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A Study of Wireless Modem Performance UsingMultiple Element AntennasbyGuillaume LebrunIngénieur de l'Ecole Nationale Supérieure des Télécommunications (2000)Submitted in ful�llment of the requirements for the degree ofDoctor of PhilosophyVICTORIA UNIVERSITYMarch 2004

Thesis Supervisor Pr. Michael FaulknerCentre for Telecommunications and Micro-ElectronicsVictoria UniversityFootscray Park CampusP.O. Box 14428 MCMCMelbourne 8001AUSTRALIA

ContentsDeclaration of originality ixAcknowledgements xiList of acronyms xiiiNotations xvVariables xixAbstract xxiiiI Introduction 1I.1 History of telecommunications . . . . . . . . . . . . . . . . . . . . . . . . . 1I.1.1 Origin of telecommunications . . . . . . . . . . . . . . . . . . . . . 1I.1.2 Evolution and revolutions of telecommunications . . . . . . . . . . . 2I.1.3 Telecommunication: a multi-faceted giant . . . . . . . . . . . . . . 4I.2 Recent trends in telecommunications . . . . . . . . . . . . . . . . . . . . . 5I.2.1 Current telecommunications . . . . . . . . . . . . . . . . . . . . . . 5I.2.2 Future of telecommunications . . . . . . . . . . . . . . . . . . . . . 6I.2.3 Towards understanding the Shannon capacity . . . . . . . . . . . . 6I.2.4 Multiple antennas wireless systems . . . . . . . . . . . . . . . . . . 7I.3 Statement of signi�cance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8I.4 Contribution to knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . 8I.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10I.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12iii

II Channel capacity 15II.1 Notion of capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15II.1.1 The sampling theorem . . . . . . . . . . . . . . . . . . . . . . . . . 16II.1.2 Elementary information theory . . . . . . . . . . . . . . . . . . . . 17II.1.3 Capacity of a channel . . . . . . . . . . . . . . . . . . . . . . . . . . 18II.1.4 Coding theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19II.1.5 Channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19II.1.6 Gaussian complex random variable . . . . . . . . . . . . . . . . . . 21II.1.7 Capacity of the �at-fading channel . . . . . . . . . . . . . . . . . . 22II.1.8 Capacity of the non-ergodic channel . . . . . . . . . . . . . . . . . . 23II.1.9 Capacity of the ergodic channel . . . . . . . . . . . . . . . . . . . . 23II.2 MIMO Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24II.2.1 MIMO channel model . . . . . . . . . . . . . . . . . . . . . . . . . 25II.2.2 Circularly symmetric complex Gaussian distribution . . . . . . . . . 26II.2.3 Capacity of the MIMO �at-fading channel . . . . . . . . . . . . . . 26II.3 Discussion of the assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 40II.3.1 Coherent detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 41II.3.2 Time and frequency properties of the channel . . . . . . . . . . . . 42II.3.3 Independence of the noise entries . . . . . . . . . . . . . . . . . . . 44II.3.4 Single user channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 44II.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45IIIUsing the large capacity of MIMO channel 47III.1 Review of MIMO transmission techniques . . . . . . . . . . . . . . . . . . 48III.1.1 Uncoded systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48III.1.2 Space-time coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54III.1.3 BER simulation of MIMO transmission techniques . . . . . . . . . . 56III.2 Multiplexing vs diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56III.3 Capacity of correlated channels . . . . . . . . . . . . . . . . . . . . . . . . 58III.3.1 Ricean channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59III.3.2 Capacity bounds for the Ricean channel . . . . . . . . . . . . . . . 61III.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 63III.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66iv

III.4 MIMO systems in highly specular channel . . . . . . . . . . . . . . . . . . 67III.5 In�uence of CSI at the transmitter . . . . . . . . . . . . . . . . . . . . . . 68III.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71IVOptimum structure: the SVD 73IV.1 Introduction to the SVD structure . . . . . . . . . . . . . . . . . . . . . . . 74IV.2 Optimality of the SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75IV.2.1 Optimal linear precoder and decoder . . . . . . . . . . . . . . . . . 76IV.2.2 Optimal space-time structure . . . . . . . . . . . . . . . . . . . . . 77IV.3 Practical SVD architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 80IV.3.1 Performance loss of the practical architecture . . . . . . . . . . . . 81IV.3.2 Notion of system capacity . . . . . . . . . . . . . . . . . . . . . . . 81IV.4 Channel estimation and SVD architecture . . . . . . . . . . . . . . . . . . 82IV.4.1 Channel estimation accuracy . . . . . . . . . . . . . . . . . . . . . . 83IV.4.2 Uniqueness of the SVD . . . . . . . . . . . . . . . . . . . . . . . . . 84IV.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88V SVD architecture in TDD environment 91V.1 SVD architecture on reciprocal channels . . . . . . . . . . . . . . . . . . . 92V.2 Hardware calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . 93V.2.1 E�ect of hardware errors on channel reciprocity . . . . . . . . . . . 93V.2.2 Calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . 96V.2.3 E�ects of the calibration . . . . . . . . . . . . . . . . . . . . . . . . 98V.2.4 Choice of calibration matrices . . . . . . . . . . . . . . . . . . . . . 99V.3 SVD-TDD system and singular values swapping . . . . . . . . . . . . . . . 99V.3.1 Matrix perturbation theory, singular values . . . . . . . . . . . . . . 100V.3.2 Matrix perturbation theory, singular vectors . . . . . . . . . . . . . 101V.3.3 Probability of robust transmission . . . . . . . . . . . . . . . . . . . 103V.3.4 Singular value crossing . . . . . . . . . . . . . . . . . . . . . . . . . 105V.3.5 Subspace swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 108V.3.6 Correction of singular value crossing . . . . . . . . . . . . . . . . . 109V.4 SVD architecture modi�ed for TDD environment . . . . . . . . . . . . . . 110V.4.1 Correction of the CSI impairment . . . . . . . . . . . . . . . . . . . 111V.4.2 Comparison with reference systems . . . . . . . . . . . . . . . . . . 113v

V.4.3 Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . 114V.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120V.5 CSI improvement through channel tracking . . . . . . . . . . . . . . . . . . 121V.5.1 Tracking of the CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . 122V.5.2 Tracking of the precoding and decoding matrices . . . . . . . . . . 124V.5.3 Correlation of pilots . . . . . . . . . . . . . . . . . . . . . . . . . . 127V.5.4 Simulation results, decoding matrix . . . . . . . . . . . . . . . . . . 127V.5.5 Simulation results, precoding matrix . . . . . . . . . . . . . . . . . 132V.5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134V.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135VIConclusion 137VI.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137VI.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140VI.2.1 Pilot symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140VI.2.2 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141VI.2.3 System level design . . . . . . . . . . . . . . . . . . . . . . . . . . . 141VI.2.4 Study of the wireless channel . . . . . . . . . . . . . . . . . . . . . 142VI.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A Ricean channel capacity bounds 143I.1 Study of the Eigenvalues of F . . . . . . . . . . . . . . . . . . . . . . . . . 143I.1.1 Ricean channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143I.1.2 Singular Value Decompositions . . . . . . . . . . . . . . . . . . . . 143I.1.3 Eigenvalues of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144I.1.4 Asymptotic eigenvalues of F . . . . . . . . . . . . . . . . . . . . . . 145I.2 Capacity Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146I.3 Capacity Upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148B IEE Electronics Letters 151C IEEE Global Telecommunications Conference 2002 153D IEEE Vehicular Technology Conference 2003 155vi

E IEEE International Conference on Communications 2004 157F IEEE International Conference on Communications 2004 159G IEEE Transactions on wireless communications 161H IEEE Transactions on wireless communications 163

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Declaration of originalityI hereby declare that the research in this thesis itself was entirely composed and com-piled by myself in the Centre for Telecommunications and Micro-Electronics at VictoriaUniversity.

Guillaume LEBRUN

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AcknowledgementsThough I am the author of this thesis, my supervisor Pr. Michael Faulkner is, in manyways, the driving force behind it. I was very fortunate to walk in his footsteps through-out this research project. Few PhD students receive support and guidance on both themethodology and the content of their project; I received them in abundance. Pr. MichaelFaulkner not only showed me how to become a researcher but also taught me invaluablelessons on how to become a respectable human being.I am deeply grateful to Pr. Mansoor Shaa�, Pr. Peter Smith and Pr. Jugdutt Singh fortheir time and their advice to improve the quality of my work as well as my understandingof telecommunications systems.The present thesis represents three years of work that I shared with colleagues in theSchool of Electrical Engineering. I received much help from the group and would especiallylike to acknowledge Melvyn Pereira, Shane Spiteri, Dr. Tan Ying and Dr. Jason Gao fortheir support. I owe many thanks to Mrs Noreen Wheaton for accepting to patiently guideme through the subtleties of the English language.This thesis is not only the result of the research conducted in the past three yearsbut more generally the result of twenty seven years of education. I was lucky to meetinspiring teachers throughout high school and university. I would especially like to thankmy teachers from the Ecole Nationale Supérieure des Télécommunications and from theLycée Saint Louis.Last but not least, I owe my happiness to my wonderful family and my faithful friends.No achievement would be possible without their love and support.

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List of acronymsAD Anno DominiAWGN Additive White Gaussian NoiseBC Before Christbit Binary DigitBLAST Bell laboratories Layered Space-Timebps bits per secondsCCDF Complementary Cumulative Density FunctionCDMA Code Division Multiple AccessCSI Channel State InformationCSMA-CD Carrier Sense Multiple Access - Collision DetectiondB decibelsD-Blast Diagonal BlastFER Frame Error RateFIR Finite Impulse ResponseGHz GigahertzHz HertzIBM International Business Machines corporationi.i.d. Independent and Identically DistributedISI Inter-Symbol-InterferenceISM Industrial, Scienti�c and Medicalkm kilometersLAN Local Area NetworkMAC Medium Access Control

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MHz MegahertzMIMO Multiple Input Multiple OutputMMSE Minimum Mean Squared Errorms millisecondsµs microsecondsOFDM Orthogonal Frequency Division MultiplexingPDF Probability Density FunctionPSAM Pilot Symbol Assisted ModulationQAM Quadrature Amplitude ModulationRF Radio Frequencys SecondsSISO Single Input Single OutputSINR Signal to Interference and Noise RatioSNR Signal to Noise RatioST Space-TimeSVD Singular Value DecompositionTDD Time Division DuplexTDMA Time Division Multiple AccessUSA United States of AmericaUSSR Union of Soviet Socialist RepublicsV-Blast Vertical BlastZF Zero-Forcing

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Notationsloga(.) base a logarithmexp(.) exponential functione exponential of 1, exp(1)

Γ(.) gamma functionJ0(.) zeroth-order Bessel function of the �rst kindδ(., .) Kronecker functionF (w) fourier transform of f(t)

N semi-ring of the positive integer numbersZ ring of integer numbersQ �eld of rational numbersR �eld of real numbersC �eld of the complex numbersRe(.) real part functionIm(.) imaginary part functionCi vector space of dimension i over C

x scalarx vectorxi ith entry of x

X matrixX i,j entry of the ith line, jth column of X

X :,i ith column of X

xv

x† transpose of x

x∗ transpose-conjugate of x

x decomposition of x into its real and imaginary parts(.) slicing (hard decision) operationVect(xi) vector of zeros (length of x) except the ith entry xi

x estimate of x

x average of all the possible x

~X i,j(lT ) vector of time samples 0 to lt of X i,j

(.)+ only non-negative values are acceptable|.| absolute value‖.‖2 norm 2‖.‖F Frobenius norm‖.‖u unitary invariant normSV D(.) SVD functionfu(.) output subspace function (component of the SVD)f v(.) input subspace function (component of the SVD)R(H) rank of a matrix H

X+ pseudo-inverse of X

Ik Identity matrix of dimension k

0 matrix of zeros1 matrix of ones.Π permutation of a matrixP(.) probability functionx, x, x random variable (scalar, vector and matrix)µx mean of the random variable x

σ2x variance of the random variable xpdfx probability density function of x

γ Gaussian probability density functionχ2

k chi-square variate with k degrees of freedomxvi

Epdfx [.] denotes the expectancy with probability density func-tion pdfxHa(.) = − loga(.) entropy functionH(.) = H2(.) entropy function (in bits)I(.) mutual information function

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Variablesx, x, X transmitted signalx signal transmitted on the transmission eigenmodesy, y, Y received signaly signal received on the transmission eigenmodesn, n, N noisen noise on the transmission eigenmodesN est estimation noiseN tvc time-varying channel perturbation (pseudo-noise)P pilot symbolsh,h,H wireless channelHA→B wireless channel from transceiver A to transceiver BHB→A wireless channel from transceiver B to transceiver AHsp specular component of a Ricean channelHsc scattering component of a Ricean channelK Ricean K-factor expressed in dBMT number of transmitting antennasMact

T number of active transmitting antennasMR number of receiver antennasM = min(MR,MT ) minimum number of antennas at either the transmitteror the receiverα = MT /MR ratio between the number of transmitting antennas andthe number of receiving antennasβ = MR/MT ratio between the number of receiving antennas and thenumber of transmitting antennas

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MA number of antennas at transceiver AMB number of antennas at transceiver Bp probability of an eventpe probability of errorpmin

e minimum probability of errorpout outage probabilityC capacityCout outage capacityCergodic ergodic capacityP total transmitted powerPwf level chosen to satisfy the water�lling requirementsQ correlation of the transmitted signalQ correlation of the signal transmitted on the transmissioneigenmodesSNRdB SNR expressed in dBSNRest SNR in the channel estimationA,B decoding matrices for MIMO receiversa decoding vectors for MIMO receiverswopt optimum linear FIR �lter (Wiener �lter)W bandwidth of the signalw angular frequencyTs sampling periodτ maximum delay of multipathFd Doppler frequencyFsvc frequency of singular value crossingsX MIMO constellationS SISO constellation

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s symbol of a SISO constellationR rate of a codeLt length of a space-time coded diversity of a MIMO systemt time indexδt time delaytc time index of a singular value crossingi, j, k, l indexesr real numberc complex numberε perturbationκ chi-square variateΠ permutation matrixU , V , W , Z unitary matricesUT , V T precoding and decoding matrices derived at the trans-mitter (when speci�c)UR, V R precoding and decoding matrices derived at the receiver(when speci�c)D, Σ diagonal matrix with non-negative entriesΦ diagonal matrix with complex entries of norm 1ErT,A, ErR,A matrices of errors in respectively the RF transmitter andreceiver chains, transceiver AErT,B, ErR,B matrices of errors in respectively the RF transmitter andreceiver chains, transceiver BR upper triangular matrix with real elements on the diag-onalW matrix from the Wishart ensemble

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AbstractNext generation wireless modems are likely to be based on multiple antenna hardwareplatforms, following the demand for high data rate. Multiple antenna at both end ofthe link can potentially increase the spectrum e�ciency of wireless systems since thecapacity of Multiple Input Multiple Output (MIMO) systems is generally higher than thecapacity of their Single Input Single Output (SISO) counterparts. Most proposed MIMOarchitectures assume that the Channel State Information (CSI) is available at the receiveronly, or not available at all. This assumption is justi�ed by the fact that CSI is di�cultto obtain. Furthermore the gain in capacity due to available CSI at the transmitter is notlarge on heavy multipath channels.However, in correlated channels, CSI at the transmitter provides a signi�cant capacityincrease over conventional MIMO systems. For example, the capacity of MIMO Riceanchannels reduces to the capacity of their Rayleigh component in the asymptotic limit ofa large number of antennas when the CSI is not available at the transmitter. This resultremains largely valid, even for a small number of antennas. In other words, MIMO systemswith no CSI at the transmitter do not use the antenna gain at the transmitter, whichis detrimental in correlated propagation environment. Besides, under any propagationenvironment, the optimal capacity is obtained with CSI at the transmitter. Finally the CSIcan be easily obtained at the transmitter in Time Division Duplex (TDD) channels. Thesethree reasons justify a thorough study of MIMO architecture with CSI at transmitter. Insuch a case, the Singular Value Decomposition (SVD) of the channel matrix gives theoptimal precoder and decoder.This thesis studies the performance of the SVD architecture under varying propagationenvironments, as well as its robustness to various impairments, e.g. incorrect channelestimation, hardware errors. The aim is to provide a comprehensive understanding of theadvantages and weaknesses of the SVD-based transmission architecture.xxiii

The precoding and decoding matrix of the SVD architecture are in theory perfectlymatched with the CSI matrix. The channel estimation requirement are shown to be mildwhen the system is synchronized, i.e. the same CSI is used at both the transmitter and thereceiver. This is due to the fact that unitary matrices are well conditioned, allowing theerror in the estimation to be considered as additional noise. However, in practice, mostsystems obtain the CSI at transmitter and receiver separately. In such a case, the non-linearity of the SVD imposes stringent constraints on the channel estimation. Furthermore,the SVD architecture relies on the possibility for transmitter and receiver to agree on aunique SVD of the channel matrix. The unicity of the SVD is studied theoretically andan upper bound on the probability of the SVD not being unique is derived, demonstratingthat a better than 20 dB of SNR channel estimation is required for SVD system to operatecorrectly.In Time Division Duplex (TDD) systems, the channel is reciprocal, allowing the trans-mitter to obtain the CSI through standard channel estimation on the reverse link. Thoughthe wireless channel is reciprocal, the transmitter and receiver electronics are usually notmatched, destroying the reciprocity of the overall transmission channel. SVD systemscannot operate in the presence of such hardware errors. A calibration procedure is pro-posed to insure that the overall channel is reciprocal and that hardware errors do nota�ect the system. Delayed CSI at the transmitter is another source of performance loss.MIMO systems without CSI at transmitter outperform their SVD-based counterpart atFdδt = 0.038, where Fd is the Doppler frequency, δt the time delay on the CSI at thetransmitter, assuming Jake's fading on an i.i.d. channel, SNR=20dB, and perfect channelestimation. Fdδt = 0.038 corresponds to walking speed (2m/s) at 5.725 GHz (802.11afrequency band) and CSI delayed by 1ms at the transmitter. The loss of performance ofSVD systems can be reduced by �ltering the CSI at the transmitter, i.e. prediction of theCSI to match it with the channel. Another identi�ed source of performance loss for SVDsystems is the occurrence of rare channel events named singular value crossings, whichresult in bursts of errors. Singular value crossing can be detected and corrected at thetransmitter if the channel is tracked over time. Finally, a new scheme is proposed, com-bining the advantages of systems with and without CSI at the transmitter: by includingthe precoding matrix in the channel, the receiver can both recover the CSI for the nexttransmission in the reverse link and decode the stream without harmful e�ect of incorrectCSI at the transmitter. This proposed transmission scheme does shift seamlessly from anxxiv

SVD system at low mobility to a standard MIMO system for higher mobility.

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Chapter IIntroductionMost of us consider telecommunications as nothing more than a useful tool. Calling aperson on the other side of the planet hardly surprises us. In an age where technical featsare mere consumer products, what are the goals and challenges of telecommunications?I.1 History of telecommunicationsTelecommunication has become a part of our everyday life in the past two centuries, somuch so that the very meaning of the word itself has been blurred and is mistaken as beingsynonymous with mobile phones. The word telecommunications conveys information aboutits own etymology. The pre�x tele- comes from the Greek τηλη, and means "remote" or"afar". Communication comes from the Latin communio, which means "to impart" or"to share", literally "to make common". Telecommunication means, strictly speaking, toshare something over a distance.However, the word communication has evolved to the present day meaning of the ex-change of information. Hence, telecommunications are adequately de�ned as the exchangeof information over a distance, or over time. As such, telecommunications are the foun-dation of any society which requires people to send and receive information outside of the"here and now" realm.I.1.1 Origin of telecommunicationsObviously, telecommunications were born with the very �rst human civilizations. The bestexample of simple telecommunications is writing, which dates back to 3500 BC. A written1

message can be sent to a recipient in a remote place, or can be transmitted to a reader ata later time. Messengers are the second example of telecommunications, e.g. the famousmarathon runner Philippides who, in 490 BC, prevented Athen's destruction by deliveringa warning message that the Persian army was arriving. Together with these two inventions,writing and messengers, came the �rst two requirements of telecommunications, namelyspeed and reliability. Thereafter all telecommunications would be measured in terms ofspeed of transfer and accuracy in the transmission. Even today's telecommunicationsindustry still focuses on satisfying these two fundamental requirements.In ancient times, systems were designed to transmit information faster than a humancould move, using either sound (drum telegraph) or visible signals (�re and/or smoke). Bythe year 150 AD, the Roman Empire was covered with a telecommunications network whichhad a total length of approximately 4500 kilometres. The network consisted of closelyspaced towers which enabled them to exchange visual information through predeterminedsmoke signals. During the French Revolution Claude Chappe rediscovered the idea of anoptical telegraph and built a line 240km long which could transfer 196 di�erent signals.Ultimately, telecommunications were to be linked with electricity and really developedin the middle of the XIXth century. Samuel Morse invented the electromagnetic telegraphin 1837. Elisha Gray and Alexander Graham Bell took out patents for telephones in1876 and by 1895 Guglielmo Marconi was demonstrating wireless transmission of signals.The engineers involved with early telecommunications innovations were mostly inventorsinterested in practical progress and consumer products. This explains the rapid impactof telecommunications on society: by 1902 worldwide communication was available onocean ships; by 1903 more than 3 million phones were installed in the USA alone, andbroadcasting stations were commercialised as early as 1922 in Russia, France, Englandand the USA. Thereafter, progress in telecommunications equipment would mirror theadvances in electronics, starting with the invention of the vacuum tube in 1910.I.1.2 Evolution and revolutions of telecommunicationsWhile telecommunications engineers were focusing on incremental advances until the sec-ond world war, a major paradigm shift was soon under way, leading to modern telecom-munications theory: the shift towards digital communication. Telecommunication theorywas founded by Nyquist [1, 2] in his pioneering work on the maximum signalling rate2

Figure I.1: Chappe's telegraph is an early example of digital optical communicationusing source coding. A message could be transmitted over 240 km in less than half anhour... that is, when the weather was not foggy.achievable on a telegraph line of a given bandwidth. He was soon followed by Hartley[3] who studied the amount of data that can be transmitted when multiple amplitudelevels are used. Both proposed the use of a logarithmic measure of information. Theirwork was completed by Shannon in two papers published in 1948 [4, 5] where for the �rsttime, a mathematical theory of telecommunications was proposed. This work remains theprecursor of most telecommunications advances and de�nes in particular the capacity ofa channel (the maximum amount of information that can be transmitted per second ona channel). This work also proposes a probabilistic framework for telecommunicationstheory. Kolmogorov and Wiener contributed to the development of the new branch oftelecommunications theory known as information theory. Their work consisted of mathe-matically modelling and deriving an optimal �lter for the reception of telecommunications3

signals [6, 7]. Information theory refers to the combination of mathematical modelling andtelecommunications theory.Telecommunication was to be revolutionized once again with the development of a newtype of user: computers communicating with computers. In 1961 IBM computers startedcommunicating over the telephone line, using what was soon to be known as modula-tor/demodulators or modems. But the rapid rise of computers and the need to commu-nicate between them led to the development of packet switched networks, more adaptedto computer tra�c. Coincidently, vast projects of interconnected computers appeared,more speci�cally, Arpanet, which initially began as a network connecting universities andthe military and armament industry, and later evolved into what is known today as theInternet.The Cold War and the space race between USSR and the USA paved the way for therapid development of satellites. As early as 1960 Echo 1, the �rst communication satellite,was in orbit. It was essentially a mirror re�ecting radio waves on earth, but successfultransmissions between the USA and France demonstrated the bene�ts obtainable throughsatellite telecommunications.Quite surprisingly, 1960 was also to see the rebirth of optical telecommunications,one of the oldest forms of telecommunications. The invention of the laser ensured theavailability of high quality light sources. It triggered research into optical �bres, which wereimpractical at the time due to high loss. However, by 1980, worldwide optical networkswere deployed.I.1.3 Telecommunication: a multi-faceted giantIn the 50 years, from 1920 to 1970, telecommunications evolved from specialized electri-cal engineering into its own engineering discipline with its own theory, a wide range ofapplications and several specialised domains. Telecommunication brings together:• mathematicians for information theory, coding theory, digital signal processing andnetwork theory• electronic engineers for the development of all enabling technologies, especially chip-sets, which are at the heart of telecommunications equipment• computer scientists since most telecommunications equipment is implemented partly4

in software, allowing, among other advantages, recon�gurability.• electromagnetic researchers, for all aspects of radio wave emission, reception andpropagation• optical physicists, involved in all aspects of the development of optical communica-tions• mechanical engineers, involved in satellite communication, but also in the develop-ment of micro-electro-mechanical devices.This diversity illustrates the challenges inherent in the design of a telecommunica-tions system as well as the necessity for the telecommunications engineer to develop anunderstanding of traditionally unrelated engineering �elds.I.2 Recent trends in telecommunicationsI.2.1 Current telecommunicationsThe development of information and coding theory combined with the development ofenabling technologies (especially electronic components) allowed telecommunications tobecome an everyday commodity readily available in developed countries. Wireless com-munication has become a part of our everyday life as recently as the past 20 years (whereastelephones were already widespread more than 30 years ago). The development of secondgeneration cellular phone networks rede�ned mobile phones from a luxury item into aconsumer product.The push for wireless telecommunications was not limited to cellular phones. Rel-atively inexpensive satellite phones became a reality, allowing consumers to be reachedanywhere in the world. Cordless phones, and other low bit rate wireless devices (for ex-ample Bluetooth), contributed to proving wireless communication was a strong candidatefor future consumer products. The advantages of wireless products over their wired coun-terparts include straightforward installation, freedom of movement and cutting of all costsassociated with cables.The other major evolution of the telecommunications industry was due to the pre-vailing use of computers. Voice was no longer the main content of telecommunicationsnetworks. Telecommunication networks now had to accommodate for the massive amount5

of electronic data being transferred. Consumers are demanding ever growing data ratesto support applications such as real time video.In this context, third generation cellular networks have been designed to accommodateboth voice and data, and provide a unique, high data rate, wireless access to the telecom-munications network. The deployment of third generation equipment has been delayed dueto economic reasons and corresponds to a downturn in the telecommunications industry.I.2.2 Future of telecommunicationsFrom a technical point of view, it has become increasingly clear that it is extremelydi�cult to answer the diverse requirements of users with a single standard. Therefore,the industry is considering the possibility of developing wireless products that recon�gurethemselves depending on their environment and data transfer requirements. For example,a single device would use a low bit rate cellular standard for a phone call from a car andswitch over to a high-speed standard when the user requires internet connectivity from ano�ce. This philosophy of "best connected" is sometimes referred to as the elusive fourthgeneration.As mentioned earlier, progress in telecommunications has mostly been measured interms of achievable data rate. Focusing on the speed of the link has been questioned inthe context of cellular networks, arguing that most users do not use high data rate appli-cations. However, this focus has never been questioned concerning Local Area Network(LAN) applications because they are data rate hungry. Computer networks are becom-ing widespread, not only in business but also for private consumers. However, the costsassociated with the installation and maintenance of networking cables are high and makewireless LANs an extremely viable alternative to their wired counterparts, provided theycan o�er comparable data rates.I.2.3 Towards understanding the Shannon capacityThe wireless spectrum is a scarce resource which everybody has to share. If two userstransmit at the same frequency, they interfere with each other. The radio frequency spec-trum is regulated internationally to control user interference. In particular, �xed frequencybands are allocated to telecommunications companies. These companies, in turn, want toprovide high data rates, to a maximum number of users, over a �xed bandwidth.6

However, Shannon proved that there exists a maximum data rate for a wireless chan-nel and that this maximum data rate (or channel capacity) is a direct function of thebandwidth of the channel, the power of the signal and the amount of noise in the channel.Therefore, until recently, it was considered that the only solution to increasing the datarate of a wireless system was either to increase its bandwidth or to increase the power itradiates. Neither of these two solutions is satisfactory. Since the total bandwidth of thetelecommunications system is �xed, increasing the bandwidth of one user directly limitsthe number of users in the system. It is no more reasonable to increase the power of thewireless system. The �rst consideration is the simple fact that radiating a large power isimpractical for battery-operated devices. The second reason is concern about the e�ecton human health of intense electromagnetic �elds.Therefore, telecommunications engineers were trying to transmit as closely as possibleto the Shannon capacity, which severely limited further increases in the data rate.This conclusion was shown to be incorrect for a simple reason: the Shannon capacity ofa channel also depends on the number of transmitting and receiving antennas. This simplefact had always been overlooked since, intuitively (although incorrectly), transmittingwith two antennas at the same frequency appears to be equivalent to creating ones owninterference.I.2.4 Multiple antennas wireless systemsThe fact that the capacity of a channel depends on the number of antennas at both thetransmitter and the receiver has enormous implications. Higher data rates can be achievedby using multiple antennas without increasing the radiated power or bandwidth. Channelswith multiple antennas at both ends are usually referred to as Multiple Input MultipleOutput (MIMO) channels.Several practical systems have been proposed to bene�t from the large capacity o�eredby MIMO channels. The proposed architectures usually aim at either increasing the datarate, or reducing the power requirement of the user. Most of these systems are designed forspeci�c communication scenarios and most of them rely on pessimistic assumptions, e.g.the propagation environment is unknown at the transmitter. Because of these assumptions,most proposed architectures are transmitting at only a fraction of the capacity of thechannel. 7

Theoretical results on the capacity of MIMO channels are now well understood. Inparticular, the variation in achievable data rate when the transmitter does not knowthe channel can be well understood by using a mathematical operation: the SingularValue Decomposition (SVD) of the channel. The SVD also suggests a communicationarchitecture that allows transmission at the channel capacity. This well-known architecturehas not been studied extensively in the literature, being commonly considered too complexto be implemented.I.3 Statement of signi�canceThe typical requirements of users of a wireless LAN are reliable and high data rates. Forthe wireless propagation channel in such an environment an obvious solution is a MIMOsystem. Given the wireless LAN environment and user requirements, it is likely that theSVD architecture will be used for the implementation of such a system.The goal of this thesis is to study the advantages and drawbacks of the SVD archi-tecture. It aims at providing exact results on whether it is implementable, under whichassumptions, in which propagation environment and with what expected bene�ts.I.4 Contribution to knowledgeThe main contributions of this thesis include:-Ergodic capacity series expansions. Series expansions of the Gaussian approxima-tion of the ergodic MIMO channel capacity are presented in Section II.2.3.3 andhighlight the importance of the minimum number of antennas at either the trans-mitter ot the receiver. The series expansions are derived in the asymptotic case ofthe number of antennas at the receiver (transmitter) being much larger than thenumber of antennas at the transmitter (receiver). Simulation results support thetheoretical results of the series expansion.-Ricean channel study. Section III.5 demonstrates that the ergodic normalized capac-ity of the Ricean MIMO channel approaches the corresponding normalized capacityof the underlying scattering channel when the antenna numbers are large and noChannel State Information (CSI) is available at the transmitter. Section III.5 also8

demonstrates that the capacity variance of the Ricean channel approaches the corre-sponding variance of the underlying scattering channel when the number of antennasis large. These results highlight the importance of CSI at the transmitter. Upperand lower bounds for the ergodic capacity of the Ricean channel are derived. Theaccuracy of the bounds is con�rmed via simulation.-Channel estimation requirements for SVD systems. An analysis of the e�ect ofincorrect channel estimation on the performance of SVD transmission systems isproposed in Section IV.4. The analysis demonstrates that these e�ects are negligiblewhen the SNR on the estimation of the channel is much larger than the SNR on thereceived data. On the contrary, under channel estimation errors, the performance ofSVD systems does not increase with increasing SNR: the capacity of SVD systemsplateaus at high SNR. Simulation results support this analysis-Uniqueness of the SVD of a complex matrix. The SVD of a complex matrix is notunique, as shown in Section IV.4.2. However, it is possible to select a unique SVDfollowing some criteria when the complex matrix is square and the singular valuesof multiplicity one. Further technical considerations allow to extend this result toall matrices with singular values of multiplicity one and to handle the very rareoccurrence of channel matrices with singular values of multiplicity higher than one.Therefore the SVD of the channel matrix can be derived separately at the transmitterand the receiver.-Calibration procedure for SVD systems over reciprocal channels. MIMO SVDsystems are well suited for reciprocal channels since the CSI can be obtained at thetransmitter without overhead. However, this technique relies on symmetric receptionand transmission chains at each transceiver. This assumption is unrealistic. Acalibration procedure is proposed in Section V.2.2 which forces the channel to bereciprocal. This calibration procedure is a form of handshaking at the start of thetransmission and relies on the assumption that the imperfections of the RF chainsvary slowly in time, allowing calibration to remain valid for large periods of time.-Singular value crossing and singular value swapping. The application of matrixperturbation theory shows in Section V.3 that SVD systems are robust to imper-fect channel estimation only when the singular values of the channel matrix have9

a multiplicity of one. Imperfections of the system allow this theoretical case studyto gain practical applicability. E.g. when the transmitter and the receiver deducetheir precoding and decoding matrices from the CSI at two di�erent time-slots, thetransmission eigenmodes might be completely di�erent on both time-slots. Thisevent is referred to as a singular value crossing. In practice, singular value crossingusually involves singular subspace swapping: the data sent on subchannel 1 is re-ceived on subchannel 2 and the data sent on subchannel 2 is received on subchannel1. Therefore, simple mechanisms can mitigate the harmful e�ect of singular valuecrossings.-Modi�ed SVD architecture. Some of the practical issues occurring on SVD systemscan be mitigated by modifying the transmission architecture. In Section V.4, thepilot symbols are sent through the precoding matrix, providing the receiver withinformation on the channel matrix and the precision of the channel estimation at thetransmitter at the same time. This proposed architecture promises high performancewhen the CSI is accurate at both end of the channel and enjoys a graceful lossof performance, down to the performance of MIMO systems without CSI at thetransmitter, when the CSI is inaccurate at the transmitter. Furthermore, the newlyproposed architecture has no additional complexity.-MIMO PSAM and SVD structure. The requirement of accurate channel estimationcan be achieved through added pilot symbols (larger overhead) or added complexitythrough the natural extension of PSAM to MIMO channels. An SVD system can�lter the CSI in time or directly �lter the precoding and decoding matrices. InSection V.5, it is shown that �ltering the CSI achieves better performance.These results are, to the best knowledge of the author, either new and unpublished orpreviously published by the author.I.5 List of publicationsThe results presented in this thesis have been partly published or accepted for publicationin the following articles:• G. Lebrun, M. Faulkner, P. J. Smith and M. Shaa�, "MIMO Ricean channel ca-pacity," in Proc. IEEE International Conference on Communications (ICC 2004),10

20-24 Jun. 2004, pp. 2939-2943,• G. Lebrun, M. Faulkner, P. J. Smith and M. Shaa�, "MIMO Ricean channel capacity:an asymptotic analysis," IEEE Transactions on wireless communications, Vol. 5, No.5, May 2006,• G. Lebrun, S. Spiteri and M. Faulkner, "Channel estimation for an SVD-MIMOSystem," in Proc. IEEE International Conference on Communications (ICC 2004),20-24 Jun. 2004, pp. 3025-3029,• S. Spiteri, G. Lebrun and M. Faulkner, "Prediction for time varying SVD Systems,"in Proc. IEEE International Symposium on Personal, Indoor and Mobile RadioCommunications (PIMRC 2004),• S. Spiteri, G. Lebrun and M. Faulkner, "Capacity performance of hardware erredMIMO systems in slowly fading Rayleigh channels," in Proc. Australian Communi-cation Theory Workshop 2004 (AusCTW),• G. Lebrun, S. Spiteri and M. Faulkner, "Antenna selection and time-varying chan-nel," in Proc. Australian Telecommunication Cooperative Research Center (ATCRC)conference 2003,• S. Spiteri, G. Lebrun and M. Faulkner, "Capacity performance of hardware erredMIMO systems in slowly fading Rayleigh channels," in Proc. Australian Telecom-munication Cooperative Research Center (ATCRC) conference 2003,• G. Lebrun, S. Spiteri and M. Faulkner, "MIMO complexity reduction through an-tenna selection," Proc. Australian Telecommunications, Networks and ApplicationsConference (ATNAC) 2003, Dec. 8-10, 2003, Melbourne, Australia,• S. Spiteri, G. Lebrun and M. Faulkner, "E�ects of hardware induced errors on TDD-based SVD systems," Proc. Australian Telecommunications, Networks and Applica-tions Conference (ATNAC) 2003, Dec. 8-10, 2003, Melbourne, Australia,• G. Lebrun, J. Gao and M. Faulkner, "MIMO transmission over a time-varying chan-nel using SVD," IEEE Transactions on wireless communications, Vol. 4, No. 2,March 2005, pp. 757-764, 11

• T. Ying, G. Lebrun and M. Faulkner, "An adaptive channel SVD tracking strategyin time-varying TDD system," Proc. IEEE Vehicular technology conference 2003,(VTC 2003-Spring), Apr. 22-25, 2003 Vol. 1, pp. 769-773,• G. Lebrun and M. Faulkner, "Analysis of singular value decomposition appliedto wireless communications," Proc. Australian Telecommunication Cooperative Re-search Center (ATCRC) conference 2002,• G. Lebrun, T. Ying and M. Faulkner, "MIMO transmission over a time-varyingchannel using SVD," Proc. IEEE Global Telecommunications Conference, (Globecom'02), Nov. 17-21, 2002 pp. 414-418,• G. Lebrun, T. Ying, M. Faulkner, "MIMO transmission over a time-varying channelusing SVD," IEE Electronics Letters, vol. 37, no. 32, pp. 1363-1364, Oct. 2001.I.6 OutlineChapter II introduces the notion of capacity, as the maximum data rate that can betransferred over a channel, under given assumptions. The intimate connection betweencoding and capacity is highlighted. The impact on the capacity of the number of antennasat both ends of a wireless link is presented in details, both for the ergodic and non-ergodicchannels.Chapter III presents some well-known MIMO transmission architectures and highlightstheir speci�cities. The fundamental concepts of diversity and spatial multiplexing areintroduced. These concepts allow to answer the challenge of exploiting the high capacityof MIMO channels. Finally, the study of the capacity of Ricean channels stresses thebene�ts associated with CSI at the transmitter.The SVD transmission architecture is presented in Section IV. The optimality of theSVD is demonstrated. Two issues linked with the implementation of the SVD are studiedin detail: the channel estimation requirements of the system and the uniqueness of theSVD of a complex matrix.Chapter V analyses the SVD system in a Time Division Duplex (TDD) environment.The impact of delayed and incorrect CSI at the transmitter is examined. The degradationof performance due to singular value crossings is assessed and a practical solution tomitigate this loss of performance is proposed, through the discovery of singular subspace12

swapping. A novel transmission architecture is proposed to prevent catastrophic lossof performance when the CSI is inaccurate. The new SVD architecture combines theperformance of SVD systems when the CSI is accurate with the performance of MIMOsystems without CSI at the transmitter when the CSI is inaccurate. Finally, an extensionof PSAM to MIMO channel is proposed to improve the accuracy of channel estimationwithout increasing the overhead.

13

Chapter IIChannel capacityII.1 Notion of capacityThe theoretical study of telecommunications systems relies on the choice of a theoreticalmodel to represent the systems. Telecommunication (wired or wireless) consists usuallyof the emission of a physical signal that varies in time, denoted in the following x(t),where t represents the time index. In most telecommunications systems, this signal isan electromagnetic wave. This signal is received as another signal y(t). The relationshipbetween x(t) and y(t) depends only on the channel, i.e. the medium over which the signalis transmitted. Telecommunication theory is concerned with:

• determining for a given channel how much information can be transmitted over thechannel, i.e. the capacity of the channel,• designing x(t) to transmit information in a fast and reliable way,• reliably recovering the information transmitted x(t) from the received signal y(t).Though very general, this model is di�cult to analyse. Further assumptions enablesimpli�cation of this model, speci�cally the sampling theorem allows to restrict the analysisto discrete time systems when the signals are bandlimited (Section II.1.1). Informationtheory provides the theoretical framework to de�ne the notion of information (SectionII.1.2) as well as the capacity of the channel (Section II.1.3).Chapter II provides a literature review of the main results concerning the capacityof both SISO and MIMO single user channels in a variety of situations: CSI at the re-ceiver/transmitter, ergodic or block fading channel. MIMO channel capacity is shown to15

increase with the number of antennas at both ends of the wireless link in Section II.2. Anovel series expansion of the capacity of the ergodic channel with no CSI at the transmit-ter is presented in Section II.2.3.3. The series expansion is new and unpublished to theknowledge of the author. The series expansion shows that symmetric antenna allocation(same number of antennas at the transmitter and the receiver) maximizes the capacity ofa MIMO channel with a given total number of antennas. Finally, Section II.3 shows thatthe assumptions are compatible with wireless LANs standards 802.11a and Hyperlan 2.II.1.1 The sampling theoremThe sampling theorem is central in telecommunications since it demonstrates that analogand digital communications are equivalent, provided the analog transmission is band lim-ited. Though the sampling theorem has been applied to telecommunications by Nyquist[1], it had been demonstrated previously in other forms by mathematicians [8]. The formgiven here corresponds to the theorem stated by Shannon [9].Theorem 1. Sampling Theorem If a function f(t) contains no frequencies higher than WHz, it is completely determined by giving its ordinates at a series of points spaced 1/2Wseconds apart.The intuitive justi�cation is that, if f(t) contains no frequencies higher than W , itcannot change to a substantially new value in a time less than one-half cycle of the highestfrequency, that is, 1/2W . The exact mathematical proof follows.Let F (w) be the spectrum of f(t). Thenf(t) =

1

2π

∫ ∞

−∞F (w) exp(

√−1wt)dw =

1

2π

∫ 2πW

−2πW

F (w) exp(√−1wt)dw, (II.1)since F (w) is assumed zero outside the band W . This relationship is veri�ed at thesampling points t = k

2Wwhere k is any integer (k ∈ Z):

f(k

2W) =

1

2π

∫ 2πW

−2πW

F (w) exp(√−1w

k

2W)dw. (II.2)The integral on the right is the kth coe�cient in the Fourier-series expansion of the function

F (w), taking the interval −W to +W as a fundamental period. So, the samples f( k2W

)determine the Fourier coe�cients in the series expansion of F (w). Furthermore, sinceF (w) is zero outside [−W,W ], F (w) is uniquely de�ned by the Fourier coe�cients of its16

series expansion and, in turn, the samples f( k2W

). Therefore, the samples determine thefunction f(t) completely, since F (w) determines f(t).Additionally, f(t) can be reconstructed from its samples:f(t) =

∞∑

k=−∞

f(k

2W)sin(2πW (t − k

2W))

2πW (t − k2W

). (II.3)Finally, it is possible to prove a similar result if the signal is not limited to the frequency

[0,W ] but bandlimited to the band [Wc − W/2,Wc + W/2] [9].Most wireless communication systems are bandlimited, due to bandwidth allocation,as explained in Section I.2.3. Therefore, applying the sampling theorem, the study oftelecommunications systems can be limited to the study of sampled systems without lossof generality. The transmitted signal and received signals can be restricted to x(kTs) andy(kTs) respectively, with k ∈ Z.II.1.2 Elementary information theoryIt was shown in Section I that the goal of a telecommunications system is to transmitinformation. However, no precise de�nition of information was given. A practical examplecan help to clarify the concept of information.Consider a system transmitting, every �ve minutes, the answer to the question:"Isthere a �re in John's house?" Most of the time, the system indicates that John's houseis �ne, which John naturally discards as a lack of information. However, if the systemraises the alarm about a �re, it is natural to consider that very important information hasbeen transmitted. Intrinsically, this event carries more information because it happensless often. Thus, information is somehow related to the inverse of the probability ofoccurrence. For our communication system, consider the event x = s1, where s1 is anysymbol, i.e. any value of the physical signal used to carry the message. The informationcarried by the event x = s1 depends on the probability of this event p1 = P(x = s1). Itis, therefore, useful to de�ne a function H(p), which measures the amount of informationin the occurrence of an event of probability p, and has the following properties:

• H(p) ≥ 0, the measure of information is real, non-negative,• H(p1p2) = H(p1) + H(p2) for independent events, i.e. the amount of informationcarried by two unrelated events is the sum of the amount of information carried byeach event, 17

• H(p) is a continuous function of p.The only functions following these requirements are Ha(p) = − loga(p), where loga(.)denotes the logarithm in base a [10]. The base 2 logarithm is usually used, and theresulting unit of information is called a bit (binary digit). In the following, it is consideredthat a = 2, and the corresponding index is dropped.Assume that the transmitted signal x(kTs) can take the values s1, s2, . . . , sq with re-spective probabilities p1, p2, . . . , pq. x(kTs) is a random variable denoted x(kTs). Theaverage information transmitted H(x) is called the entropy of the source, withH(x) =

q∑

i=1

piH(pi) =

q∑

i=1

pi log2(1

pi

). (II.4)II.1.3 Capacity of a channelThe transmission of information can now be de�ned precisely. Consider x = [x(0), x(Ts),

. . . , x(kTs)] the overall transmitted signal and y = [y(0), y(Ts), . . . , y(kTs)] the overall re-ceived signal. The actual transmission can be modelled as the set of conditional probabili-ties P(y|x), the probability of receiving y, having transmitted x. This set of probabilitiesdetermines a channel over which transmission occurs.Prior to reception, the probability of x is P(x). After reception of y, the probabilitythat the input symbol was x becomes P(x|y). The change in probability measures howmuch the receiver learned from the reception of y, and is called the mutual information,de�ned asI(x; y) = log2(

1

P(x)) − log2(

1

P(x|y)) = log2(

P(x|y)

P(x)) (II.5)It is interesting to determine the average mutual information (also called system mutualinformation):

I(x; y) =∑

x P(x = x)I(x = x; y)

=∑

x P(x = x)∑

y P(y = y|x = x)I(x = x; y = y)

=∑

x

∑

y P(x = x,y = y)I(x = x; y = y),

(II.6)where x and y are random variables. In a slight abuse of notation, P(x) denotes P(x = x)in the following.The natural question that arises is, given the conditional probabilities P(y|x), what isthe maximum amount of information that can be transmitted over the channel? The only18

parameter that can be chosen is P(x) the set of probabilities of the possible transmittedsymbol. The capacity of the channel is de�ned asC = max

P(x)I(x; y), (II.7)the maximum (over every set of possible transmitted signals) average mutual information,and is expressed either in bits per second per Hertz (bps/Hz) or alternatively in bits perchannel use.II.1.4 Coding theoremThe importance of the capacity of channels in telecommunications is due to the Shannontheorem, which links this theoretical tool with transmission devices.Theorem 2. Shannon Theorem For any R < C and any probability of error pe, given achannel of capacity C, there exists a code with rate R that can transmit with a probabilityof error smaller than pe.On the contrary, for any R > C, there exists a minimum probability of error pmin

e , i.e.no code with R > C has a probability of error smaller than pmine .The demonstration of the Shannon theorem is given in [9]. However it is quite technicaland is not reproduced here. The main insights from the Shannon theorem are:

• optimal codes are of in�nite length,• random codes are good candidates to be optimal, i.e. random codes can be as closeto optimal as desired by increasing the code's length.Because of the �rst property, it is impractical to use a random code to transmit closeto the capacity. A �nite delay in the decoding is a requirement of telecommunications sys-tems. Therefore, the capacity of the channel is considered as an upper bound on achievablerate. It is useful to understand how e�ective a code is and how much improvement can beachieved on a given system by improving the coding.II.1.5 Channel modelFollowing the results from the previous section, it is necessary to obtain the set of proba-bilities P(x,y) to study the telecommunications link. This set of probabilities is directly19

linked to the physical properties of the propagation medium, e.g. the optic �ber, thewireless spectrum, the copper cable. It is also linked to the equipment in the telecommu-nications system. Both the propagation medium and the telecommunications equipmentdistort the signal. The channel is de�ned as the system formed by the combination of theequipment and the medium.Distortion occurs due to several factors:• non-linear distortion due to non-linearities in transmission equipment,• frequency o�set, which results from the use of a carrier,• phase jitter, a low frequency modulation due to coupling of the power line,• impulse noise, due to equipment switching,• thermal noise, due to thermal agitation of electrons in the receiving device• interference noise, due to other transmissions• fading, due to the attenuation of the signal, depending on the distance, the frequencyand the propagation environment.However, all impairments due to equipment imperfection can be reduced through betterengineering design, with an associated cost. The trade-o� between performance and costof the equipment is a complex topic which would require a thorough study and therefore iskept out of the scope of this thesis. The only impairment that cannot be reduced throughbetter engineering design is thermal noise, since it is inherent to electronic equipment. Ifthe model is limited to the intrinsic distortion of the transmission medium and equipment,the channel can be modeled as a linear �lter that introduces amplitude and delay distortionand adds thermal noise. Speci�cally the channel is composed of a time-variant impulseresponse h(kTs) = [h0(kTs), h1(kTs), . . . , hτ (kTs)] and additive noise n(kTs). Therefore,the transmission equation becomes

y(kTs) = [τ∑

j=0

hj((k − j)Ts)x((k − j)Ts)] + n(kTs). (II.8)Obviously, the set of probabilities P(x,y) is entirely de�ned by the set of h(kTs) andn(kTs) for all k.Additional assumptions allow us to further simplify the model.20

• Thermal noise can be modelled as Additive White Gaussian Noise (AWGN) withmean zero, and a chosen variance, depending on the Signal to Noise Ratio (SNR).• The channel frequency response can be considered �at over the bandwidth of interest,i.e. hl = 0,∀l 6= 0. The assumption of �at-fading is further justi�ed in Section II.3.• The channel response h(kTs) can be modelled as a random variable. The randomvariable is generally assumed to be zero mean complex Gaussian (Rayleigh fadingchannel) or non-zero mean complex Gaussian (Ricean fading channel).The transmission equation becomes

y = hx + n, (II.9)where h and n are complex Gaussian random variables.Finally, the detection is assumed coherent, i.e. the receiver knows h the realization ofthe channel h, also referred to as the CSI. The practicality of this assumption is discussedin Section II.3.II.1.6 Gaussian complex random variableThe Gaussian complex distribution has a central role in telecommunications and is de�nedby the Probability Density Function (PDF) of the random variable y with mean µy andvariance σ2y

γ(y) =1

πσ2y

exp

(

−(y − µy)

∗(y − µy)

σ2y

)

, (II.10)where y∗ denotes the transpose conjugate of y. Therefore,H(y) = Eγ[−log2(γ(y))]

= log(πσ2y) + log2(e)Eγ

[(y−µy)∗(y−µy)

σ2y

]

= log2(πσ2ye),

(II.11)where Eγ[.] denotes the expectancy with γ. The importance of the complex Gaussiandistribution comes from the following theorem.Theorem 3. The complex Gaussian distribution is the distribution of which entropy ismaximal. 21

Proof: let pdfy be any density function satisfying ∫Cpdfy(y)yy∗dy = σ2

y . Clearlylog2(γ(y)) is a linear function of yy∗. Furthermore,

∫

Cpdfy(y)yy∗dy = σ2

y =∫

Cγ(y)yy∗dy

⇒∫

Cpdfy(y)(c1yy∗ + c2)dy =

∫

Cγ(y)(c1yy∗ + c2)dy ∀c1, c2 ∈ C,

(II.12)implying thatEpdfy [log2(γ(y))] = Eγ[log2(γ(y))]. (II.13)Then

H(pdfy) −H(y) = −∫

Cpdfy(y) log2(pdfy(y))dy +

∫

Cγ(y) log2(γ(y))dy

= −∫

Cpdfy(y) log2(pdfy(y))dy +

∫

Cpdfy(y) log2(γ(y))dy

=∫

Cpdfy(y) log2(

γ(y)pdfy(y))dy,

(II.14)and log2 is a concave function, solog2(pdfy(y)) − log2(γ(y)) ≥ 1pdfy(y)

× (pdfy(y) − γ(y)) ∀y

⇒∫

Cpdfy(y) log2(

pdfy(y)

γ(y)))dy ≥

∫

Cpdfy(y)dy −

∫

Cγ(y)dy = 0

(II.15)Combining (II.14) and (II.15) concludes the demonstration.II.1.7 Capacity of the �at-fading channelWith the previous de�nitions, it is possible to derive the capacity of the channel. Consid-ering h �xed and known at the receiver, the channel is e�ectively similar to the well-knownAWGN channel. The de�nition of the capacity combined with (II.5) leads toC(h) = maxpdfx I(x; y) = H(y) −H(y|x) = H(y) −H(n), (II.16)since the receivers knows h. Thus, maximising I(x; y) is equivalent to maximizing H(y).That implies that y has to be a complex Gaussian random variable. Therefore, observing(II.9), the transmitted signal has to be Gaussian to achieve the capacity of the channel.To achieve the capacity, x is a complex Gaussian random variable with variance σ2

x,n is a complex Gaussian random variable with variance σ2

n and y is a complex Gaussianrandom variable with variance σ2n + hh∗σ2

x, justifyingC(h) = H(y) −H(n)

= log2(πe(σ2n + hh∗σ2

x)) − log2(πeσ2n)

= log2(1 +σ2

x

σ2nhh∗).

(II.17)22

II.1.8 Capacity of the non-ergodic channelSuppose h is chosen randomly at the beginning of the transmission and �xed thereafter.This assumption corresponds to a system transmitting frames, with coding/decoding per-formed on a frame by frame basis and the length of the frame smaller than the coherencetime of the channel, i.e. the channel is �xed for the length of a frame.

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

SNR=0dB SNR=5dB SNR=10dBSNR=15dBSNR=20dBSNR=25dBSNR=30dB

Figure II.1: CCDF of SISO Rayleigh channelThe capacity depends on h, the realization of the channel and h is a random vari-able. Therefore, the capacity itself is a random variable. The Complementary CumulativeDensity Function (CCDF) of the channel is plotted in Fig. II.1 for di�erent SNRs.It is possible to de�ne Cout(p), the outage capacity, as the maximum capacity achieved100 × p% of the time. Using a code designed for a channel capacity of Cout(p) directlyleads to a frame error rate of 1 − p.II.1.9 Capacity of the ergodic channelWhen either coding occurs on a large number of frames, or the coherence time of thechannel is much smaller than the length of the frame, the channel is ergodic. The capacityof the channel can then be expressed as: 23

Cergodic = Eh[log2(1 +σ2

x

σ2n

hh∗)]. (II.18)The ergodic capacity of the channel is plotted as a function of the SNR in Fig. II.2.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Cap

acity

(bp

s/H

z)

SNR in dBFigure II.2: Capacity of the ergodic SISO Rayleigh channel vs SNR in dB.Interestingly, at high SNRs, the capacity increases linearly with the SNR expressed indecibels (dB). Therefore, the power at the transmitter has to be doubled to gain an extrabit of capacity.II.2 MIMO CapacityMost of the research e�ort in the �eld of telecommunications has been spent on improvingcoding techniques to achieve data rates close to the capacity at acceptable probability oferror (depending on the application). The discovery of turbo-codes by Berrou et al. [11]was a considerable step in this regard. Therefore, it became increasingly clear that furthermodi�cation of the coding would not result in a signi�cant improvement of the transmissionlink. It is clear from the previous section that for a given h and n, the only other solutionfor increasing the capacity of the channel was to increase the power of the signal σ2x. This24

is undesirable since it also increases the interference to other co-channel users. Higherdata rates could only be achieved by discovering channels with higher capacity.Section II.1 implicitly assumes that the transmitter and the receiver use only oneantenna. Several practical systems implemented in cellular networks proved that usingseveral antennas could improve the quality of the wireless link. This directly triggeredinterest in MIMO channels.II.2.1 MIMO channel modelConsider a system with MT antennas at the transmitter and MR antennas at the receiver.Transmission is over a �at fading channel. Therefore the time index is dropped and theanalysis is conducted on the transmission of a single sample.x is a vector of the input symbols (x ∈ CMT ), i.e. x = [x1, x2, . . . , xMT

] with x1 thesymbol transmitted on the �rst antenna.H the channel matrix (H ∈ CMR×MT ) is the realization of a random variable matrixassumed to have complex Gaussian random variable entries with zero mean (Rayleighchannel) or non-zero mean (Ricean channel). The entries of H can be correlated randomvariables (correlated channel) or Independent and Identically Distributed (i.i.d. channel).n is a vector of AWGN on the receiving antennas (n ∈ CMR). n is complex Gaussian,with zero mean and equal variance in the independent real and imaginary components. Itis also assumed that the noise at each receiving antenna is independent and the transmittedpower is normalized by the noise power at a single receiving antenna. This can be writtenas:

E[nn∗] = IMR, (II.19)where IMR

is the identity matrix of dimension MR.Assumptions are similar for Single Input Single Output (SISO) or MIMO channels.The only signi�cant di�erence between the two cases is the possibility in the MIMO casefor the entries of the vector or matrices to be correlated. The in�uence of this parameteris discussed in Section II.3.The vector of received symbols can be expressed as:y = Hx + n. (II.20)25

II.2.2 Circularly symmetric complex Gaussian distributionThe vector equivalent of the Gaussian distribution is a circularly symmetric Gaussiandistribution, i.e. a vector x with entries being complex Gaussian random variable and,de�ning x =

Re(x)

Im(x)

,E[(x − E[x])(x − E[x])∗ =

1

2

Re(Q) −Im(Q)

Im(Q) Re(Q)

, (II.21)where Re(c) is the real part of c, Im(c) the complex part of c and Q an hermitian positivede�nite matrix. Therefore a circularly complex Gaussian random vector is speci�ed byprescribing E[x] and E[(x − E[x])(x − E[x])∗ = Q.The probability density of a circularly symmetric complex Gaussian with mean µx andcovariance Q is given byγ(x) =

1

det(πQ)exp(−(x − µx)∗Q−1(x − µx)). (II.22)The di�erential entropy of a circularly symmetric complex Gaussian with mean zero andcovariance Q is given by

H(γ) = Eγ[− log2 γ(x)]

= log2 det(πeQ).(II.23)As in the SISO case, the importance of the Gaussian distribution comes from thefollowing theorem.Theorem 4. Suppose the complex random vector x ∈ CMR is zero-mean and satis�es

E[xx∗] = Q, then the entropy of x satis�es H(x) ≤ log2 det(πeQ) with equality if andonly if x is a circularly symmetric complex Gaussian.The demonstration is similar to its equivalent in the SISO case [12].Finally, consider A a complex matrix, A ∈ Ci×j, if x is circularly symmetric complexGaussian, so is Ax [12]. If x and n are circularly symmetric complex Gaussian, so isx + n [12].II.2.3 Capacity of the MIMO �at-fading channelAssume H �xed and known at the receiver. The capacity of the channel is given by

C(h) = maxpdfx I(x; y) = H(y) −H(y|x) = H(y) −H(n), (II.24)26

since the receiver knows h. Thus, maximizing I(x; y) is equivalent to maximizing H(y).Theorem 4 implies that the capacity is reached if and only if y is circularly symmetriccomplex Gaussian, which implies in turn, from (II.20), that x is also circularly symmetriccomplex Gaussian.The transmitted signal is usually constrained by a power limitation, i.e. E[x∗x] ≤ P .This, in turn, helps specifying the nature of x: if x satis�es the power limitation, so doesx − E[x]. Thus, we can restrict our attention to zero-mean x since a non-zero meanwould only result in a non-zero mean y which has the same entropy as the correspondingzero-mean y.Therefore, if x is zero-mean with covariance E[xx∗] = Q then y is zero-mean withcovariance E[yy∗] = HQH∗ + IMR

. The mutual information of the channel is given byI(x; y) = log2 det(IMR

+ HQH∗). (II.25)The exact formula for the capacity of the channel depends on further assumptions,which in turn determine the matrix Q providing the capacity achieving distribution.II.2.3.1 Ergodic Rayleigh i.i.d. channel, no CSI at the transmitterAssume the entries of H are independent and zero-mean Gaussian with independent realand imaginary parts, each with variance 1/2. Therefore each entry of H has uniformlydistributed phase and Rayleigh distributed magnitude, with expected magnitude squareequal to unity.The mutual information is given byI(x; (y,H)) = I(x; H) + I(x; y|H)

= I(x; y|H)

= EH[I(x; y|H = H)].

(II.26)Since Q is positive de�nite, ∃U unitary such that Q = UDU ∗ where D is non-negativeand diagonal. Therefore the mutual information can be expressed asI(x; (y,H)) = EH[log2 det(IMR

+ HQH∗)]

= EH[log2 det(IMR+ (HU )D(HU )∗)].

(II.27)Theorem 5. Suppose H is a complex Gaussian i.i.d. matrix, each entry with zero-meanand equal variance. The distribution of H is the same as the distribution of UHV ∗ forany unitary matrices U and V . 27

The theorem is proved in [12]. Combining the theorem with (II.27) leads toI(x; (y,H)) = EH[log2 det(IMR

+ (H)D(H)∗)]. (II.28)Therefore the optimal Q is non-negative diagonal. Given any permutation matrix Π,consider QΠ = ΠQΠ∗. Since HΠ has the same distribution as H ,

EH[log2 det(IMR+ (H)Q(H)∗)] = EH[log2 det(IMR

+ HQΠH∗)]. (II.29)For any H , if Q is positive de�nite, then IMR+HQH∗ is positive de�nite, and log2 det(.)is concave on the set of positive de�nite matrices. Thus, de�ning

Q =1

MT !

∑

Π

QΠ (II.30)gives usEH[log2 det(IMR

+ HQH∗)] ≥ EH[log2 det(IMR+ HQH∗)], (II.31)and Q is a multiple of the identity matrix. Therefore, the optimal Q must be of the form

αI. Clearly, the maximum is achieved when α is the largest possible, i.e. α = P/MT sinceQ is constrained to trace(Q) ≤ P . It is interesting to note that P is the average SNR ateach receiving antenna.The capacity of the channel is given by

C = EH[log2 det(IMR+

P

MT

HH∗)]. (II.32)An analytical expression of the capacity can be derived by identifying W = HH∗when MR ≤ MT (or W = H∗H when MR > MT ) as a random matrix following aWishart distribution [12].The capacity of the channel is plotted against the SNR in Fig. II.3. Obviously, thecapacity of the channel increases with the number of antennas at both ends of the link. Thecapacity of SISO and MIMO channels is linear, for high SNRs, with the SNR expressedin dB, but the slope of the curve depends on the number of antennas.The capacity of the channel at 20 dB of SNR is plotted against the number of antennasin Fig. II.4. The capacity of the channel increases linearly with the number of antennasat both ends of the link, i.e. with the minimum of either the number of antennas at thetransmitter or the number of antennas at the receiver. This point is made clear by theexample of a system where MT increases and MR = dMT /2e. When the system gains anadditional antenna at the transmitter, the capacity hardly changes. However, when both28

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Cap

acity

(bp

s/H

z)

SNR in dB

MR

=MT=4

MR

=MT=3

MR

=MT=2

MR

=MT=1

Figure II.3: Capacity of the Rayleigh channel vs SNR, ergodic Rayleigh i.i.d. channel, noCSI at the transmitterthe transmitter and the receiver obtain an additional antenna, the capacity of the systemincreases signi�cantly.Note that the channel is not symmetric. The capacity of the channel with MR > MTis always higher than the capacity with MR < MT . This is simply due to the fact that theCSI is available at the receiver but not the transmitter. This point is further discussed inSection III.5.II.2.3.2 Non-ergodic Rayleigh i.i.d. channel, no CSI at the transmitterThe mutual information of the channel is given byI(x; y) = log2 det(IMR

+ HQH∗), (II.33)and is a random variable.Therefore, however small the rate we attempt to communicate at, there is a non- zeroprobability that the realized H is incapable of supporting it, no matter the length of thecode applied. A good example is the case where the entries of H are zeros (however thiscase in itself has a probability of zero).It is possible to examine the trade-o� between the outage probability and the supported29

1 2 3 4 5 6 7 8 9 105

10

15

20

25

30

35

40

45

50

55

Cap

acity

(bp

s/H

z)

n, number of antennas

MR

=MT=n

MR

=n,MT= n/2

MR

= n/2 ,MT=n

MR

=n,MT=1

MR

=1,MT=n

Figure II.4: Capacity of the Rayleigh channel vs number of antennas, SNR=20dB,ergodic Rayleigh i.i.d. channel, no CSI at the transmitterrate. Namely, given a rate R and a power P , one can �nd pout(R,P ) such that R is less thanthe capacity of H with total transmitted power P , except on a set of H with probabilityless than pout.From theorem 5, for any unitary matrix U ,log2 det(IMR

+ HQH∗) (II.34)andlog2 det(IMR

+ HUQU ∗H∗) (II.35)have the same distribution. Hence, it is only necessary to examine diagonal Q. The choiceof a diagonalQ is not only a theoretical simpli�cation, but also corresponds to the simplesthardware implementation possible (the symbols are independent from one antenna to theother).The symmetry of the problem suggests the following conjecture.Conjecture. The optimal Q is, up to a permutation of the indexes of the antennas, ofthe formP

MactT

diag(1, . . . , 1︸︷︷︸

MactT ones, 0, . . . , 0

︸︷︷︸

MT−MactT zeros) (II.36)30

where MactT is the number of antennas actually used for transmission (active antennas).The value of Mact

T depends on the rate: the higher the rate (i.e. the higher the outageprobability), the smaller the MactT .The ordering of the entries of Q has been applied here without loss of generality: it isnot an antenna reordering but simply corresponds to a reordering of the columns of U .

0 1 2 3 4 5 6 7 8 9 100

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Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1

MT=1

MT=10

MT=10

Figure II.5: CCDF of the capacity of the channel for varying MT , with MR = 1 andSNR=20dBThe CCDF of channels with varying number of transmitting antennas and equal powerallocation among the transmitter is presented in Fig. II.5. At low outage probability (lowdata rate), the best channel is the Multiple Input Single Output (MISO) channel. However,at high outage probability (high data rate), the SISO channel has a higher outage capacitythan the MISO channel.These results can be understood intuitively as follows: when the transmitter does nothave access to the CSI and the transmission system has only one receiver, transmitting onseveral antennas is equivalent to transmitting on an average channel, i.e. the average of theindividual paths from each transmitting antenna to the receiver. Hence, the behaviour ofthe system is likely to be more stable and the capacity of the channel does not vary much.In the asymptotic case of an in�nite number of antennas, the capacity of the channel is31

�xed. This is in contrast with a single transmitting antenna system facing in turn excellentand poor transmission environments.A theoretical justi�cation follows. The capacity of the channel can be expressed asC(h) = log2(1 +

1

MT

(

MT∑

i=1

hih∗i )) (II.37)where h = (h1,h2, . . . ,hMT

). Obviously κ =∑MT

i=1 hih∗i is a chi-square variate with 2×MTdegrees of freedom (denoted χ2

2MT) normalized so that E[hih

∗i ] = 1. The PDF of 1

MTκ isgiven by [13]pdf 1

MTκ(κ) =

MT

(√

0.5)2MT 2MT Γ(MT )(MT × κ)MT−1 exp(−MT × κ), (II.38)where Γ(.) is the gamma function de�ned as

Γ(r) =∫∞

0tr−1 exp(−t)dt, r > 0

Γ(r) = (r − 1)!, r an integer > 0

Γ(1/2) =√

π,

Γ(3/2) = 12

√π.

(II.39)The PDF of 1MT

κ is plotted in Fig. II.6 for varying MT .The analysis is more complicated when the receiver side has more than one antenna.Fig. II.7 presents CCDF of the outage capacity for varying MT and MR = 2, MR = 3,MR = 4 and MR = 5. For MT < MR, increasing the number of transmitting antennasleads to a signi�cant gain in capacity. However, for MT ≥ MR, increasing the numberof transmitting antennas only leads to a limited gain in outage capacity for low outageprobability. Intuitively, MR receiving antennas can separate MR spatially independentsignals, e.g. independent signals coming from MR transmitting antennas. Therefore, forMT < MR, increasing the number of transmitting antennas creates a new spatial orthogo-nal mode of excitation [14], whereas for MT ≥ MR, any new transmitting antenna can onlybe used to increase the reliability of the existing MR channel transmission eigenmodes.These results are discussed in Section III.2 and further justi�ed in the following section.II.2.3.3 Non-ergodic Rayleigh i.i.d. channel, �xed power allocation at thetransmitterMost papers in the literature (e.g. [15]) assume that the power allocation at the transmitteris of the form

E[xx∗] = Q =P

MT

IMT. (II.40)32

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

p(k)

k

MT=1

MT=2

MT=5

MT=10

Figure II.6: PDF of 1MT

κ with κ a chi-square random variable with 2 × MT degrees offreedom, normalized so that E[χ22] = 1.This assumption is rarely discussed and is assumed "reasonable" or "intuitive".However, this particular choice of Q is not always optimal: in Fig II.5 the capacity ofthe channel is higher for MT = 1 than for MT > 1 for a high outage probability. Q can bechosen from the ensemble of diagonal matrices as explained in Section II.2.3.2. Howeverthis choice of Q can be justi�ed by examining the results of Fig. II.7. For MR > 1, thecapacity of the MIMO channels increases with MT . Allocating more power to a speci�cantenna, rather than allocating power uniformly on all antenna is an intermediate casewhere the diversity of the channel is not fully exploited.Finally, this speci�c choice of Q leads to the simplest MIMO hardware implementationat the transmitter.In such a case, the capacity is given by

C(H) = log2 det(IMR+

P

MT

HH∗). (II.41)When H is Rayleigh and the number of antennas is large, the normalized capacity( C(H)min(MT ,MR)

) can be approximated by a Gaussian random variable [16]. The importanceof this result comes from the fact that this approximation remains accurate, even for asmall number of antennas. 33

0 5 10 15 200

0.2

0.4

0.6

0.8

1

Capacity (bps/Hz), MR

=2

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1 M

T=10

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1


=3

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1

MT=2

MT=10

0 10 20 300

0.2

0.4

0.6

0.8

1


=4

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1

MT=2

MT=3

MT=10

0 10 20 30 400

0.2

0.4

0.6

0.8

1


=5

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1

MT=2

MT=3

MT=4

MT=10

Figure II.7: CCDF of the capacity of the channel for varying MT , with MR = {2, 3, 4, 5}and SNR = 20dBSuppose MR → ∞, MT → ∞ with MT /MR = α, then the mean is given by [17]E[C/ min(MT ,MR)] = (log2(r+P ) + (1 − α) log2(1 − r−) − r−α

ln 2) max(1, 1/α). (II.42)where

r± , (r ±√

r2 − 4/α)/2 (II.43)andr , 1 +

1

α+

1

P. (II.44)The variance of C is also given in [17] as,

σ2C = − log2 e log2 |1 − α × (P

4)2 × (1/α − 1 − 1/P +

√

(1/α − 1 − P )2 + . . .

4/(αP ))2 × (1 − 1/α − 1/P +√

(1 − 1/α − P )2 + 4/P )2|,(II.45)where |.| denotes the absolute value. 34

Fig. II.8, presents the CCDF of the capacity for MT = MR, as well as the correspond-ing Gaussian approximation. The capacity is well approximated by a Gaussian random

0 5 10 15 20 25 30 35 400

0.1

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1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=M

R=1 2 3 4 5 6

CCDF(C) Gaussian approximation

Figure II.8: CCDF of the capacity of the channel for varying MT = MR and SNR= 20dBvariable. The two curves are only easy to distinguish for the special cases MT = MR = 1and MT = MR = 2. The accuracy of the Gaussian approximation increases with thenumber of antennas.Fig II.9 and II.10 present both the CCDF of the capacity and its Gaussian approxi-mation to illustrate the accuracy of the model for asymmetrical channels. In [17], it isproven that in the asymptotical case of a large number of antennas, the CCDF becomesGaussian.The Gaussian model for the capacity provides us with a good understanding of thebehaviour of the capacity for varying parameters MR, MT and P . The e�ect of an in-crease in the number of antennas can be understood by examining (II.42) in asymptoticsituations.Telatar proved in [12] that the capacity of the i.i.d. MIMO channel increases linearlywith min(MT ,MR) and logarithmically with P . Novel series expansions of the Gaussianapproximation of the mean capacity are presented in the following to highlight the impor-tance of the minimum number of antennas at either the transmitter or the receiver.35

Suppose MT � MR, β = 1/α is close to zero and for reasonable variations of MT andMR, β remains close to zero. The series expansion of (II.42) with respect to β around thepoint β = 0 is given by:

E[C/MR] = log2(P + 1) − 12× P 2

(P+1)2 ln(2)β + O(β2)

= log2(P + 1) + O(β).(II.46)Increasing MR leads to a nearly linear gain in capacity. Increasing MT only varies themagnitude of the terms factor of β and those terms are negligible. Therefore increasing

MT leads to a negligible increase in capacity. CCDF of the capacity in this asymptoticsituation is given in Fig. II.11.Suppose MR � MT , α is close to zero and for reasonable variations of MT and MR, αremains close to zero. The series expansion of (II.42) with respect to α around the pointα = 0 is given by:

E[C/MT ] = log2(P ) − log2(α) − 12× P−2

P ln(2)α + O(α2)

= log2(P ) − log2(MT ) + log2(MR) + O(α).(II.47)Increasing MR only leads to a logarithmic increase in capacity. Furthermore, since MR �

MT , MR � 1. Therefore increasing MR leads to a negligible increase in capacity. Increas-

0 5 10 15 20 25 300

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Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1 2 3 4 5 6

CCDF(C) Gaussian approximationFigure II.9: CCDF of the capacity of the channel for varying MT , MR = 4 andSNR= 20dB36

0 5 10 15 20 25 30 350

0.1

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1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1 2 3 4 5 6

CCDF(C) Gaussian approximationFigure II.10: CCDF of the capacity of the channel for varying MR, MT = 4 andSNR= 20dB

ing MT does lead to an increase in capacity (since log2(MR) > log2(MT )). The CCDF ofthe capacity in this asymptotic situation is given in Fig. II.12.The outage probability might be imposed by requirements of the system, e.g. theMedium Access Control (MAC) layer might require a Frame Error Rate (FER) betterthan a given constant. If it is not the case, designing the system to allow for maximumthroughput is a sensible design criterion. The throughput is the rate of data actuallyreceived at the transmitter. Considering that every outage leads to a lost frame, an upperbound of the throughput consists of the product of the attempted data rate multiplied bythe outage probability. Fig. II.13 presents the throughput of MIMO systems versus theassumed capacity of the channel.Fig. II.13 further justi�es the choice of a small outage probability for practical telecom-munications systems. The maximum throughput of most systems corresponds to a smalloutage probability, e.g. pout = 0.245 for MR = MT = 1 and SNR= 20dB, pout = 0.125 forMR = MT = 2 and SNR= 20dB, and pout = 0.028 for MR = MT = 5 and SNR= 20dB.37

6 8 10 12 14 16 18 20 220

0.1

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0.3

0.4

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0.8

0.9

1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=100,M

R=1

MT=100,M

R=2

MT=100,M

R=3

MT=120,M

R=1

Figure II.11: CCDF of the capacity for varying MR,MT , MT � MR and SNR= 20dBII.2.3.4 Non-ergodic channel, CSI at the transmitterConsider a channel matrix H �xed and known at both sides of the link. The mutualinformation is given byI(x; y) = log2 det(IMR

+ HQH∗). (II.48)The SVD of H is de�ned byH = UΣV ∗, (II.49)where U and V are unitary and Σ is diagonal non-negative. Therefore,

I(x; y) = log2 det(IMR+ ΣV ∗QV Σ

∗). (II.50)Observe that Q = V ∗QV is non-negative de�nite if and only if Q is non-negative de�nite,and trace(Q) = trace(Q). Hence, the maximization can be carried over Q.Note that for any non-negative de�nite A, det A ≤∏i Ai,i. Hence,det(IMR

+ ΣQΣ∗) ≤

∏

i

(1 + Qi,iΣ2i,i) (II.51)with equality when Q is diagonal. Thus the maximising Q is diagonal with entries foundby water�lling power allocation [13]:

Qi,i = (Pwf − Σ−2i,i )+,∀i ∈ {1, . . . ,MT}, (II.52)38

10 15 20 25 30 35 400

0.1

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1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=1,M

R=100

MT=2,M

R=100

MT=3,M

R=100

MT=1,M

R=120

Figure II.12: CCDF of the capacity for varying MT ,MR, MR � MT and SNR= 20dBwhere Pwf is chosen to satisfy∑i Qi,i = P and (.)+ indicates that only non-negative valuesare acceptable. The capacity of the channel is given byC(H) =

∑

i

(log2(PwfΣ2i,i))+. (II.53)The CCDF of the capacity of the channel is given for varying MR = MT with andwithout CSI at the transmitter in Fig. II.14. The capacity of the channel is higher when theCSI is available at the transmitter. This is easily justi�ed since the transmitter allocatesmore power to the best spatial transmission eigenmodes, resulting in better transmission.However, the capacity gained by providing CSI to the transmitter is negligible when thechannel is i.i.d.. This fact is further discussed in Section III.5.The CCDF of the capacity of the channel is given for varying MR and MT in Fig.II.15. The other major di�erence when CSI is available at the transmitter is that a(MT = i,MR = j) channel has the same capacity as a (MR = i,MT = j) channel, ∀i, j ∈ N.II.2.3.5 Ergodic channel, CSI at the transmitterThe capacity of the ergodic channel is simply given by

C = EH[∑

i

(log2(Pwf (H)Σ2i,i)], (II.54)39

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

pout

=0.24512

pout

=0.125

pout

=0.076172

pout

=0.048828

pout

=0.03418

pout

=0.02832

Capacity (bps/Hz)

Thr

ough

put (

(1−

p out)

× C

apac

ity)

MT=M

R=6

MT=M

R=5

MT=M

R=4

MT=M

R=3

MT=M

R=2

MT=M

R=1

Figure II.13: Maximum throughput of the system for varying MR = MT and SNR= 20dBwhere Pwf (H) is the level determined by water�lling of the singular values of H and Σis a diagonal matrix with entries the singular values of H .Therefore the capacity of the ergodic channel is simply the mean corresponding to theCCDF of Fig. II.14 and II.15. The capacity of the ergodic channel for varying MR, MT isgiven in Fig. II.16. Obviously, a (MT = i,MR = j) ergodic channel has the same capacityas a (MR = i,MT = j) ergodic channel, ∀i, j ∈ N.Finally, at high SNR, the capacity of the ergodic channel is increasing linearly withthe SNR expressed in dB, and the slope of the curve depends on MR, MT , as shown inFig. II.17.II.3 Discussion of the assumptionsThe assumptions taken in Sections II.1 and II.2 are discussed here, to gain insight intotheir respective meaning. 40

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=M

R=6

MT=M

R=5

MT=M

R=4

MT=M

R=3

MT=M

R=2

MT=M

R=1Figure II.14: CCDF of the capacity of the channel with CSI at the transmitter (solidlines) and without CSI at the transmitter (dotted lines) for varying MR = MT andSNR= 20dBII.3.1 Coherent detectionThe detection is assumed coherent, i.e. the receiver knows the CSI. Measuring the channelat the receiver is actually a complicated task and the accuracy of the CSI estimationis critical in the performance of the system as is further discussed in Section IV.4. Mostsystems use pilot symbols (known symbols sent by the transmitter) to estimate the channel.Pilot symbols can be inserted at the beginning of the frame (802.11a), in the middle ofthe frame (GSM) or as a separate signal (IS-95). Blind channel estimation can be appliedto avoid the overhead due to pilot symbols.Coherent detection o�ers a gain of 3dB in performance over di�erential detection whenthe channel estimation is perfect. Non-coherent detection is usually applied in the formof di�erential coding schemes and unitary space- time codes [18].Overall, coherent detection is a �tting assumption for most wireless LAN standards(including 802.11a and Hyperlan 2). However, imperfect CSI estimation invariably reducesthe performance of the system and so special emphasis must be put on the design of thechannel estimation techniques. 41

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bps/Hz)

Pro

babi

lity(

Cap

acity

>A

bsci

ssa)

MT=6,M

R=4

MT=4,M

R=6

MT=5,M

R=4

MT=4,M

R=5

MT=4,M

R=4

MT=4,M

R=4

MT=3,M

R=4

MT=4,M

R=3

MT=2,M

R=4

MT=4,M

R=2

MT=1,M

R=4

MT=4,M

R=1

Figure II.15: CCDF of the capacity of the channel with CSI at the transmitter forvarying MR, MT and SNR= 20dBII.3.2 Time and frequency properties of the channelThe three main assumptions about the channel are:• �at frequency fading,• slow fading,• block fading.The channel is assumed �at over the frequency bandwidth of interest, i.e. the distortionof the signal due to the channel is simply a multiplication by a complex number. It isobvious that the wideband channels used in modern wireless communication are not �atover the whole frequency bandwidth (20 MHz for 802.11a). However, several telecommuni-cations standards, including 802.11a and Hyperlan 2, use Orthogonal Frequency DivisionMultiplexing (OFDM), a modulation technique which divides the available bandwidthinto subchannels of reduced bandwidth. Hence, the assumption of �at-fading is justi�edin these cases.The channel is assumed slow fading, i.e. the CSI is constant over the length of a symbol.OFDM modulation increases the length of symbols, placing more stress on the assumption42

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

Cap

acity

(bp

s/H

z)

n, number of antennas

MR

=MT=n

MR

=n,MT= n/2

MR

= n/2 ,MT=n

MR

=n,MT=1

MR

=1,MT=n

Figure II.16: Capacity of the channel with CSI at the transmitter vs number ofantennas, at SNR= 20dBof slow fading. The combination of the slow fading and coherent demodulation impliesthat the channel must be really slow fading in time. Speci�cally, the time coherence ofthe channel has to be several times longer than a symbol period, since the CSI estimatedby the pilot symbols is applied to decode a number of surrounding data symbols, e.g. forthe 802.11a standard, the time coherence of the channel should be at least an order ofmagnitude higher than the 4µs duration of an OFDM symbol. Finally, the transmissionmodel chosen is relevant and applicable to OFDM since OFDM eliminates Inter-Symbol-Interference (ISI).The channel is assumed block fading, i.e. the channel fading is independent from oneblock to the next. Though a poor approximation in practice since the channel is contin-uously fading, most communication devices recover this property of uncorrelated fadingon adjacent blocks through the use of interleavers. An interleaver simply permutes thetransmitted symbols, while at the receiver the corresponding de-interleaver regeneratesthe original order. Interleavers allow fading to be independent on consecutive symbols.43

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Cap

acity

(bp

s/H

z)

SNR in dB

MR

=MT=4

MR

=MT=3

MR

=MT=2

MR

=MT=1

Figure II.17: Capacity of the channel with CSI at the transmitter vs SNRII.3.3 Independence of the noise entriesThe noise is assumed uncorrelated at the receiver antennas. This is equivalent to discardingall other sources of noise than thermal noise in the receiver front ends. In particular, theassumption cannot account for interference due to other users on adjacent frequency bands,or even on the same band in the case of unlicensed frequency bands such as the Industrial,Scienti�c and Medical (ISM) band used in the 802.11x and Hyperlan 2 standards. Thetheoretical capacity of MIMO systems in the interference regime is still an open question.The distribution of the noise is assumed identical on each receiver antenna. Thisassumption suggests that the receiver antennas' front ends are supposed identical.II.3.4 Single user channelThroughout Section II.1 single user channels are considered. The capacity of multiuserchannels is higher than the capacity of single user channels due to the multiuser diversity,i.e. the ability to allocate the channel to the user facing the best propagation conditions.Insights on the capacity of MIMO multiuser channels are presented in [19]. However,several practical systems, including 802.11a, 802.11g and Hiperlan 2, use time divisionmultiple access, where a single user transmits during a time slot. Therefore, on each time44

slot, the relevant capacity corresponds to the capacity of single user MIMO channels.II.4 ConclusionThe capacity of a channel is a fundamental concept in telecommunications since it deter-mines how much information can be transmitted. It is possible to increase the capacityof a channel by increasing the number of antennas at the receiver and transmitter. Thenon-ergodic channel can be studied in term of outage capacity which is well approximatedby a Gaussian variable. The ergodic capacity of a Rayleigh i.i.d. channel (with CSI atthe receiver only) increases linearly with the minimum number of antennas at the trans-mitter or at the receiver. These results are demonstrated for �at-fading channels underthe assumptions of slow fading, block fading, perfect channel estimation and spatial andtemporal independence of the noise at the receivers.

45

Chapter IIIUsing the large capacity of MIMOchannelTelecommunications theory is concerned with determining the capacity of the channel,designing the transmitted signal x and reliably recovering the information transmitted inx from the received signal y. Chapter II has demonstrated that MIMO channels have alarge capacity, but has not indicated how to exploit this large capacity.Chapter III aims at providing an overview of some well-known techniques to transmitover the MIMO channel and detect the transmitted symbols. Systems with no spatialcoding at the transmitter are introduced in Section III.1.1 with the corresponding decod-ing algorithms. Space-time codes are introduced in Section III.1.2. The performance ofuncoded and coded systems are compared in terms of BER in Section III.1.3. The resultsvary greatly with the simulation parameters and the number of parameters is large. Asimple comparison of MIMO schemes is therefore impractical. The concepts of diversityand multiplexing are introduced in Section III.2 to gain insight into the performance ofMIMO transmission schemes.Most theoretical results are only valid when the channel is i.i.d.. Section III.3 presentssome new results concerning the capacity of the Ricean channel. These results havebeen partly accepted for publication [20] and submitted for publication [21]. SectionIII.3 demonstrates that the ergodic normalized capacity of the Ricean MIMO channelapproaches the corresponding normalized capacity of the underlying scattering channelwhen the antenna numbers are large and there is no CSI available at the transmitter.Theoretical bounds of the normalized capacity of the Ricean channel are proven. These47

bounds allow the capacity of the Ricean channel to be estimated in the asymptotic limitof a large number of antennas without recourse to simulation.BER simulations in Section III.4 demonstrate that uncoded MIMO transmission tech-niques are badly designed for highly specular MIMO channels. On the contrary, space-timecodes can be robust to channel correlation. However, most space-time codes do not bene�tfrom spatial multiplexing and therefore need to use high order constellations to achieve anon-negligible portion of the channel capacity when the number of antennas grows large.The shortcomings of both uncoded and coded MIMO transmission schemes are partiallydue to the fact that the CSI is not available at the transmitter. CSI at the transmitterincreases the capacity of the MIMO channel and is especially bene�cial when the channelis highly specular as demonstrated in Section III.5. The capacity gain when providing theCSI at the transmitter grows linearly with the number of antennas.III.1 Review of MIMO transmission techniquesThe large capacity o�ered by MIMO channels triggered numerous attempts to designtransmission techniques using several antennas. Some of the most widespread techniquesare presented in this section.III.1.1 Uncoded systemsThe simplest method to transmit over the MIMO channel is to send uncorrelated datastreams to each transmitter with equal power. The resulting transmission equation is(II.20) with E[xx∗] = P/MT × IMT. The transmitter does not require the CSI and thetransmission architecture can easily be implemented: the MIMO encoding simply consistsof a serial to parallel converter, as presented in Fig. III.1. The decoder uses a MIMOdetection block either to separate the transmitted symbols (which are then detected bySISO detectors) or to fully detect the transmitted symbols. The MIMO detection block isa critical component of the transmission system and is likely to be di�cult to implementdue to its high complexity. MIMO uncoded systems su�er from the disadvantage that

MR has to be greater than or equal to MT for linear MIMO detection schemes to operate.This point is further discussed in Section III.2. Though other detection schemes, e.g.the maximum likelihood scheme presented in Section III.1.1.1, are applicable even for MT48

greater than MR, their complexity is usually higher than the complexity of linear detectionschemes.4

x3

x2

x1

4y3y2y1y

ConverterParallel

ToSerial

H

4

3

2

n

n

n

n1

MIMO

Detection

x

SerialTo

ParallelConverter

y1

y2

y3

y4

1x

2x

3x

4x

1n

n

n

n

2

3

4

HMIMO

Detection

SISO Detector

SISO Detector

SISO Detector

SISO Detector

Converter

ParallelTo

Serial

Figure III.1: Block diagrams of the two possible uncoded MIMO transmissionarchitectures.Several classical MIMO detection schemes for uncoded transmission are presented inthe following. All of them assume perfect CSI at the receiver.III.1.1.1 Maximum-likelihood detectionThe MIMO detection algorithms' role is to detect the transmitted symbols on each of thetransmitting antennas. To this e�ect, the decoder not only knows the channel, but alsothe constellation used at each transmitting antenna S = {s1, s2, . . . , sk} for a constellationtransmitting log2(k) bits per symbol. Therefore, the transmitter knows the set of possibletransmitted symbols, i.e. the MIMO constellationX =

x1 =

s1

s1...s1

,x2 =

s2

s1...s1

, . . . ,xkMT =

sk

sk...sk

. (III.1)Note that the cardinality of the MIMO constellation X is kMT if the constituent constel-lation S is of size k. The receiver can determine the set of noiseless received symbols{y1,y2, . . . ,ykMT } = {Hx1,Hx2, . . . ,HxkMT } and deduce the probability

∀i P(xi|H ,y) = P(n = y − yi)

= 1πMR

exp(−(y − yi)∗(y − yi))

(III.2)since n is AWGN with variance one and mean zero on each receiving antenna.49

Therefore it is straightforward to determine the likelihood of each possible transmittedsymbol xi and to select the most likely (hard decision) or to send forward the probabil-ity of each symbol (soft decision). The Maximum Likelihood (ML) detector is the bestdetector, under the assumption of perfect channel estimation. However, the MIMO con-stellation rapidly grows in size with the number of antennas and the size of the componentconstellation. Therefore the complexity of the ML detector becomes prohibitive. For eachreceived symbol, the ML detector has to obtain kMT distances and select the smallest. Ad-ditionally, for each frame, the ML detector has to determine the set of noiseless receivedsymbols {yi}, i.e. compute kMT (MR,MT ) × (MT , 1) complex matrices multiplications.Alternative algorithms have been proposed to perform ML detection with reducedcomplexity, e.g. the sphere decoder [22], [23], [24].III.1.1.2 Linear detectionA linear detector obtains an estimate of the transmitted symbols by a linear combinationof the received symbols. Speci�cally, the detector obtainsx = Ay + B, (III.3)where x is an estimate of x and A and B are two matrices assumed �xed with H .The main advantage of linear detectors is their reasonable complexity. The complexitycan be separated into the computation of A and B on a frame by frame basis (i.e. foreach realisation of H) and the MIMO detection on a symbol per symbol basis. Thecomplexity of the computation of A and B varies depending on the speci�c linear receiverconsidered. The complexity of the MIMO detection is low, i.e. a matrix multiplication anda matrix addition. Another advantage of the linear receiver is that the symbols are detectedsimultaneously, on the contrary of some non-linear techniques. The main disadvantage ofthe linear receiver is that higher performance can be obtained with non-linear detection.The choice of A and B is discussed in the following. The detector aims at estimatingthe symbols as well as possible, i.e. minimising the expectancy of the power of the error

E[‖x − x‖2F ] = E[(Ay + B − x)∗(Ay + B − x)]

= E[(Ay − x)∗(Ay − x)] + . . .

E[B∗(Ay − x) + (Ay − x)∗B] + E[B∗B],

(III.4)where ‖.‖F denotes the Frobenius norm. The best choice of the additive component isB = 0, a matrix of zeros, since E[Ay − x] = 0.50

The Zero-Forcing (ZF) linear detector is de�ned by A = H+, where .+ denotes thepseudo-inverse of a matrix. The estimated symbols arex = x + H+n. (III.5)The ZF receivers su�er from noise ampli�cation when the channel matrix is ill-conditioned.The Minimum Mean Squared Error (MMSE) detector aims at minimising the ex-pectancy of the power of the error (mean-squared error). This criteria is equivalent toinsuring that the error is orthogonal to each received symbol

E[(x − x)y∗] = 0

⇔ E[xy∗] − E[xy∗] = 0

⇔ E[x(Hx + n)∗] − A × E[yy∗] = 0

⇔ E[xx∗]H∗ − A × E[(Hx + n)(Hx + n)∗] = 0

⇔ PMT

H∗ − A × HE[xx∗]H∗ − A × E[nn∗] = 0

⇔ PMT

H∗ − PMT

A × HH∗ − A = 0

⇔ A × ( PMT

HH∗ + IMR) = P

MTH∗

⇔ A = PMT

H∗(IMR+ P

MTHH∗)−1,

(III.6)where the inverse is justi�ed by the fact that IMR

+ PMT

HH∗ is hermitian positive de�nite.III.1.1.3 Vertical Bell laboratories layered space-timeThe Bell laboratories Layered Space-Time (BLAST) architecture encompasses a seriesof systems proposed by the Bell Laboratories, some including coding at the transmittersuch as Diagonal BLAST (D-BLAST) and some using uncorrelated data streams on thetransmission antennas (uncoded MIMO system) such as Vertical BLAST (V-BLAST) [25].The V-BLAST detection algorithm follows the logical steps:1. deciding to estimate xi, choosing index i according to some criterion,2. �nding the vector a(i) of minimum norm such that (a(i))∗H :,i = 1 and (a(i))∗H :,j =

0∀j 6= i with H :,i the ith column of H ,3. estimate xi = ((a(i))∗y), where (.) is a slicing (hard decision) operation,4. subtract the estimated e�ect of xi from the received symbol, i.e. y = y−HVect(xi),where Vect(xi) is the MT × 1 vector with xi at the ith line,51

5. reduce the transmission equation to the non-detected symbols, i.e. replace the ithcolumn of H with zeros,6. reiterate until all symbols are decoded.The �rst step determines in which order the symbols are detected. The last symbol tobe detected is facing no interference from other symbols if the other symbols have beencorrectly detected. This is due to the fact that the e�ect of previously detected symbolsis removed from the received signal. Therefore the last symbol is likely to be detectedcorrectly. On the other hand, the �rst symbol has MT − 1 other symbols interfering whilebeing detected, making it the most di�cult symbol to detect. Furthermore, the correctestimation of the �rst symbol is critical in the correct operation of the system: if thesymbol is not correctly detected, the fourth step of the detection algorithm corresponds toan increase of the noise for the other symbols. The result is an event named error propa-gation, which usually creates errors in the detection of subsequent symbols. Therefore itis reasonable to select the received symbol with the highest SNR as the �rst symbol to bedetected. This criterion is optimum in maximising the post-detection SNR [26].The advantage of V-BLAST is that it out-performs linear detectors. The disadvantageof V-BLAST is its complexity. For each frame, the receiver has to compute MT pseudo-inverses of matrices of respective sizes MR × MT ,MR × MT − 1, . . . ,MR × 1. For eachsymbol, the linear detection of the symbol is followed by a recoding of the symbol and itssubtraction from the received vector (step 4). The complexity is large compared with thelinear receiver.III.1.1.4 QR detectionThe QR detector is based on the QR-decomposition of the channel matrix H . Given anycomplex matrix H of dimension MR×MT , there exist two matrices U and R of dimensionsrespectively MR × MR and MR × MT , such that H = UR, U is unitary and R is uppertriangular with real elements on the diagonal, i.e. Ri,j 6= 0 ⇒ i ≤ j and Ri,i ∈ R.Now consider the partially detected symbolsypd = U ∗y

= U ∗(Hx + n)

= Rx + U ∗n.

(III.7)52

Since U is unitary, so is U ∗ and the distribution of n = U ∗n is identical to the distributionof n. The symbol xMTcan be detected since y

pdMT

= RMT ,MTxMT

+ nMT. Following thesame procedure as in the V-Blast detection algorithm, the symbol MT is detected and itscontribution is removed from the received symbol. Then, the algorithm detects iterativelythe symbols MT − 1 to 1.The advantage of QR detection is its moderate complexity. The QR decompositionof the channel matrix, computed on a frame by frame basis, is far less complex thanthe inversion of the channel matrix (inversion of matrices are commonly implemented asiterative applications of the QR algorithm). The complexity on a symbol by symbol basisis modest. The disadvantage of the QR algorithm lies in its modest performance. TheQR su�ers, as V-BLAST does, from error propagation and the optimal ordering of thedecoding (linked to the indexing of the columns of H) is di�cult to obtain.III.1.1.5 BER of decoding algorithmsBit Error Rate (BER) simulation results of the detection schemes previously presented areplotted in Fig. III.3.As expected,

• the ML detector has lowest BER at any SNR,• the performance of each of the detectors increases with the number of antennas atthe receiver,• the MMSE detector performs better than the ZF detector, but the performance gapbetween MMSE and ZF receivers reduces when the SNR increases,• the V-Blast detector outperforms linear receivers at high SNR but not at low SNR(due to error propagation),• the QR detector is outperformed by the V-blast detector at any SNR.Unexpected results include:• at high SNR, log10(BER) decreases linearly with the SNR expressed in dB. Fur-thermore, the asymptotic slope of the ML detector depends only on the number orreceiving antennas. The asymptotic slopes of the other detectors depends on the53

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

log 10

(BE

R)

SNR in dB

ML, MR

=MT=4

ML, MR

=5, MT=4

ML, MR

=6, MT=4

ZF MMSE Blast QR

Figure III.2: BER of several decoding algorithms for uncoded MIMO transmission overi.i.d. channel, rate of 8 bits/s/Hz (4-QAM on each transmitting antenna). Legend: plain,dot-dashed and dotted lines denote respectively MR = 4, MR = 5 and MR = 6 cases,whereas stars, squares, circles and crosses denote respectively ML,ZF, MMSE, Blast andQR detection algorithms.di�erence between the numbers of receiving and transmitting antennas, MR − MT .This point is further discussed in Section III.2.• The QR detector outperforms linear receivers at high SNR.III.1.2 Space-time codingSISO systems use channel coding to lower the probability of error of received symbols.Transmission errors occur when the channel is in deep fade or when the AWGN is large.The correct transmitted symbol can be recovered if the symbol is transmitted again ata later time. Retransmission, in essence, is channel coding. Channel coding can detecterrors or correct them, depending on the code design. In SISO systems, symbols are codedin time. MIMO channels o�er another degree of freedom since coding can occur acrossthe antennas. A code operating on MIMO systems combines bits in space and time, andtherefore is referred to as a Space-Time (ST) code.54

SISO error correction codes are usually binary codes followed by a mapping operation.ST codes are usually not binary: words of ST codes are matrices of symbols (with therows representing transmit antennas and columns the transmission time).Similarly to their SISO counterparts, ST block codes are �nite in time, whereas STtrellis codes are semi-in�nite: transmission has a start but no end. In practice, ST trelliscodes are terminated at the end of the transmission. Therefore, ST trellis and block codesmainly di�er regarding implementation issues and theoretical analysis, but not in essence.The transmission of a �nite time ST code (length Lt) is represented by the transmissionequationY = HX + N , (III.8)where X and N are matrices of size MT ×Lt and Y is a matrix of size MR×Lt. A simpleexample of ST block code is the Alamouti scheme, a ST code with MT = 2 and Lt = 2[27]

X =

s0 s1

−s∗1 s∗0

, (III.9)where s0 and s1 are the two symbols transmitted by the system. The Alamouti schemeachieves a rate of 1 (1 symbol transmitted per channel usage), and o�ers a diversity of 2,since every symbol is transmitted over both channels available. The concept of diversityis further discussed in Section III.2.In coherent detection, the metric considered for the decoding of ST codes is similarto (III.2), i.e. P(X|H ,Y ). However, direct application of a ML decoder is unlikely,due to its extreme complexity. Most space-time codes are designed to obtain a simpledecoding algorithm, e.g. the main attraction of the Alamouti scheme remains its verysimple decoding process. Both symbols are estimated ass0 = h∗

0y0 + h1y1 = (h∗0h0 + h1h

∗1)s0 + h∗

0n0 + h1n∗1

s1 = h∗1y0 − h0y

∗1 = (h∗

0h0 + h1h∗1)s1 − h0n

∗1 + h∗

1n0.(III.10)Bit Error Rate (BER) simulation results of the Alamouti code are presented in Fig.III.3. The mapping is either 4-Quadrature Amplitude Modulation (QAM, 2 bits/s/Hz) or16-QAM (4 bits/s/Hz). 55

0 2 4 6 8 10 12 14 16 18 20−3

−2.5

−2

−1.5

−1

−0.5

Pro

babi

lity

of e

rror

SNR

MT=2 M

R=1 2bps/Hz

MT=2, M

R=2, 2bps/Hz

MT=2, M

R=3, 2bps/Hz

MT=2, M

R=1, 4bps/Hz

Figure III.3: BER of Alamouti code over i.i.d. channelIII.1.3 BER simulation of MIMO transmission techniquesSimulation results of MIMO transmission techniques are presented in Fig. III.4. TheBER of transmission techniques is plotted for varying SNRs over the i.i.d. Rayleigh fadingchannel. Parameters include data rate and number of antennas. Perfect channel estimationis assumed.Fig. III.4 illustrates the fact that it is di�cult to compare MIMO transmission tech-niques. The di�culty is due to the large number of parameters involved. The relativeperformance of the transmission techniques depends heavily on the choice of those pa-rameters and general results cannot be obtained. A better understanding can be gainedthrough the introduction of the theoretical concepts of multiplexing and diversity. Theseconcepts allow us to predict the e�ect of a parameter on the BER performance of MIMOtransmission systems.III.2 Multiplexing vs diversityAs presented in the previous section, popular MIMO transmission systems can be clas-si�ed as coded or uncoded. This classi�cation reveals a major di�erence in the way the56

0 5 10 15 20−3

−2.5

−2

−1.5

−1

−0.5

MT=2,M

R=2, 4 bits/s/Hz

Pro

babi

lity

of e

rror

SNR0 5 10 15 20

−3

−2.5

−2

−1.5

−1

−0.5

MT=2,M

R=3, 4 bits/s/Hz

Pro

babi

lity

of e

rror

SNR

0 5 10 15 20−3

−2.5

−2

−1.5

−1

−0.5

MT=2,M

R=2, 2 bits/s/Hz

Pro

babi

lity

of e

rror

SNR0 5 10 15 20

−3

−2.5

−2

−1.5

−1

−0.5

MT=2,M

R=3, 2 bits/s/Hz

Pro

babi

lity

of e

rror

SNR

AlamoutiBlast ML MMSE QR

Figure III.4: BER of MIMO transmission techniques over i.i.d. channellarge capacity of MIMO channels is exploited, i.e. using multiple antennas to increase thereliability of the transmission (decrease the probability of error) or to increase the datarate (throughput).Coded multiple antennas systems increase diversity to combat channel fading. Di-versity consists in the transmission of a signal through di�erent means to increase thereliability of the transmission: if one copy of the signal is poorly received, it can be re-covered by observing the other copies of the signal. It is well-known that if the fading isindependent across antenna pairs (i.i.d. channel assumption), a maximum diversity orderof MR × MT can be achieved, i.e. every symbol can be transmitted through MR × MTindependent SISO channels for every realisation of the MIMO channel [28]. The averageerror probability of a system achieving a diversity order of d can be made to decay like1/SNRd at high SNR, which corresponds to a linear variation of the BER with SNRdB, theSNR expressed in dB. The slope of the linear variation is −d × SNRdB [28]. Most MIMO57

systems use reception diversity: each receive antenna obtains a di�erent signal when thereceive antennas are uncorrelated. This result is clearly illustrated in Fig. III.2 and Fig.III.4: increasing the number of receiving antennas leads to a direct increase of the slopeof the BER curves. As illustrated by Fig. III.2, not all detectors exploit the maximaldiversity available: while the ML detector enjoys the maximum diversity available, otherdetectors lose diversity order. ST-codes further increase the diversity by sending eachsymbol on several transmit antennas. Therefore coded MIMO systems achieve a higherdiversity gain than uncoded systems, justifying the steeper slope of the Alamouti scheme'sBER curves (when compared to those of uncoded systems).Uncoded MIMO systems consider fading as bene�cial: fading increases the number ofdegrees of freedom available for communication since i.i.d. fading increases the probabilityof a high rank channel matrix (well-conditioned matrix). Uncoded MIMO systems operateat a higher rate than ST-codes since they transmit MT symbols every time-slot. ThereforeST-codes need to operate with higher order constellations to transmit at the same datarate. This point is clearly demonstrated by Fig. III.4: to achieve high spectral e�ciency(4 bits/s/Hz), the Alamouti scheme is using 16-QAM, whereas uncoded systems achievethe same e�ciency with 4-QAM. As a result, the linear portion of the performance curvestarts approximately at 6 dB of SNR for uncoded systems but at 9 dB of SNR for theAlamouti scheme.The Alamouti scheme and uncoded systems are two extremes: the Alamouti schemeachieves maximum diversity with no multiplexing, whereas uncoded systems obtain max-imum multiplexing at the expense of diversity. The choice of a diversity order and amultiplexing level is a trade-o� and an optimal trade-o� curve can be derived, providingthe optimal diversity order achievable for each multiplexing level [28].III.3 Capacity of correlated channelsDecoding algorithms of uncoded MIMO systems all assume that the channel matrix is well-conditioned. If the matrix is not full rank, the correct reception of the MT transmittedsymbols is not possible using a linear receiver. This is simply due to the fact that thechannel projects a vector space of dimension MT into a vector space of dimension R(H),where R(H) is the rank of the channel matrix. Any information contained in the nullspace of the channel matrix cannot be retrieved at the receiver by a linear receiver.58

The BER performance degradation (see Section III.4) is not linked to the decodingalgorithms. It corresponds to the fact that correlated channels have a lower capacity thantheir full rank counterparts. A large body of knowledge has been compiled in recent yearson the capacity of MIMO correlated channels.• Several measurement campaigns have been conducted to determine the correlationproperties of real-life MIMO channels. The impact and extent of correlation onindoor MIMO channels is discussed in [29, 30, 31, 32]. In [33] it is shown thatthe MIMO capacity increases as the signal correlation decreases but the SNR has agreater impact than correlation on the MIMO capacity, as shown in [34].• Several theoretical correlation models have been proposed to simulate correlatedchannels. Though the most widespread correlation model is arguably the one ringmodel [35], other models include the exponential correlation matrix [36, 37] and thevirtual representation channel matrix [38, 39, 40].• The e�ects of fading correlation have been studied for the Rayleigh non-i.i.d. channelthrough simulation [41] and analysis [35, 42, 43, 44]. Further results investigated theimpact of correlation on the variations of the capacity [45]. An upper bound onmean capacity is given in [46, 47, 48].The e�ects of correlation on MIMO channels and systems are di�cult to analyse dueto the complexity of correlation models. Meaningful insight can be gained through thestudy of a simpli�ed model, such as the Ricean channel model. It is shown in the followingthat a fully correlated channel has a negligible capacity compared with the capacity of ani.i.d. channel when the number of antennas grows large.III.3.1 Ricean channelThe e�ect of correlation can be easily understood by studying the Ricean channel. Inthe Ricean case, the �at-fading channel is composed of a Line Of Sight (LOS) componentand a Rayleigh component. The choice of the Ricean KdB-factor (expressed in dB) variesthe Ricean channel from a Rayleigh channel (KdB → −∞ dB) to a pure LOS channel(KdB → +∞ dB). 59

III.3.1.1 Literature reviewSimulation results of the capacity of the Ricean channel are presented in [49]. A geomet-ric approach to interpret the capacity of Ricean channels is described [50]. Simulationindicated that the capacity of the Ricean channel can be approximated by a Gaussianrandom variable [16]. Finally, analytical results on the capacity of Ricean channel are nowemerging for �nite numbers of antennas [51] or in the special case of the low power regime[52].III.3.1.2 Ricean channel modelIn Ricean fading the elements of H are non-zero mean complex Gaussians. Hence we canexpress H in matrix notation as [53]H =

√

10KdB/10

1 + 10KdB/10Hsp +

√

1

1 + 10KdB/10Hsc (III.11)where the specular and scattered components of H are denoted by superscripts and KdBis the Ricean K-factor expressed in dB. The entries of Hsc = (hi,j) are independent andidentically distributed (i.i.d.) complex Gaussian random variables with zero mean andunit magnitude variance. It is assumed that Hsp = 1, where 1 is an MR × MT matrix ofunit entries.The correlation of the MIMO Ricean channel di�ers slightly from the widespread idea ofcorrelation because the entries of the channel matrix are not zero-mean Gaussian randomvariable. The correlation between the entries of the channel matrix is solely due to thenon-zero mean of the subchannels.III.3.1.3 Pure LOS channel: KdB → +∞In general, a MIMO LOS channel has a capacity of

C(KdB = +∞,MT ,MR, P ) = log2(1 + PMR). (III.12)Since the channel is not random, the capacity is �xed and the ergodic capacity and thecapacity are equal. It should be noted that the capacity does not depend on the number oftransmit antennas, and only increases logarithmically with the number of receive antennas.In the special case MT = MR = 1, the channel reduces to a Single Input Single Output(SISO) Additive White Gaussian Noise (AWGN) channel.60

III.3.1.4 Pure Rayleigh channel: KdB → −∞, MR = 1 or MT = 1For the Rayleigh channel, |H i,j|2 is a χ22 variate (chi-squared variate with two degrees offreedom) but normalized so that E[|H i,j|2] = 1. For one transmit antenna, the channelcapacity is [15]

C(KdB → −∞,MT = 1,MR, P ) = log2(1 + Pχ22MR

), (III.13)and using one receive antenna the channel capacity is [15]C(KdB → −∞,MT ,MR = 1, P ) = log2(1 + (P/MT )χ2

2MT). (III.14)Notice that

E[1 + (P/MT ) χ22MT

] = (1 + P )

E[1 + Pχ22MR

] = (1 + PMR),(III.15)and log2(·) is a convex function, that is ∀r > 0, E[log2(r)] ≤ log2(E[r]). Therefore

E[C(KdB → −∞,MT ,MR = 1, P )] ≤ E[C(KdB → +∞,MT ,MR = 1, P )] (III.16)andE[C(KdB → −∞,MT = 1,MR, P )] ≤ E[C(KdB → +∞,MT = 1,MR, P )], (III.17)Hence, for a Single Input Multiple Output (SIMO) or Multiple Input Single Output(MISO) channel, the ergodic capacity is greater in a LOS case than in a Rayleigh case (seeFig. III.5).III.3.2 Capacity bounds for the Ricean channelFor large numbers of antennas, it has been suggested in the literature that the capacityof the Ricean channel tends to the capacity of its Rayleigh component [54] and that thecapacity can be upper bounded by the sum of the capacities of the Rayleigh and LOScomponent matrices [55].This section studies the limiting case where MR → ∞, MT → ∞ and MT /MR = α.Lower and upper bounds are derived for the Ricean channel capacity. The normalizedRicean ergodic capacity is de�ned as the ergodic capacity of the Ricean channel dividedby min(MT ,MR). Since the Rayleigh capacity grows linearly with min(MT ,MR) and theLOS capacity only grows logarithmically, it is easily deduced that the normalized Ricean61

ergodic capacity approaches that of the underlying Rayleigh channel when the number ofantennas (MT ,MR) grows large. Also, the Ricean ergodic capacity should be greater thanthat of the underlying Rayleigh channel. Both results are demonstrated in this section.In order to understand the asymptotic capacity behaviour for the Ricean channel it isinstructive to study the eigenvalues of H .III.3.2.1 Towards the capacity of the Ricean channelWhile the capacities of LOS and Rayleigh channels are well understood, the capacity of theRicean channel is not straightforward to study since the capacity is not a linear operator.To begin, note thatlog2 |IMR

+P

MT

HH∗| = log2 |IMR+

1

1 + 10KdB/10

P

MT

Hsc(Hsc)∗ +P

MT

F |, (III.18)where F is the MR × MR hermitian matrix,F =

√10KdB/10

1 + 10KdB/10(Hsc(Hsp)∗ + Hsp(Hsc)∗) +

10KdB/10

1 + 10KdB/10Hsp(Hsp)∗. (III.19)From Appendix I.1, F is a matrix of maximum rank two, with one negative eigenvalueand one positive eigenvalue. The positive eigenvalue tends to

λ1(F ) → 10KdB/10

1 + 10KdB/10MRMT . (III.20)Though the positive eigenvalue of F grows quadratically with the number of antennas,

F has a �xed number of eigenvalues (2) when the number of antennas increases. Therefore,F has a negligible e�ect on the normalized capacity of the Ricean channel when the numberof antennas tends to ∞, since it contributes approximately as log2(M2

T )

MT. This insight isdemonstrated in the following section.III.3.2.2 Asymptotic capacity of a Ricean channelThis section states the main contribution of this chapter: for a Ricean channel that is notpure LOS (KdB 6= +∞), when the number of antennas grows large, the normalized capacityof the Ricean channel tends to the normalized capacity of its scattering component, i.e.when MR,MT → ∞,MT /MR = α,

E

[C(KdB,MT ,MR, P )

min(MT ,MR)

]

→ E

[C(KdB = −∞,MT ,MR, 1

1+10KdB/10 P )

min(MT ,MR)

]

. (III.21)62

III.3.2.2.1 Lower bound Appendix I.2 proves that ∀MR,MT , P,KdB,E

[C(KdB,MT ,MR, P )

min(MT ,MR)

]

≥ E

[C(KdB = −∞,MT ,MR, 1

1+10KdB/10 P )

min(MT ,MR)

]

. (III.22)III.3.2.2.2 Higher bound Appendix I.3 proves that as MT ,MR → ∞,E

[C(KdB,MT ,MR, P )

min(MT ,MR)

]

≤ E

[C(KdB = −∞,MT ,MR, 1

1+10KdB/10 P )

min(MT ,MR)

]

+ ∆, (III.23)with ∆ → 0 as MT ,MR → ∞.III.3.3 Results and discussionIII.3.3.1 Mean and variance of capacity for the Ricean channelFig. III.5 plots the average normalized capacity with α = 1 for an increasing number ofantennas and di�erent K-factors. As indicated in our analysis, for MT = 1, the meancapacity of Ricean channels is higher than the mean capacity of Rayleigh channels. Thistrend is inverted for MT ,MR > 1. This result is in contrast with the outage capacity ofthe Ricean channel which remains higher than the corresponding capacity of the Rayleighchannel for small number of antenna (i.e. up to four antennas) when the targeted outageprobability is very small (i.e. 0.01) [49]. Such a result is not surprising since the capacityof the Ricean channel is a random variable with a smaller variance than the capacity ofthe Rayleigh channel [16].For KdB = −1000, the capacity converges rapidly to a limit as MT → ∞, as indicatedin [17]. For other values of KdB, the capacity decreases with the number of antennas overthis range. For increasing MT (as soon as MT ≥ 2), the capacity of the Ricean channel isa decreasing function of KdB. The results shown in Fig. III.5 are for MT = MR.Fig. III.6 plots the variance of the total capacity for di�erent KdB values versus thenumbers of transmit and receive antennas when SNR= 20 dB and α = 1. For the purposesof comparison, the corresponding analytical results in [17] are also shown for the Rayleighchannel. From these results it is clear that the Ricean channel conditions do not impact thecapacity variance beyond the corresponding Rayleigh values for large antenna conditions.63

1 2 3 4 5 6 7 8 9 102.5

3

3.5

4

4.5

5

5.5

6

6.5

7

MT

E(C

/MT)

KdB

=−1000K

dB=1

KdB

=5K

dB=7

KdB

=12

Figure III.5: Normalized mean capacity per antenna with MT /MR = 1, Ricean fadingand SNR= 20dBIII.3.3.2 Asymptotic mean and variance of the capacity for the Ricean chan-nelFig. III.7 shows the behaviour of the normalized capacity of a Ricean channel as thenumber of antennas grows large. For all values of KdB, the normalized capacity of theRicean channel tends to the normalized capacity of its scattering component (the lowerbound on the capacity). This lower bound is tighter when KdB is smaller, and for KdB =

−1000 it is impossible to discern the simulation from the lower bound.The upper bound converges slowly to the lower bound and is tight for large valuesof KdB. An explanation for the slowness of convergence can be found in (A.31) whereit is shown that ∆ (see Section III.3.2.2.2) tends to zero like log(MR)/MT , which itselfconverges very slowly. Although the upper bound is only strictly valid for a large numberof antennas (see the assumptions in (A.32)), the simulations show excellent correlation forvalues of MT as low as 20, and for KdB ≤ 12.The capacity bounds for values of α other than 1 are shown in Fig. III.8. The tightnessof both bounds appears to remain una�ected by the value of α.The results shown in Fig. III.6 are further reinforced by Fig. III.9. This shows the64

0 50 100 1500

0.5

1

1.5

2

2.5

3

3.5

4

MT

Cap

acity

var

ianc

e [(

bit/s

/Hz)

2 ]

KdB

=−1000 simulationK

dB=−1000, Rayleigh component, analytical

KdB

=1K

dB=5

KdB

=7K

dB=12

Figure III.6: Capacity variance with MT /MR = 1, Ricean fading and SNR= 20dBbehaviour of the variance of the capacity of a Ricean channel as the number of antennasgrows large. For all values of KdB, the variance clearly tends to the corresponding vari-ance of its scattering component, the variance is maximum for α = 1 and the maximumdecreases as the channel approaches LOS condition. We note from results not reportedhere that this asymptotic behaviour of the capacity variance is only achieved for very largenumber of antennas MT ,MR(MT = MR) = 100 or more.III.3.3.3 Capacity boundsFig. III.8 shows the behaviour of the normalized capacity, for varying α, in the asymptoticcase of a large number of antennas (min(MT ,MR) = 100). As in Fig. III.7, the lowerbound is tight for small KdB, whereas the upper bound is tight for large KdB. Notethat the tightness of the bounds appears uniform across all α values, indicating that thetightness depends on the ratio MT /MR, and not on their actual values.The results demonstrate that the upper and lower bounds provide a fast and reliableway to bound the mean capacity of a Ricean channel for large min(MT ,MR), withoutextensive simulations. Furthermore, depending on the K-factor, it is straightforward todeduce which of the bounds is the tightest. 65

0 20 40 60 80 100 120 140 160 180 2002

2.5

3

3.5

4

4.5

5

5.5

6

KdB

=−1000

KdB

=1

KdB

=5

KdB

=7

KdB

=12

MT

E(C

/MT)

Upper BoundSimulationLower Bound

Figure III.7: Ergodic capacity per antenna with MT ,MR → ∞, Ricean fading andSNR=20dBIII.3.4 ConclusionThe capacity of the Rayleigh and LOS channels have been studied extensively and arewell-known, both for a small number of antennas and in the asymptotic case of a largenumber of antennas. For a Ricean channel, the capacity is more di�cult to derive.For a large number of antennas, the normalized mean capacity of a Ricean channeltends to the normalized mean capacity of its Rayleigh component. Precisely, the capacityof the Ricean channel is lower bounded by the capacity of its Rayleigh component andupper bounded by a quantity that tends to the capacity of its Rayleigh component as thenumber of antennas grows large.The lower bound is valid for any number of antennas, and depending on the choice ofa constant min(MT ,MR), the upper bound can be valid for any number of antennas, oronly for a large number of antennas (in which case the upper bound becomes tighter asthe number of antennas grows large). The lower bound is tighter when the K-factor issmaller, whereas the upper bound is tighter with increasing K. The two bounds allow usto estimate the capacity of a Ricean channel without extensive simulations.A very similar behaviour is observed for the capacity variance that also tends to the cor-66

0.5 1 1.5 22

3

4

5

6

7

8

KdB

=−1000

KdB

=1

KdB

=5

KdB

=7

KdB

=12

α

E(C

/min

(MT,M

R))

Upper BoundSimulation, min(M

T,M

R)=100

Lower Bound

Figure III.8: Ergodic capacity per antenna versus α, Ricean fading and SNR=20dBresponding variance of the scattering component especially when the number of antennasgrows very large.III.4 MIMO systems in highly specular channelAs mentioned earlier, decoding algorithms of uncoded MIMO systems typically assumethat the channel matrix is well-conditioned. Any system with a MIMO multiplexing gaincannot operate properly over a channel of rank one, since the vector space of the receivedsymbol is of dimension one. Furthermore, the capacity of highly specular MIMO channelsis usually lower than the capacity of i.i.d. MIMO channels. The in�uence of channelcorrelation on the performance of MIMO systems is further evaluated in this Sectionthrough BER simulation results.Linear detection schemes presented in Section III.1.1 all assume that the channel ma-trix is full rank. The QR detection algorithm produces a division by zero when the channelmatrix is not full rank, whereas other detection schemes can be applied but cannot decodecorrectly the transmitted signals, regardless of the SNR. It is reasonable to assume thatlimited correlation leads to a loss of performance for all detection schemes. Fig. III.1067

0.5 1 1.5 20.5

1

1.5

2

2.5

3

3.5

4

KdB

=−1000

KdB

=1

KdB

=5

KdB

=7

KdB

=12

α

Cap

acity

var

ianc

e [(

bit/s

/Hz)

2 ]

Simulation, min(MT,M

R)=100

Asymptotic result

Figure III.9: Asymptotic capacity variance, Ricean fading and SNR= 20dBpresents BER simulation results for several uncoded MIMO systems over the Ricean chan-nel. For all systems, an increase of the K-factor directly leads to a loss of performancewhich can be related to the loss of power of the i.i.d. component of the channel matrix.Fig. III.11 presents BER simulation results of the Alamouti scheme over the Riceanchannel. The performance of the Alamouti scheme does not degrade with channel corre-lation since no spatial multiplexing is used. The two transmitted signals are coded to beorthogonal in space and time, regardless of the rank of the channel. Therefore, increasingthe K-factor of the Ricean channel leads to a constant transmission environment. Thisexplains the lower BER at high SNR: at high SNR, errors only occur when the channelis in deep fade, which is less likely for a highly specular Ricean channel than for an i.i.d.channel.III.5 In�uence of CSI at the transmitterAs discussed previously, correlated channels have generally a lower capacity than i.i.d.channels. As a result, uncoded schemes su�er from a loss of performance when the channelis highly specular. ST-codes do not su�er from the same loss of performance since they donot take advantage of the multiplexing gain available in i.i.d. MIMO channels. However,68

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

0lo

g 10(B

ER

)

SNR in dB

ML

KdB

=−∞K

dB=1

KdB

=5K

dB=7

KdB

=12K

dB=+∞

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

0

log 10

(BE

R)

SNR in dB

BLAST

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

0

log 10

(BE

R)

SNR in dB

QR

0 5 10 15 20 25−3

−2.5

−2

−1.5

−1

−0.5

0

log 10

(BE

R)

SNR in dB

MMSE

Figure III.10: BER of several decoding algorithms for uncoded MIMO transmission overRicean channel, MT = 4, MR = 5, rate of 8 bits/s/Hz (4-QAM on each transmittingantenna).ST-codes do not use the multiplexing gain. As a consequence, high-order constellationsare required to transmit at a rate close to the channel capacity, especially if the numberof antennas grows large. For example, the Alamouti scheme using a 16-QAM should becompared with an uncoded scheme using a 4-QAM. However, if the number of transmittingantennas is increased to three, a ST-code (without multiplexing gain) has to use a 64-QAMto be compared with an uncoded scheme using a 4-QAM. For MT = 4, an uncoded systemusing a 4-QAM should be compared with a ST-code (without multiplexing gain) usinga 256-QAM. This comparison highlights the di�culty in designing space-time codes toachieve a non-negligible fraction of the capacity when the number of antennas grows large.Systems with CSI at the transmitter are potentially able to• achieve a non-negligible fraction of the capacity even when the number of antennas69

0 5 10 15−3

−2.5

−2

−1.5

−1

−0.5

log 10

(BE

R)

SNR in dB

MR

=1, KdB

=−∞M

R=1, K

dB=1

MR

=1, KdB

=5M

R=1, K

dB=7

MR

=1, KdB

=12M

R=2

Figure III.11: BER of Alamouti scheme over Ricean channel.is large,• operate robustly over correlated channels.However, these are not the only bene�ts linked with providing the CSI at the transmit-ter. Systems with CSI at the transmitter can bene�t from the array gain at the transmitter.Hence, they transmit over a channel with higher capacity. Fig. III.12 presents simulationresults of the capacity of Ricean channels with and without CSI at the transmitter.The normalized capacity gain due to CSI at the transmitter is constant when thenumber of antennas grows large, which means that the capacity gain due to CSI at thetransmitter increases linearly with the number of antennas. Furthermore, the gain ishigher for highly specular channels than for the i.i.d. channel.Systems with CSI at the transmitter transmit over a channel with higher capacity,especially when the number of antennas grows large. Furthermore, providing the CSI atthe transmitter limits the performance loss due to highly specular channels.70

0 20 40 60 80 100 120 140 160 180 2002

2.5

3

3.5

4

4.5

5

5.5

6

KdB

=−1000

KdB

=1

KdB

=5

KdB

=7

KdB

=12

MT

E[C

/MT]

no CSI at TxCSI at Tx

Figure III.12: Normalized capacity of Ricean channels with and without CSI at thetransmitter.III.6 ConclusionMIMO transmission techniques without CSI at the transmitter can be classi�ed as uncodedor space-time coded. Their performances depend heavily on the trade-o� chosen by systemdesigners between diversity and multiplexing gains. Systems with diversity gain are robust,but are di�cult to design for a large number of antennas. Speci�cally, a large constellationis required to achieve a non-negligible fraction of the capacity of the channel. Systemswith multiplexing gain can achieve a non-negligible fraction of the capacity but su�er fromperformance loss when the channel is highly specular. This is due to the fact that highlyspecular channels have a lower capacity than i.i.d. channels. Speci�cally, the capacityof Ricean channels tends to the capacity of their i.i.d. component when the number ofantennas grows large.Finally, providing the CSI at the transmitter can simplify the system design when alarge number of antennas is used, while mitigating the e�ect of channel correlation.71

Chapter IVOptimum structure: the SVDThe capacity of the MIMO channel is higher when CSI is available at the transmitter. Apractical coding system is required to turn the promises of high capacity into improvedperformances. The new system has to take advantage of the knowledge of the channelat the transmitter. Information theory suggests an architecture based on the SVD ofthe channel matrix [56, 57]. This architecture decomposes the MIMO channel into SISOtransmission eigenmodes and allocates power to the eigenmodes following a water�llingalgorithm.Chapter IV introduces the SVD transmission architecture both theoretically and prac-tically to demonstrate the relevance of this transmission technique.The SVD structure corresponds to the optimal jointly designed linear precoder anddecoder following the criterion of maximum capacity. The same structure can be modi�edto be optimal under other criteria by simply applying appropriate power allocation algo-rithms, as detailed in Section IV.2.1. The SVD structure is designed to be optimal overthe �at-fading channel. Section IV.2.2 indicates that the SVD structure combined withOFDM is also an optimal space-time modulation in terms of information rate.Practical SVD structures are introduced in Section IV.3, where coding is applied sep-arately on each transmission eigenmode to reduce the complexity of the transmissionsystem. The notion of system capacity is introduced to free the analysis from assumptionson coding.The SVD transmission architecture combined with water�lling is optimal, i.e. canachieve the channel capacity with perfect coding (as proven in [12]), when perfect CSI isavailable at both end of the link. Imperfect CSI can lead to noise enhancement, especially73

at low SNR. Usually, CSI is obtained through channel estimation. Channel estimationrequirements of the SVD structure are analysed in Section IV.4. Simulation results showthat the estimation noise should be smaller than the noise in the transmission to avoid aloss of performance. These results have been partly published in [58].The channel estimation is usually performed at the receiver. The receiver can feedthe explicit precoding matrix or the complete CSI back to the transmitter. When thereceiver feeds the complete CSI back to the transmitter, it is necessary to insure that thetransmitter and the receiver use the same SVD of the channel matrix, which is possible asdemonstrated in Section IV.4.2. These results are, to the knowledge of the author, newand unpublished.Transmission architectures based on SVD have been proposed and studied extensivelyin the literature [59, 60, 61, 62]. Though SVD-based transmission devices have beenproposed for multiuser channels [63], Section IV focuses on single user channels.IV.1 Introduction to the SVD structureSection II.2.3.4 derived the capacity of the non-ergodic MIMO channel with CSI at thetransmitter. The capacity is reached when the transmitted signal is circularly symmetriccomplex Gaussian with correlation Q chosen so thatQ = V ∗QV (IV.1)is diagonal (see (II.52)), where V is derived from the SVD of the channel matrixH = UΣV ∗, (IV.2)where U and V are unitary and Σ is diagonal real such that Σ1,1 ≥ Σ2,2 ≥ . . . ≥ ΣM,M ,where M = min(MR,MT ). IV.1 is equivalent toQ = V QV ∗, (IV.3)which is achieved by transmitting

x = V x (IV.4)where x has independent real and imaginary Gaussian entries andE[x2

i ] = (Pwf − Σ−2i,i )+,∀i ∈ {1, . . . ,MT} (IV.5)74

1

DetectorMIMO

+Decoder

1x

2x

3x

4x

y1

y2

y3

y4

x1~

2x~

3~x

x~4

Power allocationBits Allocation

Mapping

Mapping

Mapping

Mapping

ConverterParallel

ToSerial

+Coding

VH

4

3

2

n

n

n

n

Figure IV.1: SVD transmission architecturewhere Pwf is chosen to satisfy ∑i E[x2i ] = P . This transmission architecture is presentedin Fig. IV.1Considery = U ∗y. (IV.6)The relationship between y and the transmitted symbols can be expressed as

y = U ∗y

= U ∗(Hx + n)

= U ∗HV x + U ∗n

= U ∗UΣV ∗V x + U ∗n

= Σx + n

(IV.7)where

n = U ∗n. (IV.8)Since n is assumed zero-mean Gaussian with i.i.d. real and imaginary entries and U ∗ isunitary, n and n follow the same distribution.Equation (IV.7) clearly indicates that the MIMO channel has been decomposed intoparallel SISO virtual channels over which the power allocation is conducted. These SISOvirtual channels are commonly referred to as transmission eigenmodes. The complete SVDtransmission architecture and its equivalent model are presented in Fig. IV.2.IV.2 Optimality of the SVDThe SVD transmission structure is deduced from the analysis of the capacity achievingcorrelation of the transmitted signals over the �at-fading channel. However, this structureis optimal with other assumptions and other design criteria. The optimality of the SVDarchitecture is discussed in this section, with the following results:75

1

DetectorMIMO

+Decoder

Decoder+

MIMODetector

Coding+

SerialTo

ParallelConverter

ConverterParallel

ToSerial

+Coding


Mapping

Mapping

Mapping

Mapping


Mapping

Mapping

Mapping

Mapping

1x

2x

3x

4x

y1

y2

y3

y4

x1~

2x~

3~x

x~4

4~x

x~3

~x2

~1x

~y4

y~3

~y2

y~1

~y4

~y3

~y2

~y1

~n4

~n3

~n2

~n1

1,1Σ

3,3Σ

2,2Σ

4,4Σ

Equivalent Channel

VH

n

U*

4

3

2

n

n

n

Figure IV.2: SVD transmission architecture with decoding matrix• the SVD architecture is the optimal jointly designed precoder and decoder architec-ture over the �at-fading channel under several design criteria (Section IV.2.1),• the SVD architecture combined with OFDM is equivalent to the optimal spatio-temporal coding over a general fading channel (Section IV.2.2).IV.2.1 Optimal linear precoder and decoderLinear �lters are relatively easy to implement and are well-studied. It is natural to try todetermine the optimal, jointly designed, linear precoding and decoding �lters for a givencriterion.The SVD architecture consists of a linear precoding �lter (V ) and a linear decod-ing �lter (U ∗). Furthermore, the SVD transmission architecture is potentially capacityachieving, i.e. the system is able to transmit information at a rate as close as desired tothe capacity of the channel when the symbols on each transmission eigenmode follow aGaussian PDF, water�lling power allocation is applied and perfect coding is achieved. Itis straightforward to deduce that the SVD architecture is the optimum linear precoderand decoder architecture under the criterion of data rate.Surprisingly, the SVD architecture is also optimum under a variety of criteria throughmodi�cation of the power allocation algorithm [64]. Some of the criteria for which theSVD is optimum are detailed in the following.76

IV.2.1.1 Maximum information rateAs discussed previously, the SVD architecture combined with water�lling power allocationmaximises the information rate [64].IV.2.1.2 Relative SNR designIt is possible to achieve any set of relative SNRs on streams of data transmitted in parallel(the number of streams cannot exceed the rank of the channel matrix) by assigning streamsto transmission eigenmodes and applying adequate power allocation [64].IV.2.1.3 Equal error designThe system achieving equal error on each parallel stream can be considered as a specialcase of the previous criterion: the equal error rate criterion is ful�lled when the SNR isthe same on each parallel substream [64].IV.2.1.4 Minimum mean square error designThe system achieving the minimum mean square error in the estimation of the symbolscombines the SVD architecture with appropriate power allocation. The power allocationalgorithm corresponding to this criterion does not guarantee equal mean square error oneach eigenmode and might not transmit on the weaker eigenmodes [64].IV.2.2 Optimal space-time structureThe SVD architecture is potentially capacity achieving over the �at-fading channel, whichcorresponds to a subcarrier in an OFDM transmission (Section II.3). However, it is unclearwhether the SVD-OFDM-MIMO architecture is optimal over a general fading channel.The optimal spatio-temporal coding architecture over a fading channel can be derived asfollows.Consider the SISO transmission of a block of Lt symbols over a time-invariant fadingchannel with impulse response h = [h0, h1, . . . , hτ ], transmission equation (II.8) becomesy(kTs) = [

τ∑

i=0

hix((k − i)Ts)] + n(kTs). (IV.9)where Ts is the sampling period and is discarded in the following. Though (IV.9) isessentially the convolution of h and x = [x(0), x(1), . . . , x(Lt)], it is possible to modify it77

into a matrix multiplication (since the transmitted block is �nite in time) as follows:

y(0)

y(1)...y(Lt + τ)

=

h0 0 . . . 0

h1 h0. . . 0... ... . . . ...

hτ hτ−1. . . h0

0 hτ. . . ...... ... . . . hτ−1

0 0 . . . hτ

x(0)

x(1)...x(Lt)

+

n(0)

n(1)...n(Lt + τ)

. (IV.10)This SISO model is naturally extended to MIMO transmission by de�ning

y = [y1(0), . . . , y1(Lt + τ), y2(0), . . . , yMR(Lt + τ)]†,

x = [x1(0), . . . , x1(Lt), x2(0), . . . , xMR(Lt)]

†,

n = [n1(0), . . . , n1(Lt + τ), n2(0), . . . , nMR(Lt + τ)]†,

(IV.11)where (.)† is the transpose non-conjugate operation andH =

H1,1 . . . H1,MT... . . . . . .

HMR,1 . . . HMR,MT

(IV.12)where ∀i < MR, j < MT , Hi,j is a matrix of the Toeplitz form as appears in (IV.10), withentries the elements of the sampled channel impulse response from transmitting antenna jto receiving antenna i. The transmission equation of the MIMO space-time transmissioncan be expressed as:y = Hx + n. (IV.13)The capacity of a system following this transmission equation has been studied extensivelyin Section II.2. Obviously, the capacity of this space-time MIMO channel with CSI atthe transmitter can be found by decomposing the channel into space-time transmissioneigenmodes and applying water�lling power allocation. The block diagram of the suggestedtransmission architecture is represented in Fig. IV.3.This architecture is quite complex since the SVD of a very large matrix has to becomputed. The space and time dimensions are coupled, resulting in complex processing atboth transmitter and receiver. This complexity has to be compared with that of the SVD-OFDM-MIMO architecture. As illustrated in its simpli�ed block diagram (Fig. IV.4),the time processing and the space processing of the signal are separate since the space78

τ

SerieTo

Parallel

Converter

Converter

ParallelTo

Serie

U*

ConverterParallel

ToSerial

+Coding


Mapping

Mapping

Mapping

Mapping

nMR

n1

x (0)1

x (L )1 t

ConverterParallel

ToSerial

SerialTo

ParallelConverter

x (0)MT

Tx (L )

MM t

y (0)1

RMy (0)

t1y (L + )τ

Decoder+

MIMODetector

V ......

...

......

...

τττ

1,1H

...

...

H

τ τ τ

τττ

H

...

T

M ,MR T

...

H

τ τ τ

RM ,1

1,M

MM ty (L + )

R

Figure IV.3: SVD Space-Time MIMO transmission architecturedimension is exploited through the SVD approach, whereas the time dimension is handledthrough OFDM. This decoupling of the processing in space and time leads to a signi�cantreduction in complexity since• the OFDM modulation requires little adaptive processing (only the power allocationto the subcarriers) and can be e�ciently implemented,• computing Lt SVDs of (MR,MT ) matrices is much simpler than computing the SVDof one (Lt ×MR, (Lt + τ)×MT ) since the complexity of the SVD operation is muchgreater than linear.Counter-intuitively, the reduction in complexity linked with choosing a SVD-OFDM-MIMO architecture rather than a SVD-Space Time-MIMO architecture is not obtainedat the expense of performance. It is proved in [56], [65] that in the limit Lt → ∞, thecapacity of both the SVD-OFDM-MIMO architecture and the SVD-Space Time-MIMOarchitecture converge to the continuous frequency channel capacity. This result can beeasily understood by noting that the basis functions of OFDM are complex exponentials,which are also (for in�nite duration) the eigenfunctions of convolution. When the OFDMsymbols are much longer than the delay spread, SVD-OFDM-MIMO provides orthogonaldecomposition in space and time, and the decoupling arises naturally.The bene�ts associated with the SVD-OFDM-MIMO architecture (low complexity andequal performance) justi�es that the remaining of this thesis is focused solely on this79

...

ConverterParallel

ToSerial

SerialTo

ParallelConverter

1n

RMn

SerieTo

Parallel

Converter

Converter

ParallelTo

Serie

IFFT

IFFT

V

V U*

*U

DetectorMIMO

+Decoder

Mapping

Mapping

Mapping

Mapping

Bits Allocation Power allocation

Coding+

SerialTo

ParallelConverter

FFT

FFT

1,M

M ,1R

τττ

H

...

TRM ,M

T

...

H

τ τ τ

τττ

H

...

...

H 1,1

τ τ τ

...

Figure IV.4: SVD-OFDM-MIMO transmission architecturearchitecture. As a result, �at-fading is assumed for the remaining of the thesis.IV.3 Practical SVD architectureIt has been shown that the SVD architecture enables the decomposition of the MIMOchannel into transmission eigenmodes, allowing the use of a separate SISO modulation oneach transmission eigenmode without capacity penalty. However, coding across the trans-mission eigenmodes as well as joint demodulation and decoding of the received symbolsacross the eigenmodes are still assumed. The complexity of such a system is usually notacceptable. Therefore, a simpli�ed SVD architecture uses the transmission eigenmodes asparallel Gaussian channels, as shown in Fig. IV.5.

V

1~y

2y~

3~y

4y~4~x

x~3

~x2

~1x

4y3y2y1y

x4

x3

x2

x1Mapping

Mapping

Mapping

Mapping

Bits Allocation Power allocation

SISO Detector/Decoder



SISO Detector/DecoderSerialTo

Parallel

Converter

Coding

Coding

Coding

Coding

SerialTo

ParallelConverter

1

n

n

n

2

3

4

*U

n

H

Figure IV.5: A practical SVD transmission architecture80

IV.3.1 Performance loss of the practical architectureThe loss of performance due to the practical structure is detailed in the following. Assum-ing a soft output demodulator, SISO demodulators can be applied without penalty, sincemodulation is applied independently on the transmission eigenmodes. The practical struc-ture su�ers from a loss of performance through the application of separate SISO coding.This is simply due to the fact that the capacity is only achieved in the asymptotic case ofin�nite length coding. The capacity of the channel is equal to the sum of the capacities ofthe eigenmodes. However, when separate coding is applied to each eigenmode, the lengthof the codes applied is approximately min(MT ,MR) times shorter than the length of a codethat would be applied across all min(MT ,MR) eigenmodes. In practice, binary coding canbe applied (before modulation) to each eigenmode, or to all eigenmodes. Using coding overall eigenmodes allows us to use longer binary codes for a given block size. It is well-knownthat the performance of good codes (codes transmitting close to the capacity) is linked totheir length. For example, the performance of turbo-codes is closely related to the designof interleavers, and the performance of interleavers increases with their length [66]. Codingover all the eigenmodes leads to a faster convergence to the asymptotic channel capacitylimit as the interleaving length is increased.IV.3.2 Notion of system capacityThe performance of the system depends heavily on the coding strategy, as does the robust-ness of the system to various impairments (e.g. imperfect channel estimation). However,the study of speci�c coding strategies is not the purpose of this thesis. Therefore, it isnecessary to free the analysis from various assumptions related to coding strategies.The practical SVD architecture considers each transmission eigenmode as a separatechannel. It is possible to obtain the Signal to Interference and Noise Ratio (SINR) on eachtransmission eigenmode. From the SINR, it is straightforward to deduce the capacity ofthe eigenmode when no joint decoding is applied. The notion of capacity of a transmissionsystem is then easily de�ned as the sum of the capacities of the transmission eigenmodes.The 'system capacity' corresponds to an upper bound on the data rate achievable whencoding and decoding are applied separately on each transmission eigenmode.The 'system capacity' is generally not equal to the capacity of the channel, as demon-strated by the following example: consider the SVD transmission systems where impair-81

ments forced the decoding matrix to be mistakenly taken as equal to the identity matrix(Fig. IV.5 with U ∗ replaced with the identity matrix). The output of the decoding matrixis exactly the signal on the receiving antennas. In most cases (with a probability equalto one), the signal on each receiving antenna is the weighted sum of signals from eachtransmission eigenmode plus Gaussian noise. Therefore, the SINR on the output of theeigenmodes of this system is poor and the 'system capacity' is low (always lower than thecapacity of the channel). This is in sharp contrast to the maximum information rate sup-ported by this architecture when coding and decoding is applied across the eigenmodes:the erroneous decoding matrix does not limit the maximum data rate achievable if a jointdetector were applied on the output of the decoding matrix (Fig. IV.2 with U ∗ replacedwith the identity matrix), potentially allowing transmission at a rate equal to the capacityof the channel.In the remainder of the thesis the in�uence of various practical impairments on the 'sys-tem capacity' of SVD systems is analysed and compared with the corresponding channelcapacity.IV.4 Channel estimation and SVD architectureBoth the transmitter and receiver of SVD systems require, at least, the partial knowledgeof the CSI: it is necessary for the transmitter to obtain V , the precoding matrix, and forthe receiver to obtain U ∗, the decoding matrix.Usually, communication systems obtain the CSI through pilot based channel estima-tion, i.e. measurements of the channel response to the transmission of known symbols(pilot symbols). The insertion of pilot symbols corresponds to a loss of transmission slots,i.e. data rate, and a loss of transmission power. However, increasing the number of pi-lot symbols can lead to more accurate channel estimation. Determining the appropriatenumber of pilot symbols is a trade-o� between the accuracy of the channel estimation andthe loss related to pilot symbol insertion. This trade-o� has been studied in the literature[67, 68].Blind channel estimation [69, 70] can also be applied, where only the statistics of thesymbols is known and CSI can be recovered without any overhead (transmission of pilotsymbols).Regardless of the speci�c channel estimation technique applied, the channel estimation82

requirements are discussed in the following.IV.4.1 Channel estimation accuracyConsider the system presented in Fig. IV.6. The channel matrix is estimated throughmeasurements. However, noise in the measurements leads to inaccurate CSI. The esti-mated CSI is decomposed through the SVD and the precoding and decoding matrices aremodi�ed accordingly.n

SVD

1nx1~

2x~

3~x

x~4

1x

2x

3x

4x

y1

y2

y3

y4~y4

y~3

~y2

y~1

Estimation

estn

VH U*

4

3

2

n

n

Figure IV.6: CSI estimation and SVD architecture.The degradation of performance due to incorrect estimation of the CSI is analysed inthe following. The channel matrix H is incorrectly estimated asH =

H + N est

r, (IV.14)where N est is the channel estimation noise and the entries of N est are assumed i.i.d.complex Gaussian and r is a normalization factor de�ned as

r =√

(E[‖H‖2F ] + E[‖N est‖2

F ])/(MR × MT ). (IV.15)Normalization is required to maintain the equality between the total transmitted powerand the received SNR. The following SVD applies: H = UΣV∗. The transmissionequation becomes

y = U∗(HV x + n)

= U∗(r × HV x) − U

∗N estV x + U

∗n

= r × Σx − U∗N estV x + U

∗n.

(IV.16)The received signal consists of the signal (r× Σx), an interference term (U ∗N estV x) and�nally the noise term (U ∗

n). The analysis can be conducted by carefully examining thedi�erent terms. 83

• H is a complex i.i.d. Gaussian matrix as the sum of two Gaussian i.i.d. matrices.Furthermore H is normalized to have entries with variance one. Therefore, Σ andΣ follow the same distribution.

• U∗N estV follows the same distribution as N est since U and V are unitary and N estis an i.i.d. Gaussian matrix.

• U∗n follows the same distribution as n since U

∗n is unitary.An approximation of the SINR is given bySINR =

(E[‖H‖2F ] + E[‖N est‖2

F ])E[‖x‖2F ]

E[‖N est‖2F ]E[‖x‖2

F ] + MT × E[‖n‖2F ]

. (IV.17)When the interference is negligible compared to both the noise and the signal, theSINR tends to the SNR of the system with no interference:SINR → (E[‖H‖2F ]

MR × MT

× E[‖x‖2F ]

E[‖n‖2F ]/MR

. (IV.18)On the contrary, when the noise is negligible compared to the interference the SINRbecomes SINR =(E[‖H‖2

F ] + E[‖N est‖2F ])

E[‖N est‖2F ]

. (IV.19)Obviously, in this situation, the SINR does not depend on the SNR. Increasing the powerof the signal does not improve the transmission.Simulation results of the system capacity with varying estimation errors are presentedin Fig. IV.7. Equal power and water�lling power allocation di�er only at low SNRs.De�ne SNRest as the signal (H) to estimation noise (N est) ratio. When SNRest � SNR,the system capacity equals the channel capacity: the imperfect channel estimation doesnot reduce the performance of the system. On the contrary, when SNRest � SNR, theperformance plateaus with increasing SNR and the performance of the system dependsonly on the SNRest.IV.4.2 Uniqueness of the SVDThe architecture presented in Fig. IV.6 includes an estimation block and a SVD block. Inpractical systems, channel estimation is performed through measurements of pilot symbols.Therefore, the estimation block is usually located at the receiver. It is necessary to feedbackthe CSI to the transmitter. Two solutions are available:84

−10 −5 0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

Cap

acity

(bp

s/H

z)

SNR in dB

Capacity of the channel, waterfilling Capacity of the channel, equal power Capacity of the system, SNR

est=40dB

Capacity of the system, SNRest

=35dBCapacity of the system, SNR

est=30dB



est=20dB



est=10dB


=5dB Capacity of the system, SNR

est=0dB

Figure IV.7: Capacity of the SVD system with imperfect channel estimation over thei.i.d. channel, MR = MT = 4. Plain lines and dotted lines denote respectively equalpower and water�lling power allocation.• either the receiver performs the SVD and feeds the precoding matrix and the matrixof the singular values back to the transmitter,• or the receiver feeds the CSI back to the transmitter.In the second case, the SVD of the channel matrix is derived separately at the trans-mitter and at the receiver. This solution relies implicitly on the uniqueness of the SVDof the channel matrix: if the SVD is not unique, then the transmitter and receiver mightdecompose the channel in two di�erent ways, H = UTΣT V ∗

T and H = URΣRV ∗R. Thetransmitter �lters the transmitted signal by V T and the receiver �lters the received signalsby U ∗

R. The transmission equation becomes:y = U ∗

R(HV T x + n) = U ∗RUTΣT x + U ∗

Rn. (IV.20)In such a situation, the MIMO channel is no longer decomposed into parallel channels.Theorem 6. The SVD of a matrix H ∈ CM×M is unique, up to a multiplication of theinput and output eigenvectors by a complex number of norm 1, if the following assumptionshold: 85

• the singular values are sorted in descending order,• the singular values are of multiplicity 1, i.e. ∀i 6= j,Σi,i 6= Σj,j.IV.4.2.1 Proof of the uniqueness theoremIt has been shown that the set of singular values of a matrix is unique [71]. Consider

H ∈ CM×M such that:∀i 6= j, Σi,i 6= Σj,j, (IV.21)with singular values of multiplicity one and singular values indexes chosen so that

Σ1 > Σ2 > Σ3 . . .ΣM,M . (IV.22)Suppose H has two SVD decompositions, then:∃(U ,V ,Σ,W ,Z), H = UΣV ∗ = WΣZ∗,with U ,V ,W ,Z unitary matrices.Obviously V 6= Z, otherwise if V = Z and U 6= W , then ∃i with U :,i 6= W :,i. Thisis impossible for Σi,i 6= 0 since

W :,i =1

Σi,i

HZ∗:,i =

1

Σi,i

HV ∗:,i = U :,i (IV.23)Furthermore, Σi,i = 0 is only possible for i = M . If U :,i = W :,i for all i 6= M , then

U :,M = cW :,M where c is a complex number of norm one since U and W are unitarymatrices.Then ∃i with ∀j < i,V :,j = Z :,j and V :,i 6= Z :,i. Then ‖HV ∗:,i‖2 = Σi,i. Furthermore,

∃(c1, . . . , cM) ∈ CM such thatV :,i =

M∑

k=1

ckZ :,k (IV.24)since the vectors of Z form a base of CM .Since V is unitary, and ∀j < i,V :,j = Z :,j, it is straightforward to show thatV :,i =

M∑

k=i

ckZ :,k. (IV.25)Then‖HV ∗

:,i‖2 =n∑

k=i

|ck|Σk,k. (IV.26)86

Remember that‖HV ∗

:,i‖2 = Σi,i, (IV.27)andM∑

k=i

|ck|2 = 1, (IV.28)then |ci| = 1 since ∀i < j,Σi,i > Σj,j.This implies that V :,i = Z :,i up to a complex scalar multiplication and concludes theproof.IV.4.2.2 Application to practical systemsUnder the previous assumptions, the SVDs of the matrix H are all of the form SVD(H) =

(UΦ,Σ,V Φ), where Φ is a diagonal matrix with complex entries of norm 1. A simpleway to use the same SVD at both ends of the transmission chain is to choose Φ accordingto a given criterion, e.g. Matlab chooses Φ such that the �rst row of elements in V arereal numbers.IV.4.2.3 Discussion of the assumptionsThe assumptions of Theorem 6 are restrictive and do not always apply to practical systems.However, the channel matrix can be manipulated to verify the assumptions with a highprobability.The �rst assumption of Theorem 6 is that MR = MT . If MR 6= MT , both transmitterand receiver restrict the CSI to the min(MR,MT ) transmission eigenmodes, suppressing therequirement for the system to have the same number of antennas at both the transmitterand receiver sides of the link.Therefore, the transmission system can operate when the singular values are of mul-tiplicity 1. On the contrary, the system cannot operate when several singular values areequal to zero (highly correlated channel) or when two non-zero singular values are equal.Matrices with several singular values equal to zero can be treated by removing theeigenmodes associated with the zero singular values. This is justi�ed by the fact thateigenmodes with a gain equal to zero cannot transmit information (have a capacity of 0bps/Hz), and therefore can be simply discarded.Following the assumption that the channel matrix is i.i.d. Gaussian, the joint PDF of87

the singular values is [72]pdfΣ

= 2MT

ΠMj=1

(MT−j)!(MR−j)!exp(−ΣM

j=1Σ2j,j)×

(ΠMj=1Σ

2j,j)

max(MR,MT )−MΠi<j(Σ2i,i − Σ

2j,j)

2.(IV.29)The last term in the expression implies the probability for two equal singular values is 0.The probability becomes �nite when the singular value are quantized (see Section V.3).This occurrence is rare but practical systems need to handle such cases. Possiblesolutions include

• considering that the event is so rare that the error bursts it creates can be tolerated,• avoiding transmission on equal gain eigenmodes,• transmitting additional pilots on the equal gain eigenmodes to provide the receiverwith information on which SVD the transmitter selected.IV.4.2.4 ConclusionThough the SVD of a complex matrix is never unique, the SVD can be applied to wirelesssystems if its usage is restricted to the non-zero transmission eigenmodes. In such a case,the transmitter and the receiver can insure implicitly that they use the same SVD at bothends of the link.The only di�culty arises with the case of a channel matrix where two eigenmodes havethe same gain. In such a case, it is not possible to agree implicitly on a single SVD atboth ends of the link but practical solutions exist to correct the e�ects of this problem.IV.5 ConclusionThe optimal structure, in terms of information rate, for MIMO transmission over a �atfading channel consists of a precoding matrix and a decoding matrix which decomposethe MIMO channel into parallel transmission eigenmodes. Power allocation has to beapplied on the transmission eigenmodes to obtain the optimal performance. The SVDtransmission system actually corresponds to the optimal solution under a wide range ofcriteria, e.g. the SVD structure combined with OFDM is equivalent to the optimum spacetime coding technique in the limit of large OFDM blocks.88

Practical SVD systems apply coding and decoding to each transmission eigenmode sep-arately to reduce the complexity of the system. A loss of performance results. It is possibleto de�ne the 'system capacity' of such architectures and this de�nition allows analysis ofvarious system impairments without placing assumptions on the coding techniques.The precoding and decoding matrices, as well as the power allocation, need to bematched with the channel matrix. Errors in the estimation of the channel matrix induce aloss of performance which is negligible if the channel estimation inaccuracy is much smallerthan the inaccuracy in the detection of the transmitted signal. Finally, practical systemscan decompose the channel matrix separately at the transmitter and receiver, which isnot straightforward since the SVD of a complex matrix is not unique. Problems, such asequal singular values, have already been �agged and resolved in the case of a static channelwith perfect channel estimation. This problem is further discussed in Section V.3 undermore realistic assumptions. Other problems include getting the CSI to the transmitterand handling the dynamics in the channel. These issues are discussed in the next chapter.

89

Chapter VSVD architecture in TDD environmentThe MIMO channel has a large capacity, however low complexity solutions without CSI atthe transmitter do not take full advantage of that fact. The SVD architecture is optimalto transmit over the MIMO channel but requires the CSI at both the transmitter andthe receiver. This might result in signi�cant system overhead to transmit the CSI fromthe receiver back to the transmitter when the channel is time-varying. In practice, thisoverhead might be unacceptably large.It is possible to suppress this overhead when the channel is reciprocal. In such a casethe transmitter and the receiver can estimate the channel separately. TDD channels arepractical examples of reciprocal channels. Chapter V introduces the SVD architectureover a reciprocal channel. The impact of the errors due to channel estimation (time delayand channel estimation noise) on the system capacity is analysed and practical solutionsare proposed.TDD channels are reciprocal in essence, but mismatched transmitter and receiverchains can remove the reciprocity of the channel. In such a case, hardware calibrationhas to be applied to recover the reciprocity property of the channel. The e�ect of mis-matched transmitter and receiver chains is studied in Section V.2. A calibration procedureis proposed which completely corrects the e�ect of mismatched chains without requirementfor additional hardware. The calibration procedure relies on a handshake at the beginningof the transmission, as well as the hypothesis that the impairments of the chains are static.These results have been partly published in [73] and the calibration procedure has beengranted a provisional patent.The channel estimation is usually conducted through measurement of pilot symbols.91

Therefore, the channel estimation is not perfect. The e�ect of incorrect channel esti-mation on SVD-TDD systems is analysed in Section V.3 through the theory of matrixperturbation. An event named 'singular value crossing', which prevents the possibility ofrobust transmission, is exhibited. The probability of occurrence of this event is studiedthrough matrix perturbation theory as well as system simulation. These results are, tothe knowledge of the author, new and unpublished.The e�ects of imperfect channel estimation on system capacity are extensive, as ex-plained in Section V.4. However, a new architecture, with limited added complexity,obtains the bene�t of the SVD architecture when CSI is precisely known at both ends ofthe link while seamlessly shifting to a non-precoded system when the channel estimationprecision deteriorates at the transmitter. The new architecture transmits the pilot sym-bols through the transmitting matrix, which allows the receiver to gain knowledge of boththe CSI and the transmitting matrix with a single set of pilot symbols. These results havebeen partly published in [58], [74] and [75]Though the proposed architecture is robust to imperfect channel estimation at thetransmitter, the system capacity is always higher when a better channel estimate can beobtained. It is possible to improve the channel estimation through increasing the density ofpilot symbols. However, this results in additional overhead. Section V.5 proposes to reducethe errors due to the time-variation of the channel and noisy estimate through �ltering ofthe CSI estimate in time. This proposed scheme consists simply in an extension to MIMOchannels of Pilot Symbol Assisted Modulation (PSAM) [76]. The �ltering can be appliedto either the CSI or the precoding/decoding matrices, though superior performance isusually obtained by �ltering the CSI (see Section V.5). These results have been partlyaccepted for publication in [77].V.1 SVD architecture on reciprocal channelsConsider a MIMO reciprocal �at-fading channel. The channel from antenna i of transceiverA to antenna j of transceiver B is represented by the complex coe�cient Hj,i. For areciprocal channel, the channel from antenna j of transceiver B to antenna i of transceiverA is Hj,i. Therefore, if the MIMO channel from A to B is the matrix H , the channel fromB to A is H†.Ideal SVD based systems have knowledge of the CSI at the transmitter and receiver92

simultaneously. SVD systems over reciprocal channels can obtain the CSI at both ends ofthe link without explicit feedback of the CSI. Both transmitter and receiver estimate thechannel through measurement of pilot symbols (Fig. V.1), as detailed in the following.Channel

1

(U*)

V

V

H

Transceiver A Transceiver B

SVD

U*

SVD

1Pilots, PB

H

Pilots, PAFigure V.1: SVD transmission over reciprocal channelIt is straightforward to obtain the CSI at the receiver. Pilot symbols PA1 are trans-mitted through the channel H prior to data transmission (Fig. V.1).To obtain the CSI at the transmitter, the reciprocity properties of the channel areused. If transceiver A wishes to send data to transceiver B, it requests B to send pilotsymbols PB1 to obtain the CSI, H†, and then transmits data through the V matrix thathas been derived from the CSI.The necessity for pilot symbols can be suppressed if blind channel estimation is applied.V.2 Hardware calibration procedureV.2.1 E�ect of hardware errors on channel reciprocityFor a TDD system, the wireless channel is reciprocal. However, the channel does notinclude the transmitter and receiver chains. So far, it has been assumed that the ampli-tude and phase shift of the transmitters' analog electronic (between the digital to analogconverters and the antennas) is the same as that of the receivers' electronics (between theantennas and the analog to digital converters) for a given antenna. That is, the transmit-ter and receiver Radio Frequency (RF) chains are perfect. When the transmitter chainincludes a phase or amplitude error that is not replicated in the receiver chain, the channelis not reciprocal. 93

The errors introduced by transmitter A, receiver A, transmitter B and receiver B arerepresented respectively by the matrices ErT,A, ErR,A, ErT,B, ErR,B. Each of thesematrices are diagonal, with complex elements on the diagonal, if it is assumed that thereis no leakage between the hardware chains. In this case, ErT,A1,1 is the error introduced bythe RF section of the transmitter connected to antenna 1 of transceiver A.The overall channel from transceiver A to transceiver B is the matrixHA→B = ErR,B×

H × ErT,A whereas the the channel from transceiver B to transceiver A is the matrixHB→A = ErR,A ×H† ×ErT,B. Clearly, HA→B is not always the transpose of HB→A. Inmost cases, HA→B and HB→A are uncorrelated.Usually, transceiver B estimates the channel HA→B through reception of pilot symbols.Transceiver B performs the SVD of HA→B, and uses this results to transmit on the reverselink, i.e.over the channel HB→A, resulting in a loss of performance of the system. The overalltransmission relationship becomes:

y = (UB→A)∗(HB→AV A→Bx + n). (V.1)Simulation results of a MT = MR = 4 SVD system are presented in the following toassess the e�ect of hardware errors on the system capacity. The MIMO channel is assumedto be i.i.d. Rayleigh distributed.V.2.1.1 Amplitude errorTo measure the sensitivity of the system to an amplitude error in the hardware chains,the system was simulated over an i.i.d. Gaussian channel with ErT,A, ErR,A, ErT,B andErR,B being diagonal matrices with real entries. The system capacity (de�ned in SectionIV.3.2) was obtained.A system with ideal RF chains corresponds to the system with ErT,A = ErR,A =

ErT,B = ErR,B = I. The amplitude error introduced by each hardware chain is consideredto be uniform and is measured by the ratio of the maximum amplitude error introduced,with respect to the ideal amplitude, i.e. 1. Therefore, if the system su�ers a 20% error,the entries of the diagonal of ErT,A, ErR,A, ErT,B, ErR,B are random variables with anamplitude uniformly distributed between 0.8 and 1.2. Results are presented in Fig. V.2.Every transmission and reception chain has to be accurate in the limit of 4% (equivalentto 28 dB) to limit the capacity loss to a maximum of 2 bits.94

0 5 10 15 20 25 308

10

12

14

16

18

20

22

24

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

Amplitude Error (%)

Equal power allocation, SNR=20dB

MIMO Capacitystandard SVDFigure V.2: System capacity of a SVD based system with hardware amplitude errorsV.2.1.2 Phase errorTo measure the sensitivity of the system to a phase error in the hardware chains, thesystem was simulated with ErT,A, ErR,A, ErT,B and ErR,B being diagonal matrices withcomplex entries with amplitude 1.The phase error introduced by each hardware chain is considered to be uniform, andis measured by the maximum phase error introduced. Therefore, if the system su�ers a20 degrees error, the entries of the diagonal of ErT,A, ErR,A, ErT,B, ErR,B are randomvariables with an amplitude of 1 and a phase uniformly distributed between −20 and 20degrees. System capacity (de�ned in Section IV.3.2) results are presented in Fig. V.3. Theplot shows that the system is particularly sensitive to hardware phase errors. Consideringthe previous �gure of 2 bits of loss of capacity, the phase error has to remain smallerthan 5 degrees. This requirement is particularly unrealistic. For example OFDM systemsexperience �at fading channel on each tone. However the system is wideband, since OFDMuses a number of subcarriers, at di�erent center frequencies, to reach commercially viabledata rates. It would be di�cult to design transmission and reception hardware chains withso small phase tolerance over the entire bandwidth.95

0 20 40 60 80 100 120 140 160 1800

5

10

15

20

25

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

Phase Error (degrees)

Equal power allocation, SNR=20dB

MIMO Capacitystandard SVDFigure V.3: System capacity of a SVD based system with hardware phase errors(MT = MR = 4, SNR = 20dB, i.i.d. Rayleigh channel).V.2.2 Calibration procedureConsidering the results aforementioned, it is unrealistic to rely on hardware design toprovide RF transmission chains with performance close to the requirements of a SVDbased MIMO system.The most common solution presented in the literature to combat these e�ects in an-tenna arrays, is to conduct a calibration procedure. Calibration usually consists of mea-suring the distortion introduced on a test signal by the di�erent RF chains taking part inthe antenna array. However, this type of procedure usually relies on using a test receiving(transmitting) antenna, to measure (transmit) a signal emitted (received) by each of thetransmitting (receiving) antennas [78]. This approach is only viable in the case of a basestation, where only one end of the link is using an antenna array and where complexity canbe added to the base station in order to increase the performance of the system. However,in the case of wireless modems and full MIMO systems (as opposed to Multiple InputSingle Output (MISO) systems, e.g. antenna arrays), this solution is di�cult, for obviousreasons of size and cost of the modems. 96

It is nevertheless possible to perform some type of calibration on the system. Asin a standard calibration procedure, it is assumed that the hardware components have�xed amplitude and phase response in time, or that these characteristics drift very slowly,compared to an average transmission duration. Only sparse calibration runs are requiredin time. Consider a scenario where transceiver A and transceiver B initiate a data transferby some kind of 'handshaking'. It is assumed that the wireless channel (including the RFchains to transmit and receive the signals) can be perfectly measured through transmissionand reception of pilot signals. Also the channel is assumed �xed during the calibrationprocedure. At the start of a transmission from transceiver A to transceiver B, a calibration

B

Transceiver BTransceiver A

T,A

Er

Pilots

PilotsEstimation

H

R,B

Er

T,B

ErR,A

Er

ErR,BT,A

Er

HB−>A

ErR,A

Estimation

ErT,B

Estimation

Estimation

A−>B

H

CorA Time

Channel

H

CorFigure V.4: Calibration procedure.procedure takes place, as presented in Fig. V.4. Assume transceiver A (transceiver B) hasMA (MB) antennas. Transceiver A sends pilot symbols to transceiver B, enabling B tomeasure HA→B = ErR,B ×H ×ErT,A. Transceiver B immediately replies to A, sendingpilot symbols to enable A to measure HB→A = ErR,A ×H† ×ErT,B. Transceiver B alsotransmits back to A the CSI of the forward link HA→B. Likewise, A transmits back to Bthe CSI of the return link HB→A.The knowledge of both HA→B and HB→A is enough to deduce the correction matricesthat can make the links symmetrical. The aim of the calibration procedure is to producetwo matrices CorA and CorB such that HA→B = CorB × (HB→A)† × CorA. It is alsonecessary that the reciprocity of the channel is maintained as the transmission channelchanges in time. For example, it is not possible to choose the correction matrices as97

CorB = I and CorA = ((HB→A)†)−1HA→B.A possible choice to achieve correction isCorA

i,i =HA→B

1,i

HB→Ai,1

∀i ∈ (1, ..,MA)

CorAi,j = 0 ∀i 6= j

(V.2)andCorB

i,i =HA→B

i,1

HB→A1,i

× HB→A1,1

HA→B1,1

∀i ∈ (1, ..,MA)

CorBi,j = 0 ∀i 6= j

. (V.3)V.2.3 E�ects of the calibrationIt is not possible to deduce from the two channels' matrices the parameters ErT,A, ErR,A,ErT,B and ErR,B. Therefore, it is not possible to correct explicitly the errors introducedby the hardware chains.The �rst row of HA→B consists of the �rst row of H , with every element being mul-tiplied by the error introduced by the �rst receiver chain of B, and the element in eachcolumn being multiplied by the error introduced by the corresponding transmitter chain ofA. In a similar manner, the �rst row of (HB→A)† consists of the �rst row of H with everyelement being multiplied by the error introduced by the �rst transmitter chain of B, andthe element in each column being multiplied by the error introduced by the correspondingreceiver chain of A.Therefore, it is easily observed that, in both cases, dividing the entries of each columnby the entry of the �rst column produce two e�ects:

• it cancels the in�uence of the errors introduced by transceiver B,• it prevents us from measuring the errors introduced by each transmission (reception)chain of A on its own, but contains information on the ratio of the error introducedby a transmission (reception) chain, compared with the �rst transmission (reception)chain.The calibration process ensures that the ratio of errors to signal introduced by transmitter

i and j of transceiver A is identical to the ratio introduced by receiver i and j of transceiverA. The calibration does not correct the errors, it just ensures that they are symmetric onthe transmission and reception chains of a transceiver. Therefore, it is self evident thatusing these �xed correction matrices, symmetry in the system remains, even when the98

wireless channel varies in time. It is nonetheless crucial that the channel remains staticduring the calibration handshaking.Finally, since the correction explicitly makes the transmission and reception chainssymmetric for a transceiver, transceiver A does not need to recalibrate its hardware chainsfor a transmission to another completely di�erent transceiver C.V.2.4 Choice of calibration matricesIt is important to note that an in�nite number of correction matrices that would correct(HB→A)† into HA→B exists. Every matrix c × CorA and 1

c× CorB, where c is anynon-zero complex number, provides the adequate correction.There is no advantage in choosing the phase of the correction matrices. However,there is a signi�cant advantage in modifying the amplitude of the correction matrices: itmay be interesting to transfer some workload from the transmitter to the receiver chains.This feature may be helpful to avoid saturation of either the transmitter or the receiverampli�ers.V.3 SVD-TDD system and singular values swappingIt is possible to obtain the CSI at both ends of the link when the channel is reciprocal aspresented in Section V.1. This relies on the following assumptions:

• pilot symbols provide perfect channel estimation,• the channel is static.Neither of these assumptions applies to practical systems. The complete system, in-cluding errors in the channel estimation, over a TDD time varying channel is presented inFig. V.5.Consider the transmission from transceiver A to transceiver B at the time slot (k+1)t.The precoding matrix is derived from the CSI available at transceiver A, i.e. the CSIestimated from pilot symbols PB1 at time slot kt. This precoding matrix is denoted as

V (kt). The decoding matrix, denoted U∗((k + 1)t), corresponds to the CSI estimatedfrom the reception at time slot (k + 1)t of pilot symbols PA1. The complete transmissionequation becomes

y = U∗((k + 1)t)(HV (kt)x + n), (V.4)99

Channel

*U((k+1)t)

V(kt)

V(kt)

H(kt)

1Pilots, PB

SVD

SVD


Pilots, PA1

(U )((k−1)t)*

H((k+1)t)

Time

n

nFigure V.5: SVD transmission over a TDD channel with channel estimation error.where the precoding and decoding matrices are neither matched with the channel matrixnor with each other.The impact of the estimation errors (errors due to noise in the estimation or errorsdue to the time-varying nature of the channel) can be analysed according to matrix per-turbation theory, assuming that the channel matrix H as been estimated incorrectly asH = H + N est.V.3.1 Matrix perturbation theory, singular valuesSeveral powerful results exist concerning the perturbation of singular values. The mainresult is probably the following theorem, stating that the singular values of a matrixare perfectly conditioned, i.e. no singular value can move more than the norm of theperturbations.Theorem 7. Mirsky Let H and H be matrices of the same dimensions with singularvalues

Σ1,1 ≥ Σ2,2 ≥ . . . ≥ ΣM,M ,

Σ1,1 ≥ Σ2,2 ≥ . . . ≥ ΣM,M .(V.5)Then for any unitary invariant norm ‖.‖u

1‖diag(Σi,i − Σi,i)‖u ≤ ‖H − H‖u. (V.6)The proof of the theorem is given in [79]. Two immediate consequences are1A unitary invariant norm ‖.‖u is a norm such that for all matrix H (dimension M×N) and all unitarymatrices U (dimension M × M) and V (dimension N × N), ‖UH‖u = ‖H‖u and ‖HV ‖u = ‖H‖u.100

Theorem 8. Weyl|Σi,i − Σi,i| ≤ ‖N est‖2, i = 1, . . . ,M, (V.7)andTheorem 9. Mirsky√∑

i

(Σi,i − Σi,i)2 ≤ ‖N est‖F s (V.8)as shown in [79], where ‖.‖2 is the matrix 2-norm2 and ‖.‖F is the Frobenius norm3.It is notable that:• there is no restriction on the size of the error,• the ordering of the singular values by magnitude provides a natural pairing betweenthe singular values of the channel matrix and the estimation of the channel matrix.V.3.2 Matrix perturbation theory, singular vectorsThe perturbation of singular vectors is di�cult to analyse or bound, due to several reasons:• arbitrarily small perturbations can completely change singular vectors,• it is di�cult to de�ne a meaningful distance between vectors.V.3.2.1 Example of catastrophic perturbationConsider the matrix

H =

1 0

0 1 + ε

, (V.9)with precoding matrixV =

1 0

0 1

. (V.10)If H is estimated asH =

1 ε

ε 1

, (V.11)2The matrix 2-norm is de�ned as the largest singular value of the matrix.3The Frobenius norm of the matrix H is de�ned as the square root of the sum of the absolute squaresof its elements, i.e. ‖H‖F =∑

i

∑

j h2

i,j . 101

then the corresponding precoding matrix isV =

1√2

1 1

1 −1

, (V.12)without assumptions on the size of the perturbation ε. Therefore it is impossible to deriveperturbation bounds on singular vectors.Consider such a catastrophic perturbation, the transmitter transmits the �rst datastream on the �rst antenna and the second data stream on the second antenna, accordingto (V.10). The receiver tries to receive the �rst stream by adding the symbol sent onthe �rst antenna and the symbol sent on the second antenna since the receiver believesthat both data streams where transmitted according to (V.12). Therefore, in the case ofcatastrophic perturbation, the smallest channel estimation error results in a bit error rateof 0.5. This example, combined with the results of Section IV.4, highlights the fact thatthe primary problem is mismatch of U and V , rather than mismatch of the channel to Uand V when the latter are matched.V.3.2.2 Subspaces perturbation theorySeveral results on subspaces perturbation theory presented in [79] are mentioned in thefollowing.To analyse the perturbation of singular subspaces, it is necessary to de�ne a meaningfuldistance between two vectors. The distance usually used in perturbation theory is mean-ingless for singular vectors. E.g. consider V a precoding matrix, estimated as V = −V .Obviously the transmission eigenmodes have not been perturbed, whereas ‖V − V ‖F islarge.A meaningful distance between two singular subspaces is given by the canonical anglesbetween the two subspaces [79]. Bounds on the perturbation of singular subspaces canbe found but the accuracy of the bound depends on the inverse of the distance betweensingular values. The bound is not applicable when two singular values are equal.An expansion of the perturbations of the singular value of a matrix is also derived in[79], however the expansion only applies to the subspaces with non-zero singular valuesof multiplicity one. It is clear, with this result, that the transmitter and receiver cannotunambiguously pair input and output singular subspaces simply on the basis of size or-dering of measured singular values, and that signi�cant mutual interference among suchsubspaces may result. 102

V.3.3 Probability of robust transmissionFrom the previous results, it is clear that the SVD transmission system cannot operateproperly when some of the channel matrix singular values are zero or of multiplicity higherthan one.If some of the singular values are zero or multiplicity higher than one:• the SVD of the channel matrix cannot be determined uniquely,• the perturbation of the corresponding singular subspaces cannot be bounded.Therefore, it is necessary to determine the probability for the channel matrix to havesingular values that are non-zero and of multiplicity one. In Section IV.4.2, it was men-tioned that the probability of these events is one. This result is only true if it is assumedthat the CSI is known with in�nite precision. However, in practical systems, the CSI isonly estimated through measurement of pilot symbols. Noise in the estimation, estima-tion delay and limitations of the system (e.g. quantization noise) can limit the channelestimation accuracy.In such a case the singular values Σi,i are known up to a con�dence interval. Twosingular values are always distinct when their respective con�dence intervals are not joined.Furthermore, a singular value whose con�dence interval includes zero might not correspondto an actual transmission eigenmode. Therefore an SVD transmission system needs todetermine the singular values of the channel matrix but also the con�dence interval length,i.e. how precisely these singular values are known.Applying Theorem 8, it is possible to obtain a con�dence interval:

|Σj,j − Σj,j| ≤ ‖N est‖2. (V.13)Furthermore, applying Theorems 8 and 9, Σj and Σi correspond to two non-equalsingular values if|Σj,j − Σi,i| ≥ min(2 × ‖N est‖2,

√2 × ‖N est‖F ), (V.14)where the second term is derived from Theorem 9 as explained in the following. FromTheorem 9

√

(Σi,i − Σi,i)2 + (Σj,j − Σj,j)2 ≤√∑

k

(Σk,k − Σk,k)2 ≤ ‖N est‖F (V.15)103

and for all d1, d2 real positive numbers√

(d1 + d2)2 ≤√

2√

d21 + d2

2. (V.16)Therefore|Σi,i − Σj,j| > |Σi,i − Σi,i| + |Σj,j − Σj,j| (V.17)if

|Σi,i − Σj,j| >√

2‖N est‖F . (V.18)Obviously the length of the con�dence intervals depends on the realization of theestimation noise. One solution is to apply a probabilistic con�dence interval: the intervalthat guarantees to contain the perturbed singular value x% of the time. Another solutionis to use the average con�dence interval.Given a con�dence interval, it is straightforward to deduce the probability for allsingular values to be non-zero and of multiplicity one, by just integrating pdfΣover thedomain corresponding to non-overlapping con�dence intervals. Results are shown in Fig.V.6 where SNRconf refers to the ratio between the power of an element of the channelmatrix and the square of the size of the con�dence interval. The case of distinct singularvalues with all singular values being non-zero corresponds to the curve 'well-separated'.It is possible to reduce the requirements on the system following Section IV.4.2.3,i.e. non-zero eigenmodes are required to be of multiplicity one and zero eigenmodes arediscarded for transmission. The curve labelled 'Zero eigenmode' corresponds to the proba-bility for some eigenmodes to have a negligible gain while the other eigenmodes correspondto singular values of multiplicity one. Finally, the curve labelled 'Correct SVD' correspondsto the probability for the SVD transmission system to be robust, i.e. non-zero eigenmodescorrespond to singular values with a multiplicity of one. The curves are worst case becauseof the inequalities in (V.13) and (V.18).Wireless channels face fading in time, i.e. are time varying. If two singular values areequal to each other at one point in time, they are likely to separate later, due to channelfading. As explained earlier, when two singular values of the channel are equal the systemis not robust since a small variation of the channel might result in a large variation of theprecoding and decoding matrices. Therefore, it is of interest to determine the probabilityof occurrence of such an event which is referred to as 'singular value crossing' in thefollowing. The probability of 'singular value crossing' is investigated through computersimulation in the next Section. 104

0 5 10 15 20 25 300

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

SNRconf

in dB

Pro

babi

lity

Correct SVD Zero EigenmodeWell Separated

Figure V.6: Probability of robust transmission vs amplitude of the estimation con�denceinterval, MT = MR = 4, i.i.d. channel of complex Gaussian entries with mean 0 andvariance 1.V.3.4 Singular value crossingConsider a time-varying 4x4 MIMO channel treated as 16 independent SISO time-varyingRayleigh channels. Each SISO channel follows the Jake's time-varying channel model[80, 81], i.e. the autocorrelation of the time-varying channels is J0(2πFdδt), where Fddenotes the Doppler frequency, δt the time delay and J0(.) is the zeroth-order Besselfunction of the �rst kind. In simulations involving Fdδt, a value of Fdδt = 0.038 wasassumed. This represents a realistic value expected in the 802.11a and Hiperlan 2 standardswith a Doppler frequency of 38 Hz (corresponding to a terminal velocity of 2 m.s−1 for acarrier frequency of 5.725 GHz) and a typical time-duplex delay of 1 ms. The amplitudesof the singular values of the time-varying MIMO channel are plotted versus time in Fig.V.7. A 'singular value crossing' is exhibited.A 'singular value crossings' corresponds to the event where Σi,i − Σi+1,i+1 = 0. Deter-mining the probability of this event is similar to the well known issue of determining thezero-crossing rate of a random variable. Consider a real valued random variable x(kTs),the zero crossing rate of x for k ∈ [0, N ] is de�ned as the number of indexes k such that105

3640 3650 3660 3670 3680 3690 37000

0.5

1

1.5

2

2.5

3

3.5

4

Am

plitu

de o

f the

sin

gula

r va

lue

Time in Samples

1st singular value2nd singular value3rd singular value4th singular value

Figure V.7: A clear singular value crossing between the second and third singular valuesat time sample 3667.x(kTs)x((k + 1)Ts) is negative. Zero-crossing rate theory is used in communications, e.g.to determine the velocity of a mobile [82]. However, determining the level crossing rate ofsingular values of a matrix with random entries is di�cult, due to the following reasons:

• The singular values of a matrix are highly non-linear and non-monotonous functionsof the entries of the matrix. Singular values are determined, in practice, throughrecursive algorithms, not analytical functions.• The SVD of the matrix involves a sorting function to sort the singular values indecreasing order. The sorting function implies that, strictly speaking, Σi,i > Σi+1,i+1

∀i, i.e. singular values are never crossing.• Level crossing rate have been determined for Gaussian random variables and simplefunctions of Gaussian random variables such as monotone transformation of randomprocesses [83] or mixtures and products of Gaussian processes [84]. However, tothe knowledge of the author, the level crossing rate of the result of such a complexfunction as the SVD on Gaussian random variables has not been determined in theliterature. 106

It is unclear whether the level crossing rate of the singular values of a matrix with Gaussianentries can be determined analytically. Due to the lack of results in the literature, anad-hoc criterion is introduced in the following to estimate through simulations the levelcrossing rate of singular values.The criterion chosen to de�ne 'singular value crossings' follows. Σi,i and Σi+1,i+1 arecrossing each other at tc if:• |Σi,i(tc) − Σi+1,i+1(tc)| ≤ 0.1,• |∂Σi,i

∂t(t < tc) − ∂Σi,i

∂t(t > tc)| ≥ |∂Σi,i

∂t(t < tc) − ∂Σi+1,i+1

∂t(t > tc)|,

• |∂Σi+1,i+1

∂t(t < tc) − ∂Σi+1,i+1

∂t(t > tc)| ≥ |∂Σi+1,i+1

∂t(t < tc) − ∂Σi,i

∂t(t > tc)|.This criterion corresponds to the intuitive idea of observing two curves that cross eachother at one point: their value is the same at the crossing point and the gradient of bothcurves remains approximately constant around the crossing point. The choice of the value

0.1 is empirical and was chosen by the author through a trial and error process on thesimulation results.50 × 16384 time samples of a MR = 4, MT = 4 time-varying MIMO channel (16 i.i.d.SISO channels, Jake's fading, Fdδt = 0.038) were simulated. 622 crossings were detectedbetween the �rst and second singular values, 921 crossings between the second and thethird and 1000 between the third and the fourth. This represents a total of 2543 crossings,for 50 × 16384 time samples. At Fdδt = 0.038, there is less than 0.32% probability that a'singular value crossing' occurs from one sample to the next. A 'singular value crossing'is very unlikely under the given assumptions.Moreover, the frequency of 'singular value crossings' is independent of the samplingrate for high sampling rate, i.e. for a sampling rate much higher than the 'singular valuecrossing' rate. The results above can be restated as: at a doppler frequency of Fd = 0.038,if the channel is sampled every second (Ts = 1), there will be on average a crossingapproximately every 322 samples (seconds). The frequency of the crossings is Fsvc =

1/322Hz. Therefore, an empirical general relationship can be deduced: Fsvc/Fd = 0.08.This relationship is applicable for any MT = 4,MR = 4 i.i.d. Rayleigh fading channelfollowing Jake's fading.The results of the simulation also indicate that the �rst and second singular valuescross each other less often than the others. Due to bits (and possibly power) allocation,107

the channel corresponding to the �rst singular value is the most important: it is the sub-channel carrying the most information. This sub-channel is less subject than the othersto 'singular value crossings', o�ering a reliable, high SNR, channel for transmission.V.3.5 Subspace swappingIt has been shown in Section V.3.4 that 'singular value crossings' are rare, but do occur.'Singular value crossings' can a�ect the throughput of a SVD-based system. If the trans-mitter measures the channel before the 'singular value crossing', and the receiver afterthe 'singular value crossing', the result of the SVD performed on their respective channelmatrices might be very di�erent, leading to a burst of errors in the transmission.Therefore, it is of interest to determine the behaviour of the singular subspaces beforeand after a 'singular value crossing'. Following the SVD of the channel matrix, the �rstcolumn of the V matrix and the �rst column of the U matrix are respectively the inputsubspace and output subspace corresponding to the �rst eigenvalue.To determine the behaviour of the subspaces around a 'singular value crossing' point,the autocorrelation of the �rst input subspace (i.e. the autocorrelation of the the �rstcolumn of the V matrix) is plotted, as well as the cross-correlation between the �rst andsecond input subspaces (i.e. the cross-correlation between the �rst and second columns ofthe V matrix). Results (corresponding to the 'singular value crossing' presented in Fig.V.7) are presented in Fig. V.8 for the input subspace, and Fig. V.9 for the output subspace.The autocorrelation stays high until the crossing point and then drops suddenly. On thecontrary the cross-correlation is very low, but increases suddenly after the crossing point.This clearly indicates that the subspaces are linked with the singular value, and when the�rst singular value becomes lower than the second, the corresponding columns of the Vmatrix swap position at the same time as the singular values swap position.The swapping of the subspaces is simply due to the ordering function performed duringthe SVD. If the transceiver measures the channel before the crossing and the receiver afterthe crossing, the information sent on sub-channel 1 is received on sub-channel 2, and theinformation sent on sub-channel 2 is received on sub-channel 1.Supposing that it is possible to detect 'singular value crossings', either the transmitteror the receiver might be able to handle it by simply swapping the sub-channels at either thetransmitter or the receiver. This result is of great interest for the design of a wireless SVD-108

3664 3665 3666 3667 3668 3669 36700

0.1

0.2

0.3

0.4

0.5

0.6

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0.8

0.9

1

Am

plitu

de o

f the

cor

rela

tion

Time in Samples

autocorrelation of V1crosscorrelation V1−V2

Figure V.8: Correlation of the input subspacesbased system: singular subspaces are related before and after a 'singular value crossing'.'Singular subspace swapping' can be used as a criterion to detect a 'singular valuecrossing'. It is possible to determine whether two singular values become close to eachother with or without crossing by checking the amplitude of the autocorrelation and crosscorrelation of both the input and output subspaces. If both of them display a crossing ata point where the singular values are close in amplitude, then a 'singular value crossing' isdetected. This new criterion produces the following results: 50× 16384 time samples weresimulated. 823 crossings were detected between the �rst and second singular values, 1149crossings between the second and the third and 1065 between the third and the fourth.This represents a total of 3037 crossings, for 50 × 16384 time samples and Fsvc/Fd = 0.1.Obviously this criterion indicates a higher probability of singular value swapping. Howeverit corresponds better to the reality of a MIMO-SVD transmission system and therefore ismore representative of the expected behaviour of the system.V.3.6 Correction of singular value crossingA practical way to correct the e�ect of singular value crossing involves transmitting asecond set of pilots through the precoding matrix. Therefore the receiver can correct any109

3664 3665 3666 3667 3668 3669 36700

0.1

0.2

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0.4

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1

Am

plitu

de o

f the

cor

rela

tion

Time in Samples

autocorrelation of U1crosscorrelation U1−U2

Figure V.9: Correlation of the output subspacesmismatch between transmitting and decoding matrices. This method is, however, notlimited to the correction of 'singular value crossings' as explained in the following Section.V.4 SVD architecture modi�ed for TDD environmentAs presented in Section V.3, the precoding and decoding matrices can be completelymodi�ed when the channel is perturbed. However, these catastrophic events, referred toas 'singular value crossings' are rare.Regardless of 'singular value crossings', the precoding matrix of SVD-TDD systems isgenerally not matched with the channel matrix (see (V.4)). This leads to a degradationof the system capacity (de�ned in Section IV.3.2) of SVD-TDD systems. A severe 4 dropin capacity results.The delay between channel measurement in one direction and transmission in the othermay vary due to time slot allocation in the TDD protocol. Furthermore, this delay maynot be identical for symbols inside a frame, depending on whether the symbols are at the4For a SVD-MIMO system with 4 transmitting and 4 receiving antennas, over a �at fading i.i.d.channel, at SNR=20dB, assuming Jake's fading, the capacity drops from more than 21 bits/s/Hz forperfect CSI to less than 8 bits/s/Hz for Fdδt > 0.1, see Section V.4.3.2110

start or the end of the frame. A conservative approach in assessing the performance ofthe system, identi�es δt with the worst case delay. In the following, the capacity of asystem with �xed Fdδt is considered and then the capacity is analysed for varying Fdδt.It is considered in Section V.4 that pilot symbols provide perfect CSI, i.e. the channelestimation noise is considered negligible.V.4.1 Correction of the CSI impairmentV.4.1.1 Proposed ArchitectureSystem capacity drops when the transmitter �lters data through a V (t−δt) matrix, whichcorresponds to an outdated CSI. However, the receiver can mitigate the e�ect of thisimpairment through processing of the received signals. The receiver can be provided withinformation on the outdated V matrix (V (t − δt)) if the transmitter sends an additionalset of pilot signals PA2 through V (t−δt), the channel and U (t) (Fig. V.10). A correctionEstimation

PA1PA

* δ(U )(t−2 t)

1Pilots, PB

SVD


Pilots, PA1Pilots, PA2

V(t− t)

*U(t)

H(t− t)

SVD

Correction, B(t)

Time

Channel

H(t)V(t− t)

δ

δ

δ

2Figure V.10: SVD transmission over a TDD channel with channel estimation errorrecovery pilot symbols PA2.matrix, B(t) can then be derived to partially compensate the e�ect of the outdated V (t−δt) matrix.

y = B(t)U(t)(H(t)V (t − δt)x + n) (V.19)The requirement of two sets of pilots in the one direction (PA1 to calculate U (t) and PA2to calculate B(t)) adds to the signaling overhead in the system. A new architecture isproposed that does not use the set of pilot signals PA1. In the new architecture, one setof pilot symbols (PA2) is sent from the transmitter through V (t − δt) and the channelH(t), to the receiver (Fig. V.11). In this situation, the matrix H(t)V (t − δt) is known111

at the receiver and linear processing can be applied:y = C(t)(H(t)V (t − δt)x + n) (V.20)

H(t)4

y 2

3y

1yx 1

2x

x 3

4xV(t− t)δ

2PAPA2

n3

4n

2nn1

C(t)yFigure V.11: Correction of the CSI Impairmentwhere C(t) is the matrix of the linear correction. Since U(t) is unitary, the performanceof equations (V.19) and (V.20) is identical for a given selection criteria of C(t) and B(t).Therefore, from here on, we only consider the system that uses one set of pilots. Twocriteria for the selection of C(t) are considered: zero-forcing (ZF) and minimum meansquare error (MMSE). The ZF correction is applied by setting

C(t) = (H(t)V (t − δt))+. (V.21)This method forces the interference to zero, but can lead to noise enhancement. It isreferred to in the remainder of the paper as a 'SVD+ZF' system.The MMSE correction is applied by settingC(t) = Q(H(t)V (t − δt))∗ (IMR

+ (H(t)V (t − δt))Q(H(t)V (t − δt))∗)−1 . (V.22)It is referred to in the remainder of the paper as a 'SVD+MMSE' system. It is assumedthat Q is a real diagonal matrix, i.e. the input symbols are independent and the entriesof Q represent the power allocated to the sub-channels. The receiver needs to have priorknowledge of the power allocation applied by the transmitter to perform MMSE decoding.From an implementation perspective the new architecture only requires the set of pilotsPA2. This applies to the reverse link (transceiver B to transceiver A), where (U (t)∗)† isrequired for transmission because (U (t)∗)† can be obtained from the SVD of H(t)V (t−δt),measured by pilots PA2 only. The fact that V (t− δt) does not match the channel has noe�ect on the calculated value of U (t), since any error in V (t − δt) only a�ects the inputsubspaces (V is a unitary matrix). 112

It should also be noted that the so-called 'SVD transmission' system over a reciprocalchannel performs the eigenvalue decomposition of the channel, but does not need to per-form a full SVD. Transceiver B requires knowledge of U (t) to receive data transmitted overH(t), and (U (t)∗)† to transmit data over (H(t + δt))†, but does not require knowledge ofV (t).The new architecture solves some implementation issues of the SVD architecture whilekeeping the desirable features of SVD architectures that have been outlined in SectionIV.2.1.V.4.2 Comparison with reference systemsThe system capacity (see Section IV.3.2) of the 'SVD+ZF' and 'SVD+MMSE' systemswas compared with the following reference systems:

• The theoretical capacity of the channel (named 'MIMO Capacity'), which providedthe upper bound of the systems capacities.• A standard SVD-based system (named 'Standard SVD') which provided a referencefor the capacity gains provided by both 'SVD+ZF' and 'SVD+MMSE' systems.• A MIMO system with no precoding which included �ltering of the received symbols(Fig. V.12).V.4.2.1 Theoretical capacityThe theoretical capacity of the channel is independent of Fdδt. It should be noted thattwo di�erent capacity formulas exist, depending on whether the CSI is available at thetransmitter or not. Systems implementing water�lling were compared with the theoreticalcapacity of the channel with CSI at the transmitter [12]. Systems with equal powerallocation were compared with the theoretical capacity of the channel without CSI at thetransmitter [15].V.4.2.2 Standard SVD systemA standard SVD system refers to a system based on (V.4) (named 'Standard SVD') whichincluded CSI impairment. 113

V.4.2.3 MIMO systems without precodingA MIMO system with no precoding and receiver �ltering (Fig. V.12), can be representedby the transmission equation:y = D(t)(H(t)x + n). (V.23)MIMO systems with no precoding have a low complexity: the SVD does not have to be

H(t)nn3

2nn1

1y

3yy 2

y 44xx 3

2xx 1

1PAPA1

D(t)4Figure V.12: Transmission system with no precodingperformed and the pilots PB1 are not needed. The ZF linear receiver (system named 'ZFAlone', [85]) was simulated by setting

D(t) = (H(t))+ (V.24)This method forces the interference to zero, but can lead to noise enhancement. TheMMSE linear receiver (system named 'MMSE Alone') was simulated by settingD(t) = QH(t)∗ (IMR

+ H(t)QH(t)∗)−1 . (V.25)In both cases (ZF and MMSE), equal power allocation was used (Q = I), because theCSI was not available at the transmitter. Their capacity is una�ected by Fdδt.V.4.3 Simulation and resultsTo validate the proposed architecture described in Section V.4.1 a MT = MR = 4 MIMOsystem was simulated (16 i.i.d. SISO channels, Jake's fading, Fdδt = 0.038). The timevarying MIMO channels were simulated for independent variables SNR and Fdδt.The simulated systems and their corresponding transmission equations are shown inTable V.4.3. The 'system capacity' of each system, for a given channel matrix H(t),is obtained from its transmission equation as explained in IV.3.2. The average 'systemcapacity' is obtained by repeating the process over a large number of realizations of thechannel and averaging the results. 114

System Transmission Equation'SVD+MMSE' (V.20)'SVD+ZF' (V.20)'Standard SVD' (V.4)'MMSE Alone' (V.23)'ZF Alone' (V.23) .Table V.1: Simulated systems equation referencesV.4.3.1 System capacity with equal power allocation for varying SNRThe system capacity (see Section IV.3.2) of the systems against SNR with Fdδt = 0.038are shown in (Fig. V.13).

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

AvSNR in dB

MIMO Capacitystandard SVDSVD+ZFSVD+MMSEZF AloneMMSE Alone

Figure V.13: System capacity of equal power systems with delayed CSI (Fdδt = 0.038) atthe transmitterAll systems collapse at low SNRs. However, at high SNRs, the capacity of the MIMOchannel increases linearly with the SNR when expressed in dB, this has been demonstratedin [15].The systems employing linear processing at the receiver ('SVD+ZF', 'SVD+MMSE',115

'ZF Alone' and 'MMSE Alone') exhibit a linear increase in system capacity. However, the'SVD+ZF' and 'SVD+MMSE' systems display a system capacity several bits higher thanthe 'ZF Alone' and 'MMSE Alone' systems. The di�erence is approximately 5 bits/s/Hzat high SNRs. The 'MMSE Alone' outperforms the 'ZF Alone' at low SNRs, and equals'ZF Alone' performance at high SNRs. This trend is identical when coupled with the SVD('SVD+MMSE' and 'SVD+ZF' systems).The 'Standard SVD' system with the outdated V (t− δt) matrix does not bene�t fromthe MIMO e�ect, its system capacity reaches a ceiling at high SNRs. This is due to thecross talk between the sub-channels, which dominate the SINR. The cross talk createsan interference noise �oor, that grows proportionally to the signal power increase. Thisinterference noise �oor does not fall even if the receiver noise is lowered.V.4.3.2 System capacity with equal power allocationThe system capacity (see Section IV.3.2) of the three systems against Fdδt with SNR=10dBis shown in Fig. V.14.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

2

3

4

5

6

7

8

9

10

11

12

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

FdT

MIMO Capacitystandard SVDSVD+ZFSVD+MMSEZF AloneMMSE AloneFigure V.14: System capacity of equal power systems with delayed CSI at thetransmitter, average SNR=10dBWhen Fdδt is small, capacities for all systems implementing SVD converge towardsthe theoretical capacity of the channel. This is to be expected, since the channel varies116

slowly and the transmitter can estimate CSI accurately, in this situation SVD systemsallow transmission at a rate equal to the theoretical capacity of the channel [12].The 'MMSE Alone' system is 2.5 bits below the theoretical capacity of the channelwhereas the 'SVD+MMSE' system approaches the theoretical bound. This demonstratesthe e�ectiveness of the SVD approach and the importance of the processing by the Vmatrix at the transmitter. The 'ZF Alone' system su�ers an additional 2.5 bits degradationin system capacity, due to noise enhancement when the channel matrix is ill-conditioned.The system capacity of the 'Standard SVD' system deteriorates rapidly and its systemcapacity at Fdδt = 0.038 is nearly 2 bits below the theoretical capacity of the channel.The 'SVD+ZF' system reduces the e�ect of the CSI impairment at the transmitter, butleads to noise enhancement, hence outperforming the 'Standard SVD' system by less thana bit at Fdδt = 0.038. On the other hand, the 'SVD+MMSE' system mitigates the e�ectsof CSI impairment without su�ering from noise enhancement. This results in a systemcapacity drop by only 0.3 bits at Fdδt = 0.038.When Fdδt is large (fast fading), the estimate of the CSI at the transmitter is incorrect.The 'SVD+ZF' and 'SVD+MMSE' systems compensate for this incorrect processing atthe transmitter, by achieving similar system capacity to the 'ZF Alone' and the 'MMSEAlone' systems respectively. The precoding by a completely incorrect matrix does notchange the capacity of the channel, since the precoding matrix is unitary. The 'StandardSVD' system collapses due to the incorrect CSI at the transmitter, and has a systemcapacity (1.3 bits/s/Hz) lower than the capacity of a SISO channel at SNR=10dB (3.4594bits/s/Hz).The system capacity of the three systems against Fdδt with SNR=20dB is shown inFig. V.15. When Fdδt is small or large, the conclusions presented above are applicable,excluding the fact that the capacities are higher, since SNR=20dB.At high SNRs, the ZF solution becomes closer to the MMSE solution, since the noisebecomes negligible when compared with the interference.The range of Fdδts over which precoding provides an improvement ranges from zero toapproximately 0.25. The range of improvement is therefore similar at high or low SNR.V.4.3.3 System capacity of systems with water�lling power allocationThe system capacity (see Section IV.3.2) of the three systems against Fdδt with SNR=10dBis shown in Fig. V.16. 117

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

FdT

MIMO Capacitystandard SVDSVD+ZFSVD+MMSEZF AloneMMSE AloneFigure V.15: System capacity of equal power systems with delayed CSI at thetransmitter, average SNR=20dBAs discussed in Section V.4.1, the receiver requires prior knowledge of the power al-location applied by the transmitter to perform MMSE decoding. This is derived by thetransmitter using its best estimate of the channel ( H(t − δt)) and therefore is not theoptimal power allocation. It was assumed for the 'SVD+MMSE' system that the receiverhad access to this power allocation information.When the receiver does not have access to this information it can estimate the appliedpower allocation using the current CSI (H(t)) and applying the water�lling power alloca-tion algorithm. A drop in system capacity will occur, since the power allocation appliedby the transmitter, will di�er from that calculated by the receiver to derive the MMSE�lter. The new system, named 'SVD+MMSE+I', is compared with the 'SVD+MMSE'system in Fig. V.16.At low Fdδts (slow fading) the capacities of 'SVD+MMSE' and 'SVD+MMSE+I' aresimilar. As Fdδt increases the capacities diverge. However, the di�erence remains minimal,particularly over the Fdδt values the 'SVD+MMSE' scheme is likely to be used. Thesmall system capacity variation indicates the high stability of the channel singular values(corresponding to the eigenmodes' gains) over time. This is in marked contrast to the Vmatrix which �uctuates rapidly as is evident by the sudden reduction in system capacity118

of the 'standard SVD' curve.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.42

3

4

5

6

7

8

9

10

11

12

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

FdT

MIMO Capacitystandard SVDSVD+ZFSVD+MMSE+ISVD+MMSEZF AloneMMSE AloneFigure V.16: System capacity of water�lling systems with delayed CSI at thetransmitter, average SNR=10dBThe observations reported in Section V.4.3.2 at low Fdδt are relevant to this section.However the capacity of the channel is higher with water�lling power allocation. At high

Fdδt, the 'SVD+ZF' and 'SVD+MMSE' systems not only su�er from CSI impairment butalso from incorrect power allocation. This is con�rmed in Fig. V.16, where the 'SVD+ZF'and 'SVD+MMSE' system capacity curves drop below the 'ZF Alone' and 'MMSE Alone'system capacity boundaries. The 'SVD+MMSE' system is particularly sensitive to powerallocation errors, with performance dropping below the 'MMSE Alone' line at Fdδt = 0.15.The crossover point for the 'SVD+ZF' and 'ZF Alone' systems is at Fdδt = 0.22.At Fdδt = 0.038, the capacities of both the 'SVD+ZF' and 'SVD+MMSE' using wa-ter�lling are marginally better to the corresponding values with equal power allocation.However, the system capacity of the 'Standard SVD' system degrades faster with wa-ter�lling compared to equal power allocation, with system capacity half a bit lower (8.5compared to 9 bits/s/Hz) at Fdδt = 0.038 at SNR=10dB.In an ideal situation (low Fdδt, slow fading), the system capacity of water�lling powerallocation systems is only marginally higher than the system capacity of their equiva-lent equal power allocation systems. However, the simulation was based on i.i.d. MIMO119

channels. The i.i.d. channel corresponds to a high scattering environment (numerous mul-tipaths), and the gain of the sub-channels seldom becomes small. Other MIMO channelswould produce di�erent results. For example, in a line of sight (LOS) channel, a singlepositive singular value for the LOS path and null singular values for the other eigenmodeswould result. Hence an equal power allocation system would only use a transmitting powerof PMT

, whereas the water�lling power allocation system would allocate all the availablepower to the single non zero transmission eigenmode. Caution should be applied whencomparing equal power allocation systems with water�lling power allocation systems.V.4.4 ConclusionMIMO systems based on SVD algorithms produce transmission rates close to theoreticalcapacities, provided the transmitter has an accurate estimate of the CSI. When a TDDsystem is considered, the CSI can be obtained by sending pilot symbols from the receiverto the transmitter. However, frequent CSI updates are required at the transmitter, sincethe performance of an SVD based system severely degrades when the CSI is incorrect.The 'Standard SVD' system was shown to be unsuitable for Fdδts greater than 0.03 (asystem capacity loss of approximately 5 bits at SNR=20dB). The loss gets even larger athigher SNRs because the performance plateaus rather than linearly increasing. There isno bene�t in implementing an SVD algorithm alone if the V matrix is outdated.A new architecture was proposed to counter the e�ects of incorrect CSI at the trans-mitter. It uses the outdated V matrix of the 'Standard SVD' system in combination withlinear �ltering at the receiver. Firstly, pilot tones were sent through the V (t− δt) matrixand channel prior to transmission and secondly, ZF or MMSE processing was implementedat the receiver for good MIMO performance. Finally, the pre�ltering matrix, (U (t)∗)†, fortransmission in the reverse direction is obtained by taking the SVD (or eigen decomposi-tion) of H(t)V (t − δt). This architecture was shown to mitigate the e�ects of incorrectCSI at the transmitter, without increasing pilot overhead. Problems with respect to non-uniqueness of the SVD, subspace swapping, and sub-channel cross talk are all suppressedby linear �ltering based on a channel that includes the outdated pre-coding matrix.Results showed that the 'SVD+MMSE' system always outperforms the 'MMSE Alone'(no pre�ltering) system with equal power allocation. Its performance is bounded by thechannel capacity at Fdδt = 0 and approaches the 'MMSE Alone' performance at large120

Fdδts. In fact at a useable Fdδt of 0.038 and SNR=20dB the scheme has an operatinglimit less than 1 bit below the channel capacity. This compares to a 5 bit loss for the'MMSE Alone' system and a 7 bit loss for the 'SVD Alone' system. In addition thereis no performance plateau with increasing SNR. Similar trends apply to the 'SVD+ZF'system and this con�rms that improved system capacity is possible by combining SVDpre�ltering and linear post processing.When water�lling power allocation is added the performance of the 'SVD+linear �lter-ing' schemes is no longer lower bounded by the linear �ltering alone limit at large Fdδts.Water�lling should therefore not be used when power allocation is based on an overlyoutdated channel. At low Fdδts there is some system capacity improvement, but for fori.i.d. channels this is small and again water�lling would not be recommended.In conclusion systems using both SVD transmission and linear processing at the receiveroutperform systems with no CSI at the transmitter by several bits at reasonable Fdδt(<0.1), whether water�lling is used or not. The addition of some precoding by the Vmatrix, regardless if it is erroneous (outdated), was shown to be the source of majorcapacity gains.V.5 CSI improvement through channel trackingIt is shown in Section V.4 that the SVD architecture can be modi�ed to see its performancedegrade gracefully to the performance of an uncoded system as the coding matrix Vbecomes less accurate. It is assumed in Section V.4 that pilot symbols provide perfectCSI, i.e. the channel estimation noise is negligible. In practice, the channel estimationnoise is usually not negligible and degrades the accuracy of the precoding and decodingmatrices.The architecture presented in Section V.4 insures that a catastrophic perturbation ofthe precoding matrix does not have a catastrophic result on the performance of the system.However, the proposed architecture does not prevent errors in the decoding matrix tosigni�cantly reduce the performance of the system.Decoding matrices are obtained through applying the SVD on the CSI. However, theSVD is a non-linear function and a slight error in the estimated CSI can result in a largevariation of the decoding matrix. Therefore channel estimation has to be more accuratethan in a standard MIMO system. 121

In a slow fading channel, tracking of the CSI in time can improve the accuracy ofthe channel estimation. Another solution consists of tracking the precoding and decodingmatrices. These two solutions are compared in the following when tracking is performedusing linear Finite Impulse Response (FIR) Wiener �lters.V.5.1 Tracking of the CSIThe estimation of Rayleigh fading channels through PSAM has been analysed in [76].Estimation is achieved by linearly combining measured pilot symbols that have been time-multiplexed with data.This technique cannot be applied when the channel is quasi-static, i.e. the channel is�xed for one frame and consecutive frames are subject to independent fading. In sucha case, the CSI estimated on previous frames cannot be used to improve the channelestimation for the current frame.In indoor wireless LAN systems, such as modems following the 802.11a standard, thesituation is likely to be a combination of both previous cases. The channel can be consid-ered constant over one frame and correlated with channels at previous time slots.V.5.1.1 Forward estimationIn a slow fading environment, the pilot symbols measured at previous time-slots can beused to improve the accuracy of the CSI estimate. The channel estimate measured frompilot symbols can be de�ned as:

P (t) = H(t) + N est(t). (V.26)De�ning δt as the time between two received frames, using an FIR Wiener �lter (theoptimum linear FIR �lter) leads to:H i,j(lδt) = w

Hi,j

opt × ~P i,j(lδt), (V.27)122

where l + 1 is the length of the �lter, and~P i,j(lδt) =

P i,j(0)

P i,j(δt)...P i,j(lδt)

wHi,j

opt = (R−1p cp)

∗

Rp = E[~P i,j(lδt)~P∗i,j(lδt)]

cp = E[H i,j(lδt)~P∗i,j(lδt)].

(V.28)From this de�nition, the optimum �lter is di�erent for all MR × MT paths of the MIMOchannel. However, if all paths have the same statistics, the �lters are identical, i.e.w

Hi,j

opt = wHopt. It is assumed the channels time variation follows Jake's model, i.e.

∀i, j E[H i,j(0)H∗i,j(δt)] = J0(2πFdδt). Then obviously (Rp)i,j = J0(2πFd(i − j)δt) +

E[nn∗]δ(i, j) and (cp)i = J0(2πFd(l − i)δt), where δ(i, j) is the Kronecker function.V.5.1.2 Forward-backward estimationThe previous channel estimation method only accounts for pilot symbols in the past.Channel estimation using both past and future pilot symbols is likely to provide a betterchannel estimation. In such a case, the system needs to bu�er the received frame andprocess it later (when the pilot symbols required for CSI estimation have been received).Storing l/2 frames is only possible when a large memory bu�er is available at the receiver.Furthermore, delay sensitive applications might forbid such a solution.However, in most fading scenarios (such as the Jake's fading assumed here), most ofthe improvement in the channel estimation is due to the pilots immediately preceding andfollowing the desired frame. This is because the channel time correlation tends to decreasewith increasing delay. Therefore, waiting for the next pilot before processing the framemight not only provide a signi�cant improvement of the channel estimation but also keepthe hardware requirements reasonable and satisfy the delay constraints of the system.123

V.5.2 Tracking of the precoding and decoding matricesInstead of tracking the channel, it is possible to track the precoding and decoding matrices.With (U ,Σ,V ) = SV D(H), de�ne the functions fu(.) and f v(.) asfu

i,j(H) = U i,j,

f vi,j(H) = V i,j.

(V.29)For each received frame (at transceiver A or transceiver B), pilot symbols are measured,and the SVD of the resulting channel matrix is calculated.The decoding matrices fu(P (t)) can be �ltered in time to improve their accuracy.De�ning~fui,j(P (lδt)) =

fui,j(P (0))...

fui,j(P (lδt))

, (V.30)the elements of this vector are combined in the following way:U i,j(lδt) = w

Ui,j

opt × ~fui,j(P (lδt)) (V.31)where w

Ui,j

opt is the FIR Wiener (optimum) �lter.w

Ui,j

opt = (R−1fu

i,j(P )cui,j)∗

Rfui,j(P ) = E[fu

i,j(P (lδt))fui,j(P (lδt))∗]

cui,j= E[U i,j(lδt)~fu

i,j(P (lδt))∗].

(V.32)Similarly, the precoding matrices f v(P (t)) can be �ltered in time. In the following,there is no attempt to correct for subspace swapping before the interpolation. Additionalperformance gain may be obtained by correcting for subspace swapping.V.5.2.1 Correlation of the decoding matrixDerivation of the Wiener �lter presented in the previous section implies explicit knowledgeof the correlation in time of the elements of the decoding matrix. This correlation, unlikethe correlation of the elements of the channel matrix, is not well known.Consider H(1) and H(2) = rH(1) +√

1 − r2N tvc, where H(1) and N tvc are i.i.d.complex Gaussian matrices with zero mean and unit variance. The mathematical deriva-tion of the correlation between U (1) = fu(H(1)) and U(2) = fu(H(2)), is still an openproblem, to the author's knowledge. The expectancy of the correlation between the el-ements of U(1) and the elements of U (2) is presented in Fig. V.17. As expected, the124

elements of a column of U have the same expected correlation when perturbed, since thechannel is i.i.d. and a permutation of the label of the antennas at the receiver permutesthe lines of U accordingly. All correlations have a maximum value of 0.25 since the ele-ments of U are equally likely and the matrix is unitary. The correlation of the channel isalways higher in absolute value than the correlation of the singular vectors: as mentionedin [58], a small perturbation of the channel may result in a large change in the singularvectors. The singular vectors corresponding to the largest singular value shows the highestcorrelation.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

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0.2

0.25

r

E(U

1 U2* )

E((U1):,1

(U2):,1* )

E((U1):,2

(U2):,2* )

E((U1):,3

(U2):,3* )

E((U1):,4

(U2):,4* )

E(hi,j

h*i,j

)

Figure V.17: Correlation of the decoding matrix:E(fu(H(1))fu(rH(1) +

√1 − r2N tvc)

∗)

V.5.2.2 Correlation of the precoding matrixFig.V.18 shows the correlation between elements of a column of V under similar as-sumptions as in the previous section. The lower curves represent the correlation betweenelements of the precoding matrix not on the main diagonal5. They have the same corre-lation when perturbed, since the channel is i.i.d. and a permutation of the labels between5The SVD in this sense is a modi�ed version to standard literature where columns of the V matrixare rotated such that the main diagonal are positive real values. Columns of U must be rotated by thesame amount. This does not a�ect generalisation. 125

antennas at the transmitter permutes the columns of V accordingly. The upper curves ofFig.V.18 are the correlation between elements along the main diagonal. They have di�er-ent correlations since each correlation corresponds to a di�erent singular value (sorted indescending order). All correlations have a maximum value of 0.25 since the elements of Vare equally likely and the matrix is unitary.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

E[V

ijVij* ]

r

E[V111

V211* ]

E[V122

V222* ]

E[V133

V233* ]

E[V144

V244* ]

E[V1:1

V2:1* ]

E[V1:2

V2:2* ]

E[V1:3

V2:3* ]

E[V1:4

V2:4* ]

Figure V.18: Correlation of the Precoding Matrix:E[f v(H(1))f v(rH(1) +

√1 − r2N tvc)

∗]V.5.2.3 General SVD correlation resultsTwo further theoretical results are required to obtain the coe�cients of the �lter of thedecoding matrix.Consider H(1), H(2) two matrices. ∀r ≥ 0, fu(rH(2)) = fu(H(2)). ThereforeE((fu

i,j(H(1)))(fui,j(H(2)))∗) = E((fu

i,j(H(1)))(fui,j(rH(2)))∗). (V.33)The correlation between the decoding matrices of two channel matrices are not a�ectedwhen the power of the channel matrices is modi�ed.Consider N tvc a perturbation matrix, H(1) and H(2) are i.i.d. complex Gaussianmatrices with zero mean and same variance. Then

E(fui,j(H(1) + N tvc)f

ui,j(H(2))∗) = E(fu

i,j(H(1))fui,j(H(2) + N tvc)

∗). (V.34)126

This is simply because ∀c1, c2 ∈ C, E(c1c∗2) = E(c2c

∗1)

∗ and E(fui,j(H(1))fu

i,j(H(2))∗) isreal. Exchanging the labels c1 and c2 completes the proof.The two previous results can be obtained in a similar manner for the precoding matrix.V.5.3 Correlation of pilotsFrom the general results presented in Sections V.5.2.1 and V.5.2.3, the correlation betweenpilot symbols at time 0 and at time δt can be derived for all δt.Then,E(fu

i,j(P (0))fui,j(P (δt))∗)

= E(fui,j(H(0) + N(0)√SNRest )fu

i,j(H(δt) + N(δt)√SNRest )∗)= E(fu

i,j(H(0))fui,j(rH(0) +

√1 − r2N eq),

(V.35)where N (0), N (t) and N eq are complex i.i.d. Gaussian random matrices with zero meanand unit variance,r =

J0(2πFdδt)

1 + 1SNRest . (V.36)Combining (V.35) and the results in Fig. V.17, it is straightforward to obtain thecorrelation between the elements of the decoding matrices derived from pilot symbols atdi�erent time slots, for varying SNRs. Finally, the taps of the FIR Wiener �lter for eachelement of the precoding matrix are deduced from the correlation, as in Section V.5.1.Similar results can be obtained for the precoding matrix. The prediction on the precod-ing matrix results in a matrix that, due to some predictive error, is not necessarily unitary.The precoding matrix is required to be unitary in a double pilot architecture (Section V.4).The predicted matrix, V predV (δt), can be projected onto an orthonormal basis. Based onthe correlation curves of Fig. V.18, the �rst column VpredV:,1 (δt) is normalized and thenmade orthogonal to V

predV:,4 (δt), the column with the next strongest correlation. The resul-tant matrix V ON(δt) is the orthonormal basis matrix of the predicted precoding matrix,

V predV (δt).V.5.4 Simulation results, decoding matrixTo validate the proposed �ltering methods, a MT = MR = 4 MIMO system was simulated(16 i.i.d. SISO channels, Jake's fading). The time varying MIMO channels were simulatedusing independent variables of SNR and Fdδt. The in�uence of the �lter length on thesystem capacity was also investigated. 127

Fig. V.19 presents the block diagrams of the various simulated systems. The precodingmatrix is assumed perfect.Pilot System

Filtered pilots


U(t)*

Channel

Time

δPilots

δH(t− t)Pilots

Pilots

Pilots δ

Pilots δ

H(t−k t)

H(t)

H(t+ t)

H(t+k t)

......

*U(t)

Predicted pilots


...H(t)

H(t−k t)

Pilots

Pilots H(t− t)δ

Pilots δChannel

Time

Simple Pilot System


U(t)*

Channel

Time

Pilots H(t)

*U(t)

Predicted pilots + 1 step forward


...

H(t+ t)

H(t)

H(t−k t)

δPilots

Pilots

Pilots H(t− t)δ

Pilots δ

Time

Channel

Predicted + 1 (H) Filtered Pilots (H)

Predicted Pilots (H)

Figure V.19: The CSI at the receiver can be improved using previous and/or future pilotsymbols to improve the accuracy of the channel estimation.V.5.4.1 System capacity for varying FdδtThe system capacity (de�ned in Section IV.3.2) of the systems against Fdδt with anSNR=20dB, is shown in Fig. (V.20).The capacity of the MIMO channel (curve 'MIMO Capacity') as well as the capacityachieved by a using only the pilots of the current frame (curve 'Pilot System') are givenas a reference. The capacity of the MIMO channel is not a�ected by Fdδt. Neither is thesystem capacity of the 'Pilot System' under the block fading assumption.Obviously the performance of these two systems is not a�ected by Fdδt.At all Fdδt, �ltering the CSI (curve 'Filtered Pilots (H)') or �ltering the decodingmatrix (curve 'Filtered Pilots (U)') leads to a signi�cant improvement of the capacity ofthe system.Filtering the H matrix rather than the U matrix with the same �ltering technique128

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.115

16

17

18

19

20

21

22

23

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

FdT

MIMO Capacity Pilot system Predicted pilots (H)Predicted + 1 (H) Filtered pilots(H) Predicted pilots (U)Predicted + 1 (U) Filtered pilots(U)

Figure V.20: Performance of MIMO-PSAM systems, SNR=20dB, Equal Power, �lterlength of 31 taps(e.g. curve 'Filtered Pilots (H)' rather than curve 'Filtered Pilots (U)') leads to a betterestimation of the decoding matrix for all Fdδt. Filtering the decoding matrix directly ise�ective at low Fdδt but does not perform well at high Fdδt. The correlation betweenconsecutive decoding matrices decreases rapidly with Fdδt.For both methods (�ltering the CSI or �ltering the decoding matrix), a balanced �lter(curves 'Filtered Pilots'), which �lters past and future channel estimates, provides the bestperformance. However, a balanced �lter requires a large memory bu�er and introduceslarge delay in the reception chain, as discussed in Section V.5.1. An unbalanced �lterbu�ering for only one time-slot (curves 'Predicted + 1'), can achieve a similar perfor-mance with less stringent hardware requirement at high Fdδt. This can be explained byobservation of the correlation curves of Fig. V.17. At high Fdδt little correlation existsbeyond the �rst time-slot and hence minimal performance improvement is obtained bywaiting for further pilots. However, using only the information about past pilots (curves'Predicted Pilots') leads to a severe degradation in performance at all FdδtV.5.4.2 System capacity for varying SNRThe capacity of the systems against SNR with a Fdδt = 0.04 is shown in Fig. V.21.129

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

SNR


Figure V.21: Performance of MIMO-PSAM systems, FdT = 0.04, equal power allocation,�lter length of 31 tapsFiltering on H provides better performance than �ltering on U , at all SNR.At low SNR, the noise dominates the time variation of the channel and therefore the�lters mainly average out the noise. Therefore the performance of the �lters do not dependon their position.On the contrary, at high SNR, the �lters are disregarding the noise and focus onpredicting the variations of the channel. In such a case, a balanced �lter performs betterthan a �lter with a delay of one or a predicting �lter.V.5.4.3 System capacity for varying �lter lengthThe capacity of the systems against the length of the �lter with SNR=20dB and Fdδt =

0.04, is shown in Fig. V.22.For a �lter of length 3, the balanced (curves 'Filtered pilots') and unbalanced (curves'Predicted + 1') �lters are the same, which explains why they provide exactly the samesystem capacity.Filters of reasonable length (L = 3) achieve a large part of the system capacity gainachievable through �ltering. This is mainly due to the fact that the time-slots mostcorrelated with the time-slot of interest are its immediate neighbours. This is especially130

0 10 20 30 40 50 6015

16

17

18

19

20

21

22

23

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

Length of the Wiener Filter


Figure V.22: Performance of MIMO-PSAM systems, SNR=20dB, Fdδt = 0.04, equalpower allocationcrucial when �ltering is performed on the decoding matrix. The correlation of the decodingmatrix is so weak that the decoding matrices more than two frames away provide littleinformation about the desired frame. Using them provides negligible system capacityimprovement.V.5.4.4 Orthonormalization of the �ltered decoding matrixWhen �ltering the decoding matrix directly, the �ltering process alters the unicity of thedecoding matrix. It is possible to follow the �ltering with a Gramm-Schmidt orthonormal-ization, to retrieve the interesting unicity properties of the decoding matrix. Simulationresults are presented in Fig. (V.23).At all FdT , for all three �lters, the orthonormalization provides a slight improvementof the performance. Orthonormalizing actually corresponds to further �ltering, since itis simply a projection of the estimation of the decoding matrix on the unitary matricesensemble. 131

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.115

16

17

18

19

20

21

22

23

Ave

rage

Cap

acity

(bi

ts/s

/Hz)

FdT

MIMO Capacity Pilot system Predicted pilots (U) Predicted + 1 step filtered pilots (U)Filtered pilots (U) Orthonormalized Predicted Pilots (U) Orthonormalized Predicted + 1 (U) Orthonormalized Filtered pilots (U)

Figure V.23: Performance of PSAM systems, SNR=20dB, Equal PowerV.5.5 Simulation results, precoding matrixThe system capacity of SVD system with �ltering of the CSI and �ltering of the precodingmatrix at the transmitter is presented in this section through simulation results, withidentical parameters as in Section V.5.4. A perfect decoding matrix (U) is assumed. Theprecoding matrix can only be estimated using past pilot symbols.V.5.5.1 System capacity vs. FdδtThe system capacity against Fdδt is determined through simulations (Fig.V.24). As ex-pected, the system capacity performance reduces for faster moving channels for all cases.The curve labelled 'Prev V' corresponds to a system where no prediction is performedand the precoding matrix V p is used. Signi�cant performance improvement results whenl previous CSI estimates are linearly combined to estimate the CSI on the downlink asH (Section V.5.1). The corresponding system capacity curve is labelled 'Pred H'. Theperformance improvement (when compared with a system using no prediction) o�eredthrough prediction of the precoding matrix, is about half that of the gain o�ered fromprediction of the CSI. A further small performance increase can be achieved by projectingthe predicted precoding matrix onto an orthonormal basis, V ON , curve 'Pred V+ON'.132

For a static channel channel, Fdδt = 0, all prediction �lters become averaging �lters,i.e. all taps are equal. It is seen that noise dominates when no prediction is performed.System performance is almost ideal when averaging is performed on the CSI dependingon the length of the �lter. Filtering the precoding matrix has a similar system capacityas �ltering the CSI on very slow fading channels.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

5

10

15

20

25

Cap

acity

(b/

s/H

z)

fdT

MIMOPred on HPred on VPred V+ONPrev V

Figure V.24: System capacity vs. Fdδt (SNR=20dB, L = 30 taps)V.5.5.2 System capacity vs. SNRHigher SNR means that the pilot measured CSI, P is more accurate. Therefore lessinterference occurs as a result of incorrectly matched precoding and decoding matrices.Higher system capacity is obtained for all systems as shown in Fig. V.25. It can beseen that when no �ltering is applied (curve 'Prev V'), an improvement in SNR increasesthe system capacity but an error �oor exists due to the channel fading. For a systemcapacity of 10 bps/Hz, a gain of slightly more than 7dB can be achieved by �ltering theCSI (curve 'Pred H'). When prediction is performed on the precoding matrix (curve 'PredV'), a gain of about 5dB can be achieved. An additional gain of almost 1dB can beachieved by projecting the predicted precoding matrix onto an orthonormal basis (curve'Pred V+ON'). 133

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Cap

acity

(b/

s/H

z)

SNR

MIMOPred on HPred on VPred V+ONPrev V

Figure V.25: System capacity vs. SNR (Fdδt = 0.040, L = 30 taps)V.5.6 ConclusionMost practical communication systems deduce the CSI through measurement of pilotsymbols multiplexed with the data. In a MIMO-SVD system, the channel needs to beestimated accurately since the SVD is a non-linear function. Therefore, estimation of thechannel on a frame by frame basis might not provide su�cient accuracy in the estimationof the channel. The channel correlation from frame to frame can be exploited to improvethe channel estimation without additional pilot overhead. This is achieved through FIRlinear Wiener �ltering. Filtering can be applied on the CSI or directly on the precodingand decoding matrices.Properties of the correlation of the precoding and decoding matrices when the channelis perturbed were presented, as well as a generic method to derive the optimal �ltercorresponding to the precoding and decoding matrices.Estimation and �ltering of the precoding and decoding matrices were shown to bepossible through simulation results. However, the best performance was achieved whenestimation and �ltering was performed on the CSI. At the receiver side, when delays aretolerable and hardware complexity is not an issue, the best overall performance is obtainedwith a balanced �lter. For most practical scenarios it was shown that only a small number134

of taps is necessary to achieve most of the performance gain. Most importantly, it is notnecessary to wait for more than one additional pilot symbol before estimating the channel.Filtering the precoding and decoding matrices should be limited to cases where theFdδt is small. In such a case, orthonormalization of the estimated precoding and decodingmatrices provides additional �ltering and leads to performance improvement.V.6 ConclusionThe SVD transmission architecture requires knowledge of the CSI at both the transmitterand receiver. It is possible to provide the transmitter with the CSI without additionalsignaling overhead when the channel is reciprocal. However, the CSI is usually obtainedthrough pilot symbols. Therefore the CSI is imperfect at both the transmitter and thereceiver.The errors due to imperfect CSI can create a large loss of system capacity. Speci�cally,when catastrophic events named 'singular value crossings' occur, a small perturbation inthe channel(however small) can create a large perturbation of the precoding and decodingmatrices. In such a case, the transmission is no longer robust to noisy channel estimates.The analysis of 'singular value crossings' can be conducted through the theory of matrixperturbation. However the time correlation of the channel prevents the direct applicationof the theory to fading channels, to the knowledge of the author. Simulation resultsindicate that 'singular value crossings' create 'singular subspace swappings' in Rayleighfading: two singular values vary in amplitude to the point were they are crossing eachother, but their corresponding subspaces are stable through the process. Results showthat the probability of 'singular value crossings' is small and the e�ects of 'singular valuecrossings' can be corrected.Incorrect CSI reduces the system capacity of SVD systems. When the precoding matrixis in error, the capacity of the system can drop below the system capacity of uncodedsystems. In a TDD environment where the channel is a�ected by Doppler spread, theSVD system was shown to be unsuitable for FdT s greater than 0.03 (a system capacityloss of approximately 5 bits at SNR= 20dB). The loss gets even larger at higher SNRsbecause the performance plateaus rather than linearly increases. There is no bene�tin implementing an SVD algorithm alone if the precoding matrix is outdated. A newarchitecture is proposed which allows the system to bene�t from the high system capacity135

of SVD systems when the channel estimation is correct and seamlessly transmit withthe performance of an uncoded system when the precoding matrix is in error. The newarchitecture considers the precoding matrix as part of the channel for channel estimationpurposes. Therefore, errors in the precoding matrix can be corrected in the decodingmatrix using a simple zero-forcing or MMSE process. This new architecture does notrequire additional pilots and improves the useable FdT from 0.03 to in�nity: even at veryhigh FdT the performance of the system is better or equal to the performance of systemswithout linear precoding.Finally, the estimation of the channel on a frame by frame basis might not providesu�cient accuracy in the estimation of the channel. The channel correlation from frameto frame can be exploited to improve the channel estimation without additional pilotoverhead. This is achieved through FIR linear Wiener �ltering. Filtering can be appliedon the CSI or directly on the decoding matrix. Properties of the correlation of the decodingmatrix when the channel is perturbed were demonstrated, as well as a generic method toderive the optimal �lter corresponding to the decoding matrix. Estimation and �lteringof the decoding matrix was shown to be possible through simulation results. The bestperformance was achieved when estimation and �ltering was performed on the CSI. Whendelays are tolerable and hardware complexity is not an issue, the best overall performanceis obtained with a balanced �lter. However for most practical scenarios it was shown thatonly a small number of taps (i.e. 3) is necessary to achieve most of the performance gain.Particularly, it is not necessary to wait for more than one additional pilot symbol beforeestimating the channel.

136

Chapter VIConclusionVI.1 SummaryWireless LANs are both user-friendly and cost-e�ective. These advantages explain thecurrent interest in Wireless LAN telecommunications equipment. Transmission techniquesbased on transceivers with multiple element antennas enable higher spectrum e�ciencythan is currently obtained using existing wireless LAN products. Therefore, it is of interestto apply MIMO transmission techniques to the wireless LAN environment.MIMO telecommunications theory indicates that the capacity of a wireless link in-creases linearly with the number of antennas at both ends of the channel for constantbandwidth and transmitted power (Chapter II) when the channel is uncorrelated. Fur-thermore, a series expansion of the capacity of the ergodic channel with no CSI at thetransmitter indicates that symmetric antenna allocation (same number of antennas at thetransmitter and the receiver) maximizes the capacity of a MIMO channel with a giventotal number of antennas.The capacity of correlated MIMO channels is lower than that of uncorrelated channels.Speci�cally, the capacity of Ricean channels tends to the capacity of their i.i.d. componentwhen the number of antennas grows large (Chapter III). Theoretical bounds of the nor-malized capacity of the Ricean channel are proven in Chapter III. These bounds allow thecapacity of the Ricean channel to be estimated in the asymptotic limit of a large numberof antennas without recourse to simulation.Practical transmission techniques are required to turn the promises of high capacityinto high performance transmission devices. Chapter III presents an overview of some137

well-known techniques to transmit over the MIMO channel and detect the transmittedsymbols. Robust systems use diversity to obtain reliable transmission. However, theyrequire large constellations to achieve a non-negligible fraction of the capacity of the chan-nel. Systems with a large multiplexing gain are bandwidth e�cient but only transmit wellon uncorrelated channels.Therefore, transmission systems with no CSI at the transmitter are designed eitherfor the correlated channel or for the uncorrelated channel. Wireless LANs experience alarge variety of propagation environments and so su�er from performance loss when theCSI is not available at the transmitter. CSI at the transmitter increases the capacityof the MIMO channel and is especially bene�cial when the channel is highly correlatedas demonstrated in Section III.5. The capacity gain with CSI at the transmitter, growslinearly with the number of antennas.Information theory suggests a transmission architecture based on the SVD of the chan-nel matrix to bene�t from the CSI at the transmitter. The SVD architecture decomposesthe MIMO channel into SISO transmission eigenmodes and allocates power to the eigen-modes following a water�lling algorithm. This architecture is the optimal linear precoderand decoder under a variety of criteria, as detailed in Section IV.2.1. Furthermore, theSVD structure combined with OFDM is also an optimal space-time modulation in termsof information rate (Section IV.2.2).The complexity of the SVD structure can be reduced by considering each transmissioneigenmode as a separate channel, which leads to the notion of system capacity (SectionIV.3). However, this reduction of complexity is obtained through a reduction of therobustness of the system to various impairments. E.g. the channel estimation noise shouldbe smaller than the noise in the transmission to avoid a loss of performance.As explained earlier, SVD systems bene�t from the availability of CSI at the transmit-ter. The estimation of an accurate CSI at the transmitter is the key challenge faced bySVD system designers. TDD channels enable the estimation of the CSI at the transmitterwithout overhead data transmission from the receiver to the transmitter. This is due tothe fact that TDD channels are theoretically reciprocal. In practical TDD channels, aloss of accuracy in the estimation of the CSI at the transmitter occurs, due to severalimperfections of the channel and the transmission system:• The SVD of the channel matrix is not unique. Matched transmitting and decod-138

ing matrices are required. The transmitter and the receiver can implicitly choosematched transmitting and decoding matrices as presented in Section IV.4.2.• Mismatched transmitter and receiver RF chains cancel the reciprocity of the chan-nel. A calibration procedure is presented in Section V.2 to cancel the e�ects ofmismatched transmitter and receiver RF chains. The calibration procedure relies ona handshake at the beginning of the transmission, as well as the hypothesis that theimpairments of the chains are stationary.• The performance of practical channel estimation algorithms is limited and the CSIis quantised in practical systems. The theory of matrix perturbation highlightsthe e�ect of imperfect channel estimation on SVD systems (Section V.3). A smallperturbation of the CSI can result in a large performance loss when a 'singular valuecrossing' occurs, i.e. the channel matrix has two equal singular values. Simulationresults indicate that 'singular value crossings' create 'singular subspace swappings'in Rayleigh fading: two singular values vary in amplitude to the point were theyare crossing each other, but their corresponding subspaces are stable through theprocess. Results show that the probability of 'singular value crossings' is small andthe e�ects of 'singular value crossings' can be corrected.• Practical channels are time-varying. In Section V.4, the SVD system is shown tobe unsuitable for FdT s greater than 0.03 (a capacity loss of approximately 5 bitsat SNR= 20dB). The loss gets even larger at higher SNRs because the performanceplateaus rather than linearly increases. There is no bene�t in implementing an SVDalgorithm alone if the precoding matrix is outdated.It is possible to mitigate the e�ects of both imperfect channel estimation and channelfading: FIR linear Wiener �lters can exploit the channel correlation from frame to frame toimprove the channel estimation without additional pilot overhead, as presented in SectionV.5. The correlation of the CSI degrades rapidly in time for a Rayleigh fading channel.Therefore, �lters of reasonable length (3 taps) achieve near optimum performance. At thereceiver, the estimated CSI should include knowledge of the future pilot symbols whenallowed by the hardware, i.e. when large memory bu�ers are available and the applicationis not time-delay sensitive. The CSI is usually more accurate at the receiver than atthe transmitter since the CSI of the current time slot (and possibly future time slots) is139

available. This is not the case for the transmitter, when the CSI for the current time slotis not available and must be predicted from past estimates.It is possible to mitigate the e�ects of imperfect CSI at the transmitter if the systemincludes the precoding matrix in the channel estimation. This allows the receiver tomitigate the e�ect of the incorrect precoding matrix. This new architecture, with limitedadded complexity, obtains the bene�t of the SVD architecture when CSI is precisely knownat both ends of the link while seamlessly shifting to a non-precoded system (such asZF or MMSE linear decoder) when the channel estimation precision deteriorates at thetransmitter. The SVD architecture adapts dynamically to the propagation environmentto obtain very high spectrum e�ciency under varying channel conditions.VI.2 Future workThe results presented in the previous Chapters create several research opportunities in thefollowing key areas: pilot symbol theory, error correction codes, wireless channel modelsand complexity issues.VI.2.1 Pilot symbolsSection V.5 introduces a proposal to improve the performance of pilot symbol assistedsystems. The proposed systems combines the noisy channel estimate at several pointsin time to obtain an accurate estimate of the current channel. Practically, the solutionis implemented through �ltering of the noisy channel estimate. It is implicitly assumedthat the time samples of the channel (the transmission slots) are evenly spaced in time.This assumption is realistic for Time Division Multiple Access (TDMA) systems such asHiperlan 2. However, wireless LAN standards such as 802.11 (a, b and g) are CarrierSense Multiple Access - Collision Detection (CSMA-CD) systems where the assumption isno longer realistic. The current proposal requires to derive the coe�cients of the �lter foreach set of time-intervals between the transmission time slots. Therefore, it is unrealisticto expect the current proposal to be implemented as is. Further research is required toadapt the proposal to CSMA-CD systems.The proposed system takes advantage of the time-correlation properties of the time-varying wireless channel. Further performance gains are expected through adequate con-140

sideration of the frequency and space correlation properties of the channel. Further re-search is required to determine the most e�ective pilot allocation in time/space/frequency.Finally, a novel algorithm is required to combine the estimates derived from each pilottransmission (i.e. at a precise time slot, from a single transmitting antenna, on a givensubcarrier) into a global estimate (i.e. the channel estimate at every time slot, from eachtransmitting antenna to each receiving antenna, on every subcarrier).VI.2.2 CodingThe notion of 'system capacity' (the sum of the capacity of each transmission eigenmodefor equal power systems) has been introduced to free the analysis from assumptions oncoding. However, the coding strategy is an important part of any practical transmissionsystem. To achieve the promises o�ered by system capacity results, further research needsto be conducted in the following areas:• Power and bit allocation. Several algorithms are required to allocate power to eachtransmission eigenmode on each subcarrier. Rate adaptation (bit allocation to eachtransmission eigenmode on each subcarrier) is likely to improve the performance ofthe system.• Level crossing rates. In practical systems, the capacity of each transmission eigen-mode on each subchannel is time-varying. Practical systems require an estimate ofthe speed of capacity variation to determine the required update frequency of thebit and power allocation. Further research is required to determine the level crossingrate of the eigenmode capacities.• Practical coding schemes. It is necessary to choose or develop adequate codingschemes for SVD-MIMO-OFDM modems over the time-varying channel.VI.2.3 System level designThe results introduced in this thesis focus on the optimization of a single wireless link.Further research is required to determine:• the possibilities o�ered by Spatial Division Multiple Access. Multiple point to pointlinks can possibly transmit simultaneously at the same frequency when each point141

to point link has a transmission eigenmode with limited interference from the otherpoint to point links.• the impact of multiple antenna modems on multiple access techniques. CurrentOFDM systems allow multiple access to the transmission medium through TDMA(Hiperlan 2) or CSMA-CD (802.11 a and g). Code Division Multiple Access (CDMA)is also a multiple access technique candidate for OFDM systems. Multiple antennamodems are likely to modify the respective performance of these multiple access tech-niques. E.g., multiple antenna modems require a large overhead when pilot symbolassisted channel estimation is implemented. Therefore, multiple access techniquesallowing longer frames are likely to perform better with multiple antenna modems.VI.2.4 Study of the wireless channelThe results presented in this thesis were derived under precise assumptions on the wirelesschannel. Further research is required to improve the MIMO wireless channel models. Thetime, frequency and space correlation of the channel is of particular interest. These prop-erties of the channel can be obtained through extensive channel measurement campaignsof typical propagation environments.VI.2.5 ImplementationThe advances in the communications theory o�er a wide range of possibilities to thetransmission system designer. As presented throughout this thesis, the introduction ofmodems with multiple antennas allows an increase in the transmission data rate at theexpense of complexity (further processing). However, wireless transmission devices shouldremain small in size and power e�cient. Further research is required to:• estimate and reduce the complexity of MIMO transmission algorithms,• identify and solve the implementation issues linked with multiple antenna wirelessmodems.

142

Appendix ARicean channel capacity boundsI.1 Study of the Eigenvalues of FI.1.1 Ricean channelThe Ricean channel is de�ned as

H = aHsp + bHsc (A.1)where a =√

10K/10

1+10K/10 and b =√

11+10K/10 .

F is de�ned byP

MT

HH∗ = b2Hsc(Hsc)∗ +P

MT

F . (A.2)Therefore,F = ab(Hsc(Hsp)∗ + Hsp(Hsc)∗) + a2Hsp(Hsp)∗. (A.3)

F consists of two parts: a cross product term due to the specular and scattering channelgains and a specular term. The former is itself a sum of two terms. The eigenvalues ofF provide useful insights to understanding the MIMO Ricean capacity. It is assumedthroughout this analysis that K 6= −∞.I.1.2 Singular Value DecompositionsThe matrices Hsp(Hsp)∗ and Hsc(Hsp)∗ can be written,

Hsp(Hsp)∗ = MT × (1)MR,MR. (A.4)and 143

Hsc(Hsp)∗ = (

MT∑

k=1

hi,k)i=1..MR,j=1..MR. (A.5)Both are rank one matrices and have the following singular value decompositions,

Hsp(Hsp)∗ = (~v1)MRMT (~v1∗) (A.6)

Hsc(Hsp)∗ = ( ~u1)σ(~v1∗). (A.7)The singular values are MRMT and σ and the singular vectors are ~v1 and ~u1. These arede�ned below,

~v1 =1√MR

(1)MR,1. (A.8)The singular vector ~u1 is given by ~u1 = ~x1/‖x1‖, where~x1(

MT∑

k=1

h1,k,

MT∑

k=1

h2,k, . . . ,

MT∑

k=1

hMR,k)†, (A.9)

σ =

√√√√MR ×

MR∑

i=1

‖MT∑

k=1

hi,k‖2, (A.10)and (.)† denotes the transpose.I.1.3 Eigenvalues of FUsing the singular value decompositions above, F in (III.19) can be written as F =

a2MRMT (~v1)(~v1∗) + abσ((~v1)( ~u1

∗) + ( ~u1)(~v1∗)). Hence rank(F ) ≤ 2 since F is the sum oftwo rank one matrices, a2MRMT (~v1)(~v1

∗) + abσ(~v1)( ~u1∗) and abσ( ~u1)(~v1

∗). By construc-tion, it follows that any eigenvector, ~k, of F , associated with the non-zero eigenvalue κsatis�es the following,

∃β1, β2 such that ~k = β1 ~v1 + β2 ~u1

F~k = κ~k,(A.11)Substituting for F and ~k and equating coe�cients in (A.11) gives:

β1abσ( ~u1∗ ~v1) + β2abσ + β1a

2MRMT + β2a2MRMT (~v1

∗ ~u1) = κβ1

β1abσ + β2abσ(~v1∗ ~u1) = κβ2

(A.12)144

De�ning o = ~v1∗ ~u1, and solving (A.12) for κ gives,

κ = a2MRMT +abσ(o+o∗)2

± ...√(a2rt+abσ(o+o∗))2+4(abσ)2(1−oo∗)

2.

(A.13)which de�nes the two possibly non-zero eigenvalues of F .I.1.4 Asymptotic eigenvalues of FEquation (A.10) indicates that σ ≥ 0 and E(σ2) = r2 × t. Furthermore, ‖o‖ ≤ 1, so forMR,MT → ∞, (a2MRMT +abσ(o+o∗))2 � 4(abσ)2(1−oo∗) and a2MRMT � ‖abσ(o+o∗)‖with probability 1. Hence, one solution of (A.13) is positive and the other negative. Since,all other eigenvalues are zero we have the ordered eigenvalues denoted by λMR

(F ) < 0 =

λMR−1(F ) = . . . = λ2(F ) < λ1(F ). Taking the positive square root in (A.13) givesλ1(F ) ∼ a2MRMT + abσ(o + o∗) and in the limit

λ1(F )/(MRMT ) → a2. (A.14)From equation (A.13), F is a matrix of maximum rank two, with one negative and onepositive eigenvalue. The positive eigenvalue, denoted λ1(F ), behaves as belowλ1(F )/(MT MR) → a2. (A.15)Hence the positive eigenvalue of F grows quadratically with the number of antennas( when MR = MT ). Despite this, F is expected to have a negligible e�ect in (III.18) forlarge numbers of antennas, since F only has two eigenvalues whereas the scattering termhas min(MRMT ) with a probability of one. The two eigenvalues of F are shown in Fig.A.1 and Fig. A.2.Now the following further observations can be made:

• The positive eigenvalue is several orders of magnitude larger than the magnitude ofthe negative eigenvalue. Its growth with MT ,MR (MT = MR) is quadratic.• This disparity between the positive and negative eigenvalues increases even furtherwhen the K value is such that the channel is e�ectively a LOS channel.Results not reported here show that the eigenvalues of the sum of the cross productterms in (III.19)is a matched pair of positive and negative terms . The e�ect of the145

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

MT

Eig

enva

lue

K=1, Analytical K=1, Simulation K=5 K=7 K=12 Hsp(Hsp)*

Figure A.1: Positive eigenvalue of F versus antenna numbers for various K valuesterm Hsp(Hsp)∗ in (III.19) is to present a large increase to the positive eigenvalue that isobvious in A.1.Whilst the positive eigenvalue may seem quite large, its contribution to capacity isrelatively small due to the logarithmic operation.I.2 Capacity Lower boundWe now derive the capacity lower bound.Since HH∗ is a non-central complex Wishart matrix we can use Bartlett's decompo-sition [86] to giveHH∗ = b2L∗L (A.16)where L is upper triangular with diagonal elements denoted L1, L2, . . . Lr which are inde-pendent of all other elements. It is assumed that MR ≤ MT but the proof can easily beadapted to MR > MT . The distribution of L2

1 is non-central chi-squared, L21 ∼ χ2

2MT(δ)146

0 10 20 30 40 50 60 70 80 90 100−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

MT

Eig

enva

lue

K=1 K=5 K=7 K=12

Figure A.2: Negative eigenvalue of F versus antenna numbers for various K valueswith δ = (a2/b2)trace(Hsp(Hsp)∗). For j > 1 the distributions are central chi-squared,L2

j ∼ χ22MT−2j+2. Hence,

D = |IMR+

ρ

MT

HH∗| =

∣∣∣∣∣∣

[IMR

√

b2ρ

MT

L∗]

IMR√

b2ρMT

L

∣∣∣∣∣∣

(A.17)Using the Cauchy-Binet theorem givesD =

∑

γ

|Aγ||Aγ|∗ =∑

γ

|Aγ|2, (A.18)where Aγ is an MR × MR submatrix of [IMR

√b2ρMT

L∗] and γ is a subset of MR columnsfrom (1, 2, . . . , 2MR).Now the summation is split into two parts, over γ1 where the determinants |Aγ1| donot involve L1 and over γ2 where the determinants |Aγ2

| do involve L1. HenceD =

∑

γ1

|Aγ1|2 +

∑

γ2

|Aγ2|2. (A.19)147

The only choice of columns which gives determinants involving L1 are those where columnMR + 1 is selected and column 1 is omitted. Hence the Aγ2

matrices are of the formAγ2

=

0 . . . 0√

b2PMT

L1 0 . . . 0

Dγ21

0...0

Dγ22

. (A.20)Hence |Aγ2

|2 = b2PMT

L21|Dγ2

|2 where Dγ2= [Dγ21

Dγ22] and

D =∑

γ1|Aγ1

|2 + b2ρMT

L21

∑

γ2|Dγ2

|2

= X + L21Y.

(A.21)The exact same analysis holds for the Rayleigh case, except L21 ∼ χ2

2MT.To summarize,

DRicean = X + χ22MT

(δ)Y

DRayleigh = X + χ22MT

Y(A.22)where X,Y are positive random variables with X,Y independent of the χ2 variables.Hence

E[C(H)] = E[log2(X)] + E[log2(1 + χ22MT

(δ)Y/X)]

E[C(bHsc)] = E[log2(X)] + E[log2(1 + χ22MT

Y/X)].(A.23)Now χ2

2MT(δ) is stochastically greater than χ2

2MT. Hence E[f(χ2

2MT(δ))] ≥ E[f(χ2

2MT)] forany increasing function f and E[C(H)] ≥ E[C(bHsc)] as required. A lower bound of thecapacity can now be written ∀MR,MT , P,K,

E

[C(K,MT ,MR, P )

min(MT ,MR)

]

≥ E

[C(K = −∞,MT ,MR, b2P )

min(MT ,MR)

]

. (A.24)I.3 Capacity Upper boundThe capacity upper bound is derived in this Section. De�ningA = IMR

+b2P

MT

Hsc(Hsc)∗, (A.25)the normalized capacity becomesC

MT

=1

MT

log2(|A + F |) =1

MT

log2(

MR∏

i=1

λi(A + F )), (A.26)148

where F = PMT

F and λi(A + F ) are the eigenvalues of the hermitian positive de�nitematrix A + F , ordered so that 0 ≤ λMR(A + F ) ≤ . . . ≤ λ1(A + F ). Combining Weyl'stheorem [87] and results from Appendix I.1 leads to

λMR(A + F ) ≤ λMR−1(A) + λ2(F ) = λMR−1(A)

λMR−1(A + F ) ≤ λMR−2(A) + λ2(F ) = λMR−2(A)...λ2(A + F ) ≤ λ1(A) + λ2(F ) = λ1(A)

λ1(A + F ) ≤ λ1(A) + λ1(F )

(A.27)Therefore,

CMT

≤ 1MT

log2(λMR−1(A) . . . λ1(A)(λ1(A) + λ1(F ))

= 1MT

log2(∏MR

i=1 λi(A)) + 1MT

log2(λ1(A)+λ1(F )

λMR(A)

).(A.28)Now write

∆ =1

MT

log2

(

λ1(A) + λ1(F )

λMR(A)

)

, (A.29)and, since λj(A) ≥ 1 for any j,∆ ≤ 1

MT

log2(λ1(A) +P

MT

λ1(F )). (A.30)It is known that the eigenvalues of A are bounded as MR,MT → ∞ [72]. Therefore, ∃Msuch that λ1(A) ≤ M and when MT → ∞,∆ ≤ 1

MT

log2(M + PMR(λ1(F )/(MRMT ))) → 0, (A.31)since λ1(F )/(MRMT ) → a2. This concludes the demonstration. From [72], it is knownthatλ1(A) → 1 + b2P (1 +

√

min(MT ,MR)/ max(MT ,MR))2, (A.32)as MT ,MR → ∞ with MT /MR = α. This provides the smallest value for M that can beused and gives the bound that is used in the simulations.Therefore the capacity upper bound as MT ,MR → ∞ can be written,E

[C(K,MT ,MR, P )

min(MT ,MR)

]

≤ E

[C(K = −∞,MT ,MR, b2P )

min(MT ,MR)

]

+ ∆, (A.33)where ∆ → 0 as MT ,MR → ∞.Hence, for Ricean channels that are not pure LOS (K 6= +∞), the normalized ergodiccapacity tends to the normalized ergodic capacity of the scattering component. Hence,E

[C(K,MT ,MR, P )

min(MT ,MR)

]

→ E

[C(K = −∞,MT ,MR, b2P )

min(MT ,MR)

]

. (A.34)149

Appendix BIEE Electronics LettersG. Lebrun, T. Ying, M. Faulkner, "MIMO transmission over a time-varying channel usingSVD," IEE Electronics Letters, vol. 37, no. 32, pp. 1363-1364, Oct. 2001.

151

Appendix CIEEE Global TelecommunicationsConference 2002G. Lebrun, T. Ying and M. Faulkner, "MIMO transmission over a time-varying channelusing SVD," Proc. IEEE Global Telecommunications Conference, (Globecom '02), Nov.17-21, 2002 pp. 414-418.

153

Appendix DIEEE Vehicular Technology Conference2003T. Ying, G. Lebrun and M. Faulkner, "An adaptive channel SVD tracking strategy intime-varying TDD system," Proc. IEEE Vehicular Technology Conference 2003, (VTC2003-Spring), Apr. 22-25, 2003 Vol. 1, pp. 769-773.

155

Appendix EIEEE International Conference onCommunications 2004G. Lebrun, M. Faulkner, P. J. Smith and M. Shaa�, "MIMO Ricean channel capacity," inProc. IEEE International Conference on Communications (ICC 2004), 20-24 Jun. 2004,pp. 2939-2943.

157

Appendix FIEEE International Conference onCommunications 2004G. Lebrun, S. Spiteri and M. Faulkner, "Channel estimation for an SVD-MIMO System,"in Proc. IEEE International Conference on Communications (ICC 2004), 20-24 Jun.2004, pp. 3025-3029.

159

Appendix GIEEE Transactions on wirelesscommunicationsG. Lebrun, J. Gao and M. Faulkner, "MIMO transmission over a time-varying channelusing SVD," IEEE Transactions on wireless communications, vol. 4, no. 2, pp. 757-764,Mar. 2005.

161

Appendix HIEEE Transactions on wirelesscommunicationsG. Lebrun, M. Faulkner, P. J. Smith and M. Shaa�, "MIMO Ricean channel capacity: anasymptotic analysis," IEEE Transactions on wireless communications, Vol. 5, No. 5, May2006,

163

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