Pilot Power Control and Assignment Games in MU-MIMO Systems

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IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2018,

Pilot Power Control and Assignment Games in MU- MIMO Systems

ZHONGYAN BI

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

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Pilot Power Control and Assignment Games in MU-MIMO Systems

ZHONGYAN BI

Master Thesis

Date: August 11, 2018 Supervisor: Gábor Fodor Examiner: Mikael Johansson

KTH School of Electrical Engineering and Computer Science

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iii

Acknowledgement

First of all, I would like to thank my supervisor Prof. Gábor Fodor who provided me with the opportunity to do this project. Through the process of researching and writing this thesis, he guided me to the right track and made many precious suggestions which helped to improve the results. It is my honour to work with him.

I would also like to thank all the people in the Network and Sys- tems Engineering department where I sat for last six months. Espe- cially, I want to thank Peiyue Zhao who helped me to settle down in the beginning.

Finally, I want to express my profound gratitude to my parents and family members for supporting and encouraging me throughout my years of study.

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iv

Abstract

In this work, we consider a MU-MIMO communication system for both single cell and multi-cell cases. Other than the fundamental Rayleigh channel model which has been analysed intensively in previous works, we also adopt the Rician channel to model different types of wireless links. Then we study the impact of data and pilot power on the system performance in a single cell for both channel models. The simulation results show that the characteristics of these two channel models are similar. Therefore, we modify an existing power allocation algorithm and implement it. The characteristics of a small scale multi-cell system are also investigated. In addition, based on the derived channel estimation error formulations, a pilot sequence assignment game is proposed. The simulation results show that the algorithm can mitigate the pilot contamination and has a fast converge speed.

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v

Sammanfattning

I detta arbete, anser vi en MU-MIMO kommunikationssystem för både enkla cell och multicellfall. Annat än det grundläggande Rayleigh ka- nalmodell som har analyserats intensivt i tidigare verk, vi också anta Rician kanal för att modellera olika typer av trådlösa länkar. Då stu- derar vi effekterna av data och pilotstyrka på systemet prestanda i en enda cell för båda kanalmodellerna. Simuleringen resultaten visar att egenskaperna hos dessa tvåkanalmodeller är liknande. Därför ändrar vi en befintlig algoritm för algoritmer och implementera det. Egen- skaperna hos ett småskaligt multicellsystem undersöks också. Dess- utom baseras på den härledda kanaluppskattningen felformuleringar, ett pilot-sekvensuppdragsspel föreslås. Simuleringsresultaten visar att algoritmen kan mildra pilotföroreningen och har en snabb konverge- ringshastighet.

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Chapter 1 Introduction

1.1 Background

With the advent of 5G technology, Vehicle-to-Everything (V2X) communication has been made possible. Vehicular communication systems enable services that help intelligent transportation systems to en- hance traffic safety, traffic management, reduce fuel consumption and become an integral part of smart city solutions. Furthermore, due to the high mobility of vehicles, a great number of V2X-specific requirements have to be satisfied. Maintaining end-to-end latencies below a predefined threshold, providing ultra-high reliability to the mobile stations (MSs), or handling a large density of connected vehicles pro- vide examples on such requirements.

When vehicles are equipped with multiple transmit and receive antennas, they maintain communication links with both the cellular and road side infrastructure and with other vehicles and vulnerable road users. Indeed, multiple-antenna vehicles use multiuser multiple input multiple output (MU-MIMO) technologies as a prime enabler for spectral and energy efficient communication with one another and with cellular base station (BS). In the uplink of MU-MIMO systems, the BS typically acquires channel state information (CSI) by means of uplink pilot or reference signals that are orthogonal in the code domain.

In Long Term Evolution (LTE) systems, for example, MSs, including vehicles, use cyclically shifted Zadoff-Chu sequences to form demod- ulation reference signals allowing the BS to acquire channel state information at the receiver (CSIR), which is necessary for uplink data reception [1]. In general, in systems employing pilot aided channel es-

1

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2 CHAPTER 1. INTRODUCTION

timation, the number of pilot symbols and the PDPR play a crucial role in optimizing the system performance in terms of spectral and energy efficiency. As the number of antennas and the number of simultaneously served MSs by a single BS increases, employing decentralized power allocation algorithms for MU-MIMO systems becomes important, since they can reduce the required processing power in a fashion that is scalable with the number of antennas.

In addition, small cells become a crucial component of 5G net- works, since they have the ability to significantly increase network ca- pacity, density and coverage, especially in urban environments where the coverage of a BS is constrained by the buildings and other construction. In small cell systems, from an overall system performance point of view, coordinating and managing highly mobile MSs and ac- quiring CSI in the multi-cell dense system becomes non-trivial.

1.2 Related Work

Recognizing the impact of pilot and data signals on the system performance, several research works have been dedicated to this area. In the seminal work by [2], the difference between the mutual information when the receiver has only a channel estimation and when it has perfect knowledge of the channel is evaluated. The work reported in [3] reveals the optimal number of training symbol and its relation to the power allocation. Furthermore, it also shows that training-based schemes can be optimal at high signal-to-noise ratio (SNR), but sub- optimal at lower SNR. Subsequently, in [4] and [5], the trade off between pilot and data power with different pilot patterns and receiver structures is studied. The results of [5] indicate that the optimal PDPR provides 2-3 dB gain compared with the equal power for pilot and data symbols. Aiming at minimizing the mean squared error (MSE) of received data symbol, the work of [6] reveals the impact of power allocation on system performance. Then in its sequential work [7], a closed-form expressions for achieved MSE and Squared Error (SE) using different channel estimation techniques and receivers, accounting for PDPR and antenna correlation, is derived.

Along another line of the research, there is an increasing interest in decentralized optimization schemes for multi-user (MU)-multiple input multiple output (MIMO) systems. A number of papers pro-

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CHAPTER 1. INTRODUCTION 3

posed a game theoretic approach for power control and resource allocation in MU-MIMO system in which performance coupling exists among the users. A number of related works focused on this area, that address various technical aspects of Single-User Multiple Input Multiple Output (SU-MIMO) and MU-MIMO systems, including spec- trum management [8], channel estimation, data reception, power control, resource allocation [9], Quality of Service (QoS) management [10], device-to-device communications [11], [12] and aspects related to PDPR setting. Reference [13] models a Gaussian interference relay game, in which instead of allocating the power budget across the set of sub- channels, each player aims at decide the optimal power control strat- egy over across a set of hops. For cooperative cognitive radio net- works, a coalitional game theoretic approach is proposed in [14]. A non-cooperative feedback-rate control game with pricing is considered in [15], as a model of the downlink transmission of a closed- loop wireless network, in which a multi-antenna BS utilizes CSI feedback to properly set linear precoders to communicate with multiple users. Game theory is also applied to decentralize an energy efficiency optimization problem subject to signal-to-interference-plus-noise ratio (SINR) and power constraints of each user for a multi-cell system in [16]. In [17], a decentralized PDPR allocation algorithm which aims at minimizing MS’s individual MSE of received data symbol, assuming imperfect CSI and implementing MMSE receiver based on game theory is proposed.

For multi-cell systems, most literature mainly focuses on pilot signal decontamination. Reference [18] studies the fundamental problem of pilot contamination in multi-cell systems and develops a new multi-cell MMSE-based precoding method that mitigates that problem. In [19], a novel channel estimation scheme for multi-cell system with Rician fading channel model is proposed. Assuming the number of antennas at BS is large, this estimation scheme utilizes the pilot contamination free property of Line of Sight (LOS) component of channel and uses LOS component to detect data.

1.3 Objective

The main objective of the present thesis project is to develop algorithms for power allocation in single cell MU-MIMO systems, and for

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4 CHAPTER 1. INTRODUCTION

pilot sequence assignment in multi-cell MU-MIMO system. Since previous works have already studied the case where the channel is assumed to be Rayleigh fading channel intensively, in this project we will focus on the general Rician fading channel. To verify the oper- ation of the proposed algorithms and to obtain engineering insights in various use cases, we will use Monte Carlo simulations to examine and evaluate the performance of these algorithm.

1.4 Thesis Outline

This thesis is divided into 5 chapters. Chapter 2 explains how the MU-MIMO system and the wireless communication channels are modelled. It also discusses important assumptions that we use throughout the whole project. Chapter 3 presents how the channel estimation and receiver construction are done for the single cell system. Furthermore, a centralized power control algorithm and a decentralized power control algorithm are developed. Chapter 4 focuses on studying the multi- cell MU-MIMO system, and explains the approach of a pilot sequence assignment algorithm. Chapter 5 draws conclusions, and discusses future work.

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Chapter 2 System and Channel Model

2.1 System Model

In this project, we consider the uplink transmission of a multi-antenna wireless system, in which users are scheduled on orthogonal frequency channels and transmit their signals simultaneously. This is a common assumption in massive MU MIMO system in which a single MS may have a single antenna [20]. The uplink transmission consists of two stages: training stage and data transmission stage. In the training stage, the MS transmits a pilot signal to the BS such that the BS can use that signal to estimate the state information of the channel between that MS and itself in each coherence time interval. In the data transmission stage, the BS receives the data signal from all the users and implements receivers which are constructed based on the CSI that it derived during the training stage to estimate the data symbols that each MS transmits. We also suppose that in our scenario the communication bandwidth is much smaller than the reciprocal of the delay spread, such that the channel is non-frequency selective.

There are three different pilot arrangements: (1) block type arrangements which dedicates all frequency channels within a given time slot to either channel estimation or data transmission, (2) comb pilot pat- tern employs pilot and data symbol mixed in the frequency domain within a single time slot and (3) mixed type arranges the pilot and data subcarriers discontinuously in time and frequency domains [21].

These three patterns are shown in the following figure.

5

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6 CHAPTER 2. SYSTEM AND CHANNEL MODEL

Figure 2.1: Different pilot patterns

We focus on the comb type pilot arrangement in this project, since it is a suitable for non-frequency selective channels [21]. Specifically, given F subcarriers in the coherence bandwidth, a fraction of τp subcarriers are assigned to pilot and τd= F −τ_pare used for data transmission. The power that the MS can use is constrained at a constant value P_tot, but it can be distributed unequally to pilot and data subcarriers which can be expressed as

τ_pP^p+ τ_dP = P_tot, (2.1) where P^pand P are the power distributed to pilot and data subcarriers respectively and 1 ≤ τp, τ_d < F.

2.2 Channel Model

The channel is characterized by two sets of parameters, such as (1) the large scale fading parameters which include Pathloss (PL) and shadowing, since these parameters change slowly and thus can be ac- quainted by BSs (the study of which is out of the scope of this project), and (2) the small scale fading parameters which capture the effects of the additive and subtractive interference of multiple signal paths. We refer to this as small scale fading and its associated parameters in the

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CHAPTER 2. SYSTEM AND CHANNEL MODEL 7

sequel. Figure 2.2 illustrates the impact of the channel on the signal power as a MS moves away from the BS.

Figure 2.2: The effect of PL, shadowing and multipath on signal power Since, according to the assumption, the communication bandwidth is much smaller than the reciprocal of the delay spread, the complex base band channel can be represented by a single tap at each time [22].

The two simplest probabilistic channel model that can be used in our MIMO system are: (1) the Rayleigh fading channel, and (2) the Rician fading channel. Depending on the considered scenarios and prevail- ing channel conditions, different channel models can be more appro- priate.

2.2.1 Rayleigh Fading Channel

In the Non-Line of Sight (NLOS) case, there are a larger number of statistically independent reflected and scattered paths with random amplitudes in the delay window corresponding to a single tap [22].

Therefore, it is reasonable to assume that the channel coefficients h are circularly symmetric complex Gaussian random variables

h ∼ CN (0, C). (2.2)

Since the power of channel is normalized to one in our system model, the covariance matrix C is set to identity matrix IN where N is the number of antennas. As the power of the channel in this form ||h|| follows the Rayleigh distribution, it is referred to as the Rayleigh fading channel.

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2.2.2 Rician Fading Channel

In the LOS case, other than random scattered and reflected component from NLOS paths, there is a specular path which has a known and constant magnitude. For single antenna case, this type of model can be modelled as Rician fading channel as follow.

h =

r K

K + 1σ_le^jθ

| {z }

hLOS

+

r 1

K + 1CN (0, σ_l²)

| {z }

hN LOS

=

r K

K + 1σ_l(cos θ + j sin θ) +

r 1

K + 1(x + jy)

=

r K

K + 1σlcos θ +

r 1

K + 1x + j(

r K

K + 1σlsin θ +

r 1

K + 1y), (2.3) where K is the Rician factor indicating the ratio between the power of LOS and NLOS components and θ is the phase of the LOS component.

The power of the LOS and NLOS component are P_LOS = E{hLOSh^∗_LOS} = K

K + 1σ_l²cos²θ + K

K + 1σ_l²sin²θ = K K + 1σ²_l,

(2.4) P_{N LOS} = E{hN LOSh^∗_{N LOS}} = 1

K + 1E{x²} + 1

K + 1E{y²} = 1 K + 1σ_l².

(2.5) The total power of the channel is

P = P_LOS+ P_{N LOS} = σ_l², (2.6) and it can be deduced that the parameters for the Rician distribution are

ν =

r K

K + 1σ_l, σ = s

1

2(K + 1)σ_l (2.7) . Thus the channel h derived from equation (1) follows |h| ∼ Rice(ν, σ) and its expected value is

E{h} = ν cos θ + jν sin θ. (2.8) The channel also follows the complex normal distribution

h ∼ CN (ν cos θ + jν sin θ, Γ), (2.9)

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CHAPTER 2. SYSTEM AND CHANNEL MODEL 9

where Γ is the covariance,

Γ =Cov{h} = E{(h − ¯h)(h − ¯h)^∗} = 2σ². (2.10) In our system model, the power of the channel is normalized to unit, and therefore we have σl = 1. Thus, assuming the pseudo- covariance of the Rician channel C to be 0 and given the covariance of the channel Γ, the parameters of Rician fading channel can be determined by σ =

qΓ

2, ν = √

1 − Γ. When ν = 0, the LOS component disappears and the channel turns to Rayleigh fading channel, which implies that the Rayleigh channel is a special case of Rician channel.

In the case when the system has multiple receiver antennas, the Rician channel can be modelled in the following way as shown in [19],

h(θ, d, N, K, C) =

r K

K + 1h_LOS+

r 1

K + 1h_{N LOS}, (2.11) where

h_LOS =1, e^{j2πd cos θ}, ..., ej2π(N −1)d cos θ^T

, (2.12)

in which θ is the angle of arrival of the LOS component and d is the antenna spacing in the wavelength. And hN LOSis a column of complex Gaussian random variable, following hN LOS ∼ CN (0, C). The Rician channel follows the complex normal distribution as in:

h ∼ CN (

r K

K + 1hLOS, Γ), (2.13) where Γ = _K+1¹ C.

2.2.3 Channel Model for Different Scenario

In this project, we study both single and multi-cell systems. As the geometric size of the system is different, the type of link (i.e. the wireless communication channel) between MSs and BSs also gets affected.

From the 3GPP technical report [23], we know that the LOS probability functions for Urban Micro (UMi) and Urban Macro (UMa) scenario are:

Pr^UMi_LOS =

1 d_2D ≤ 18m

18

d_2D + (1 − _d¹⁸

2D)exp(−^d₃₆^2D) d_2D > 18m, Pr^UMa_LOS =

( 1 d_2D ≤ 18m

18

d_2D + (1 − _d¹⁸

2D)exp(−^d₃₆^2D)

1 + 1.25C⁰(h_UT)(^d₁₀₀^2D)³exp(−^d₁₅₀^2D)

d_2D > 18m,

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where

C⁰(h_UT) =

( 0 h_UT ≤ 13m

h_UT−13 10

1.5

13m < hUT ≤ 23m.

Then we can plot the LOS probability with the 2D distance between BS and MS when setting the height of the BS’s antennas hUT to 20m.

Form the above figure, we can see that the LOS probability decreases dramatically with the increase of the distance between BS ans MS in both scenarios. Especially in the UMi scenario, where the LOS possi- bility drops to 10% when MS is only 200m away from BS. Therefore, it is reasonable to make the assumption that the type of link within a cell is always LOS, while the link between the cells is always NLOS.

Figure 2.3: The LOS probability vs the 2D distance between BS and MS (m) when the height of BS’s antennas set at 20m.

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Chapter 3 Power Control and Game in Sin- gle Cell System

3.1 Channel Estimation

Based on the assumption made above, we can now model the signal that the BS receives. For single cell, the pilot signal model transmitted by MS k is

Y_κ = α_κp

Pκ^ph_κs^T_κ + N, (3.1) where αk accounts for the PL in linear unit, and ακ = 10^−PL^κ^/10. sκ = [s₁, ..., s_τ_p]^T is the pilot sequence assigned to MS k, and each symbol of it is scaled as |si|² = 1, ∀i. N ∈ C^N^r^×τ^p denotes the spatially and temporally additive white Gaussian noise (AWGN) with element-wise variance σ²_p. Note that, for all MSs in the single cell case, their assigned pilot sequences are orthogonal to each other. We use two estimation techniques to estimate the channel hκ, i.e., the LS and the MMSE channel estimation.

3.1.1 LS Estimation

Utilizing the pilot sequence orthogonality, the BS can estimate the channel for each MS based on (3.1) assuming:

h =ˆ 1 α√

P^pYs^∗(s^Ts^∗)⁻¹ = h + 1 α√

P^pτ_pNs^∗, (3.2) where (·)^∗denotes the conjugation, and the index κ is dropped for sim- plicity. By considering the Rician fading channel h ∼ CN (q

K

K+1h_LOS, Γ),

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12 CHAPTER 3. POWER CONTROL AND GAME IN SINGLE CELL SYSTEM

the estimated channel ˆhfollows

h ∼ CN (¯ˆ h, R_LS), (3.3) where ¯h = E{h} =q

K

K+1h_LOS and RLS= Γ +_Pp^σα^p²²τpI. From the above, the channel estimation error w = h − ˆhfollows

w ∼ CN (0, C_w), (3.4)

where Cw = ^σ

2 p

P^pα²τpI. The covariance of the estimation error w is only the function of pilot power P^p which implies that the covariance matrix of channel Γ does not affect the estimation accuracy. It can be seen that ˆh and h are jointly Gaussian, then we are able to compute the following conditional distributions.

Given a random channel realization h, the distribution of the estimated channel conditioned to h can be shown to be

(ˆh|h) ∼ CN (h, σ_p²

P^pα²τ_pI), (3.5) and the distribution of channel h conditioned to a channel estimation hˆ can be derived as:

(h|ˆh) ∼CN (¯h + Γ(Γ + σ²_p

α²P^pτ_pI)⁻¹(ˆh − ¯h), Γ − Γ(Γ + σ_p²

α²P^pτ_pI)⁻¹Γ)

∼CN (ΓR⁻¹_LSh + (I − ΓRˆ ⁻¹_LS)¯h, Γ − ΓR⁻¹_LSΓ).

(3.6) Proof. According to [24], we know that for jointly Gaussian distributed variables, the conditional distribution ˆh|his still Gaussian, and it follows that

E{h|h} = E{ˆˆ h} + Chhˆ C⁻¹_hh(h − E{h}) = ¯h + ΓΓ⁻¹(h − ¯h) = h, Cˆh|h= Chˆˆh− Chhˆ C⁻¹_hhC_hˆ_h = Γ + σ_p²

α²P^pτ_pI − ΓΓ⁻¹Γ = σ_p² α²P^pτ_pI.

Similarly, the conditional distribution h|ˆh is also Gaussian and it follows that

E{h|h} = E{h} + Cˆ hˆhC⁻¹_ˆ

hˆh

ˆh − E{ˆh}

= ¯h + Γ(Γ + σ_p²

α²P^pτ_pI)⁻¹(ˆh − ¯h), C_h|ˆ_h = C_hh− C_hˆ_hC⁻¹_ˆ

hˆhC_ˆ_hh = Γ − Γ(Γ + σ²_p α²P^pτp

I)⁻¹Γ.

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CHAPTER 3. POWER CONTROL AND GAME IN SINGLE CELL SYSTEM 13

3.1.2 MMSE Channel Estimation

For the MMSE channel estimation, the received signal model can be transferred into

Y = αpP˜ _pSh + ˜N, (3.7)

where S = s ⊗ INr ∈ C^τ^p^N^r^×N^r, ˜Y =vec(Y) and ˜N =vec(N) ∈ C^τ^p^N^r^×1. After transferring the pilot signal (3.1) into vector form (3.7), the stan- dard linear MMSE estimator [25] can be implemented, and it is

h = Cˆ _{h ˜}_YCY˜

−1( ˜Y − E{ ˜Y}) + E{h}. (3.8) Substituting C_{h ˜}_Y = α√

P ΓS^H and CY˜ = α²P SΓS^H+ σ²_pIinto (3.8), we have

h = (ˆ σ_p²

α²P^pτ_pI + Γ)⁻¹Γ (h − ¯h) + 1 α√

P^pτ_pS^HN˜

!

+ ¯h. (3.9)

Note that S^HN ∼ CN (0, τ_pσ_p²I), thus the channel estimation ˆhis also a complex normal distributed vector which follows

h ∼ CN (¯ˆ h, R_MMSE). (3.10) The covariance matrix of the channel estimation ˆhis

R_MMSE = Γ²( σ²_p

α²P^pτ_pI + Γ)⁻¹, (3.11) where we consider covariance matrix Γ is Hermitian and applied the commutativity of Γ and I to substitute

( σ²_p

α²P^pτ_pI + Γ)⁻¹Γ = Γ( σ_p²

α²P^pτ_pI + Γ)⁻¹. The channel estimation error w = ˆh − hfollows

w ∼ CN (0, C_w), (3.12)

where

Cw = ( ˜R⁻¹_MMSEΓ − I)Γ( ˜R⁻¹_MMSEΓ − I)^H + σ²_p α²P^pτ_p

R˜⁻¹_MMSEΓ²R˜⁻¹_MMSE,

in which ˜R_MMSE = Γ + ^σ

2p

Ppα²τpI.

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Then, analogously to the LS estimation, the distribution of channel hconditioned to a channel estimation ˆhfollows

E{h|h} = E{h} + Cˆ hˆhC⁻¹_ˆ

hˆh(ˆh − E{ˆh}), (3.13) C_h|ˆ_h = C_hh− C_hˆ_hC⁻¹_ˆ

hˆhChhˆ , (3.14) where

C_hˆ_h = Chˆˆh = Chhˆ = R_MMSE, (3.15) thus

(h|ˆh) ∼ CN (ˆh, Γ − R_MMSE). (3.16)

3.2 Linear Receiver

The MU-MIMO received data signal at the BS for single cell can be written as:

yκ = ακ

pPκhκxκ+

K

X

k6=κ

αk

pPkhkxk+ nd, (3.17)

where ndis the noise on the received data signal, K is the total number of MSs in the cell and we suppose that the data symbol x is random and normalized so that |x|² = 1. Then the naive receiver G^naiveis

G^naive_κ = α_κp

P_κhˆ_κ(α²_κP_κhˆ_κhˆ^H_κ +

K

X

k6=κ

α²_kP_kC_k+ σ_d²I)⁻¹, (3.18)

where Ck = E{hkh^H_k} = Γ_k+ E{hk}E{hk}^H.

Proof. The naive receiver is derived under the assumption that the perfect CSI is available. In other words we suppose that hκ is known to the BS. The goal is to minimize the estimation error between the estimated signal Gκy_κ where Gκ is the receiver and the transmitted data signal xκ, and the problem can be written as

minimize

G E{|Gκy_κ− x_κ|²}. (3.19) The objective function V can also be written as

N →∞lim 1

N(GYκ− x_κ)(GYκ− x_κ)^H,

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where Yκ is the realization matrix (y⁽¹⁾κ , ..., y^{(N )}κ )and xκ is the realization row vector (x⁽¹⁾κ , ..., x^{(N )}κ ). The optimal estimator can derived by solving ∂V /∂G = 0,

∂V

∂Gκ

= 1 N

∂

∂Gκ

(GκY_κY_κ^HG^H_κ − 2x_κY^H_κG^H_κ + x_κx^H_κ)

= −2

Nx_κY_κ^H + 2

NGκY_κY_κ^H = 0, and the optimal estimator Gκis given by

Gκ = lim

N →∞( 1

Nx_κY^H_κ)(1

NY_κY_κ^H)⁻¹, (3.20) where

N →∞lim 1

Nx_κY_κ^H = α_κp

P_κh^H_κcov(xκ) = α√

Ph^H, (3.21) and

N →∞lim 1

NY_κY_κ^H = α_κ²P_κhκh^H_κ +

K

X

k6=κ

α²_kP_k(Γ_k+ E{hk}E{hk}^H) + σ_d²I.

(3.22) By substituting (3.21) and (3.22) into (3.20), the optimal receiver which gives MMSE is

G_κ = α_κp

P_κh^H_κ(α²_κP_κh_κh^H_κ +

K

X

k6=κ

α²_kP_k(Γ_k+ E{hk}E{hk}^H) + σ_d²I)⁻¹. (3.23) Assuming the channel estimate ˆhκ is the actual channel and letting C_k = E{hkh^H_k} = Γ_k+ E{hk}E{hk}^H, the naive MMSE receiver (3.23) turns to

Gκ = α_κp

P_κhˆ^H_κ(α²_κP_κhˆκhˆ^H_κ +

K

X

k6=κ

α²_kP_kC_k+ σ²_dI)⁻¹.

The MMSE receiver minimizes the same objective function as the naive receiver, but it takes into account that it has access only to the estimated channel ˆh_κ. It is given by

G^?_κ = α_κp

P_κu^H_κ α²_κP_κ Q_κ+ u_κu^H_κ +

K

X

k6=κ

α²_kP_kC_k+ σ_d²I

!⁻¹

, (3.24)

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where we let uκ = D_κhˆ_κ+ (I − D_κ)¯h_κ, and

D =ΓR⁻¹_LS for LS channel estimation

IN for MMSE channel estimation, (3.25)

Q =Γ − ΓR⁻¹_LSΓ for LS channel estimation

Γ − R_MMSE for MMSE channel estimation, (3.26) and ¯hκ = E{h^κ} =q

Kκ

Kκ+1hLOS,κ, Ck= E{h^kh_k^H} = Γk+ ¯hkh¯^H_k.

Proof. The MSE of the received data symbol given receiver Gκ, and all the channel realization {hi} is

MSE(Gκ, {h_i}) =Exκ,nd{|G_κy_κ− x_κ|²}

=|α_κp

P_κG_κh_κ− 1|²+

K

X

k6=κ

|α_κP_κG_κh_κ|²+ σ_d²G_κG^H_κ. (3.27) The MSE of the received data symbol given only the receiver Gκ, and the channel realization of user κ hκis

MSE(Gκ, h_κ) =Eh1,...,hκ−1,hκ−1,...,hK{MSE(Gκ, {h_i})}

=α²_κP_κG_κh_κh_κG^H_κ − α_κp

P_κ(G_κh_κ+ h^H_κG^H_κ)+

K

X

k6=κ

α_k²P_kG_κ(Γ_k+ ¯h_kh¯^H_k )G^H_κ + σ_d²G_κG^H_κ + 1.

(3.28)

The MSE of the received data symbol as a function of the estimated channel ˆhκis

MSE(Gκ, ˆh_κ) =Ehκ|ˆhκ{MSE(Gκ, h_κ)}

=α²_κP_κG_κ(Q_κ+ u_κu^H_κ)G^H_κ − α_κp

P_κ(G_κu_κ+ u^H_κG^H_κ) +

K

X

k6=κ

α²_kP_kG_κ(Γ_k+ ¯h_kh¯^H_k)G^H_κ + σ²_dG_κG^H_κ + 1,

(3.29) where we let uκ = D_κhˆ_κ+(I−D_κ)¯h_κ. By solving ∂MSE(Gκ, ˆh_κ)/∂G_κ = 0, we can get the MMSE receiver.

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3.3 A Centralized Power Control Method

Before we start looking into the power control game for the single cell system with the Rician channel model, we first review the fundamental case where the channels are assumed to be Gaussian and the antennas are assumed to be uncorrelated. The characteristics of the system with Rayleigh channel and its decentralized power control method has been intensively investigated and studied in [21], [17], [7] and [6]. In this section, a centralized power control algorithm is proposed.

From [6], we know that the unconditional MSE of the received data symbol of user i when the BS uses the optimal receiver

G_i = αi

√Pidi

α²_iP_i(d²_i||ˆh_i||²+ q_i) +PK

k6=iα²_kP_kc_k+ σ²_d hˆ^H_i

is

MSEi(P) = µ_ie^µⁱE_in(N_r, µ_i), (3.30) where

µ_i = q_iα²_iP_i+PK

k6=iα_k²P_kc_k+ σ_d²

d²_iα²_iP_ir_i . (3.31) And for LS and MMSE channel estimation we have

r^LS_i = c_i+ σ²_p

α_i²(P_tot− τ_dP_i), r^MMSE_i = c²_i

σ_p²

α²_i(P_tot− τ_dP_i) + c_i

⁻¹

and the parameter qi, diare

d_i =

(c/r^LS_i for LS estimation 1 for MMSE estimation,

q_i =

(c_i− c²_i/r^LS_i for LS estimation c_i− r_i^MMSE for MMSE estimation, and we also assume that σd= σ_pand drop the index.

We are now ready to study the centralized power control optimization problem, in which we want to minimize the maximum of the MSE over all the users in a single cell. Note that BPA [17] focuses on minimizing MSs’ individual MSE, since it is a non-cooperative game. Note

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that this problem ignores fairness constraints, which aims at providing similar quality of service characteristics to all MSs in the cell.

minimize

{P_i} max

i {MSEi(P)}

s.t. 0 < P_i < P_tot τ_d , ∀i τ_dP_i+ τ_pP_i^p = P_tot, ∀i.

(3.32)

From [7], we know that ∂MSE(µi)/∂µ_i is positive for all µi > 0 which is always satisfied in our case. This implies that the system performance metric MSEi is an monotonic increasing function in µi, thus it is equivalent to choose the objective function as max

i {µ_i(P)}

and the original optimization problem (3.32) can be converted to minimize

{Pi} max

i {µ_i(P)}

s.t. 0 < Pi < P_tot τ_d , ∀i τ_dP_i+ τ_pP_i^p = P_tot, ∀i.

(3.33)

We observe that when the system uses MMSE receiver with either MMSE or LS channel estimation, the µican be written as

µ_i = q_iα²_iP_i+PK

k6=iα²_kP_kc_k+ σ²_d

d²_iα²_iP_ir_i = −1 + PK

k=1α²_kP_kc_k+ σ²_d

P_iγ_i , (3.34) where

γ_i = α²_ir_i = α²_ic²_i c_i+_α2^σ²^p

iτpP_i^p

= α⁴_iτ_pP_i^pc²_i

α²_iτpP_i^pci+ σ_p². (3.35) This result can be easily proven. For MMSE channel estimation case, di = 1and qi = c_i− r_i, thus we have

µ_i =q_iα²_iP_i+PK

k6=iα_k²P_kc_k+ σ_d² d²_iα_i²P_ir_i

=q_iα²_iP_i+PK

k6=κα_k²P_kc_k+ σ_d²+ α²_iP_ic_i− α²_iP_ic_i d²_iα²_iP_ir_i

=(q_i− c_i)α_i²P_i+PK

k=1α²_kP_kc_k+ σ_d² d²_iα²_iP_ir_i

=−r_iα²_iP_i+PK

k=1α²_kP_kc_k+ σ_d² d²_iα²_iP_κr_i

= − 1 + PK

k=1α²_kP_kc_k+ σ_d² P_iα_ir_i .

(3.36)

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Similarly, for LS channel estimation, di = c_i/r_i and qi = c − c²_i/r_i. Sub- stituting them into the third line of equation (3.36), the result follows.

As a by-product, we find that for this specific system setup, the performance of the MMSE receiver does not depend on the channel estimation technique the BS uses.

Then the problem (3.32) is equivalent to

minimize

{P_i} max

i

(PK

k=1α²_kP_kc_k+ σ_d² P_iγ_i

)

(3.37)

Taking the reciprocal of the objective function of (3.37), the problem (3.32) is also equivalent to

maximize

{P_i} min

i

( P_iγ_i PK

k=1α²_kP_kc_k+ σ_d² )

which is in the same form of the Max-Min SINR for Maximum Ratio Combining (MRC) optimization problem in [26]. Thus the method, lemmas and propositions used to solve that problem can be also implemented to this problem. First, we implement lemma that at optimal point all MSs have same MSE value. Since the denominator is same for all the users, at optimal point we could let

Piγi = x, ∀i. (3.38)

Substituting equation (3.35) into (3.38), we get P_iα⁴_i(P_tot− τ_dP_i)c²_i

α²_i(P_tot− τ_dP_i)c_i+ σ_p² = x. (3.39) Writing it in quadratic form, we have

c_i²α_i⁴τ_dP_i²− (c_ixα²_iτ_d+ c_i²α_i⁴P_tot)P_i+ (c_iα²_iP_tot+ σ²_p)x = 0, (3.40)

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and the solution of this quadratic equation given x is

P_i =

c_ixα_i²τ_d+ c²_iα⁴_iP_tot±q

(c_ixα²_iτ_d+ c_i²α⁴_iP_tot)²− 4c²_iα⁴_iτ_d(c_iα_i²P_tot+ σ²_p)x 2c²_iα⁴_iτ_d

=

xτd+ ciα²_iPtot ±q

x²τ_d²+ c²_iα⁴_iP_tot² − 2τdx(ciα_i²Ptot+ 2σ_p²) 2ciα²_iτd

.

(3.41) If the roots (3.41) are real-valued, the sum and products of the roots are positive, therefore both roots are positive. When Piγ_i is fixed, smaller P_i gives lower MSE. Therefore, the optimal solution given x is

P_i =

xτ_d+ c_iα²_iP_tot−q

x²τ_d²+ c²_iα⁴_iP_tot² − 2τ_dx(c_iα²_iP_tot+ 2σ²_p)

2c_iα²_iτ_d . (3.42)

Substituting equation (3.42) into the objective function of problem (3.37), the problem (3.37) can be reduced to

minimize

x

σ²_d

x + P_tot 2xτ_d

K

X

i=1

c_iα²_i − 1 2τ_d

K

X

i=1

s

τ_d²+c²_iα⁴_iP_tot²

x² − 2τ_d(c_iα²_iP_tot+ 2σ²_p) x

s.t. x ∈

K

\

i=1

Range {Piγ_i} ,

(3.43) where x is constrained to be achievable for all users. By substituting ¹_x with y, we get

minimize

y y(σ_d²+P_tot 2τ_d

K

X

i=1

c_iα²_i) − 1 2τ_d

K

X

i=1

q

τ_d²+ c²_iα⁴_iP_tot² y²− 2τ_d(c_iα_i²P_tot+ 2σ_p²)y

s.t. y ∈

K

\

i=1

Range

1 Piγi

.

(3.44) The first term of the objective function is linear, and the second term is concave in y ≥ 0 element-wisely if

2τ_d(c_iα²_iP_tot+ 2σ²_p)2

− 4c²_iα⁴_iP_tot² τ_d² = 16τ_d²(c_iα²_iP_totσ²_p+ σ_p⁴) > 0, ∀i, (3.45)

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which is always satisfied. Since the nonnegative weighted sum of concave functions is still concave [27], the second term is strictly concave and thus the negative of it is strictly convex. The sum of an affine function and a convex function is still convex, therefore we can find the unique and optimal y^∗ by setting the first derivative of the objective function to zero, which yields a monotonically increasing function f (y)

f (y) = σ_d²+P_tot 2τ_d

K

X

i=1

c_iα_i²− 1 2τ_d

K

X

i=1

c²_iα⁴_iP_tot² y − τ_d(c_iα²_iP_tot+ 2σ_p²) q

τ_d²+ c²_iα⁴_iP_tot² y² − 2τ_d(c_iα²_iP_tot+ 2σ²_p)y

= 0.

(3.46) And the root of this equation can be derived numerically. We choose to use bisection method for which the bounds of y have to be determined first. We can show that _P¹

iγi is convex in Pi within (0, Ptot/τd).

Proof. The variable for each MS can be written as (3.47) 1

P_iγ_i = 1

α²_ic_iP_i + σ²_p

α⁴_ic²_i(P_tot− τ_dP_i)P_i. (3.47) We observe that the first term is convex, and the denominator of second term is concave and positive which implies that the second term is convex. Thus the convexity of 1/Piγ_ifollows.

Since y is convex in Pi, y has only lower bound and it can be derived by setting the first order derivative of 1/Piγ_i to zero, and for each MS i the data power corresponding to the lowest value is

P_i =

P_totα²_ic_i+ σ_p²− σ_pq

σ²_p+ P_totα²_ic_i

α_i²c_iτ_d . (3.48)

Substituting it into _P_i¹_γ_i, we can get the lower bound of y

y_l =max

i





 τ_d

2σ_pq

(σ_p²+ P_totα²_ic_i)³+ P_tot² α_i⁴c²_i + 2σ_p⁴+ 3P_totα²_ic_iσ_p² P_tot² α⁴_ic²_i(σ_p²+ P_totα²_ic_i)





 . (3.49) Therefore, using the bisection method to find the optimal y^∗ is viable.

The following pseudo code illustrates the whole procedure of solving the original Min-Max MSE optimization problem. The algorithm

Pilot Power Control and Assignment Games in MU-MIMO Systems

Pilot Power Control and Assignment Games in MU- MIMO Systems

ZHONGYAN BI

Pilot Power Control and Assignment Games in MU-MIMO Systems

ZHONGYAN BI

Acknowledgement

Abstract

Sammanfattning

Contents

Chapter 1 Introduction

1.1 Background

1.2 Related Work

1.3 Objective

1.4 Thesis Outline

Chapter 2

System and Channel Model

2.1 System Model

2.2 Channel Model

2.2.1 Rayleigh Fading Channel

2.2.2 Rician Fading Channel

2.2.3 Channel Model for Different Scenario

Chapter 3

Power Control and Game in Sin- gle Cell System

3.1 Channel Estimation

3.1.1 LS Estimation

3.1.2 MMSE Channel Estimation

3.2 Linear Receiver

3.3 A Centralized Power Control Method