AntoniosPitarokoilis OntheperformanceofMassiveMIMOsystemswithsinglecarriertransmissionandphasenoise

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Link¨oping Studies in Science and Technology Thesis No. 1618

On the performance of Massive

MIMO systems with single carrier

transmission and phase noise

Antonios Pitarokoilis

Division of Communication Systems Department of Electrical Engineering (ISY) Link¨oping University, SE-581 83 Link¨oping, Sweden

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This is a Swedish Licentiate Thesis.

The Licentiate degree comprises 120 ECTS credits of postgraduate studies.

On the performance of Massive MIMO systems with single carrier transmission and phase noise

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

ISSN 0280-7971

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Abstract

In the last decade we have experienced a rapid increase in the demand for high 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 multiple antennas and the users share the time-frequency resources. However, modern communication systems are highly power inefficient. Further, the increase in demand for higher data rates is expected to accelerate in the years to come due to the popularity of mobile devices like smartphones and tablets. Hence, next generation cellular systems are required to exhibit high energy efficiency as 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 to the served users can be a promising candidate to meet these contradictory requirements. These systems are termed as Massive MIMO. When the num-ber of BS antennas grows large, the channels between different users become orthogonal and low complexity transceiver processing exhibits sum-rate per-formance that is close to optimal. In order to realize the promised gains of Massive MIMO systems, it is required that power efficient and inexpen-sive components are used. In contemporary cellular systems, multi-carrier transmission is used since it facilitates simple equalization at the receiver side. 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 single carrier signals exhibit lower PAPR and are suitable for signal design that is more robust to non-linear power amplifiers. Further, single-carrier signals are less vulnerable to hardware impairments, such as phase noise. In this thesis we study the fundamental limits of Massive MIMO systems in terms of sum-rate performance with single-carrier transmission and phase noise and provide important insight on the design of Massive MIMO under these scenarios.

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Acknowledgments

I would like to thank my supervisor, Prof. Erik G. Larsson, for offering me the opportunity to pursue the PhD degree at Link¨oping University. I am grateful to Erik because he provided his guidance to challenging research ideas 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 research and provided me with insightful feedback that helped me to stay focused on important research questions. However, he also encouraged me to think independently 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 this initial part of my PhD studies. We have cooperated closely while he was here and I had the chance to learn a lot from him especially in the area of information theory. Further, his contribution to improving my writing style was particularly appreciated. I am also grateful to a former member of the group, Eleftherios (Lefteris) Karipidis. Lefteris has helped me, not only through his technical expertise, but also he has offered his generous assis-tance to adjust to the new environment when I first arrived in Link¨oping. Finally, I would like to thank all the PhD students, Chaitanya, Johannes, Reza, Mirsad, Hien, Victor, Marcus and Christopher and the seniors of the group, Peter, Danyo, Mikael, Daniel and Vladimir, for making the everyday life 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 I would have achieved much less in my life.

Link¨oping, October 2013 Antonios Pitarokoilis

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Introduction

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

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

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4 Introduction

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

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

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

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5

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

In order to achieve the multiplexing gains of MU-MIMO systems, acquiring accurate channel knowledge at the transmitter is crucial. In [9] it was shown that in a single-cell scenario increasing the number of base station antennas is always beneficial in obtaining an accurate channel estimate. In the large array 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 noise also vanish and significant throughput can be achieved by use of very simple channel estimation techniques and rudimentary signal processing at the BS. The only remaining hindrance is pilot contamination. Pilot contamination is the degradation of the channel estimate acquired during channel training due 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 assume narrowband channels. That is, the bandwidth of the transmitted signal is small enough compared to the coherence bandwidth of the channel. This implies that all the frequency components of the signal are affected in the same way by the channel. Hence the output of the channel depends solely on the current state and the current input, rendering it memoryless. In case of 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 channels is valid. This thesis focuses mainly on broadband single carrier communica-tion systems. There are several reasons for this. Firstly, this thesis searches to determine performance bounds for single-carrier communication systems and to which extent these bounds differ to the simpler case of narrowband channels. Additionally, OFDM was promoted in fourth generation cellular systems because equalization of broadband signals in the time domain is prohibitively complex when the length of the channel impulse response is large or when dense modulation schemes are used. However, OFDM has various disadvantages as well. For instance, OFDM signals suffer from high peak-to-average-power-ratio (PAPR). High PAPR signals require expensive

1_{Initially, pilot contamination during uplink training was observed due to the use of}

the same pilot sequence in neighboring cells. However, in [12] the authors show that pilot contamination occurs even when the users in neighboring cells transmit data instead of pilots.

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6 Introduction

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

1 The Intersymbol Interference Channel

In the following we introduce the channel model adopted in this thesis. Even though 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-input single-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, the channel impulse response (CIR) varies in time. In this thesis we assume that the channel changes slowly compared to the symbol interval, Ts. The time

interval that the channel remains constant is the channel coherence time. As a result, a block of N symbols experience the same CIR. The realization of the CIR is assumed to change independently between the blocks of symbols. This assumption is commonly known as the block fading assumption. The fading process of the CIR is also assumed to be stationary and ergodic. Modern wireless communication systems are required to support high data transmission rates. Therefore, it is common that the transmitted signals oc-cupy large bandwidth in the frequency domain. However, due to the physical phenomena of reflection, refraction and random scattering, the signals ar-rive at the receiver through various paths. As a result, they are delayed and attenuated in a random way. The symbol interval, Ts, of broadband signals

is small. If the excess delay of the channel, TD, is comparable to the symbol

interval, Ts, substantial amount of the transmitted symbol energy interferes

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1. The Intersymbol Interference Channel 7 Ts τ R el at iv e p ow 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 intersymbol interference (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 communication channel 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 of

Npost 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 information}

symbols and the rest are added to eliminate interblock interference (IBI). There are various choices for the symbols that constitute the preamble and the postamble, such as zero symbols or cyclic repetition of information sym-bols [14]. For the sake of simplicity of the exposition, we consider that the preamble 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 that}

contain contributions from the input information symbols are given by

y[i] =

L−1

X

l=0

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8 Introduction

where n[i] are assumed to be independent, identically distributed (i.i.d.) zero mean circularly symmetric complex Gaussian (ZMCSCG) random variables with 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 not have 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. These coefficients vary slowly in comparison to the fast fading coefficients, hl. As

a consequence, they can be considered accurately known to both ends of the communication link. Further, the channel impulse response is normalized such that the total received power does not depend on the length of the channel echo, L. This normalization can be expressed as

E   L−1 X l=0 gl 2 = L−1 X l=0 d2l = 1 (4) 8

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1. The Intersymbol Interference Channel 9

1.1 Detection in ISI channels

In the following we introduce the basic principles of detection in channels with intersymbol interference. We start with the optimal detection in ISI channels 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 the symbol to be detected with a number of previously transmitted symbols. Due to the memory introduced by the channel, it is more suitable to detect the transmitted sequence of symbols rather than the transmitted symbols separately. In a SU-SISO with L channel taps, decoding a sequence of N symbols selected from a finite alphabetA, requires the calculation of the like-lihood of |A|N L _{different sequences and select the one with the maximum}

likelihood, 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 dense constellations and large L, the complexity of this solution is still prohibitive and 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 symbol x[i], i.e. we wish to derive an estimate ˆx[i] of x[i]. Since there are contribu-tions of the symbol x[i] into the observacontribu-tions y[i], y[i+1], . . . , y[i+L−1], it is reasonable to combine the observations appropriately in order to extract the maximum possible energy of symbol x[i] from the observations. Assuming that the receiver has full knowledge of the channel impulse response, g, the solution that maximizes the signal-to-noise-ratio (SNR) is the Time-Reversal Maximum Ratio Combining scheme, given by

ˆ x[i] = L−1 X l=0 gl∗y[i + l]. (5)

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

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10 Introduction

(L = 1) TR-MRC reduces to the well-known matched filter. The TR-MRC filter is very simple in its implementation. Further, when the thermal noise is dominant over ISI the performance of TR-MRC is close to optimal. In addition, the complexity of the scheme is minimal and introduces a decoding delay that is proportional to L. However, since interference is treated as noise, the performance saturates as the power of interference increases. In the 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 be done 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 processing that suppresses the interference is called zero-forcing and is given by the Moore-Penrose pseudoinverse matrix of Gtoep, GHtoepGtoep

−1

GH_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 in the high SNR regime but it can perform worse than time-reversal maximum ratio combining in the low SNR regime, where thermal noise dominates over interference.

2 Multi-user MIMO Channels with Intersymbol

Interference

The main contributions of this thesis is on MU-MIMO channels. In this section 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 node transmits 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 transmitted symbol vector is therefore xBC_{[i] = [x}BC

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

be-tween the m-th BS antenna and the k-th user is an FIR filter with channel impulse response gBC_m,k = [g_m,k,0BC ,· · · , gBC

m,k,L−1]T. The channel impulse

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

yBCk [i] = p PBC M X m=1 L−1 X l=0 gm,k,lBC xBCm [i− l] + nBCk [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] to}

be 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 precoding operation, where a set of information symbols s[i] = [s1[i], . . . , sK[i]]T is

transformed into xBC_{[i]. Achieving optimal performance in the BC can}

be very complex which is unsuitable for Massive MIMO [15], [16]. Linear precoding instead has an affordable complexity in the large array regime. When linear precoding is used, the information vector, s[i], is mapped to the transmitted vector, xBC[i], via a linear map W ∈ CM ×K_{, i.e. x}BC_{[i] =}

1

γBCW s[i]. The coefficient γBCis 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 set of users communicate with the base station. Let xMAC

k [i] be the symbol

transmitted by the k-th user and gMAC_m,k = [g_m,k,0MAC,· · · , gMAC

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12 Introduction

coefficients of the channel impulse response between the k-th user and the m-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] = p PMAC K X k=1 L−1 X l=0

gm,k,lMACxMACk [i− l] + nMACk [i] (8)

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

Eh_xMAC_k _[i]2 i

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

With the constraint (9) user k can transmit with power PMACon the average

in the i-th channel use.

2.3 Channel Estimation in MU-MIMO Channels

The advantages offered by MIMO systems (multiplexing gain, diversity gain and array power gain) can only be realized when coherent combination of the received signals is possible. In order to perform coherent detection, accurate knowledge of the channel impulse responses is required. Such knowledge can be acquired either by exploiting the structure of the received signal (blind methods) or by using dedicated known symbols that are transmitted at 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, pilot signaling 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 estimate to 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 bandwidth is required so that these coefficients are fed back error-free. Given that the signaling overhead scales with the number of antennas it is argued that Mas-sive MIMO cannot operate using FDD without a substantial reduction of the signaling overhead.

In time division duplex (TDD) systems the channel coherence time is split into two parts. In the first part the users transmit their data and possibly a set of pilot symbols. The BS calculates an estimate of the channel impulse response based on the received pilots and uses these estimates to detect the received data. Assuming that the downlink channel is reciprocal with the estimated uplink channel, the BS can use these estimates to precode the signals to be transmitted on the downlink.

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

We consider a single-cell MU-MIMO uplink channel as given by (8). It is required that we estimate KL channel coefficients. Therefore, the first KL channel uses of the coherence interval are used for training. During the training interval the users transmit their pilot sequences, which consist of a single pulse at a predetermined channel use and remain idle during the rest of the training interval. In particular, user k transmits a single pulse of amplitudepPpKL 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 channel use during the training interval is Pp. Hence, the received signal at BS

antenna m at time (k− 1)L + l, 0 ≤ l ≤ L − 1 is given by

ym[(k− 1)L + l] =

p

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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 this work we use two different estimates due to their mathematical tractability. The first, which is the maximum likelihood (ML) estimate, is derived simply by scaling (10) by the amplitude of the training pulses, i.e.

ˆ

gm,k,l= gm,k,l+p 1 PpKL

nm[(k− 1)L + l] (11)

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

ˆ gm,k,l=

p

PpKLdk,l

PpKLdk,l+ σ2ym[(k− 1)L + l] (12)

The MMSE estimate is sometimes preferred because ˆgm,k,land ˆgm,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 to downconvert the received passband signal to the baseband at the receiver side. Ideally, an oscillator produces a perfect sinusoidal waveform that has a stable frequency and phase reference. However, practical oscillators in-clude passive elements, such as resistances, capacitors and inductances, as well as active elements, such as transistors. The random movement of elec-trons and holes in these elements introduces thermal, shot and flicker noise in the circuit. This noise translates at the output of the oscillator into ran-dom fluctuations of the phase of the produced sinusoidal waveform. One can 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,

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3. Phase Noise 15

the spectrum of an ideal oscillator should exhibit a Dirac pulse precisely at the resonance frequency, fc. In practice, however, the output of a real

oscillator 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, it has a multiplicative effect on the signal. Hence, it increases proportionally with the signal power. When a transmitter with high phase noise transmits it signal at high power levels, significant interference is introduced in the adjacent channels. Consequently, weak signals in the adjacent channels can experience very high levels of interference. At the receiver side a similar phe-nomenon can occur. When a phase noise impaired receiver downconverts a weak 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 the quality of the channel estimates. This is particularly important in coherent communication systems, where a small discrepancy between the estimated channel from the actual channel realization can have detrimental effect in the effective SNR.

3.1 Time-Domain Characterization of Phase Noise

A complete characterization of phase noise is difficult due to the complexity of 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-established model of phase noise in oscillators. It describes accurately the phase noise of free-running oscillators. The output of an oscillator with phase noise can be expressed by

a(t) = cos(2πfct + φ(t)), (13)

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

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16 Introduction

φ(t) = Z t

0

N (τ )dτ, (14)

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

∆φ(∆t) = φ(t + ∆t)− φ(t) = Z t+∆t

t

N (τ )dτ. (15)

An important property of the Wiener process is that ∆φ(∆t) is a zero-mean Gaussian random variable with variance proportional to ∆t, independent of the past realization of the process. In this work we are particularly interested in the discrete time model for Wiener phase noise. This model can be derived directly from (14) by sampling every Tsseconds

φ[i]= φ(iT ) =∆ Z iTs 0 N (τ )dτ = Z (i−1)Ts 0 N (τ )dτ + Z iTs (i−1)Ts N (τ )dτ (16) = φ[i− 1] + w[i], where w[i] ∼ N (0, σ2

φ). The variance σφ2 of the discrete time increments

w[i] depends on the quality of the oscillator, the oscillation frequency and is proportional to the sampling time, Ts. In [22] the authors rigorously

derive a stochastic characterization of the random time shift α(t). Phase noise is related to α(t) at carrier frequency, fc, by φ(t) = 2πfcα(t) They

prove that α(t) becomes asymptotically in time a Gaussian random variable with 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 )]}_{∝ c min(t, t + τ)} ₍₁₇₎

where c is an oscillator dependent parameter. In this thesis, the variance of the phase noise increments is given by σ_φ2= 4π2fc2cTs.

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3. Phase Noise 17

L(fm)

f1/f3 f_c/(2Q) log f_m

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

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 spectral broadening around the nominal oscillator frequency. One of the most com-mon measures of phase noise in the frequency domain is the single sideband noise spectral density,L(fm). L(fm) is defined as the ratio of the phase noise

power at an offset fmfrom the carrier frequency measured at a bandwidth of

1 Hz over the power at the carrier frequency. In [22] the authors show that the power spectral density of a sinusoidal waveform perturbed with Wiener phase noise has Lorentzian shape

L(fm)≈      10 log10 _f2 cc π2_f2 cc2+fm2 , 0≤ fm≪ fc 10 log10 _f c fm 2 c , πf2 cc≪ 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 spectral density of Wiener phase noise. That is, when the noise is generated by

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18 Introduction

white and modulated-white noise sources. In this thesis, we will consider this model for consistency with the time domain characterization of phase noise. However, phase noise is introduced due to colored or modulated-colored noise, such as 1/f noise. This results in a region of the power spectral density that scales as 1/f3 at frequency offsets sufficiently close to the carrier frequency. Further, at large frequency offsets the power spectral density eventually flattens out rather than continuing to drop at a rate of 20 dB/decade. An empirical model proposed by Leeson [24], captured these effects and the phase noise power spectral density is given by

L(∆ω) = 10 log10 " 2F kT Psig ( 1 + fc 2Qfm 2) 1 +f1/f3 |fm| # , (19)

where k is the Boltzmann constant, T is the absolute temperature, Psigis the

signal power and Q is the quality factor of the resonator. The parameters F 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 and it can degrade the performance of coherent communication systems consid-erably. Therefore, there has been vigorous research interest in calculating the capacity of channels with phase noise. However, this problem turns out to be challenging even in simplified scenarios. In [26] the behavior of the capacity of a non-fading SU-SISO channel with phase noise is studied asymptotically in the high SNR regime. In [27] the authors show that the capacity achieving input distribution of a discrete-time noncoherent additive white Gaussian noise channel subject to an average power constraint has in-finitely many discrete mass points and they calculate a tight lower bound on the capacity. In [28] a block memoryless channel is studied and capacity bounds are derived that are tight asymptotically as SNR goes to infinity. In [29] the authors study a single user deterministic MIMO channel with Wiener phase noise and calculate numerically upper and lower bounds on the capacity.

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

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4. Contributions of the Thesis 19

Capacity Hardware

Bounds impairments

Flat Fading/OFDM [9] Paper C

Single 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 of hardware impairments in Massive MIMO.

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

4 Contributions of the Thesis

In this thesis we study the fundamental limits of Massive MIMO systems with single carrier transmission and phase noise. For the sake of clarity of exposition we provide tabular overview of previous work in the topic and we place our contributions in the corresponding slots. This overview however is not intended to be exhaustive of the literature on Massive MIMO or phase noise. In particular, in Table 1 we include studies of the fundamental limits of Massive MIMO under realistic scenarios and we extend this survey to the study of Massive MIMO with hardware impairments. Further, in Table 2 we provide an indicative overview of the literature on phase noise in various communication scenarios.

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20 Introduction

Capacity Performance Estimation

Bounds 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 in

Large 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 selective multi-user (MU) multiple-input single-output (MISO) Gaussian Broadcast Channel (GBC) with a base station (BS) having M antennas and K single antenna users. The proposed transmission scheme has low complexity and for M ≫ K it is shown to achieve near optimal sum-rate performance at low transmit power to receiver noise power ratio. Additionally, the proposed transmission 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 shown to be independent of the channel power delay profile (PDP). In terms of power 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 of the conference paper [35].

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4. Contributions of the Thesis 21

Multi-user multiple-input multiple-output (MU-MIMO) cellular systems with an excess of base station (BS) antennas (Massive MIMO) offer un-precedented multiplexing gains and radiated energy efficiency. Oscillator phase noise is introduced in the transmitter and receiver radio frequency chains and severely degrades the performance of communication systems. We study the effect of oscillator phase noise in frequency-selective Massive MIMO systems with imperfect channel state information (CSI). In particu-lar, we consider two distinct operation modes, namely when the phase noise processes at the BS antennas are identical (synchronous operation) and when they are independent (non-synchronous operation). We analyze a linear and low-complexity time-reversal maximum-ratio combining (TR-MRC) recep-tion strategy. For both operarecep-tion modes we derive a lower bound on the sum-capacity and we compare the performance of the two modes. Based on the derived achievable sum-rate, we show that with the proposed receive processing 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 rates provide insights into the optimum choice of the data interval length and the number of scheduled users.

Paper C Achievable Rates of ZF Receivers in Massive MIMO

with 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, and Computers, Pacific Grove, CA, USA, Nov. 2013.

The effect of oscillator phase noise on the sum-rate performance of large multi-user multiple-input multiple-output (MU-MIMO) systems, termed as Massive MIMO, is studied. A Rayleigh fading MU-MIMO uplink channel is considered, 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 lower bound on the sum-rate performance is derived. It is shown that the proposed receiver structure exhibits an O(√M ) array power gain. Additionally, the proposed receiver is compared with earlier studies that employ maximum ratio 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.

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22 Introduction

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

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Link¨oping Studies in Science and Technology Licentiate Theses, Division of Communication Systems

Department of Electrical Engineering (ISY) Link¨oping University, Sweden

Erik Axell, Topics in Spectrum Sensing for Cognitive Radio, Thesis No. 1417, 2009. Johannes Lindblom, Resource Allocation on the MISO Interference Channel, Thesis No. 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 in Wireless Systems, Thesis No. 1494, 2011.

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

AntoniosPitarokoilis OntheperformanceofMassiveMIMOsystemswithsinglecarriertransmissionandphasenoise

On the performance of Massive

MIMO systems with single carrier

transmission and phase noise

Antonios Pitarokoilis

Abstract

Acknowledgments

Contents

I

Introduction

1

II

Included Papers

29

Part I

Introduction

1

The Intersymbol Interference Channel

1.1

Detection in ISI channels

2

Multi-user MIMO Channels with Intersymbol

Interference

2.1

MIMO Broadcast Channel

2.2

MIMO Multiple Access Channel

2.3

Channel Estimation in MU-MIMO Channels

3

Phase Noise

3.1

Time-Domain Characterization of Phase Noise

3.2

Frequency-Domain Characterization of Phase Noise

3.3

Previous Work on Phase Noise Channels

4

Contributions of the Thesis

Paper A On the Optimality of Single-Carrier Transmission in

Large Scale Antenna Systems

Paper B Uplink Performance of Time-Reversal MRC in

Mas-sive MIMO Systems subject to Phase Noise

Paper C Achievable Rates of ZF Receivers in Massive MIMO

with Phase Noise Impairments

Bibliography