Linkoping Studies in Science and TechnologyThesis No. 1618

On the performance of MassiveMIMO systems with single carrier

transmission and phase noise

Antonios Pitarokoilis

Division of Communication SystemsDepartment of Electrical Engineering (ISY)

Linkoping University, SE-581 83 Linkoping, Swedenwww.commsys.isy.liu.se

Linkoping 2013

This is a Swedish Licentiate Thesis.

The Licentiate degree comprises 120 ECTS credits of postgraduate studies.

On the performance of Massive MIMO systems with singlecarrier transmission and phase noise© 2013 Antonios Pitarokoilis, unless otherwise noted.

LIU-TEK-LIC-2013:52ISBN 978-91-7519-513-1

ISSN 0280-7971Printed in Sweden by LiU-Tryck, Linkoping 2013

Abstract

In the last decade we have experienced a rapid increase in the demand forhigh data rates over cellular networks. This increase has been partly satis-fied by the introduction of multi-user multiple-input multiple-output (MU-MIMO). In such systems, the base station (BS) is equipped with multipleantennas and the users share the time-frequency resources. However, moderncommunication systems are highly power inefficient. Further, the increase indemand for higher data rates is expected to accelerate in the years to comedue to the popularity of mobile devices like smartphones and tablets. Hence,next generation cellular systems are required to exhibit high energy efficiencyas well as low power consumption. Recently, it has been shown that the de-ployment of a large excess of base station (BS) antennas in comparison tothe served users can be a promising candidate to meet these contradictoryrequirements. These systems are termed as Massive MIMO. When the num-ber of BS antennas grows large, the channels between different users becomeorthogonal and low complexity transceiver processing exhibits sum-rate per-formance that is close to optimal. In order to realize the promised gainsof Massive MIMO systems, it is required that power efficient and inexpen-sive components are used. In contemporary cellular systems, multi-carriertransmission is used since it facilitates simple equalization at the receiverside. However, multi-carrier signals exhibit high peak-to-average-power-ratio(PAPR) and require expensive highly linear power amplifiers. Power ampli-fiers in this regime are also very power inefficient. On the other hand singlecarrier signals exhibit lower PAPR and are suitable for signal design that ismore robust to non-linear power amplifiers. Further, single-carrier signalsare less vulnerable to hardware impairments, such as phase noise. In thisthesis we study the fundamental limits of Massive MIMO systems in termsof sum-rate performance with single-carrier transmission and phase noiseand provide important insight on the design of Massive MIMO under thesescenarios.

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Acknowledgments

I would like to thank my supervisor, Prof. Erik G. Larsson, for offering methe opportunity to pursue the PhD degree at Linkoping University. I amgrateful to Erik because he provided his guidance to challenging researchideas on a topic that was just starting to appear in the research community.He has constantly been showing his interest in the progress of my researchand provided me with insightful feedback that helped me to stay focusedon important research questions. However, he also encouraged me to thinkindependently and look for new research challenges.

I would also like to thank Dr. Saif Khan Mohammed, now in Indian In-stitute of Technology, Delhi, India, who has been my co-supervisor for thisinitial part of my PhD studies. We have cooperated closely while he washere and I had the chance to learn a lot from him especially in the areaof information theory. Further, his contribution to improving my writingstyle was particularly appreciated. I am also grateful to a former member ofthe group, Eleftherios (Lefteris) Karipidis. Lefteris has helped me, not onlythrough his technical expertise, but also he has offered his generous assis-tance to adjust to the new environment when I first arrived in Linkoping.Finally, I would like to thank all the PhD students, Chaitanya, Johannes,Reza, Mirsad, Hien, Victor, Marcus and Christopher and the seniors of thegroup, Peter, Danyo, Mikael, Daniel and Vladimir, for making the everydaylife enjoyable.

Finally, I am most grateful to my parents and my sister for their uncondi-tional support that has enabled me to pursue my goals. Without them Iwould have achieved much less in my life.

Linkoping, October 2013Antonios Pitarokoilis

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Contents

I Introduction 1

Introduction 31 The Intersymbol Interference Channel . . . . . . . . . . . . . 6

1.1 Detection in ISI channels . . . . . . . . . . . . . . . . 92 Multi-user MIMO Channels with Intersymbol Interference . . 10

2.1 MIMO Broadcast Channel . . . . . . . . . . . . . . . . 102.2 MIMO Multiple Access Channel . . . . . . . . . . . . 112.3 Channel Estimation in MU-MIMO Channels . . . . . 12

3 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Time-Domain Characterization of Phase Noise . . . . 153.2 Frequency-Domain Characterization of Phase Noise . 173.3 Previous Work on Phase Noise Channels . . . . . . . . 18

4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . 19

II Included Papers 29

A On the Optimality of Single-Carrier Transmission in LargeScale Antenna Systems 311 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Achievable Sum-Rate . . . . . . . . . . . . . . . . . . . . . . . 364 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 41

B Uplink Performance of Time-Reversal MRC in MassiveMIMO Systems subject to Phase Noise 471 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.1 Phase Noise Model . . . . . . . . . . . . . . . . . . . . 522.2 Received Signal . . . . . . . . . . . . . . . . . . . . . . 53

3 Transmission Scheme and Receive Processing . . . . . . . . . 543.1 Channel Estimation . . . . . . . . . . . . . . . . . . . 543.2 Maximum Ratio Combining . . . . . . . . . . . . . . . 56

vi

4 Achievable sum-rate . . . . . . . . . . . . . . . . . . . . . . . 575 Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . 666 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 687 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

C Achievable Rates of ZF Receivers in Massive MIMO withPhase Noise Impairments 831 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Transmission Scheme and ZF Receiver . . . . . . . . . . . . . 88

3.1 LMMSE Channel Estimation . . . . . . . . . . . . . . 883.2 Zero-Forcing (ZF) Equalization . . . . . . . . . . . . . 89

4 Achievable Rates . . . . . . . . . . . . . . . . . . . . . . . . . 905 Results - Discussion . . . . . . . . . . . . . . . . . . . . . . . 92

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Part I

Introduction

1

Introduction

In the last decades wireless networking has changed significantly the fieldof personal communications. Now, many people can connect to the internetand exchange large amounts of data through the cellular network. Popularapplications (social media, video streaming and sharing applications) pro-duce increased volume of traffic and advanced mobile terminals are able tohandle data from more complex applications. Hence, the increase in demandfor higher data transfer is expected to accelerate for the years to come. Inaddition, the environmental footprint of the information and communica-tion technology (ICT) is contributing a significant part to the overall carbondioxide emissions [1]. As a result, it is necessary that future cellular com-munication systems are designed such that they have improved spectral andenergy efficiency.

In the late 90s, researchers have proposed the use of multiple-input multiple-output (MIMO) systems as a way of providing spectrally efficient wirelesscommunication systems. MIMO systems have multiple antennas both atthe transmitter and at the receiver side [2], [3]. The capacity of a MIMOchannel increases linearly with the minimum number of antennas at thetransmitter and the receiver, when channel knowledge is available at bothsides of the communication link. As a result, multiple data streams can bemultiplexed in the same time-frequency resource when MIMO technologyis used. Further, MIMO systems offer improved diversity leading to morereliable communication systems [4].

3

4 Introduction

The promised multiplexing gains of MIMO systems can be realized in arich scattering propagation environment. In the presence of strong correla-tion at either side of the communication link, a strong line-of-sight (LoS)component or when the desired signal power is weak in comparison to theinterference and noise power, point-to-point MIMO systems can no longerprovide multiplexing gains over single antenna systems. Multi-user MIMOsystems (MU-MIMO) naturally generalize single-user MIMO systems andprovide an accurate abstraction of the cellular uplink and downlink chan-nel. MU-MIMO do not require of rich scattering environment to provideincreased multiplexing gain [5]. In a typical MU-MIMO system many -possibly single-antenna- users communicate over the same time-frequencyresources with the base station (BS). Given that the position of each user israndom it is unlikely that the channels of two users are strongly correlated.Further, multi-user diversity can assist so that the spatial multiplexing gainsof MU-MIMO are retained by scheduling users that are nearly orthogonal toeach other. Therefore, MU-MIMO systems appear to be suitable for moderncellular systems.

Recently, it has been advocated that the high data rates and the increasedenergy efficiency required for future wireless networks can be achieved by theuse of a large number of antennas at the BS [6]. These systems have beentermed as Massive MIMO (other terms used are Large MIMO and LargeScale Antenna Systems) and are considered to operate in a regime where anumber of -typically- single antenna users is served by a BS equipped witha multiplicity of antennas (possibly an order of magnitude more than thenumber of the served users). In order to realize such systems it is requiredthat their complexity scales reasonably with the number of BS antennas.However, the complexity of the optimal detection and precoding algorithmsin MIMO systems is prohibitive for Massive MIMO [7]. As a result, it isrequired that the processing is low-complex or even linear. Further, thecomponents that equip the transmission and reception RF chains need tobe inexpensive and power efficient. Therefore, the introduction of MassiveMIMO systems implies that a whole new design of cellular communicationnetworks is required.

When an excess of BS antennas is used, the law of large numbers kicks in andthe random phenomena observed in the small array regime start to appeardeterministic. The channel matrices are typically well conditioned and theusers are adequately spatially separated. As a consequence simple linearprocessing techniques come close to the optimal performance bounds. Afundamental assumption in Massive MIMO is that the squared norms of the

4

5

channel vectors grow at a rate that is faster compared to the inner productsof channel vectors between different users. Measurements conducted in realenvironments confirm the assumption of asymptotic orthogonality betweenchannels of different users [8].

In order to achieve the multiplexing gains of MU-MIMO systems, acquiringaccurate channel knowledge at the transmitter is crucial. In [9] it was shownthat in a single-cell scenario increasing the number of base station antennasis always beneficial in obtaining an accurate channel estimate. In the largearray limit the effects of fast fading and thermal noise vanish. Further, in [10]it was shown that in a multi-cell scenario small scale fading and thermal noisealso vanish and significant throughput can be achieved by use of very simplechannel estimation techniques and rudimentary signal processing at the BS.The only remaining hindrance is pilot contamination. Pilot contaminationis the degradation of the channel estimate acquired during channel trainingdue to the simultaneous transmission of pilots or data in neighboring cells1

[11], [12].

So far, the vast majority of the studies of Massive MIMO systems assumenarrowband channels. That is, the bandwidth of the transmitted signal issmall enough compared to the coherence bandwidth of the channel. Thisimplies that all the frequency components of the signal are affected in thesame way by the channel. Hence the output of the channel depends solelyon the current state and the current input, rendering it memoryless. In caseof broadband channels, typically orthogonal frequency division multiplexing(OFDM) is used, which is known to split a broadband channel into a set of or-thogonal narrowband subchannels and the analysis of narrowband channelsis valid. This thesis focuses mainly on broadband single carrier communica-tion systems. There are several reasons for this. Firstly, this thesis searchesto determine performance bounds for single-carrier communication systemsand to which extent these bounds differ to the simpler case of narrowbandchannels. Additionally, OFDM was promoted in fourth generation cellularsystems because equalization of broadband signals in the time domain isprohibitively complex when the length of the channel impulse response islarge or when dense modulation schemes are used. However, OFDM hasvarious disadvantages as well. For instance, OFDM signals suffer from highpeak-to-average-power-ratio (PAPR). High PAPR signals require expensive

1Initially, pilot contamination during uplink training was observed due to the use ofthe same pilot sequence in neighboring cells. However, in [12] the authors show that pilotcontamination occurs even when the users in neighboring cells transmit data instead ofpilots.

5

6 Introduction

highly linear power amplifiers that work at a significant power back-off toensure that the output signal is not distorted by the non-linear region of thepower amplifier input-output characteristic. However, these amplifiers areparticularly power inefficient. The use of low PAPR signals is essential inMassive MIMO systems so that the gains in energy efficiency can be realized.Single-carrier systems have a lower PAPR and this makes them attractive forMassive MIMO. Further, due to the fact the small inexpensive componentsare required to be used in Massive MIMO systems, the effect of hardwareimperfections, such as phase noise, is essential to be investigated. It is knownthat OFDM is more sensitive to phase and timing errors in comparison tosingle carrier transmission [13].

1 The Intersymbol Interference Channel

In the following we introduce the channel model adopted in this thesis. Eventhough the thesis is on MU-MIMO systems, for the sake of clarity and sim-plicity of the exposition we focus initially on the single-user single-inputsingle-output (SU-SISO) channel and then generalize in the MU-MIMO case.

The mobile radio channel, through which the transmitted signals propagate,can be viewed as a linear filter that distorts the transmitted signal charac-teristics. Due to the relative movement between the BS and the users, thechannel impulse response (CIR) varies in time. In this thesis we assume thatthe channel changes slowly compared to the symbol interval, Ts. The timeinterval that the channel remains constant is the channel coherence time. Asa result, a block of N symbols experience the same CIR. The realization ofthe CIR is assumed to change independently between the blocks of symbols.This assumption is commonly known as the block fading assumption. Thefading process of the CIR is also assumed to be stationary and ergodic.

Modern wireless communication systems are required to support high datatransmission rates. Therefore, it is common that the transmitted signals oc-cupy large bandwidth in the frequency domain. However, due to the physicalphenomena of reflection, refraction and random scattering, the signals ar-rive at the receiver through various paths. As a result, they are delayed andattenuated in a random way. The symbol interval, Ts, of broadband signalsis small. If the excess delay of the channel, TD, is comparable to the symbolinterval, Ts, substantial amount of the transmitted symbol energy interferes

6

1. The Intersymbol Interference Channel 7

Tsτ

Relativepow

er

Figure 1: A typical discrete time power delay profile, that shows the dis-tribution of the relative power as a function of the channel excess delay,τ .

with the succeeding symbols at the receiver. This introduces intersymbolinterference (ISI). In this thesis, the discrete time baseband equivalent rep-resentation of the CIR is described by a finite impulse response filter (FIR).The vector of the FIR coefficients is g = [g0, · · · , gL−1]

T where L is the num-ber of resolvable taps. A typical CIR of a broadband wireless communicationchannel is depicted in Fig. 1.

In order to avoid interference from previous blocks the transmitted sig-nal is augmented with a preamble of Npre symbols and a postamble ofNpost symbols, with Npre ≥ L − 1 and Npost ≥ L − 1. Let xb =[x[−Npre], . . . , x[N +Npost − 1]]T be the vector of the transmitted symbols,out of which only the symbols x = [x[0], . . . , x[N − 1]]T are informationsymbols and the rest are added to eliminate interblock interference (IBI).There are various choices for the symbols that constitute the preamble andthe postamble, such as zero symbols or cyclic repetition of information sym-bols [14]. For the sake of simplicity of the exposition, we consider that thepreamble is an all-zero prefix and the postamble an all-zero tail. The ob-servations y = [y[0], . . . , y[N + L − 2]]T at the output of the channel thatcontain contributions from the input information symbols are given by

y[i] =L−1∑

l=0

glx[i− l] + n[i], i = 0, . . . , N − 1 (1)

7

8 Introduction

where n[i] are assumed to be independent, identically distributed (i.i.d.) zeromean circularly symmetric complex Gaussian (ZMCSCG) random variableswith variance σ2. The input-output relation can be written also in vector-matrix form

y = Gtoepx+ n, (2)

where n = [n[0], . . . , n[N + L− 2]]T is the noise vector and

Gtoep∆=

g0 0 · · · · · · 0

g1 g0. . .

......

. . .

gL−1. . .

. . ....

0. . . g0 0

gL−1 · · · g1 g0... 0

. . . g1. . .

...0 · · · 0 gL−1

(3)

is a matrix containing the channel coefficients and has Toeplitz structure.

In this thesis, the tap coefficients, gl, are assumed to be ZMCSCG ran-dom numbers. Depending on the propagation conditions the taps need nothave the same variance. Therefore, we parameterize the channel taps as the

product gl∆=

√dlhl, where hl is ZMCSCG with unit variance. With dl we

describe the relative power of the l-th channel tap, l = 0, . . . , L − 1. Thesecoefficients vary slowly in comparison to the fast fading coefficients, hl. Asa consequence, they can be considered accurately known to both ends of thecommunication link. Further, the channel impulse response is normalizedsuch that the total received power does not depend on the length of thechannel echo, L. This normalization can be expressed as

E

∣∣∣∣∣

L−1∑

l=0

gl

∣∣∣∣∣

2

=

L−1∑

l=0

d2l = 1 (4)

8

1. The Intersymbol Interference Channel 9

1.1 Detection in ISI channels

In the following we introduce the basic principles of detection in channelswith intersymbol interference. We start with the optimal detection in ISIchannels and a brief complexity analysis. Optimality can be expressed un-der various criteria, such as bit-error-rate (BER), symbol-error-rate (SER),block-error-rate (BLER) or maximum a posteriori probability (MAP). How-ever, in ISI channels the observed symbol contains a superposition of thesymbol to be detected with a number of previously transmitted symbols.Due to the memory introduced by the channel, it is more suitable to detectthe transmitted sequence of symbols rather than the transmitted symbolsseparately. In a SU-SISO with L channel taps, decoding a sequence of Nsymbols selected from a finite alphabetA, requires the calculation of the like-lihood of |A|NL different sequences and select the one with the maximumlikelihood, where |A| is the cardinality of A. The well-known Viterbi algo-rithm gives the maximum likelihood sequence with complexity O(N |A|L),which is a significant reduction of the complexity. However, in case of denseconstellations and large L, the complexity of this solution is still prohibitiveand suboptimal low-complexity detection schemes are more realistic.

Time-Reversal Maximum Ratio Combining

Consider the channel model in (1). Suppose that we seek to detect symbolx[i], i.e. we wish to derive an estimate x[i] of x[i]. Since there are contribu-tions of the symbol x[i] into the observations y[i], y[i+1], . . . , y[i+L−1], it isreasonable to combine the observations appropriately in order to extract themaximum possible energy of symbol x[i] from the observations. Assumingthat the receiver has full knowledge of the channel impulse response, g, thesolution that maximizes the signal-to-noise-ratio (SNR) is the Time-ReversalMaximum Ratio Combining scheme, given by

x[i] =

L−1∑

l=0

g∗l y[i+ l]. (5)

In (5) the observed symbols are reversed in time and convolved with thecomplex conjugated channel impulse response. For the flat fading channel

9

10 Introduction

(L = 1) TR-MRC reduces to the well-known matched filter. The TR-MRCfilter is very simple in its implementation. Further, when the thermal noiseis dominant over ISI the performance of TR-MRC is close to optimal. Inaddition, the complexity of the scheme is minimal and introduces a decodingdelay that is proportional to L. However, since interference is treated asnoise, the performance saturates as the power of interference increases. Inthe high SNR regime it is desirable to suppress interference more efficiently.

Zero-Forcing Filter

In the TR-MRC case the observations y[i] are processed serially and pro-duce estimates of the transmitted symbols. However, processing can bedone in blocks. Consider a block of N + L − 1 observations, which we

stack in an (N + L − 1)-dimensional vector y∆= [y[0], . . . , y[N + L − 2]]T

and x∆= [x[0], . . . , x[N − 1]]T is the vector of transmitted information sym-

bols. The input-output relation is given by (2). The linear processingthat suppresses the interference is called zero-forcing and is given by the

Moore-Penrose pseudoinverse matrix of Gtoep,(GH

toepGtoep

)−1GH

toep. Zero-forcing suppresses the interference, however, when the matrix Gtoep is ill-conditioned, zero-forcing amplifies noise. As a result it is more effective inthe high SNR regime but it can perform worse than time-reversal maximumratio combining in the low SNR regime, where thermal noise dominates overinterference.

2 Multi-user MIMO Channels with IntersymbolInterference

The main contributions of this thesis is on MU-MIMO channels. In thissection we generalize the discussion of SU-SISO channels with ISI to multi-user channels.

2.1 MIMO Broadcast Channel

The Broadcast Channel (BC) describes the scenario where a central nodetransmits information to a set of receivers. In Massive MIMO the base sta-tion is equipped with an excess of antenna elements M and communicates

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2. Multi-user MIMO Channels with Intersymbol Interference 11

with K single antenna users in the cell. Let xBCm [i] be the symbol that is

transmitted at time i by the m-th BS antenna element. The transmittedsymbol vector is therefore xBC[i] = [xBC

1 [i], . . . , xBCM [i]]T . The channel be-

tween the m-th BS antenna and the k-th user is an FIR filter with channelimpulse response gBC

m,k = [gBCm,k,0, · · · , gBC

m,k,L−1]T . The channel impulse re-

sponse is defined similarly to the discussion in Section 1. At the k-th userthe received signal is given by

yBCk [i] =

√

PBC

M∑

m=1

L−1∑

l=0

gBCm,k,lx

BCm [i− l] + nBC

k [i], (6)

similarly to (1). There is always a constraint on the radiated power. Here,we restrict the long-term average power of the transmitted vector xBC[i] tobe constrained by

E[xBC[i]

]= 1. (7)

Therefore, the total transmit power is proportional to PBC.

The transmitted symbol vector xBC[i] is typically the result of a precodingoperation, where a set of information symbols s[i] = [s1[i], . . . , sK [i]]T istransformed into xBC[i]. Achieving optimal performance in the BC canbe very complex which is unsuitable for Massive MIMO [15], [16]. Linearprecoding instead has an affordable complexity in the large array regime.When linear precoding is used, the information vector, s[i], is mapped tothe transmitted vector, xBC[i], via a linear map W ∈ C

M×K , i.e. xBC[i] =1

γBCWs[i]. The coefficient γBC is selected such that the power constraint (7)

is fulfilled.

2.2 MIMO Multiple Access Channel

The Multiple Access Channel (MAC) describes the scenario where a setof users communicate with the base station. Let xMAC

k [i] be the symboltransmitted by the k-th user and gMAC

m,k = [gMACm,k,0, · · · , gMAC

m,k,L−1]T be the

11

12 Introduction

coefficients of the channel impulse response between the k-th user and them-th BS antenna element, defined similarly to the discussion in (1). Then,the signal received by the m-th BS antenna is given by

yMACm [i] =

√

PMAC

K∑

k=1

L−1∑

l=0

gMACm,k,lx

MACk [i− l] + nMAC

k [i] (8)

similarly to (1). In this scenario it is always assumed that the users arenon-cooperative. Consequently, per user power constraints are imposed forevery channel use and the encoding of each user’s information symbols isperformed independently. In this thesis we assume that all the users havethe same power constraint, i.e.

E

[∣∣xMAC

k [i]∣∣2]

= 1, k = 1, . . . ,K. (9)

With the constraint (9) user k can transmit with power PMAC on the averagein the i-th channel use.

2.3 Channel Estimation in MU-MIMO Channels

The advantages offered by MIMO systems (multiplexing gain, diversity gainand array power gain) can only be realized when coherent combination of thereceived signals is possible. In order to perform coherent detection, accurateknowledge of the channel impulse responses is required. Such knowledgecan be acquired either by exploiting the structure of the received signal(blind methods) or by using dedicated known symbols that are transmittedat predetermined intervals (channel training using pilots).

Communication systems divide the resources for uplink and downlink trans-mission either in frequency or in time. Depending on this partitioning, pilotsignaling is handled differently. In frequency division duplex (FDD) sys-tems, the user terminals calculate an estimate of the channel impulse re-sponse based on the received signal during training and send this estimateto the BS via a feedback channel on an other frequency band. Every user is

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2. Multi-user MIMO Channels with Intersymbol Interference 13

required to estimate M channel impulse responses and adequate bandwidthis required so that these coefficients are fed back error-free. Given that thesignaling overhead scales with the number of antennas it is argued that Mas-sive MIMO cannot operate using FDD without a substantial reduction of thesignaling overhead.

In time division duplex (TDD) systems the channel coherence time is splitinto two parts. In the first part the users transmit their data and possibly aset of pilot symbols. The BS calculates an estimate of the channel impulseresponse based on the received pilots and uses these estimates to detect thereceived data. Assuming that the downlink channel is reciprocal with theestimated uplink channel, the BS can use these estimates to precode thesignals to be transmitted on the downlink.

Pilot assisted channel estimation in the frequency selective channel is not atrivial task. In a frequency selective channel apart from the choice of trainingsequences, the precise placement of the pilots is important [17], [18], [19],[20]. Optimal designs that exist in literature are based on the maximizationof a lower bound on the channel capacity and often lead to complicatedexpressions. In this thesis, we are mainly interested in deriving lower boundson the performance of single-carrier Massive MIMO systems. In the followingwe present a simple training scheme for the single-carrier MU-MIMO uplinkthat provides a pertinent statistical characterization of the estimates of thechannel impulse responses. This characterization facilitates the derivationof simple closed-form lower bounds.

We consider a single-cell MU-MIMO uplink channel as given by (8). It isrequired that we estimate KL channel coefficients. Therefore, the first KLchannel uses of the coherence interval are used for training. During thetraining interval the users transmit their pilot sequences, which consist ofa single pulse at a predetermined channel use and remain idle during therest of the training interval. In particular, user k transmits a single pulse ofamplitude

√PpKL at channel use (k−1)L, for 1 ≤ k ≤ K. The amplitude of

the pulse is chosen such that the average per user power spent per channeluse during the training interval is Pp. Hence, the received signal at BSantenna m at time (k − 1)L+ l, 0 ≤ l ≤ L− 1 is given by

ym[(k − 1)L+ l] =√

PpKLgm,k,l + nm[(k − 1)L+ l]. (10)

13

14 Introduction

Based on (10) the BS can determine the CIR of each user. The channel esti-mates can be derived with respect to various optimality criteria [21]. In thiswork we use two different estimates due to their mathematical tractability.The first, which is the maximum likelihood (ML) estimate, is derived simplyby scaling (10) by the amplitude of the training pulses, i.e.

gm,k,l = gm,k,l +1

√PpKL

nm[(k − 1)L+ l] (11)

and is selected because of the simplicity of the estimation. The minimummean square estimate (MMSE) is also used, i.e.

gm,k,l =

√PpKLdk,l

PpKLdk,l + σ2ym[(k − 1)L+ l] (12)

The MMSE estimate is sometimes preferred because gm,k,l and gm,k,l−gm,k,l.are uncorrelated

3 Phase Noise

Oscillators are electronic circuits used in communication systems to upcon-vert the baseband signal to the passband at the transmitter side and todownconvert the received passband signal to the baseband at the receiverside. Ideally, an oscillator produces a perfect sinusoidal waveform that hasa stable frequency and phase reference. However, practical oscillators in-clude passive elements, such as resistances, capacitors and inductances, aswell as active elements, such as transistors. The random movement of elec-trons and holes in these elements introduces thermal, shot and flicker noisein the circuit. This noise translates at the output of the oscillator into ran-dom fluctuations of the phase of the produced sinusoidal waveform. Onecan distinguish a waveform without phase noise from a waveform contam-inated with phase noise by observing the zero-crossings of the waveform.In the no-phase-noise case, the zero-crossings appear periodically in time.However, in the phase noise impaired waveform the zero-crossings are ran-domly perturbed around their nominal position. In the frequency domain,

14

3. Phase Noise 15

the spectrum of an ideal oscillator should exhibit a Dirac pulse preciselyat the resonance frequency, fc. In practice, however, the output of a realoscillator widens around fc.

Phase noise is a particularly undesirable hardware imperfection in commu-nication systems. It causes a random rotation of the constellation points,thereby reducing the effective SNR and increasing the probability of er-ror. Since phase noise is a random phase rotation of the received signal, ithas a multiplicative effect on the signal. Hence, it increases proportionallywith the signal power. When a transmitter with high phase noise transmitsit signal at high power levels, significant interference is introduced in theadjacent channels. Consequently, weak signals in the adjacent channels canexperience very high levels of interference. At the receiver side a similar phe-nomenon can occur. When a phase noise impaired receiver downconverts aweak signal in the presence of strong signals at the adjacent channels, signif-icant interference can be introduced due to phase noise in the desired signal.Strong signals in neighboring bands can overwhelm the desired signal. Fi-nally, in training based communication systems phase noise degrades thequality of the channel estimates. This is particularly important in coherentcommunication systems, where a small discrepancy between the estimatedchannel from the actual channel realization can have detrimental effect inthe effective SNR.

3.1 Time-Domain Characterization of Phase Noise

A complete characterization of phase noise is difficult due to the complexityof the phenomenon. It depends on the type of the oscillator, on the compo-nents used and on imperfections introduced during the manufacturing pro-cess. In this work we consider Wiener phase noise, which is a well-establishedmodel of phase noise in oscillators. It describes accurately the phase noiseof free-running oscillators. The output of an oscillator with phase noise canbe expressed by

a(t) = cos(2πfct+ φ(t)), (13)

where φ(t) is the instantaneous phase shift of the oscillator. According to theWiener phase noise model φ(t) is described as a continuous time Brownianmotion. That is,

15

16 Introduction

φ(t) =

∫ t

0N(τ)dτ, (14)

where N(t) is a white Gaussian noise process. According to the definitionof Brownian motion the variation of ∆φ(∆t) at interval ∆t is given by

∆φ(∆t) = φ(t+∆t)− φ(t) =

∫ t+∆t

tN(τ)dτ. (15)

An important property of the Wiener process is that ∆φ(∆t) is a zero-meanGaussian random variable with variance proportional to ∆t, independent ofthe past realization of the process. In this work we are particularly interestedin the discrete time model for Wiener phase noise. This model can be deriveddirectly from (14) by sampling every Ts seconds

φ[i]∆= φ(iT ) =

∫ iTs

0N(τ)dτ =

∫ (i−1)Ts

0N(τ)dτ +

∫ iTs

(i−1)Ts

N(τ)dτ (16)

= φ[i− 1] + w[i],

where w[i] ∼ N (0, σ2φ). The variance σ2

φ of the discrete time incrementsw[i] depends on the quality of the oscillator, the oscillation frequency andis proportional to the sampling time, Ts. In [22] the authors rigorouslyderive a stochastic characterization of the random time shift α(t). Phasenoise is related to α(t) at carrier frequency, fc, by φ(t) = 2πfcα(t) Theyprove that α(t) becomes asymptotically in time a Gaussian random variablewith constant mean and variance that increases proportionally with time.Further, they calculate the auto-correlation of the process α(t),

E [α(t)α(t + τ)] ∝ cmin(t, t+ τ) (17)

where c is an oscillator dependent parameter. In this thesis, the variance ofthe phase noise increments is given by σ2

φ = 4π2f2c cTs.

16

3. Phase Noise 17

L(fm)

f1/f3 fc/(2Q) log fm

Figure 2: The phase noise power spectral density as described by the Leesonmodel.

3.2 Frequency-Domain Characterization of Phase Noise

Even though the time domain characterization is concise and easily com-prehensible, a frequency domain characterization is also necessary. The fre-quency domain description of phase noise provides a measure of spectralbroadening around the nominal oscillator frequency. One of the most com-mon measures of phase noise in the frequency domain is the single sidebandnoise spectral density, L(fm). L(fm) is defined as the ratio of the phase noisepower at an offset fm from the carrier frequency measured at a bandwidth of1 Hz over the power at the carrier frequency. In [22] the authors show thatthe power spectral density of a sinusoidal waveform perturbed with Wienerphase noise has Lorentzian shape

L(fm) ≈

10 log10

(f2c c

π2f2c c

2+f2m

)

, 0 ≤ fm ≪ fc

10 log10

((fcfm

)2c

)

, πf2c c ≪ fm ≪ fc

dBc/Hz. (18)

The unit of L(fm) is ’decibel below carrier per Hertz’ or dBc/Hz.

The model in (18) is an accurate description of the phase noise power spectraldensity of Wiener phase noise. That is, when the noise is generated by

17

18 Introduction

white and modulated-white noise sources. In this thesis, we will considerthis model for consistency with the time domain characterization of phasenoise. However, phase noise is introduced due to colored or modulated-colored noise, such as 1/f noise. This results in a region of the powerspectral density that scales as 1/f3 at frequency offsets sufficiently close tothe carrier frequency. Further, at large frequency offsets the power spectraldensity eventually flattens out rather than continuing to drop at a rate of20 dB/decade. An empirical model proposed by Leeson [24], captured theseeffects and the phase noise power spectral density is given by

L(∆ω) = 10 log10

[

2FkT

Psig

{

1 +

(fc

2Qfm

)2}(

1 +f1/f3

|fm|

)]

, (19)

where k is the Boltzmann constant, T is the absolute temperature, Psig is thesignal power and Q is the quality factor of the resonator. The parametersF and f1/f3 are determined empirically [25].

3.3 Previous Work on Phase Noise Channels

As explained earlier, phase noise is an inevitable hardware impairment andit can degrade the performance of coherent communication systems consid-erably. Therefore, there has been vigorous research interest in calculatingthe capacity of channels with phase noise. However, this problem turnsout to be challenging even in simplified scenarios. In [26] the behavior ofthe capacity of a non-fading SU-SISO channel with phase noise is studiedasymptotically in the high SNR regime. In [27] the authors show that thecapacity achieving input distribution of a discrete-time noncoherent additivewhite Gaussian noise channel subject to an average power constraint has in-finitely many discrete mass points and they calculate a tight lower boundon the capacity. In [28] a block memoryless channel is studied and capacitybounds are derived that are tight asymptotically as SNR goes to infinity.In [29] the authors study a single user deterministic MIMO channel withWiener phase noise and calculate numerically upper and lower bounds onthe capacity.

Significant research literature exists for the problem of phase noise in orthog-onal frequency division multiplexing (OFDM) systems. In [13] it is shown

18

4. Contributions of the Thesis 19

Capacity HardwareBounds impairments

Flat Fading/OFDM [9] Paper CSingle Cell

Flat Fading/OFDM [10], [11], [36], [39]Multi-Cell [37], [12], [38]

Selective Fading Paper A , [40] Paper B, [35]Single Cell

Table 1: Contributions in the topics of fundamental limits and impact ofhardware impairments in Massive MIMO.

that multi-carrier transmission is more sensitive to phase noise in comparisonto single-carrier transmission and in [30] the authors calculate the bit-error-rate of an OFDM system impaired with Wiener phase noise. In [31] theauthors study the SINR degradation due to Wiener phase noise and pro-pose a method for mitigation. In [32] the authors approximate the phasenoise realization by the first terms of its Fourier expansion and propose amethod to compensate for it. The problem of estimating the phase of aphase noise impaired system has also attracted significant research interest.In [33] various phase noise estimation schemes are derived and comparedto the Cramer-Rao lower bound and in [34] the authors propose decodingalgorithms for channels with strong phase noise based on the sum-productalgorithms on factor graphs.

4 Contributions of the Thesis

In this thesis we study the fundamental limits of Massive MIMO systemswith single carrier transmission and phase noise. For the sake of clarity ofexposition we provide tabular overview of previous work in the topic and weplace our contributions in the corresponding slots. This overview however isnot intended to be exhaustive of the literature on Massive MIMO or phasenoise. In particular, in Table 1 we include studies of the fundamental limitsof Massive MIMO under realistic scenarios and we extend this survey to thestudy of Massive MIMO with hardware impairments. Further, in Table 2we provide an indicative overview of the literature on phase noise in variouscommunication scenarios.

19

20 Introduction

Capacity Performance EstimationBounds Analysis (e.g. BER) Compensation

SISO [26], [27] [28] [34]

OFDM [13], [30] [31], [32], [42]

MIMO [29] [43] [33]

Massive Paper B,MIMO Paper C, [35]

Table 2: Contributions in area of phase noise.

This licentiate thesis contains the work reported in the following three re-search papers.

Paper A On the Optimality of Single-Carrier Transmission inLarge Scale Antenna Systems

Authored by Antonios Pitarokoilis, Saif Khan Mohammed and Erik G. Lars-son. Published in IEEE Wireless Communications Letters, vol. 1, no. 4,pp. 276–279, Aug. 2012.

A single carrier transmission scheme is presented for the frequency selectivemulti-user (MU) multiple-input single-output (MISO) Gaussian BroadcastChannel (GBC) with a base station (BS) having M antennas and K singleantenna users. The proposed transmission scheme has low complexity andfor M ≫ K it is shown to achieve near optimal sum-rate performance atlow transmit power to receiver noise power ratio. Additionally, the proposedtransmission scheme results in an equalization-free receiver and does not re-quire any MU resource allocation and associated control signaling overhead.Also, the sum-rate achieved by the proposed transmission scheme is shownto be independent of the channel power delay profile (PDP). In terms ofpower efficiency, the proposed transmission scheme also exhibits an O(M)array power gain. Simulations are used to confirm analytical observations.

Paper B Uplink Performance of Time-Reversal MRC in Mas-sive MIMO Systems subject to Phase Noise

Authored by Antonios Pitarokoilis, Saif Khan Mohammed and Erik G. Lars-son. To be submitted for journal publication. This work is an extension ofthe conference paper [35].

20

4. Contributions of the Thesis 21

Multi-user multiple-input multiple-output (MU-MIMO) cellular systemswith an excess of base station (BS) antennas (Massive MIMO) offer un-precedented multiplexing gains and radiated energy efficiency. Oscillatorphase noise is introduced in the transmitter and receiver radio frequencychains and severely degrades the performance of communication systems.We study the effect of oscillator phase noise in frequency-selective MassiveMIMO systems with imperfect channel state information (CSI). In particu-lar, we consider two distinct operation modes, namely when the phase noiseprocesses at the BS antennas are identical (synchronous operation) and whenthey are independent (non-synchronous operation). We analyze a linear andlow-complexity time-reversal maximum-ratio combining (TR-MRC) recep-tion strategy. For both operation modes we derive a lower bound on thesum-capacity and we compare the performance of the two modes. Basedon the derived achievable sum-rate, we show that with the proposed receiveprocessing an O(

√M) array gain is achievable. Due to the phase noise drift

the estimated effective channel becomes progressively outdated. Therefore,phase noise effectively limits the length of the interval used for data trans-mission and the number of scheduled users. The derived achievable ratesprovide insights into the optimum choice of the data interval length and thenumber of scheduled users.

Paper C Achievable Rates of ZF Receivers in Massive MIMOwith Phase Noise Impairments

Authored by Antonios Pitarokoilis, Saif Khan Mohammed and Erik G. Lars-son. To be presented at the Asilomar Conference on Signals, Systems, andComputers, Pacific Grove, CA, USA, Nov. 2013.

The effect of oscillator phase noise on the sum-rate performance of largemulti-user multiple-input multiple-output (MU-MIMO) systems, termed asMassive MIMO, is studied. A Rayleigh fading MU-MIMO uplink channel isconsidered, where channel state information (CSI) is acquired via training.The base station (BS), which is equipped with an excess of antenna elements,M , uses the channel estimate to perform zero-forcing (ZF) detection. A lowerbound on the sum-rate performance is derived. It is shown that the proposedreceiver structure exhibits an O(

√M) array power gain. Additionally, the

proposed receiver is compared with earlier studies that employ maximumratio combining and it is shown that it can provide significant sum-rate per-formance gains at the medium and high signal-to-noise-ratio (SNR) regime.

21

22 Introduction

Further, the expression of the achievable sum rate provides new insights onthe effect of various parameters on the overall system performance.

22

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27

Linkoping Studies in Science and TechnologyLicentiate Theses, Division of Communication Systems

Department of Electrical Engineering (ISY)Linkoping University, Sweden

Erik Axell, Topics in Spectrum Sensing for Cognitive Radio, Thesis No. 1417, 2009.

Johannes Lindblom, Resource Allocation on the MISO Interference Channel, ThesisNo. 1438, 2010.

Reza Moosavi, Aspects of Control Signaling in Wireless Multiple Access Systems,Thesis No. 1493, 2011.

Tumula V. K. Chaitanya, Improved Techniques for Retransmission and Relaying inWireless Systems, Thesis No. 1494, 2011.

Mirsad Cirkic, Optimization of Computational Resources for MIMO Detection, The-sis No. 1514, 2011.

Hien Quoc Ngo, Performance Bounds for Very Large Multiuser MIMO Systems,Thesis No. 1562, 2012.