj 2π = E e N

(B.29)

¯ is given by: and element (n, m) of matrix Ψ   o n (n − m) δF T j2π n−m k ¯ N Ψn,m = E e = ΦG 2π N

(B.30)

where ΦGj (·) and ΦG (·) are the characteristic functions of the distribution functions Gj and G of normalized frequency offsets of the user of interest and the interfering users respectively. Proof of Theorem 2.2 (p.23): 1. Define 2 H Vj,k = bH j M DN (δFk ) Mak .

(B.31)

By Equation (B.3)-(B.5), the average SINR for user j at the receiver output is SINRj =

Ej

 . Vj − Ej + σz2 tr bH b j j

(B.32)

The total power from the signal of interest, sj is: Ej = E δFj {Vj,j (δFj )} ,

(B.33)

and the total expected power of interference is: V j − Ej =

K X k=1 k6=j

E δFk {Vj,k (δFk )} .

(B.34)

158

2. Using Lemma B.3, Vj,k = 1H



H Mbj bH j M

∗

  H H Maj aH M pp 1 j

(B.35)

 T where p(δFk ) = 1, ej2πδFk T /N , · · · , ej2(N −1)πδFk T /N , is composed of the elements of the diagonal frequency offset matrix DN (δFk ). 3. Taking the expectation with respect to the frequency offset δFk yields: E δFk {Vj,k (δFk )}      H H H ∗ H p(δF )p(δF ) Ma a M 1 = 1H Mbj bH M E δF T k k k k j 4. Assuming that the frequency offsets for all interfering users k 6= j, are identically distributed, for k = 1, ..., j − 1, j + 1, K, define  H ¯ Ψ(δF . k ) = E δFk p(δFk )p(δFk ) ¯ is: Then element (n, m) of matrix Ψ ¯ n,m (δFk ) = E δF {exp(j2π(n − m)δFk T /N )} . Ψ k ¯ will depend on the distribution of frequency offsets, but it is easy to The matrix Ψ calculate, since it has the same form as the characteristic function of the random variable δFk . The matrix  ¯ j (δFj ) = E δF p(δFj )p(δFj )H Ψ j is defined similarly for the user of interest. 5. By definition, Ej

E δFj {Vj,j (δFj )}    H ∗ H H ¯ j Mbj bH = 1H Ψ M Ma a M 1 j j j =

(B.36)

and Vj

=

=

K X k=1 K X

E δFk {Vj,k (δFk )}    H H ∗ H ¯ Mbj bH 1H Ψ M Ma a M 1 k k j

k=1

    H ∗ ¯ MAAH MH Mbj bH = 1H Ψ M 1. j

(B.37)

159

which is the result in Equations (B.24)- (B.25).

B.1.3

Proof of Theorem 2.3:

The second result, Theorem 2.3 repeated from Chapter 2 p.25, relates the effect of random frequency offset to that of known, deterministic frequency offset: Theorem. 2.3. Assume that the probability distribution of the normalized frequency offset, G(δF T ), has zero mean and is symmetric about δF T = 0. Assume also that δF lies in the ¯ is bounded interval [−1/(2T ), 1/(2T )] with probability 1. Then element (n, m) of matrix Ψ from below by: ¯ n,m Ψ



(n − m) ≥ cos 2π σF T N

 (B.38)

r n o n o 2 T . As the fourth moment of δF T , E (δF T )4 approaches Define σF = E (δF T ) (σF T )4 , the distribution of δF T approaches a binomial distribution with δF = ±σF , and the above bound becomes arbitrarily close. Proof of Theorem 2.3 (p.25): 1. Assume that the probability distribution of the normalized frequency offset δF T had zero mean, and is symmetric about 0. Assume that δF T is in the interval [−0.5, 0.5] with probability 1. 2. From Theorem 2.2, ¯ k,l = E δF {exp(j2π(k − l)δF0 T /N )} . Ψ 0

(B.39)

160

3. The characteristic function for a probability distribution, G(δF T ) is defined as: ΦG (t) = F [G] (t) n o = E ejt(δF T ) o n 1 1 t E {(δF T ) } − t2 E (δF T )2 1! 2! o 1 n o 1 3 n 3 − j t E (δF T ) + t4 E (δF T )4 3! 4! o 1 n o 1 5 n + j t E (δF T )5 − t6 E (δF T )6 5! 6! o 1 n o 1 7 n 7 − j t E (δF T ) + t8 E (δF T )8 7! 8! + ...

= 1+j

4. Since the probability distribution G(δF T ) is assumed to be symmetric about the origin, and G(δF T ) is nonzero only in the interval [−1, 1], the odd moments of the random variable δF T will be zero, that is: n o n o E {(δF T )} = E (δF T )3 = E (δF T )5 = ... = 0 5. By the definition of the characteristic function for a probability distribution and Equation (B.39): n o n o n o ¯ k,l = 1 − 1 t2 E (δF T )2 + 1 t4 E (δF T )4 − 1 t6 E (δF T )6 + ... Ψ 2! 4! 6! 6. Consider the function gk (x) =

1 k 1 x − xk+1 . (2k)! (2k + 2)!

for k greater than or equal to 1. 7. The second dervative of the function gk (x) is: k(k − 1) k−2 (k + 1)k k−1 d2 gk (x) = x − x . 2 dx (2k)! (2k + 2)! If the first term of

d2 gk (x) dx2

is greater than or equal to the second term, then the

second derivative of gk (x) is positive, and thus the function is convex.

161

8. Studying the ratio of the two terms: k(k − 1)/(2k)!xk−2 (k − 1)(2k + 2)(2k + 1) 2(k − 1)(2k + 1) = = k−1 (k + 1)x x (k + 1)k/(2k + 2)!x which is greater than 1 whenever x is less than or equal to 2(k − 1)(2k + 1). For k greater than or equal to 2, the above term is greater than 1 when x is less than 10. Thus, the second derivative of gk (x) is greater than or equal to zero for x in the interval [0, 10]. This means that for k ≥ 2, and x in the interval [0, 10], the function gk (x) is convex. 9. By Jensen’s Inequality (Lemma B.5),

 E

E {gk (x)} ≥ gk ( E {x})  1 k 1 1 1 k+1 ≥ x − x E {x}k − E {x}k+1(B.40) . (2k)! (2k + 2)! (2k)! (2k + 2)!

for k ≥ 2,and 0 ≤ x ≤ 10. 10. Let x = (tδF T )2 , where t = (2π(k −l)/N ). Since x is the square of a real number, x is greater than 0. Since k, l are indices in an N -by-N matrix, k − l must be an integer between −(N − 1) and (N − 1), so that for all k, l, t is less than 2π. By assumption, δF T is in the interval [−1/2, 1/2]. Then by definition x must be  less than or equal to π 2 , which is less than 10. Thus the function gk (tδF T )2 is convex for k greater than or equal to 2, and δF in the intervalr[−1/2T, 1/2T ]. n o 11. Define the standard deviation of the frequency offset as σF = E (δF T )2 /T . Then the above inequality becomes     o o (k − l) 2k n 1 (k − l) 2k+2 n 1 2k 2π E (δF T ) − 2π E (δF T )2k+2 (2k)! N (2k + 2)! N  2k   1 (k − l) 1 (k − l) 2k+2 2k ≥ 2π (σF T ) − 2π (σF T )2k+2 . (2k)! N (2k + 2)! N 12. The characteristic function of G can be rewritten in terms of gk (x) as: n  o n  o 2 ¯ k,l = 1 − 1 (σF T )2 + E g2 (δF T )2 Ψ + E g4 (δF T )2 + ... 2! ¯ k,l ≥ 1 − 1 Ψ 2!

 2π

(k − l) N

2

  (σF T )2 + g2 (σF T )2 + g4 (σF T )2 + ...

162

Using the inequality in Equation (B.40) the result is   (k − l) 2 1 ¯ 2π (σF T )2 Ψk,l ≥ 1 − 2! N     1 (k − l) 4 1 (k − l) 6 4 + 2π (σF T ) − 2π (σF T )6 4! N 6! N (B.41) + ... 13. The right hand side of this Equation (B.41) is the Taylor series expansion of cos (2π(k − l)σF T /N ). Substituting this expression yields ¯ k,l ≥ cos (2π(k − l)σF T /N ) . Ψ which establishes the inequality in Equation (B.28) of the theorem. ¯ k,l can also be written in terms of gk (x) as: 14. Ψ n o ¯ k,l = 1 − 1 t2 (σF T )2 + 1 t4 E (δF T )4 Ψ 2! n  4! o o n  2 − E g5 (δF T )2 − ... − E g3 (δF T ) 15. Using the inequality in Equation (B.40) in the reverse direction, and substituting t = (2π(k − l)/N )2 , the result is   1 (k − l) 2 ¯ Ψk,l ≤ 1 − 2π (σF T )2 + 2! N   1 (k − l) 6 2π − (σF T )6 + 6! N − ...

1 4!



(k − l) 2π N

4

1 8!



(k − l) 2π N

8

n o E (δF T )4 (σF T )8

16. Substituting cos (2π(k − l)σF T /N ) again for its Taylor series expansion now yields ¯ k,l ≤ cos (2π(k − l)σF T /N ) + 1 Ψ 4!

 2π

(k − l) N

4  n o  E (δF T )4 − (σF T )4 .

Although the factor of N in the denominator looks promising, since k and l are indices of an N -by-N matrix, the difference k − l varies from −N to N . So there is no overt dependence on N , except for entries around the diagonal, n o ¯ k,l will approach cos (2π(k − l)σF T /N ) as E (δF T )4 goes where k − l  N . Ψ to (σF T )4 . The result holds with equality when (δF T )4 equals (σF T )4 with probability 1, i.e. for a known, fixed frequency offset.

163

17. More formally, let  =

4!N 4 ∆. (2π)4 (k−l)4

Then

¯ k,l − cos (2π(k − l)σF T /N ) ≤ ∆ Ψ with probability 1 whenever o n E (δF T )4 − (σF T )4 ≤  with probability 1, which shows that the bound in Equation (B.38) becomes aritrarily close as the probability distribution for δF approaches the binomial distribution: G(δF ) = 0.5 I{δF =σF } + 0.5 I{δF =−σF }

B.1.4

Proof of Theorem 2.4:

Theorem. 2.4. For the system described in Theorem 2.2, with integer bandwidth scaling ratio α > 1, the SINR at the output of the receiver is: SINRj =

Ej   Vj − Ej + σz2 tr bH b j j

(B.42)

where Ej represents the desired signal power:   ∗  H H H ¯ j WM Ej = 1H Ψ aj aH 1, j WM WM bj bj WM

(B.43)

and Vj represents total signal and interference power:   ∗  H H ¯ WM Vj = 1H Ψ AAH WM WM bj bH W 1 M j

(B.44)

where ¯j Ψ



 n,m

=

¯ n,m = Ψ



  (n − m) δFj T , E δFj ∆α (δFj T ) exp j2π M α     (n − m) δFk T E δFk ∆α (δFk T ) exp j2π , M α

(B.45) for k 6= j, (B.46)

and  ∆α (δF T ) =

sin (πδF T ) α sin (πδF T /α)

2 .

(B.47)

164

Proof of Theorem 2.4 (p.26): 1. By Lemma B.9 Vj,k = ∆α (δFk T ) 1H

 H WM aj aH j WM  ∗ H WM bj bH W Ψ (δF /α) 1. M M k j

(B.48)

2. Using the assumption that the frequency offsets for all users, k, are identically distributed, let ¯ = E δF {∆α (δFk T ) ΨM (δFk /α)} . Ψ k ¯ is as given in Equation (B.45). It can be shown that element m, n of matrix Ψ 3. Substituting the definition for Vj,k (δFk ) from Equation (B.8) into the definition of the SINR in Equation (B.3)-(B.5) yields the results in Theorem 2.4 Ej

= 1H



  ∗ H H H ¯ WM aj aH W W b b W Ψ 1 j M M j M j

1H



(B.49)

and Vj

=

K X

  ∗ H H H ¯ 1 WM a k aH Ψ k WM WM bj bj WM

k=1

= 1H

B.2



  ∗ H H ¯ WM AAH WM WM bj bH W Ψ 1. M j

(B.50)

Asymptotic SINR for random channels

In the previous sections, theorems were introduced which showed how frequency offset error affects the performance of linear receivers in a MC-CDMA system. The following sections contain the proofs of theorems which involve specific receiver structures, namely the Matched Filter receiver and the LMMSE receiver. The performance of these systems is studied as the number of carriers and the number of users go to infinity. The following lemmas hold true under these conditions: • The number of users per degree of freedom (the ratio of the number of users to the number of subcarriers) is constant: β = K/M .

165

• The bandwidth scaling ratio of the subcarriers is an integer constant: α = N/M . • The users’ channel spreading gains aj,k are independent and identically distributed, 2 so that aH j aj = σ j :

E a∗j,m ak,n 



  σ 2 /M j =  0

if k = j and m = n,

(B.51)

otherwise.

and n o E |ak,n |4 < ∞.

(B.52)

In addition to those defined at the beginning of this appendix, in Equations (B.6)-(B.12), the following terms are used throughout the proofs in this section: n o n o2 E |ak,n |4 − E |ak,n |2 h ˜ a1 ˜ a2 · · · ˜ aj−1 ˜ aj+1 · · · h a1 a2 · · · aj−1 aj+1 · · ·   2 ˜ jA ˜H A j + σz I  2 Aj AH j + σz I   ˜A ˜ H + σz2 I = X ˜j + ˜ A aj ˜ aH j  AAH + σz2 I = Xj + aj aH j

Ak,4 = ˜j A

=

Aj

=

˜j X

=

Xj

=

˜ = X X = B.2.1

(B.53) ˜ aK aK

i

(B.54)

i

(B.55) (B.56) (B.57) (B.58) (B.59)

Lemmas

Lemma B.10 (Strong law of large numbers [39, p. 297].). Let X1 , X2 , · · · be independent identically distributed random variables. Then n

1X Xi → µ almost surely, as n → ∞, n i=1

for some constant µ, if and only if E {|X1 |} < ∞. In this case, µ = E {X1 }. Lemma B.11. Let aj = [aj,0 , aj,1 , · · · aj,M −1 ]T , where aj,m for m = 0, 1, ..., M − 1 are independent identically distributed random variables such that E {aj } = [0, 0, ..., 0]T and n o H a , where W E aj aH = σj2 I. where σj is a constant. Define ˜ aj = WM j M is the Fourier j

166

matrix. Then, the elements of ˜ aj are also independent and identically distributed, with n o n o 4 2 I and E |˜ aj ˜ aH = σ a | < ∞. E {˜ aj } = [0, 0, ..., 0]T , E ˜ j,n j j Proof of Lemma B.11: H E {˜ aj } = WM E {aj } = [0, 0, ..., 0]T   H H aj ˜ aH E ˜ = WM E aj aH WM = σj2 WM WM = σj2 I j j

n o E |˜ aj,n |4 =

 4  −1   M X  H E WM a n,m j,m   m=0

=

M −1 M −1 M −1 M −1 X X X X

H WM

 n,m1

H WM

∗ n,m2

H WM

∗ n,m3

H WM



H WM



n,m4

×

m1 =0 m2 =0 m3 =0 m4 =0

 E aj,m1 a∗j,m2 a∗j,m3 aj,m4 =

M −1 M −1 M −1 M −1 X X X X

H WM

 n,m1

H WM

∗ n,m2

H WM

∗ n,m3

n,m4

×

m1 =0 m2 =0 m3 =0 m4 =0

 E aj,m1 a∗j,m2 a∗j,m3 aj,m4

 n o 4  |a | E  j,m 1      2 2  σj /M E aj,m1 a∗j,m2 a∗j,m3 aj,m4 =        0

if m1 = m2 = m3 = m4 if m1 = m2 , m3 = m4 and m1 6= m3 or m1 = m3 , m2 = m4 and m1 6= m2 otherwise.

Thus, n o aj,n |4 = E |˜

n o E |aj,m1 |4 + 2M (M − 1)σj2 /M 2

n o n o aj,n |4 < ∞. If E |aj,m1 |4 < ∞ and σj2 < ∞, then E |˜ Lemma B.12. Let x ∈ C N , X ∈ C N ×N , Y ∈ C N ×N . Then, xH X + xxH

−1

Y X + xxH

−1

x=

xH X−1 YX−1 x (1 + xH X−1 x)2

(B.60)

167

Proof of Lemma B.12: By the Sherman-Morrison-Woodbury formula [37, p. 50]: X + xxH

−1

−1 H −1 = X−1 − X−1 x 1 + xH X−1 x x X

Define C = xH X−1 x, D = xH X−1 YX−1 x Then, −1  xH X + xxH Y X + xxH x    −1 X−1 xxH X−1 X−1 xxH X−1 −1 H −1 Y X − x = x X − 1+C 1+C  2 C C 2D xH X−1 YX−1 x DC CD = D 1 − + = = D− 2 1+C 1+C 1+C (1 + C) (1 + xH X−1 x)2

Lemma B.13. For an MC-CDMA system with M subcarriers, integer bandwidth scaling ratio α = N/M , and K users with frequency offset errors δFk , and channel spreading gains given by ak , the SINR at the output of a linear MMSE receiver is: SINRLMMSE,j =

Ej Zj σz2 + Vj − Ej

(B.61)

where Xj

= AAH + σz2 I

(B.62)

Zj

−2 = aH j X j aj 2 −1 H = ∆α,k aH j Xj WM DM (δFj /α)WM aj

(B.63)

Ej Vj

=

K X

∆α,k

2 H −1 H aj Xj WM DM (δFk /α)WM ak

(B.64) (B.65)

k=1

where ∆α,k is defined in Equation (B.9). Proof of Lemma B.13 ˜ j , Aj , Xj , X, Vj,k and ∆α,k , are defined in Equations (B.6) 1. The terms ˜ aj , b (B.12) on page 150, and Equations (B.53) - (B.59) on page 164. P H 2. By Theorem 2.4, Ej = Vj,j , Vj = K k=1 Vj,k , and Zj = bj bj .

168

3. By Lemma B.9, 2 ˜H D (δF /α) ˜ a Vj,k (δFk ) = ∆α,k b M k k . j 4. Since the linear MMSE receiver for user j is bj = σz2 I + AAH

−1

aj ,

−2 = aH j X aj 2 −1 H Vj,k (δFk ) = ∆α,k aH j X WM DM (δFk /α) WM ak .

˜H ˜ b j bj

5. Using Lemma B.12: −1 −1 = aH j X I X aj = 

˜H ˜ b j bj

−2 aH j X j aj −1 1 + aH j X j aj

2

 2 −1 H Vj,k (δFk ) = ∆α,k aH WM DM (δFk /α) WM ak j X 2 H −1 Ha aj Xj WM DM (δFk /α) WM k = ∆α,k 2  −1 1 + aH j X j aj ˜H b ˜ 6. Since Zj = b j j , Ej = Vj,j (δFj ) and Vj =

PK

k=1 Vj,k (δFk )

and canceling out the

terms in the denominator yields Equations (B.63)-(B.65).

Lemma B.14. Let G be the set of all permutations of the numbers {1, 2, ..., M }. Define Pg to be the permutation matrix corresponding to permutation g ∈ G as defined in Proposition B.20 [37, p. 109]. Define h iT p(δF ) = 1, ej2δF T /M , · · · , ej2δF T (M −1)/M . Then, Θ =

1 X Pg p(δF )p(δF )H PTg M →∞ M ! lim

g∈G

2

 = sinc (δF T ) 1M ×M + 1 − sinc2 (δF T ) I Proof of Lemma B.14: 1. For the diagonal elements of matrix Θ, 1 X j2πδF T (g(n)−g(n)) e =1 M →∞ M !

Θn,m = lim

g∈G

(B.66)

169

2. For non-diagonal elements, when summing over all permutations, g(n) takes on all values between 1 and M an equal number of times, for all possible permutations of the remainder of the set, hence Θn,m =

1 X j2πδF T (g(n)−g(m))/M e M →∞ M ! lim

g∈G

=

M −1 M −1 X X 1 ej2πδF T (m−n)/M M →∞ M (M − 1) n=0

lim

m=0

n6=m

  2 M −1 X 1  = lim ej2πδF T m/M − M  M →∞ M (M − 1) n=0

3. Using Lemma B.4, Θn,m

! sin(πδF T ) 2 1 −M = lim M →∞ M (M − 1) sin(πδF T /M ) sin(πδF T ) 2 = sinc2 (δF T ) = lim M →∞ M sin(πδF T /M )

4. Then,  Θ = I + sinc2 (δF T )(1 − I) = sinc2 (δF T ) 1 + 1 − sinc2 (δF T ) I

Lemma B.15. In the limit as M and K go to infinity, the following convergence holds almost surely. −1 lim aH j X j aj =

M,K→∞ K/M =β

σj2  −1  tr Xj M

Proof of Lemma B.15: (Also see Tse [128] for convergence in probability.) 1. In the limit as M and K go to infinity, −1 lim aH j X j aj

M,K→∞ K/M =β

=

=

lim

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

M −1 M −1 X X

  a∗j,n X−1 j

n=0 m=0 M −1 X

  |aj,n |2 X−1 j

n=0

M −1 M −1 X X n=0 m=n+1



a∗j,n X−1 j



n,m

n,n

aj,m

+ 

n,m

aj,m + a∗j,m X−1 j

!

 m,n

aj,n

170

2. Assume that the random spreading gains aj,n and aj,m are independent. When n is different from m. The elements of matrix X−1 j are independent of both aj,n and aj,m for all values of n, m, since Xj does not include terms from the spreading vector of user j, aj and the different users’ spreading gains are assumed to be independent. So the random variables defined as:   yn,m = a∗j,n X−1 j

n,m

  aj,m + a∗j,m X−1 j

m,n

n 6= m

aj,n

for n strictly less than m are independent random variables. By the law of large numbers, lim

M,K→∞ K/M =β

M −1 M −1 X X

yn,m =

n=0 m=0

M (M − 1) E {yn,m } 2 2

= M (M − 1) | E {aj,n }| E



X−1 j



 m,n

= 0.

3. Similarly defining   yn,n = |aj,n |2 X−1 j

n,n

,

and assuming that the elements of user j’s spreading vector aj,n are independent and identically distributed, by the law of large numbers: lim

M,K→∞ K/M =β

M −1 X

yn,n = M E {yn,n } .

n=0

4. From the definition of matrix Xj and vector aj , and the assumption that the channel spreading gains are independent random variables: n o   2 −1 −1 X lim aH X a = M E |a | E j,n j j j j M,K→∞ K/M =β

=

 n,n

1  H n  −1 o E aj aj E tr Xj . M

5. Since all limits were exacted using the law of large numbers, the convergence is almost surely.

171

Lemma B.16. As the number of users, K goes to infinity, the diagonal elements of matrix 2 Xj = Aj AH j + σz I, where Aj is defined in Equation (B.55), converge almost surely to:

 lim (Xj )m,m = β E k6=j σk2 + σz2

K→∞

(B.67)

where β = K/M , σk2 is the total transmission power for user k, and σk2 are independent identically distributed random variables for all interfering users k 6= j. Proof of Lemma B.16: By definition, the diagonal elements of matrix X are: (Xj )m,m =

K X

|ak,m |2 + σz2 .

k=1 k6=j

Since the users’ effective spreading gains are independent and identically distributed, by the law of large numbers n o K −1  H E ak ak + σz2 |ak,m |2 = (K − 1) E |ak,m |2 + σz2 = K→∞ M k=1

lim (Xj )m,m = lim

K→∞

K X k6=j

almost surely. Since K  1, (K − 1)/M will approach β = K/M in the limit as K goes to infinity. Lemma B.17. As the number of users, K goes to infinity, the diagonal elements of matrix H 2 X−1 j = Aj Aj + σ I

−1

(B.68)

where Aj is defined in Equation (B.55), approach a constant, lim

K→∞



X−1 j

 m,m

=C

(B.69)

whenever Lemma B.16 holds. Proof of Lemma B.17: 1. Since the diagonal elements of X−1 j are the ratio of determinants of the matrix Xj and the matrix derived from Xj by removing row n and column n, the following will show that the determinant of matrix Xj converges to a constant.

172

2. The determinant of matrix Xj det Xj =

X

(−1)S(g) Xg(0),0 Xg(1),1 · · · Xg(M −1),(M −1)

g

where g denotes a permutation of the integers {0 · · · M −1}, S(g) is the number of inversions in the permutation g, and the summation is over the set of all possible g. This can be rewritten as: det Xj =

M −1 X q=0

X

X

v⊂{0,...,M −1}, #v=q

g:g(k)=k for k∈v g(k)6=k for k∈v /

M −1 Y

! (−1)S(g) Xg(n),n

(B.70)

n=0

to separate diagonal elements of X from off-diagonal elements. Define ! M −1 Y S(g) r(q, v, g) = (−1) Xg(n),n n=0

3. By Lemma B.16, in the limit as K goes to inifinity, the diagonal elements of ma trix Xj approach β E k6=j σk2 + σz2 almost surely. Since spreading gains are independent and identically distributed for each user k, and different users’ spreading gains are independent random variables, off-diagonal elements of matrix Xj which are not in the same column or the same row of the matrix are also independent identically distributed random variables. Thus the terms r(q, v, g) are the product of independent random variables. 4. For q equal to zero, the only possible value for v is the empty set, and there is only one permutation g. By Lemma B.16, in the limit as K goes to inifinity, the  diagonal elements of matrix Xj approach β E k6=j σk2 + σz2 almost surely, and thus their product will approach this term to the M th power. 5. Consider r(q1 , v1 , g) for given values of v1 and q1 greater than zero. This term will be the product of q1 terms on the diagonal of matrix Xj , and N − q1 terms from off the diagonal. Summing over all g yields at least M identically distributed variables. As M goes to infinity, by the law of large numbers, the following holds almost surely: det Xj =

M −1 X q=0

 q n M −q o M! β E k6=j σk2 + σz2 E Xn,m6 =n = C. q!

173

Lemma B.18. In the limit as M and K go to infinity, ˜ −1 lim ˜ aH j Xj

M,K→∞ K/M =β

  −1 ˜ ˜ ˜ aj ˜ aH I X j j aj

  σj2 ˜ −2 + tr X j M2

=

n o σj4 E |˜ aj,n |4 − 2 M

!

n  o2 ˜ −1 E tr X j

Proof of Lemma B.18: 1. Define C˜p2 as below: C˜p2 = C˜p2 =

˜ −1 lim ˜ aH j Xj

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

  −1 ˜ ˜ ˜ aj ˜ aH I X j j aj

M −1 M −1 M −1 X X X

  ˜ −1 a ˜∗j,n X j

n=0 q=n+1 m=0

n,m

  ˜ −1 +a ˜∗j,q X j +

lim

M,K→∞ K/M =β

M −1 M −1 X X n=0 m=0

  ˜ −1 |˜ aj,n |2 X j

n,m

  ˜ −1 |˜ aj,m |2 X j

q,m

m,q

  ˜ −1 |˜ aj,m |2 X j

  ˜ −1 |˜ aj,m |2 X j

m,n

a ˜j,q

m,n

a ˜j,n

174

C˜p2 =

lim

M,K→∞ K/M =β

M −1 M −1 X X

M −1 X

  ˜ −1 a ˜∗j,n X j

n=0 q=n+1 m=n+1

n,m

  ˜ −1 +a ˜∗j,m X j +

+

lim

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

M −1 M −1 X X

  ˜ −1 a ˜∗j,n X j

n=0 q=n+1 M −1 M −1 X X

M −1 X

n,n

+

lim

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

m,n

  ˜ −1 a ˜∗j,q X j

n=0 q=n+1 m=n+1

M −1 M −1 X X

  ˜ −1 a ˜∗j,q X j

n=0 q=n+1 M −1 M −1 X X n=0 m=n+1

q,m

q,m

q,n

m,q

  ˜ −1 |˜ aj,n |2 X j

  ˜ −1 |˜ aj,n |2 X j

  ˜ −1 +a ˜∗j,q X j +

  ˜ −1 |˜ aj,m |2 X j

n,q

n,q

a ˜j,q

a ˜j,q

  ˜ −1 |˜ aj,m |2 X j

  ˜ −1 |˜ aj,n |2 X j

  ˜ −1 |˜ aj,m |2 X j

a ˜j,q

m,m

m,n

n,m

a ˜j,n

a ˜j,m

a ˜j,m

 −1  2 |˜ ˜ |˜ aj,n | Xj aj,m |2 n,m 2

 −1  2 |˜ ˜ + |˜ aj,m | X aj,n |2 j m,n  M −1  2 X 4 ˜ −1 |˜ aj,n | Xj + lim M,K→∞ n,n 2

K/M =β

n=0

Each term in the above equation is a summation of independent identically distributed random variables, which by the law of large numbers, converges to the expectation of the random variable. The expectation of the random variables in the first four summation terms is zero. Then the remaining terms yield: ( 2 ) n o2   ˜ −1 C˜p2 = M (M − 1) E |˜ aj,n |2 E X j n,m6=n ( ) n o  −1  2 4 ˜ aj,n | E X + M E |˜ j n,n   ( M −1 )2 M −1 M −1  2    X X  1 X 2 −1 ˜ = E |˜ aj,n | E Xj n,m   M   n=0 m=0 n=0 m6=n ) (M −1 n o  2 X  4 −1 X ˜ + E |˜ aj,n | E j n,n n=0

175

C˜p2 =

) (M −1 M −1  2 X X   H 2 1 −1 X ˜ E ˜ aj ˜ aj E j n,m M2 n=0 m=0 (M −1  n o n o2   X  X ˜ −1 + E |˜ E aj,n |4 − E |˜ aj,n |2 j n=0

=

σj4 M2

n  o ˜ −2 + E tr X j

o n σj4 E |˜ aj,n |4 − 2 M

!

2 ) n,n

n  o ˜ −1 X ˜ −1 E tr X j j

( )  −1  2 ˜ approaches a constant as K goes to infinity, By Lemma B.17, E X j n,n therefore

(  )   −1  2  −1  2 ˜ ˜ = lim E Xj lim E Xj K→∞ K→∞ n,n n,n

and C˜p2 =

σj4 n  −2 o ˜ E tr X + j M2

n o σj4 E |˜ aj,n |4 − 2 M

!

n  o2 ˜ −1 E tr X j

Lemma B.19. In the limit as M and K go to infinity,   σj2  2  n  −2 o −1 −1 ˜ 2 ˜ ˜ ˜ lim ˜ aH X X I X ˜ a = σ + σ . β E j j k6 = j j z E tr Xj k j j M,K→∞ M K/M =β Proof of Lemma B.19: 1. Define C˜n2 =

=

  −1 ˜ ˜ ˜ −1 ˜ lim ˜ aH X X I X j j j j aj

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

−1 M −1 M −1 M X X X

  ˜ −1 a ˜H j,n Xj

n=0 m=0 q=0

 n,m

˜j X



 m,m

˜ −1 X j

 m,q

a ˜j,q

By Lemma B.16, C˜n2 =

  β E k6=j σk2 + σz2 lim

M,K→∞ K/M =β

M −1 M −1 M −1 X X X n=0 m=0 q=0

  ˜ −1 a ˜∗j,n X j

 n,m

˜ −1 X j

 m,q

a ˜j,q

176

C˜n2 =

  β E k6=j σk2 + σz2 ×  M −1 M −1   X X ˜ −1  lim a ˜∗j,n X j M,K→∞ K/M =β

+

n=0 q=n+1

lim

M,K→∞ K/M =β

M −1 M −1 X X

M −1 X

 n,n

˜ −1 X j

  ˜ −1 a ˜∗j,n X j

n=0 m=n+1 q=n+1



 n,m

  ˜ −1 +a ˜∗j,m X j +

lim

M,K→∞ K/M =β

M −1 M −1 X X

M −1 X

  ˜ −1 a ˜∗j,q X j

n=0 m=n+1 q=n+1

q,m

+

+

lim

M,K→∞ K/M =β

lim

M,K→∞ K/M =β

+ lim

M,K→∞ K/M =β

  ˜ −1 a ˜∗j,q X j

n=0 q=n+1 M −1 M −1 X X

n=0

q,n

m,n

˜ −1 X j

q,n

 n,n

m,q

˜ −1 X j

˜ −1 X j 



 n,q

 m,n

˜ −1 X j

a ˜j,q a ˜j,q

a ˜j,n

 n,m

a ˜j,m

a ˜j,n

   2 −1  2 2 ˜ −1 ˜ + |˜ aj,m | Xj |˜ aj,n | Xj n,m m,n 2

n=0 m=n+1 M −1 X



˜ −1 X j 



  ˜ −1 +a ˜∗j,q X j M −1 M −1 X X

a ˜j,q

n,q

 # −1  2 ˜ |˜ aj,n | X j n,n 2

As with C˜p2 , C˜n2 has been factored into a sum of terms which are sums of independent identically distributed variables. The first four terms are sums of random variables with a mean of zero. The remaining terms yield the result: ( 2 ) n o  −1   2  2 2 ˜ C˜n2 = β E k6=j σk + σz M (M − 1) E |˜ aj,n | E X j n,m6=n ( )! n o  −1  2 2 ˜ aj,n | E X + M E |˜ j n,n

 C˜n2 =

  β E k6=j σk2 + σz2  E

(

1 M

M −1 X

) |˜ aj,n |2

n=0

( +E

=

   2   −1 M −1 M  X X  X ˜ −1 E j n,m    n=0 m=0  m6=n

1 M

M −1 X n=0

n  o   1 ˜ −2 β E k6=j σk2 + σz2 σj2 E tr X j M

) |˜ aj,n |2

 (M −1 )  2 X  X  ˜ −1 E j n,n  n=0

177

Proposition B.20. Let g be a permutation of the numbers {1, 2, ..., M }. Let Pg be the permutation matrix corresponding to this permutation [37, p. 109] that is, the matrix such that:  T Pg [x0 , x1 , · · · xM −1 ]T = xg(0) , xg(1) , · · · xg(M −1) . Let bj , ak ∈ C M have elements bj,n , ak,n , and   Dk = diag 1, ej2πδFk T /M , · · · ej2πδFk T (M −1)/M . Then for positive δFk T less than 0.5, 2 2   H  H T ≤ 8 (πδFk T ) < bH P D P a bj Dk ak − bH g k k j g j ak = bj ak 2 (B.71) + 12 (πδFk T )2 bH j ak    H + 16 (πδFk T )3 < bH j ak = bj ak For MMSE and MF receivers, when j is equal to k, bH j ak is a real number, therefore 2 2 H 2 H 2 T P D P a bj Dj aj − bH j g j ≤ 12 (πδFj T ) bj aj j g

(B.72)

If the elements of vector aj,n are independent, identically distributed, n 2 H T 2 o =0 E bH j Dk ak − bj Pg Dk Pg ak

(B.73)

where the expectation is over all possible carrier gains for user k. Proof of Proposition B.20: 1. Begin with Equation (B.73): n 2 H T 2 o E ak bH D a − b P D P a k k k g k j j g n n 2 o 2 o H T = E ak bH D a − E b P D P a a g k k k k j j g k Since carrier gains are independent and identically distributed, the expectations of the two terms will be equal, and the difference will be zero.

178

2. Define bH j Dk ak

Br + jBi =

=

M −1 X

b∗j,n ej2πδFk T n/M ak,n

n=0 T Cr + jCi = bH j Pg Dk Pg ak =

M −1 X

b∗j,g(n) ej2πδFk T n/M ak,g(n)

n=0

3. Since |Br + jBi |2 − |Cr + jCi |2 = (Br − Cr )(Br + Cr ) + (Bi − Ci )(Bi + Ci ) 4. First, consider Bi − Ci : M −1 X

Bi − Ci =

n o = b∗j,n ak,n − b∗j,g(n) ak,g(n) cos (2πδFk T n/M )

n=0 M −1 X

+

n o < b∗j,n ak,n − b∗j,g(n) ak,g(n) sin (2πδFk T n/M )

n=0

4.1 First study the term: |K1 | =

M −1 X

n o = b∗j,n ak,n − b∗j,g(n) ak,g(n) cos (2πδFk T n/M )

n=0

Using the Taylor expansion of the cosine, 1 − x2 /2 ≤ cos(x) ≤ 1 for 0 < x < 0.5. Therefore, |K1 | ≤

M −1 X

=



b∗j,n ak,n



n o  2 ∗ (1) − = bj,g(n) ak,g(n) 1 − (2πδFk T n/M ) /2

n=0

=



M −1 o (2πδFk T )2 X n ∗ 2 = b a j,g(n) k,g(n) n 2M 2 n=0 M −1 2 X

(2πδFk T ) 2M 2

n o  = b∗j,g(n) ak,g(n) M 2 = 2 (πδFk T )2 = bH j ak

n=0

o n and |K1 | ≤ 2 (πδFk T )2 = bH a . k j 4.2 Next, consider the term |K2 | =

M −1 X n=0

n o < b∗j,n ak,n − b∗j,g(n) ak,g(n) sin (2πδFk T n/M )

179

Using the Taylor expansion of the sine, 0 ≤ sin(x) ≤ x for 0 < x < 0.5. Therefore, |K2 | ≤

M −1 X

 < b∗j,n ak,n (2πδFk T n/M )

n=0

=



M −1 (2πδFk T ) X  ∗ < bj,n ak,n n M

(2πδFk T ) M

n=0 M −1 X

 < b∗j,n ak,n M

n=0

 = (2πδFk T ) < bH j ak n o and |K2 | ≤ (2πδFk T ) < bH j ak . As a result,   H |Bi − Ci | ≤ 2 (πδFk T )2 = bH j ak + (2πδFk T ) < bj ak H = πδFk T (1 − jπδFk T ) bH j ak + πδFk T (1 + jπδFk T ) ak bj  = 2< πδFk T (1 − jπδFk T ) bH j ak

5. Next, consider Br − Cr : Br − Cr =

M −1 X

n o < b∗j,n ak,n − b∗j,g(n) ak,g(n) cos (2πδFk T n/M )

n=0 M −1 X

+

n o = b∗j,n ak,n − b∗j,g(n) ak,g(n) sin (2πδFk T n/M )

n=0

5.1 First study the term: |K3 | =

=

M −1 X n=0 M −1 X

  < b∗j,n ak,n cos (2πδFk T n/M ) − cos 2πδFk T g −1 (n)/M o n < b∗j,n ak,n − b∗j,g(n) ak,g(n) cos (2πδFk T n/M )

n=0

Using the Taylor expansion of the cosine, 1 − x2 /2 ≤ cos(x) ≤ 1 for 0 < x
Rl do

20: 21:

for n = 1 : N, do if ρn (l) = 0, then

22:

k ← kmin (n)

23:

λˆl (n) ←

24:

1 H −1 hl (n)2 f

end if

25:

end for

26:

m ← arg minn λˆl (n)

27: 28:



    2 Hf λk hk (n)2 + ∆H hhl (n) 2 (n) k

194

Algorithm C.1: Lagrangian Relaxation Algorithm (Part 2) 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40:

λl ← λˆl (m)

41:

ρl (m) ← 1

42:

for n = 1 : N , do

43:

if ρl (n) = 1 then

44:

Assign carrier to user l.

45:

Update r, ρ, R.

46:

kmin (n) ← l

47: 48: 49: 50: 51: 52: 53: 54: 55: 56:

end if end for

195

Algorithm C.1: Lagrangian Relaxation Algorithm (Part 3) PN

57:

Rl ←

58:

l if Rmin < Rl then

n=1 rl (n)

59:

Rl ← Rl − rl (m)

60:

l if Rmin > Rl then l −Rl Rmin rl (m)

61:

ρl (m) ←

62:

l rl (m) ← Rmin − Rl

63:

ρkmin (m) (m) ← 1 − ρl (m) else

64:

l while Rmin < Rl do

65: 66:

λl ← λl − δλ

67:

Update Rl , rl . end while

68:

end if

69: 70:

end if

71:

k l ← arg max1≤k≤K Rmin − Rk

72:

end while

73: 74: 75: 76: 77: 78: 79:

for n = 1 : N , do

80:

l = arg max1≤k≤ ρk (n).

81:

ρk (m) ← Ik=l for k = 1, ..., K.

82:

end for

83:

Re-calculate rk (n), Rk for final allocation vector kmin

196

Algorithm C.2: Modified Lagrangian Relaxation Algorithm (Part 1) 1:

Initialization

for k = 1 : K, do  3: λk ← max ∆, f 0 (0)/hk (n)2 , 2:

4:

Rk ← 0

5:

mk ← 0  k  Nk ← Rmin /Rmax

6: 7:

end for

8:

for n = 1 : N , do

9:

l ← arg min1≤k≤K

10:

kmin (n) ← l

11:

k2,min (n) ← −1

12:

for k = 1 : K, do

Hf (λk hk (n)2 ) hk (n)2

13:

ρk (n) = Ik=l

14:

rk (n) ← ρk (n)f 0−1 λk hk (n)2

15:

end for

16:

end for

17:

Iteration

18:

k l ← arg max1≤k≤K Rmin − Rk

19:

l while Rmin > Rl do

20: 21:

for n = 1 : N, do if ρn (l) = 0, then

22:

k ← kmin (n)

23:

λˆl (n) ←

24:

1 H −1 hl (n)2 f

end if

25:

end for

26:

m ← arg minn λˆl (n)

27:

if λˆl (m) > λmax then

28:



if mk (l) ≥ Nk (l) then

    2 Hf λk hk (n)2 + ∆H hhl (n) 2 (n) k

197

Algorithm C.2: Modified Lagrangian Relaxation Algorithm (Part 2) l while (Rmin > Rl ) and (λl < λmax ) do

29: 30:

λl ← λl + ∆λ

31:

Update rl (n), Rl for n = 1, ..., N end while

32: 33:

end if

34:

if (mk (l) < Nk (l)) or (λl ≥ λmax ) then

35:

Block user l from transmitting,

36:

Remove all entries of user l from kmin and k2,min

37:

end if

38:

Go to line 71

39:

end if

40:

λl ← λˆl (m)

41:

ρl (m) ← 1

42:

for n = 1 : N , do

43:

if ρl (n) = 1 then

44:

Assign carrier to user l.

45:

Update r, ρ, mk , R.

46:

kmin (n) ← l

47:

else if kmin (n) ≥ 0 then

48:

k ← kmin (n)

49:

if Hf (λl hl (n)2 )/hl (n)2 > Hf (λk hk (n)2 )/hk (n)2 then

50:

Assign carrier from user k to user l.

51:

Update r, ρ, mk , R.

52:

k2,min (n) = kmin (n)

53:

kmin (n) = l end if

54: 55: 56:

end if end for

198

Algorithm C.2: Modified Lagrangian Relaxation Algorithm (Part 3) 57:

Rk ←

PN

n=1 rk (n),

for k = 1, ..., K

58: 59: 60: 61: 62: 63: 64: 65: 66: 67: 68: 69: 70: 71:

k l ← arg max1≤k≤K Rmin − Rk

72:

end while

73:

for k = 1 : K, do

74:

if mk (k) < Nk (k) then

75:

Find l such that k2,min (m) = k and mk (l) > Nk (l)

76:

kmin (m) ← k

77: 78:

end if end for

79: 80: 81: 82: 83:

Re-calculate rk (n), Rk for final allocation vector kmin

199

C.2

The Bandwidth Assignment Based on SINR (BABS) Algorithm

This Section of Appendix C is concerned with proving Theorem 3.4, and showing that the BABS Algorithm solves problem (3.9). Consider an OFDMA system where K users share N channels. User k’s channel gain k on any carrier n is Hk . User k aims to transmit Rmin bits per frame and will be allocated

mk carriers to do so. A subcarrier can transmit at most Rmax bits per unit time. The aim of the BABS algorithm is to solve the following problem, from Equation (3.9), restated below. Some definitions from Section 3.4.1 are also restated below. min

s.t.

K X mk k=1 K X

Hk

k f (Rmin /mk )

(C.1)

mk = N

(C.2)

k=1

 mk ∈

  k Rmin , ..., N Rmax

1≤k≤K

(C.3)

Definition C.1. Let the bandwidth allocation, [mk ], be the set of integers {mk : 1 ≤ k ≤ K}, where mk represents the number of carriers assigned to user k. Let F denote the set of feasible bandwidth allocations, i.e. [mk ] ∈ F if and only if [mk ] satisfies Equations (C.2)-(C.3). Define the transmission power when mk carriers are allocated to user k:  k  Rmin mk f Gk (mk ) = Hk mk

(C.4)

Whenever a new carrier is allocated to user k, the power received at the base station from user k is reduced by by Fk (mk ) = Hk Gk (mk + 1) − Hk Gk (mk ).

(C.5)

Assumption C.2. Assume that for all users k, the function Fk (mk ) defined in Equation (C.5): • Negative definite Fk (mk ) < 0

∀mk ∈ N

(C.6)

200

• A monotonically increasing function of mk , ∀mk ∈ N

Fk (mk + 1) > Fk (mk )

(C.7)

Assumption C.3. Assume that the set of feasible bandwidth allocations, F, for the optimization problem described in Equations (C.1)-(C.3) is not empty, i.e. there exists at least one distribution of carriers [lk ] that satisfies Equations (C.2)-(C.3).

Algorithm C.3: BABS Algorithm 1:

mk ←

2:

while

k Rmin Rmax , PK k=1 mk

l

m

k = 1, .., K. > N do

3:

k ∗ ← arg min mk ,

4:

mk∗ ← 0,

1≤k≤K

end while PK 6: while k=1 mk < N , do  k  Rmin f 7: Gk ← mHk +1 (mk +1) − k 5:

8:

l ← arg min Gk ,

9:

ml ← ml + 1,

mk Hk f



k Rmin mk



,

k = 1, .., K

1≤k≤K

10:

end while

Theorem C.4. If Assumptions C.2 and C.3 hold, then the Bandwidth Assignment Based on SNR (BABS) Algorithm, described in Algorithm C.3 (re-stated from Chapter 3 for convenience) solves the problem stated in Equations (C.1) - (C.3). Proof of Theorem C.4: 1. Let [m0k ] denote the distribution of carriers after completing Step 1 of the BABS algorithm, where m0k

k Rmin = Rmax



is the number of carriers assigned to user k.

 (C.8)

201

2. Let [mk ] denote the distribution of carriers found by the algorithm, after finishing the last step, where mk is the number of carriers assigned to user k. 3. Distribution [mk ] satisfies Equation (C.2): Define Msum =

K X

m0k

(C.9)

k=1

After the first step, when the mk are assigned, there are three possibilities: • If Msum = N , then both conditions on lines C.3 and C.3 of the BABS algorithm (Algorithm C.3) are false, Equation (C.2) is satisfied. • If Msum > N then the first while loop (lines C.3-C.3 in Algorithm C.3) will iterate until Equation (C.2) is satisfied, the second while loop (lines C.3-C.3 in Algorithm C.3) will never be called. • If Msum < N then the first while loop (lines C.3-C.3 in Algorithm C.3) will never be activated, and the second loop (lines C.3-C.3 in Algorithm C.3)will iterate until Equation (C.2) is satisfied. In all cases, Equation (C.2) is satisfied by [mk ]. 4. Distribution [mk ] satisfies Equation (C.3): mk is always incremented by 1, so it is always integer. mk ≤ N for all k since Equation (C.2) is satisfied. After the initial step, mk is set to m0k as defined in Equation (C.8), then, one of the below is true • mk is reassigned in the first while loop (lines C.3-C.3 in Algorithm C.3), and mk = 0 • mk is incremented in the second while loop (lines C.3-C.3 in Algorithm C.3), and mk > m0k • mk is never modified in either loop and mk = m0k By Assumption C.3 the first option can never occur, and mk ≥ m0k for all k. Therefore, Equation (C.3) is satisfied by [mk ]. 5. From Steps 3-4, the result of the BABS algorithm is a feasible bandwidth allocation, [mk ] ∈ F, by Definition C.1.

202

6. Assume that there exists another feasible bandwidth allocation [nk ] ∈ F such that • Distribution [nk ] yields the lowest objective function (C.1) of all distributions [lk ] K X

Gk (nk )
mq , and at least one p ∈ Q such that np < mp . 8. Choose any q ∈ Q such that nq > mq , and p ∈ Q such that np < mp . Since [mk ] is the result of the BABS algorithm, in some step of the algorithm, the last carrier is assigned to user p. In this stage the assignment is made to user p, not user q, and at this stage, by the rules of the algorithm: Gq (mq ) − Gq (mq + 1) < Gp (mp − 1) − Gp (mp )

(C.14)

9. By Equation (C.6), since nq > mq Gq (nq − 1) − Gq (nq ) ≤ Gq (mq ) − Gq (mq + 1)

(C.15)

and since np < mp Gp (np ) − Gp (np + 1) ≥ Gp (mp − 1) − Gp (mp )

(C.16)

203

10. Combining Equations (C.14)-(C.16): Gq (nq − 1) − Gq (nq ) < Gp (np ) − Gp (np + 1) Gq (nq − 1) + Gq (np + 1) < Gq (nq ) + Gp (np )

11. Define a distribution [lk ] such that:    np − 1 if k = p,   lk = nq + 1 if k = q,     n otherwise. k

(C.17) (C.18)

(C.19)

12. Distribution [lk ] is a feasible bandwidth allocation, because • Equation (C.2) is satisfied:  K X X lk = 

 nk  + (np − 1) + (nq + 1) = N

(C.20)

1≤k≤K/ {p,q}

k=1

• Equation (C.3) is satisfied, since both [mk ] and [nk ] satisfy this equation, and: mq ≤ l k ≤ n q

if k = q,

(C.21)

n p ≤ l k ≤ mp

if k = p,

(C.22)

otherwise.

(C.23)

l k = nk

So [lk ] is a feasible bandwidth allocation ([lk ] ∈ F). 13. From Step 11 K X

Gk (lk ) =

k=1

X

Gk (nk ) + Gp (np − 1) + Gq (nq + 1)

(C.24)

1≤k≤K,k6=p,q

By Equation (C.18) K X

X

Gk (lk )
wAM (ekp ,c2 ), and the new label for edge ekp ,c2 is np−1 .

(C.40)

209

circuit c path p path p0

c2 x

l20 l10

lC0

cC

kp−1 np−1 kp k1

np kp+1

Figure C.3: Graph G(AM ), path p, circuit c0 , path p0 .

2. Node kp ∈ p is connected to node kp+1 in path p with carrier label l10 . When this assignment is implemented, user kp no longer owns carrier l10 , thus the label of link ekp ,c2 must change. In this case, unless the first option also holds true, wAM +1 (ekp ,c2 ) < wAM (ekp ,c2 ).

(C.41)

5. First, assume that there in only one node kp on both circuit c0 and path p (Figure C.3). 6. Since wAM (c0 ) > wAM +1 (c), and the only edge on circuit c0 whose weight can change is ekp ,c2 , the first condition above must hold: wAM (c0 ) = wAM +1 (c) − wAM +1 (ekp ,c2 ) + wAM (ekp ,c2 )

(C.42)

7. Consider the path p0 = (k1 , n1 , ..., kp−1 , x, c2 , l2 , ..., cC , lC , np , kp , ..., nP −1 , kP ) ∈ G(AM ) (Figure C.3). wAM (p0 ) = wAM (p) + wAM (c0 ) − wAM (ekp−1 ,kp ) − wAM (ekp ,c2 ) + wAM (ekp−1 ,c2 ) (C.43)

210

8. By Equation (C.29) wAM (ekp−1 ,c2 ) ≤ rkp−1 (np−1 ) − rc2 (np−1 )

(C.44)

rkp−1 (np−1 ) − rc2 (np−1 ) = wAM (ekp−1 ,kp ) + rkp (np−1 ) − rc2 (np−1 )(C.45) rkp−1 (np−1 ) − rc2 (np−1 ) = wAM (ekp−1 ,kp ) + wAM +1 (ekp ,c2 )

(C.46)

wAM (ekp−1 ,c2 ) ≤ wAM (ekp−1 ,kp ) + wAM +1 (ekp ,c2 )

(C.47)

9. From Equations (C.42), (C.43): wAM (p0 ) = wAM (p) + wAM +1 (c) − wAM +1 (ekp ,c2 ) − wAM (ekp−1 ,kp ) + wAM (ekp−1 ,c2 ) (C.48) 10. By Equation (C.47), −wAM +1 (ekp ,c2 ) − wAM (ekp−1 ,kp ) + wAM (ekp−1 ,c2 ) ≤ 0

(C.49)

and by assumption wAM +1 (c) < 0. Then from Equation C.48, wAM (p0 ) < wAM (p)

(C.50)

However this cannot be true because by steps C.5-C.5 in Algorithm C.5, p is the minimum cost path from node k1 to node kP on graph G(AM ). 11. The contradiction above means that the assumption in Step 5 of this proof is false. Thus if a circuit c on graph G(AM +1 ) exists such that wAM +1 (c) < 0, then there are at least two nodes which are on both path p and circuit c. 12. The nodes that are on both circuit c and path p divide the circuit c on graph G(AM +1 ) into two sets of paths: • B = {b1 , ..., bB }, such that any path bb is between two consecutive nodes on circuit c, which are also on path p, in the same direction as path p, • D = {d1 , ..., dD } such that any path dd is between two consecutive nodes on circuit c, which are also on path p, in the opposite direction to path p. The circuit c0 on graph G(AM ) is divided into two corresponding set of paths traversing the same nodes: B 0 = {b01 , ..., b0B }, and D0 = {d01 , ..., d0D }.

211

circuit c0

b01

path p

b02 p02

kq

kp

p01 d02

d01

Figure C.4: Graph G(AM ), path p, circuit c0 , and sets of paths: B 0 = {b01 , b02 }, D0 = {d01 , d02 }, P 0 = {p1 , p2 }.

13. Let P = {p1 , ..., pB }, be a set of paths such that pb has the same end points as b0b ∈ B 0 , but follows the links that are on path p in between (Figure C.4). Let Q = {q1 , ..., qD }, be a set of paths such that qd has the same end points as d0d ∈ B0 , but follows the links that are on path p in between. The path qd will then run in the opposite direction to path d0d . 14. Consider that the graph G(AM ) is partitioned into two sections, cutting across a link e ∈ p, not across any other links on path p (Figure C.5). If the circuit c0 has n edges cutting across the partition in one direction, then an equal number of edges must cut across in the opposite direction, to ensure that the circuit is closed. As a result, the two sets of paths B 0 and D0 that are the projections of circuit c0 onto path p contain the same number of passes across each edge e ∈ p. Then by definition of the previous section, D X d=1

wAM (qd ) =

B X

wAM (pb )

(C.51)

b=1

15. First, consider the path bb ∈ B0 with end nodes kp and kq (Figure C.6). Define path

212

circuit c0 path p

kq

kp np−1 l10

Partition G1

Partition G2

Figure C.5: Partition of graph G(AM ), and path p.

circuit c0 path p path p0

kp kp−1

kq

np−1 l10 b01

x c2

Figure C.6: Graph G(AM ), path p, circuit c0 , path p0 .

213

p0 = (k1 , n1 , ..., kp−1 , x, c2 , l20 , ..., kq , nq , ..., nP −1 , kP ) ∈ G(AM ) wAM (p0 ) = wAM (p) − wAM (pb ) + wAM (b0b ) − wAM (ekp−1 ,kp ) − wAM (ekp ,c2 ) + wAM (ekp−1 ,c2 )

(C.52)

16. Again using the result of Equation (C.47): wAM (p0 ) ≤ wAM (p) − wAM +1 (pb ) + wAM (b0b ) − wAM (ekp ,c2 ) + wAM +1 (ekp ,c2 )

(C.53)

17. As discussed in Step 4 of this proof, the paths bb ∈ B on graph G(AM +1 ) and b0b ∈ B 0 on graph G(AM ) can only differ in the first link, ekp ,c2 : wAM +1 (bb ) = wAM (b0b ) − wAM (ekp ,c2 ) + wAM +1 (ekp ,c2 )

(C.54)

18. By Steps C.5-C.5 in Algorithm C.5, p is the minimum cost path from node k1 to node kP on graph G(AM ), thus wAM (p) ≤ wAM (p0 ). Using this and Equations (C.53)-(C.54): wAM (bb ) ≥ wAM (pb )

(C.55)

Then all paths bb ∈ B on graph G(AM +1 ) have either the same or greater weight as a path pb following path p across those nodes. 19. Consider the circuit ld formed between path d0d = (kp , l10 , c2 , ..., kq ) ∈ D0 and qd = (kq , nq , ..., np−1 , kp ) ∈ Q, the corresponding section of path p (Figure C.7). The total weight of such a circuit is: wAM (ld ) = wAM (qd ) + wAM (d0d ) − wAM (ekp−1 ,kp ) − wAM (ekp ,c2 ) + wAM (ekp−1 ,c2 )

(C.56)

20. Again using the result of Equation (C.47), and noting that wAM (d0d ) = wAM +1 (dd ) − wAM (ekp ,c2 ) + wAM +1 (ekp ,c2 )

(C.57)

the total weight of the circuit can be shown to satisfy the following: wAM (ld ) ≤ wAM (qd ) + wAM +1 (dd )

(C.58)

214

circuit c0 path p

kq

kp−1 x

kp

np−1 l10

l20

c3

c2

Figure C.7: Graph G(AM ), path p, circuit c0 .

21. By summing over all loops ld , and using Equations (C.51)-(C.55): D X d=1 D X d=1 D X d=1 D X

wAM (ld ) ≤ wAM (ld ) ≤ wAM (ld ) ≤

D X d=1 B X b=1 B X

wAM (qd ) + wAM +1 (dd )

(C.59)

wAM (pb ) + wAM +1 (dd )

(C.60)

wAM +1 (bb ) + wAM +1 (dd )

(C.61)

b=1

wAM (ld ) ≤ wAM +1 (c)


wA (p) then

13:

p∗ ← p

14:

end if

15:

end if

16:

end for

17: 18:

end if

19:

end for

20:

for kp∗ ∈ p∗ = (k1∗ , k2∗ , ..., kP∗ ) do

21:

Akp ← Akp \ {ekp ,kp+1 }

22:

Akp+1 ← Akp+1 ∪ {ekp ,kp+1 }

23: 24:

end for end while

218

Appendix D APPENDIX FOR CHAPTER 4

In Chapter 4, two theorems are introduced which describe the effect of channel granularity in relating effort fairness to outcome fairness, and the effect of the number of carriers N and the number of time slots M on the fairness implied by the minimum transmission rate requirement: N X M X

k rk (n, m) ≥ Rmin .

n=1 m=1

This chapter includes the proofs of those theorems. D.1

OFDMA and Fairness

Theorem D.1. Consider an OFDMA system with N carriers, M time slots, and K users. k . Let r (n, m) represent the transEach user k has minimum transmission requirement Rmin k

mission rate for user k on carrier n in time slot m. Let T be the duration of a single time slot, let {0, Y1 , · · · , YQ } be the set of modulation and coding levels per carrier, and V = {v0 , v1 , · · · , vQ } where v0 = 0 and vk = Yk /(M N T ) be the set of available transmission rates per carrier. Let f (vq ) represent minimum SNR requirement for any user to transmit at rate vq on some carrier. Define ∆ = Rk =

max (Yq − Yq−1 )

1≤q≤Q

N X M X

rk (n, m).

(D.1) (D.2)

n=1 m=1

Then any resource allocation which satisfies the conditions k Rk ≥ Rmin , K X k=1

δ [rk (n)] ≤ 1

k = 1, ..., K n = 1, ..., N,

(D.3) m = 1, ..., M,

(D.4)

219

will satisfy the following fairness condition: Rk Rl ∆ 1 Rk − Rl ≤ M N T min Rk , Rl  . min min min min

(D.5)

Lemma D.2. Consider a user k in the system described in Theorem D.1. Assume without loss of generality that carriers 1 to Nk have been allocated to user k by a resource allocation algorithm during time slots 1 to Mk , and that N k Mk ≥

k Rmin . vQ

(D.6)

Then, a rate and power allocation algorithm can be found such that k Rk − Rmin ≤

∆ . MNT

(D.7)

Proof of Lemma D.2: 1. Assume that the following algorithm is used to find the power and bit loading for each carrier for user k: Single User Loading Algorithm 1. Find the set of transmission rates r˜k (n, m) such which solve the following problem: min

s.t.

Nk X Mk X f (˜ rk (n, m)) |Hk (n, m)|2 n=1 m=1 Nk X Mk X

k r˜k (n, m) ≥ Rmin

n=1 m=1

r˜k (n, m) ≤ vQ

for n = 1, · · · , Nk ,

m = 1, · · · , Mk .

2. For n = 1, · · · , Nk , m = 1, · · · , Mk , let qk (n, m) = arg min {vq ∈ V : vq ≥ r˜k (n, m)} , q

rk (n, m) = vqk (n,m) , βk,n,m = rk (n, m) − r˜k (n, m).

220

3. Let Rk =

Nk X Mk X

rk (n, m),

n=1 m=1

(˜ n, m) ˜ = arg

  f vqk (n,m) − f vqk (n,m)−1

max n,m: k ≥ v Rk −Rmin ( qk (n,m) −vqk (n,m)−1 )

β = βk,˜n,m ˜. 4. While

PNk PMk n=1

m=1 rk (n, m)

k − β ≥ Rmin

(a) Adjust the transmission rate qk (˜ n, m) ˜ = qk (˜ n, m) ˜ − 1, rk (˜ n, m) ˜ = vqk (˜n,m) ˜ , βk,˜n,m = rk (˜ n, m) ˜ − r˜k (˜ n, m), ˜ ˜ Rk =

Mk Nk X X

rk (n, m).

n=1 m=1

(b) Find the new rate to be adjusted (˜ n, m) ˜ = arg

max n,m: k ≥ v Rk −Rmin ( qk (n,m) −vqk (n,m)−1 )

  f vqk (n,m) − f vqk (n,m)−1

β = βk,˜n,m ˜. 2. The above algorithm will converge when k Rk − Rmin < vqk (n,m) − vqk (n,m)−1



for all n, m. Since ∆ = max Yk − Yk−1 1≤k≤Q

k Rk − Rmin < vqk (n,m) − vqk (n,m)−1 =

Proof of Theorem D.1:

 1 ∆ Yqk (n,m) − Yqk (n,m)−1 ≤ . MNT MNT

221

By Lemma D.2, k l Rk − Rmin Rk R − R R l l min − Rk − Rl = Rk l Rmin min min min  ! k l Rk − Rmin Rl − Rmin ≤ max , k l Rmin Rmin ≤

1 ∆ . k l M N T min Rmin , Rmin

Theorem D.3. Consider an OFDMA system with N carriers, M time slots, and K users. k . Let r (n, m) represent the transEach user k has minimum transmission requirement Rmin k

mission rate for user k on carrier n in time slot m. Let T be the duration of a single time slot, let {0, Y1 , · · · , YQ } be the set of modulation and coding levels per carrier, and V = {v0 , v1 , · · · , vQ } where v0 = 0 and vk = Yk /(M N T ) be the set of available transmission rates per carrier. Let f (vq ) represent minimum SNR requirement for any user to transmit at rate vq on some carrier. If Rk Rl Rk − Rl ≤ , min min

(D.8)

then     vM Nk vM Nl Rl vM 1 Rk − − 1 + ≤ Rk l k l v1 2 Rmin Rmin Rmin min    vM vM  + +1 + k , Rl 2 v1 min Rmin min Proof of Theorem D.3: 1. By definition, the number of carriers a user is allocated is limited from below and above by the maximum and minimum transmission rates allowed per carrier, the number of carriers that user k transmits on, Nk is bounded by:     Rk Rk Rk Rk ≤ ≤ Nk ≤ ≤ +1 vM vM v1 v1 2. Then, Rk 1 Nk Rk ≤ k ≤ k k Rmin vM Rmin Rmin



1 1 + v1 Rk



(D.9)

222

3. Since a similar equation holds for a user l, Rk 1 1 Nk Nl Rk 1 Rl 1 Rl 1 1 − l ≤ k − l ≤ k − l − l + k k Rmin vM Rmin v1 Rmin Rmin Rmin Rmin v1 Rmin vM Rmin 4. Using some algebra, Rk 1 1 Rl 1 − l − l k v v Rmin M R R 1    min   min   1 1 1 Rk Rl 1 1 1 Rk Rl 1 = − − + + + − − l k l k l 2 v1 vM 2 v1 vM Rmin Rmin Rmin Rmin Rmin        1 1 1 Rk Rl 1 1  1 1 − ≥ − − + − max + l , l k k 2 v1 vM v1 vM 2 Rmin Rmin Rmin Rmin 5. Similarly, Rl 1 1 Rk 1 − l + k k Rmin v1 Rmin vM R    min     1 1 1 Rk 1 1 1 Rk 1 Rl Rl = − + k + + − k + k l l 2 v1 vM 2 v1 vM R Rmin Rmin Rmin Rmin       min   Rk 1 1 1 1 1 Rl 1  1 + + , ≤ − + + max l k k l 2 v1 vM v1 vM 2 Rmin Rmin Rmin Rmin 6. Thus, vM Nk vM Nl − l Rk Rmin min       vM Rl 1 Rk  vM vM  −1 +1 + + l + ≤ k k , Rl v1 2 Rmin Rmin 2 v1 min Rmin min

Corollary D.4. If the conditions of Theorem 4.1 and 4.2 are both met, then     vM Nk vM Nl vM  vM vM , − ≤ − 1 + 3 − 1 + Rk l k l v 2 v R min R 1 1 min min min , Rmin

(D.10)

where =

Proof of Corollary D.4:

∆ 1 . k , Rl M N T min Rmin min

(D.11)

223

1. If the single user powerfilling algorithm in Lemma D.2 is used:  !   k l Rk − Rmin Rl − Rmin 1 Rk Rl 1 = 1+ + l + k k l 2 Rmin 2 Rmin Rmin Rmin   1 1 ∆ 1 + k ≤ 1+ l M N T 2 Rmin Rmin ∆ 1  = 1 + . ≤ 1+ l k M N T min Rmin , Rmin 2. Using Theorem D.3:     Nk Nl 1 Rl 1 Rk 1 − + l Rk − Rl ≤ k v1 vM 2 Rmin Rmin min min    1 1 1 . + + + k l 2 v1 vM min Rmin , Rmin 3. By Theorem D.1:     Nk N 1 l ≤ 1 − 1 (1 + ) +  1 + 1 +  − Rk l k , Rl v v 2 v v R min R 1 1 M M min min min min and     vM Nk vM Nl  vM vM vM  3 − l ≤ −1 + +1 + Rk k , Rl v1 2 v1 Rmin min Rmin min min where =

∆ 1  k l M N T min Rmin , Rmin

224

VITA

Didem Kivanc received her B.S. in Electrical Engineering from Bogazici University, Turkey, and her M.S. in 1995 from the University of Surrey, UK. Since 1998, she has been a member of the Wireless Information Technology (WIT) lab at the University of Washington, Seattle. Her research focuses on enabling algorithms for wireless communications.

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