Parameter Estimation of Nonlinearities in FutureWireless Systems

EHSAN OLFAT

Doctoral ThesisStockholm, Sweden 2018

TRITA-EECS-AVL-2018:66ISBN 978-91-7729-948-6

KTH Royal Institute of TechnologySchool of Electrical Engineering

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlaggestill offentlig granskning for avlaggande av teknologie doktorsexamen i electro- ochsystemteknik fredag den 19 oktober 2018 klockan 10.00 i F3, Lindstedtsvagen 26,Stockholm.

© 2018 Ehsan Olfat, unless otherwise noted.

Tryck: Universitetsservice US AB

To my beloved family

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Abstract

Nowadays, our every-day life is immersed with wireless communications.From our hand-held cell-phones to televisions to navigation systems in cars, alland all are using wireless communications. This usage will even be enormouslyexpanded due to the introduction of the era of 5G-based Internet-of-Things(IoT) which consists wearables, sensors and more smart appliances.

Orthogonal frequency division multiplexing is a very well-known commu-nication method which has been utilized in modern standards and technolo-gies due to its high spectral efficiency, simple frequency-domain equalization,and robustness against inter-symbol interference. Nevertheless, the major do-wnside of OFDM systems is the large fluctuations of the amplitudes of theirsignals causing high peak-to-average-power-ratio (PAPR). This forces the po-wer amplifier (PA) in the transmitter’s RF front-end to work in its saturationregion, hence introducing nonlinear distortion to the transmitted signal. Thisis particularly challenging in low-cost and low-power (and even low-weight)devices where a high-quality PA with a large dynamic range is not affordable,using complex digital processing techniques to mitigate the PAPR or to line-arize the PA is not computationally feasible, and introducing input back-offto change the operating point of the PA is not desirable due to decreasingthe power efficiency of the PA, which can be problematic because of the shortbattery-life. On the other hand, there are more resources available for a high-quality base station (or IoT gateway) in terms of power, budget, space andcomputational complexity, which motivates transferring all the complexityand cost to them and implement receiver-side nonlinearity estimation andcompensation algorithms.

To compensate the effects of a nonlinear PA on the transmitted signaland lastly detect them correctly, an iterative detection algorithm has beenproposed in the literature. However, to use this algorithm successfully, thereceiver first needs to estimate the nonlinearity parameters. The importanceof this is more noticeable in the 5G-based Internet-of-Things networks, inwhich presumedly, numerous low-cost and low-power devices aim to transmitdata to a base station (or an IoT gateway).

The focus of this thesis is on estimating the nonlinearity parameters al-ong with channel estimation, nonlinearity distortion mitigation, and symboldetection in future wireless systems deploying OFDM. In particular, we firstconsider an OFDM system with a limiter (clipper) communicating over anAWGN channel, and derive a maximum-likelihood estimator of the clippingamplitude. Next, we consider OFDM systems tranceiving over multi-pathfading channels, and propose a joint channel and clipping amplitude esti-mation algorithm using block-type frequency-domain pilots. Furthermore, wepropose a new packet-frame consisting time-domain and frequency-domainpilots to separately estimate channel and clipping amplitude. After, we consi-der a broader types of memoryless nonlinear PA models, and propose a jointestimation-detection algorithm to jointly estimate the nonlinearity parame-ters and channel and detect symbols. Finally, the joint channel and clippingamplitude estimation algorithm is extended to SIMO-OFDM systems. Theperformance of all of these algorithms are verified by means of simulations.

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Sammanfattning

Nu for tiden ar vart dagliga liv fyllt av tradlos kommunikation. Allt franvara mobiltelefoner till television och GPS i bilarna, alla dessa apparaterutnyttjar tradlos kommunikation. Detta anvandande kommer att okas annumycket mer med 5G-baserad Internet of Things (IoT), bestaende av aktivi-tetsarmband, sensorer alla andra sorters smarta apparater.

Ortogonal frekvensduplex (OFDM) ar en valkand kommunikationsmetodsom utnyttjas i moderna kommunikationsstandarder och teknologier, tack varsin hoga spektraleffektivitet, enkel utjamning i frekvensdomanen och robust-het mot intersymbolinterferens. Trots det, ar den storsta nackdelen de storavariationerna i signalamplitud som orsakar hogt topp till medel-forhallande(PAPR). Detta tvingar effektforstarkaren (PA) i sandarens radio-frontendatt arbeta i klippningsomradet, vilket introducerar ickelinjar distorsion i denutsanda signalen. Detta problem ar speciellt patagligt for utrustning med lagkostnad och lag energiforbrukning, dar en hogkvalitativ PA med stort dy-namiskt omrade vore alltfor dyr, avancerad digital signalbehandlingsteknikfor att motverka PAPR och linearisera PA:n inte ar implementerbar ochdar det inte ar onskvart att undvika ickelineariteterna genom att dra nedsandeffekten, eftersom det skulle minska energieffektiveteten hos forstarkarenoch darmed begransa batteritiden. A andra sidan finns det mer resursertillgangliga i en basstation (eller IoT-accesspunkt), i termer av effekt, bud-get, utrymme och berakningskomplexitet, vilket gor det motiverat att flyttakomplexitet och kostnad till dem och implementera algoritmer for att skattaoch kompensera ickelineariteter pa mottagarsidan.

En iterativ algoritm har tidigare foreslagits i litteraturen for att kom-pensera for inverkan pa den utsanda signalen av en ickelinjar PA och utforakorrekt detektion. For att kunna anvanda denna algoritm, kravs dock att mot-tagaren forst skattar de parametrar som beskriver ickelineariteterna. Dettablir mer viktigt i 5G-baserade IoT-natverk, dar foretradesvis utrustning medlag kostnad och lag effekt ska skicka data till en basstation (eller en IoT-accesspunkt).

Denna avhandling fokuserar pa skattning av ickelinearitetsparametrarna,tillsammans med kanalskattning, kompensation av ickelinearitetsdistorsionenoch symboldetektering i framtida tradlosa kommunikationssystem somanvander OFDM. Speciellt, studerar vi forst OFDM-system med en signal-begransare (klippning), som kommunicerar via en AWGN-kanal, och harlederen maximum likelihood-skattare av klippningsamplituden. Darefter betraktarvi OFDM-system som sander over en fadandeflervagsutbredningskanal, och foreslar en gemensam skattningsalgoritm forkanalen och klippningsamplituden, mha blockvisa traningssymboler i frek-vensplanet. Dessutom foreslar vi en ny ramstruktur bestaende av traningsdatai bade tids- och frekvensdomanen for att separat skatta kanal och klippnings-amplitud. Darefter riktar vi in oss pa mer allmanna typer av modeller forickelinjara PA och foreslar en gemensam algoritm for att samtidigt skatta ic-kelineariteten och kanalen samt detektera datasymbolerna. Slutligen utokasalgoritmen for gemensam kanal- och klippningsamplitudskattning till SIMO-

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OFDM-system. Prestanda for alla dessa algoritmer utvarderas med hjalp avnumeriska simuleringar.

List of Papers

The thesis is based on the following papers:

[A] E. Olfat and M. Bengtsson, "Estimation of the Clipping Level in OFDM Sys-tems," 49th Asilomar Conference on Signals, Systems and Computers, PacificGrove, CA, USA, pp. 1169-1173, 2015.

[B] E. Olfat and M. Bengtsson, "Joint Channel and Clipping Level Estimationfor OFDM in IoT-based Networks," IEEE Transactions on Signal Processing,vol. 65, no. 18, pp. 4902-4911, Sept. 2017.

[C] E. Olfat and M. Bengtsson, "Channel and Clipping Level Estimation in OFDMSystems," submitted to IEEEWireless Communications and Networking Con-ference (WCNC), Sept. 2018.

[D] E. Olfat and M. Bengtsson, "A General Framework for JointChannel-Nonlinearity Parameters Estimation and Symbol Detection forOFDM in IoT-based 5G Networks," submitted to IEEE Transactions on Sig-nal Processing, Sept. 2018.

[E] E. Olfat and M. Bengtsson, "Joint Channel and Clipping Level Estimationfor SIMO-OFDM Systems," submitted to IEEE Wireless Communicationsand Networking Conference (WCNC), Sept. 2018.

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In addition to papers A-E, the following journal and conference papers have alsobeen (co)-authored by the author of this thesis:

[1] E. Olfat and A. Olfat, "Performance of nth Best Relay Selection HybridDecode–Amplify–Forward Protocol," Wireless Personal Communications, vol.78, Issue 2, pp. 1403-1412, Sep. 2014.

[2] E. Olfat and A. Olfat, "BER Analysis of Decode–Amplify–Forward Protocolwith nth Best Relay Selection in Multi-Relay Cooperative Networks," IJECE(Iranian Journal of Electrical and Computer Engineering), vol. 13, Issue 1a,pp. 87-92, Spring 2015.

[3] E. Olfat, H. Sokri-Ghadikolaei, N. N. Moghadam, M. Bengtsson, C. Fischione,"Learning-based Pilot Precoding and Combining for Wideband Millimeter-wave Networks," 2017 IEEE 7th International Workshop on ComputationalAdvances in Multi-Sensor Adaptive Processing (CAMSAP), Curacao, 2017,pp. 1-5.

[4] H. Sokri-Ghadikolaei, E. Olfat, M. Bengtsson, C. Fischione, "Learning-basedPilot Precoding and Combining for Wideband Millimeter-wave Networks,"submitted to IEEE JSAC Special Issue on Artificial Intelligence and MachineLearning for Networking and Communications.

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Acknowledgements

First and foremost, I am sincerely grateful to my supervisor ProfessorMats Bengtsson for giving me the opportunity to pursue my doctoral studiesat the signal processing lab in KTH, and for all his support and care in allthe ups and downs during my journey. He has always been eager and openfor technical discussions, and very smart to guide me to the right direction.Besides, He has always been very supportive in other aspects of my life as aPhD student, and I have been so comfortable to talk about different issueswith him. I also would like to thank my co-supervisor Professor Joakim Jaldénfor technical discussions we had and his insightful questions in the internalseminars.

I would like to thank all my colleagues and friends for making my journeymore pleasant. In particular, I would like to thank Arash Owrang, Sina Mo-lavipour, Alireza Mahdavijavid, Hamid Ghourchian, Baptiste Cavarec, NimaNajari Moghadam, Alla Tarighati, Farshad Naghibi, M. Reza Gholami, Ma-jid Gerami, Hamed Farhadi, Nafiseh Shariati, Hadi Ghauch, Shahab Nazari,Hossein Shokri, Serveh Shalmashi, Amirpasha Shirazinia, Rasmus Brandt,Satyam Dwivedi, Vijaya Parampalli Yajnanarayana, Martin Sundin and KlasMagnusson. I would like to thank Marie Maros, Arun Venkitaraman andPol Del Aguila Pla for having great times full of enjoyment and laugh atgame/fika sessions. I would like also to acknowledge our former administra-tor Tove Schwartz for providing enjoyable moments and for all her help andsupport. I also would like to thank Sina Molavipour, Baptiste Cavarec, AllaTarighati and Nima Najari Moghadam for their help in the proofreading pro-cess and their valuable feedbacks to improve the presentation of this thesis.During my studies, I have had the chance to collaborate with Dr. Nima Na-jari Moghadam, Dr. Hossein Shokri and Prof. Carlo Fischione. I would liketo thank them for the fruitful discussions and collaboration and also theirinvaluable suggestions and feedbacks.

Finally, I would like to express my deep and heartfelt gratitude to myfamily, who have always been providing me with their endless love, encoura-gement and believing. To my beloved mother for all her continuous love andsupport throughout my whole life, to my brother, Ali, and my sister, Elham,who have always been there for me, to my nephew, Sam, whose being hasmade our lives full of joy and happiness, and to the memory of my belovedfather who has been my first teacher in life.

Ehsan OlfatStockholm, September 2018

Contents

Abstract v

Sammanfattning vii

List of Papers ix

Acknowledgements xi

Contents xiii

List of Acronyms xvii

List of Notations

I Thesis Overview

1 Introduction 11.1 Multicarrier Communications and OFDM . . . . . . . . . . . . . 3

1.1.1 OFDM in Standards . . . . . . . . . . . . . . . . . . . . . 41.1.2 Practical OFDM Transceivers . . . . . . . . . . . . . . . . 61.1.3 PAPR in OFDM Systems . . . . . . . . . . . . . . . . . . 8

1.2 Nonlinear Power Amplifiers and Mitigating Their Effects . . . . . 81.2.1 Nonlinear Power Amplifiers . . . . . . . . . . . . . . . . . 91.2.2 Mitigation Techniques . . . . . . . . . . . . . . . . . . . . 121.2.3 Transmitter Side Compensation Techniques . . . . . . . . 131.2.4 Receiver Side Compensation Techniques . . . . . . . . . . 14

1.3 General System Model . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Iterative Detection Algorithm . . . . . . . . . . . . . . . . . . . . 151.5 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5.1 Mean Square Error . . . . . . . . . . . . . . . . . . . . . . 181.5.2 Unbiased Estimators . . . . . . . . . . . . . . . . . . . . . 181.5.3 Cramér-Rao Lower Bound . . . . . . . . . . . . . . . . . . 18

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xiv Contents

1.5.4 Consistency and Efficiency of Estimators . . . . . . . . . 221.6 Linear Channel Estimation in OFDM Systems . . . . . . . . . . 23

2 Background and Contributions of the Thesis 272.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Problem I: Algorithms to estimate the nonlinearity para-meter of a limiter (along with channel estimation) . . . . 29

2.2.2 Problem II: An efficient iterative joint estimation-detectionalgorithm to estimate nonlinearity parameters and chan-nel, and detect symbols . . . . . . . . . . . . . . . . . . . 40

2.2.3 Problem III: An algorithm to jointly estimate nonlinearityparameter and channel for SIMO-OFDM . . . . . . . . . 46

2.3 Further Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Conclusion 57

References 59

II Included Papers 73

A Estimation of the Clipping Level in OFDM Systems 75A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77A.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . 79A.4 Cramér Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . 80A.5 Pilot-Based Estimation . . . . . . . . . . . . . . . . . . . . . . . . 82A.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

References 87

B Joint Channel and Clipping Level Estimation for OFDM inIoT-based Networks 89B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91B.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B.3 Joint Channel and Clipping Amplitude Estimation . . . . . . . . 95

B.3.1 Clipping Amplitude Estimation Given the Channel . . . . 96B.3.2 Channel Estimation given Clipping Amplitude . . . . . . 98B.3.3 Initialization of the Alternating Algorithm . . . . . . . . . 98B.3.4 Alternating Optimization Algorithm . . . . . . . . . . . . 99B.3.5 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 99B.3.6 Computational Complexity . . . . . . . . . . . . . . . . . 100

B.4 Cramér-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . 101

Contents xv

B.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References 113

C Channel and Clipping Level Estimation in OFDM Systems 117C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119C.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C.3 Channel and Clipping Level Estimation . . . . . . . . . . . . . . 123

C.3.1 Time-Domain Channel Estimation . . . . . . . . . . . . . 124C.3.2 Clipping Amplitude Estimation . . . . . . . . . . . . . . . 124

C.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 125C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

References 129

D A General Framework for Joint Estimation-Detection of Channel-Nonlinearity Parameters and Symbols for OFDM in IoT-based 5G Networks 131D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133D.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136D.3 Joint Channel-Nonlinearity Parameters Estimation and Symbol

Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139D.3.1 Nonlinearity Parameter Estimation Given the Channel

and Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 140D.3.2 Channel Estimation Given Pilots . . . . . . . . . . . . . . 140D.3.3 Data symbol detection given the nonlinearity parameters

and channel . . . . . . . . . . . . . . . . . . . . . . . . . . 140D.3.4 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . 141D.3.5 Joint Estimation-Detection of Channel-Nonlinearity Pa-

rameters and Symbols . . . . . . . . . . . . . . . . . . . . 143D.4 Cramér-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . 144D.5 Limiter as the Nonlinearity . . . . . . . . . . . . . . . . . . . . . 145

D.5.1 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 146D.5.2 Computational Complexity . . . . . . . . . . . . . . . . . 146D.5.3 Cramér-Rao Lower Bound . . . . . . . . . . . . . . . . . . 146

D.6 Nonlinearity Modeled by Polynomials . . . . . . . . . . . . . . . 146D.6.1 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 150D.6.2 Computational Complexity . . . . . . . . . . . . . . . . . 150

D.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 150D.7.1 Normalized Mean Square Error . . . . . . . . . . . . . . . 150D.7.2 Symbol Error Rate . . . . . . . . . . . . . . . . . . . . . . 151

D.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

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References 163

E Joint Channel and Clipping Level Estimation for SIMO-OFDM Systems 167E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169E.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171E.3 Joint Channel and Clipping Level Estimation . . . . . . . . . . . 173

E.3.1 Clipping Level Estimation Given Channel . . . . . . . . . 173E.3.2 Channel Estimation Given Clipping Level . . . . . . . . . 173E.3.3 Algorithm Initialization . . . . . . . . . . . . . . . . . . . 174E.3.4 Joint Alternating Optimization . . . . . . . . . . . . . . . 175

E.4 Cramér-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . . 175E.5 Signal Detection Method . . . . . . . . . . . . . . . . . . . . . . . 176E.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 176E.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

References 181

List of Acronyms

2G 2nd generation

3G 3rd generation

3GPP 3rd generation partnership project

4G 4th generation

5G 5th generation

A/D analog-to-digital converter

ADSL asynchronous digital subscriber line

AM amplitude modulation

AWGN additive white Gaussian noise

BIC-OFDM bit interleaved coded OFDM

CL clipping level

CO-OFDM coherent optical OFDM

CP cyclic prefix

CRLB Cramér-Rao lower bound

D/A digital-to-analog converter

DAB digital audio broadcasting

DFT discrete Fourier transform

DSP digital signal processing

DTMB digital terrestrial multimedia broadcasting

DVB digital video broadcasting

E2E end-to-end

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xviii List of Acronyms

ECMA European computer manufacturers association

EPA extended pedestrian A

FIM Fisher information matrix

FWA fixed wireless access

HDSL high-bit-rate digital subscriber line

IBO input back-off

ICI inter-channel interference

IDFT inverse discrete Fourier transform

iid independent and identically distributed

IoT Internet-of-Things

ISI inter-symbol interference

LNA low-noise amplifier

LO local oscillator

LPWA low-power wide-area

LS least-squares

LTE long-term evolution

M2M machine-to-machine

MB multiband

MIMO multiple-input multiple-output

ML maximum likelihood

mmWave millimeter Wave

MRC maximal ratio combining

MSE mean square error

MTC machine-type communications

NMSE normalized mean square error

NR new radio

OBO output back-off

List of Acronyms xix

OFDM orthogonal frequency division multiplexing

OFDMA orthogonal frequency division multiple access

P/S parallel-to-serial

PA power amplifier

PAPR peak-to-average power ratio

PDF probability density function

PM phase modulation

PN pseudo-noise

QAM quadrature amplitude modulation

RF radio frequency

S/P serial-to-parallel

SCS supported subcarrier spacing

SDMA space-division multiple access

SER symbol error rate

SIMO single-input multiple-output

SNR signal-to-noise-ratio

SSPA solid state power amplifier

TDS time-domain synchronous

TWTA traveling wave tube amplifiers

UMVUE uniform minimum variance unbiased estimator

UWB ultra wideband

V2X vehicle to everything

xDSL digital subscriber lines

List of Notations

Uppercase boldface letters and lowercase boldface letters denote matrices (e.g. X),and column vectors (e.g. x), respectively. Moreover, scalars are denoted by italicletters (e.g. x,X). The following mathematical notations are used.

Cn×m The set of complex-valued n×m matrices.Rn×m The set of real-valued n×m matrices.R+ The set of nonnegative real numbers.[X]kl The k, l-th element of the matrix X.diag(x) The diagonal matrix with x at its diagonal.XT The transpose of X.X∗ The conjugate of X.XH The Hermitian of X.X−1 The inverse of the square matrix X.X† The Moore-Penrose inverse of X.Rx The real part of x.Ix The imaginary part of x.|x| The absolute value of x.arg(x) The phase of x.loga(x) The logarithm of x with base a.log(x) The natural logarithm (the logarithm with base e) of x.O(·) Big O notation.tr(X) The trace of the square matrix X.vec(X) The vector obtained by stacking the columns of X.λmax(X) The maximum eigenvalue of the square matrix X.λmin(X) The minimum eigenvalue of the square matrix X.cond(X) The condition number of the square matrix X defined as

|λmax(X)/λmin(X)|.CN (x,Σ) The multivariate Gaussian distribution with mean X and covari-

ance matrix Σ.

List of Notations

U [x, y] The uniform distribution in the interval [x, y].R(x) The Rayleigh distribution with parameter x.Ex The statistical expectation of x.Ex|y The conditional expectation of x given y.‖x‖p The Lp-norm of x.S \ k The remaining set when the member k is removed.∀x The statement holds for all x.x ∈ S x is a member of S.X Y Means that X−Y is positive definite.X Y Means that X−Y is positive semi-definite.IN The N ×N identity matrix.0N×M The all-zero N ×M matrix.0N The all-zero N × 1 vector.1N The all-one N × 1 vector.〈·〉 The hard detection (demodulation) operator. Denotes the Hadamard (element-wise) product. Denotes the Hadamard (element-wise) division.⊗ Denotes the Kronecker product.x ∼ D Means the distribution of x is D.xn

dist.−−−→ D Means xn converges to D in distribution.

Part I

Thesis Overview

1

Introduction

New generations of wireless communication systems require reliability, high-speeddata transmission, high mobility, and spectrum and power efficiency. Bandwidth-intensive multimedia applications have boosted the evolution of wireless technology,e.g., mobile television service via digital video broadcasting (DVB) and wireless tele-conferencing. The 5th generation (5G) wireless network technology standardizationis set up by 2018, its first commercial version will be launched by 2018, and itsworldwide launch will be by 2022 and onwards [ARS16,GJ15,DLK+17]. The maingoals of 5G are improving capacity, energy efficiency, and reliability, while reducinglatency and increasing connection density massively.

In particular, the focus of 5G mobile communications is providing seamlessmachine-to-machine (M2M) communications, a type of data communication bet-ween heterogeneous devices without any human intervention [ASHAM18] for theInternet-of-Things (IoT) as well as personal communications. The IoT is a promi-sing technology which aims to revolutionize the world by run-on connectivity amongsmart devices. The current demand for machine-type communications (MTC) hasresulted in a variety of communication technologies with diverse service require-ments to achieve the vision of the IoT. Recently devised cellular standards suchas long-term evolution (LTE) have been introduced for mobile devices but are notwell suited for low-power and low data rate devices which are the building compo-nents of IoT networks [ASHAM18]. To address this issue, there are several emergingIoT standards. 5G mobile networks, specifically, try to address the limitations ofprevious cellular standards and to be a potential key player for the future IoT.

IoT deals with low-cost and low-power devices intercommunicating with each ot-her through the Internet. The concept of the IoT [PDG+16,Bor14,WSJ15,AIM10,AFGM+15, VF11] has attracted a lot of research attention whose goal has beento ensure that wearables, sensors, smart appliances, smart-phones, smart trans-portation system, and other entities be connected via a common medium withthe capability of interacting with each other. This can be achieved through analways-on communication medium [VRM+17]. IoT is expected to influence manyaspects of everyday-life and business applications and contribute towards growingthe world’s economy, through massive and critical IoT, depending on the nature

1

2 Introduction

of applications. According to [ASHAM18], massive IoT applications require thatenormous number of smart devices to be connected. It can be deployed in shippingenvironments, smart-homes (buildings) and smart-cities, smart power systems, andagricultural monitoring environments, etc., which requires frequent updates to thecloud with low end-to-end cost. Applications in this domain require low-cost devi-ces with low-power consumption, extended coverage area, and high scalability foreffective deployment of Massive IoT. Alternatively, critical IoT applications inclu-ding remote healthcare systems, traffic and industrial control (vehicles, drones, androbots) and tactile Internet etc., require higher availability, higher reliability, safetyand lower latency to guarantee high quality end-user experience due to the factthat failure of such applications can result in severe consequences.

According to forecasts made by Ericsson [Eri16], about twenty-eight billion ofsmart devices will be connected across the world by 2021, with more than fifteenbillion of these devices connected through M2M and consumer electronics devices.Anticipation has also shown that roughly seven billion of these devices will be con-nected by cellular technologies such as 2nd generation (2G), 3rd generation (3G)and 4th generation (4G) which are currently being used for IoT but not fully opti-mized for such applications and low-power wide-area (LPWA) technology [Oyj16]with a revenue of about 4.3 trillion dollars to be generated across the entire IoTsector globally [BM15]. The current demand for MTC applications such as smartcities [PSB+13], smart grid [FFK+11, YQST13, KPP14, AMOK+15, DCOAM16],smart building and surveillance [DBN14], remote maintenance and monitoring sy-stems [SFKH15,CCJ+16, PKH16], and smart water system [AMHP+16] etc., hasimposed the need of connectivity of massive number devices leading to a majorresearch issue in terms of capacity for ongoing deployed and future communicationnetworks [JJM+15].

Furthermore, new applications such as tactile Internet (A network or network ofnetworks for remotely accessing, perceiving, manipulating or controlling real or vir-tual objects or processes in perceived real time by humans or machines [MCRV16]),high resolution video streaming, tele-medicine, tele-surgery, smart transportation,and real-time control dictate new specifications for throughput, reliability, end-to-end (E2E) latency, and network robustness [ZXWL14]. In addition, intermittentor always-on type connectivity is required for MTC serving diverse applicationsincluding sensing and monitoring, autonomous cars, smart homes, moving robotsand manufacturing industries [PRG+18].

Therefore, these use cases of the next generation network push the specificati-ons of 5G in multiple aspects such as data rate, latency, reliability, device/networkenergy efficiency, traffic volume density, mobility, and connection density. In the lite-rature, surveys on 5G network including architecture [ARS16, GJ15],SDN/NFV/MEC based core network [NBGT17,TSM+17], caching [IW16,ZLZ15],backhaul [JITT16], resource management [ODK16] and data centric network[XZWX17,BBE+13] are available.

Indeed, the concept of IoT is quickly transforming an attractive reality by whichanything to anything anytime, and anywhere can be connected. Smart wearable de-

1.1. Multicarrier Communications and OFDM 3

vices (smart watches, glasses, bracelets, etc.), smart home appliances (smart meters,fridges, televisions, thermostat), sensors, autonomous cars, cognitive mobile devices(drones, robots, etc.) are connected to always-on hyper-connected world to enhanceour life style [SMK+17,PDG+16,SSP+18].

Considering the physical layer of 5G-based IoT networks, the importance ofmulticarrier communications, and in particular orthogonal frequency division mul-tiplexing (OFDM) emerges.

1.1 Multicarrier Communications and OFDM

One shortage of single-carrier communication systems is that in these systems thewhole bandwidth is just used by one symbol and other symbols are transmittedin different time-slots. When the channel changes over the transmission bandwidthdue to the multipath fading phenomenon, these systems need complex equalizers atthe receiver. To overcome this difficulty, multicarrier systems have been proposed.

The history of multicarrier communications dates from the mid-sixties, whenChang [Cha66], for the first time, proposed the concept of parallel data trans-mission. Later, Weinstein and Ebert [WE71] proposed the use of discrete Fouriertransform (DFT) for baseband modulation and demodulation, and hence, elimina-ted the need of banks of analog oscillators. They also proposed using some guardspace between symbols to avoid inter-channel interference (ICI) and inter-symbolinterference (ISI). Next, to ensure the orthogonality of the subcarriers, Peled andRuiz [PR80] introduced cyclic prefix (CP) and its utilization instead of empty guardspace.

In contrast to single-carrier systems, the available bandwidth is divided intoseveral subcarriers (or subchannels) in multicarrier systems. Each subcarrier expe-riences a (almost) flat fading channel and therefore simple equalization techniquescan be utilized at the receiver. If we properly choose the transmission parametersand the channel bandwidth, subcarriers will be orthogonal to each other.

Many of the new communication systems are using multicarrier techniques,in particular, OFDM, which are effective techniques for mitigating the impact ofmultipath fading and enabling high data rate transmissions over mobile wirelesschannels.

Although the concept of OFDM has been known for decades, it has been justcomparatively recently that it has gained popularity and become one of the mostprominent and dominant type of signal modulation in wireless communication sys-tems. This is due the technological advances in designing and implementing electro-nic elements and digital signal processing (DSP) algorithms that have eliminatedthe original obstacles of OFDM implementation. For a comprehensive history ofOFDM see, e.g., [Bin90] and [WY95].

OFDM provides high spectral efficiency and robustness against multipath fadingchannels [HWK00,NP00]. In multicarrier systems, the modulation is performed ona block-by-block basis, with a guard interval between the blocks. By transmitting

4 Introduction

several symbols in parallel, the symbol duration is increased thereby reducing theeffects of ISI caused by the multipath fading channel. By assuming that the channeldelay spread is shorter than the guard interval (e.g. CP), the convolution betweenthe transmitted data and the channel is converted to circular convolution. Thismeans that the received frequency domain signal is simply the point-wise multi-plication of the frequency-domain transmitted symbols and the frequency-domainchannel coefficients. Then, the transmitted symbols can be recovered by a sim-ple single-tap equalizer for each subcarrier leading to an easy implementation ofreceivers.

The integration of OFDM with multi-antenna systems is a promising way toefficiently use the radio spectrum to extend the capacity of mobile communica-tion systems, see, e.g., [STT+02,DPSB10]. Multiantenna OFDM technologies aresuitable for dynamic multiuser resource allocation [CSGK17] that exploits systemdiversity in time, frequency and space through an intelligent allocation of band-width, multiple access, scheduling, and power and rate adaptation.

1.1.1 OFDM in Standards

Due to its impressive merits such as high spectral efficiency, simple equalization inthe frequency domain, and robustness against ISI [Cha66,Bin90], OFDM has gai-ned considerable attention recently and been deployed in many wired and wirelesstechnologies and standards. One of the first implementations of OFDM was in thedigital subscriber lines (xDSL) standards, such as asynchronous digital subscriberline (ADSL) [ITU98], and high-bit-rate digital subscriber line (HDSL) [CTC91]. Itis also deployed in digital audio broadcasting (DAB) [ETS06], DVB [ETS04b], Wi-Fi (IEEE 802.11), WiGig (IEEE 802.11ad), and the European alternative for theIEEE 802.11 standards, i.e., HIPERLAN/2 [ETS04a]. Moreover, when OFDM is in-tegrated with binary coding, the combination is referred to as bit interleaved codedOFDM (BIC-OFDM). BIC-OFDM has attracted attention in the recent years, andhas been deployed in standards such as IEEE 802.11a/g for wireless local area net-works (WLANs) [IEE99], WiMAX standards, i.e., IEEE 802.16 (wireless broadbandaccess), and the 3rd generation partnership project (3GPP) LTE wireless cellular sy-stems. OFDM also allows the allocation of several user bands dividing the spectrumin groups, referred to as European computer manufacturers association (ECMA)multiband (MB)-OFDM. The advantages of ECMAMB-OFDM are exploited in theWiMedia MB-OFDM standard developed for ultra wideband (UWB) communica-tion systems [ECM08]. While OFDM exploits the temporal and spectral domains,multiple-input multiple-output (MIMO) technology exploits the spatial domain byusing multiple antennas at the transmitter and receiver. MIMO-OFDM systems areimplemented by evolving standards as IEEE802.11n and IEEE802.16 WMAN, andIMT-Advanced mobile cellular systems. A useful detailed table represented here asTable 1.1 about the use of OFDM and its parameters used by most popular radioaccess technologies can be found in [Wan17].

OFDM is also used in 5G new radio (NR). 3GPP release 15 focusing on 5G

1.1. Multicarrier Communications and OFDM 5

Table 1.1: OFDM usage in popular radio access technologies [Wan17].

general outline has been published in February 2018 and is mainly dedicated to5G NR eMBB and fixed wireless access (FWA) [3GP18]. The downlink transmis-sion waveform is conventional OFDM using a cyclic prefix. The uplink transmissionwaveform is conventional OFDM using a cyclic prefix with a transform precodingfunction performing DFT spreading that can be disabled or enabled [3GP18]. Themajor difference with respect to currently deployed LTE is the support of variousphysical layer numerologies (corresponds to one subcarrier spacing in the frequencydomain, which allows scalable supported subcarrier spacing (SCS) and symbol du-ration. By scaling a reference subcarrier spacing by an integer N, different nume-rologies can be defined and challenges of 5G can be properly addressed.). Makingthe physical layer scalable allows to properly address new services. By making thesymbols duration smaller, it can support Ultra Reliable and Low Latency Com-munications [LZ16] and by increasing the SCS, the robustness against the Dopplereffect is increased which is required for millimeter communications [ZLG+16]. Thefive different numerologies supported by 5G NR are shown in Table 1.2 [3GP18].

Additionally, thanks to nice capability of OFDM in dealing with chromatic dis-persion and polarization mode dispersion in the frequency domain [Wan17], OFDM

6 Introduction

µ ∆f = 2µ.15 [kHz] Cyclic prefix Supported for data Supported for synch0 15 Normal Yes Yes1 30 Normal Yes Yes2 60 Normal, Extended Yes No3 120 Normal Yes Yes4 240 Normal No Yes

Table 1.2: Supported transmission numerologies in 5G NR [3GP18].

also found its applications in optical communications, including both transmis-sion and access networks, such as coherent optical OFDM (CO-OFDM) [Arm09],and orthogonal frequency division multiple access (OFDMA) passive optical net-works [Cvi12,Gid14].

1.1.2 Practical OFDM TransceiversAn OFDM transceiver structure is illustrated in Figure 1.1, which consists of di-gital baseband components, digital-to-analog converters (D/As), analog-to-digitalconverters (A/Ds), and analog radio frequency (RF) front-ends. As depicted, theinput bits at the transmitter, which can be the output of coding, scrambling andinterleaving blocks, are mapped into modulation symbols drawn from a modulationconstellation such as QAM, then grouped together using a serial-to-parallel (S/P)convertor, and processed by the inverse discrete Fourier transform (IDFT). To makethe signal resistant against ISI the CP is pre-added to the output of the IDFT andthe result is serialized using a parallel-to-serial (P/S) convertor. In this way thedigital baseband time-domain OFDM symbols are generated. This stream of serialcomplex-valued time-domain OFDM symbols are divided into real (in-phase) andimaginary (quadrature) components (I and Q), and then using D/As convertedinto a baseband analog signal. Next, the analog signals go through low-pass filters,and up-converted to a carrier frequency to construct the pass-band signal, thenamplified, and finally transmitted.

At the receiver, basically the reverse operations are performed on the receivedsignal. In particular, after being received by the antenna, the signal is first amplifiedby a low-noise amplifier (LNA), then down-converted into baseband analog I andQ components, low-pass filtered, and next translated into digital baseband signalsusing A/Ds. After the CP removal the signal at the baseband part is equalized,demodulated, de-mapped, and decoded to retrieve the transmitted bits.

Note that, the RF front-end at the transmitter consists of several importantelements which can have huge impact on the performance of the overall OFDMcommunication system. Indeed, they introduce the main source of impairmentsand distortion to the signals, which needed to get dealt with. Here, we enumeratethe possible problems caused by these RF components:

• Imperfect D/As and A/Ds: In practice, D/As and A/Ds with finite re-

1.1. Multicarrier Communications and OFDM 7

D/A LPF

D/A LPF

PA

Transmitter RF front-endDigital baseband

I

Q

P/S

IDFT

S/P

Map

ping

A/D

900

LPF

A/D LPF

LO LNA

I

Q

S/P

DFTP/S

Dem

apping

Digital basebandReceiver RF front-end

Outputbits

900LOC

PAdd

erCP

Rem

over

Inputbits

Figure 1.1: A practical OFDM transceiver system.

solution are used. Indeed, using more quantization bits offers higher preci-sion. However, the power dissipation of D/As and A/Ds scales exponentiallywith the resolution [LS08], i.e., adding one more quantization bit doubles theenergy cost of the D/As and A/Ds. This issue is especially important forcommunications at millimeter Wave (mmWave) frequencies due to the largenumber of antennas as well as RF chains needed for digital beamforming,leading to prohibitive hardware energy cost with high-resolution D/As andA/Ds [WWD+16]. They can also introduce clipping due to insufficient dyna-mic range [Sma12]. This is usually related to the resolution and noise floor.

• I/Q imbalance: In ideal scenarios, the amplitude and phase shift of theI and Q branches are exactly equal and exactly 90, respectively. However,in practice, due to the hardware imperfection, I/Q imbalance happens. I/Qimbalance refers to the amplitude and phase mismatch between the I and Qbranches leading to possible considerable performance degradation communi-cation systems. See [SSMK16] for a survey on the effects of I/Q imbalance.

• Phase noise: Due to the non-ideality of the local oscillators (LOs), theycan suffer from random phase noise. OFDM systems are sensitive to phasenoise. The reasons are phase noise destroys the subcarriers orthogonality,causes the leakage of discrete Fourier transform (DFT), and also leads toICI. See [MB15] for a survey on phase noise mitigation techniques in OFDMsystems. Moreover, [Neg15] proposes techniques to estimate and compensate

8 Introduction

phase noise along with channel estimation and symbol detection in OFDMsystems.

• Carrier frequency offset: The emergence of the carrier frequency offset isdue to the mismatch in the carrier frequencies of the local oscillators in thetransmitter and receiver. Another reason can be the Doppler effect as a resultof the mobility of the transceivers.

• Nonlinear power amplifier (PA): Practical PAs suffer from nonlinearfunctions which introduce in-band and out-of-band distortion. The formercauses system performance degradation, and the latter creates not only per-formance reduction but interference to other adjacent channels.

The focus of this thesis is on the last item, i.e., nonlinear PAs. Note that for thesake of simplicity and mathematical tractability, we assume all other items to beperfect causing no impairments for the system.

The importance of studying the effects of nonlinear PAs in OFDM systems co-mes from one of the main drawbacks of OFDM systems; its time-domain signalssuffer from large amplitude fluctuations, and as a result, produce a high peak-to-average power ratio (PAPR). This leads to distortion introduced when the highPAPR signals are fed to a nonlinear device. Although almost all of the electroniccomponents in the analog stages of the transmitter and receiver do not behave per-fectly linear, the major source of the nonlinearity and, hence, the major source ofthe distortion, is basically caused by the transmitter PA, specially if high power effi-ciency and low-cost transmitters are desired. Indeed, the high PAPR OFDM signalsnecessitate utilizing PAs with a large linear dynamic range operating far from theirsaturation points. In other words, such transmission schemes require expensive PAsand high power consumption. Reducing the sensitivity to nonlinear amplification,in the uplink transmission, is of great significance for mobile terminals, which areusually low-power and low-cost devices especially in 5G IoT networks. For the basestation, i.e. in the downlink transmission, using low-cost and low-power PAs is notthat important due to relatively unrestricted resources such as space, power andbudget.

1.1.3 PAPR in OFDM SystemsAs mentioned earlier, the main drawback of OFDM signals is their high PAPR.Because of the presence of IDFT, the time-domain OFDM signal is a linear com-bination of many randomly scaled complex-valued exponentials with different fre-quencies. Therefore, by central limit theorem, the time-domain OFDM signal iswell approximated by a complex Gaussian distribution, hence its amplitude by aRayleigh distribution. This leads to a high PAPR input signal to the nonlinear PA.For an OFDM system with N subcarriers, the PAPR is defined as:

PAPR = max0≤n≤N−1 |xn|2

E|xn|2, (1.1.1)

1.2. Nonlinear Power Amplifiers and Mitigating Their Effects 9

where xn, n = 0, . . . , N − 1 is the time-domain OFDM signal sampled at symbolrate, and E· denotes the statistical expectation. As N increases, the probabilityof the signals being added coherently goes up, thus the PAPR increases.

1.2 Nonlinear Power Amplifiers and Mitigating TheirEffects

In this section, first the specifications of nonlinear PAs are described, and then thetechniques for mitigating (compensating) their impact in the OFDM communica-tion systems will be explained.

1.2.1 Nonlinear Power AmplifiersBased on electronic circuit design and operational configuration, PAs are catego-rized into four classes as A, B, AB, and C. This classification depends on theoperation point of the PA defined by its collector electrical current (bipolar tran-sistors) or drain electrical current (field effect transistors). These classes provide agood indication of linearity and power efficiency1 [Ken00,Gre07].

The power efficiency of PA is a measure of how effectively a PA can convertthe power drawn from the DC supply to useful power at the output [Gre07]. It isdefined as

PE ,Pout

PDC, (1.2.1)

where Pout is the output power and PDC is the power from the DC supply.

The power amplifier’s operating point

If the PA’s operating point is close to the saturation point, the input signal willbe amplified in a highly nonlinear manner. In other words, as the operating pointgets far from the saturation point, the PA becomes more linear. Nevertheless, thismakes the PA less power efficient.

The PA’s operating point is defined using the input back-off (IBO) and theoutput back-off (OBO). The IBO is defined as

IBO , 10 log10

(Pmax

inP avg

in

)[dB], (1.2.2)

where Pmaxin and P avg

in are the PA’s input saturation power and the average powerof the input signal, respectively, and the OBO is defined as

OBO , 10 log10

(Pmax

outP avg

out

)[dB], (1.2.3)

1Diving further into the details of these classifications distances us from the system-viewconsideration, and hence lies outside of the scope of this thesis. In this thesis the high-level (orbehavioral) models of PAs are considered.

10 Introduction

Pmaxout

P avgout

PmaxinP avg

in Pin

Pout

IBOOBO

Figure 1.2: The IBO and OBO.

where Pmaxout and P avg

out are the PA’s output saturation power and the average power ofthe output signal, respectively. The PA’s output saturation power is the maximumtotal power available from the amplifier and the input saturation power is thedrive power at which the output saturation power occurs. Figure 1.2 illustrates theconcepts of the IBO and OBO.

Power amplifier models

Practical and power-efficient amplifiers will have a nonlinear response unless thePA is operating far from the saturation point. Therefore, accurate nonlinear modelsfor the amplifier are important when studying the system performance.

Higher-level (or behavioral) models provide good accuracy with reasonable com-plexity for analysis purposes. These empirical models are developed based on a setof selected input-output observations and offer a compact representation of the PAcharacteristics. These models are divided into two categories as:

• Memoryless nonlinear PA models: In these models the output at asample-time of the nonlinear PA is the instantaneous nonlinear function ofthe input at that sample-time. Clipper (limiter), polynomial, traveling wavetube amplifiers (TWTA) and solid state power amplifier (SSPA) are includedin this class.

• Nonlinear PA models with memory: The memory is primarily the resultof electrical and electro-thermal effects in the electronic device [BG03], whichare frequency dependent. Therefore, the output of these nonlinear PAs at a

1.2. Nonlinear Power Amplifiers and Mitigating Their Effects 11

sample-time depends on input signal at current and previous sample-times.Volterra, Wiener (the cascade of a linear filter and a memoryless nonlinearity),Hammerstein (the cascade of a memoryless nonlinearity and a linear filter),and Wiener-Hammerstein (the cascade of a linear filter, a memoryless non-linearity and another linear filter) are examples of this class. A comparativeanalysis of these models is given in [IWR06].

In this thesis only the memoryless nonlinear PA models are considered.

Memoryless nonlinear power amplifier models

If the nonlinear PA has a flat frequency response, or the communication systemis narrow-band [Deu08a], or electro-thermal effects are neglected, the nonlinearPA can be modelled as a memoryless system. The memoryless nonlinear PAs arecharacterized by their amplitude modulation (AM)/AM and AM/phase modulation(PM) conversion characteristics, which depend only on the input signal value at thecurrent instance.

If x = rejφ denotes the polar form of the input to the nonlinear PA, the outputz can be written as

z = g(x; θNL) = F (r)ej(φ+Φ(r)) (1.2.4)

where θNL is the nonlinearity parameter vector, and F (·) and Φ(·) are the AM/AMand AM/PM characteristics, respectively.

Several memoryless PA models have been proposed to define the behavior ofmemoryless nonlinear PAs which are explained in the following.

Limiter:This model can capture the clipping effect. It has the following AM/AM andAM/PM conversion characteristics [THC03]

F [r] =r, r ≤ AA, r > A,

(1.2.5)

Φ[r] = 0, (1.2.6)

where A is the clipping amplitude. The clipping level (CL) can be defined as

CL , 10 log10

(A2

Px

), (1.2.7)

where Px = E|x|2 denotes the power of the input signal x.

Traveling wave tube amplifiers (TWTA) model:The TWTA model is proposed by Saleh [Sal81]. Its AM/AM and AM/PM charac-

12 Introduction

teristics are

F (r) = aar

1 + bar2 , (1.2.8)

Φ(r) = aΦr

1 + bΦr2 , (1.2.9)

where a, b, c, and d are positive real numbers.

Solid state power amplifier (SSPA) model:The SSPA model is proposed by Rapp [Rap91], and its AM/AM and AM/PMcharacteristics are

F [r] = r(1 + ( rA )2%

) 12%, (1.2.10)

Φ[r] = 0, (1.2.11)

where A and % are positive real numbers determining the saturation level and thesmoothness factor, respectively. Note that as %→∞, the SSPA model approachesthe limiter model.

Memoryless polynomial model:A memoryless nonlinear PA can be modeled using a Taylor series with complexcoefficients. The baseband output of the PA can be expressed as [RQZ04]

zn =K∑k=1

bk|xn|k−1xn, (1.2.12)

where bk’s are complex-valued coefficients and K denotes the order of the polyno-mial function.

1.2.2 Mitigation TechniquesThe first way to deal with the nonlinearity is to use PAs with large dynamic range.However, these linear PAs are very expensive, hence not economically viable to bedeployed in low-cost devices. The second way is to force nonlinear PAs to work intheir linear region. This is achievable by considering a high IBO, i.e. attenuatingthe input signal. By doing this, the high peaks of the high PAPR OFDM signalsare compressed, thus the PA will operate in its linear region and the distortionwill be reduced. Nevertheless, the use of IBO jeopardises the PA power efficiency,as PAs are more power efficient when they operate close to the saturation point.This again is not appropriate for low-power devices whose operation depends onthe short battery life.

Because of these reasons a large body of research has been devoted to techniquesfor compensating the nonlinear distortion [Sch08]. These compensation techniquescan be mainly divided in two categories:

1.2. Nonlinear Power Amplifiers and Mitigating Their Effects 13

• Transmitter side techniques: The aim of these techniques is to modify (oradapt) the signal before feeding it to the nonlinear PA.

• Receiver side techniques: The aim of these techniques is to somehow com-pensate the effects of the nonlinear PAs and reconstruct the original signal atthe receiver.

In the following, these two groups of compensation techniques will be briefly ex-plained.

1.2.3 Transmitter Side Compensation Techniques

There are many techniques proposed in the literature that modify the signal beforepassing it to nonlinear PAs. These techniques are mostly divided into two classes:

• PAPR reduction techniques,

• Nonlinear PA linearization or predistortion.

PAPR reduction techniques

These techniques reduce the envelope fluctuations of the OFDM signal before itbeing fed to the nonlinear PA. The simplest form of these techniques is clippingthe signal at the transmitter. By clipping, all the amplitudes of the samples greaterthan the predetermined threshold (clipping amplitude) are deliberately cut [OI02,QRZ04]. The main disadvantage of clipping is that it increases both the in-band andout-of-band distortion, hence degrading the bit error rate. It is worth mentioningthat the clipping technique is equivalent to passing the signal through a limiternonlinearity. Moreover, clipping (using the limiter nonlinearity model) can be agood approximation of other nonlinearly models.

There are other techniques for PAPR reduction such as tone reservation [BE09,HLLZ15], active constellation extension [KJ03], tone injection [HCL06], selectedmapping [BFH96], partial transmit sequence [CS00], interleaving [JT00], and coding[Wul96]. For an exhaustive survey on PAPR reduction methods, see [HL05,RM13]and the references therein.

Putting the clipping technique aside, all other mentioned PAPR methods requiresome side information, hence increasing the complexity and decreasing the datarate.

Linearization/predistortion of nonlinear power amplifiers

There are many linearization methods proposed to deal with the nonlinearity ofPAs. These methods can be categorized into three general types as: feedback, feed-forward and predistortion [Deu08b]. For a comprehensive description see [Kat01,Tei08] and references therein. These techniques require analog circuit design. Only

14 Introduction

predistortion can be implemented fully digitally. Predistortion also has other ad-vantages over the other methods: feedback features conditional stability and isnarrow-band and single carrier oriented. On the other hand, feed-forward is ratherexpensive and difficult to add to an existing PA structure [Deu08b].

The idea of digital predistortion is to modify the input signal to the PA in a waythat the output is the well-approximated linearly amplified original signal. However,since the output power from the PA is limited, linearization can only be reachedbelow the saturation point. Therefore, even in an ideal case, the cascaded blocksof predistorter and the nonlinear PA behave as a clipper (limiter). Furthermore,deploying a predistorter, which also needs to be adaptive due to the variationsof nonlinear PA parameters over time can increase the complexity and cost whichmakes it not suitable for simple low-cost devices. These variations are due to changesin temperature and humidity, unstable power supply, and the process of aging.

1.2.4 Receiver Side Compensation TechniquesConsidering low-cost and low-power devices, low power consumption and limitedhardware complexity can be an essential need for using a receiver compensationtechnique. In other words, the computational complexity and power needed can betransferred to the base station which has more resources available. The pioneer andmost well-known compensation technique at the receiver side has been proposedin [THC99, THC03], and a similar version in [CH03]. The technique is a quasi-maximum likelihood (ML) iterative detection algorithm, which will be discussedthoroughly in Section 1.4, since the compensation part of the work in this thesis isbased on this technique.

1.3 General System Model

The baseband OFDM system under consideration (depicted in Figure 1.3) has Nnumber of subcarriers. The input vector s = [s0, . . . , sN−1]T is the frequency-domainsymbol vector selected from a constellation such as quadrature amplitude modula-tion (QAM). The time-domain symbols are obtained by taking the IDFT from thefrequency-domain symbols as

xn = 1√N

N−1∑k=0

skej 2πkn

N , n = 0, . . . , N − 1. (1.3.1)

We can rewrite (1.3.1) in matrix form as:

x = FHs, (1.3.2)

where F is the N×N unitary DFT matrix. Note that xn = rnejφn , n = 0, . . . , N−1

or more compactly x = r ejφ, where denotes the Hadamard product. Thechannel is frequency-selective slow-fading with L + 1 taps (L N), denoted as

1.3. General System Model 15

sN−1

g(.; θNL)s0

xN−1

x0

xN−1

x0

xN−1

xN−Lcp+1

Channel

AWGN

uN−1

u0

uN−1

uN−Lcp+1

uN−1

u0

yN−1

y0AddCyclicPrefixIDFT DFT

RemoveCyclicPrefixP

/S

S/P

Figure 1.3: The baseband OFDM system model.

h = [h0, . . . , hL]T. g(. ; θNL) is the nonlinearity function modeling a nonlinear PA,parameterized by the vector θNL. The output of the nonlinear PA is z = g(x; θNL),in which g(. ;A) is taken element-wise. Using (1.2.4), the output of nonlinear PAcan be written as

z = g(x; θNL) = F (r) ej(φ+Φ(r)), (1.3.3)

where F (·) and Φ(·) are taken element-wise. To remove the ISI, a CP with a lengthLcp (≥ L) is pre-added to the time-domain symbols at the transmitter and later willbe removed at the receiver. After the process of adding and removing the CP, thevector-form time-domain representation of the OFDM transmission can be writtenas

u = Hz + w, (1.3.4)

where

H =

h0 0 · · · 0 hL hL−1 · · · h1

h1 h0 0 · · · 0 hL−2 · · · h2...

. . . . . . . . . . . . . . . . . ....

0 · · · 0 hL hL−1 · · · h1 h0

is an N ×N circulant matrix, which represents the circular convolution operator,and w = [w0, . . . , wN−1]T is a zero-mean circularly symmetric complex Gaussiannoise vector, i.e., w ∼ CN (0, σ2IN ).

Taking the DFT from both sides of (1.3.4), we obtain the frequency-domainrepresentation of the OFDM transmission as:

y = Fu =√NDHFz + w, (1.3.5)

where DH is a diagonal matrix with the N -point DFT of h as its diagonal elements,and w ∼ CN (0, σ2IN ) is the DFT of the time-domain noise vector. In deriving(1.3.5), we have used the fact that the circulant matrix H can be diagonalized bypre- and post-multiplication with N -point DFT and IDFT matrices, i.e., FHFH =√NDH. Let F be a semi-unitary matrix formed by the first L + 1 columns of F,

then DH = diag(Fh).

16 Introduction

1.4 Iterative Detection Algorithm

In this section, we explain the iterative detection algorithm proposed in [THC03].In [THC03], it is assumed that the parameters of the nonlinearity, i.e., θNL, andchannel are perfectly known a priori, which is not valid in real-life scenarios.

The in-band distortion created by the nonlinear PA can be expressed by thein-band intermodulation products of the subcarriers in the OFDM system. As aresult, due to the nonlinearity, the subcarrier k of an OFDM symbol will be dis-torted by the other N − 1 subcarriers in the same OFDM symbol. The optimalML solution to detect the transmitted symbols at the receiver is to make the de-cision based on all the data symbols existing in each OFDM symbol, i.e. sequencedetection. Therefore, instead of the conventional symbol by symbol decision, thedecision on the k-th subcarrier is done based on all received N subcarriers. However,this optimal sequence detection increases the computational complexity exponen-tially with respect to the number of subcarriers and the constellation size, since itrequires an exhaustive search through all possible combinations of transmitted datatones. One way to overcome this difficulty is to approximate the received OFDMsymbol as parallel distorted subcarriers. In this way, the receiver can utilize theconventional tone by tone detection, hence reducing the complexity. Nevertheless,as shown in [THC03], this can decrease the performance dramatically. As a solu-tion to these problems, [THC03] proposes a quasi-ML iterative detection technique,which uses both the intermodulation products of the subcarriers as well as paral-leled subcarriers. Before stating this technique, we need first understand how thereceived OFDM symbol can be approximated by the parallel distorted subcarriers.This is nicely done by using Bussgang theorem [Bus52,DTV00,Row82,Fri98] whichis restated here as Theorem 1.4.1.

Theorem 1.4.1 (Bussgang theorem [DTV00,Row82,Fri98]). For a complex Gaus-sian distributed input signal xn to a memoryless nonlinearity g(.; θNL), the outputcan be written as

zn = g(xn; θNL) = κxn + dn, (1.4.1)

where α is a linear gain of the nonlinearity, and dn is a zero-mean (non-Gaussian)distortion signal such that xn and dn are uncorrelated. Moreover

κ = Eznx∗nE|xn|2

, (1.4.2)

where (·)∗ denotes the complex conjugate.

As explained in Section 1.1.3, the output of the IDFT is well-approximated bya Gaussian distributed signal, hence using the Bussgang theorem seems valid forOFDM systems.

Now, we can state the proposed iterative detection algorithm proposed in [THC03]as Algorithm 1.4.1. In Algorithm 1.4.1, Nq is the maximum number of algorithm

1.5. Parameter Estimation 17

Algorithm 1.4.1 Iterative detection algorithm [THC03].1: Inputs:

y, h, F, κ, and Nq2: Initialize:

DH = diag(Fh)d(0) = 0N

3: for i = 1 to Nq do

4: s(i) =⟨

1κ

( 1√N

D−1H y− d(i−1))⟩

5: x(i) = FHs(i)

6: d(i) = F(g(x(i); θNL)− κx(i))

7: end for

iterations, κ is the optimal linear gain for the limiter, given by (1.4.2), and 〈·〉denotes element-wise hard detecting function.

Note that Algorithm 1.4.1 consists of two parts. First, the impact of the mul-tipath fading channel is eliminated and then the original signal is calculated fromits distorted version. As mentioned earlier, this algorithm assumes that the channelis perfectly known a priori while in practical situations, channel estimation is re-quired. Moreover, the nonlinearity parameters is also assumed to be known in thisalgorithm. This assumption is either not valid due to several reasons discussed later,and estimation methods should be devised to obtain these parameters. These twoproblems are the main subject of this thesis. It is worth mentioning that, later, Al-gorithm 1.4.1 has been extended to space-division multiple access (SDMA)-OFDMsystems [GLC06,GWLC07].

In the following, some background on parameter estimation methods and chan-nel estimation techniques in OFDM systems will be given.

1.5 Parameter Estimation

The content of this section is mostly from [CB02, Kay93]. Parameter estimationis the procedure of estimating the parameters of a population (distribution) usinga random sample. The parameter estimation methods split into two approachesdepending on the nature of the parameters to be estimated. If the parameters arerandom, the Bayesian estimation techniques are used. In this case, the probabilitydistribution to which the parameters belong should be known a priori, called theprior distribution. Otherwise, if the parameters are non-random (deterministic),frequentist (classical) inference or estimation techniques are useful. When applyingto communication systems, the classical approach is more desirable, since the apriori information is rarely available. Therefore, here, we only consider the classicalapproach.

18 Introduction

Mathematically, consider a random sample of size M , expressed as the vectory = [y0, . . . , yM ]T from the distribution with probability density function (PDF)p(y; θ), parameterized by θ, then the estimator of θ is a function of the sample,denoted by θ = τ(y) [CB02]. Indeed, any function of the sample is an estimatorof the parameter. However, one should find a good estimator which is optimal insome sense. The method of ML estimation is the most well-known technique to findestimators, which is defined as the maximiser of the likelihood function, when yheld fixed, as [CB02]

θ = arg maxθ∈Θ

p(y; θ), (1.5.1)

where Θ is the parameter space.Now that we could find an estimator, it is needed to somehow evaluate its perfor-

mance. The general topic of evaluating statistical procedures is part of the branchof statistics known as decision theory [CB02]. The mean square error (MSE) is anice criterion to evaluate an estimator. One class of estimators aiming to minimizethe MSE is the class of unbiased estimators.

1.5.1 Mean Square ErrorMSE can give good information about the performance of an estimator. For ageneral estimator, τ , it is defined as

MSEτ = E(τ(y)− θ)2. (1.5.2)

The beauty of MSE is that besides its tractability, it gives the following nice inter-pretation

MSEτ = varτ+ biasτ, (1.5.3)

where biasτ = Eτ− θ gives the difference between θ and the expected value ofτ . If biasτ = 0, i.e., Eτ = θ, the estimator τ is called unbiased.

Indeed, the MSE combine two important elements, one measuring the variabilityof the estimator (precision) and the other measuring the accuracy by its bias [CB02].An estimator with good MSE properties should have small combined variance andbias.

1.5.2 Unbiased EstimatorsIn this section, unbiased estimators are considered, and searching for good estima-tors is performed by trying to minimize the MSE (or equivalently variance) of theunbiased estimators.

An estimator τ is a best unbiased estimator of θ if it satisfies τ = θ for all θ ∈ Θand, for any other estimator µ with µ = θ, we have varτ ≤ varµ, ∀θ ∈ Θ. τ isalso called a uniform minimum variance unbiased estimator (UMVUE) of θ [CB02].

A best unbiased estimator does not always exist, and even it exists finding itcan be very difficult or even impossible. For this reason, we try to find a lower

1.5. Parameter Estimation 19

bound on the variance of any unbiased estimator. Then, if we are lucky to find anunbiased estimator with its variance equal to this lower bound, we have found abest unbiased estimator. This is why the Cramér-Rao lower bound comes into thepicture.

1.5.3 Cramér-Rao Lower BoundThe Cramér-Rao lower bound (CRLB) gives a lower bound on the variance of anyunbiased estimator [CB02,Kay93]. In this section, first the CRLB for a real-valuedscaler parameter is discussed. Then, since frequently there are a number of para-meters to be estimated, the CRLB for a real-valued parameter vector is expressed.Occasionally, it happens that the parameterizing vector is complex-valued or con-tains the combination of real-valued and complex-valued parameters. In this case,an extension of the results for CRLBs for real-valued parameters is needed. Thiswill be discussed next. Finally, we consider cases in which the parameters or someof them are bounded (or constrained), or the parameter space contains regions, inwhich the regularity conditions needed to derive CRLB does not hold.

Cramér-Rao lower bound (real-valued scalar parameter)

The CRLB for a real-valued scaler parameter is expressed as Theorem 1.5.1.

Theorem 1.5.1 (Cramér-Rao inequality [CB02]). Let y = [y0, . . . , yM ]T be a sam-ple with PDF p(y; θ), and let τ(y) with Eτ(y) = α(θ) be any estimator satisfyingthe regularity conditions

d

dθEτ(y) =

∫Y

∂

∂θ(τ(y)p(y; θ))dy, (1.5.4)

where Y is the sample space, and varτ(y) <∞, then

varτ(y) ≥(ddθα(θ)

)2E(

∂∂θ log p(y; θ)

)2 . (1.5.5)

The denominator, E(

∂∂θ log p(y; θ)

)2, is called the information number, orFisher information. Lemma 1.5.1 gives an equivalent quantity to the informationnumber [CB02].

Lemma 1.5.1 (Information number equivalence [CB02]). If p(y; θ) satisfies

d

dθE∂

∂θlog p(y; θ)

=∫Y

∂

∂θ

[(∂

∂θlog p(y; θ)

)p(y; θ)

]dy, (1.5.6)

then

E

(∂

∂θlog p(y; θ)

)2

= −E∂2

∂θ2 log p(y; θ). (1.5.7)

20 Introduction

Note that (1.5.6) is true for exponential families such as Gaussian. Lemma 1.5.1can ease the computations.

Cramér-Rao lower bound (real-valued parameter vector)

Occasionally, we face a set of real-valued parameters θ1, . . . , θK from a population.By putting them in a vector, we have a real-valued parameter vector, denoted asθ = [θ1, . . . , θK ]T, and the parameterized PDF is denoted by p(y; θ). In this case,the CRLB for each parameter is given by Theorem 1.5.2.

Theorem 1.5.2 (CRLB for real-valued parameter vector [Kay93]). Lety = [y0, . . . , yM ]T be a sample with PDF p(y; θ), and let τ(y) with Eτ(y) = α(θ)be any estimator satisfying the regularity conditions

E∂ log p(y; θ)

∂θ

= 0K , ∀θ ∈ Θ, (1.5.8)

then, the covariance matrix Cτ of τ(y) satisfies

Cτ −∂α(θ)∂θ

J−1(θ)∂α(θ)T

∂θ 0K×K , (1.5.9)

where 0K×K denotes the matrix is non-negative semidefinite, and ∂α(θ)/∂θis the M × K Jacobian matrix. The matrix J(θ) is called the Fisher informationmatrix (FIM), and for 1 ≤ i, j ≤ K given by

[J(θ)]ij = −E∂2 log p(y; θ)

∂θi∂θj

= E

∂ log p(y; θ)

∂θi

∂ log p(y; θ)∂θj

(1.5.10)

In particular, when α(θ) = θ, then Cτ J−1(θ), or equivalently for 1 ≤ i ≤ K,

varθi = [Cτ ]ii ≥ [J−1(θ)]ii. (1.5.11)

Cramér-Rao lower bound (mixed real-complex-valued parametervector)

When facing with such problems, the traditional approach used to be forming areal-valued parameter vector consisting of the real and imaginary parts of the ori-ginal complex-valued parameter vector, and then using the results given in Theo-rem 1.5.2. However, in this way the computations are tedious and messy. In [NK92],an attractive direct approach using the original mixed real-complex-valued para-meter vector is proposed, whose main results are given here. Interested readers arereferred to [NK92] for comprehensive details, proofs, and examples.

The mixed real-complex-valued parameter vector is denoted by

θ =[

θ1

θ2

]∈ Cp1 × Rp2 , (1.5.12)

1.5. Parameter Estimation 21

where θ1 and θ2 denote the p1 × 1 complex-valued and the p2 × 1 real-valuedparameters, respectively. Theorem 1.5.3 gives the CRLB for this case

Theorem 1.5.3 (CRLB for mixed real-complex-valued parameter vector [NK92]).Let y = [y0, . . . , yM ]T be a sample with PDF p(y; θ), where θ in given by (1.5.12),and let τ(y) with Eτ(y) = α(θ) be any estimator satisfying the usual regularityconditions, then the covariance matrix Cτ of τ(y) satisfies

Cτ −∂α(θ)∂θ

Υ−1(θ)∂α(θ)H

∂θ 0K×K , (1.5.13)

where (·)H denotes the Hermitian, and Υ(θ) denotes the complex FIM, given by

Υ(θ) = E∂ log p(y; θ)

∂θ∗∂ log p(y; θ)H

∂θ∗

, (1.5.14)

or equivalently for 1 ≤ i, j ≤ 2p1 + p2,

[Υ(θ)]ij = −E∂2 log p(y; θ)∂θ∗i ∂θj

. (1.5.15)

Particulary, when α(θ) = θ, then Cτ Υ−1(θ).

Moreover, Lemma 1.5.2 gives a nice structure for the complex FIM and complexCRLB, which comes in handy from the computational point of view.

Lemma 1.5.2 (Simplified CRLB for mixed real-complex-valued parameter vector[NK92]). The complex FIM can be partitioned as

Υ(θ) =

C1 C∗2 PC2 C∗1 P∗

PH Pᵀ Q

, (1.5.16)

where

C1 = E

∂ log p(y; θ)

∂θ∗1

∂ log p(y; θ)∂θ∗1

H, (1.5.17)

C2 = E∂ log p(y; θ)

∂θ1

∂ log p(y; θ)∂θ1

ᵀ, (1.5.18)

P = E∂ log p(y; θ)

∂θ∗1

∂ log p(y; θ)∂θ2

ᵀ, (1.5.19)

Q = E∂ log p(y; θ)

∂θ2

∂ log p(y; θ)∂θ2

ᵀ, (1.5.20)

22 Introduction

then, for the particular case when α(θ) = θ,

CRLB(θ2) =[Q− 2<

PH∆P + PᵀΛP

]−1, (1.5.21)

CRLB(θ1) = ∆ + (∆P + ∆∗P∗) CRLB(θ2)(PH∆H + Pᵀ∆ᵀ

), (1.5.22)

where

∆ =(C1 −C∗2C∗1

−1C2

)−1, (1.5.23)

Λ = −C∗1−1C2C1

−1. (1.5.24)

Cramér-Rao lower bound (constrained real-valued parameters)

Finally, in this section, the problem of calculating the CRLB for constrained (boun-ded) parameters is discussed. Occasionally, in some applications, we encounter pa-rameters belonging to a parameter space which contains regions (or points) in whichthe regularity condition required to derive the CRLB does not hold. However, asshown by [GH90], the constrained CRLB is identical to the unconstrained CRLBat the regular points. In particular, we have Lemma 1.5.3.

Lemma 1.5.3 (CRLB for constrained real-valued parameters [GH90]). When theconstraint set Θc can be expressed as a smooth functional inequality constraint,i.e., Θc = θ ∈ RK : ξ(θ) ≤ 0Q, where ξ = [ξ1, . . . , ξQ]T is a continuouslydifferentiable vector function on ξ : RK → RQ, then at the regular points of Θc,i.e. where no equality constraints are active, the constrained CRLB is identicalto the unconstrained CRLB. However, at the points of active equality constraints,the full-rank FIM in the unconstrained CRLB must be replaced by a rank-reducedFIM obtained by projecting the full-rank FIM onto the tangent hyperplane of theconstraint set at θ.

1.5.4 Consistency and Efficiency of EstimatorsIn this section, the asymptotic behaviour of estimators, i.e, when the size of therandom sample tends to infinity, is discussed. In practice, this asymptotic beha-viour can be very insightful. Two important asymptotic behaviour properties areconsistency and efficiency of estimators.

To be able to talk about the properties of consistency and efficiency of anestimator, we first need to introduce the concept of a sequence of estimators. Assumethat we observe a random sample y1, y2, ... from a distribution with PDF p(y; θ)with an increasing sample size, i.e., the sample size approaches infinity, and for eachsample size we have an estimator for θ as

τn = τ(y1, . . . , yn), n = 1, 2, . . . , (1.5.25)

therefore we have a sequence of estimators.

1.5. Parameter Estimation 23

Consistency

The sequence of estimators, given by (1.5.25) is consistent if [CB02]

limn→∞

P|τn − θ| < ε, ∀ε > 0, θ ∈ Θ, (1.5.26)

where P(·) denotes the probability operator with respect to the PDF p(y; θ). In otherwords, (1.5.26) says that as the sample size increases, the consistent estimator getscloser and closer to the true value of θ. Theorem 1.5.4 gives sufficient conditionsfor an estimator to be consistent.

Theorem 1.5.4 (Consistency [CB02]). If τn, a sequence of estimators of a para-meter θ, for all θ ∈ Θ, satisfies

1. limn→∞ varτn = 0,

2. limn→∞ biasτn = 0,

then τn is consistent.

One of the important and nice features of ML estimators is that they are con-sistent under some regularity conditions [CB02].

Efficiency

As the consistency property gives useful information about the accuracy of theestimator under consideration, the property of efficiency expresses the asymptoticprecision or the asymptotic variance. Without going into the details, the definitionof efficiency is given here.

The sequence of estimators, given by (1.5.25) is asymptotically efficient for aparameter α(θ) if [CB02],

√n[τn − α(θ)] dist.−−−→ N (0, υ(θ)), as n→∞ (1.5.27)

where dist.−−−→ denotes the convergence in distribution, and

υ(θ) =(ddθα(θ)

)2E(

∂∂θ log p(y; θ)

)2 , (1.5.28)

which states that the asymptotic variance of τn achieves the CRLB.In addition to being consistent, another important and attractive feature of ML

estimators is that they are efficient under some regularity conditions [CB02].

24 Introduction

1.6 Linear Channel Estimation in OFDM Systems

In this section, linear channel estimation methods for OFDM systems are reviewed.By linear, we simply mean that the effect of nonlinearity is neglected, which ofcourse is unrealistic. As it mentioned in Section 1.5, from the statistical point ofview, channel estimation methods, as examples of parameter estimation, are gene-rally divided into the two categories of classical and Bayesian estimation approa-ches. While the Bayesian channel estimation methods are useful for performanceevaluations and simulations, they are rarely used in practical algorithm implemen-tations since they impose a prior knowledge on the wireless channels, which is notavailable. Therefore, the focus in this section is merely on classical channel esti-mation methods, in which the channel is assumed to be a deterministic unknownparameter.

From the perspective of communication engineering, the OFDM channel esti-mation methods are divided into three classes as blind [HFBC00, SHP07], semi-blind [LMAMB17], and pilot-aided channel estimation methods. We also concen-trate on the pilot aided techniques due to their massive use in modern OFDM-basedcommunication systems [LTH+14].

The pilot-aided OFDM channel estimation refers to techniques in which thechannel is estimated using pilots, which are known to both the transmitter andreceiver a priori. These techniques can primarily be divided into using time-domainpilots and frequency-domain pilots for channel estimation. There are several worksin the literature using time-domain pilots for OFDM channel estimation [YL99,LCH12a,LCH12b]. These works have mostly used pseudo-noise (PN) sequences aspilots due to their attractive cross-correlation properties [SP80]. The most popularone is time-domain synchronous (TDS)-OFDM which has been used by the Chinesedigital terrestrial multimedia broadcasting (DTMB) standard [Chi06].

The techniques using frequency-domain pilots are mainly split into two classesdepending on the arrangement of the pilots as

• Block-type pilots: In this type, all the subcarriers in an OFDM symbolare dedicated to pilots. This OFDM symbol is resent periodically over time,depending on the rate of channel variations. Block-type pilot arrangementsare usually used in the systems, in which the channel is static over severalOFDM symbols.

• Comb-type pilots: In this type, the pilot subcarriers are often inserted inevery OFDM symbol. This arrangement is more suitable for systems, in whichthe rate of channel variations is fast.

The concept of block-type and comb-type pilots is illustrated in Figure 1.4. In thefollowing, channel estimation methods using block-type and comb-type pilots arebriefly described. For a comprehensive survey on OFDM channel estimation met-hods, see [OA07]. Later, these methods are combined with nonlinearity parameterestimation, and iterative symbol detection.

1.6. Linear Channel Estimation in OFDM Systems 25

frequency

time

pilotdata

(a) The block-type pilot arrangement.

frequency

time

(b) The comb-type pilot arrangement.

Figure 1.4: The block-type and comb-type pilot arrangements.

Channel estimation using block-type pilots

Consider the system model depicted in Figure 1.3, when there is no nonlinearity,i.e. ideally g(x; θNL) = x. In this case, we have a linear OFDM system. Therefore,the received signal after the DFT at the receiver, given by (1.3.5) can be written as

y =√NDHs + w =

√Ndiag(s)Fh + w. (1.6.1)

Using frequency-domain block-type training pilots, the problem of ML estimationof the channel can be formulated by least-squares (LS) as

minh

∥∥y−√Ndiag(s)Fh∥∥2

2, (1.6.2)

where ‖ · ‖2 denotes the Euclidean norm. Since (1.6.2) is an unconstrained convexoptimization problem, its solution can be calculated by taking the derivative of thecost function with respect to h and setting it to zero. Therefore, the channel LSestimation can be derived as

hLS = 1√N

(FHdiag(s)Hdiag(s)F

)−1FHdiag(s)Hy. (1.6.3)

Note that in derivation of (1.6.3), it is assumed that the channel length is known,which improves the accuracy of the estimate by performing noise reduction.

Channel Estimation using Comb-type Pilots

Now, consider that just a set of subcarriers in each OFDM symbol are dedicatedto pilots. More specifically, Np pilots are inserted in s in such a way that the totalN subcarriers are divided into Np groups with length Lp = N/Np, and in each,

26 Introduction

the first subcarrier is allocated by a pilot. Therefore, for the k-th subcarrier, thefrequency-domain input signal can be written as

s[k] = s[k1Lp + k2], k1 = 0, . . . , Np − 1

=

sp[k1], k2 = 0sd[k1(Lp − 1) + k2 − 1], k2 = 1, . . . , Lp − 1

(1.6.4)

where sp and sd denote the pilots and data tones, respectively. We can construct apermutation matrix P, such that

s = P(

sp

sd

), (1.6.5)

where PPT = PTP = IN , and IN is the N ×N identity matrix.To perform the channel estimation, the receiver can estimate the channel by

separating the observations for pilot and data subcarriers as(yp

yd

)= Pᵀy, (1.6.6)

Using LS, the channel estimates at pilot subcarriers are obtained as

ˆhp = 1√N

yp sp, (1.6.7)

where denotes the Hadamard (element-wise) division. To find the channel esti-mates at data subcarriers, we use the DFT interpolation technique, by which theestimated channel can be calculated as [OA07]

h = N

NpFH

pˆhp, (1.6.8)

where Fp is a semi-unitary matrix formed by the first L + 1 columns of a unitaryDFT matrix with the rows corresponding to the subcarrier indexes of the pilottones. Note that one can also use other interpolation methods, e.g., spline or low-pass filter [OA07]. However, the DFT interpolation technique not only gives lowerestimation MSE, but is more computationally efficient, and can be most efficientlyimplemented using the FFT. Furthermore, it has been shown that the DFT inter-polation technique gives much better performance than linear, second order, cubicand low-pass interpolations [RMdS07].

2

Background and Contributions ofthe Thesis

In this chapter, we first describe the problem formulation and the motivation behindit. Then, the previous works done in the literature will be reviewed. Finally, themain contributions of the thesis will be presented.

2.1 Background

OFDM is one of the most popular multi-carrier transmission methods that has beenrecognized as a highly effective technique for high-speed wireless communications.It enjoys having immunity to multipath fading channels, having maximal spectralefficiency, and having the capability of being implemented on low-cost embeddeddevices. Due to these attractive features, OFDM has been adopted in several techno-logies and standards such as 4G (LTE and its evolutions so far) and IEEE 802.11(WiFi). More recently, OFDM using cyclic prefix (CP-OFDM) which is the mainfocus of this thesis, has been selected in the race to 5G NR [3GP18] over otherrivals [ZBT+16].

Despite its impressive characteristics, OFDM suffers from a high PAPR. Oneimmediate solution to this problem is designing and deploying PAs with high dy-namic range. However, manufacturing such PAs is expensive making their use inlow-cost devices economically infeasible. It will also be more problematic in futurewireless networks, particularly in case of applying OFDM to mmWave spectrumin IoT-based 5G NR, since manufacturing efficient PAs is much more difficult andcostly at those frequencies [ABC+14]. The other way to overcome the problem ofOFDM’s high PAPR is forcing the PA to work in its linear region by introducinga high input IBO. However, this not only leads to low signal-to-noise-ratio (SNR)at the receiver side, but reduces the PA energy efficiency, and hence results inincreased energy consumption. This situation prevents the use of OFDM in IoT-based communication networks such as MTC [ADF+13], which presumably consistof low-power devices working on years-long button batteries, or for vehicle to ever-

27

28 Background and Contributions of the Thesis

ything (V2X) services [CHS+17]. The third approach which is also the underlyingassumption of this thesis, is to let the PA’s operating point be close to its saturationlevel and the high PAPR OFDM signal get distorted because of the nonlinearityintroduced by the PA. This way allows the PA to be more efficient, and hence lesspower consuming which makes it usable in low-cost and low-power devices. Nevert-heless, we need to devise solutions to deal with the distorted transmitted signalsince it causes inter-subcarrier interference, and hence degrades the performance ofthe system.

As mentioned in Section 1.2.2, there are many works in the literature that haveproposed solutions to deal with the PA nonlinearity distortion at the transmitterside. However, these approaches suffer from increased complexity and cost at thetransmitter. The other approach which is considered here is to leave the distortionat the transmitter side and compensate it at the receiver side. It is a particularlyattractive solution in the uplink transmission, as we can transfer all the complexityand cost to the receiver side which is a high-quality receiver such as a base stationand make the transmitter as simple and cheap as possible. To this regard, the firstwork was introduced in [THC03], which is an iterative symbol detection algorithmthat aims to estimate the distortion and remove it from the received symbols in eachiteration at the receiver side. However, in [THC03], it is assumed that the channeland nonlinearity are known a priori, which is not the case in practical situations,especially in IoT-based MTC. In particular, the receiver needs to estimate (and re-estimate) the nonlinearity parameters especially in IoT-based MTC because thereare a lot of sensors that need to send some information occasionally. Then, at eachtime of transmission, the receiver (e.g. a fusion center or a base station) needs toestimate the nonlinear parameters to obtain an updated value to be deployed inthe detection stage. Those nonlinear parameters for different nodes may changeduring time because of environmental conditions such as temperature, humidity,and unstable power supply. Furthermore, for low-cost devices, the manufacturingvariations from unit to unit may be fairly large and it would be too expensiveto measure and calibrate each unit separately. Moreover, electronic devices andcomponents suffer from process of aging, so even if the nonlinearity parameterswere known at the beginning, it would not later. Therefore, these nonlinearityparameters will become unknown to the transmitter over time and hence there isan essential need to estimate them at the receiver side occasionally.

There are several works in the literature that aim to integrate channel ornonlinearity parameters estimation with the iterative symbol detection algorithm[DF11,DF13,DF14,GWCL06,GWCW07,GWC+11]. In [DF11] the ML estimationof the parameters of the SSPA has been proposed, when the channel is additivewhite Gaussian noise (AWGN), i.e., the multi-path fading channel is neglected.In [DF13,DF14], the authors consider presence of the multi-path fading channel,and proposed the ML estimation of the nonlinearly parameters, in particular theTWTA model. Nevertheless, they assumed that the channel is known a priori.In [GWCL06], assuming that the nonlinearity parameters are known a priori, acombination of frequency-domain and time-domain channel estimation is used to

2.2. Contributions of the Thesis 29

enhance the channel estimation performance. In [GWCW07,GWC+11], the authorshave used the iterative detection algorithm proposed in [THC03] for the broadbandnonlinear PAs, e.g., PAs with memory. In particular, they have considered a Ham-merstein model in [GWCW07], and the Wiener-Hammerstein model in [GWC+11]for the nonlinear PA, and defined an effective channel by convolving the Hammer-stein’s linear filter (resp., Wiener-Hammerstein’s linear filters) and the channel.Then, they have proposed a technique to separately estimate the effective channeland the nonlinearity parameters. To estimate the channel, they have considered alow PAPR OFDM symbol, designed by activating just a set of subcarriers. For thenonlinearity parameters estimation part, they have considered a polynomial model,given by (1.2.12), and dedicated another OFDM symbol, designed in a way to haveuniformly distributed amplitudes in time-domain. Then, using LS the nonlinearityparameters can be estimated. For the subsequent OFDM symbols, they have used atracking method to update the values of the effective channel and the nonlinearityparameters.

In the following, the contributions of this thesis is presented.

2.2 Contributions of the Thesis

The work in this thesis is mainly focused on proposing estimation methods fornonlinearity parameters along with channel estimation and symbol detection atthe receiver in OFDM communication systems. Of special interest is the clipper(limiter) nonlinearity. The importance of this model comes from the fact that itcan be used in different scenarios. As mentioned in Section 1.2.3, one way to reducethe PAPR at the transmitter side is the intentional clipping of the OFDM signal.To model this clipping, a limiter is used in the literature. Moreover, a limiter itselfcan be considered as a simplified yet useful model of nonlinear PAs. Furthermore,in Section 1.2.3, we saw that even if a pre-distorter is used at the transmitter sidethe cascaded combination of that and the PA often well approximated as a clipper.Considering low-cost devices in IoT-based networks, clipping the high PAPR signalby a limiter can be a very promising approach to reduce the cost of having highlyefficient PAs. Furthermore, clipping the high PAPR signal increases the batterylife, thanks to the resulting increased power efficiency of the PA.

2.2.1 Problem I: Algorithms to estimate the nonlinearityparameter of a limiter (along with channel estimation)

The contribution of this section belongs to Papers A, B, and C.

Paper A: Estimation of the Clipping Level in OFDM Systems [OB15]

In this paper, the problem of estimating the clipping level of the limiter model atthe receiver is considered. In particular, a blind method and a pilot-based method

30 Background and Contributions of the Thesis

are proposed. For the blind estimator, the probability distribution of the time-domain received signal at the receiver is computed to find the ML estimate of theclipping level. On the other hand, for the pilot-based method, the pilot symbols aredesigned in a way to fully exploit the nonlinearity characteristic. By this design,the maximum possible number of samples can be used estimate the clipping level.The main contributions of this work is as follows:

• Calculating the exact PDF of the output of a limiter, when the input sam-ples are independent and identically distributed (iid) zero-mean circularlysymmetric complex Gaussian random variables,

• Calculating the CRLB of the clipping amplitude in an OFDM system.

System model

The system model here is depicted in Figure 1.3 with the additional assumption ofneglecting the multi-path fading channel, i.e., h[n] = δ[n], therefore there is just anAWGN channel in the system. The nonlinearity function is a limiter parameterizedby the clipping amplitude, i.e.,

g(u;A) =u, |u| ≤ AAej arg(u), |u| > A.

(2.2.1)

where A is the clipping amplitude.

The estimation of the clipping amplitude

To estimate the clipping amplitude, a blind method and and a method using pilotsare proposed.

ML estimation

We have used that if the number of subcarriers is sufficiently high the time-domainsignal is well approximated by a zero mean circularly symmetric complex Gaussianrandom vector. Then, the joint PDF of the real part and imaginary part of outputof the limiter is derived analytically. Since the output signal of the nonlinearityand the noise are independent stochastic signals, the joint PDF of the real andimaginary parts of their addition is equal to the two-dimensional convolution oftheir joint PDFs. Then this PDF is maximized over the nonlinearity parameter Aby taking the derivative with respect to A and setting it to zero.

Moreover, the CRLB is derived as a benchmark for the ML estimator, and itis shown by simulation that the normalized mean square error (NMSE) of the MLestimator achieves the CRLB very fast.

2.2. Contributions of the Thesis 31

Estimation using pilots

The ML estimator has a high computational complexity. In fact, in practice, a less-complex estimation method using pilots is more desired. To do so, a training OFDMsymbol is designed to satisfy the transmitting power constraint as well as havingthe maximum possible number of clipping amplitude hits, therefore the receiver canexploit the possible highest number of training samples to perform the estimation.

Paper B: Joint Channel and Clipping Level Estimation for OFDM inIoT-based Networks [OB17]

In this paper, we propose algorithms to jointly estimate channel and clipping am-plitude, when a limiter (clipper) is deployed as the nonlinearity model. The clippingamplitude is assumed to be unknown to the transmitter and the receiver a priori,and the receiver uses an alternating optimization algorithm to jointly estimate thechannel taps and the clipping amplitude. Once the clipping amplitude and the chan-nel has been estimated, the receiver uses them to detect the transmitted symbolsby deploying the iterative detection method proposed in [THC03].

The main contributions of this work is as follows:

• Proposing a low-complexity optimization algorithm to efficiently estimate theclipping amplitude. Note that outside the context of wireless communications,this algorithm also can be used in the fields of automatic control, and systemverification [Bai02].

• In contrast to other previous works [GWCW07,GWC+11], in this work theclipping amplitude is unknown to the transmitter too. This assumption isimportant in practice since the clipping amplitude as a parameter created byelectronic circuits is exposed to changes over time. Due to this phenomenon,one can not design pilots that are surely below the saturation point of thenonlinear PA.

• Proposing an iterative optimization algorithm to jointly estimate channel andclipping level using block-type pilots. In contrast to the works [GWCW07,GWC+11], this algorithm just uses one OFDM symbol. Moreover, the pilotscan be arbitrarily chosen, and there is not any need to design them.

• In contrast to other previous works, in the estimation procedure, no approx-imation such as the assumption of Gaussianity of the time-domain OFDMsample, or using the Bussgang theorem has been considered.

• Calculating the CRLB for the channel and clipping level, which is the appli-cation of the combination of the concept of CRLBs for mixed-real-complex-valued and constrained parameters as explained in Section 1.5.3.

32 Background and Contributions of the Thesis

System model

The system model is explained in Section 1.3, and is depicted in Figure 1.3. Thesystem uses block-type pilots, therefore the whole vector s is known to the trans-mitter as well as the receiver. The nonlinearity function is a limiter parameterizedby the clipping amplitude, given by (2.2.1).

Working directly with the output of the limiter is mathematically difficult. The-refore, we invoke the technique described in [DG00], where the output of the limiteris represented in a linear fashion by introducing N augmented binary variables cnfor n ∈ 0, . . . , N − 1 indicating whether the sample at time n has been clipped(cn = 1) or not (cn = 0), i.e.

cn =

1 rn > A

0 rn ≤ A,(2.2.2)

which leads us to

zn = (1− cn)xn +Acnejφn , n = 0, . . . , N − 1, (2.2.3)

where xn = rn exp(jφn), and in vector form as:

z = (1N − c) x +Ac ejφ, (2.2.4)

where 1N denotes an all-one vector, ejφ = [ejφ0 , . . . , ejφN−1 ]T denotes the phasevector of x, and denotes the Hadamard product. Also, note that x = r ejφ.

Joint Channel and Clipping Amplitude Estimation

In this section an alternating optimization algorithm is proposed to jointly estimatechannel and clipping amplitude. This joint estimation algorithm can be used oncein a while, when there is a need to have the updated estimated value of the clippinglevel. Otherwise, having the estimate of the clipping amplitude the channel canbe estimated using conventional channel estimation methods. The intuition behindthe proposed algorithm is that clipping level is a slowly time-varying parameter,indeed much slower in comparison to channel variations. Moreover, wireless channelsare usually slowly time varying, therefore the block-fading channel model, whichremains the same during the transmission of several OFDM blocks is reasonable.This algorithm is using the first OFDM symbol, entirely dedicated to pilots, of ablock of symbols during which the channel is static.

First, an optimization algorithm is proposed to estimate the clipping level, whenthe channel is fully known as Algorithm 2.2.1. This algorithm minimizes the costfunction, given by

J(A) = ‖y−√NDHFz‖22, (2.2.5)

over the clipping level A, when the channel is given. The cost function (2.2.5) isnon-convex and non-smooth. The trick used in solving this optimization problem,

2.2. Contributions of the Thesis 33

and finding its global minimizer is that if we sort the magnitudes of x, i.e., r in anascending manner, and denote the sorted version by r, in each region (rk−1, rk), k =0 . . . N − 1, the cost function is a convex quadratic function of A. Therefore, overeach interval, the minimizer can be found by taking the first derivative of (2.2.5)with respect to A and setting it to zero, then checking if the solution lies in thatinterval. If it lies in that interval, the minimizer has already been found, otherwisethe minimum takes place at the interval’s corner-point with the smaller value of thecost function. To find the global minimizer, we find the minimizer for each interval,and then find the global minimizer among those minimizers.

Algorithm 2.2.1 Estimation of clipping amplitude (A).1: Inputs:

y, r and exp(jφ)F and DHck for k = 0, . . . , N − 1

2: Initialize:[r,Ps] = sort(r)ejφ = Pse

jφ

B =√NDHFPᵀ

sr−1 = 0

3: for k = 0 to N − 1 do

4: Ak =<

[y−B(x¯ck)]HB(ckejφ)

‖B(ckejφ)‖22

5: if rk−1 ≤ Ak ≤ rk then6: Ak = Ak7: else8: Ak = rk9: end if

10: Jk = Jk(Ak)11: end for12: k = arg min

0≤k≤N−1Jk

13: A = Ak

In Algorithm 2.2.1, x = Psx, c = Psc, φ = Psφ, and Jk(A) denotes the costfunction in interval rk−1 ≤ A ≤ rk, k = 0 . . . N − 1. Moreover, ck is an all-onevector except its first k elements which are 0, and ¯ck = 1N − ck. Then, an iterativeoptimization algorithm with two different ways of initializations is proposed forjointly channel and clipping amplitude estimation as Algorithm 2.2.2, and Algo-rithm 2.2.3. In Algorithm 2.2.2 the initializing channel estimate is given by (1.6.3),in which the effect of the nonlinearity is neglected. In parallel, in Algorithm 2.2.3

34 Background and Contributions of the Thesis

the initializing clipping amplitude estimate is calculated by solving the optimiza-tion problem for h, and substituting the solution into the cost function, then usinga grid search method.

Algorithm 2.2.2 Alternating optimization with initializing the channel.1: Inputs:

y, r and exp(jφ)Fck for k = 0, . . . , N − 1

2: Initialize:[r,Ps] = sort(r)ejφ = Pse

jφ

D(0)H = diag(Fh(0)), h(0) is given by (1.6.3)

r−1 = 0i = 1

3: while (convergence criteria not met) do

4: B =√ND(i−1)

H FPᵀs

5: calculate A(i−1) and k(i−1) using Algorithm 2.2.16: z = x (1N − ck(i−1)) + A(i−1)ejφ ck(i−1)

7: V = diag(FPᵀs z)F

8: h(i) = (VHV)†VHy9: D(i)

H = diag(Fh(i))10: i = i+ 111: end while

2.2. Contributions of the Thesis 35

Algorithm 2.2.3 Alternating optimization with initializing the clipping level.1: Inputs:

y, r and exp(jφ)Fck for k = 0, . . . , N − 1

2: Initialize:[r,Ps] = sort(r)ejφ = Pse

jφ

A(0) using grid searchr−1 = 0i = 1

3: while (convergence criteria not met) do

4: V = diag(FPᵀ

s z(i−1))F5: h(i−1) = (VHV)†VHy6: D(i−1)

H = diag(Fh(i−1))7: B = D(i−1)

H FPᵀs

8: calculate A(i) and k(i) using Algorithm 2.2.19: i = i+ 1

10: end while

From the computational complexity point of view, both Algorithms 2.2.2 and2.2.3 have the same order of complexity per iteration, which is dominated by Al-gorithm 2.2.1 which has complexity of order O(N2 log2N). Therefore, both Algo-rithms 2.2.2 and 2.2.3 have the same worse case asymptotic order of complexity.

Cramér-Rao lower bound

Using the results explained in Section 1.5.3, the CRLB for A and h can be calculatedas

CRLB(A) = 1q − 2pHC−1

1 p, (2.2.6)

andCRLB(h) = C−1

1 + C−11 ppHC−H

1 CRLB(A), (2.2.7)

where

C1 = N

σ2 FHdiag(Fz)Hdiag(Fz)F, (2.2.8)

p = N

σ2 FHdiag(Fz)HDHF(c ejφ), (2.2.9)

q = 2Nσ2

∥∥DHF(c ejφ)∥∥2

2. (2.2.10)

36 Background and Contributions of the Thesis

Simulation results

In simulations, the subcarrier modulation of 16QAM is chosen. The Rayleigh fadingchannel model is the exponential channel model (IEEE 802.11b model) [SHW00],which has L + 1 complex Gaussian distributed taps hl having a mean power ofσ2l = E|hl|2 = σ2

0e−Ts/τrmsl for l = 0, . . . , L in which σ0 is chosen such that∑L

l=0 σ2l = 1, Ts = 50ns is the sampling rate, and τrms = 30ns and 150ns are the

RMS delays leading to channel lengths L + 1 = 7 and 31, respectively. Moreover,the channel model follows the slow-time varying condition, in which the channel isfixed during several OFDM blocks. Algorithm 1.4.1 is used for symbol detection,but with the estimated channel and clipping amplitude.

0 5 10 15 20 25 30 35 4010

-6

10-5

10-4

10-3

10-2

10-1

100

NMSE

CRLB(A)

Initialized by h(0) (Alg. 3)Initialized by A(0) (Alg. 4)

σ−2 [dB]

Figure 2.1: NMSE performance of the clipping amplitude estimation using Algo-rithms 2.2.2 and 2.2.3, when L+ 1 = 7, N = 128 and CL = 1 dB.

Figures 2.1 and 2.2 show the NMSE performance of estimating clipping am-plitude (A) and channel taps (h), when L + 1 = 7, N = 128 and CL = 1 dB,respectively. As observed from the figures the NMSEs are decreasing by increasingthe SNR and both of them reach CRLB in a medium SNR regime. Figure 2.3shows the symbol error rate (SER) performance of Algorithms 2.2.2 and 2.2.3, inte-grated with the iterative detection method (Algorithm 1.4.1), when L+ 1 = 7, andCL = 1 dB for N = 128. Two maximum number of iterations for Algorithm 1.4.1as Nq = 2 and 3 are used. As illustrated, the SER performance of different schemeswhen clipping level and channel are estimated using the proposed algorithms almost

2.2. Contributions of the Thesis 37

0 5 10 15 20 25 30 35 4010

-5

10-4

10-3

10-2

10-1

100

NMSE

CRLB(h)

Initialized by h(0) (Alg. 3)Initialized by A(0) (Alg. 4)

σ−2 [dB]

Figure 2.2: NMSE performance of the channel estimation using Algorithms 2.2.2and 2.2.3, when L+ 1 = 7, N = 128 and CL = 1 dB.

0 5 10 15 20 25 30 35 4010

-5

10-4

10-3

10-2

10-1

100

SER

Initialized by h(0) (Alg. 3) with Nq = 2

Initialized by A(0) (Alg. 4) with Nq = 2

Initialized by h(0) (Alg. 3) with Nq = 3

Initialized by A(0) (Alg. 4) with Nq = 3Clipping without any CompensationLinear without Clipping

σ−2 [dB]

Genie-aided with Nq = 2

Genie-aided with Nq = 3

Figure 2.3: The SER performance of the iterative detection (Algorithm 1.4.1) vs.σ−2, when L+ 1 = 7, N = 128 and CL = 1 dB.

38 Background and Contributions of the Thesis

0 5 10 15 20 25 30 35 4010

-6

10-5

10-4

10-3

10-2

10-1

100

σ−2 [dB]

Genie-aided

4

Figure 2.4: The SER performance of the iterative detection (Algorithm 1.4.1) withNq = 3 vs. σ−2, when L+ 1 = 31, N = 128 and CL = 1 dB.

perfectly coincide with the genie-aided scenario, where the channel and clipping le-vel are perfectly known at the receiver side. Figure 2.4 shows the SER performanceof the proposed algorithms, integrated with Algorithm 1.4.1 with Nq = 3, whenL+1 = 31, CL = 1 dB for N = 128. Note that the channel length L+ 1 = 31 is themaximum allowed length for Lcp = 32 which is used conventionally for an OFDMsystem with N = 128 number of subcarriers. In this case the channel frequencyselectivity is higher. As can be seen from the figure, both estimation algorithmsprovide really good results.

Paper C: Channel and Clipping Level Estimation in OFDMSystems [OB18a]

In this paper, a method to estimate channel and clipping level for OFDM systemsusing time-domain and frequency pilots is proposed. In particular, by proposing anew OFDM packet frame, we combine TDS-OFDM and classical CP-OFDM sys-tems. This new packet frame, as illustrated in Figure 2.5, consists of time-domainpilots and frequency-domain pilots, each preceded by a separate CP. For the time-domain pilots, PN sequences are deployed due to their attractive cross-correlationproperties [SP80]. Moreover, the amplitudes of these time-domain pilots are cho-sen conservatively small in a way that they do not hit the clipping threshold. Theintuition behind using this new frame is that the problem of channel and clip-

2.2. Contributions of the Thesis 39

ping level estimation is divided into two sub-problems. In the first one, channelis estimated using the time-domain pilots, then they are plugged in to the secondsub-problem in which the clipping amplitude is estimated using Algorithm 2.2.1.Finally, the resulting estimates of channel and clipping amplitude are deployed inAlgorithm 1.4.1.

M0

L(t)cp Lcp

N

Figure 2.5: The proposed OFDM packet frame.

System model

The system model is depicted in Figure 1.3. The difference in this work is thatthe packet frame is different. Therefore, by invoking [WG00], a more descriptivevectorized system model is depicted in Figure 2.6. where s[i] is the i-th N × 1

1NFH

HP×P0 + HP×P

1 z−1

s[i]

zc[i] uc[i]

wc[i]t[i]

x[i]

ut[i]

u[i]

T(c)cp

xc[i] R(c)cpg(·)

y[i]F

ChannelEstimation

ClippingAmplitudeEstimation

A

xc[i] uc[i]

Figure 2.6: The vector based system model.

symbol block vector dedicated to frequency-domain pilots, and t[i] is the i-th M0×1 PN-sequence vector used as the time-domain pilots. These PN-sequences areconstructed directly in the time-domain and must satisfy the power constraint ofthe transmitter. Here, the CP adder and remover are denoted by matrix operations.In particular, the CP adder matrices for the time-domain and frequency-domainpilots are T(t)

cp = [I(t)cpT

ITM0]T , and Tcp = [ITcp ITN ]T , respectively, where I(t)

cp is thematrix of the last L(t)

cp rows of theM0×M0 identity matrix IM0 , and Icp is the matrixof the last Lcp rows of the N ×N identity matrix IN , and the CP remover matricesfor the time-domain and frequency-domain pilots are R(t)

cp = [0M0×L(t)

cpIM0 ] and

Rcp = [0N×Lcp IN ], respectively. Therefore, the whole CP adder and CP removercan be constructed as

T(c)cp =

[T(t)

cp 0M×N0Q×M0 Tcp

], (2.2.11)

R(c)cp =

[R(t)

cp 0M0×Q

0N×M Rcp

], (2.2.12)

40 Background and Contributions of the Thesis

Moreover, for l = 0, 1, the P × P matrices HP×Pl are defined to have the (i, j)-th

entry as h[lP + i− j], and M = M0 + L(t)cp , Q = N + Lcp and P = M +Q.

Therefore, the channel output for the i-th packet can be written as

uc[i] = HP×P0 zc[i] + HP×P

1 zc[i− 1] + wc[i] (2.2.13)

where zc[i] = [zt[i]T z[i]T ]T = [xt[i]T z[i]T ]T . Since the PN-sequence sampleshave small enough amplitudes not hitting the clipping threshold, we can rewrite(2.2.13) as

uc[i] =[

HM×M0 xt[i] + HM×Q

1 z[i− 1]HQ×M

1 xt[i] + HQ×Q0 z[i]

]+[

wt[i]w[i]

](2.2.14)

Using (2.2.11) and (2.2.14), we have

uc[i] =[

ut[i]u[i]

]=[

HM×M0 t[i]

HQ×Q0 g(x[i])

]+[

wt[i]w[i]

](2.2.15)

where g(·) is the function defined in (2.2.1) and x[i] is equal to x[i] after removingthe CP. Moreover, the matrices HM×M

0 and HQ×Q0 areM×M and Q×Q circulant

matrices, respectively.

Channel and clipping level estimation

Now, using (2.2.15), first the channel can be estimated using LS and then theclipping amplitude can be estimated by using the channel estimates and Algo-rithm (2.2.1).

2.2.2 Problem II: An efficient iterative jointestimation-detection algorithm to estimate nonlinearityparameters and channel, and detect symbols

The contribution of this section belongs to Paper D.

Paper D: A General Framework for Joint Estimation-Detection ofChannel-Nonlinearity Parameters and Symbols for OFDM inIoT-based 5G Networks [OB18b]

In this work, a joint estimation-detection iterative algorithm to jointly estimatechannel and nonlinearity parameters and detect symbols for a general memorylessnonlinear PA model, for an OFDM system which uses comb-type pilots is proposed.Next, to demonstrate the applicability of this general proposed algorithm, we applyit to a limiter and polynomial nonlinearity models in the transmitter. Note thatalthough the work here considers memoryless nonlinearities, the extension to the

2.2. Contributions of the Thesis 41

Hammerstein models as nonlinearity models with memory is straightforward sincethe linear filter of these models can be dissolved into the channel. Then interestingly,we show, using simulations, that the limiter-based algorithm also works quite goodfor a SSPA model when used as the nonlinearity in the system. Furthermore, notethat the reason that we consider comb-type pilots is two-fold. The first is that it canbetter track and estimate channels in fast-fading situations, in which the channelremains the same just for a few OFMD symbols [CEPB02,NC98]. Moreover, it iscompatible with the existing 4G LTE standards [ZM07] and 5G NR [3GP18]. Themain contributions of this work is as follows:

• It is the first algorithm proposed to jointly perform the estimation and de-tection for channel-nonlinearity parameters and symbols, respectively in oneOFDM symbol consisting of comb-type pilots and data subcarriers.

• The algorithm can be generally used for every memoryless nonlinearity model.

• It is shown that the limiter-based algorithm can work flawlessly for the SSPAmodel, too. It is particularly important since there is no need for the receiverto change its estimation-detection algorithm for different (typically low-costand low-power) transmitters deploying limiters and SSPAs at their RF front-end.

System model

The system model is explained in Section 1.3, and is depicted in Figure 1.3. Thesystem uses comb-type pilots, therefore a set of subcarriers of the vector s is knownto the transmitter as well as the receiver. The allocation of pilots is given by (1.6.4).The nonlinearity function is in general a memoryless nonlinear PA model.

As introduced in Section 1.2.1, the polynomial model description is given by(1.2.12), which can be rewritten as

zn =K∑k=1

bkωk(xn), (2.2.16)

where ωk(xn) = |xn|k−1xn, k = 1, . . . ,K, is a set of basis functions, by which wecan represent the output of a nonlinearity. In vector form, (2.2.16) can be writtenas

z =K∑k=1

bkωk(x) = Ω(x)b, (2.2.17)

where ωk(·) is taken element-wise, Ω(x) = [ω1(x), . . . , ωK(x)], and b = [b1, . . . , bK ]T.To estimate the parameters bk’s using the measurements from the output of thenonlinearity, LS can be used as

bLS =(ΩH(x)Ω(x)

)−1ΩH(x)z, (2.2.18)

42 Background and Contributions of the Thesis

However, the inversion of ΩH(x)Ω(x) can experience a numerical instability pro-blem [RQZ04]. To deal with this problem a set of orthogonal basis functions areintroduced in [RQZ04] for xn with independent magnitude with |xn| ∼ U [0, 1], andphase with arg(x) ∼ U [0, 2π] as

ψk(xn) =k∑l=1

Ulk|xn|l−1xn, (2.2.19)

whereUlk = (−1)l+k (k + l)!

(l − 1)!(l + 1)!(k − l)! , (2.2.20)

by which the output of the nonlinearity can be written as

z =K∑k=1

βkψk(x) = Ψ(x)β. (2.2.21)

The joint estimation-detection algorithm

In this section, an iterative estimation-detection algorithm to jointly estimate chan-nel and nonlinear parameters and detect symbols with the help of frequency-domaincomb-type pilots for a general memoryless PA nonlinearity model in an OFDM sy-stem is proposed. In particular, we propose an iterative alternating optimizationmethod to solve the LS problem as

minθNL,h,sd

∥∥∥y−√Ndiag(Fh)F g(FHP

( spsd)

; θNL

)∥∥∥2

2, (2.2.22)

which is non-convex combinatorial and in general very difficult to solve.To develop the joint estimation-detection algorithm, we first start by minimizing

the cost function in (2.2.22) over the nonlinearity parameters when channel and allthe subcarriers are given. If the nonlinearity is a limiter, Algorithm 2.2.1 can beused, and if it is a polynomial model, given by (1.2.12), the optimization problemconverts to an unconstrained convex problem whose closed-form solution can beobtained.

Then, we consider the channel estimation when the nonlinearity parameters andall the subcarriers are given. In this case, again (2.2.22) is an unconstrained convexproblem whose closed-form solution is given by the LS solution.

Finally, we use these estimates of the nonlinearity parameters and channel in themodified version of Algorithm 2.2.1, adapted in a way to use the side informationavailable as comb-type pilots.

For initializing the proposed algorithm, we neglect the effect of the nonlinearityand estimate the channel using the method described in Section 1.6, and in parti-cular (1.6.8). Then using the estimated channel for data subcarriers ˆhd, the data

2.2. Contributions of the Thesis 43

subcarriers can be detected as

sd =⟨

1√N

yd ˆhd

⟩. (2.2.23)

Now, the joint estimation-detection algorithm can be described as Algorithm 2.2.4.

Algorithm 2.2.4 Joint estimation-detection algorithm.1: Inputs:

y, sp

2: Initialize:ˆhp = 1√

Nyp sp

ˆh = NNp FFH

pˆhp

ˆhd ← last N −Np elements of PT ˆhsd =

⟨1√N

yd ˆhd

⟩s← P

[sᵀp, s

ᵀd]T

x← IFFT[s]dd = 0N

3: while Stopping criterion not met do

4: θNL ← arg minθNL

∥∥y−√Ndiag(ˆh)F g (x; θNL)

∥∥22

5: κ← Eznx∗n6: z← g(x; θNL)7: Z = diag

(FFT[z]

)F

8: h← 1√N

(ZHZ

)−1ZHy

9: ˆhd ← last N −Np elements of PTFFT[h]10: sd =

⟨1κ ( 1√

Nyd ˆhd − dd)

⟩11: s← P

[ spsd

]12: x← IFFT[s]13: d← FFT[g(x; θNL)− κx]14: dd ← last N −Np elements of PTd15: end while

Cramér-Rao lower bound

Here, we have a hybrid estimation and detection problem. To deal with this, wefind the CRLB for estimation parameters under the assumption that the detection

44 Background and Contributions of the Thesis

parameters are perfectly known a priori. The estimation parameters comprise ofcomplex-valued channel taps and real-valued nonlinearity parameters. Using theresults explained in Section 1.5.3, the CRLB for θNL and h can be calculated as

CRLB(θNL) =[Υ− 2<

ΞHΣ−1Ξ

]−1, (2.2.24)

CRLB(h) = Σ−1 +(Σ−1Ξ + Σ−∗Ξ∗

)CRLB(θNL)

(ΞHΣ−H + ΞᵀΣ−ᵀ

),

(2.2.25)

where (·)−∗ = (·)−1∗, (·)−ᵀ = (·)−1ᵀ, (·)−H = (·)−1H,

Σ = N

σ2 FHdiag(Fz)Hdiag(Fz)F, (2.2.26)

Ξ = N

σ2 FHdiag(Fz)HDHFdiag(ejφ)Γ, (2.2.27)

Υ = 2Nσ2 <Γ

H,diag(ejφ)HFHDHHDHFdiag(ejφ)Γ, (2.2.28)

where

Γ =[∂g(r,θNL)∂θ

(1)NL

, . . . ,∂g(r,θNL)∂θ

(M)NL

], (2.2.29)

∂g(r,θNL)∂θ

(m)NL

=[∂g(r0,θNL)∂θ

(m)NL

, . . . ,∂g(rN−1,θNL)

∂θ(m)NL

]T, (2.2.30)

for m = 1, . . . ,M , and θNL =[θ

(1)NL, . . . , θ

(M)NL]T, where M is the number of nonli-

nearity parameters.

Simulation results

Here, the performance of the proposed estimation-detection algorithm is investiga-ted using simulations. The subcarrier modulation is 16QAM. The Rayleigh fadingchannel model used here is the exponential channel model (IEEE 802.11b mo-del) [HHW00], which has L + 1 complex Gaussian distributed taps hl having amean power of σ2

l = E|hl|2 = σ20e−Ts/τrmsl for l = 0, . . . , L in which σ0 is chosen

such that∑Ll=0 σ

2l = 1, Ts = 50 ns is the sampling rate, and τrms = 30 ns and 80 ns

are the RMS delays leading to channel lengths L+ 1 = 7 and 17, respectively.Figures 2.7 and 2.8 show the NMSE performance of estimating the clipping

amplitude (A) and channel taps (h), when L + 1 = 7, N = 128 and CL = 1 dB,respectively. As we can see from the figures the NMSEs are decreasing by increasingσ−2 and both of them attain the CRLB. It is an interesting result since the CRLBis derived for the case that block-type pilots are used, i.e., the whole OFDM symbolis dedicated to pilots. Moreover, we achieve good results for both cases where thenumber of pilots are Np = 32 and 16 out of N = 128, which correspond to the pilotto subcarrier ratios L−1

p = 25% and 12.5%, respectively.

2.2. Contributions of the Thesis 45

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

NMSE

Normalized CRLBInitialized by h [OB17]Alg. 2.2.4, L−1

p = 25%Alg. 2.2.4, L−1

p = 12.5%

Figure 2.8: The NMSE performance of h for L+ 1 = 7, N = 128, and CL = 1 dB.

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

NMSE

Normalized CRLBInitialized by h [OB17]Alg. 2.2.4, L−1

p = 25%Alg. 2.2.4, L−1

p = 12.5%

Figure 2.7: The NMSE performance of A for L+ 1 = 7, N = 128, and CL = 1 dB.

Figure 2.9 shows the SER performance of the proposed algorithm in comparison

46 Background and Contributions of the Thesis

0 5 10 15 20 25 30 35 4010−7

10−6

10−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

SER

LinearClipping w/o compensationPerfect knowledge [THC03]Alg. 2.2.4, L−1

p = 25%Alg. 2.2.4, L−1

p = 12.5%Alg. 2.2.4, L−1

p = 6.25%

Figure 2.9: The SER performance for the limiter when L + 1 = 7, N = 128, andCL = 1 dB.

with the genie-aided scenario, when L + 1 = 7, N = 128 and for CL = 1 dB.Figure 2.10 shows the SER performance of the proposed algorithm designed for alimiter, but used for a SSPA in comparison to the genie-aided scenario. As can beseen from the figures, although the algorithm has been designed for a limiter, itworks really well for the SSPA. Figure 2.11 shows the SER performance when L+1 = 17 andN = 256 for the TWTA with parameters aa = 1.6623, ba = 0.0552, aΦ =0.1533, bΦ = 0.3456, when K = 4 is in (2.2.21). As can be seen from the figure, thealgorithm works well when there is a non-zero AM/PM conversion characteristic.When L−1

p = 6.25%, the performance is not good because the frequency-selectivityof the channel in this case is higher than that the channel interpolation can give agood initial estimation.

2.2.3 Problem III: An algorithm to jointly estimatenonlinearity parameter and channel for SIMO-OFDM

The contribution of this section belongs to Paper E.

2.2. Contributions of the Thesis 47

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

SER

LinearSSPA w/o compensationPerfect knowledge [THC03]Alg. 2.2.4, L−1

p = 25%Alg. 2.2.4, L−1

p = 12.5%Alg. 2.2.4, L−1

p = 6.25%

Figure 2.10: The SER performance for the SSPA when L + 1 = 7, N = 128,CL = 1 dB and % = 2.

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

SER

LinearTWTA w/o compensationPerfect knowledge [THC03]Alg. 4, L−1

p = 25%Alg. 4, L−1

p = 12.5%Alg. 4, Lp = 6.25%

Figure 2.11: The SER performance for the TWTA when L + 1 = 17, N = 256,aa = 1.6623, ba = 0.0552, aΦ = 0.1533, bΦ = 0.3456, and K = 4.

48 Background and Contributions of the Thesis

Paper E: Joint Channel and Clipping Level Estimation forSIMO-OFDM Systems [OB18c]

In this work, a method to jointly estimate channel and clipping level for single-inputmultiple-output (SIMO)-OFDM systems is considered. The importance of this workis that in practical scenarios, in comparison to the transmitter the receiver is usuallyof a higher quality which is equipped with multiple antennas. This can be consideredas a first step of studying the joint channel and clipping level estimation and symboldetection in multi-user MIMO-OFDM systems.The contributions of this work are as follows:

• The joint nonlinearity parameter and channel estimation algorithm is exten-ded to the SIMO-OFDM scenario.

• The iterative detection algorithm (i.e., Algorithm 1.4.1) is extended to max-imal ratio combining (MRC) detectors.

• It is numerically shown that the CRLB for the nonlinearity parameter isbetter than the single-antenna case by the factor of the number of receiveantennas.

System model

The transmitter has a single antenna, while the receiver has Nrx antennas. Thereceiver uses MRC for the symbol detection. The SIMO-OFDM is depicted in Fi-gure 2.12. The received signal at receive antennam and subcarrier k, after removingthe CP, can be written as

ym[k] =√Nhm[k]DFTz[n][k] + wm[k], (2.2.31)

where hm[k] = DFThm[n], z[n] = g(x[n];A), DFTz[n][k] is the DFT of z[n]calculated at subcarrier k, and wm[k] is the DFT of the additive zero-mean complexGaussian noise, with variance σ2, at receive antenna m and subcarrier k. Therefore,we can derive the received signal at subcarrier k for all the receive antennas as

y[k] =√N h[k]DFTz[n][k] + w[k], (2.2.32)

where

y[k] = [y1[k], . . . , yNrx [k]]T, (2.2.33)h[k] = [h1[k], . . . , hNrx [k]]T, (2.2.34)w[k] = [w1[k], . . . , wNrx [k]]T. (2.2.35)

Collectively, (2.2.32) can be rewritten as

Y =√NHdiag(Fz) + W, (2.2.36)

2.2. Contributions of the Thesis 49

P/S

IDFTs

CP

adde

r

S/P

DFT

CP

rem

overAWGN

g(.; θNL)

S/P

DFT

CP

rem

overAWGN

MRC

persubc

arrie

r

MRC

yMRC

h1[n]

hNrx [n]

1

Nrx

Figure 2.12: The baseband SIMO-OFDM system model.

where

Y = [y[0], . . . ,y[N − 1]], (2.2.37)H = [h[0], . . . , h[N − 1]], (2.2.38)W = [w[0], . . . , w[N − 1]]. (2.2.39)

By transposing and vectorizing both sides of (2.2.36), we can have another usefulform as

yvec =√Nvec

(diag(Fz)HT)+ wvec, (2.2.40)

where yvec = vec(YT) =[yT

1 , . . . ,yTNrx

]T, wvec = vec(WT) =[wT

1 , . . . , wTNrx

]T,and

HT =

h[0]T...

h[N − 1]T

= [h1, . . . , hM ] = F[h1, . . . ,hM ]. (2.2.41)

We can rewrite (2.2.40) as

yvec =√N(INrx ⊗ diag(Fz)F)h + wvec, (2.2.42)

where ⊗ denotes the Kronecker product,

h = [hT1 , . . . ,hT

Nrx ]T, (2.2.43)

and we used vec(ABC) = (CT ⊗A)vec(B). Note that (2.2.42) is equivalent to aset of Nrx equations corresponding to each receive antenna as

ym =√Ndiag(Fz)Fhm + wm,m = 1, . . . , Nrx, (2.2.44)

Yet another way to write (2.2.36) is as

y =√NDHFz + w, (2.2.45)

50 Background and Contributions of the Thesis

where y =[y[0]T, . . . ,y[N − 1]T

]T, DH = diag(h[0], . . . , h[N − 1]

), and w =[

w[0]T, . . . , w[N − 1]T]T. Note that yvec = Ky, where K is the commutation ma-

trix used for transforming the vectorized form of a matrix into the vectorized formof its transpose.

Joint Channel and Clipping Level Estimation

We use block-type pilots, in which the whole OFDM symbol is dedicated to frequency-domain pilots, and assume that the channel is unchanged during several OFDMsymbols.

The joint channel and clipping level estimation can be expressed as Algo-rithm 2.2.5. The value of clipping level is unknown at the beginning, therefore thefirst estimate of the channel can be done using the unclipped version of transmittedtime-domain symbols. The initial channel estimates are

h(0)m = 1√

N

(FHdiag

(s)Hdiag

(s)F)−1

FHdiag(s)Hym, m = 1, . . . , Nrx. (2.2.46)

Algorithm 2.2.5 Joint channel and clipping level estimation.1: Inputs:

ym, m = 1, . . . , Nrxy, r and ejφFck for k = 0, . . . , N − 1

2: Initialize:[r,Ps] = sort(r)ejφ = Pse

jφ

h(0)m ,m = 1, . . . , Nrx given by (2.2.46)

D(0)H = diag

(h(0)[0], . . . , h(0)[N − 1]

)r−1 = 0i = 1

3: while (convergence criteria not met) do

4: B =√ND(i−1)

H FPᵀs

5: calculate A(i−1) and k(i−1) using Algorithm 2.2.16: z = x (1N − ck(i−1)) + A(i−1)ejφ ck(i−1)

7: V = diag(FPᵀs z)F

8: h(i)m = (VHV)−1VHym,m = 1, . . . , Nrx

9: D(i)H = diag

(h(i)[0], . . . , h(i)[N − 1]

)10: i = i+ 111: end while

2.2. Contributions of the Thesis 51

Cramér-Rao lower bound

Using the results explained in Section 1.5.3, the CRLB for A and h as defined in(2.2.43), can be calculated as

CRLB(A) = 1β − 2αHΓ−1α

, (2.2.47)

and

CRLB(h) = Γ−1 + 1β − 2αHΓ−1α

Γ−1ααHΓ−H, (2.2.48)

where using (2.2.42) and (2.2.45)

Γ = N

σ2 (INrx ⊗ diag(Fz)F)H(INrx ⊗ diag(Fz)F), (2.2.49)

α = N

σ2 (INrx ⊗ diag(Fz)F)HKDHF(c ejφ), (2.2.50)

and

β = 2Nσ2

∥∥DHF(c ejφ)∥∥2

2. (2.2.51)

Signal detection method

We extend Algorithm 1.4.1 to the case of using MRC receiver. Here, we use thesignal model defined in (2.2.36) to derive the MRC receiver at each subcarrier k as

yMRC[k] = h[k]Hy[k]‖h[k]‖22

, k = 0, . . . , N − 1, (2.2.52)

and hence collectively as

yMRC = [yMRC[0], . . . , yMRC[N − 1]]T. (2.2.53)

Now, we can use the iterative detection algorithm for the SIMO-OFDM system asrepresented by Algorithm 2.2.6.

52 Background and Contributions of the Thesis

Algorithm 2.2.6 Iterative Detection Algorithm for MRC.1: Inputs:

y, F, and Nqh and A calculated by Algorithm 2.2.5

2: Initialize:ˆH = [ˆh[0], . . . , ˆh[N − 1]]κ = 1− e−A2 + A

√π

2 erfc(A)Compute yMRC using (2.2.52) and (2.2.53)d(0) = 0N

3: for i = 1 to Nq do

4: s(i) =⟨

1κ

( 1√N

yMRC − d(i−1))⟩5: x(i) = FHs(i)

6: d(i) = F(g(x(i); A)− κx(i))

7: end for

Simulation results

Here, for the simulations, the subcarrier modulation is 16QAM. The fading chan-nels’ paths are generated using Jake’s Doppler spectrum with power loss of thechannel taps are [0,−1,−2,−3,−8,−17.2,−20.8] dB with the delay profiles equal to[0, 30, 70, 90, 110, 190, 410] ns, that corresponds to an extended pedestrian A (EPA)scenario [3GP11]. The maximum Doppler shift is 5 Hz, and the sampling frequencyis 30 MHz. This model provides a good extent of frequency selectivity. The spatialmodel is iid fading. Furthermore, the model follows the slow-time varying conditionin which the channel is fixed during several OFDM blocks.

Figures 2.13 and 2.14 show the NMSE performance of estimating the clippingamplitude (A) and channel taps (h) for different number of receive antennas, whenN = 128 and CL = 1 dB, respectively. As we can see from the figures the NMSEsare decreasing by increasing the SNR and both of them reach CRLB in a me-dium SNR regime. Moreover, from the figures we can observe that the CRLB forclipping level estimation for the multiple antenna receiver is better than the oneof a single antenna receiver by the factor of the number of receive antennas, i.e.,CRLB(Nrx=1) [dB](A) = CRLB(Nrx=M)(A) [dB] + log10(M). However, the CRLBfor estimating the channels is the same for multiple antenna and single antennareceivers, which is justifiable by the fact that estimating the whole channels here isequivalent to Nrx separate channel estimations.

2.2. Contributions of the Thesis 53

0 4 8 12 16 2010−5

10−4

10−3

10−2

10−1

100

σ−2 [dB]

NMSE

CRLB of A

NMSE of A

Nrx = 1

Nrx = 2

Nrx = 4

Nrx = 8

Figure 2.13: The NMSE and CRLB of A for the limiter when N = 128, andCL = 1 dB for different number of receive antennas.

54 Background and Contributions of the Thesis

0 4 8 12 16 2010−4

10−3

10−2

10−1

σ−2 [dB]

NMSE

CRLB of h

NMSE of h

Figure 2.14: The NMSE and CRLB of h for the limiter when N = 128, andCL = 1 dB for different number of receive antennas.

0 4 8 12 16 20 2410−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Nq = 2

Nq = 3

σ−2 [dB]

SER

NL w/o compensationLinearGenie-aided with Alg. 2.2.6Algs. 2.2.5 and 2.2.6

Figure 2.15: The SER performance for the limiter when N = 128, CL = 1 dB,and Nrx = 4.

2.3. Further Discussion 55

Figure 2.15 shows the SER performance of Algorithm 2.2.5 integrated withthe iterative detection method (Algorithm 2.2.6), when N = 128, Nrx = 4, andCL = 1 dB. We have used two maximum number of iterations for Algorithm 2.2.6as Nq = 2 and 3. As illustrated in the figures, the SER performance of differentschemes when we estimate the clipping level and the channel taps almost perfectlycoincide with a genie-aided scenario, where the channel and clipping amplitude areperfectly known at the receiver side.

2.3 Further Discussion

In this thesis, the nonlinearity is affecting the time-domain OFDM signal on theNyquist sampling rate. This results in the distortion to fall all in-band. However,when the nonlinearity is performed on an over-sampling rate, it creates out-of-band distortion, which requires additional bandpass filter and may also cause peakregrowth. In addition, if the over-sampling factor is denoted by M , it requireslarger size (MN -point) DFT/IDFT blocks, which increases not only computationalcomplexity, but also hardware costs [KKI09]. Figure 2.16 shows the out-of-banddistortion when the Nyquist rate is 50 MHz, the over-sampling factor M = 8, thecarrier frequency is 2.1 GHz, and raised-cosine pulse shape is used.

1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3

Frequency (GHz)

-140

-120

-100

-80

-60

-40

Pow

er s

pect

ral d

ensi

ty (

dBm

/ H

z)

LinearCL = 1 dBCL = 3 dB

Figure 2.16: The out-of-band distortion when over-sampling is used and clippinglevels are 1 dB and 3 dB.

3

Conclusion

In this thesis, we have discussed the problem of nonlinear PAs and their effectson the system, and the ways to estimate and compensate these effects. The focushas been on receiver-side compensation methods, which are sensible when we face5G-based IoT networks, in which many low-cost, low-power and low-weight devicesneed to seamlessly transmit their data.

We have proposed different methods and algorithms to estimate nonlinearityparameters along with channel estimation and symbol detection. The estimationof these nonlinearity parameters are important, particularly in 5G-based IoT andMTC networks because there are a lot of sensors that need to send informationoccasionally. Then, at each time of transmission, the receiver (e.g., a fusion centeror a base station or an IoT gateway) needs to estimate the nonlinearly parame-ters to have updated values which will be deployed in the detection stage. Thosenonlinearly parameters for different nodes may change during time due to envi-ronmental conditions such as temperature, humidity, and unstable power supply.Furthermore, for low-cost devices, the manufacturing variations from unit to unitmay be rather large and it would be too expensive to measure and calibrate eachunit separately. Moreover, electronic devices and components suffer from processof aging and changing over time. Therefore, these nonlinearity parameters will be-come unknown to the transmitter over time and hence estimating them at thereceiver-side occasionally seems essential.

We have studied the ML estimation of clipping level for a AWGN channel. Next,we have discussed joint ML estimation of channel and clipping level in multi-pathfading OFDM systems. In particular, we have proposed an alternative optimizationalgorithm which uses frequency-domain block-type training symbols. An efficientalgorithm to estimate the clipping level has been devised. We have also used thisalgorithm in TDS-OFDM systems by proposing a new packet frame consisting oftime-domain and frequency-domain pilots for channel and clipping amplitude, re-spectively. Thereafter, we have studied a joint estimation-detection algorithm forjointly ML channel-nonlinearity parameters estimation, and symbol detection usingfrequency-domain comb-type pilots in multi-path fading OFDM systems, and pro-posed an iterative optimization algorithm to solve it. Finally, we have extended the

57

58 Conclusion

joint channel and nonlinearity parameter estimation to the SIMO-OFMD systems.Moreover, We have provided numerical results and curves to verify the performanceof these algorithms.

Moreover, as a future work, there are several research directions one can taketo extend the work in this thesis.

1. The symbol detection part of this thesis has been based on the quasi-MLiterative detection method. One can aim to devise and propose new detectionalgorithms and combine it with the estimation algorithms proposed here.

2. The focus in this thesis has been on memoryless nonlinearity models. Onefuture work can be on extending the results to more broad types of nonline-arities which exhibit memory effects.

3. The work in this thesis has been mainly on single antenna OFDM systems andSIMO-OFDM systems. One can aim to extend the results to MIMO-OFDMsystems and also multiuser MIMO-OFDM systems such as SDMA-OFDM.

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