researchdirect.westernsydney.edu.auresearchdirect.westernsydney.edu.au/islandora/object/uws:11188... · DECLARATION Date: JULY 2011 Author: Pushpika Wijesinghe Title: BIASED MONTE

BIASED MONTE CARLO METHODSFOR

EFFICIENT SIMULATIONOF

COMMUNICATION SYSTEMS

Pushpika Wijesinghe

A thesis submitted for the degree of

Doctor of Philosophy in Engineering

SCHOOL OF ENGINEERING

UNIVERSITY OF WESTERN SYDNEY

AUSTRALIA

JULY 2011

c© Pushpika Wijesinghe, 2011

To my Parents and my beloved Husband

DECLARATION

Date: JULY 2011

Author: Pushpika Wijesinghe

Title: BIASED MONTE CARLO METHODS FOR EFFICIENTSIMULATION OF COMMUNICATION SYSTEMS

Degree: Ph.D.

I certify that the work presented in this thesis is, to the best of my knowledgeand belief, original, except as acknowledged in the text, and that the material has notbeen submitted, either in full or in part, for a degree at this or any other institution.

I certify that I have complied with the rules, requirements, procedures and policyrelating to my higher degree research award of the University of Western Sydney.

Author's Signature

ACKNOWLEDGEMENTS

It is with great pleasure that I express my sincere gratitude to my principalsupervisor, Dr. Upul Gunawardana, for his continuous encouragement, adviceand guidance. He has been a source of generosity, insight and inspiration; guidingme in all my eorts throughout my candidature. I owe my research achievementsto his enthusiastic supervision.

I extend my profound gratitude to Dr. Ranjith Liyanapathirana, who wasthe principal supervisor for the rst year of my candidature and a co-supervisorthereafter. He provided me with the uninching encouragement and support Ihave needed most since I decided to apply for Postgraduate studies at UWS.Successful completion of this thesis would not have been possible without hisinvaluable insights and comments on my work.

I am also thankful to my co-supervisor Dr. Qi Cheng for his support andvaluable advice.

I gratefully acknowledge the University of Western Sydney and the AustralianGovernment for granting me the Endeavour International Postgraduate ResearchScholarship, which gave me the opportunity to be exposed to a new knowledgebase. I appreciate the travel support given by the School of Engineering for myattending national and international conferences.

The motivation provided by my Honours and Masters Degree Supervisor,Professor Dileeka Dias of University of Moratuwa, Sri Lanka guided me towardsthe accomplishment of a Doctoral Degree. I have beneted extensively from herdedication to the intellectual and personal growth of her students. I also gratefullyremind all my teachers from pre-school to high-school for their valuable guidance.

I would like to thank all current and previous technical sta, general sta andacademics of School of Engineering who directly or indirectly helped me duringmy candidature. My gratitude also goes to all my colleagues for their supportand friendship. I specially thank Robert, Madhuka and Prasanna for allowing me

iv

access to their PCs to run my simulations. Further, I extend my gratitude to Dr.Kim Nguyen for her assistance during my rst three months of the candidature.

As always, my heartfelt gratitude goes to my Mother, Father and Grandpar-ents for their love and constant support throughout my life and for inspiring meto pursue an academic career. I can never forget their warmth and inspiration.

Finally, my most tender and sincere thanks go to my loving husband, SumuduWijetunge, who has been a shadow behind all my success during the last veyears. His understanding throughout these years has meant more than I can everexpress.

ABSTRACT

This thesis investigates biased Monte Carlo (MC) methods for ecient simulationof orthogonal frequency division multiplexing (OFDM), multiple-input multiple-output (MIMO) and coded communication systems. Analytical complexity ofexact performance evaluation of modern communication systems demands MCsimulations, which estimate the performance metrics by statistical sample aver-aging. Even though MC is generally applicable for arbitrarily complex systems,it requires large sample sizes and extensive computational time to estimate lowprobabilities with a high degree of accuracy. This motivates the development ofecient simulation techniques. Statistical eciency of simulations can be im-proved by properly biasing the MC simulations to encourage the occurrence ofimportant events. This thesis investigates such biasing strategies for communi-cation system simulation.

First, optimal importance sampling (IS) methods are proposed for ecientsimulation of OFDM and MIMO-OFDM systems operating over frequency se-lective fading channels. Optimum IS parameters that minimise the statisticalvariance of the estimator are derived. The proposed methods provide increasedsample size reductions with decreasing probability. Next, at histogram MonteCarlo (FHMC) techniques, developed for statistical physics systems, are inves-tigated in a communication engineering context. The potential of FHMC tech-niques to accelerate communication system simulations is explored. Simulationresults show that sample size reductions in the order of 105 can be achievedin estimating probabilities in the order of 10−10 with a relative error of ±10%.Moreover, improved MC estimators are derived and these estimators are used toenhance the eciency of FHMC algorithms. The improved FHMC algorithms aresuccessfully applied to capacity estimation of MIMO systems whose analytical so-lutions are intractable. Of these derived estimators, the improved combination ofWang-Landau (WL) and transition matrix Monte Carlo (TMMC) methods shows

vi

the best performance while all of them can be generally applied to probabilitydensity function (pdf) estimation problems. Finally, IS and FHMC techniquesare applied to ecient simulation of coded systems with Viterbi and maximuma posteriori probability (MAP) decoders. A novel algorithm, consisting of twophases that use FHMC concepts, is proposed for ecient simulation of Viterbiand MAP decoders. The proposed algorithm demonstrates substantial samplesize reductions in estimating low probabilities with a relative error of ±10%.

This thesis has laid the foundation for further studies on using FHMCmethodsfor performance evaluation of complex decoders and multiuser communicationsystems.

Contents

Acknowledgement iii

Abstract v

Contents vii

Abbreviations xi

Notation xiv

List of Figures xvi

List of Tables xix

List of Algorithms xxi

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Major Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Thesis Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Literature Review 122.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 12

CONTENTS viii

2.3 Ecient Simulation Techniques . . . . . . . . . . . . . . . . . . . 162.3.1 Tail Extrapolation . . . . . . . . . . . . . . . . . . . . . . 172.3.2 pdf Estimation . . . . . . . . . . . . . . . . . . . . . . . . 172.3.3 Importance Sampling . . . . . . . . . . . . . . . . . . . . . 182.3.4 Flat Histogram Monte Carlo Methods . . . . . . . . . . . . 24

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Ecient Simulation of OFDM Systems 283.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Importance Sampling for OFDM Systems . . . . . . . . . . . . . . 29

3.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Variance Scaling and Mean Translation of Noise . . . . . . 33

3.2.2.1 Variance Scaling . . . . . . . . . . . . . . . . . . 343.2.2.2 Mean Translation . . . . . . . . . . . . . . . . . . 35

3.2.3 Variance Scaling of Fading Coecients . . . . . . . . . . . 373.3 Importance Sampling for OSTBC-OFDM Systems . . . . . . . . . 39

3.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.2 Variance Scaling . . . . . . . . . . . . . . . . . . . . . . . 423.3.3 Mean Translation . . . . . . . . . . . . . . . . . . . . . . . 433.3.4 VS-Rayleigh Technique . . . . . . . . . . . . . . . . . . . . 45

3.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . 473.4.1 OFDM System . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2 OSTBC-OFDM System . . . . . . . . . . . . . . . . . . . 55

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Flat Histogram Monte Carlo Methods 624.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 FHMC techniques in Communications Context . . . . . . . . . . . 63

4.2.1 Generating Biased Random Samples . . . . . . . . . . . . 714.2.2 Multicanonical Monte Carlo Technique . . . . . . . . . . . 73

CONTENTS ix

4.2.3 Transition Matrix Monte Carlo Technique . . . . . . . . . 764.2.4 Wang-Landau Algorithm . . . . . . . . . . . . . . . . . . . 79

4.3 Characteristics and Challenges of FHMC Algorithms . . . . . . . 814.4 FHMC Techniques for Communication Systems . . . . . . . . . . 844.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Ecient Simulation of MIMO Systems 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 MIMO-MRC System Model . . . . . . . . . . . . . . . . . . . . . 915.3 Ecient FHMC Algorithms . . . . . . . . . . . . . . . . . . . . . 93

5.3.1 Smooth-MMC . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.2 Optimal-MMC . . . . . . . . . . . . . . . . . . . . . . . . 975.3.3 TMMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.4 WL-TMMC . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . 1045.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Ecient Simulation of Coded Systems 1146.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3 Error Event Simulation of Terminated Convolutional Codes . . . . 1196.4 FHMC Techniques for Viterbi and MAP Decoders . . . . . . . . . 1216.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . 128

6.5.1 Modied-EES . . . . . . . . . . . . . . . . . . . . . . . . . 1286.5.2 Two-Phase Algorithm . . . . . . . . . . . . . . . . . . . . . 131

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Conclusion 1367.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . 1367.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

CONTENTS x

References 140

Appendices 158

A Derivations of Biasing Parameters for OFDM andOSTBC-OFDM Systems 158A.1 Optimum β for Variance Scaling of Noise in OFDM System . . . . 158A.2 Optimum Mean Value for Mean Translation of Noise in OFDM

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.3 Optimum βR for Variance Scaling of Fading Coecients in OFDM

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161A.4 Optimum βoo

R for VS-Rayleigh Technique in OSTBC-OFDMSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

B Derivations of Variance of FHMC Methods 165B.1 Variance of WL Algorithm . . . . . . . . . . . . . . . . . . . . . . 165B.2 Variance of TMMC Method . . . . . . . . . . . . . . . . . . . . . 166

Abbreviations

AP acceptance probabilityAWGN additive white Gaussian noiseBEP bit error probabilityBPSK binary phase-shift-keyingCC convolutional codeCCI cochannel interferenceCP cyclic prexCSCG circular symmetric complex GaussianCSI channel state informationDAIS dual-adaptive importance samplingECC error control codingEES error event simulationFFT fast Fourier transformFHMC at histogram Monte CarloFIR nite impulse responseHSDPA High-Speed Downlink Packet Accessi.i.d. independent, identically-distributedIFFT inverse fast Fourier transformIS importance sampling

ABBREVIATIONS xii

ISI inter-symbol interferenceITU International Telecommunication UnionLDPC low-density parity checkLDT large deviation theoryLTE long-term evolutionMAP maximum a posteriori probabilityMC Monte CarloMCMC Markov chain Monte CarloMIMO multiple-input multiple-outputML maximum-likelihoodMMC multicanonical Monte CarloMMSE minimum mean-square errorMPSK M -ary phase-shift-keyingMQAM M -ary quadrature amplitude modulationMRC maximal ratio combiningMT mean translationOFDM orthogonal frequency division multiplexingOSTBC orthogonal space-time block codespdf probability density functionPMD polarisation-mode dispersionQPSK quadrature phase-shift-keyingRV random variableSEP symbol error probabilitySISO single-input single-outputSNR signal-to-noise ratioTE tail extrapolation

ABBREVIATIONS xiii

TMMC transition matrix Monte CarloTPA two-phase algorithmVA Viterbi algorithmVS variance scalingWL Wang-LandauZF zero-forcing

Notation

(.) element-wise matrix multiplication(.)† conjugate transpose operatorC95% relative error for 95% condence intervalE[.] expectation operatorE∗[.] expectation with respect to biased pdfNMC sample size of MC simulationPe symbol error probabilityW (v) IS weight function of v

[.]T matrix transposeΓ(.) Gamma functionℑC imaginary part of complex number C

ℜC real part of complex number C

αij acceptance probability from state i to state j of Markov chainδxi

(x) Dirac-delta mass located at xi

ǫr normalised error or relative errorηMC/IS sample size reduction factor of IS over MCR realT transition matrix of Markov chainH(v) function that describes the error region of v

NOTATION xv

Nc0, 2σ2z CSCG random variable with zero-mean and variance σ2

z per dimensionRV range of variable V

S signal constellationconj(.) complex conjugate operationIFFT. inverse fast Fourier transform operatorξMC/IS run-time gain or computational eciency gain of IS over MCζMC/IS variance reduction factor of IS over MCxiN

i=1 random sample sequence of length N

f ∗V (v) biased pdf of v

nR number of receive antennasnT number of transmit antennas

List of Figures

2.1 Relative error of the MC estimator for 95% condence interval. . . 16

3.1 OFDM transmitter and receiver structure. . . . . . . . . . . . . . 303.2 OSTBC-OFDM transmitter and receiver structure. . . . . . . . . 393.3 SEP estimation results of Simulation I with QPSK modulation

using 300 OFDM blocks. . . . . . . . . . . . . . . . . . . . . . . . 493.4 SEP estimation results of Simulation I with 16-QAM modulation

using 300 OFDM blocks. . . . . . . . . . . . . . . . . . . . . . . . 503.5 Run-time gain (ξMC/IS) as a function of SEP in Simulation I with

QPSK modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6 SEP estimation results of Simulation II with QPSK modulation. . 543.7 SEP estimation results of Simulation II with 16-QAM modulation. 543.8 Run-time gain (ξMC/V S−Rayleigh) as a function of SEP in Simulation

II with QPSK modulation. . . . . . . . . . . . . . . . . . . . . . . 573.9 Run-time gain (ξMC/IS) as a function of SEP in Simulation I with

QPSK modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 593.10 Run-time gain (ξMC/IS) as a function of SEP in Simulation II with

16-QAM modulation. . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Mapping of output bins to input domains. . . . . . . . . . . . . . 644.2 Adaptation of FHMC algorithms towards a at visits histogram

[166]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

LIST OF FIGURES xvii

4.3 Functional block diagram of the MMC algorithm. . . . . . . . . . 734.4 pdf of decision variable V for BPSK system, Inset panel: corre-

sponding visits histogram. . . . . . . . . . . . . . . . . . . . . . . 754.5 Comparison of pdf estimation using basic update and Berg's update. 764.6 (a) Transitions between states i, j and k in TTT identity. (b)

Moves between transition matrix elements that satisfy TTT iden-tity [110]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.7 Example of a transition matrix that satises TTT identity. . . . . 794.8 QPSK system operating over an AWGN channel . . . . . . . . . . 844.9 Probability distribution of V at SNR = 17dB estimated by MC

and MMC. N = 9, NI = 2 × 105 . . . . . . . . . . . . . . . . . . 87

5.1 Block diagram of MIMO-MRC system. . . . . . . . . . . . . . . . 915.2 Comparison of κopt with approximation . . . . . . . . . . . . . . . 1045.3 Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 over

Nakagami-m fading channels, m = 0.5; Curves from left to rightcorrespond to SNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b)Optimal-MMC (c) TMMC (d) WL-TMMC. . . . . . . . . . . . . 105

5.4 Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 overNakagami-m fading channels, m = 1; Curves from left to rightcorrespond to SNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b)Optimal-MMC (c) TMMC (d) WL-TMMC. . . . . . . . . . . . . 106

5.5 Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 overNakagami-m fading channels, m = 2; Curves from left to rightcorrespond to SNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b)Optimal-MMC (c) TMMC (d) WL-TMMC. . . . . . . . . . . . . 107

5.6 Outage probability estimation of MIMO-MRC system with nT = 4

and nR = 4 over Nakagami-m fading channels, m = 0.5. . . . . . . 108

LIST OF FIGURES xviii

5.7 Outage probability estimation of MIMO-MRC system with nT = 4

and nR = 4 over Nakagami-m fading channels, m = 1. . . . . . . . 1095.8 Outage probability estimation of MIMO-MRC system with nT = 4

and nR = 4 over Nakagami-m fading channels, m = 2. . . . . . . . 1105.9 Relative error comparison of proposed FHMC estimators; MIMO-

MRC system with nT = 4 and nR = 4 over Nakagami-m fadingchannels, m = 0.5, SNR = 6 dB. . . . . . . . . . . . . . . . . . . . 111

5.10 Relative error comparison of proposed FHMC estimators; MIMO-MRC system with nT = 4 and nR = 4 over Nakagami-m fadingchannels, m = 1, SNR = 6 dB. . . . . . . . . . . . . . . . . . . . . 112

5.11 Relative error comparison of proposed FHMC estimators; MIMO-MRC system with nT = 4 and nR = 4 over Nakagami-m fadingchannels, m = 2, SNR = 6 dB. . . . . . . . . . . . . . . . . . . . . 112

6.1 Structure of K = 5, rate-1/2 convolutional encoder. . . . . . . . . 1156.2 Estimating BEP of the Viterbi decoder in AWGN channel at SNR

= 6 dB. K = 3, L = 13. . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Simulation results of code C2. EES, modied-EES and MC results

coincide after 3 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.4 Run-time gain Vs BEP for EES and modied-EES. . . . . . . . . 1316.5 Bit error probability of convolutional-coded system in AWGN chan-

nel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.6 Bit error probability of convolutional-coded system in Rayleigh

fading channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

List of Tables

3.1 Parameters of ITU Pedestrian-A and Vehicular-A channel models[160]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Average power relative to the strongest path of Rayleigh fadingmultipath channels. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 VS and MT estimator accuracy and sample size reduction factorof Simulation I with QPSK over Pedestrian Channel-A; total sim-ulated OFDM blocks = 300. . . . . . . . . . . . . . . . . . . . . . 51

3.4 VS and MT estimator accuracy and sample size reduction factor ofSimulation I with QPSK over Vehicular Channel-A; total simulatedOFDM blocks = 300. . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 VS and MT estimator accuracy and sample size reduction factorof Simulation I with 16-QAM over Pedestrian Channel-A; totalsimulated OFDM blocks = 400. . . . . . . . . . . . . . . . . . . . 51

3.6 VS and MT estimator accuracy and sample size reduction factor ofSimulation I with 16-QAM over Vehicular Channel-A; total simu-lated OFDM blocks = 400. . . . . . . . . . . . . . . . . . . . . . . 52

3.7 VS-Rayleigh estimator accuracy and sample size reduction factor ofSimulation II with QPSK over Channel-1; total simulated OFDMblocks = 300. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

LIST OF TABLES xx

3.8 VS-Rayleigh estimator accuracy and sample size reduction factorof Simulation II with QPSK over Equal-power Channel; total sim-ulated OFDM blocks = 300. . . . . . . . . . . . . . . . . . . . . . 55

3.9 VS-Rayleigh estimator accuracy and sample size reduction factorof Simulation II with 16-QAM over Channel-1; total simulatedOFDM blocks = 300. . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.10 VS-Rayleigh estimator accuracy and sample size reduction factorof Simulation II with 16-QAM over Equal-power Channel; totalsimulated OFDM blocks = 300. . . . . . . . . . . . . . . . . . . . 56

3.11 VS and MT estimator accuracy and sample size reduction factorof OSTBC-OFDM Simulation I over Pedestrian Channel-A; totalsimulated OFDM blocks = 300. . . . . . . . . . . . . . . . . . . . 57

3.12 VS and MT estimator accuracy and sample size reduction factorof OSTBC-OFDM Simulation I over Vehicular Channel-A; totalsimulated OFDM blocks = 300. . . . . . . . . . . . . . . . . . . . 58

3.13 VS-Rayleigh estimator accuracy and sample size reduction factorof OSTBC-OFDM Simulation II with 16-QAM over Channel-1;total simulated OFDM blocks = 300. . . . . . . . . . . . . . . . . 59

3.14 VS-Rayleigh estimator accuracy and sample size reduction factorof OSTBC-OFDM Simulation II with 16-QAM over Equal-powerChannel; total simulated OFDM blocks = 300. . . . . . . . . . . . 60

4.1 Comparison of the MMC estimator with the MC estimator in termsof the sample size reduction factor and run-time gain. . . . . . . . 87

6.1 Estimator accuracy and sample size reduction factor of the EESand modied-EES techniques for code C1. . . . . . . . . . . . . . 130

6.2 Estimator accuracy and sample size reduction factor of the EESand modied-EES techniques for code C2. . . . . . . . . . . . . . 130

6.3 Simulation run-time comparison of EES and modied-EES. . . . . 130

LIST OF TABLES xxi

6.4 Estimator accuracy and sample size reduction factor of TPA; C1

code in AWGN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5 Estimator accuracy and sample size reduction factor of TPA; C3

code in AWGN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.6 Estimator accuracy and sample size reduction factor of TPA; C1

code in Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . . . 1346.7 Estimator accuracy and sample size reduction factor of TPA; C3

code in Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . . . 134

List of Algorithms

4.1 MMC algorithm for estimating probability distribution of V. . . . 865.1 TMMC algorithm for estimating probability distribution of C. . . 1005.2 WL-TMMC algorithm for estimating probability distribution of C. 1136.1 Phase 1 of two-phase algorithm. . . . . . . . . . . . . . . . . . . . 1266.2 Phase 2 of two-phase algorithm. . . . . . . . . . . . . . . . . . . . 127

Chapter 1

Introduction

Simulation attempts to imitate dierent aspects of real-world processes in a re-alistic manner. The need for the ways to assess systems before their actualimplementation is the motivation behind the concept of simulation. Moreover,simulation is a valuable tool to gain insight into system behaviour. A well de-veloped simulation model allows to evaluate the performance of the system moreeasily and economically than by implementing hardware prototypes.

Nowadays, computer has become the implicit medium used to carry out themajority of simulations. In a computer simulation, a model of the actual systemis numerically evaluated by gathering data to estimate true characteristics of thesystem. Initially, analog computers were used for waveform level simulation ofthe control systems in aircrafts and weapons systems [1]. Later, the developmentof high-speed digital computers and large-capacity memory paved the way forthe development of numerical analysis, modelling and programming tools such asSPICE and MATLAB.

Traditionally, performance evaluation of communication systems was accom-plished by analytical methods. Analytical methods develop mathematical rela-tionships between various system parameters and performance metrics of interest.Exact performance evaluation using analytical techniques is intractable for mostpractical communication systems, and varying degrees of assumptions, approxi-

1. INTRODUCTION 2

mations and simplications have to be made to render the analysis tractable. Onthe other hand, simulation is not aected by the factors that increase analyticalcomplexity. This is the driving force behind the wide spread use of computersimulation for performance analysis of communication systems.

Simulation of communication systems is involved with estimating performancemetrics using properly designed simulation models. The classical method used forcommunication system simulations is the Monte Carlo (MC) technique [2], whichis a numerical method of solving mathematical problems by means of randomsampling [3]. MC is a general method that has the ability to cope with arbitrarilycomplex systems. However, it requires long pseudo-random sequences to simulaterare events and estimate the performance metrics down to very low probabilitieswith a high degree of accuracy. In fact, MC simulation requires a sample sizeof 100/Pe to estimate the probability of Pe with a relative error of about 20%

with 95% condence [4]. Moreover, the run-time of a simulation depends on thecomplexity of the system, and modern communication systems are excessivelycomplex. This along with large sample size requirements lead to prohibitively longMC simulations. This provided the motivation to investigate ecient simulationtechniques for communication systems.

The eciency of a simulation mandates careful programming of the computa-tion algorithm to accelerate execution and minimise storage requirements. How-ever, this alone is not sucient to speed up highly-complex simulations unless thestatistical eciency of the simulation is improved. The statistical eciency of asimulation is measured by the variance of the estimated outputs of the simulation.If the variance of an estimated output such as the bit error probability (BEP) orsymbol error probability (SEP) can be reduced without altering its expectation, agreater accuracy can be achieved for the same sample size; alternatively, a desiredaccuracy can be achieved with a smaller sample size [5]. Variance reduction tech-niques may highly speed-up the simulations depending on the particular model ofinterest. Most variance reduction techniques involve more computing than MC;

1. INTRODUCTION 3

however, this reduction in computational eciency can be compensated by thepotential gain in statistical eciency [5].

Estimator variance reductions can be achieved by modifying the MC simu-lation process to enhance the occurrence of the important system outputs. Im-portance sampling (IS) [6] is one such modied MC technique, which reduces theestimator variance by sampling more from the important regions of the inputspace. The important system outputs are encouraged by appropriately biasingthe probability density function (pdf) of the inputs. The amount of bias is cor-rected at the output using suitable weights called IS weights to obtain an unbiasedestimator.

The eciency of IS techniques relies on the biasing of the input pdf. Optimalinput biasing requires the knowledge of the input space that contributes to theimportant outputs. While a properly biased input pdf can enhance the eciencyof the simulation, incorrect biasing may drastically increase the sample size re-quirement. Therefore, a deep knowledge on the system and high analytical skillsare essential to gain most out of IS techniques.

IS techniques have been widely used in communication system simulations.Perhaps, they have the most potential for improved statistical eciency. Apartfrom the conventional IS techniques, other forms of modied MC methods havebeen developed in dierent disciplines. Flat histogram Monte Carlo (FHMC) isone such family of ecient simulation techniques developed by physicists [7]-[9].FHMC methods can estimate the entire distribution of a system output variablewith approximately equal degree of accuracy. They oer new powerful tools forsimulation of systems which are dicult to solve by analytical methods.

The potential of FHMC methods for accelerating communication systems sim-ulations has not yet been fully explored. A vast array of applications ranging fromcomplex decoder simulations through to multiuser detection are awaiting to beexplored using FHMC techniques. This thesis serves as a foundation for suchinvestigations.

1. INTRODUCTION 4

1.1 Motivation

Investigation of ecient simulation techniques has been a major topic in the eldof communication system modelling. Numerous techniques and tools have beendeveloped for fast simulation of complex communication systems such as optical,multiuser, spread spectrum and error control coded systems [10, and referencesthere in]. Despite the amount of research done on the subject, the applicationspecic nature of these techniques necessitates further investigations.

Rapidly increasing demand for high data rate wireless broadband services callfor the development of new standards to achieve increased data rates over exist-ing radio interfaces. Achieving high data rates over wireless links is an enormouschallenge due to channel impairments (multipath fading, noise and interference),limited frequency spectrum and limited power in mobile handsets. Current stan-dards employ adaptive modulation, error control coding (ECC) and diversitytechniques to mitigate the eects of fading, noise and interference thereby im-proving the reliability of communication. Diversity transmission and receptionwith multiple antennas has become a clever strategy in modern wireless systems.These multiple-input multiple-output (MIMO) techniques can drastically improvethe capacity of the wireless transmission without bandwidth expansion [11][12].In addition, multicarrier modulation techniques such as orthogonal frequency di-vision multiplexing (OFDM) has gained prominence in mitigating the eects offrequency-selective fading [13]. A lack of knowledge on ecient simulation ofthese new technologies provides a strong motivation for the subject of this thesis.

Despite the work of Ammari et al. [14] on ecient simulation of OFDMsystems, important aspects of the problem have been overlooked. This providedthe inspiration to develop a theoretical framework for applying IS techniques toOFDM system simulations.

Capacity estimation of MIMO systems has been analytically solved only forcertain fading models such as Rayleigh and Rician fading. Simulation is the

1. INTRODUCTION 5

only choice available for performance evaluation of other channel models such asNakagami-m. Hence, investigation of ecient methods to simulate MIMO systemperformance is an important contribution to the eld of communication systemmodelling.

Conventional IS techniques have been the most widely used mechanisms foraccelerating simulations of communication systems. Some simulation methodssuch as at histogram techniques [7] used in statistical physics have good potentialfor achieving improved eciency. This provided the motivation to investigate athistogram methods in application to communication system simulation.

ECC techniques are essential for controlling errors in digital communicationsystems. A range of coding techniques including convolutional, turbo and low-density parity check (LDPC) codes have been developed for achieving improvederror control capability. These coding techniques employ very complex decoderswhose performance is analytically intractable. Hence, nding ecient means forperformance evaluation of such decoders has been an active research problem formany decades.

Thus, the main purpose of this thesis is to investigate modied MC methodsfor ecient simulation of OFDM, MIMO and error control coded communicationsystems.

1.2 Major Contributions

This thesis has resulted in several contributions to the eld of ecient simulationin relation to communication systems. The major contributions are as follows:

• A comprehensive review of theory and principles of biased MC methodsfor ecient simulation of communication systems is presented. IS is identi-ed as the prominent method used for accelerating communication systemsimulations. FHMC methods are recognised as potential candidates forimproving the eciency of communication system simulations.

1. INTRODUCTION 6

• Importance sampling analysis for ecient simulation of OFDM systemsis presented. In particular, optimum IS parameters are derived for bias-ing input densities of OFDM and MIMO-OFDM systems operating overfrequency-selective fading channels. The proposed biasing strategies canachieve increasing eciency gains with decreasing SEP.

• FHMCmethods, which have revolutionised the eld of ecient simulation instatistical physics, are investigated in a communication engineering context.The potential of FHMC methods for ecient simulation of communicationsystems is explored. Simulation results show that these methods providesignicant eciency gains compared to the conventional MC method.

• New pdf estimators are derived for improved statistical eciency of FHMCalgorithms. The rst estimator, named as smooth-MMC, uses moving av-erage ltering and linear interpolation to reduce statistical variance andestimate lower probabilities than conventional FHMC estimators. Anotherapproach that optimally combines the current estimate and a previous es-timate results in the optimal-MMC estimator. Furthermore, two dierentFHMC techniques are eciently combined such that the statistical varianceof the nal estimator is minimised. This leads to an algorithm called WL-TMMC. While these algorithms show signicant sample size reductions inestimating capacity and outage probability of MIMO systems they can begenerally applied to any pdf estimation problem.

• Ecient methods for Viterbi and maximum a posteriori probability (MAP)decoder simulation are proposed. One method is an extension of the errorevent simulation technique. The other method applies FHMC techniques intwo dierent phases to estimate the performance of the decoders. The latterprovides signicant computational eciency gains in simulating Viterbi andMAP decoders in additive white Gaussian noise (AWGN) channels while

1. INTRODUCTION 7

moderate sample size reductions are achieved for Rayleigh fading environ-ments.

1.3 Publications

The following collection of papers, which have either been published in, acceptedby or submitted to peer-reviewed journals or conferences, describes outcomes ofthis thesis.

1. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Ecient Algo-rithm for Capacity and Outage Probability Estimation in MIMO Channels,Accepted for publication in the IEEE Communications Letters.

2. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Ecient Sim-ulation of Orthogonal Frequency Division Multiplexing Systems using Im-portance Sampling, IET Communications, vol. 5, no. 3, pp. 274283, 2011.

3. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Improved Per-formance Upper Bounds for Terminated Convolutional Codes in RayleighFading Channels, ECTI Transactions on Electrical Engineering, Electron-ics, and Communications, ISSN 1685-9545, vol. 8, no. 1, 2010.

4. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Transition Ma-trix Monte Carlo Technique for Outage Probability Estimation in MIMOChannels, in Proceedings of the Australian Communications Theory Work-shop (AusCTW), pp. 130135, Jan. 2011, Melbourne, Australia.

5. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Ecient Sim-ulation of OFDM and OSTBC-OFDM Systems over Multipath RayleighFading Channels, in Proceedings of the Australian Communications The-ory Workshop (AusCTW), pp. 136141, Jan. 2011, Melbourne, Australia.

1. INTRODUCTION 8

6. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, MulticanonicalEstimation of Outage Probabilities in MIMO Channels, in Proceedings ofthe IEEE Global Communications Conference (GLOBECOM), pp. 15,Dec. 2010, Miami, Florida.

7. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Improved Mul-ticanonical Algorithm for Outage Probability Estimation in MIMO Chan-nels, in Proceedings of the Asia Pacic Conference on Communications(APCC), pp. 297301, Nov. 2010, Auckland, New Zealand.

8. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Fast Simulationof OFDM Systems over frequency-selective Fading Channels, in Proceed-ings of the International Conference on Signal Processing and Communica-tions (SPCOM), pp. 15, Jul. 2010, Bangalore, India.

9. P. Wijesinghe, U. Gunawardana, R. Liyanapathirana, Improved Error Boundsfor Rate -1/2 Terminated Convolutional Codes with QPSK in Nakagami-mFading Channels, in Proceedings of the International Conference on SignalProcessing and Communications (SPCOM), pp. 14,Jul. 2010, Bangalore,India.

10. P. Wijesinghe, U. Gunawardana, R. Liyanapathirana, Low-ComplexityMAP Decoding of Tailbiting Convolutional Codes, in Proceedings of theInternational Conference on Signal Processing and Communications (SP-COM), pp. 14, Jul. 2010, Bangalore, India.

11. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Ecient Sim-ulation of Terminated Convolutional Codes, in Proceedings of the IEEERegion 10 Conference (TENCON), pp. 13, Nov. 2009, Singapore.

12. P. Wijesinghe, U. Gunawardana, R. Liyanapathirana, Tighter PerformanceUpper Bounds for Terminated Convolutional Codes over Rayleigh Fading

1. INTRODUCTION 9

Channels, in Proceedings of the Sixth International Conference on Elec-tronics, Computer, Telecommunications and Information (ECTI-CON), pp.800803, May 2009, Pattaya, Thailand.

13. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, MulticanonicalMonte Carlo Techniques for Fast Estimation of Outage Probabilities inMIMO Channels, Submitted to IET communications, 2010.

14. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Flat HistogramTransition Matrix Method for pdf Estimation in Communication Systems,Submitted to IEEE Global Communications Conference (GLOBECOM),2011.

15. P. Wijesinghe, U. Gunawardana, and R. Liyanapathirana, Fast Simulationof Error Control Coded Systems using Flat Histogram Monte Carlo Meth-ods, Submitted to IEEE Global Communications Conference (GLOBE-COM), 2011.

1.4 Thesis Organisation

The remainder of the thesis is organised as follows:In Chapter 2, dierent simulation methods including MC and biased MC

techniques are reviewed. The drawbacks of applying the MC method to moderncommunication systems are discussed, and the need for more ecient simulationtechniques is elaborated. Ecient simulation methods including tail extrapola-tion, pdf estimators and IS are reviewed. Of these techniques, IS is identied asthe most promising technique for communication system simulations. An emerg-ing technique known as at histogram Monte Carlo is identied as a potentialcandidate for fast simulation of communication systems. The developments ofFHMC techniques for statistical physics systems are investigated, and the poten-tial applications in communication systems are identied.

1. INTRODUCTION 10

Chapter 3 presents the IS simulation of OFDM and MIMO-OFDM systems.Two IS techniques, variance scaling (VS) and mean translation (MT), are appliedto derive the optimum biasing densities for both systems operating over frequency-selective fading channels. The proposed biasing strategies improve eciency ofthe simulation process by signicantly reducing the variance of the estimator.

In Chapter 4, the theory behind FHMC techniques is investigated in a com-munication engineering context. FHMC is viewed as a form of adaptive IS thatiterates to the optimal biasing density by using a blind adaptation procedure.Markov chain Monte Carlo (MCMC) techniques provide the underlying dynam-ics for the iterative adaptation procedure. Dierent forms of FHMC algorithmsand MCMC algorithms are investigated. Finally, the application of FHMC tech-niques to communication system simulation is illustrated.

Chapter 5 develops ecient FHMC algorithms for performance evaluation ofMIMO systems. In particular, the capacity and outage probability estimationof a MIMO maximal ratio combining (MRC) system operating over Nakagami-m fading channels is considered. Four dierent estimators are developed forachieving improved eciency and reduced relative error. The proposed estimatorscan estimate probabilities down to very low values (< 10−10) with a realisablesample size.

In Chapter 6, ecient techniques for performance evaluation of Viterbi andMAP decoders are investigated. New weight enumerators of terminated convolu-tional codes [15] are exploited to extend the error event simulation (EES) tech-nique for ecient simulation of terminated convolutional codes. The extended-EES method performs better than the conventional EES method in AWGN chan-nel conditions. Further investigation on the same topic leads to an ecient algo-rithm based on FHMC techniques. The algorithm estimates error performance ofthe decoders using two phases (Two-Phase algorithm) where both phases employWang-Landau (WL) at histogram technique. It is shown that the Two-Phasealgorithm performs well in AWGN as well as in Rayleigh fading channels.

1. INTRODUCTION 11

Chapter 7 concludes the thesis by presenting a summary of the investigations,research outcomes, and recommendations for future research.

Chapter 2

Literature Review

2.1 Introduction

This chapter reviews simulation techniques that have been applied to commu-nication systems. An overview of the traditional MC simulation technique isfollowed by a discussion of the drawbacks of applying MC simulations to mod-ern communication systems. The review focuses on applications and drawbacksof tail extrapolation and pdf estimation techniques in communication systemsimulations. The use of IS techniques for communication system simulations isdiscussed and unexplored gaps in knowledge identied. Finally, the potential ofFHMC techniques for ecient simulation of communication systems is discussed.

2.2 Monte Carlo Simulation

MC simulation has been successfully applied to gain insight into a vast rangeof scientic problems over the past sixty years. The method, which is based onstatistical sample averaging, was originally developed in the Los Alamos NationalLaboratory during the post-World War II era [2]. However, evidence for the ap-plication of similar techniques to determine the value of π during the second halfof the nineteenth century has been described in [16]. Ulam's idea of random sam-

2. LITERATURE REVIEW 13

pling, as described by the quote in [17], appears to have triggered the spark thatled to the development of the MC technique. In 1947, John von Neumann wrotethe rst proposal on the method, outlining the possible statistical approach tosolve the problem of neutron diusion in ssionable materials [18]. The rst pub-lication about the MC method itself appeared in 1949 [2]. Apparently, Metropolisnamed the method after the famous Casino in Monaco. The method has provento be a powerful and useful tool for a vast array of scientic problems, rangingfrom neutron diusion through to the evaluation of multi-dimensional integralsand the exploration of properties of high-temperature plasmas [17].

MC simulation draws an independent, identically-distributed (i.i.d.) set ofsamples xiN

i=1 from a target probability distribution p(x) dened on a highdimensional space Γ and uses these N samples to approximate the target distri-bution with the empirical point-mass function given by [19]

pN(x) =1

N

N∑

i=1

δxi(x) (2.1)

where δxi(x) denotes the Dirac-delta mass located at xi. This approach can be

used to approximate the integrals I(f) with tractable sums IN(f) that convergeas follows:

IN(f) =1

N

N∑

i=1

f(xi), limN→∞

I(f) =

∫

Γ

f(x)p(x) dx. (2.2)

When p(x) has a standard form (Uniform, Gaussian), it is straightforward tosample from it using the available methods such as the Marsaglia-Zaman algo-rithm [20][21], Ziggurat method [22] and Box-Müller Method [1]. However, whenp(x) has a non-standard form, more sophisticated techniques such as rejectionsampling and MCMC should be employed.

Due to its generality and the ability to cope with arbitrarily complex systems,the MC simulation method has been the major tool used to estimate the perfor-


mance of communication systems [1][23]. Consider the estimation of SEP Pe ofa simple communications system. Let the receiver output (decision variable) ofthe system be v. Then, Pe is given by [4]

Pe =

∫

Γ

H(v)fV (v) dv (2.3)

where fV (v) is the pdf of v and H(v) describes the error region as

H(v) =

1, v ∈ Γ0

0, v 6∈ Γ0

(2.4)

with Γ0 being the region of v that corresponds to an error.The corresponding MC estimator for Pe, denoted by Pe, is given by

Pe =1

NMC

NMC∑

i=1

H(vi) =Ne

NMC

(2.5)

where viNMC

i=1 is the output sequence, NMC is the sample size of the MC simula-tion and Ne is the number of symbol errors. By construction, H(vi) is a Bernoullidistributed random variable with the probability of success Pe. If the error eventsare independent, Ne is binomially distributed with parameters NMC and Pe. Us-ing the mean and variance of Ne, the mean and variance of the MC estimatorcan be derived as

E[Pe] = Pe (2.6)

and

σ2MC =

Pe(1 − Pe)

NMC

≈ Pe

NMC

, (2.7)

respectively. In (2.6), E[.] denotes the expectation operator. It follows from (2.6)


that the MC estimator of (2.5) is unbiased and on average provides the correctresult. According to (2.7), the estimator is consistent, since the variance decreasesas the number of simulation samples increases (NMC → ∞).

The normalised error or relative error ǫr of the MC estimator is given by [4]

ǫr =σMC

Pe

. (2.8)

This is also known as the accuracy of the MC estimator. For large NMC , theCentral Limit Theorem gives that Pe is Gaussian distributed, with the mean valuegiven by (2.6) and variance given by (2.7). Consequently, the 95% condenceinterval of the estimator Pe can be written as

[Pe ± 2σMC ] = [Pe(1 ± 2ǫr)] = [Pe(1 ± C95%)] (2.9)

where C95% is the relative error of the MC estimator for 95% condence interval.From (2.8) and the approximation of (2.7), C95% can be written as

C95% = 2ǫr =2√

NMCPe

=2

√

E[Ne]. (2.10)

Figure 2.1 shows the variation of C95% with NMCPe, which is the expected value ofthe number of errors. Accordingly, NMC = 1010 samples are required to estimatea probability Pe = 10−8 with C95% = 20% or ǫr = 10%.

From (2.10), the number of samples required to estimate Pe with a given ǫr

can be deduced as

NMC ≈ 1

ǫ2rPe

. (2.11)

This reveals that, if Pe is very small, NMC must be very large to ensure accu-rate estimation of Pe. Hence, the fundamental problem associated with the MCsimulation is that the run-time required to obtain a reliable estimate for low prob-


10−1

100

101

102

103

104

105

0

1

2

3

4

5

X: 1Y: 2

C9

5%

NMC

Pe=E[N

e]

X: 10Y: 0.6325

X: 100Y: 0.2

X: 1Y: 2

X: 10Y: 0.6325

X: 100Y: 0.2

Figure 2.1: Relative error of the MC estimator for 95% condence interval.

abilities is prohibitively long. This makes the MC method impractical, and haspaved the way for the development of more ecient simulation techniques.

2.3 Ecient Simulation Techniques

Simulation techniques that aim at overcoming the lengthy run-time requirementsof the MC method are collectively known as ecient simulation techniques. Sev-eral approaches have been developed for reducing the run-time of simulations.

This section briey reviews dierent ecient simulation approaches. The rstapproach, known as tail extrapolation (TE), involves curve tting. The secondtechnique is based on estimating the pdf of a decision metric using a posterioritechniques [24]. The third approach, importance sampling, involves modifyingthe statistical properties of the input sequence to encourage the occurrence ofimportant events (errors) in the simulation process. Of these three approaches, ISappears to be the most widely accepted approach for communication systems. Afourth approach known as FHMC, which can be considered as a family of adaptive


IS techniques, is also reviewed briey. The theory behind FHMC techniques isdiscussed in detail in Chapter 4.

2.3.1 Tail Extrapolation

The method of tail extrapolation was originally suggested by Weinstein [25] forthe simulation of binary signalling. The method involves executing a numberof MC simulations for a number of signal-to-noise ratio (SNR) values, for whichreliable estimates can be obtained with reasonable run-times. The results areextrapolated for values of SNR where MC simulations are not practical. Extremecare must be taken when extrapolating results into regions where experimental orsimulation results are not available [25]. This method also requires knowledge ofthe pdf in the tail region so that it can be described by a member of the generalisedexponential class [1]. Hence, TE is only accurate if the analyst is skilled enoughto select the appropriate family of pdf for a given system. It may not give validresults, especially for systems that exhibit a oor of the performance curve.

2.3.2 pdf Estimation

pdf estimation uses the deterministic relationship between the system perfor-mance and the pdf of the decision variable. The pdf of the decision variableis estimated using a posteriori techniques such as Parzen's estimator [24] andGram-Charlier series approximation [24]. Parzen's technique is a part of a broadclass of pdf estimators known as kernel estimators. It provides a smoothed esti-mator of the pdf which is asymptotically unbiased. It can also be proved that theParzen's estimator is consistent [23]. On the other hand, Gram-Charlier seriesis not asymptotically unbiased, which means that it does not uniformly convergeto the true pdf. In addition, the Gram-Charlier series approximation can havenegative values in the tails of the distribution when the target pdf is not nearlyGaussian, as is the case with multipath fading and cochannel interference (CCI).


Since the SEP and BEP depend on the tails of the pdf, this leads to unreliableestimates.

Despite its drawbacks, the Gram-Charlier series approximation has showngood results in BEP estimation of a 1-bit dierential demodulator with a veryshort observation interval in the AWGN channel and AWGN with multipathfading. Parzen's estimator has also shown better performance in similar scenariosand also in AWGN with CCI [24].

2.3.3 Importance Sampling

Importance sampling [6] falls into a category of ecient simulation methodsknown as variance reduction techniques [26]-[28]. Variance reduction techniquesdevelop an unbiased estimator that reduces the estimator variance or the sim-ulation run-time compared with MC simulation. IS reduces the sample size re-quirement of MC simulations by modifying the simulation in a controlled manner,which encourages the occurrence of important events (errors) in the process. Thisis normally achieved by varying (biasing) the statistical properties of the systeminputs. Knowledge of each biased input sample is exploited at the output to makethe estimator unbiased.

IS can be explained using the scenario discussed in Section 2.2. It introducesa new (biased) pdf f∗

V (v) that is preferable for ecient sampling. By using thenew pdf, (2.3) is written as

Pe =

∫

Γ

H(v)fV (v)

f∗V (v)

f∗V (v) dv

=

∫

Γ

H(v)W (v)f∗V (v) dv (2.12)

where W (v)= fV (v)f∗

V(v)

is called the IS weighting function. The biased output pdff∗

V (v) is obtained by appropriately warping the input samples of the system, and


Pe is now estimated as

P ∗e =

1

NIS

NIS∑

i=1

H(vi)W (vi) (2.13)

where P ∗e is the IS estimator and NIS is the sample size of IS simulation. The

IS estimator of (2.13) is unbiased [29]. The variance of the IS estimator can bederived as [30]

σ2IS =

1

NIS

∫

Γ

H(v)fV (v)[W (v) − Pe] dv. (2.14)

This formulation of IS was later named as the òutput version' of IS, as it isconstructed based on the knowledge of the system output pdf fV (v). The outputversion of IS is rarely used since fV (v) is not known or not easily deducible [30].

Generally, the statistical properties of the system inputs are known to thesimulation designer. Therefore, reformulation of Equations (2.12) - (2.14) interms of the input densities is advantageous. This formulation is known as theìnput version' of IS [30].

Let g(.) be the transfer function of the system that relates the system outputv to input x through v = g(x). The input version of (2.12) is

Pe =

∫

Γx

H[g(x)]W (x)f∗X(x) dx (2.15)

where f ∗X(x) is the biased input simulation density, W (x) = fX(x)

f∗

X(x)

, Γx is the inputspace and fX(x) denotes the pdf of the input. The empirical counterpart of (2.15)is

P ∗e =

1

NIS

NIS∑

i=1

H[g(xi)]W (xi) (2.16)


whose variance is given by

σ2IS =

1

NIS

∫

Γx

H[g(x)]fX(x)W (x) dx − P 2e

NIS

. (2.17)

The signicance of IS rests on how small σ2IS can be made. This depends on the

selection of the biased input density f∗X(x). While a well-designed biased density

can oer massive run-time savings, a poorly designed biased density can increasethe variance of the estimator. Hence, a deep system knowledge is essential todesign optimal biased densities, and this makes IS more system specic.

Optimal biased density that achieves zero variance can be derived from (2.17)as

f ∗opt(x) =

H[g(x)]fX(x)

Pe

. (2.18)

The optimal biased density is not realisable due to the presence of the unknownquantity Pe. However, it provides some useful hints for designing good biaseddensities. The optimal biased density puts all of its mass in the important (error)region where it is proportional to the original density. The important events arethose for which the original density is relatively large. In other words, they arethe most likely events that can be observed under the original density. Thus,good IS designs should increase the occurrence of these important events [10].

The eciency of IS techniques can be described using three measures. Therst of these is `variance reduction factor', denoted by ζMC/IS, which is denedas the ratio between the variance of MC and IS estimators for a given sample sizeNMC = NIS.

ζMC/IS ,σ2

MC

σ2IS

∣

∣

∣

∣

NMC=NIS

(2.19)

The second performance measure is the `sample size reduction factor' ηMC/IS,which is dened as the ratio between the sample size of MC and IS required to


achieve a given relative error ǫr.

ηMC/IS ,NMC

NIS

∣

∣

∣

∣

ǫr

(2.20)

The third measure is `run-time gain', denoted by ξMC/IS, which provides theexact simulation run-time gain or computational eciency gain for achieving agiven relative error ǫr of the estimator. This is dened as

ξMC/IS ,NMC

NIS

TMC

TIS

∣

∣

∣

∣

ǫr

(2.21)

where TMC and TIS denote the time required to complete a simulation using MCand IS methods, respectively.

Initial work on IS related to communication systems simulation proves thatIS is a powerful rare event simulation tool that has the most potential for sig-nicant computational eciency gain [4][29][32][33]. IS has been successfullyapplied in many digital communication systems including optical communicationsystems [34]-[36], multiuser communication systems employing spread spectrumtechniques [37]-[41], detection systems [42]-[44], satellite communication systems[45]-[48] and general communication systems [29][30][49]-[57]. IS has also shownsubstantial run-time savings for coded communication systems with sequentialdecoders [58][59] and Viterbi decoders [60]-[62], block-coded systems [63]-[65],space-time coded systems [66][67], concatenated systems [68][69], trellis-codedsystems [70]-[73], turbo-coded systems [74]-[77] and LDPC coded systems [78][79].A comprehensive review of the use of IS techniques in the simulation of digitalcommunication systems can be found in [10].

VS has been applied in the original approaches for IS simulation of commu-nication systems [4][29][32], and these techniques are now called conventional IStechniques. In VS, the biased pdf f ∗

X(x) is chosen to have the same form asfX(x) with a higher variance. When this is applied to bias the noise distribution


of communication systems, it results in heavy tails and more errors. For systemswith memory or multiple dimensions, uniform VS is applied in each dimension.This approach was largely dominated by the MT technique [50][56][80], whichwas proposed by Lu and Yao [50]. The MT technique is now called improved IS.It shifts the mean of the original distribution towards the important region asf∗

X(x) = fX(x − mx). The mean mx is selected so that the mode of the biaseddistribution lies around the mid point between the transmitted symbol and thetarget error event. MT has proven to be more ecient than VS [50]; however,implementation of MT is more complex than VS. Also, VS has been shown tosuer severely from the dimensionality eect arises with the system memory [56].

Viterbi decoder simulation using IS techniques was rst addressed by Herroand Nowack [60]. Their method is known as the block method and its computa-tional eciency gain is moderate. Later, Chen et al. [61] proposed an improvedVS technique called Burst IS with better performance. A more powerful IS tool,named as EES, for Viterbi decoder simulation was proposed by Sadowsky [62].In the Viterbi decoder, decision error occurs in bursts called error events. Anerror event is a partial sequence of incorrectly decoded branches that begins ata correct path node, terminates at a correct path node, and connects no correctpath nodes in the interim [62]. The essence of EES is to overcome the eectof memory length by isolating the simulation of error events in such a way thateach simulation run produces exactly one error event. EES technique has shownsignicant run-time gains for convolutional-coded systems operating over AWGNchannels and CCI environments [62][81].

The EES concept has also been described as `conditioning' [82]-[84]. Thecrucial advantage of conditioning is that it divides a complex simulation probleminto much simpler subproblems for which the IS techniques can be applied to gainlarge run-time savings. Conditioning has been applied to trellis-coded systems inwhich conditioning is done on the correct trellis path [70][71]. In addition, con-ditioning has been extended for systems with other coding techniques including


block codes [64][65], turbo codes [76][77] and concatenated codes [69].Another IS technique applied to communication systems uses a biased dis-

tribution completely dierent from the original distribution [51][52][85]. Wangand Lu [85] used the Weibull distribution while Beaulieu [52] used Gaussian andRayleigh tail distributions. A composite technique that uses a folded distributionand mean translation was proposed by Beaulieu [51], and it can oer a samplesize saving of 2.5 compared with the MT technique.

The use of large deviation theory (LDT) to design and evaluate IS simulationscan be rst seen in [86]. LDT has been developed into a coherent simulationstrategy in [87]-[89]. The essence of LDT in relation to communication systemsimulation is that for low error probabilities, errors are often caused by unusuallylarge noise terms that may contain a large deviation of noise process [10]. Theasymptotic eciency of IS has also been studied using LDT [88]. However, forcomplex systems, mathematical derivations involved with LDT may prove to beintractable.

Adaptive IS is a further improvement to IS techniques [90]-[95] in which thesimulation itself is used to derive good biased distributions. It reduces the burdenof deriving optimal biased distribution by using an approximate initial guess andimproving it adaptively. The adaptive IS method proposed by Stadler and Roy[90] converges to the parameters of optimal biased distribution by using groupsof simulations to update the distribution. Each group estimates the mean andmode of the distribution conditioned on the occurrence of an error. The originaldistribution is then moved to the estimated mean or mode. Al-Qaq et al. [93]minimised a cost function based on the estimator variance using a stochastic gra-dient descent algorithm to estimate parameters for the biased distribution. Thisapproach has demonstrated run-time savings of up to ten orders of magnitude forproblems in 30-dimensional space.


2.3.4 Flat Histogram Monte Carlo Methods

FHMC methods perform a random walk on the input space such that the outputalmost uniformly visits the entire range. These methods have been successfullyused in many scientic computing problems. The idea of at histogram samplingemerged after the invention of the multicanonical Monte Carlo (MMC) methodby Berg and Neuhaus in 1991 [7]. The MMC technique revolutionised the eldof ecient simulation in statistical physics and has been widely applied to manystatistical physics problems since then [96]-[99]. The technique adaptively con-verges to the optimal biased density using an iterative procedure. Hence it can beconsidered as a member of adaptive IS techniques. The MMC technique employsMCMC algorithms [100]-[104] to perform the random walk in the input spacewhile generating a visits histogram for the corresponding outputs. The pdf of theoutput is estimated at the end of each iteration and the estimated pdf is thenused to drive the random walk for the next iteration.

A variant of the MMC technique, the transition matrix Monte Carlo (TMMC),was later proposed by Swendsen et al. [8][105]-[109]. TMMC generates a tran-sition matrix for output states based on the random walk of the input space.The transition matrix and the properties of the Markov chain are exploited toestimate the output pdf at each iteration. Similar to the MMC technique, theestimated pdf is then used to drive the random walk for the next iteration. Thetransition matrix generated during one iteration can be used to drive the randomwalk for the next iteration [108]. The TMMC method is statistically accuratewhen it begins with a good initial estimate for the transition probabilities [110].

Another at histogram method was developed by Wang and Landau in 2001[9]. The WL algorithm is based on the observation that if a random walk isperformed in the input space with a probability inversely proportional to theprobability of occurrence of the output, then a at histogram is generated to theoutput distribution. This is the key mechanism of any at histogram method. The


WL algorithm accomplishes this by modifying the estimated output probabilityin a systematic way to produce a at histogram over the entire output range. Itsimultaneously converges the estimated probability to the true value [9]. The WLalgorithm has been successfully used in numerous problems of statistical physics,biophysics and others [111]-[114].

Numerous improvements have been proposed for the WL algorithm [115]-[120],and the eciency and convergence of the algorithm have been studied in detail[117]-[119]. Zhou and Bhatt [117] presented a mathematical analysis of the WLalgorithm together with a proof of convergence. They also discussed the sourcesof error for the WL algorithm and proposed strategies for improvement. Lee etal. [118] presented an error analysis for the WL algorithm. They found that,at steady state, uctuations in the accumulated histogram saturates at valuesproportional to [log(f)]−1/2 where f is the modication factor used in the WLalgorithm. This implies that the estimation obtained from the WL algorithmreaches a limiting statistical accuracy that cannot be improved in the course offurther simulations.

A combination of the TMMC method and the WL algorithm to improve thestatistical properties of the calculation of the density of states has been proposedby Shell et al. [121], where the strengths of both algorithms are exploited. TheWL-TM algorithm runs a pure WL simulation while accumulating a transitionmatrix. At the end of each iteration, the accumulated transition matrix is used toobtain a new estimate (called refreshing), which is used to drive the random walkfor the next iteration. In this combination, the WL algorithm enforces a broaddistribution of the output states while the TMMC method reduces the statisticalerror of the estimate.

It took more than ten years for the idea of at histogram techniques to es-cape from the physics community and to propagate to the statistics community[122][123]. Atchade and Liu [122] generalised the WL algorithm for general statespace. They also discussed the issues in application and convergence of WL


algorithm [123]. At the same time, at histogram techniques have gained theattention of optical communications researchers. Yevick [124], and Holzlöhnerand Menyuk [125] applied MMC techniques to nd the outage probability andthe bit error rate due to ber polarisation-mode dispersion (PMD). Later, a largenumber of publications appeared on the application of the MMC technique forvarious problems in optical communications [126]-[133]. The TMMC algorithmhas been employed to estimate the distribution of the outage times of an opticalber system impaired by stochastically-varying PMD. However, so far the WLalgorithm has not been applied to optical systems.

Application of MMC techniques for communication systems other than opticalsystems can be found in [134]-[137]. Ecient simulation of coded communicationsystems has been addressed in [134]-[136]. An adaptive algorithm was proposed byHolzlöhner et al. [134] for error performance evaluation of LDPC codes operatingover AWGN channels. The control variable that drives the proposed adaptivealgorithm was dened as a function of the noise samples and it could not directlydetermine the error performance. Thus, a second constrained simulation cyclewas required to evaluate the true error performance, giving it the name dual-adaptive importance sampling (DAIS). This technique potentially works for allcoded systems operating over AWGN channels; however, the requirement of twosimulation cycles somewhat decreases the eciency of the algorithm.

Another MMC-based method for evaluating the performance of Viterbi de-coders has been proposed by Secondini et al. [136]. They dened a controlvariable that can directly determine the system performance so that a single cy-cle of MMC simulation is sucient. However, this control variable is specicto the Viterbi decoder and it only considers the continuous mode of decoding.Therefore, it is neither generally applicable for any coding type nor it can be ap-plied for block based decoding encountered in practice. In addition, Dong [137]has applied MMC techniques to estimate the performance of a communicationsystem with co-channel interference and fading. The same problem was further


investigated using the TMMC method by Yevick and Reimer [138].The aforementioned discussion demonstrates that the at histogram concept

has been applied to a limited set of problems in communication systems. Infact, at histogram methods have a potential for achieving high computationaleciency gains as ecient pdf estimators. This exposes dierent avenues for theapplication of these techniques to communication systems.

2.4 Conclusion

The widespread use of computer simulations, combined with the drawbacks ofthe traditional MC method, has paved the way for the development of ecientsimulation techniques. IS is the most common ecient simulation technique ap-plied to communication systems. IS requires a deep knowledge of the system todesign an ecient simulation density and is application specic. FHMC tech-niques have revolutionised the eld of ecient simulation in statistical physics.FHMC, as a form of adaptive IS, shows good potential for ecient simulationof communication systems. Yet, little work has been done on the application ofFHMC techniques to communication systems. This provides the motivation toinvestigate FHMC techniques in a communications engineering context.

Chapter 3

Ecient Simulation of OFDMSystems

3.1 Introduction

OFDM has gained prominence during the last two decades, because it is robustagainst frequency-selective fading and it can be implemented eciently using fastFourier transform (FFT) [13]. OFDM has been applied to a diverse range of wiredand wireless applications, including digital audio and video broadcasting [139],digital subscriber lines using discrete multitone [140] and the current generationof wireless LANs [141]-[143]. Latest applications of OFDM includes xed-wirelessbroadband services (e.g., WiMax) [144] and long-term evolution (LTE) of cellularsystems (LTE and LTE-Advanced) [145].

Ever increasing demand for high-rate data services through wireless channelsmotivated the design of multiple antenna transmission to increase data rates with-out bandwidth expansion. Orthogonal space-time block codes (OSTBC) is onesuch transmit diversity technique which was rst proposed by Alamouti [146] fortwo transmit antennas and then generalised by Tarokh et al. [147][148] for arbi-trary number of transmit antennas. However, single-carrier OSTBC transmission

3. EFFICIENT SIMULATION OF OFDM SYSTEMS 29

over frequency-selective fading channels destroys the orthogonality crucial for lin-ear maximum-likelihood (ML) decoding of OSTBC. Owing to the fact that theOFDM technique can transform a frequency-selective fading channel into paral-lel at fading channels, the combination of OSTBC and OFDM [149][150] hasbecome an eective strategy to combat multipath fading eects.

Ecient simulation of OFDM systems has so far been overlooked. Ammari etal. [14] presented some initial work on the IS simulation of subcarrier synchro-nised OFDM systems; however, they did not provide evidence for the variancereduction gain and accuracy of the IS estimator. A recent work has consideredthe problem of estimating the SEP of OFDM receivers with non-linear distortions[151]; however, it has assumed an ideal communication channel. The lack of workon ecient simulation of OFDM systems provided the motivation for the analysispresented in this chapter.

This chapter analyses the IS simulation of OFDM and OSTBC-OFDM sys-tems operating over frequency-selective fading channels. Two IS techniques, VSand MT, described in Section 2.3.3, are employed in this analysis. The optimumbiasing parameters of VS and MT techniques applied to bias the time domainnoise samples and fading coecients are derived. A signicant improvement inthe computational eciency and accuracy of the estimator is achieved using theproposed methods.

3.2 Importance Sampling for OFDM Systems

In this section, IS analysis of OFDM systems is presented. In particular, VS andMT techniques are applied to bias the system inputs, and the optimal biasingparameters for VS and MT are derived.


Figure 3.1: OFDM transmitter and receiver structure.

3.2.1 System Model

Consider an OFDM system with N subcarriers and multilevel baseband modula-tion, operating over a frequency-selective fading channel as shown in Figure 3.1.The incoming information bits are rst modulated in baseband and the modu-lated symbols with symbol time Ts are then grouped into blocks of N symbols.Such a baseband modulated data symbol block can be denoted by a column vector

X = [X0, X1, · · · , XN−1]T (3.1)

where Xi is chosen from the set S= s0, s1, · · · , sM−1 of M distinct elements suchthat S represents the signal constellation, [.]T denotes the matrix transpose and M

represents the order of the signal constellation. For example, M -ary phase-shift-keying (MPSK) and M -ary quadrature amplitude modulation (MQAM) constel-lations can be described as follows:

SMPSK =

exp

[

j

(

2π

Md

)]

, d = 0, 1, · · · ,M − 1

, (3.2)

SMQAM = [±(2d1 + 1) ± j(2d2 + 1)], d1 = d2 = 0, 1, · · · , log2(M) − 1 .


The signal energy E[|Xi|2] is normalised to one using appropriate normalisationfactors for each signal constellation. It is assumed that the total transmit power isdivided evenly across subcarriers, which is optimal when the channel is unknownto the transmitter [152].

The baseband modulated symbols are then OFDM modulated and transmit-ted over the fading channel. Low-pass equivalent of the OFDM transmitted signalcan be written as

x(t) =N−1∑

i=0

Xiej2πi t

T , − Tg ≤ t < T (3.3)

where T = NTs is the OFDM symbol duration and Tg is the cyclic prex (CP)interval introduced between consecutive OFDM symbols to prevent inter-symbolinterference (ISI) and to preserve the orthogonality between OFDM subcarriers.

The communication channel is assumed to be a multipath fading channel withfrequency-selective fading. The fading process is assumed to be stationary andslowly varying compared with the OFDM block duration such that it is approxi-mately constant during one block length. With these assumptions, the complexbaseband representation of the multipath fading channel impulse response h(τ)

is given by [153]

h(τ) =L−1∑

l=0

hlδ(τ − τl) (3.4)

where hl is the attenuation factor for the signal received on the l-th path, τl is thepropagation delay for the l-th path and L is the total number of received paths.

After passing through the above channel, the low-pass equivalent of the re-ceived OFDM signal y(t) can be written as

y(t) =L−1∑

l=0

hlx(t − τl) + z(t), − Tg ≤ t < T (3.5)


where z(t) is a complex Gaussian random process with zero-mean and varianceσ2

z per dimension.Assuming Tg is larger than the maximum delay spread of the channel, the

output symbol of the subcarrier i, after demodulation, is obtained as

Yi =1

T

∫ T

0

r(t)e−j2π iT

t dt, i = 0, 1, · · · , N − 1. (3.6)

After some manipulations, (3.6) can be reduced to [154]

Yi = Hi.Xi + Zi (3.7)

where Zi is a circular symmetric complex Gaussian (CSCG) random variable withzero-mean and variance σ2

z per dimension (Zi ∼ Nc0, 2σ2z) and Hi is the channel

frequency response at subcarrier i given by

Hi =L−1∑

l=0

hle−j2πiτl/T , i = 0, 1, · · · , N − 1. (3.8)

Coherent detection at the receiver requires the knowledge of channel stateinformation (CSI). Let Hi be the estimated CSI for the i-th subcarrier. Thefrequency domain equaliser weights each element of the received vector Y =

[Y0, Y1, · · · , YN−1]T by the elements of a weight vector G = [G1, G2, · · · , GN−1],

which depends on the type of the equaliser. For instance, a zero-forcing (ZF)equaliser [139] has weighting factors described by

Gi =1

conj(Hi), i = 0, 1, · · · , N − 1 (3.9)

while weighting factors of a minimum mean-square error (MMSE) equaliser [139]


are given by

Gi =conj(Hi)

|Hi|2 + σ2z/σ

2s

, i = 0, 1, · · · , N − 1 (3.10)

where conj(.) denotes the complex conjugate operation and σ2s is the transmitted

symbol variance. The received samples, after channel equalisation, are expressedas

Vi = Gi.Yi

= Gi.Hi.Xi + Z ′i, i = 0, 1, · · · , N − 1

(3.11)

where Z ′i is a CSCG variable Nc0, 2σ2

i with σ2i = |Gi|2σ2

z . Decisions on thetransmitted symbols are made using V = [V0, V1, · · · , VN−1]

T .

3.2.2 Variance Scaling and Mean Translation of Noise

According to Section 3.2.1, the eect of channel memory is cancelled out withTg larger than the maximum delay spread of the channel, and the input-outputrelationship per subcarrier is identical to that of a at fading single-input single-output (SISO) channel. This simplies the IS simulation to a system withoutchannel memory.

Let S = G.H where (.) represents the element-wise multiplication. The ISestimator for the SEP of the i-th subcarrier P ∗

i can be written from (2.12) as

P ∗i =

∫ ∞

−∞

H[vi]W (vi)fVi(vi) dvi. (3.12)

For given Si, assuming that the input noise samples are biased, (3.12) can bewritten as

P ∗i =

∫ ∞

−∞

∫ ∞

−∞

H[Si.xi + z′i]W (z′i)fXi(xi)f

∗Z′

i(z′i) dx dz′

i (3.13)


where f∗Z′

i(z′i) is the biased noise density function in the frequency domain. Ac-

cording to (2.14), the estimator variance can be expressed as

σ2IS ≈

∫ ∞

−∞

∫ ∞

−∞

H[Si.xi + z′i]fXi(xi)fZ′

i(z′i)W (z′i) dxi dz′i. (3.14)

3.2.2.1 Variance Scaling

Let σ∗zibe the standard deviation of the biased noise of subcarrier i and βi be the

VS factor of subcarrier i for 0 ≤ i ≤ N − 1, which are related as

σ∗zi

= βiσz. (3.15)

The corresponding IS weighting function can be written as

WV S(z′i) = β2i exp

(

−|z′i|22σ2

i

(

1 − 1

β2i

))

. (3.16)

Considering a binary phase-shift-keying (BPSK) system and the fact that fXi(xi) =

12[δ(xi − A) + δ(xi + A)] where A is the BPSK signal amplitude, the VS estima-

tor variance σ2V S can be derived as (see Appendix A.1)

σ2V S =

β2i

(

2 − 1β2

i

)Q

(√

(

2 − 1β2

i

)

)

ℜSiA

σi

(3.17)

whereQ(t) =

1√2π

∫ ∞

t

e−t2

2 dt,

and ℜC represents the real part of the complex number C.Since the exact solution for βi which minimises σ2

V S is analytically insolvable,the approximation [4]

Q(t) ≈ e−t2

2

t√

2π


is used to nd a sub-optimum value for βi as

β2i =

(5σ2i + 4ℜSiA2 + (25σ4

i + 8σ2i ℜSiA2 + 16ℜSiA4)0.5)

8σ2i

. (3.18)

By substituting (3.18) in (3.17), it is observed that σ2V S is minimum when σi

is maximum. Hence, assuming that the signal power of dierent subcarriers areequal and all subcarriers are identically biased, a global β value for the systemcan be found as

β2 =(5σ2 + 4D2 + (25σ4 + 8σ2D2 + 16D4)0.5)

8σ2(3.19)

where σ = maxi|Gi|σz, D = miniℜSiA for BPSK and D = A. miniℜSi±ℑSi for MPSK and MQAM and A = dmin

2with dmin being the minimum Eucle-

dian distance between two symbols in the signal constellation. ℜC and ℑCrepresent the real part and imaginary part of the complex number C, respectively.

3.2.2.2 Mean Translation

Let the mean of the biased pdf of frequency domain noise samples Z ′i be m′

zi=

m′x + jm′

y. The corresponding IS weighting function can be written as

WMT (z′i) = exp

(

−|z′i|2 − |z′i − m′zi|2

2σ2i

)

(3.20)

with the biased pdf

f ∗Z′

i(z′i) =

1

2πσ2i

exp

(

−|z′i − m′zi|2

2σ2i

)

. (3.21)

Considering the BPSK system with −A transmitted, the MT estimator vari-


ance σ2MT can be derived as (see Appendix A.2)

σ2MT ≈ σi√

2π(ℜSiA + m′x)

exp

(

−3m′2

x + 2m′2y + 2ℜSiAm′

x + ℜSiA2

2σ2i

)

.

(3.22)

Minimising (3.22) with respect to m′x and m′

y yields the optimum mean forBPSK modulation with all zero transmission as

m′zi(opt) =

2ℜSiA + (ℜSiA2 − 3σ2i )

0.5

3. (3.23)

For ℜSiA2 ≫ σ2i , the optimum simulation mean can be approximated as

m′zi(opt) ≈ ℜSiA. (3.24)

The optimum mean given by (3.24) would shift the mean value of the decisionvariable Vi to the mid point between the transmitted symbol and the error symbol,thereby generating errors with a probability of 1/2.

It should be noted that m′zi(opt) is the optimum mean value for the biased

density f ∗Z′

i(z′i). In simulation, time domain noise pdf fz(z) should be biased such

that the mean of Z ′i given by (3.24) is obtained. Accordingly, the optimum mean

value for time-domain biased noise pdf can be obtained as

mzi(opt) = IFFTℜHiA (3.25)

where IFFT. represents the inverse fast Fourier transform operation.Application of the MT technique for higher order modulation techniques can

be accomplished by aiming specic error centres of the signal constellation in eachsimulation run. Assuming that the all-zero message s0 is transmitted, the SEP


of error centre method can be computed by

Pe =M−1∑

k=1

P (sk|s0) (3.26)

where sk is the target error centre and P (sk|s0) is the estimated probability ofdecoding sk given that s0 is transmitted. For each error centre, NIS simulationruns are performed. In this approach, the decision variable Vi should be biasedsuch that E∗[Vi] = Gi.Hi(sk + s0)/2, i.e., the mid point between the transmittedsymbol and the error centre. E∗[.] represents the expectation with respect to thebiased pdf. Thus, time domain noise zi should be sampled such that E∗[zi] =

mzi= IFFTHi(sk − s0)/2.

3.2.3 Variance Scaling of Fading Coecients

The VS technique can be applied to bias the pdf of fading coecients as fol-lows. Let the multipath coecients hl in (3.4) be mutually independent complexrandom variables (RVs) represented as

hl = |hl|ejφl (3.27)

where |hl| and φl are the amplitude and phase of hl, respectively. Amplitude |hl|is modelled as Rayleigh RVs with pdf

f|hl|(R) =R

σ2l

exp

(

− R2

2σ2l

)

(3.28)

where 2σ2l = E[|hl|2] is the power of the l-th path, and φl is uniformly distributed

over [0, 2π]. Then, Hi of (3.8) can be written as

Hi =L−1∑

l=0

|hl|ejθl =L−1∑

l=0

(Xl + jYl) (3.29)


where θl =[

φl − 2πilN

]

mod(2π) is also uniformly distributed over [0, 2π] and (Xl+

jYl) takes a joint complex Gaussian distribution. Application of FFT representsa linear transformation of jointly Gaussian RVs and yields a jointly Gaussian RV.Therefore, |Hi| takes a Rayleigh distribution with E[|Hi|2] =

∑L−1l=0 2σ2

l = 2σ2Hi.

Suppose that the variance of the Rayleigh coecients are scaled by a factorβ2

R. This method is called VS-Rayleigh. This type of VS has been applied toa coherent BPSK system [155] and a MIMO system with orthogonal space-timeblock coding [66]. They have obtained empirically optimum values for the VSfactor whereas an exact expression for the optimum VS factor is derived in thissection.

According to VS-Rayleigh technique, the biased variance of hl is β2Rσ2

l andthat of Hi is β2

Rσ2Hi. As a result, the IS weighting function WV S−R(R) is given by

WV S−R(R) = β2R exp

(

− R2

2σ2Hi

(

1 − 1

β2R

))

. (3.30)

Considering a BPSK system with −A transmitted and the output symbol ofsubcarrier i given by (3.7), the VS-Rayleigh estimator variance can be writtenaccording to the output version of IS [30] as (see Appendix A.3)

σ2V S−R =

∫ ∞

R=0

∫ ∞

y=0

fYi(y)WV S−R(R)f|Hi|(R) dy dR

=β3

Rσ2z

2AσHi(A2β2

Rσ2Hi

+ 2σ2zβ

2R − σ2

z)0.5

. (3.31)

Minimising σ2V S−R with respect to βR yields the optimum value for βR as

β2R =

3σ2z

2A2σ2Hi

+ 4σ2z

. (3.32)


Figure 3.2: OSTBC-OFDM transmitter and receiver structure.

3.3 Importance Sampling for OSTBC-OFDMSystems

In this section, three IS techniques VS, MT and VS-Rayleigh developed inSection 3.2 are extended for OSTBC-OFDM systems. Using an equivalent SISOmodel per OFDM subcarrier, the noise density function is biased optimally us-ing VS and MT techniques. Additional processing is needed to derive optimumMT of time-domain noise samples for OSTBC-OFDM systems. The applicationof VS-Rayleigh technique to bias Rayleigh distributed fading coecients is alsodiscussed. A closed-form solution for the VS factor of VS-Rayleigh technique is in-tractable for OSTBC-OFDM systems and the optimal biasing parameters shouldbe found numerically. This reduces the eciency of the proposed technique.Therefore, an approximate biasing parameter, which is analytically tractable, isproposed.


3.3.1 System Model

Consider an OSTBC-OFDM system with nT transmit antennas, nR receive an-tennas and N OFDM subcarriers as shown in Figure 3.2. Let the duration ofan arbitrary OSTBC-OFDM codeword be nB OFDM time slots. The KN infor-mation symbols taken from an M -ary signal constellation dened in (3.2) can beorganised into vectors as

Xk = [Xk,0, Xk,1, · · · , Xk,N−1]T , k = 0, 1, · · · , K − 1. (3.33)

The symbol sequence Xk is arranged into nB × nT column orthogonal trans-mission matrices Ci, (i = 0, 1, · · · , N − 1), which are linear combinations of(X1,i, · · · , XK,i, conj(X1,i), · · · , conj(XK,i)) [147]. This results in the STBC rateRc = K/nB. After OFDM modulation, xq

i [t] denotes a coded symbol transmittedfrom the q-th (q = 1, 2, · · · , nT ) transmit antenna over i-th (i = 0, 1, · · · , N − 1)subcarrier at t-th (t = 1, 2, · · · , nB) OFDM time slot. Accordingly, NnBnT sym-bols are transmitted from nT antennas over nB OFDM time slots.

Suppose that the frequency-selective fading channel between the q-th transmitantenna and the r-th (r = 1, 2, · · · , nR) receive antenna is modelled as a niteimpulse response (FIR) lter with taps hr,q

l , l = 0, 1, · · · , L − 1 where L is thetotal number of channel taps. Throughout this section, the channel is assumedto be constant over nB OFDM slots.

At the receiver, the OFDM demodulated signal at time slot t, receive antennar and subcarrier i, denoted by Y r

i [t], is given by

Y ri [t] =

nT∑

q=1

Hr,qi Xq

i [t] + Zri [t] (3.34)

where Zri [t] is zero-mean CSCG noise with variance σ2

z per dimension (Zri [t] ∼

Nc0, 2σ2z) and Hr,q

i is the frequency response of the channel between the q-th


transmit antenna and the r-th receive antenna on the i-th subcarrier given by

Hr,qi =

L−1∑

l=0

hr,ql e−

j2πilN , i = 0, 1, · · · , N − 1. (3.35)

From (3.34), the input-output relationship of the OSTBC-OFDM system atthe i-th subcarrier in the t-th time slot can be expressed as

Yi[t] = HiXi[t] + Zi[t] (3.36)

whereYi[t] = [Y 1i [t], Y 2

i [t], · · · , Y nR

i [t]]T , Xi[t] = [X1i [t], X2

i [t], · · · , XnT

i [t]]T , Zi[t] =

[Z1i [t], Z2

i [t], · · · , ZnR

i [t]]T and Hi is nR × nT matrix with rq-th element of Hi be-ing Hr,q

i . Analogous to SISO channels, MIMO-OFDM decomposes the frequency-selective fading channel into N orthogonal at fading MIMO channels [156].

Assuming perfect CSI and employing the ML decoding [157] with the squaringmethod [158], an equivalent SISO-OFDM model with channel gain a‖Hi‖2

F of thei-th subcarrier can be obtained as

Vk,i = a‖Hi‖2F Xk,i + Z ′

k,i, k = 0, 1, · · · , K − 1 (3.37)

where Vk,i is the output of the equivalent SISO-OFDM model, Z ′k,i is the zero-

mean complex Gaussian noise with variance σ2z′i

= a‖Hi‖2F σ2

z per dimension,a = 1/Rc, and ‖.‖2

F represents the squared Frobenius norm.From (3.36), assuming that the transmit and receive antennas are perfectly

synchronised, every OFDM subcarrier can be treated as a separate MIMO chan-nel. Further, those MIMO channels can be reduced to equivalent SISO-OFDMchannels as in (3.37). Then, using the analogy between (3.11) and (3.37), ISweighting functions and biasing parameters of VS, MT and VS-Rayleigh tech-niques can be derived for the OSTBC-OFDM system.


3.3.2 Variance Scaling

Let σ∗zibe the standard deviation of the biased noise of the i-th subcarrier and

βooi be the VS factor of the i-th subcarrier for 0 ≤ i ≤ N − 1, which are related

as

σ∗zi

= βooi σz. (3.38)

Here, the superscript oo is used to denote the OSTBC-OFDM system.Considering Vk,i of (3.37) as the decision variable and using an approach simi-

lar to that of Section 3.2.2.1, an optimum value for βoo, which biases all subcarriersidentically, can be derived as

βoo =(5σ2 + 4D2 + (25σ4 + 8σ2D2 + 16D4)0.5)0.5

21.5σ(3.39)

where σ = maxi√

a‖Hi‖F σz, D = minia‖Hi‖2FA and A = dmin

2with dmin

being the minimum Euclidean distance between two symbols of the signal con-stellation.

The corresponding weighting function used to unbias the error count of sub-carrier i at the receiver can be written as

W ooV S(Z, i) =

nB∏

τ=1

nR∏

r=1

1

2πσ2z

exp

− |Zri [τ ]|2

2σ2z

1

2πσ∗2

z

exp

− |Zri [τ ]|2

2σ∗2

z

= (βoo)nBnR exp

[

− 1

2σ2z

(

nB∑

τ=1

nR∑

r=1

|Zri [τ ]|2

)

(

1 − 1

(βoo)2

)

]

.(3.40)


3.3.3 Mean Translation

Let the mean of the noise samples Z ′k,i of (3.37) after MT biasing be moo

k,i. Usingan approach similar to that of Section 3.2.2.2, the optimum value for moo

k,i can bederived as

mook,i(opt) ≈ a‖Hi‖2

F A. (3.41)

This translates the mean value of the decision variable Vk,i to the mid pointbetween the transmitted symbol and the error centre, which gives

E∗[Vk,i] = a‖Hi‖2F

Xk,i + X ′

2(3.42)

where X ′ is the targeted error centre. Thus, Z ′k,i should be sampled such that

E∗[Z ′k,i] = a‖Hi‖2

F

X ′ − Xk,i

2. (3.43)

Since E∗[Z ′k,i] is known, the optimum MT for frequency domain noise samples,

Zri [t] of (3.34), denoted by E∗[Zr

i [t]], can be derived. It is also known that thenoise term Z ′

k,i is a function of Zri [t] and Hr,q

i . The derivation is only possiblewhen the number of Zr

i [t] components are equal to the number of Z ′k,i components,

which means nB = K and the code is full-rate. Therefore, this technique can beapplied only to full-rate OSTBC systems.

Consider a full-rate OSTBC Ci and let nR = 1. Dene Vi = [V1,i, · · · , VK,i]

and Xi = [X1,i, X2,i, · · · , XK,i]. Equation (3.37) can be written as [159]

Vi = a‖Hi‖2FXi + ZiΩ

Hi (3.44)

where Zi = [Z1i [1], · · · , Z1

i [K2], conj(Z1

i [K2

+1]), · · · , conj(Z1i [K]]) and Ωi is a func-


tion of Hr,qi such that [159],

HiCTi = [X1,i, X2,i, · · · , XK,i]Ωi. (3.45)

For example, Ωi for the Alamouti code G2 [146][148] is given by

Ωi(G2) =

H1,1i conj(H1,2

i )

H1,2i conj(−H1,1

i )

.

From (3.44), let Z′i = ZiΩ

Hi . Then,

E∗[Z′i] = E∗[Zi]Ω

Hi

E∗[Z′i]Ωi = E∗[Zi]Ω

Hi Ωi (3.46)

It can be shown that [159]

ΩHi Ωi =

K∑

k=1

|H1,ki |2IK (3.47)

where IK is the identity matrix of size K. From that, the optimum MT for thefrequency domain noise samples, denoted by E∗[Zi], can be expressed as

E∗[Zi] =1

∑Kk=1 |H

1,ki |2

E∗[Z′i]Ωi = Ψi (3.48)

where Ψi = [Ψri [1], · · · , Ψr

i [K2], conj(Ψ1

i [K2

+ 1]), · · · , conj(Ψ1i [K])] with Ψr

i [k] =

E∗[Zri [k]].

For nR > 1, (3.48) gives the optimum MT of frequency domain noise samplesat each receive antenna, which biases the received symbol at each receive antennatowards the targeted error centre. It is important to note that (3.48) gives theoptimum MT for the frequency domain noise samples of (3.34). In simulations,noise samples are biased in time-domain; in which the optimum MT for time-


domain noise samples can be obtained by taking the N point inverse fast Fouriertransform (IFFT) of (3.48).

The corresponding weighting function that unbiases the error count of the i-thsubcarrier can be written as

W ooMT (Z, i) = exp

−

∑nB

τ=1

∑nR

r=1 |Zri [τ ]|2 − |Zr

i [τ ] − Ψri [τ ]|2

2σ2z

. (3.49)

3.3.4 VS-Rayleigh Technique

The VS-Rayleigh technique introduced in Section 3.2.3 is extended for OSTBC-OFDM systems. Let the multipath coecients hr,q

l be mutually independentcomplex RVs represented as

hr,ql = |hr,q

l |ejφr,ql (3.50)

where |hr,ql | and φr,q

l are the amplitude and the phase of hr,ql , respectively. Am-

plitude |hr,ql | is modelled as Rayleigh RVs with the pdf given by (3.28). Then,

|Hr,qi | is also Rayleigh distributed with E[|Hr,q

i |2] =∑L−1

l=0 2σ2l = 2σ2

Hr,qi. Let

||Hi||2F =∑nT

q=1

∑nR

r=1 |Hr,qi |2 = Ri. Ri is Gamma distributed with parameters

nd = nT nR and 2B2 = 2σ2Hr,q

i. The decision variable of (3.37) can be written as

Vk,i = aRiXk,i + Z ′k,i with Ri being distributed according to

fRi(nd, 2B

2) =1

(2B2)ndΓ(nd)Rnd−1

i exp

(

− Ri

2B2

)

(3.51)

where Γ(.) is the Gamma function.Suppose that the variance of hr,q

l is scaled by a factor βooR . The corresponding

IS weighting function can be written as

WV S−R(Ri) = (βooR )nd exp

[

− Ri

2B2

(

1 − 1

βooR

)]

. (3.52)


Considering a BPSK system with −A transmitted, the VS-Rayleigh estimatorvariance can be derived as (see Appendix A.4)

σ2V S−R =

∫ ∞

0

(βooR )ndσz′i

aA(2B2)ndΓ(nd)(2π)0.5Rnd−2

i exp(

υ1υ22 − υ1(Ri + υ2)

2)

dRi

(3.53)

where υ1 = a2A2

2σ2z′i

and υ2 = 14υ1B2

(

2 − 1βoo

R

)

.Since a closed-form solution for (3.53) is intractable numerical solutions are

sought. Consider the case of nT = 2 and nR = 1 (corresponding to Alamoutischeme with one receive antenna). Equation (3.53) can be evaluated as (seeAppendix A.4)

σ2V S−R =

∫ ∞

0

(βooR )2σz′i

4aAB4(2π)0.5exp

(

υ1υ22 − υ1(Ri + υ2)

2)

dRi

=25(βoo

R )3σ2z′

2aAB2[41σz′(2βooR − 1) + 5(76σ2

z′(βooR )2 − 76σ2

z′βooR + 19σ2

z′ + 800(βooR )2a2A2B4)0.5]

.

(3.54)

By substituting the values for A, a, B and σ2z′ = minσ2

z′i and minimising (3.54)

with respect to βooR , optimum values for βoo

R are found numerically for each SNR.The drawback of this technique is that the βoo

R values should be found for eachsimulation run per SNR since σ2

z′ depends on the frequency response of the channel|Hr,q

i | per subcarrier. This consumes a considerable amount of time making theVS-Rayleigh technique computationally inecient. In fact, the run-time requiredto complete one simulation of VS-Rayleigh with aforementioned βoo

R calculationis on average 770 times higher than that required to complete one simulation ofMC. Even though the VS-Rayleigh technique provides higher variance reductiongains, according to (2.21) the actual run-time gain reduces by a factor of 770.Meanwhile, it was observed that the average βoo

R computed by minimising (3.54)


for dierent channels is comparable with that of (3.32) derived for the OFDMsystem. Hence, an approximate value for βoo

R is obtained from (3.32) as

βooR ≈ 3σ2

z

2aA2B2 + 4σ2z

. (3.55)

Simulation results of Section 3.4.2 show that this approximation performs well inimproving the eciency of the simulation.

3.4 Numerical Results and Discussion

3.4.1 OFDM System

Performance of the proposed IS techniques for OFDM systems operating overdierent frequency-selective fading channels was evaluated. The system param-eters were set according to the LTE standards [145] with coherent quadraturephase-shift-keying (QPSK) and 16-QAM modulations. It was observed that theIS techniques derived in Section 3.2.2 are best suited for OFDM systems operat-ing over frequency-selective fading channels modelled according to InternationalTelecommunication Union (ITU) standards [160] while that derived in Section3.2.3 is more suitable for fading channels with Rayleigh distributed multipathcomponents. Therefore, two simulation scenarios have been chosen as follows:

• Simulation I - an OFDM system operating over a 20 MHz bandwidth withN = 2048 subcarriers and a guard interval length G = 512 was consid-ered. With a sampling rate of 1.536, the OFDM symbol duration becameTs = 66µs while the guard interval was Tg = 16.66µs. The system wassimulated over ITU Pedestrian-A channels and Vehicular-A channels whosetapped-delay-line parameters (time delay relative to the rst path and theaverage power relative to the strongest path) are shown in Table 3.1. MMSEequalisation was employed in frequency domain at the receiver by assum-


ing that the perfect CSI is available. The SEP of the system is evaluatedusing two IS techniques proposed in Section 3.2.2, and the computationaleciency of those techniques over MC technique was computed.

• Simulation II - The OFDM system in Simulation II has a bandwidth of5 MHz, N = 512 subcarriers and a guard interval length G = 128. Thesystem was simulated over two Rayleigh fading multipath channels whoseparameters are shown in Table 3.2. ZF frequency domain equalisation wasemployed at the receiver. The VS-Rayleigh technique proposed in Section3.2.3 was used to evaluate the SEP of the system. The MC simulationresults were also obtained for comparison.

The SEP performance of the system in Simulation II is comparable to that of asingle-channel system operating over a Rayleigh (at) fading channel with fadingpower equal to the total power of all multipaths. Therefore, theoretical SEPperformance of this system can be evaluated by an approach similar to that in[161, (8.107) and (8.113)] by substituting correct fading power for each channel.This implies that the optimum β2

R of (3.32) is also valid for single channel-systemsoperating over Rayleigh fading channels.

Table 3.1: Parameters of ITU Pedestrian-A and Vehicular-A channel models [160].

Path Pedestrian Channel-A Vehicular Channel-ARelative delay Average power Relative delay Average power

(ns) (dB) (ns) (dB)

1 0 0.0 0 0.02 110 -9.7 310 -1.03 190 -19.2 710 -9.04 410 -22.8 1090 -10.05 - - 1730 -15.06 - - 2510 -20.0

Figures 3.3 and 3.4 show results of Simulation I for QPSK and 16-QAM mod-ulations, respectively. A total of 300 OFDM blocks were simulated using MC and


Table 3.2: Average power relative to the strongest path of Rayleigh fading mul-tipath channels.

Path Channel-1 Equal-power Channel(dB) (dB)

1 -2.5 02 0.0 03 -12.8 04 -10.0 05 -25.2 -6 -16.0 -

two IS techniques. Both IS techniques can accurately estimate the SEP in theorder of 10−12 while MC simulation technique can provide accurate results onlyup to 10−5 using 300 OFDM blocks.

10 12 14 16 18 20 22 2410

−10

10−8

10−6

10−4

10−2

100

Eb/N

0 (dB)

Sym

bol E

rror

Pro

babi

lity

Monte−CarloVariance ScalingMean Translation

Vehicular − A

Pedestrian − A

Figure 3.3: SEP estimation results of Simulation I with QPSK modulation using300 OFDM blocks.

To verify the estimation accuracy or relative error (ǫr) and the sample sizereduction factor of VS (ηMC/V S) and MT (ηMC/MT ) over MC, the accuracy and


12 14 16 18 20 22 24 2610

−12

10−10

10−8

10−6

10−4

10−2

100

Eb/N

0 (dB)

Sym

bo

l Err

or

Pro

ba

bili

ty

MCVSMT

Vehicular − A

Pedestrian − A

Figure 3.4: SEP estimation results of Simulation I with 16-QAM modulationusing 300 OFDM blocks.

variance of VS and MT estimators were calculated by running the simulation 100

times for each bit-energy-to-noise density ratio Eb/N0. 300 OFDM blocks weresimulated at each Eb/N0 for the QPSK system while 400 OFDM blocks weresimulated for the 16-QAM system. The number of symbols needed to obtain thesame SEP with the MC technique was estimated from (2.11). For a given samplesize N = NV S = NMT , the variance reduction gain achieved by the MT techniqueover VS technique (ζV S/MT ) was also calculated as in (2.19). These results arepresented in Tables 3.3 - 3.6 for QPSK and 16-QAM systems operating over twoITU channel conditions.

According to Table 3.3 and Table 3.4, VS and MT show a signicant samplesize reduction over MC for the QPSK-OFDM system of Simulation I. Simulationgain is higher with MT than with VS in a QPSK-OFDM system. The relativeerror of the MT estimator is much improved than that of the VS estimator forthe same sample size. This behaviour is consistent with the previous results [50]


Table 3.3: VS and MT estimator accuracy and sample size reduction factor ofSimulation I with QPSK over Pedestrian Channel-A; total simulated OFDMblocks = 300.

Eb/N0 Variance Scaling Mean Translation ζV S/MT

(dB) Mean-SEP ǫr(%) ηMC/V S Mean-SEP ǫr(%) ηMC/MT

17 2.583e-06 3.17 6.27e+02 2.595e-06 1.08 5.31e+03 8.4518 2.270e-07 4.43 3.66e+03 2.260e-07 1.33 4.08e+04 11.2019 1.114e-08 4.59 6.93e+04 1.108e-08 1.27 9.12e+05 13.7020 2.616e-10 5.94 1.76e+06 2.628e-10 1.41 3.09e+07 18.6021 2.489e-12 6.77 1.42e+08 2.494e-12 1.42 3.24e+09 22.80

Table 3.4: VS and MT estimator accuracy and sample size reduction factor ofSimulation I with QPSK over Vehicular Channel-A; total simulated OFDM blocks= 300.




Table 3.5: VS and MT estimator accuracy and sample size reduction factor ofSimulation I with 16-QAM over Pedestrian Channel-A; total simulated OFDMblocks = 400.





Table 3.6: VS and MT estimator accuracy and sample size reduction factor ofSimulation I with 16-QAM over Vehicular Channel-A; total simulated OFDMblocks = 400.




published for non-OFDM systems. In fact, a sample size reduction factor close to70, 000 is observed with the VS technique in estimating a SEP in the order of 10−8

whereas the sample size reduction observed with the MT technique for the sameSEP is higher than 900, 000 with an improvement in the order of 13.7 over VS.Table 3.5 and Table 3.6 show a similar improvement in sample size reduction withVS and MT over MC while a higher sample size reduction gain is obtained fromMT than VS for 16-QAM-OFDM system operating over Pedestrian Channel-Aand Vehicular Channel-A.

The simulation run-time of the VS technique is on average 1.4 times higherthan that of the MC technique, whereas the MT technique consumes on average3.6 times more time than the MC technique. The exact run-time gain ξMC/IS ofboth IS techniques is computed from (2.21). Figure 3.5 illustrates the ξMC/IS ratioversus SEP for QPSK system over two ITU channel conditions. Figure 3.5 showsan approximately linear variation of run-time gain versus SEP in logarithmicscale. It is clear that both techniques can provide substantial run-time savingsover MC while the run-time saving of MT is higher than that of VS. Since therelative error values achieved by MT is higher than that of VS for the same samplesize, MT can estimate with the same accuracy as VS using fewer samples thanVS. This further increases the run-time gain of MT over VS.


0 2 4 6 8 10 12 14 1610

0

102

104

106

108

1010

1012

−log10(SEP)

ξ MC

/IS

Pedestrian−A ChannelVehicular−A Channel

MT

VS

Figure 3.5: Run-time gain (ξMC/IS) as a function of SEP in Simulation I withQPSK modulation.

Figures 3.6 and 3.7 show results of Simulation II for QPSK and 16-QAMmodulations, respectively. A total of 300 OFDM blocks were simulated usingMC and VS-Rayleigh techniques. The VS-Rayleigh technique can accuratelyestimate the SEP in the order of 10−12 while MC technique can provide accurateresults only up to 10−4. The accuracy and the simulation gain of VS-Rayleighover MC (ηMC/V S−Rayleigh) were evaluated as explained in Simulation I. Theseresults are presented in Tables 3.7-3.10. Clearly, VS-Rayleigh technique showsextensive sample size reductions in simulating OFDM systems. The accuracy ofthe VS-Rayleigh estimator is always less than 5% for both modulation techniquesover both channel conditions.

The simulation run-time of the VS-Rayleigh technique is on average 1.2 timeshigher than that of the MC technique. The exact run-time gain ξMC/V S−Rayleigh

computed from (2.21) is illustrated in Figure 3.8. An approximately linear vari-ation of run-time gain versus SEP in logarithmic scale can be observed as in


20 30 40 50 60 70 80 90 10010

−12

10−10

10−8

10−6

10−4

10−2

100

Eb/N

0 (dB)

Sym

bol E

rror

Pro

babi

lity

MCVS−RayleighTheory

Equal−power Channel

Channel−1

Figure 3.6: SEP estimation results of Simulation II with QPSK modulation.

20 30 40 50 60 70 80 90 10010

−12

10−10

10−8

10−6

10−4

10−2

100

Eb/N

0 (dB)

Sym

bol E

rror

Pro

babi

lity

MCVS−RayleighTheory

Channel−1

Equal−power Channel

Figure 3.7: SEP estimation results of Simulation II with 16-QAM modulation.


Table 3.7: VS-Rayleigh estimator accuracy and sample size reduction factor ofSimulation II with QPSK over Channel-1; total simulated OFDM blocks = 300.

Eb/N0 SEP SEP(dB) Theory VS-Rayleigh ǫr(%) ηMC/V S−Rayleigh

44 1.04e-05 1.04e-05 2.29 1.19e+0356 6.55e-07 6.53e-07 2.10 2.25e+0468 4.13e-08 4.13e-08 2.14 3.43e+0580 2.61e-09 2.61e-09 2.20 5.17e+0692 1.65e-10 1.65e-10 2.10 8.96e+07100 2.61e-11 2.61e-11 2.05 5.94e+08

Table 3.8: VS-Rayleigh estimator accuracy and sample size reduction factor ofSimulation II with QPSK over Equal-power Channel; total simulated OFDMblocks = 300.



Simulation I. The computational eciency gain of the VS-Rayleigh techniqueincreases as the SEP decreases.

3.4.2 OSTBC-OFDM System

Performance of the proposed IS techniques for the OSTBC-OFDM system is eval-uated using two simulation scenarios as in Section 3.4.1. Simulation I employs anOSTBC-OFDM system with QPSK modulation, Alamouti code and nT = 2 andnR = 1 antennas operating over a 5 MHz bandwidth with N = 512 subcarriersand G = 128 guard interval. This system was simulated over ITU Pedestrian-A


Table 3.9: VS-Rayleigh estimator accuracy and sample size reduction factor ofSimulation II with 16-QAM over Channel-1; total simulated OFDM blocks =300.



Table 3.10: VS-Rayleigh estimator accuracy and sample size reduction factor ofSimulation II with 16-QAM over Equal-power Channel; total simulated OFDMblocks = 300.


52 1.02e-05 1.02E-05 2.88 7.71e+0264 6.45e-07 6.48E-07 2.62 1.46e+0476 4.07e-08 4.10E-08 2.70 2.18e+0588 2.57e-09 2.56E-09 2.93 2.96e+06100 1.62e-10 1.62E-10 2.95 4.62e+07112 1.02e-11 1.02E-11 2.97 7.21e+08

and Vehicular-A channels whose parameters are shown in Table 3.1.The OSTBC-OFDM system of Simulation II employs 16-QAM modulation,

H3 code [148] with nT = 3 and nR = 1 antennas. It also operates over a 5 MHzbandwidth with N = 512 subcarriers and G = 128 guard interval. The systemwas simulated over two Rayleigh fading multipath channels whose parametersare shown in Table 3.2. Theoretical performance of this system was calculated asproposed in [162].

Tables 3.11 and 3.12 show the estimated SEP, estimation accuracy and samplesize reduction factor of VS and MT techniques applied to Simulation I. A total of


4 5 6 7 8 9 10 1110

2

103

104

105

106

107

108

109

−log10(SEP)

ξ MC

/VS

−R

ayle

igh

Channel−1Equal−power Channel

Figure 3.8: Run-time gain (ξMC/V S−Rayleigh) as a function of SEP in SimulationII with QPSK modulation.

Table 3.11: VS and MT estimator accuracy and sample size reduction factor ofOSTBC-OFDM Simulation I over Pedestrian Channel-A; total simulated OFDMblocks = 300.

SNR Variance Scaling Mean Translation ζV S/MT


18 9.508e-05 7.03 1.39E+01 9.400e-05 5.15 2.61e+01 1.8819 1.904e-05 10.18 3.30e+01 1.870e-05 8.45 4.88e+01 1.4820 2.656e-06 10.63 1.08e+02 2.642e-06 9.43 1.39e+02 1.2721 2.330e-07 14.12 7.01e+02 2.346e-07 10.12 1.35e+03 1.9322 1.143e-08 17.87 6.69e+03 1.162e-08 13.47 1.16e+04 1.7323 2.641e-10 26.47 1.32e+05 2.631e-10 18.47 2.72e+05 2.06

300 OFDM blocks were simulated in each case. Accordingly, the optimal biasingparameters of VS and MT techniques are capable of properly biasing the systemsuch that the estimator variance is reduced. The MT technique has a highersample size reduction gain as well as improved accuracy than the VS technique.


Table 3.12: VS and MT estimator accuracy and sample size reduction factor ofOSTBC-OFDM Simulation I over Vehicular Channel-A; total simulated OFDMblocks = 300.

SNR Variance Scaling Mean Translation ζV S/MT


21 2.296e-05 11.79 1.02E+01 2.280e-05 6.49 3.38e+01 3.3222 4.094e-06 13.13 3.46e+01 4.110e-06 9.32 6.83e+01 1.9823 4.922e-07 18.40 1.46e+02 4.872e-07 10.40 4.63e+02 3.1624 3.705e-08 24.69 1.08e+03 3.650e-08 15.69 2.72e+03 2.5125 1.364e-09 32.99 1.64e+04 1.384e-09 22.89 3.37e+04 2.0526 2.306e-11 40.97 6.31e+05 2.306e-11 29.84 1.19e+06 1.88

The sample size reduction gain obtained by both IS techniques for the OSTBC-OFDM system is lower than that of the OFDM system of Chapter 3. This canbe explained as an eect of the dimensionality, added to the OFDM system byspace-time block coding.

Figure 3.9 illustrates the exact run-time gain obtained by the VS and MTtechniques over MC technique. An approximately linear variation of the run-time gain versus SEP can be observed in the logarithmic scale. Clearly, both IStechniques provide substantial run-time savings over the MC technique while therun-time saving of the MT technique is higher than that of the VS technique.

Tables 3.13 and 3.14 show the performance of VS-Rayleigh technique forOSTBC-OFDM system with H3 code and 16-QAM modulation operating overmultipath Channel-1 and Equal-power channel, respectively. The approximatebiasing factor of VS-Rayleigh provides sample size reduction gains in the order of105 in estimating probabilities of 10−9 with a relative error less than 15%. Eventhough this is slightly less than the gain of the same technique for the OFDMsystem in Section 3.4.1, it is still a signicant saving in computational cost.

Figure 3.10 depicts the exact run-time gain of the VS-Rayleigh technique. Italso demonstrates an approximately linear variation of the run-time gain versus


4 5 6 7 8 9 10 11 1210

0

101

102

103

104

105

106

107

108

−log10(SEP)

ξ MC

/IS

Pedestrian−A ChannelVehicular−A Channel

MT

VS

Figure 3.9: Run-time gain (ξMC/IS) as a function of SEP in Simulation I withQPSK modulation.

Table 3.13: VS-Rayleigh estimator accuracy and sample size reduction factorof OSTBC-OFDM Simulation II with 16-QAM over Channel-1; total simulatedOFDM blocks = 300.

SNR SEP SEP(dB) Theory VS-Rayleigh ǫr(%) ηMC/V S−Rayleigh

26 1.47e-05 1.45e-05 12.5 2.87e+0128 3.81e-06 3.72e-06 12.8 1.07e+0230 9.75e-07 9.54e-07 12.9 4.08e+0232 2.48e-07 2.40e-07 12.3 1.78e+0334 6.28e-08 6.10e-08 12.9 6.46e+0336 1.58e-08 1.57e-08 13.2 2.38e+0438 3.99e-09 3.82e-09 14.9 7.68e+0440 1.00e-09 9.99e-10 13.7 3.48e+05

SEP in the logarithmic scale. This proves the generality of the derived optimumVS factor for both OFDM and OSTBC-OFDM systems.


Table 3.14: VS-Rayleigh estimator accuracy and sample size reduction factor ofOSTBC-OFDM Simulation II with 16-QAM over Equal-power Channel; totalsimulated OFDM blocks = 300.

SNR SEP SEP(dB) Theory VS-Rayleigh ǫr(%) ηMC/V S−Rayleigh

26 7.32e-05 7.14e-05 9.22 1.07e+0128 1.94e-05 1.88e-05 10.4 3.20e+0130 5.04e-06 4.77e-06 10.7 1.19e+0232 1.29e-06 1.26e-06 10.3 4.93e+0234 3.29e-07 3.20e-07 12.0 1.42e+0336 8.34e-08 8.09e-08 9.26 9.38e+0338 2.11e-08 2.06e-08 10.8 2.70e+0440 5.31e-09 5.16e-09 11.1 1.02e+05

3 4 5 6 7 8 910

−1

100

101

102

103

104

105

106

−log10(SEP)

ξ MC

/VS

−R

ayl

eig

h

Channel−1Equal−power Channel

Figure 3.10: Run-time gain (ξMC/IS) as a function of SEP in Simulation II with16-QAM modulation.

According to Figures 3.5, 3.8 3.10, the run-time gain varies with the SEP as

ξMC/IS =10p2

(SEP)p1(3.56)


where p1 and p2 denote the slope and intercept of extrapolated lines. This deducesthat the run-time gain obtained from the proposed IS techniques would increasewith decreasing SEP for increasing values of the slope. This demonstrates thepotential of IS that is achieved through proper biasing of system parameters foraccelerating simulations.

Outcomes of this chapter have been published in [163]-[165].

3.5 Conclusion

Importance sampling analysis of OFDM and OSTBC-OFDM systems is presentedfor ecient simulation of these systems. VS and MT techniques can be exploitedfor ecient sampling of OFDM systems operating over frequency-selective fadingchannels. The proposed VS and MT approaches and derived optimal biasingparameters that bias the time-domain noise density and fading density improvethe eciency of the sampling process. The equivalent SISO model of the OSTBCsystem is exploited to derive the optimum biasing parameters for OSTBC-OFDMsystems. Approximations for the VS factor of fading coecients are drawn toovercome the analytical intractability of exact solutions. Sample size reductionfactor, computational eciency gain and accuracy of the estimators are used asthe gures of merit to compare the proposed techniques with the conventionalMC technique. Simulation results demonstrate that the proposed methods areaccurate and computationally ecient in estimating very low error probabilities.

Chapter 4

Flat Histogram Monte CarloMethods

4.1 Introduction

FHMC methods, which have been developed by physicists, have demonstrated apowerful tool for ecient simulation of statistical physics problems. They have agood potential for accelerating communication system simulations that are di-cult to evaluate using analytical techniques.

FHMC methods estimate the distribution of the system output variables byimplementing a random walk in the multi-dimensional distribution of the systeminput variables. The ultimate goal of FHMC methods is to obtain a at visitshistogram over the entire range of the system output, thus providing approxi-mately equal estimation accuracy. FHMC methods are related to IS in the waythe weights are calculated and in some cases the way the at output histogramis weighted to obtain the nal estimate for the output distribution. IS requiresa deep knowledge of the system to derive an optimal biased distribution andconsumes more time and eort in the design phase. FHMC methods provideinnovative, adaptive algorithms, which iteratively search for the optimal biased

4. FHMC SIMULATION TECHNIQUES 63

input distribution. The success of FHMC methods mostly relies on their ease ofimplementation.

Many dierent algorithms, which are able to generate at visits histograms,have been invented in relation to statistical physics applications. Of these algo-rithms, MMC method [7][96], TMMC method [108, and references there in] andWL algorithm [9] have taken prominence in the literature. This chapter presentsthe theory of FHMC methods in relation to communication systems simulationand discusses the aforementioned FHMC algorithms by considering a basic sim-ulation problem in telecommunications.

4.2 FHMC techniques in CommunicationsContext

Consider the estimation of the SEP Pe of a simple communication system operat-ing over an AWGN channel. Let the decision variable of the system be V = g(X)

where g : Γ →R is a real valued function of a random vector X dened over theinput space Γ. The SEP of this system can be determined as in (2.3), given thatthe pdf of V fV (v) is known in its entire range. Even though fV (v) is analyticallytractable for simple communication systems, it is often analytically intractablefor complex systems. Hence, the probability distribution of V is estimated usingsimulations.

To estimate the pdf fV (v) of the continuous output variable V over the rangeRV at particular SNR, RV is divided into M bins of width ∆v such that the m-thbin is dened as Bm ,

[

vm − ∆v2

, vm + ∆v2

]

with vm being the centre of the m-thbin. The probability of V falling into the m-th bin is dened as Pm , P (V ∈Bm) = fV (vm)∆v where fV (vm) is the pdf in the discretised m-th bin of V . Thisbinning process implies a division of the input space Γ in to M domains DmM

m=1


Figure 4.1: Mapping of output bins to input domains.

whereDm = X ∈ Γ : V ∈ Bm

is the domain in the input space that maps into the m-th bin [166]. This isillustrated in Figure 4.1. Accordingly, Pm can be expressed as

Pm =

∫

Dm

fX(x) dx =

∫

Γ

Hm[g(x)]fX(x) dx = E[Hm[g(x)]] (4.1)

where Hm[g(x)] is the indicator of event V ∈ Bm given by

Hm[g(x)] =

1, V ∈ Bm

0, otherwise

and E[.] is the expectation operator.In the conventional MC technique, Pm is estimated by drawing NMC samples

from the input pdf fX(x), evaluating them through the system g(.), counting thesamples falling into each bin (forming the visits histogram) and calculating thenormalised histogram which gives the sample mean for the expectation in (4.1).


The MC estimator for Pm, denoted by PMCm , is written as

PMCm =

1

NMC

NMC∑

i=1

Hm[g(xi)]. (4.2)

The mean and variance of the estimator can be evaluated as in (2.6) and (2.7),respectively. The relative error of the estimator in the m-th bin can be obtainedas

ǫMCm =

√

1 − Pm

NMCPm

≈ 1√NMCPm

. (4.3)

In MC simulations, most of the samples fall in the vicinity of the mode of V , andless samples are obtained from the tail region. This increases the relative errorof the estimator at the tails of the distribution.

In IS, the input pdf is biased such that more samples fall into the bins of thetail region, and Pm is correctly estimated by suitably weighing the count in eachbin. Let the biased input pdf be f∗

X(x). Then, (4.1) can be written as

Pm = E∗[Hm[g(x)]W (x)] (4.4)

where the expectation is with respect to the biased pdf and W (x) , fX(x)/f ∗X

(x)

is the IS weight. The expectation in (4.4) can be estimated by the sample meanas

P ISm =

1

NIS

NIS∑

i=1

Hm[g(xi)]W (xi). (4.5)


Assuming that there are N∗m visits to the m-th bin, (4.5) can be written as [166]

P ISm =

N∗m

NIS

[

1

N∗m

N∗

m∑

j=1

Wm(xj)

]

= H∗m.Wm

= P ∗m.Wm (4.6)

where H∗m = N∗

m/NIS = P ∗m is the normalised histogram entry or the probability

estimate at the m-th bin of the biased system and Wm is the average weightof all samples that fall into the m-th bin of the biased system. Equations (4.5)and (4.6) provide the rule to convert the visits histogram of the biased systemto an estimate of the true histogram [4]. Using (4.4), it can be proved that theIS estimator of (4.5) is unbiased. The variance and the relative error of the ISestimator can be derived as

V ar[P ISm ] =

V ar[Hm[g(x)]W (x)]

NIS

=Pm[Wm − Pm]

NIS

(4.7)

and

ǫISm =

√

[Wm − Pm]

NISPm

, (4.8)

respectively. According to (4.8), if the biased distribution is chosen such thatWm ≪ 1, then ǫIS

m ≪ ǫMCm for the same number of samples. This also implies

a massive reduction in the estimator variance of (4.7). Such a biasing requiresa deep knowledge on the domain Dm associated with the bin Bm. If the inputdistribution is biased blindly, it may decrease the weights in some parts of Dm

while increasing them on the other parts, which may lead to an increased esti-mator variance. This is the major drawback that prevents IS being applied in


complex system simulations.If the biased distribution is chosen as [168]

f ∗X

(x) =

fX(x)Pm

for x ∈ Dm

0 otherwise,(4.9)

then,

P ISm =

1

NIS

NIS∑

i=1

Pm = Pm, (4.10)

and all xi samples fall into Dm. This provides a zero estimator variance because(4.10) is deterministic. The biased distribution corresponding to the zero-varianceestimator is called the optimum IS biased distribution. It is not realisable as itrequires the knowledge of Pm which is unknown beforehand. Nevertheless, itreveals an important fact that the weights should be constant within Dm.1

The zero-variance estimator provides the best possible estimation for the m-th bin when the knowledge of Dm in Γ is available. Such a biasing is useless forthe other bins because all the samples fall into the m-th bin and no samples aregenerated for other bins. However, the perception of the zero-variance estimatorcan be exploited for optimum biasing of all bins, if the knowledge of Dm and theassociated output probabilities Pm for all bins m = 1, · · · ,M are available. Inthat case, the input pdf is biased according to the zero-variance biasing of (4.9)for all domains Dm.

The resultant biased distribution can be written as [167]

f∗X

(x) =1

M

M∑

m=1

Hm[g(x)]

Pm

fX(x). (4.11)

In (4.11),∑M

m=1Hm[g(x)]

Pm= 1

P [g(x)]with P [g(x)] being the probability of the

1It can be shown that Wm = Pm


bin where v = g(x) falls. Accordingly, (4.11) can be re-written as

f∗X

(x) =fX(x)

MP [g(x)]. (4.12)

The output visits histogram generated from the biased input pdf of (4.12) canbe obtained as

H∗m =

∫

Dm

f ∗X

(x) dx =1

M

∫

Dm

fX(x)

Pm

dx =1

M, (4.13)

which is constant for all bins providing a Flat Histogram.Input biasing of (4.12) yields a at visits histogram over the entire range of V .

This is also a form of constant weight biasing as W (x) = MPm for all x ∈ Dm.In this case the number of samples that fall into all bins are approximately equaland, as will be shown, the variance of the estimator in each bin is not zero.

Consequently, the estimator for FHMC techniques can be derived from (4.5)as

P FHMCm =

1

NIS

NIS∑

i=1

Hm[g(xi)]fX(xi)

f∗X

(xi)

=

(

N∗m

NIS

)

MPm. (4.14)

The estimator of (4.14) is unbiased because

E[P FHMCm ] =

M

NIS

Pm.E[N∗m] =

M

NIS

Pm.NISH∗m

(4.13)= Pm. (4.15)


The variance of the FHMC estimator can be derived as [168]

V ar[P FHMCm ] =

(

MPm

NIS

)2

V ar[N∗m]

=

(

MPm

NIS

)2NIS

M

(

1 − 1

M

)

=P 2

mM

NIS

(

1 − 1

M

)

. (4.16)

The relative error of the FHMC estimator of the m-th bin can be written as [168]

ǫFHMCm =

√

(M − 1)

NIS

M≫1∼= 1√

N b

(4.17)

where N b = NIS

Mis the average number of samples per bin. The relative error is

the same for all bins and all bins are estimated with equal accuracy.The optimum biased density of (4.12) for at histogram techniques is also

intractable due to the presence of the unknown quantity P [g(x)]. FHMC algo-rithms surmount this diculty by iteratively estimating P [g(x)] such that thebiased input density adaptively converges to the optimal biased density of (4.12).At the beginning, P [g(x)] is initialised to a constant value for all bins. Let theestimated value for P [g(x)] at the n-th iteration be P n[g(x)]. Starting fromthe known input pdf fX(x), FHMC algorithms generate the biased pdf for the(n + 1)-th iteration, denoted by f

(n+1)X

(x), as

f(n+1)X

(x) =fX(x)

qnP n[g(x)]n = 0, 1, . . . (4.18)

where qn is the normalising constant. In (n+1)-th iteration, samples are generatedfrom f

(n+1)X

(x) to form a new estimate for P (n+1)[g(x)]. The formation of the newestimate at the end of each iteration is known as the pdf update procedure. The


Figure 4.2: Adaptation of FHMC algorithms towards a at visits histogram [166].

histogram entry for the m-th bin at the (n + 1)-th iteration can be computed as

H(n+1)m =

∫

Dm

fX(x)

qnP n[g(x)]dx =

Pm

qnP nm

(4.19)

where P nm is the estimated probability of the m-th bin at the n-th iteration. Ac-

cording to (4.19), more samples fall into the bins where P nm ≪ Pm whilst less

samples fall into the bins with P nm ≫ Pm. This eventually attens the visits his-

togram whereas an appropriately dened pdf update procedure drives the systemto the convergence where qn → M and P n[g(x)] → P [g(x)] , PmM

m=1. Fig-ure 4.2 illustrates the adaptation of FHMC algorithms towards a at histogram.

Accordingly, FHMC algorithms contain two important steps:

1. Drawing samples from the biased input pdf fnX

(x) that may have a non-standard from in a high dimensional space

2. pdf update procedure.


In step 1, random samples are generated from the biased distribution at eachiteration using MCMC methods, which will be discussed in Section 4.2.1. Thepdf update procedure drives the algorithm towards convergence and is the mostimportant part of FHMC algorithms. All FHMC algorithms that will be discussedin Sections 4.2.2, 4.2.3 and 4.2.4, dier in their pdf update procedure.

4.2.1 Generating Biased Random Samples

Sample generation from the biased input distribution that may take a non-standard form in high dimensional space, is achieved from the MCMC methods[100][101][103][104]. MCMC methods construct a Markov chain such that its sta-tionary distribution π precisely converges to the distribution of interest, fn

X(x).

Markov chain is characterised by its transition Matrix T with transition probabil-ity from state xi to state xj, dened as Tji, such that its stationary distributionsolves the equation,

π = πT. (4.20)

MCMC considers nding a matrix T that satises (4.20) for a known π. Thisrequires the Markov chain to be ergodic [104]. Among the innitely many ergodicmatrices that satisfy (4.20), a unique solution is found by imposing the Markovchain to be time-reversible. Time-reversibility states that

πiTji = πjTij, (4.21)

which is called the detailed balance condition of the Markov chain.Metropolis et al. [100] and Hastings [101] have described an algorithm to

implement a reversible Markov chain in order to achieve the desired stationarydistribution. In the Metropolis-Hastings algorithm, given the current state xi, thenext state xj is chosen by sampling a candidate point from a proposal distribution


Q(xj|xi). For an arbitrary choice of Q, it may nd that

πiQ(xj|xi) > πjQ(xi|xj). (4.22)

In this case, the chain moves from i to j very often and j to i vary rarely. Inorder to satisfy the detailed balance condition, the number of moves from i to j isreduced by introducing a probability αij < 1 that the move is made. If the moveis not made with the probability αij (if the move is rejected), the chain remainsin the same state i. Hence, the transitions from i to j are made according to

Tji = αijQ(xj|xi). (4.23)

Inequality (4.22) also tells that the movements from j to i are not made oftenenough. Therefore, the movements from j to i should always be accepted givingαji = 1. The acceptance probability (AP) αij is found from the detailed balancecondition as, αij = πjQ(xi|xj)/πiQ(xj|xi) [102]. If the inequality of (4.22) isreversed, the roles of i and j are swapped in the above argument. Thus, the APthat satises the detailed balance condition can be obtained as

αij = min

(

1,πjQ(xi|xj)

πiQ(xj|xi)

)

= min

(

1,fnX

(xj)Q(xi|xj)

fnX

(xi)Q(xj|xi)

)

. (4.24)

Since only the ratio of the densities at two states is needed, the knowledge of thedensities up to a normalisation constant is sucient to drive the algorithm.

Most FHMC techniques employ the `Random Walk Metropolis' algorithm,which considers symmetric proposal distributions of the form Q(xj|xi) = Q(xi|xj) =

Q(|xi − xj|) for all xi and xj. This reduces the AP to

αij = min

(

1,fnX

(xj)

fnX

(xi)

)

= min

(

1,P n[g(xi)]fX(xj)

P n[g(xj)]fX(xi)

)

. (4.25)

Equation (4.25) says that the movements from state i to state j are always ac-


cepted if the probability of state j is lower than that of state i. In other words,the proposals that move the system towards the low probability region are alwaysaccepted. The samples that satisfy the reverse of this condition are accepted witha probability less than one. This process enhances sample generation from the tailof the probability distribution while reducing the samples in the modal region.As a result, the histogram eventually becomes at.

4.2.2 Multicanonical Monte Carlo Technique

MMC simulation technique is the rst at histogram algorithm developed by Bergand Neuhaus [7] in 1991. The functional block diagram of the MMC algorithmis illustrated in Figure 4.3. As stated before, all FHMC methods dier in theirpdf update procedure. In its original version, Berg and Neuhaus proposed a pdfupdate procedure, which can be derived from the constant weight biasing of (4.14)

Figure 4.3: Functional block diagram of the MMC algorithm.


as

P (n+1)m =

(

N(n+1)m

NI

)

1

N(n+1)m

N(n+1)m∑

j=1

W (n+1)m (xj)

= H(n+1)m .qnP n

m

= qnH(n+1)m P n

m (4.26)

where N(n+1)m is the number of samples fall into the m-th bin at (n+1)-th iteration,

NI is the total number of samples generated per iteration and H(n+1)m is the

histogram entry for the m-th bin at (n + 1)-th iteration. At the beginning, P(0)m

is initialised to 1/M for all bins m = 1, . . . ,M . The histogram is reset after eachiteration by

Hnm = max (Nn

m, 1) (4.27)

to avoid division by zero in (4.18). This pdf update procedure is called the `basicupdate'.

Figure 4.4 shows the pdf of the decision variable in a BPSK system operatingover an AWGN channel estimated using the basic update. pdf estimations fordierent iterations are shown. The number of samples per iteration is NI =

5×104. A oor of the pdf curve in the unexplored bins occur due to the conditionimposed by (4.27). Also Figure 4.4 shows the corresponding visits histogram,which eventually tends towards a at histogram. The visits histogram suersfrom stochastic uctuations due to the nite sample size per iteration. This hasresulted in an un-even pdf curve.

Berg [97] proposed a smoothing strategy that exploits bin smoothing usingmore than two bins to mitigate the uctuations in the estimated pdf. The new


Figure 4.4: pdf of decision variable V for BPSK system, Inset panel: correspond-ing visits histogram.

pdf update proposed by Berg is now called `Berg's update', which is given by

P (n+1)m =

P(n+1)m−1 P n

m

P nm−1

(

H(n+1)m

H(n+1)m−1

)gm,(n+1)

gm,n =gm,n

∑nl=1 gm,l

gm,l =H l

m−1Hlm

H lm−1 + H l

m

. (4.28)

Figure 4.5 shows a comparison of the pdfs estimated by the basic update andthe Berg's update for the same sample size. Berg's update can estimate withgreater smoothness; however, it can explore less than the basic update for thesame number of samples.


−4 −3 −2 −1 0 1 2

10−15

10−10

10−5

100

Decision variable (V)

Pro

babi

lity

dens

ity fu

nctio

n of

V

Basic update lawBerg’s update

Figure 4.5: Comparison of pdf estimation using basic update and Berg's update.

4.2.3 Transition Matrix Monte Carlo Technique

The TMMC technique has been proposed for calculating density of states in sta-tistical physics applications by Swendsen et al. [8][105]-[109], and later extendedfor optical communication systems by Yevick and Reimer [169]. In contrast to theMMC technique, whose pdf update procedure is based on the visits histogram,the TMMC technique exploits the transitions from one state of the system outputto another in the Markov chain to update the pdf at the end of each iteration.Hence, it does not accumulate a visits histogram, instead it accumulates a matrixwith relative transitions from one state to the other.

The TMMC algorithm accumulates a transition matrix T whose m1m0-thelement, denoted by Tm1m0 , contains the frequency of all proposed (accepted orrejected) transitions from the m0-th bin to the m1-th bin. At the end of eachiteration, the normalised transition matrix T

n, where n denotes the iteration


number, is obtained as

Tnm1m0

=Tm1m0

∑

m1 Tm1m0

. (4.29)

Equation (4.29) is obtained from the fact that the summation of the transitionprobabilities from one state should be equal to unity. The pdf update is formu-lated using the detailed balance condition of the Markov chain as

P nm = P n

m−1

Tnm(m−1)

Tn(m−1)m

, for m = 1, . . . , M (4.30)

with P n1 = 1, P n

m = P nm−1 for which T(m−1)m = 0 or Tm(m−1) = 0 and P n is

normalised to unity. It is a fact that P n is an eigenvector of the transition matrixT

n with unit eigenvalue [109]. Hence, an improved estimate for P n is obtained as

P nm =

∑

k

P nk T

nmk for m = 1, . . . , M. (4.31)

The transition matrix accumulates transitions from all iterations (i.e., T isnever zeroed across iterations) providing a better estimation for the transitionprobability towards the end of the simulation. Another interesting property of theTMMC algorithm is that the elements of the transition matrix should satisfy totaltransition matrix (TTT) identities, if the detailed balance condition is satisedduring the simulation [170].

The TTT identity and its relation to the detailed balance condition can beexplained as follows. Consider three distinct states i, j and k for which thetransition matrix elements of all possible moves are non-zero. For each pair of


Figure 4.6: (a) Transitions between states i, j and k in TTT identity. (b) Movesbetween transition matrix elements that satisfy TTT identity [110].

mutual moves, the detailed balance condition can be written as

P ni T

nji = P n

j Tnij

P nj T

nkj = P n

k Tnjk

P nk T

nik = P n

i Tnki. (4.32)

By multiplying the left and right sides of (4.32) and assuming non-zero probabil-ities, the TTT identity is derived as [170]

TnjiT

nkjT

nik = T

nkiT

njkT

nij. (4.33)

Figure 4.6(a) illustrates the moves between states i, j and k in the TTT iden-tity while Figure 4.6(b) illustrates the moves between transition matrix elementsthat satisfy the detailed balance condition and hence the TTT identity. Ac-cording to Figure 4.6(b), the transition matrix that satises the detailed balancecondition broadens along the diagonal. This serves as a measure of the detailedbalance condition, which in turn relates to the quality of the nal estimate.


Figure 4.7: Example of a transition matrix that satises TTT identity.

An example for pdf estimation using the TMMC algorithm is shown in Fig-ure 4.7 where the development of the transition matrix across the iterations isillustrated. The transition matrix clearly follows the TTT identity and broadensalong the diagonal.

4.2.4 Wang-Landau Algorithm

Another at histogram algorithm that has gained prominence in statistical physicswas proposed by Wang and Landau [9] in 2001. They observed that if a randomwalk is performed in the input space with a probability inversely proportional to


the probability of occurrence of the output, then a at histogram is generated tothe output distribution. This is the basic dynamic behind all FHMC methodsand it is achieved collectively by both random walk algorithm and pdf updateprocedure. The pdf update procedure of the WL algorithm diers drasticallyfrom that of the other FHMC algorithms. The WL algorithm updates its pdfafter each move of the Markov chain whereas the MMC and TMMC algorithmsperform the pdf update procedure at the end of each iteration.

In the WL algorithm, when the random walk proposes a new state m1 fromthe current state m0, the proposal is accepted with the probability αm0m1 , givenby (4.25). If the move is accepted, the probability of the m1-th bin is updatedby a modication factor f as

Pm1 → Pm1f. (4.34)

If the move is rejected, the probability of the m0-th bin is updated by the samef . Initially, f is set to a large value as f = f1 = e ∼= 2.71828 . . ., which allowsto reach all possible states quite fast. This random walk and the pdf updateprocedure is continued until the accumulated histogram is àpproximately at'.At this stage, the modication factor is reduced to a ner one according to afunction that monotonically decreases to one, and the histogram is reset to zero.Wang and Landau proposed to use the square root function for reducing themodication factor across iterations as fn =

√

fn−1. Small values of f lead toslow sampling among states. This is surmounted by increasing the sample sizeper iteration as f decreases. For instance, sample size per iteration NI can beincreased by NI/2 after each iteration. The atness condition for the histogramis dened as the time when each histogram entry is greater than or equal to80% of the average histogram; however, this may vary depending on the accuracyrequired for the application.

Since the probability density is changed at every step of the random walk, the


WL algorithm violates the detailed balance condition for all fn > 1. However,the renement process of f from one iteration to another eventually reduces thevalue of f to one, and the detailed balance is satised towards the end of thesimulation.

In practice, the pdf update equation of (4.34) is implemented in a logarithmscale in order to t all possible density values into double precision numbers[111]. One constraint of the WL algorithm is that the statistical error of thenal estimate converges to ln(f) that can not be reduced by further simulationbecause ln(f) = 0 does not update the pdf at all.

4.3 Characteristics and Challenges of FHMCAlgorithms

The important characteristics of FHMC algorithms and challenges associatedwith the application of these algorithms in practical communication systems arediscussed in this section. Most of the characteristics are common to the threeFHMC algorithms described in this chapter while some of them depend on thealgorithm itself.

1. Acceptance probability of the Markov chainThe reversible Markov chain that is used to generate samples from thebiased input distribution is one of the key elements of all FHMC algorithms.The reversibility condition is achieved through the AP αij associated withthe movement from state i to state j. The AP highly depends on thebin width ∆v and the proposal distribution Q(xj|xi). If the bin width istoo large, the dierence between the probability of two adjacent bins maybecome very large. This would result in very small AP (high rejectionrate), and the chain moves very slowly in the input space aecting theinput exploration. The inuence of the proposal distribution on the AP is


discussed later in item 3 of this list.

2. Dimensionality of the output spaceThe implementation of FHMC algorithms dier from system to system.When the system has a one-dimensional output (real component of the de-cision variable), e.g., decision variable of a BPSK system, this output can bedescretised to obtain a one-dimensional histogram and one-dimensional pdf.However, for higher level modulation systems, the decision on the transmit-ted symbol depends on both real and imaginary components of the decisionvariable. This requires the generation of histograms and pdfs for two vari-ables and increases the computational burden. When the system has severalvariables that gives the decision at a given instant, e.g., coded communi-cation systems, FHMC algorithms require one control variable that candirectly determine the performance of interest. In this case, the histogramis generated for the new control variable and the performance is estimatedusing the pdf of the new control variable. The selection of a control variablethat can directly determine the system performance is a real challenge inapplying FHMC algorithms to the systems with multi-dimensional outputs.This hinders the application of FHMC techniques for coded communicationsystems. Nonetheless, some researchers have succeeded in selecting propercontrol variables for coded communication systems [134][136].

3. Input space explorationAll FHMC algorithms employ MCMC techniques to perform the randomwalk on the input space. The Metropolis random walk algorithm [100] thatuses a symmetric proposal distribution of the form Q(xj|xi) = Q(xi|xj) =

Q(|xj − xi|), is the popular MCMC technique used for FHMC algorithms.With this distribution, the new sample xj is generated from the currentsample xi as xj = xj + U where U is a uniform random variable over[−∆U, +∆U ]. ∆U is dened as the perturbation step size and is an im-


portant parameter for proper exploration of the input space. If ∆U is toolarge, the new state may fall into a far away bin from the current bin. Thiswould lead to a small AP, and consequently the chain hardly moves. If ∆U

is too small, the new state would fall closer to the current state providinga reasonable AP; however, the step size would be very small resulting aslowly moving chain. This would require a large number of samples forproper input exploration.

Another MCMC method is called the independence chain algorithm [104],whose transition probability depends only on the nal state xj. The pro-posal distribution is same as the original distribution and the new sampleis drawn from the original distribution independent of the current sample,i.e., Q(xj|xi) = fX(xj). This would lead to too many rejections in highdimensional input space [166]. A strategy to improve the acceptance ratein this case is to use the one-variable-at-a-time Markov chain [104], whichmodies only one variable of the current input vector to propose the newinput vector.

4. Number of iterations and sample size per iterationTheoretically, FHMC estimators based on (4.14) have a constant relativeerror for all bins given by ǫ ≥

√

MNI

(Equation (4.17)) where NI denotes thesample size per iteration. Therefore, at least M/ǫ2 number of samples areneeded per iteration to estimate with a relative precision of ǫ. Empiricalresults for the number of iterations (N) and sample size per iteration havesuggested that the product N × ln(K1/(NI − N0)) must remain constantwith K1 a suitable constant and NI > N0 = M/ǫ2. The optimal NI thatminimises the total cost NNI is close to the lower bound N0 [166]. It wasalso observed that it is not always necessary to have a large NI to obtaina desired accuracy of the estimator. A smaller NI with more iterations canachieve the same accuracy level with a lower total cost. Alas, it is dicult to


Figure 4.8: QPSK system operating over an AWGN channel

derive hard constraints for NI and N as they are highly problem dependent.The analyst must be skilled enough to determine the appropriate values forthese variables depending on the expected accuracy levels, desired level ofpdf estimation and system congurations. Hence, the selection of NI andN is a kind of trial and error procedure.

In the WL algorithm, the estimate is updated dynamically within eachiteration and the modication factors (f) are changed from one iteration tothe other. In such situations, NI should also be changed from one iterationto the other to cope with the changed modication factors. How NI ischanged is dependent on the function that changes the modication factor.It was observed that when f is decreased according to the square rootfunction, incrementing NI by half of its previous value (NI ← 1.5NI) is agood choice to obtain good estimates.

4.4 FHMC Techniques for CommunicationSystems

In this section, the application of the FHMC methods for performance evaluationof communication systems is illustrated using a simple baseband communicationsystem.

Figure 4.8 shows the block diagram of a simple baseband communicationsystem that employs QPSK modulation and operates over an AWGN channel.


The SEP of this system is estimated using the MMC algorithm along with theBerg's update. In particular, the MMC algorithm is applied to estimate the pdf ofthe decision variable and thereby the probability distribution of the error region.

In a baseband QPSK system, the decision on the transmitted symbol St de-pends on both the real and the imaginary components of the received signal Sr.Therefore, direct FHMC simulation requires the estimation of two pdfs, one pereach component, to obtain the probability of the error region. The high compu-tational complexity involved with this procedure can be overcome by dening acontrol variable that is directly related to the SEP. In the QPSK constellation,decision boundaries are π/2 rad apart from each other and a decision error oc-curs if the angle dierence ∆θ between the transmitted signal St and the receivedsignal Sr is greater than π/4 rad (450). This decision rule can be exploited todene a simple control variable V as

V = ∆θ = |∠St − ∠Sr| (4.35)

where ∠ denotes the angle, and angles are measured in degrees. Then, the SEPPe can be calculated as

Pe = PrV > V0 (4.36)

where V0 = 450 is the decision threshold.The input of this system consists of the transmitted symbols and the Gaussian

noise samples. The input symbols are randomly generated while the noise spaceis explored using Metropolis random walk algorithm. The MMC algorithm forestimating the pdf of V is summarised in Algorithm 4.1. Here, Z represents thenoise sample and fZ(z) is the pdf of the noise sample.

Figure 4.9 shows the estimated probability distribution of V at SNR = 17 dBusing MMC and MC techniques. The MMC algorithm used N = 9 iterations


Algorithm 4.1 MMC algorithm for estimating probability distribution of V.Purpose: This algorithm estimates the probability distribution of the system outputvariable V using MMC method. It performs a random walk in the input noise spaceusing Metropolis random walk algorithm. The pdf is estimated using an iterative pdfupdate procedure given by (4.28).Require: divide the range of V into M bins with V0 coincides with an edge of a bin1: initialise: n ⇐ 0, P 0

m ⇐ 1/MMm=1

2: while n < N do3: initialise: k ⇐ 04: generate Z0 from fZ(z)5: nd v0 and corresponding bin m0

6: while k < NI do7: generate ∆z from a zero-mean symmetric pdf8: Z1 ⇐ Z0 + ∆z9: generate u from U [0, 1]

10: if u ≤ min[

fZ(Z1)fZ(Z0)

, 1]

then11: nd v1 and corresponding bin m1

12: generate u from U [0, 1]

13: if u ≤ min

[

P(n−1)m0

P(n−1)m1

, 1

]

then14: m0 ⇐ m1, Z0 ⇐ Z1

15: end if16: else17: Hn

m0⇐ Hn

m0+ 1

18: end if19: end while20: update pdf using (4.28)21: end while22: PN

m Mm=1 is the nal estimate for the probability distribution of V

and NI = 2 × 105 samples per iteration. The estimated SEP using MMC at thisSNR is 1.01×10−12 while the theoretical SEP is 1.44×10−12. Table 4.1 comparesthe MMC estimations with MC estimations in terms of the sample size reductionfactor ηMC/MMC of (2.20) and run-time gain ξMC/MMC of (2.21). The results wereobtained by running the MMC algorithm 100 times per each SNR, computing theSEP using (4.36) and taking the average over 100 estimations. Accordingly, theMMC estimator can estimate very low probabilities with an acceptable level ofaccuracy using a smaller sample size compared with the MC technique.


0 20 40 60 80 10010

−25

10−20

10−15

10−10

10−5

100

Control variable (V)

Pro

babi

lity

dist

ribut

ion

of V

MMCMC

Figure 4.9: Probability distribution of V at SNR = 17dB estimated by MC andMMC. N = 9, NI = 2 × 105

Table 4.1: Comparison of the MMC estimator with the MC estimator in termsof the sample size reduction factor and run-time gain.

SNR (dB) Theoretical SEP Mean-SEP ǫr(%) ηMC/MMC ξMC/MMC

13 7.94e-06 5.82e-06 7.49 2.56e+01 1.02e+0114 5.39e-07 3.90e-07 11.96 1.50e+02 5.98e+0115 1.87e-08 1.37e-08 11.40 4.69e+03 1.87e+0316 2.80e-10 2.05e-10 14.41 1.96e+05 7.85e+0417 1.45e-12 1.01e-12 17.67 1.77e+07 7.07e+06

4.5 Conclusion

FHMC techniques have provided ecient tools for rare event simulation in statis-tical physics systems. The probability theory behind FHMC that was hidden bythe excessive statistical physics details can be well described using IS concepts.FHMC techniques form an adaptive IS procedure that eventually converges to theoptimal sampling of rare events. All FHMC algorithms share a common approach


for optimally exploring the input space while they mainly dier in their pdf esti-mation mechanism. Simulation results demonstrate that the FHMC methods canbe strategically applied to accelerate communication system simulations. Appar-ently, a sample size reduction in the order of 107 can be achieved in estimatingprobabilities less than 10−12 using FHMC methods. A vast array of rare eventsimulation problems in communication systems including MIMO systems, multi-user systems and coded communication systems provides potential candidates tobe dealt with the FHMC techniques.

Chapter 5

Ecient Simulation of MIMOSystems

5.1 Introduction

The discovery of a third dimension, space, as another degree of freedom, apartfrom time and frequency, expanded the range of choices available for wirelesscommunication systems design. The space diversity was achieved by increas-ing the number of transmit and receive antennas, introducing the concept ofMIMO transmission [152]. Such systems have shown improved spectral eciencyof wireless communication systems in fading environments [12][11]. The discov-ery of receiver architectures that can mitigate spatial interference with limitedcomplexity [171] and space-time coding strategies that can approach desired ca-pacity [146][172] made MIMO transmission a useful addition that can improvetransmission quality of the existing systems.

Evaluation of channel capacity of MIMO systems is of great importance toappreciate their limiting performance under dierent channel conditions. Fadingchannel models used to analyse MIMO systems are mainly two fold; ergodic andnonergodic. In ergodic channels, a transmitted code word experiences essentially

5. EFFICIENT SIMULATION OF MIMO SYSTEMS 90

all states of the channel (channel changes in each channel use) and hence itaverages out the channel randomness. The maximummutual information betweenthe channel input and its output yields the ergodic capacity of the channel. Innonergodic channels, the channel state is chosen randomly at the beginning of thetransmission and held unchanged for the entire duration of the communication.The mutual information is represented as a random variable, and the probabilityof the mutual information falling below a certain transmission rate is known asthe outage probability. The maximum rate that can be supported by the channelwith a given outage probability is known as the outage capacity [173].

Analytical evaluation of the capacity and outage probability of nonergodicMIMO channels has been extensively studied in the literature [174]-[177]. How-ever, these studies are limited to Rayleigh or Rician fading conditions, and toapproximations and lower or upper bounds. The analytical complexity of exactsolutions restricts the performance evaluation to MC simulations. This chapterproposes ecient MC algorithms based on FHMC concepts described in Chapter4 for capacity and outage probability estimation of nonergodic MIMO channels.A MIMO MRC system [178] operating over Nakagami-m fading channels, whichclosely model the multipath fading in wireless transmission, is considered.

Four dierent FHMC algorithms are proposed with improved characteristics:

1. Smooth-MMC - This algorithm is based on the MMC method with an im-proved pdf update procedure. It uses the moving average ltering to reducethe uctuations in the visits histogram while the pdf update procedure usesthe resulting smoothed histogram to produce an improved estimate.

2. Optimal-MMC - This algorithm also shares the basics of the MMC methodwith a new pdf update procedure that reduces the statistical error of theestimator. The new procedure is derived by optimally combining the esti-mate of the previous iteration and an intermediate estimate of the currentiteration.


Figure 5.1: Block diagram of MIMO-MRC system.

3. TMMC - The third algorithm is based on the TMMC technique and itexploits the transition matrix obtained from the current iteration to biasthe input pdf of the next iteration.

4. WL-TMMC - This is an ecient and accurate combination of the WLalgorithm and the TMMC method to improve the statistical properties ofthe estimator.

The proposed algorithms are validated using a MIMO-MRC system described inthe next section.

5.2 MIMO-MRC System Model

A MIMO-MRC system equipped with nT transmit and nR receive antennas isconsidered as shown in Figure 5.1. This is a close-loop scheme, which requiresthe channel knowledge at the transmitter that steers the transmit power in thereceivers direction to maximise the SNR after applying MRC at the receiver. Thisform of spatial diversity is also known as transmit-beamforming. The channel isassumed to be nonergodic. The amplitudes of channel coecients are assumedto be Nakagami-m distributed while the phase is assumed to be uniformly dis-


tributed over [0, 2π]. The channel matrix is represented by

H = [hr,q]nR,nT

r,q=1 (5.1)

where hr,q is the channel coecient between the q-th transmit and r-th receiveantennas. For a given channel matrix, the nR × 1 received signal vector y can beexpressed as

y = Hwtx + z (5.2)

where x is the transmitted symbol with average power PT , wt denotes the nT × 1

unit energy beamforming vector and z is the nR × 1 noise vector with elementsi.i.d. CSCG random variables with variance N0. Given the transmit beamformingvector wt, the signals from the nR received antennas are combined according tothe MRC principle using the nR ×1 received weight vector wr = Hwt. Then, theestimated signal at the receiver is given by

x = w†ry

= w†rHwtx + w†

rz (5.3)

where (.)† denotes the conjugate transpose operator. The output SNR condi-tioned on H and wt can be derived as [178]

γTB = γw†tH

†Hwt (5.4)

where γ = PT /N0 is the average SNR per receive antenna and the subscript TB

represents transmit beamforming.The beamforming vector wt is chosen to maximise this instantaneous output

SNR thereby minimising the error probability. It is well known that the opti-mum beamforming vector is the unit eigenvector corresponding to the maximum


eigenvalue λmax of H†H.In this case, the output SNR of the MIMO-MRC system is given by [178]

γTB = γλmax. (5.5)

The performance of the MIMO-MRC system depends directly on the statisticalproperties of λmax.

The nonergodic capacity C of the MIMO-MRC system can be deduced as[179]

C = log2

(

1 + γTB)

= log2 (1 + γλmax) . (5.6)

The outage probability associated with a given transmission rate R, denoted byPout (R), is given by

Pout (R) = P (C ≤ R) =

∫ R

0

fC(c) dc (5.7)

where fC(c) is the pdf of the capacity C. Evaluation of (5.7) requires the knowl-edge of the pdf of C or the pdf of γTB. In [179], the pdf of γTB has been derivedfor a Rayleigh fading system using a nite-series representation and a numeri-cal algorithm developed for determining the coecients. However, no analyticalsolutions exist for Nakagami-m fading channels. Hence, ecient algorithms areproposed for estimating the capacity pdf and outage probability of MIMO-MRCsystems operating over Nakagami-m fading channels. The proposed algorithmsare generally applicable for any fading scenario encountered in practice.

5.3 Ecient FHMC Algorithms

As mentioned before, four ecient estimators for capacity and outage probabilityestimation of the MIMO-MRC system are proposed. The rst two estimators,


smooth-MMC and optimal-MMC, provide improved pdf update procedures andshare the underlying principles of the MMC algorithm. The third estimatoris based on the TMMC algorithm with a dierent biasing approach. The WL-TMMC estimator combines the properties of WL and TMMC algorithms to derivean ecient estimator with improved statistical properties.

The four estimators described above share a common mechanism, i.e., theMetropolis-Hastings algorithm, to implement the random walk in the input space.The input of this system is the nR × nT dimensional channel matrix H and theoutput is the corresponding capacity C computed from (5.6). The interestedrange of C is selected to be R′

c, and it is divided into M bins with bin width∆c. The proposed algorithms estimate the pdf of C conditioned on R′

c, denotedby fC(c|c ∈ R′

c). If a simulated sample cj 6∈ R′c, then that sample is discarded,

and a counter maintained for counting outliers N ′ is increased by one. Theunconditional estimate of the pdf is obtained as

fnC(c) = fn

C(c|c ∈ R′c)P

n(c ∈ R′c)

= fnC(c|c ∈ R′

c)(NI − N ′)

NI

(5.8)

where NI is the sample size per iteration and n denotes the iteration number.Since the components of H are assumed to be independent, the component-

wise Metropolis accept or reject mechanism [125] is employed. In this mechanism,a new input sample H1 is generated from the previous input sample H0 using asymmetric proposal distribution. Each component of H1 is accepted or rejectedindependently with the probability min [fi(h

1i )/fi(h

0i ), 1] for i = 1, . . . , nT nR

where fi is the pdf of the i-th component. If a component of H1 is rejected,the corresponding component of H0 is retained. This results in a new input vec-tor H1 in which only some of the components are dierent from their previousvalues of H0. If the previous output sample corresponding to H0 is c0 and the newoutput sample corresponding to H1 is c1, the transition from c0 to c1 is accepted


with the probability min[

P(n−1)m0 /P

(n−1)m1 , 1

]

where m0 is the bin corresponding toc0, m1 is the bin corresponding to c1 and P (n−1) is the estimated probability atthe (n − 1)-th iteration. The composite AP for this system can be written as

α(H0,H1) =

[

nT nR∏

i=1

min

(

fi(h1i )

fi(h0i )

, 1

)

]

min

(

P(n−1)m0

P(n−1)m1

, 1

)

. (5.9)

The AP given by (5.9) would accept H1 if c1 falls into a bin with lower probabilitythan the probability corresponding to the m0-th bin and remain in H0 otherwise.This would move the chain towards the low probability region of the desired pdf.

This random walk is employed in the four algorithms described in the followingsections.

5.3.1 Smooth-MMC

The MMC algorithm estimates the pdf at the end of each iteration based on thevisits histogram Hn of that iteration. The basic pdf update procedure suers fromthe stochastic uctuations of the visits histogram, and this results in an un-evenpdf curve. The Berg's update reduces this eect by bin smoothing over more thantwo bins; however, it explores less than the basic update for the same sample size.Here, an improved update procedure called smooth-MMC that can estimate lowerprobabilities than the Berg's update with a comparable smoothness is proposed.

The smooth-MMC estimator uses moving average ltering to smooth out thevisits histogram by replacing each histogram bin entry with the average of theneighbouring bin entries within a dened span. The histogram entry of the m-thbin at the n-th iteration after averaging is given by

Hn

m =Hn

m+L + Hnm+L−1 + · · · + Hn

m−L

2L + 1(5.10)

where 2L + 1 is the neighbourhood span.Consider the logarithm of the estimate at the n-th iteration log(P n) plotted


versus C. The slope of such a curve at point cm can be written as

δn(cm) ,log

(

P nm

)

− log(

P nm−1

)

∆c,

=log

(

P(n−1)m

)

− log(

P(n−1)m−1

)

∆c+

log(

Hn

m

)

− log(

Hn

m−1

)

∆c. (5.11)

From (5.11), the new estimator is derived as

P nm

P nm−1

=P

(n−1)m

P(n−1)m−1

Hn

m

Hn

m−1

. (5.12)

After incorporating the moving average ltering into the estimator, (5.12) can bewritten as

P nm =

P nm−1P

(n−1)m

P(n−1)m−1

∑Lk=−L Hn

m+k∑L

k=−L Hnm−1+k

. (5.13)

The smooth-MMC estimator in (5.13) reduces the eect of random uctua-tions in the visits histogram by exploiting the bin smoothing over more than twobins and it can explore lower probabilities than that can be estimated by theBerg's estimator. When Hn

m is zero, the Berg's estimator [97] updates the proba-bility of the m-th bin by the probability of the (m− 1)-th bin, which results in aoor of P n after m-th bin. In contrast, the smooth-MMC estimator can providean estimation up to the (m + L)-th bin, at which the oor of P n occurs, therebyexploring lower probabilities.

Equation (5.11) can be re-written as

δn(cm) = δ(n−1)(cm) +log[H

n

m] − log[Hn

m−1]

∆c. (5.14)

The variance of δn(cm) depends on the variance of δ(n−1)(cm) and the variance oflog[H

n] quantities. The variance of δ(n−1)(cm) can be neglected as it is a xed

function used in the n-th iteration [98].


From (5.10),

Hn

m =

∑Lk=−L Hn

m+k

2L + 1. (5.15)

Assuming independent histogram bins, the variance of Hn

m, denoted by σ2[Hn

m],can be written as

σ2[Hn

m] =1

(2L + 1)2

L∑

k=−L

σ2[Hnm+k]. (5.16)

Because the uctuations of the histogram bins are known to grow with the squareroot of the number of entries, σ2[Hn

m] ≈ Hnm [98, p.12]. Hence, (5.16) can be

written asσ2[H

n

m] =1

(2L + 1)2

L∑

k=−L

Hnm+k =

Hn

m

(2L + 1). (5.17)

Then, the variance of log[Hn

m], denoted by σ2[log[Hn

m]], can be expressed as [128]

σ2[log[Hn

m]] =

(

σ[Hn

m]

Hn

m

)2

=1

(2L + 1)Hn

m

. (5.18)

This implies that the variance is innite when the histogram bin is empty. Accord-ingly, the variance of the logarithm of the estimator P n

m, denoted by σ2[log[P nm]],

can be obtained as

σ2[log[P nm]] = σ2[log[P n

m−1]] +1

(2L + 1)

[

1

Hn

m

+1

Hn

m−1

]

. (5.19)

It can be seen from (5.19) that the variance of the smooth-MMC estimatorreduces as the neighborhood span increases; however, the increased span is morecomputationally intensive. Thus, there always exists a trade-o between theestimator accuracy and the computational complexity.

5.3.2 Optimal-MMC

Optimal-MMC combines the estimation obtained from the previous iteration andan intermediate estimation of current iteration so that the statistical error of the


nal estimation is minimised. This resembles the method used in [97] to derive arecursion relationship to estimate canonical expectation values of the Potts modeland that applied in [128] to estimate the pdf of polarisation mode dispersion inoptical bre systems.

Let the intermediate estimate of pdf obtained at iteration n be P n. It isobtained from the smoothed histogram of (5.10) and the basic pdf update law of(4.26) as

P nm = qnH

n

mP (n−1)m for m = 1, . . . , M. (5.20)

The nal estimate for P nm is obtained as

log[P nm] = (1 − β) log[P (n−1)

m ] + β log[P nm] (5.21)

where β is the reliability factor. The value of β in (5.21) is derived to minimisethe variance of log[P n

m], which can be written as

σ2[log[P nm]] = (1 − β)2σ2[log[P (n−1)

m ]] + β2σ2[log[P nm]]. (5.22)

Setting ∂(σ2[log[P nm]])/∂β = 0, the optimum value for β is obtained as

βopt =σ2[log[P

(n−1)m ]]

σ2[log[P(n−1)m ]] + σ2[log[P n

m]]

=λn

m

λ(n−1)m + λn

m

(5.23)

where λnm is the statistical weight of log[P n

m], which is inversely proportional toits variance as λn

m = 1/σ2[log[P nm]]. Hence, the optimal-MMC estimator of (5.21)

can be re-written as

P nm = [P (n−1)

m ](1−βopt)[P nm]βopt for m = 1, . . . , M. (5.24)


Substituting βopt in (5.22), a recursion relationship for λn can be obtained as

λnm = λ(n−1)

m + λnm for m = 1, . . . , M (5.25)

with λ0m = 0 and λn

m ≈ (2L+1)Hn

m, which can be obtained from (5.18). Equation(5.25) implies that the statistical weight of the estimator increases graduallyfrom zero with increasing number of iterations that results in a monotonicallydecreasing variance.

5.3.3 TMMC

The proposed TMMC algorithm for capacity and outage probability estimation isbased on the original TMMC technique described in Section 4.2.3. The proposedalgorithm uses a dierent biasing approach that uses the normalised transitionmatrix T

n generated at the end of the n-th iteration to bias the input for the(n + 1)-th iteration. The biased input density for the (n + 1)-th iteration iswritten as

f(n+1)H

(h) =fH(h)

qnTn. (5.26)

The composite acceptance probability for this system varies from (5.9) as

α(H0,H1) =

[

nT nR∏

i=1

min

(

fi(h1i )

fi(h0i )

, 1

)

]

min

(

Tnm0m1

Tnm1m0

, 1

)

. (5.27)

The proposed TMMC algorithm for capacity pdf estimation can be summarisedas in Algorithm 5.1.

5.3.4 WL-TMMC

The WL algorithm and TMMC algorithm have received much attention as e-cient MC techniques for the simulation of density of states. Both of those have


Algorithm 5.1 TMMC algorithm for estimating probability distribution of C.Purpose: This algorithm estimates the probability distribution of C using transitionmatrix method. It performs a random walk in the input space using Metropolis randomwalk algorithm. The pdf is estimated using the transition matrix generated throughthe random walk. The input for the n-th iteration is biased using the transition matrixof the (n − 1)-th iteration.Require: divide the desired range of C, R′

c into M bins1: initialise: n ⇐ 0, T0

l,r ⇐ 1/MMl,r=1, Tl,r = 0M

l,r=1

2: while n < N do3: initialise: k ⇐ 04: generate H

0 from fH(h)5: nd capacity c0 corresponding to H

0 and corresponding bin m0

6: while k < NI do7: generate ∆H from a zero-mean symmetric pdf8: H

1 ⇐ H0 + ∆H

9: for r = 1 : nR do10: for l = 1 : nT do11: generate u from U [0, 1]

12: if u > min

[

fh(H1r,l

)

fh(H0r,l

), 1

]

then13: H

1r,l ← H

0r,l

14: end if15: end for16: end for17: nd capacity c1 corresponding to H


18: Tm1m0 = Tm1m0 + 119: generate u from U [0, 1]

20: if u ≤ min

[

T(n−1)m0m1

T(n−1)m1m0

, 1

]

then21: m0 ⇐ m1, H

0 ⇐ H1

22: end if23: end while24: form T

n using (4.29)25: nd Pn using (4.30) and (4.31)26: end while27: PN

m Mm=1 is the nal estimate for the probability distribution of C

their advantages and drawbacks. Initial iterations of the WL algorithm (largef) achieves broad sampling; however, the statistical error of the WL algorithmconverges to some value that cannot be reduced by further simulations. Theadvantage of the TMMC algorithm is that the statistical errors of the nal es-timation is small if the simulation is started with a good initial guess. Hence,


these two methods can be combined to sample the input space eciently and toobtain estimates with high accuracy.

Shell et al. [121] proposed to combine the WL and TMMC methods by run-ning a pure WL simulation rst, and using the TMMC estimation when f be-comes closer to one. They also proposed to use the transition matrix elementsin calculating AP towards the end of the simulation. Later, Ghulghazaryan etal. [110] proposed a more ecient combination of these two methods to calculatethe density of states for peptide in gas phase and in the presence of water. Theircombination adds a fraction (set to 0.1) of the dierence between the TMMCestimation and WL estimation to the WL estimation.

Here, an improved combination of WL and TMMC algorithms is proposedto minimise the statistical error of the estimator. According to the properties ofWL and TMMC methods, it is desirable to have WL estimates to be dominant atthe initial stages of the combined method to achieve broad sampling across stateswhile TMMC estimates to be dominant thereafter to reduce the statistical error.The proposed combination achieves this through an optimisation parameter thatis changed systematically across iterations.

Thus, the combined WL-TMMC algorithm is based on the original WL algo-rithm with an additional step of accumulating the relative transitions among thestates. When the WL algorithm periodically checks for the atness of the vis-its histogram, the current WL estimation is updated by an intermediate TMMCestimation, obtained according to (4.29), (4.30) and (4.31), to generate a newpdf estimation. This new pdf estimation is then used to compute the acceptanceprobability of the chain until it is refreshed at the next atness check. Since newpdf estimation is unchanged for a certain period of the random walk, the detailedbalance condition is not violated compared with the WL algorithm.

Let the WL estimation of the m-th bin at the n-th iteration be P nm(WL) and

the corresponding TMMC estimation be P nm(TM). These two estimations can be


combined as

log [P nm] = (1 − κ) log [P n

m(WL)] + κ log [P nm(TM)] (5.28)

where 0 < κ < 1 is the optimisation parameter for WL-TMMC combination.When κ ≈ 1 the estimation resembles the TMMC estimation, which is desiredtowards the end of the simulation.

The value of κ is derived to minimise the statistical variance of log [P nm], which

can be written as

σ2[log [P nm]] = (1 − κ)2σ2[log [P n

m(WL)]] + κ2σ2[log [P nm(TM)]]. (5.29)

Setting ∂(σ2[log [P nm]])/∂κ = 0, the optimum value for κ is obtained as

κopt =σ2[log [P n

m(WL)]]

σ2[log [P nm(WL)]] + σ2[log [P n

m(TM)]]. (5.30)

Variance of theWL estimator of the m-th bin at the n-th iteration σ2[log [P nm(WL)]]

can be derived as (Appendix B.1)

σ2[log [P nm(WL)]] =

n∑

k=1

Hkm log(fk)

2 = σ2[log [P (n−1)m (WL)]] + Hn

m log(fn)2

(5.31)

where fk is the modication factor of the WL algorithm at the k-th iteration.Variance of the TMMC estimator of the m-th bin at the n-th iteration, denotedby σ2[log [P n

m(TM)]], can be derived as (Appendix B.2)

σ2[log [P nm(TM)]] = σ2[log [P n

m−1(TM)]] +Tm−1,m + Tm,m−1

Tm−1,mTm,m−1

+

∑

k Tm−1,k +∑

k Tm,k∑

k Tm−1,k

∑

k Tm,k

(5.32)


where Tm,k is the transition matrix entry corresponding to the transition fromthe k-th bin to the m-th bin and σ2[log [P n

1 (TM)]] = 0.The WL-TMMC algorithm for capacity pdf estimation is summarised in Al-

gorithm 5.2, which is shown at the end of this chapter. This algorithm achievesfast convergence to the true probability distribution with small statistical errors.It was observed that the value of κopt per histogram bin monotonically increasesto one with iterations. During initial iterations, variance of both estimates islarge; hence, (5.30) results in small values of κopt. This allows the WL estimateto be dominant in these stages. As the number of iterations increases, varianceof the WL estimate reaches a limiting value while that of the TMMC estimatekeeps on reducing. Thus, κ → 1 towards the end of the simulation. This allowsthe TMMC estimate to be dominant, which is desired towards the end of thesimulation to reduce the statistical error of the nal estimation. Furthermore,WL-TMMC method uses the new pdf estimation to compute the detailed bal-ance condition of the Markov chain. Since this pdf estimation remains unchangedfor some period of each iteration, a reduced detailed balance violation is achievedcompared with the WL algorithm.

Periodic calculation of κopt involves more computations that aect the e-ciency of the algorithm. This can be overcome by approximating κ to a functionthat monotonically increases to one as the iteration number increases. It was ob-served that for the n-th iteration κn = (1 − exp(−(n − c1)/c2) with c1 = −1 andc2 = e is a good approximation to κopt across iterations. Figure 5.2a shows thecomparison of κopt with the approximated values across iterations for the capac-ity pdf estimation of MIMO-MRC system operating over a Nakagami-m fadingchannel with m = 1 at SNR = 6 dB. The values of κopt were obtained by running10 independent simulations and taking the mean of κopt per iteration in each run.Figure 5.2b depicts the relative error of the pdf estimation of the same scenario.The proposed approximation to κ exhibits comparable error performance withκopt, which validates the use of the approximation to improve the eciency of the


0 5 10 15 200.4

0.5

0.6

0.7

0.8

0.9

1

Iteration number

κ

κopt

κ − Approximate

(a) κopt and approximation to κopt

1 2 3 4 5 6 7 80

10

20

30

40

50

Re

lativ

e e

rro

r ε (%

)

Capacity (C) b/s/Hz

κopt

κ − Approximate

(b) Relative error distribution

Figure 5.2: Comparison of κopt with approximation

WL-TMMC algorithm.


The proposed FHMC algorithms were used to estimate the capacity pdf of theMIMO-MRC system with nT = 4 and nR = 4 antennas, operating over Nakagami-m fading channels. Three fading scenarios were considered as m = 0.5, m = 1

and m = 2. m = 1 corresponds to Rayleigh fading channels for which analyticalresults for outage probability can be found in [179]. Smooth-MMC, optimal-MMC and TMMC algorithms used ve iterations (N = 5) and 105 samples periteration (NI = 105). The WL-TMMC algorithm employed the approximatevalues for κopt. It was started with NI = 400, and NI was increased by NI/2

after each iteration. The WL-TMMC estimation was obtained using 12 suchiterations (total simulation samples = 1.03 × 105).

Figures 5.3-5.5 show the capacity pdf estimations using proposed algorithmsfor m = 0.5, m = 1 and m = 2, respectively. The pdfs estimated by the proposedalgorithms are compared with that estimated by the MMC algorithm with Berg's


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

ba

bili

ty d

en

sity

fu

nct

ion

of C

(a)

Berg’s UpdateSmooth−MMCMC

0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

ba

bili

ty d

en

sity

fu

nct

ion

of C

(b)

Berg’s UpdateOptimal−MMCMC

0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

ba

bili

ty d

en

sity

fu

nct

ion

of C

(c)

Berg’s UpdateTMMCMC

0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

ba

bili

ty d

en

sity

fu

nct

ion

of C

(d)

Berg’s UpdateWL−TMMCMC

Figure 5.3: Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 overNakagami-m fading channels, m = 0.5; Curves from left to right correspond toSNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b) Optimal-MMC (c) TMMC(d) WL-TMMC.

update and MC technique. The MMC algorithm with Berg's update used veiterations with 105 samples per iteration, and the MC technique used a total of5 × 105 samples.

According to Figures 5.3(a), 5.4(a) and 5.5(a), smooth-MMC algorithm canestimate lower probabilities than the Berg's update with a comparable smooth-ness. Figures 5.3(b), 5.4(b) and 5.5(b) prove that the optimal-MMC algorithm,which is optimal in terms of error performance, requires more iterations to esti-mate probabilities comparable to that estimated by the Berg's estimate. This is


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(a)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(b)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(c)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(d)


Figure 5.4: Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 overNakagami-m fading channels, m = 1; Curves from left to right correspond toSNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b) Optimal-MMC (c) TMMC(d) WL-TMMC.

due to the behaviour of βopt with increasing iteration number. For a given SNR,the value of βopt reduces with increasing iteration number rendering the updateprocess of (5.24) very slow. Similar reasoning applies to gm,n of the Berg's updateof (4.28) that causes the lowest probability estimated by the Berg's update to behigher than that estimated by the smooth-MMC and WL-TMMC estimators.Nevertheless, the Berg's update exploits interpolation over two bins that allowsto produce a lower probability estimate than the optimal-MMC estimator.

The TMMC algorithm shows a comparable performance to the MMC algo-rithm with the Berg's update. The smoothness of the pdf curves estimated by


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(a)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(b)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(c)


0 5 10 15

10−15

10−10

10−5

100

Capcity (C) b/s/Hz

Pro

babi

lity

dens

ity fu

nctio

n of

C

(d)


Figure 5.5: Capacity pdf of MIMO-MRC system with nT = 4 and nR = 4 overNakagami-m fading channels, m = 2; Curves from left to right correspond toSNR (dB) = 0, 3, 6, 9, 12, 15. (a) Smooth-MMC (b) Optimal-MMC (c) TMMC(d) WL-TMMC.

the smooth-MMC, optimal-MMC and TMMC algorithms is somewhat degradedfor severe fading environments with m = 0.5 while the smoothness improves forgentle fading conditions with m = 2. The proposed combination of the WL andTMMC algorithms exhibits a better performance in all three fading environmentscompared with other three algorithms. It can estimate lower probabilities thanthe Berg's update with improved smoothness.

Figures 5.6-5.8 depict the outage probability estimations for MIMO-MRCsystem in Nakagami-m fading channels with m = 0.5, m = 1 and m = 2, respec-tively. Clear agreement between the estimations of proposed algorithms and that


0 2 4 6 8 1010

−15

10−10

10−5

100

R b/s/Hz

Out

age

Pro

babi

lity

Smooth−MMCOptimal−MMCTMMCWL−TMMCMC

SNR = 0,3,6,9,12,15 dB

Figure 5.6: Outage probability estimation of MIMO-MRC system with nT = 4and nR = 4 over Nakagami-m fading channels, m = 0.5.

of the MC method in high probability range proves the accuracy of the proposedalgorithms. The WL-TMMC algorithm outperforms the other three algorithmswhile all four algorithms can estimate lower probabilities than the MC techniquewith the same sample size.

Figure 5.7 also plots the analytical results, which are obtained using a niteseries representation and a numerical algorithm for determining the coecients,for Rayleigh fading channels [179]. The estimations of proposed algorithms clearlyfollow the analytical solution up to 10−13 range where the analytical solutionstarts to deviate. This deviation occurs due to the weak convergence of thenumerical algorithm used to calculate the coecients in this region.

Figures 5.9-5.11 compare the relative error of proposed FHMC estimatorsusing the capacity pdf estimation of the MIMO-MRC system with nT = 4 and


0 2 4 6 8 10

10−15

10−10

10−5

100

R b/s/Hz

Ou

tag

e P

rob

ab

ility

Smooth−MMCOptimal−MMCTMMCWL−TMMCMCTheory

SNR = 0,3,6,9,12,15 dB

Figure 5.7: Outage probability estimation of MIMO-MRC system with nT = 4and nR = 4 over Nakagami-m fading channels, m = 1.

nR = 4 over Nakagami-m fading channels with m = 0.5, m = 1 and m = 2 at SNR= 6 dB, respectively. The relative error was computed by running 100 indepen-dent simulations per method, calculating the variance and mean of the estimationof each bin using 100 samples and evaluating equation (2.8). Accordingly, theoptimal-MMC method shows the best error performance in high probability re-gion; however, the relative error gradually increases in the low probability region.Meanwhile, WL-TMMC algorithm exhibits a relative error less than 30% in theentire range of pdf estimation.

The WL-TMMC algorithm, which shows better performance in terms of thesmoothness and statistical properties, can estimate probabilities less than 10−15

with a relative error of 30% using a sample size of 5× 105. To estimate the sameprobability with the same relative error, MC would require at least 1016 samples.


0 2 4 6 8 1010

−15

10−10

10−5

100

R b/s/Hz

Ou

tag

e P

rob

ab

ility

Smooth−MMCOptimal−MMCTMMCWL−TMMCMC

SNR = 0,3,6,9,12,15 dB

Figure 5.8: Outage probability estimation of MIMO-MRC system with nT = 4and nR = 4 over Nakagami-m fading channels, m = 2.

Accordingly, the WL-TMMC algorithm provides a sample size reduction factorgreater than 1010, which is a signicant saving in computational cost.

Results of this chapter have been either published or accepted for publicationin [180]-[183].

5.5 Conclusion

This chapter presented ecient Monte Carlo algorithms for capacity and outageprobability estimation of MIMO systems whose exact analytical solutions are in-tractable. Of these algorithms, the proposed combination of the WL and TMMCmethods outperforms the other three algorithms by estimating lower probabili-ties with a greater smoothness and lower relative error. The proposed algorithms


2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

100

Capacity (C) bits/s/Hz

Re

lativ

e e

rro

r (ε) %

Smooth−MMCOptimal−MMCTMMCWL−TMMC

Figure 5.9: Relative error comparison of proposed FHMC estimators; MIMO-MRC system with nT = 4 and nR = 4 over Nakagami-m fading channels, m = 0.5,SNR = 6 dB.

provide ecient tools to verify analytical results in low probability regions thatare not reachable by the MC technique. They can be generally applied to anypdf estimation problem of interest.


2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

100


Rel

ativ

e er

ror

(ε) %


Figure 5.10: Relative error comparison of proposed FHMC estimators; MIMO-MRC system with nT = 4 and nR = 4 over Nakagami-m fading channels, m = 1,SNR = 6 dB.

2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

100


Re

lativ

e e

rro

r (ε) %


Figure 5.11: Relative error comparison of proposed FHMC estimators; MIMO-MRC system with nT = 4 and nR = 4 over Nakagami-m fading channels, m = 2,SNR = 6 dB.


Algorithm 5.2 WL-TMMC algorithm for estimating probability distribution ofC.Purpose: This algorithm estimates the probability distribution C using the denedcombination of WL and TMMC algorithms.Require: divide the desired range of C, R′

c into M bins1: initialise: n ⇐ 0, log Pm ⇐ log 1/MM

m=1, log Pm(WL) ⇐ log 1/MMm=1, Tl,r =

0Ml,r=1, f0 = 2.71828, NI = 400

2: while fn > fend do3: initialise: k ⇐ 0, Tl,r = 0M

l,r=1, flag = 14: n ← n + 15: generate H

0 from fH(h)6: nd capacity c0 corresponding to H


7: while (flag == 1)||(k < NI) do8: generate ∆H from a zero-mean symmetric pdf9: H

1 ⇐ H0 + ∆H

10: for r = 1 : nR do11: for l = 1 : nT do12: generate u from U [0, 1]

13: if u > min

[

fh(H1r,l

)

fh(H0r,l

), 1

]

then14: H

1r,l ← H

0r,l

15: end if16: end for17: end for18: nd capacity c1 corresponding to H


19: Tm1m0 = Tm1m0 + 120: generate u from U [0, 1]

21: if u ≤ min[

exp(log(Pm0) − log(Pm1)), 1]

then22: m0 ⇐ m1, H

0 ⇐ H1

23: end if24: Hn

m0← Hn

m0+ 1, log(Pm0(WL)) ← log(Pm0(WL)) + log(f0)

25: if mod(k,100)==0 then26: form T using (4.29)27: form log(Pn

m(TM)) using (4.30)28: nd κopt using (5.30), (5.31) and (5.32)29: update log(Pm) using (5.28)30: if histogram is approximately at then31: flag == 032: end if33: end if34: end while35: fn+1 ←

√fn, increase NI (NI ← 1.5NI)

36: end while37: PmM

m=1 is the nal estimate for the probability distribution of C

Chapter 6

Ecient Simulation of CodedSystems

6.1 Introduction

ECC, also known as channel coding, is widely used for controlling bit errors indigital communication systems. ECC techniques provide reliable transmissionover impaired channels by introducing redundant bits that allow the receiverto detect and correct bit errors in the received data. The two major types ofECC techniques include block coding and convolutional coding. Block codes usespecic generator polynomials to convert a block of k information bits to a longersequence of n coded bits, called a codeword. A codeword depends only on thecurrent input message.

In contrast to block codes, convolutional codes (CCs) use both current andpast input bits (memory) to generate the output codeword. CCs were rst pro-posed by Elias [184] in 1955. Since then, CCs have been used in numerouscommunication applications as it outperforms block codes of the same order ofcomplexity. A convolutional encoder is a linear nite-state machine consistingof M shift registers and n linear algebraic function generators. The input data,

6. EFFICIENT SIMULATION OF CODED SYSTEMS 115

Figure 6.1: Structure of K = 5, rate-1/2 convolutional encoder.

which is usually binary, is shifted along the register, k bits at a time. The rateR of such a code is dened as R = k/n and the constraint length of the code isdened as K = M + 1. The structure of a convolutional encoder with K = 5,n = 2 and k = 1 is shown in Figure 6.1.

The optimum decoder for convolutional codes is a ML sequence decoder [153].Viterbi algorithm (VA) [186] is the well known decoding algorithm for convolu-tional codes that yields a ML decision [187]. VA minimises the sequence errorprobability by performing an ecient trellis search that selects the path throughthe trellis with the minimum path metric.

Another decoding algorithm, which minimises the symbol (or bit) error prob-ability, is based on the MAP decoding principle. The symbol-by-symbol MAPdecoding algorithm was introduced by Bahl, Cocke, Jelinek and Raviv givingthe name BCJR algorithm [188]. The BCJR algorithm computes the a posteri-ori probability for each transmit symbol based on the received sequence. UnlikeViterbi decoding, MAP decoding requires a large memory and a large number ofoperations involving multiplications and exponentiations, making it more com-putationally complex. Reduced complexity versions of MAP decoding algorithmsinclude log-MAP and Max-Log-MAP [189].

In most practical communication systems CCs are terminated by encoding anite length information sequence followed by additional tail bits (zeros). Theoverhead introduced by this termination can be reduced by increasing the framelength of the code. However, many applications of terminated CCs use short


frame lengths resulting in a rate loss. For example, the control channel of thecdma2000 system uses a CC of constraint length 9 with a frame of 40 inputinformation bits while the control channel of the High-Speed Downlink PacketAccess (HSDPA) system uses a constraint length 9 code with 8 and 29 inputinformation bits [190].

The performance of Viterbi decoded convolutional codes can be estimatedfrom the conventional union bound derived from the transfer function [192]. Thisbounding technique assumes an innitely long input sequence and the same errorprobability for each bit regardless of the bit position within the frame. Theconventional union bound becomes loose for terminated CCs due to the violationof innitely long input assumption. The exact analytical performance of theMAP decoded convolutional codes has also not been addressed in the literature.Therefore, the exact performance evaluation of Viterbi and MAP decoding isdone through MC simulations.

The widespread acceptance of convolutional codes paved the way for the de-velopment of parallel concatenated convolutional codes known as turbo codes[193]. These codes have shown to perform close to the Shannon limit in AWGNchannels. The concept behind turbo codes is the parallel concatenation of twoidentical recursive systematic convolutional encoders separated by an interleaver.The same information bits are fed to both encoders except that they are inter-leaved before feeding to the second encoder. A large coding gain is achieved byusing a long interleaver. Iterative decoding is the other key feature related toturbo codes. The iterative decoder consists of two component decoders seriallyconcatenated via an interleaver that is identical to the one in the encoder. Thecomponent decoders are based on either the MAP algorithm [188] or soft outputViterbi algorithm [194]. The performance of Turbo codes is also estimated usingupper bounds, which are quite loose in some cases [195].

MC simulation of Viterbi and MAP decoding requires excessive run-times toestimate low bit error probabilities with an acceptable level of accuracy. This is


due to the complexity of these decoders and large sample sizes required to producerare error events. This chapter proposes methods for reducing the lengthy run-time requirement of simulating Viterbi and MAP decoders.

Ecient simulation of Viterbi decoders has been addressed several times inthe literature. First, the existing methods are discussed and their drawbacksidentied. Then, the EES technique is extended for ecient simulation of Viterbidecoded terminated CCs by exploiting the new weight enumerator of terminatedCCs proposed in [15]. Moreover, an ecient simulation algorithm, based on theWL at histogram method described in Section 4.2.4, is proposed for MAP andViterbi decoder simulation.

6.2 Related Work

The rst eort in ecient simulation of Viterbi decoders was based on IS tech-niques, and the method was called the `block method' [60]. This method appliesvariance scaling to noise samples and computes the IS weight per output sampleas the multiplication of the weights of all samples that inuence the particularoutput. It has shown moderate run-time savings for convolutional-coded systemsoperating over AWGN channels. Later, Sadowsky [62] proposed an IS techniquecalled EES for Viterbi decoder simulation. EES has shown to be more ecientthan the block method.

In the Viterbi decoder, decision errors occur in bursts called error events. Theessence of EES is to overcome the eect of memory length by isolating the sim-ulation of these error events in a way that each simulation run produces exactlyone error event. The EES technique has shown signicant run-time savings forconvolutional-coded systems operating over AWGN channels and radio frequencyinterference environments [62][81]. However, to our knowledge, there is no ev-idence for the EES technique providing substantial run-time savings in fadingenvironments.


A dierent approach for reducing the sample size requirement of coded com-munication system simulations is known as DAIS [134]. DAIS uses the MMCsimulation technique described in Section 4.2.2 to approximate the pdf of a con-trol variable that is a suitable function of the noise samples in an AWGN en-vironment. The control variable of DAIS is not directly related to the SEP ofthe system; hence, a second cycle of MMC simulation is required to determinethe error performance of the system [134]. This second cycle causes deteriorationof the eciency of the algorithm up to some extent. DAIS potentially worksin all scenarios where the error probability is signicantly correlated with thenoise power. It has been applied to turbo-coded systems operating over AWGNchannels where moderate simulation gains have been obtained [196].

Another MMC-based simulation technique for Viterbi decoders was proposedby Secondini et al. [136]. The control variable that drives the MMC algorithmof [136] can promptly determine the error performance so that a single cycleof MMC simulation is sucient. However, this control variable is specic tothe Viterbi decoder and applicable only for the continuous mode of decoding.Therefore, it is neither generally applicable for any coding type nor it can beapplied for terminated codes encountered in practice. Both [134] and [136] havenot provided evidence for the computational eciency gain and the accuracy ofthe estimator.

As mentioned before, this chapter discusses ecient simulation methods forViterbi and MAP decoders. The rst method extends the well known EES tech-nique for terminated CCs by using a new weight enumerator for terminated CCsproposed by Moon and Cox [15]. The extended-EES technique shows improvedperformance over the EES technique for terminated CCs operating over AWGNchannels. The second method is the two-phase algorithm (TPA) that is based onthe WL at histogram method for Viterbi and MAP decoder simulations. TPAuses the Euclidean distance between the transmitted sequence and the receivedsequence as the control variable for the WL algorithm. Simulation results show


signicant sample size reductions for Viterbi and MAP decoder simulations inAWGN and fading environments.

6.3 Error Event Simulation of TerminatedConvolutional Codes

Consider the estimation of the BEP Pe of Viterbi decoded terminated CCs withconstraint length K, memory order m, code rate R = k/n and input infor-mation sequence length L [197]. Assume that an information sequence u =

(u0, · · · ,uL−1) of length L′ = kL is encoded into a codeword v = (v0,v1, · · · ,vM+L−1)

of length N = n(L+M), and a sequence r = (r0, r1, · · · , rM+L−1) is received overa binary-input, continuous-output channel. Also assume that the channel inputs0 and 1 are represented by the BPSK symbols −1 and +1 normalised to thesignal energy, respectively. The Viterbi decoder that produces an estimate v ofthe codeword v based on the received sequence r is a ML decoder, which choosesv as the codeword v that maximises the log-likelihood function P (r|v).

As mentioned before, the performance of the above system can be approxi-mated by the conventional union bound derived from the transfer function; how-ever, this approximation becomes loose when the frame length becomes short.Moon and Cox [15] have presented a new weight enumerator that can be usedto derive tighter performance upper bounds for terminated CCs. This weightenumerator can be exploited to improve the performance of EES of terminatedCCs.

In EES, each simulation run produces precisely one simulated Viterbi decodererror event in a manner analogous to the way the simple binary decisions aresimulated. All-zero transmission u is assumed without loss of generality. Theprobability of decoding an error event u′ using NEES simulation samples per


error event is given by

P ∗(u′|u) =1

NEES

NEES∑

n=1

W (u(n)|u)Hu′(u(n)) (6.1)

where u(1),u(2), . . . ,u(NEES) are NEES i.i.d. random samples that are gener-ated from the biased simulation density, W (.) is the IS weighting function andHu′(u(n)) is the indicator function of decoding the error event u′, which is givenby

Hu′(u(n)) =

1 if u(n) = u′

0 otherwise.(6.2)

The EES estimator for BEP P ∗b for a rate-k/n convolutional code is given by,

P ∗b =

1

k

∑

u′∈Ξ

nb(u,u′)P ∗(u′|u) (6.3)

where Ξ is the set of all error events u′ of the code being considered and nb(u,u′)

is the number of post decoding information bit errors caused by decoding u′

instead of u.The new weight enumerator W (B,D) for terminated convolutional codes is

expressed as [15]W (B, D) =

∑

i

∑

j

ci,j.BiDj (6.4)

where ci,j is the number of error events composed of a single error event with inputHamming distance i and output Hamming distance j. The process of obtainingci,j and thereby W (B,D) is clearly described in [15] by using a modied trellisapproach.

Equation (6.4) accurately represents the relationship between the input andoutput Hamming distances of terminated CCs. Therefore, it is proposed to sim-ulate only one error event per input Hamming distance i and output Hamming


distance j of the CC, and to use the new weight enumerator to approximatethe event error probability for i-th input Hamming distance and j-th outputHamming distance. Accordingly, the event error probability for input Hammingdistance i and output Hamming distance j, denoted by P ∗

i,j(u′i,j|u), is dened as

P ∗i,j(u

′i,j|u) , ci,j

1

NEES

NEES∑

n=1

W (u(n)|u)Hu′

i,j(u(n)) (6.5)

where u′i,j is the error event corresponding to input Hamming distance i and

output Hamming distance j. There exist more than one error event with inputHamming distance i and output Hamming distance j. The proposed methodsimulates only one such error event per input Hamming distance i and outputHamming distance j.

Then, the new estimator for BEP of a rate-k/n CC terminated by L informa-tion bits is dened as

P ∗b ,

1

kL

∑

j

∑

i

i.P ∗i,j(u

′i,j|u). (6.6)

This estimator is referred as the `modied-EES' estimator through out this chap-ter. The performance of modied-EES in comparison with EES is presented inSection 6.5.1.

6.4 FHMC Techniques for Viterbi and MAPDecoders

Biasing input densities of coded systems using IS techniques requires a goodknowledge of the input realisations that cause errors at the decoder. Therefore,designing biased densities for complex decoders is dicult when the codewordsare correlated with the input distributions in a complex manner. This can beovercome by using a biasing procedure that adaptively approaches the optimal


biasing distribution.As described in Chapter 4, FHMCmethods provide ecient means of adaptive-

biasing of input distributions to speed-up simulations. The advantage of FHMCmethods is that they adaptively iterate to the optimal biased density with little apriori knowledge of how to bias. The iterative procedure uses a control variable(such as V in Section 4.4) to update the next iteration's biased density so that,as the iteration number increases, an approximately equal number of hits in eachhistogram bin of the control variable is obtained [96].

The selection of the control variable that drives the FHMC algorithm whiledetermining the error performance is a challenging task. As described in Sec-tion 6.2, Holzlöhner et al. [134] and Secondini et al. [136] have derived controlvariables for ecient simulation of the sum-product decoder and Viterbi decoderusing MMC algorithm, respectively. The control variable in [134] requires twoMMC cycles to determine the error performance while that in [136] is specic tothe continuous mode of decoding of the Viterbi decoder.

The proposed algorithm employs a dierent control variable and the WL athistogram algorithm to eciently estimate the performance of the Viterbi de-coder and MAP decoder. Viterbi decoder and MAP decoder use the signal spacedistance or the Euclidean distance between the transmitted codeword and thereceived sequence to determine the most-likely transmitted sequence. Therefore,the performance of the decoder can be characterised using the Euclidean dis-tance. Thus, the control variable V that drives the algorithm can be selected asthe Euclidean distance dE between the transmitted codeword and the receivedsequence, which is given by

V = dE = |r − v|. (6.7)


With this control variable, the BEP of the decoder is expressed as

Pb =

∫ ∞

o

P (e|dE)fS(dE) d(dE) (6.8)

where P (e|dE) denotes the probability of a decoded bit being in error given thatthe Euclidean distance is dE, and fS(dE) is the probability density function ofthe Euclidean distance at the target SNR S expressed in dB.

Consider the desired range RE of the Euclidean distance dE. RE is dividedinto M bins of width ∆dE such that the m-th bin is dened as Bm , [dEm

−∆dE

2, dEm

+ ∆dE

2] with dEm

being the centre of the m-th bin. Then, the BEP of(6.8) can be estimated as

Pb =M

∑

m=1

P (e|dEm)PS(dEm

) (6.9)

where PS(dEm) is the probability of dE falling into the m-th bin, which is dened

as PS(dEm) , P (dE ∈ Bm) = fS(dEm

)∆dE.The major advantage of selecting dE as the control variable for the FHMC

algorithm results from the fact that P (e|dEm) is independent of the value of SNR.

Therefore, P (e|dEm) can be estimated once and used later to estimate BEP at

dierent SNRs. In fact, P (e|dEm) is computed accurately at a low SNR value, and

it is then used to estimate the BEP at high SNR values where the computationof P (e|dEm

) requires large simulation trials.Accordingly, the algorithm consists of two phases. In the rst phase, P (e|dEm

)

is estimated at a low SNR. The second phase involves estimating the pdf ofdE at a desired high SNR and computing the BEP using P (e|dEm

) estimatedduring the rst phase. The main steps of this algorithm can be summarised asfollows:

1: Divide the desired range of dE, RdEinto M bins

2: Phase 1: estimate P (e|dEm) at a low SNR


3: Phase 2: estimate PS(dE) at desired high SNR S

4: Use (6.9) to estimate BEP at S

Figure 6.2 shows an example of estimating BEP of the Viterbi decoder in anAWGN channel at SNR = 6 dB. Here, P (e|dEm

) was estimated at SNR = 2 dB,using the MC technique with 107 samples. As can be seen, the estimation is notaccurate in the range of dE ∼ [3, 3.5]. In fact, this is the dominant region ofP (e|dEm

)P6dB(dE) that is used to compute BEP at 6 dB. Therefore, P (e|dEm)

should be very accurate in this region to obtain reliable estimates for BEP at 6

dB.A similar constraint is encountered in DAIS, which employs the MMC al-

gorithm with Berg's update to nd P (e|V ). The MMC method estimates thepdf at the end of each iteration using a histogram accumulated during respectiveiterations. Current iteration's random walk completely depends on the pdf esti-mated during the previous iteration. Therefore, MMC starts sampling from thelow probability region of the pdf only towards the end of the simulation. Thisleads to insucient sampling of the input realizations that lead to dominant er-rors. Conversely, the WL algorithm employs a dynamic pdf update procedure,which updates the pdf estimation after every move of the random walk in eachiteration. Even though this violates the detailed balance condition of the Markovchain during the early stages of the simulation, it helps to achieve broad samplingover the entire range from the beginning of the simulation. This allows one tosample realizations of V that lead to dominant errors with sucient statistics.Therefore, WL algorithm is employed in Phase 1 of the proposed algorithm toovercome the insucient sampling of dominant error events.

While the WL algorithm estimates P (dE) at a low SNR value by updatingthe visits histogram H(n) for each iteration n, another histogram G

(n)m for m =

1, . . . , M is updated to count the bit errors, given that dE falls into the m-th bin.At the end of N iterations that satises the termination conditions of the WL


2 2.5 3 3.5 4 4.5 510

−12

10−10

10−8

10−6

10−4

10−2

100

dE

Pro

ba

bili

ty

P(e|dE) − Phase 1

P6dB

(dE) − Phase 2

P(e|dE).P

6dB(d

E) − Phase 2

Figure 6.2: Estimating BEP of the Viterbi decoder in AWGN channel at SNR =6 dB. K = 3, L = 13.

algorithm, P (e|dEm) is estimated as [134]

P (e|dEm) =

∑Nn=1 G

(n)m

∑Nn=1 H

(n)m

. (6.10)

The process that drives the Phase 1 of the proposed method is summarisedin Algorithm 6.1. Using Algorithm 6.1, a reliable estimate for P (e|dEm

) can beobtained with a maximum sample size of 2 × 105.

Phase 2 of the algorithm consists of estimating PS(dE) for the target SNRof S dB. When S increases, the PS(dE) should be estimated down to very lowprobabilities to obtain a reliable estimate. This is also accomplished by theWL algorithm. In Phase 2, the knowledge of the Euclidean distance betweenthe transmitted sequence and received sequence is sucient to estimate PS(dE).Thus, the decoding step, which consumes more time, can be omitted in thisphase. This is an added advantage that further reduces the computational timeand complexity of the proposed algorithm.


Algorithm 6.1 Phase 1 of two-phase algorithm.Purpose: This algorithm estimates P (e|dEm) that requires to estimate the performanceof Viterbi or MAP decoder. It uses the dynamics of WL algorithm along with theMetropolis random walk to explore the input space.Require: divide the desired range of dE , RdE

into M bins1: initialise: n ⇐ 0, log (Pm) ⇐ log (1/M)M

m=1,, f0 = 2.71828, NI = 3002: while fn > 1.0001 do3: n ← n + 14: initialise: k ⇐ 0, G(n)

m = 0Mm=1, H

(n)m = 0M

m=1, flag = 15: generate input bit sequence and AWGN samples Z

0 from fZ(z)6: nd dE , number of bit errors b0 and corresponding bin m0

7: while (flag == 1)||(k < NI) do8: generate ∆Z from a zero-mean symmetric pdf9: Z

1 ⇐ Z0 + ∆Z

10: for r = 1 : L do11: generate u from U [0, 1]

12: if u > min[

fz(Z1r)

fz(Z0r)

, 1]

then13: Z

1r ← Z

0r

14: end if15: end for16: nd dE for Z

1, bit errors b1 and corresponding bin m1


18: if u ≤ min[


then19: m0 ⇐ m1, b0 ⇐ b1, Z0 ⇐ Z

1

20: end if21: H

(n)m0 ← H

(n)m0 + 1, G

(n)m0 ← G

(n)m0 + b0, log(Pm0) ← log(Pm0) + log(f0)

22: if mod(k,100)==0 then23: if histogram is approximately at then24: flag == 025: end if26: end if27: end while28: fn+1 ←


29: end while30: compute P (e/dEm) =

∑Nn=1 G

(n)m

∑Nn=1 H

(n)m


Algorithm 6.2 Phase 2 of two-phase algorithm.Purpose: This algorithm estimates PS(dE) and thereby the error performance of theViterbi and MAP decoders. It also uses the WL dynamics along with the Metropolisrandom walk algorithm.Require: divide the desired range of dE , RdE

into M bins1: initialise: n ⇐ 0, log (Pm) ⇐ log (1/M)M

m=1,, f0 = 2.71828, NI = 3002: while fn > fend do3: n ← n + 14: initialise: k ⇐ 0, Hm = 0M

m=1, flag = 15: generate input bit sequence and AWGN samples Z

0 from fZ(z)6: nd dE and corresponding bin m0

7: while (flag == 1)||(k < NI) do8: generate ∆Z from a zero-mean symmetric pdf9: Z

1 ⇐ Z0 + ∆Z

10: for r = 1 : L do11: generate u from U [0, 1]

12: if u > min[

fz(Z1r)

fz(Z0r)

, 1]

then13: Z

1r ← Z

0r

14: end if15: end for16: nd dE for Z



18: if u ≤ min[


then19: m0 ⇐ m1, Z0 ⇐ Z

1

20: end if21: Hm0 ← Hm0 + 1, log(Pm0) ← log(Pm0) + log(f0)22: if mod(k,100)==0 then23: if histogram is approximately at then24: flag == 025: end if26: end if27: end while28: fn+1 ←


29: end while30: PS(dE) = exp(log(P ))31: nd Pe using PS(dE), P (e/dE) of Phase 1 and (6.9).


Algorithm 6.2 summarises the procedure of Phase 2. In this phase, the ter-mination condition is determined by the value fend, which should be selectedaccording to the range of the pdf and the desired level of accuracy. For example,when S increases, fend should be decreased.

Algorithms 6.1 and 6.2 can be used to estimate the BEP of both Viterbi andMAP decoders operating over AWGN channel conditions. They can be easilyextended to estimate the performance of the same decoders in dierent fadingenvironments as well. In that scenario, the input consists of separate AWGN andfading vectors, both of which should be perturbed for proper input exploration.

The advantage of the proposed algorithm over DAIS is two fold. Firstly,though the proposed algorithm has two phases only Phase 2 should be executedfor all target SNRs while Phase 1 should be executed only once for all SNRs. Incontrast, DAIS requires both simulations to be executed for all SNRs consumingmore time than the proposed algorithm. Secondly, Phase 2 of the proposedalgorithm does not require the decoding operation that incur more time in thesimulation process, while decoding is required only for Phase 1, which is executedonce for all SNRs. In contrast, DAIS requires to decode the codewords in bothsimulation stages, which increases the simulation run-time.

Following section validates the proposed algorithms using simulations.


6.5.1 Modied-EES

The numerical results are based on a typical convolutional-coded system operatingover an AWGN channel. Two convolutional codes C1 and C2 with constraintlengths 3 and 9, and generator polynomials given by G1 = [5 7]8 and G2 =

[561 753]8, respectively were employed. Input information length L for C1 was13 while that for C2 was 37. The number of error events simulated for C1 was 31


0 1 2 3 4 5 6 710

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

SNR (dB)

Bit

erro

r pr

obab

ility

Modified−EESEESMCUpper bound

Figure 6.3: Simulation results of code C2. EES, modied-EES and MC resultscoincide after 3 dB.

with EES and only 5 with modied-EES while that for C2 was 100 with EES and20 with modied-EES. Figure 6.3 shows MC, EES and modied-EES simulationresults of C2. Compared with the EES technique, the modied-EES techniquecan estimate probabilities closer to the MC technique in low SNR region. Thisis because the modied EES technique uses the coecients of the new weightenumerator, which can provide a better approximation than the general weightenumerator of the CCs. The modied-EES technique also has the ability tosimulate very low bit error probabilities accurately.

The sample size reduction factor η obtained from the EES and modied-EES over MC is illustrated in Tables 6.1 and 6.2, respectively. The tables alsoshow the variance reduction factor ζ of the modied-EES technique over the EEStechnique. Accordingly, the modied EES technique provides a higher samplesize reduction factor than the EES technique while that factor increases withdecreasing BEP for both methods. The variance reduction gain of modied-EESis also higher than that of EES.


Table 6.1: Estimator accuracy and sample size reduction factor of the EES andmodied-EES techniques for code C1.

SNR EES Modied EES(dB) Mean BEP ǫ(%) η Mean BEP ǫ(%) η ζ

3 3.28e-03 1.98 1.67e+01 2.22e-03 1.51 4.25e+01 3.695 8.17e-05 3.62 2.01e+02 6.15e-05 1.74 1.15e+03 7.547 3.89e-07 5.66 1.73e+04 3.18e-07 2.42 1.15e+05 7.6410 8.04e-13 9.20 3.16e+09 7.00e-13 3.44 2.60e+10 9.42

Table 6.2: Estimator accuracy and sample size reduction factor of the EES andmodied-EES techniques for code C2.

SNR EES Modied EES(dB) Mean BEP ǫ(%) η Mean BEP ǫ(%) η ζ

3 3.16e-05 2.92 4.12e+01 3.02e-05 2.25 7.27e+01 1.844 1.07e-06 3.64 7.84e+02 9.77e-07 2.66 1.61e+03 2.265 1.61e-08 4.68 3.15e+04 1.42e-08 3.24 7.45e+04 2.686 9.16e-11 5.76 3.66E+06 8.02e-11 3.87 9.25e+06 2.88

Table 6.3: Simulation run-time comparison of EES and modied-EES.

SNR Run-time for Code C1 Run-time for Code C2

EES Modied-EES Gain EES Modied-EES Gain(s) (s) (s) (s)

3 36.70 5.87 6.25 367.95 62.93 5.845 36.55 5.87 6.22 424.83 62.86 6.757 36.70 5.86 6.26 464.66 62.81 6.7610 36.22 5.83 6.21 378.90 63.09 6.00


2 4 6 8 10 12 1410

0

102

104

106

108

1010

1012

−log10(BEP)

ξ MC

/EE

S

EESModified−EES

C1

C2

Figure 6.4: Run-time gain Vs BEP for EES and modied-EES.

To compare the computational time of both techniques, NEES was kept con-stant at 1000 for C1 and C2 while keeping all other conditions the same. Table 6.3compares the run-time of EES and modied-EES. Modied-EES requires approx-imately six times lesser run-time than EES because the modied-EES techniquesimulates lesser number of error events than EES.

Figure 6.4 depicts the computational eciency gain ξ obtained by the modied-EES technique and EES technique over MC for codes C1 and C2. The run-timegain increases linearly with the logarithm of the BEP for both techniques; how-ever, it reduces for high constraint lengths. The computational eciency gain ofthe modied-EES technique is higher than that of the EES technique for termi-nated CCs operating over AWGN channels.

6.5.2 Two-Phase Algorithm

A convolutional-coded system operating over AWGN and frequency-at Rayleighfading channels was considered for the simulation. The code C1 of Section 6.5.1


2 3 4 5 6 7 8 9 1010

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

SNR (dB)

Bit

erro

r P

roba

bilit

y

Two−Phase Algorithm−ViterbiTwo−Phase Algorithm−MAPMC−ViterbiUpper Bound

C3

C1

Figure 6.5: Bit error probability of convolutional-coded system in AWGN channel.

and the K = 7 code C3 with generator polynomial G3 = [133 171]8 were em-ployed. Input information length L for C1 was 13 while that for C3 was 29. Thisselection makes the block length equal to 5K for both codes. The system wassimulated using both Viterbi and MAP decoders.

Figures 6.5 and 6.6 illustrate the bit error probabilities of Viterbi and MAPdecoders estimated by TPA in AWGN and Rayleigh fading channels, respectively.The results are compared with the MC simulation results and the conventionalupper bound obtained from the transfer function. The TPA can accurately es-timate error probabilities in the order of 10−10 using a realisable sample size.Further, it performs well in fading scenarios where most of the other algorithmsfail.

Tables 6.4 and 6.5 show the sample size reduction factor obtained from theTPA over MC in AWGN channels for codes C1 and C3, respectively. Comparable


4 6 8 10 12 1410

−9

10−8

10−7

10−6

10−5

10−4

10−3

SNR (dB)

Bit

erro

r P

roba

bilit

y

Two−Phase Algorithm−ViterbiTwo−Phase Algorithm−MAPMC−ViterbiUpper Bound

C1

C3

Figure 6.6: Bit error probability of convolutional-coded system in Rayleigh fadingchannel.

Table 6.4: Estimator accuracy and sample size reduction factor of TPA; C1 codein AWGN.

Viterbi decoder MAP decoderSNR fend BEP ǫr(%) ξMC/TPA BEP ǫr(%) ξMC/TPA

4.0 1.000300 4.22e-04 5.27 1.45 4.16e-04 4.34 2.145.0 1.000100 5.86e-05 5.43 3.61 5.85e-05 4.93 4.516.0 1.000010 5.14e-06 7.97 4.73 5.12e-06 8.77 4.037.0 1.000010 2.81e-07 12.12 38.84 2.39e-07 11.66 34.877.5 1.000010 5.22e-08 14.24 163.41 4.07e-08 13.33 138.428.0 1.000003 7.84e-09 15.86 453.01 5.13e-09 15.60 283.23


Table 6.5: Estimator accuracy and sample size reduction factor of TPA; C3 codein AWGN.


3.5 1.0002000 4.49e-05 6.66 5.14 4.23e-05 5.38 5.084.0 1.0001000 1.00e-05 6.96 7.78 8.29e-06 6.73 5.724.5 1.0000100 1.93e-06 9.28 22.97 1.23e-06 11.12 13.545.0 1.0000050 2.81e-07 9.74 53.00 2.30e-07 9.77 38.985.5 1.0000010 3.63e-08 11.03 105.24 2.23e-08 14.23 90.186.0 1.0000005 3.15e-09 14.14 1037.52 2.11e-09 15.95 951.84

Table 6.6: Estimator accuracy and sample size reduction factor of TPA; C1 codein Rayleigh fading.


9 1.000300 6.59e-05 4.13 5.11 6.89e-05 3.99 3.9110 1.000010 3.42e-05 4.89 4.16 2.59e-05 5.37 7.1711 1.000005 1.06e-05 5.21 8.54 9.76e-06 5.29 11.0212 1.000003 3.58e-06 6.47 9.44 2.60e-06 7.65 9.3913 1.000001 9.15e-07 9.83 16.18 8.23e-07 9.19 21.1714 1.000001 3.50e-07 10.19 22.06 2.38e-07 10.06 23.86

Table 6.7: Estimator accuracy and sample size reduction factor of TPA; C3 codein Rayleigh fading.


6 1.000050 8.50e-05 3.84 3.07 8.42e-05 4.14 1.677 1.000030 1.51e-05 3.68 4.79 1.24e-05 6.15 1.708 1.000010 2.13e-06 4.64 8.76 1.99e-06 6.27 6.189 1.000005 3.98e-07 4.60 12.01 3.53e-07 6.63 9.3310 1.000003 5.94e-08 6.17 20.89 5.50e-08 7.72 17.5711 1.000001 8.25e-09 10.91 39.38 8.17e-09 7.91 65.52


sample size reductions and estimator accuracies are obtained for both decoders.In addition, the performance of the TPA does not degrade with the increase ofthe constraint length of the CC, which is an advantage of the TPA.

The sample size reduction gains obtained from the TPA over MC in Rayleighfading channels are shown in Tables 6.6 and 6.7 for codes C1 and C3, respec-tively. The sample size reduction in fading channels is lower than that in AWGNchannels. Due to the higher dimensionality of the fading scenario than the noiseonly channel, TPA requires more samples to explore the input space for ecientsampling.

Results of this chapter have been either published or are under review in [198]and [199]

6.6 Conclusion

The diculty of deriving ecient IS biasing densities for coded systems can beovercome by using iterative, adaptive biasing strategies such as at histogramtechniques. A novel algorithm is proposed for ecient simulation of terminatedCCs using iterative at histogram techniques. Euclidean distance between thetransmitted codeword and the received sequence is selected as the control variablethat drives the at histogram algorithm. The proposed algorithm consists of twophases with the WL algorithm employed in both phases. It provides signicantcomputational eciency gains in simulating Viterbi and MAP decoders in AWGNchannels while moderate sample size reductions are achieved for Rayleigh fadingenvironments.

Chapter 7

Conclusion

7.1 Summary and Conclusion

This thesis described dierent biased MC methods for ecient simulation ofOFDM, MIMO and coded communication systems. The major investigationsand their outcomes can be summarised as follows:

• The concept of biased MC for ecient simulation is described, and a com-prehensive review of biased MC techniques that have been developed for e-cient simulation is presented. IS is identied as the most prominent ecientsimulation technique used for communication system simulations. Never-theless, the system dependent nature of IS hinders it being more widelyapplied to evaluate complex system architectures. FHMC is a novel biasedsampling approach invented by physicists. FHMC methods use innovativealgorithms that can be considered as forms of adaptive IS. The potentialof FHMC techniques for ecient simulation of communication systems isidentied.

• IS techniques are successfully applied for ecient simulation of OFDMsystems operating over frequency-selective fading channels with SISO andMIMO transmission. Optimum parameters for biasing the time-domainnoise density function and fading coecients using VS and MT are derived

7. CONCLUSION 137

for SISO systems. An equivalent SISO model of the MIMO-OFDM sys-tem is successfully applied to derive the IS parameters for biasing inputsof the MIMO-OFDM system. The optimum biasing of noise density of theMIMO system with OSTBC using MT is only possible for full-rate codes.In addition, the optimum biasing of fading coecients of the MIMO systemis computationally intensive due to the requirement of numerical integra-tion of complex equations. Therefore, the biasing parameters of the MIMOsystem are approximated using those of the SISO system. The proposed bi-asing strategies signicantly improve the eciency of the sampling process.In fact, the run-time gain obtained from the proposed IS methods linearlyincreases with the decreasing SEP in logarithmic scale. This demonstratesthe power of IS that can be achieved through proper biasing of systemparameters.

• FHMC methods are investigated in a communication engineering context.The probability theory behind FHMC that was hidden by the excessivestatistical physics details has been described using IS concepts. MCMCmethods provide the underlying dynamics for proper exploration of the in-put space such that at visits histograms are obtained. Dierent FHMCalgorithms and their characteristics are discussed. The challenges of apply-ing FHMC to communication system simulations are also argued. Theseinvestigations provide a basis for ecient simulation of complex communi-cation systems which are dicult to solve by analytical methods.

• Improved FHMC algorithms are proposed for pdf estimation in communi-cation systems. Four dierent algorithms named as smooth-MMC, optimal-MMC, TMMC andWL-TMMC with improved characteristics are proposed.They are successfully applied to estimate capacity and outage probabilityof a MIMO-MRC system operating over Nakagami-m fading channels. TheWL-TMMC algorithm, which is a combination of WL and TMMC at his-

7. CONCLUSION 138

togram techniques, shows better performance by estimating lower probabil-ities with greater smoothness. The proposed algorithms provide powerfultools to verify analytical results in very low probability regions that are notnormally reached by MC simulations. They have the added advantage ofgeneral applicability to any pdf estimation problem of interest.

• IS and FHMC methods are investigated for ecient simulation of codedcommunication systems employing Viterbi and MAP decoders. The newweight enumerators of terminated CCs are exploited to extend the EEStechnique for fast simulation of terminated CCs. The extended-EES methodis more ecient than the EES method in AWGN channels. An ecientalgorithm, which employs FHMC methods in two dierent phases, is pro-posed for ecient performance evaluation of Viterbi and MAP decoders.The proposed Two-Phase algorithm provides signicant computational ef-ciency gains over AWGN channels while moderate sample size reductionsare obtained for Rayleigh fading environments.

7.2 Future Work

This section provides recommendations for possible directions for future researchon the subject of this thesis.

• Ecient simulation techniques for OFDM and MIMO-OFDM systems oper-ating over frequency-selective fading channels were addressed in Chapter 3.An interesting extension to this work includes the investigation of the eectsof channel correlations on the performance of the proposed IS techniques.

• A vast range of applications is yet to be investigated in terms of FHMC tech-niques. One interesting application includes multiuser detection in OFDMand MIMO systems. The optimal joint multiuser detection, equalisation

7. CONCLUSION 139

and decoding problems are excessively complex, and simulation of such sys-tems involve prohibitively long run-times. FHMC techniques may providebetter ways to speed up these simulations.

• FHMC techniques provide ecient means of pdf estimation. This can beexploited in designing iterative receivers for multiuser MIMO systems withunknown CCI and background noise.

• Estimation of very low probabilities is a crucial requirement in optical com-munication system simulation. A natural extension to the work presentedin Chapter 5 is to investigate the performance of the proposed estimatorsin relation to optical communication systems.

• The proposed Two-Phase algorithm improves the eciency of Viterbi andMAP decoder simulations. This algorithm can be investigated in relationto iterative decoders such as turbo decoders and sum-product decoders forLDPC codes used in modern wireless communication systems.

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Appendix A

Derivations of Biasing Parametersfor OFDM and OSTBC-OFDMSystems

A.1 Optimum β for Variance Scaling of Noise inOFDM System

The estimator variance of VS technique can be written as

σ2V S ≈

∫ ∞

−∞

∫ ∞

−∞

H[Si.xi + z′i]fXi(xi)fZ′

i(z′i)W (z′i) dxi dz′i (A.1.1)

where

fXi(xi) =

1

2[δ(xi − A) + δ(xi + A)] ,

fZ′

i(z′i) =

1

2πσ2i

exp

(−|z′i|22σ2

i

)

,

APPENDIX 159

W (z′i) = β2i exp

(

−|z′i|22σ2

i

(

1 − 1

β2i

))

,

and A is the BPSK signal amplitude.The inner integral can be evaluated as

∫ ∞

−∞H[Si.xi + z′i]fXi

(xi) dxi =

∫ ∞

−∞

1

2H[ℜSi.xi + z′i] [δ(xi − A) + δ(xi + A)] dxi

=

12U [ℜ−SiA + z′i] forXi = −A

12U [ℜ−SiA − z′i] forXi = A

(A.1.2)

where U [.] is the unit step function and ℜ. is the real part of the complexnumber.

Assuming that Xi = −A is transmitted (A.1.1) can be written as

σ2V S ≈

∫ ∞

−∞

U [ℜ−SiA + z′i]fZ′

i(z′i)W (z′i) dz′i. (A.1.3)

Using the change of variables, above equation can be evaluated as

σ2V S ≈

∫ ∞

v=−∞

∫ ∞

u=ℜSiA

β2i

2πσ2i

exp

(

−(u2 + v2)

2σ2i

(

2 − 1

β2i

))

≈ β2i

(

2 − 1β2

i

)Q

(√

(

2 − 1β2

i

)

)

ℜSiA

σi

. (A.1.4)

A.2 Optimum Mean Value for Mean Translationof Noise in OFDM System

Let the mean of the biased pdf of frequency domain noise samples be m′zi

=

m′u + m′

v and assume that Xi = −A is transmitted. The MT estimator varianceσ2

MT can be written as

APPENDIX 160

σ2MT ≈

∫ ∞

ℜSiA

∫ ∞

−∞

1

2πσ2i

exp

(

−(m′u)

2 + (m′v)

2

σ2i

)

× exp

(

−(u + m′u)

2 + (v + m′v)

2

2σ2i

)

du dv

≈ exp

(

−(m′u)

2 + (m′v)

2

σ2i

)

Q

(ℜSiA + m′u

σi

)

(A.2.1)

where

Q(t) =1√2π

∫ ∞

t

e−t2

2 dt.

Using the Q-function approximation given by [200]

Q(t) ≈ e−t2

2

t√

2π(A.2.2)

σ2MT can be written as

σ2MT =

σi√2π(ℜSiA + m′

u)exp

(

−(3(m′u)2 + 2(m′

v)2 + 2ℜSiAm′

u + ℜSiA2)

2σ2i

)

.

(A.2.3)

Taking the derivative of σ2MT with respect to m′

u and m′v and equating to zero

yields the optimum values for m′u and m′

v as

m′u =

2ℜSiA +√

ℜSiA2 − 3σ2i

3

m′v = 0. (A.2.4)

APPENDIX 161

A.3 Optimum βR for Variance Scaling of FadingCoecients in OFDM System

Using the output version of IS [30] and the output symbol Yi given by (3.7), theVS-Rayleigh estimator variance can be written as

σ2V S−R =

∫ ∞

−infty

H(Yi)fYi(Yi)W (Yi) dYi. (A.3.5)

Let, |Hi| = R and Xi = −A. The pdf of Yi can be written as

fYi(x, y, R) =

1

2πσ2z

exp

(

−(x + AR)2 + y2

2σ2z

)

f|Hi|(R) (A.3.6)

where f|Hi|(R) is the pdf of |Hi| which is Rayleigh distribution with E[|Hi|2] =

2σ2Hi. The indicator function H(Yi) is

H(Yi) =

1 x > 0,∀y

0 elsewhere

and the weighting function W (Yi) = WV S−R(R) given by (3.30). Then, (A.3.5)can be written as

σ2V S−R =

∫ ∞

0

∫ ∞

0

∫ ∞

−∞

Rβ2R

2πσ2zσ

2Hi

e

(

−(x+AR)2+y2

2σ2z

)

e− R2

2σ2Hi

(

2− 1

β2R

)

dy dx dR

=

∫ ∞

0

Q

(

AR

σ2z

)

Rβ2R

σ2Hi

e− R2

2σ2Hi

(

2− 1

β2R

)

dR. (A.3.7)

Using the Q-function approximation of (A.2.2), σ2V S−R can be simplied as

σ2V S−R =

β3Rσ2

z

2A(A2β2Rσ4

Hi+ 2σ2

zβ2Rσ2

Hi− σ2

zσ2Hi

)0.5. (A.3.8)

APPENDIX 162

Dierentiating with respect to βR yields

∂

∂βR

(

σ2V S−R

)

=3β2

Rσ2z(A

2σ2Hi

β2R + 2σ2

zβ2R − σ2

z) − β3Rσ2

z(A2σ2

HiβR + 2σ2

zβR)

2AσHi(A2σ2

Hiβ2

R + 2σ2zβ

2R − σ2

z)1.5

.

(A.3.9)

Solving (A.3.9) for βR yields

β2R =

3σ2z

2A2σ2Hi

+ 4σ2z

. (A.3.10)

A.4 Optimum βooR for VS-Rayleigh Technique in

OSTBC-OFDM System

Using the output version of IS [30] and the decision variable Vk,i given by (3.37),the VS-Rayleigh estimator variance can be written as

σ2V S−R =

∫ ∞

−∞

H(Vk,i)fVk,i(Vk,i)W (Vk,i) dVk,i. (A.4.11)

Let, ||Hi||2F =∑nT

q=1

∑nR

r=1 |Hr,qi |2 = Ri and Xk,i = −A. The pdf of Vk,i can be

written as

fVk,i(x, y, Ri) =

1

2πσ2z′i

exp

(

−(x + aARi)2 + y2

σ2z′i

)

fRi(nd, 2B

2) (A.4.12)

where fRi(nd, 2B

2) is the pdf of Ri which is Gamma distribution given by (3.51).The indicator function H(Vk,i) is

H(Vk,i) =

1 x > 0,∀y

0 elsewhere

and the weighting function W (Vk,i) = WV S−R(Ri) given by (3.52). Then, (A.4.11)

APPENDIX 163

can be written as

σ2V S−R =

∫ ∞

0

∫ ∞

0

∫ ∞

−∞

(βooR )nd

(2B2)ndΓ(nd)(2π)0.5σz′i

Rnd−1i exp

(

−(x + aARi)2 + y2

2σ2z′i

)

× exp

[

− R

2B2

(

2 − 1

βooR

)]

dy dx dRi

=

∫ ∞

0Q

(

aARi

σz′i

)

(βooR )nd

(2B2)ndΓ(nd)Rnd−1

i exp

(

− R

2B2

(

2 − 1

βooR

))

dRi.

(A.4.13)

Using the Q-function approximation of (A.2.2), σ2V S−R can be written as

σ2V S−R =

∫ ∞

0

(βooR )ndσz′i

aA(2B2)ndΓ(nd)(2π)0.5Rnd−2

i exp(

υ1υ22 − υ1(Ri + υ2)

2)

dRi

(A.4.14)

where υ1 = a2A2

2σ2z′i

and υ2 = 14υ1B2

(

2 − 1βoo

R

)

.For nT = 2 and nR = 1, (A.4.14) can be simplied as

σ2V S−R =

∫ ∞

0

(βooR )2σz′i

4aAB4(2π)0.5exp

(

υ1υ22 − υ1(Ri + υ2)

2)

dRi. (A.4.15)

Using the solution (2.33-16.) of [201], (A.4.15) can be evaluated as

σ2V S−R =

(βooR )2σz′

4aAB4(2π)0.5exp(υ1υ

22)Q

(

(2υ1)0.5υ2

)

. (A.4.16)

Using a tighter approximation to Q-function given by [200]

Q(t) ≈exp

(

− t2

2

)

1.64t + (0.76t2 + 4)0.5

APPENDIX 164

σ2V S−R can be simplied as

σ2V S−R =

25(βooR )3σ2

z′(2aAB2)−1

[41σz′(2βooR − 1) + 5(76σ2

z′(βooR )2 − 76σ2

z′βooR + 19σ2

z′ + 800(βooR )2a2A2B4)0.5]

.

(A.4.17)

Since a closed-form solution for βooR which minimizes (A.4.17) is intractable, opti-

mum values for βooR are found numerically by substituting values for A, a, B and

σ2z′ = minσ2

z′i for each SNR.

Appendix B

Derivations of Variance of FHMCMethods

B.1 Variance of WL Algorithm

The logarithm of the estimate of the WL algorithm can be written as

log [P nm(WL)] =

n∑

k=1

Hkm log(fk). (B.1.1)

Assuming independent histogram bin entries, the variance σ2[log [P nm(WL)]] can

be expressed as

σ2[log [P nm(WL)]] =

n∑

k=1

σ2[Hkm] log(fk)

2. (B.1.2)

Since the uctuations of the histogram bins are known to grow with the squareroot of the number of entries [98]

σ2[Hkm] ≈ Hk

m. (B.1.3)

APPENDIX 166

Therefore, (B.1.2) can be written as

σ2[log [P nm(WL)]] ≈

n∑

k=1

Hkm log(fk)

2. (B.1.4)

B.2 Variance of TMMC Method

The logarithm of the estimate of the TMMC method can be written as

log [P nm(TM)] = log [P n

m−1(TM)] + log

[

Tnm−1,m

Tnm,m−1

]

. (B.2.1)

The variance of the estimator σ2[log [P nm(TM)]] is


m−1(TM)]] + σ2[log[Tnm−1,m]] + σ2[log[Tn

m,m−1]]

(B.2.2)

where Tnm,k =

Tm,k

Tm

with Tm =∑

k Tm,k. Hence, σ2[log[Tnm,k]] can be written as

σ2[log[Tnm,k]] = σ2[log[Tm,k]] + σ2[log[Tm]]

=

(

σ[Tm,k]

Tm,k

)2

+

(

σ[Tm]

Tm

)2

=1

Tm,k

+1

Tm

. (B.2.3)

From that, (B.2.2) can be written as


m−1(TM)]] +Tm−1,m + Tm,m−1

Tm−1,mTm,m−1

+

∑

k Tm−1,k +∑

k Tm,k∑

k Tm−1,k

∑

k Tm,k

. (B.2.4)

Documents

researchdirect.westernsydney.edu.auresearchdirect.westernsydney.edu.au/islandora/object/uws:11188... · DECLARATION Date: JULY 2011 Author: Pushpika Wijesinghe Title: BIASED MONTE