大型天線陣列系統的秩值與複合通道估計及閉迴路傳收機之設計

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國立交通大學

電信工程研究所

碩士論文

大型天線陣列系統的秩值與複合通道估計

及閉迴路傳收機之設計

Rank Determination, Composite Channel Estimation and

Closed-loop Transceiver Design for Massive MIMO Systems

研究生：陳科夆

指導教授：蘇育德教授

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大型天線陣列系統的秩值與複合通道估計

及閉迴路傳收機之設計

Rank Determination, Composite Channel Estimation and

Closed-loop Transceiver Design for Massive MIMO Systems

研究生：陳科夆 Student：Ko-Feng Chen

指導教授：蘇育德 Advisor：Dr. Yu T. Su

國立交通大學

電信工程研究所

碩士論文

A Thesis

Submitted to the Institute of Communications Engineering College of Electrical and Computer Engineering

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master of Science

in Communications Engineering

July 2013 Hsinchu, Taiwan

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大型天線陣列系統的秩值與複合通道估計及閉迴路傳收機之設計

學生：陳科夆

指導教授：蘇育德

國立交通大學

電信工程研究所碩士班

摘

要

我們探討一個裝有大型天線陣列基地台(BS)來服務多個單天線用戶(UE)的

多輸入多輸出(MIMO)分時多工系統。此系統藉由上行(UL)的領航信號(pilot)來估

計包含大尺度衰減係數(LSFC)及小尺度衰減係數(SSFC)的通道狀態資訊(CSI)。

雖然在MU-MIMO或是分散式MIMO系統之運作，LSFC資訊是不可或缺的，然而

有關MIMO通道估計之研究往往被假設為已知或是被忽略。我們利用大型天線陣

列之通道硬化(channel-hardening)並能同時收到大量空間樣本的特性，在不需

SSFC資訊的前提下能有效的利用相對少量的領航符元精準地估計LSFC。

至於SSFC的估計，我們利用降秩(rank-reduced, RR)通道模型來完成。由於這

種方法之降秩效應需選擇適當的基底並有準確的秩值估計，後者又需先知道通道

的空間相關矩陣。針對這三項議題我們首先分析了最佳先設(predetermined)基底

的選擇，證明兩種常用的基底之近優性(near-optimality)。接著我們探討秩值對

SSFC估計的性能影響、設計一套秩值決定的演算法，最後並發展了估計空間相

關矩陣的演算法。這些成果乃是以我們對SSFC估計法的均方誤差(mean squared

error, MSE)性能的詳細分析為基礎。我們結合了大、小尺度衰減係數的估計並證

明在接收信號之入射角度擴散(AS)不大時，還可利用適當的RR模型來一併估計

平均的接收角度(AoA)。比起使用不含AoA資訊的模型之通道估計法，這種方法

不但可降低MSE而且所估得的角度資訊可用來形成下傳鏈路的波束。

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最後，我們推導了上行領航信號的設計方式，並且在含括前述的複合通道估

計器後，提出了分別適用於分時多工和分頻多工模式的閉迴路傳收機(closed-loop

transceiver)設計流程與細部演算法。由電腦實驗結果可以看出我們的複合通道估

計器、秩值決定及空間相關矩陣估計等演算法在大型天線陣列系統中的優異表

現。

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Rank Determination, Composite Channel

Estimation and Closed-loop Transceiver Design

for Massive MIMO Systems

Student : Ko-Feng Chen Advisor : Yu T. Su

Institute of Communications Engineering National Chiao Tung University

Abstract

We consider a multiuser (MU) multiple-input multiple-output (MIMO) time-division duplexing (TDD) system in which the base station (BS) is equipped with a large num-ber of antennas for communicating with single-antenna mobile users. In such a system the BS has to estimate the channel state information (CSI) that includes large-scale fading coefficients (LSFCs) and small-scale fading coefficients (SSFCs) by uplink pi-lots. Although information about the former FCs are indispensable in a MU-MIMO or distributed MIMO system, they are usually ignored or assumed perfectly known when treating the MIMO CSI estimation problem. We take advantage of the large spatial samples of a massive MIMO BS to derive accurate LSFC estimates in the absence of SSFC information and with a training overhead no larger than that required by con-ventional LSFC estimators. With estimated LSFCs, SSFCs are then obtained using a rank-reduced (RR) channel model.

We analyze the mean squared error (MSE) performance of the proposed compos-ite channel estimator and prove that the separable angle-of-arrival (AoA) information provided by the RR model is beneficial for enhancing the estimator’s performance, espe-cially when the mean angle spread of the uplink signal is not too large. To fully exploit

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the RR channel model, we have to select a proper basis and determine associated rank (modeling order). We solve these two issues by developing a rank-determination al-gorithm based on two popular bases and verify the near-optimality through computer simulations. We discuss uplink pilot design and suggest closed-loop transceiver design flows for both TDD and FDD modes using the estimated AoA for downlink beamforming and the LSFC information for power allocation.

Some computer experiment results are provided to validate the efficiencies of the proposed estimators and the rank determination method.

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誌

謝

對於論文得以順利完成，首先感謝指導教授蘇育德博士。老師的諄諄教誨

使我對於通訊領域的研究有更深入的了解，也教導我們許多書上學不到的知識與

人生道理，讓我受益良多。並感謝口試委員蘇賜麟教授、祁忠勇教授、趙啟超教

授及林大衛教授給予的許多寶貴意見，以補足這份論文的缺失與不足之處。

也由衷感謝實驗室的劉彥成學長，在我研究上有問題時，能給予我建議以及

討論，使我的研究能夠順利完成，從中學習到的經驗是很珍貴的。另外也感謝實

驗的學長姐、同學、及學弟妹們，在這兩年內的互相支持與鼓勵。

最後，要感謝我的家人及朋友，他們總是在背後默默的關心與支持，使我有

動力可以努力往前進，在此僅獻上此論文，以代表我最深的敬意。

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List of Tables

3.1 Simulation parameters . . . 30

4.1 Convergence speed and accuracy of proposed IMOD algorithm for

differ-ent choice of η assume DCT-II basis is used and SNR= 10dB. . . 70

4.2 Effect of large system to optimal modeling order found by IMOD

algo-rithm given DCT-II or KLT basis is used. Assume SNR= 10dB, AS= 7.2◦

and η = 1. . . 76

4.3 Effect of large system to optimal modeling order found by IMOD

algo-rithm given DCT-II or KLT basis is used. Assume SNR= 10dB, AS= 15◦

and η = 1. . . 76

4.4 Effect of imperfect spatial correlation matrix estimated by ML estimator

(4.59) to optimal modeling order found by IMOD algorithm when using

KLT, DCT-II or polynomial basis. Assume that SNR= 10dB, AS= 7.2◦,

η = 1 and that there are total n subcarriers. . . 76

4.5 Effect of imperfect spatial correlation matrix estimated by ML estimator

(4.59) to optimal modeling order found by IMOD algorithm when using

KLT, DCT-II or polynomial basis. Assume that SNR= 10dB, AS= 15◦,

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List of Figures

2.1 A co-located MU-MIMO system with an M -antenna BS and K

single-antenna MSs. . . 7

2.2 “One-ring” model with M transmit antennas and a single-antenna MS.

The angle spread (AS) and mean angle of arrival (AoA) is depicted. . . . 8

3.1 MSE performance of the conventional and proposed LSFC estimator with

perfect SSFC knowledge assumed for the former, AS= 15◦. . . 31

3.2 MSE performance of the conventional and proposed LSFC estimator with

perfect SSFC knowledge assumed for the former, AS= 15◦. . . 32

3.3 MSE performance comparison between the proposed estimators (LSFC,

SSFC) and the EM-based estimators (LSFC, SSFC) versus iteration

num-ber of EM-based estimators, where AS= 7.2◦, SNR=10dB, and full

mod-eling order is used. Initial b_{β is chosen as E {β}. . . .} 32

mod-eling order is used. Initial bβ is chosen as 1₂1K. . . 33

mod-eling order is used. Initial bβ is chosen as 1

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3.6 MSE performance comparison between the proposed estimators (LSFC, SSFC) and the EM-based estimators (LSFC, SSFC) versus iteration

num-ber of EM-based estimators, where AS= 15◦, SNR=10dB, and full

mod-eling order is used. Initial bβ is chosen as 1

21K. . . 34

num-ber of EM-based estimators, where AS= 15◦, SNR=0dB, and full

model-ing order is used. Initial bβ is chosen as 1₂1K. . . 34

3.8 MSE performance of the proposed LSFC and SSFC estimator versus

num-ber of BS antennas and received SNR, where AS= 7.2◦, and full modeling

order is used. . . 35

3.9 MSE performance of the proposed LSFC and SSFC estimator versus

num-ber of BS antennas and received SNR, where AS= 15◦, and full modeling

3.10 MSE performance of the proposed LSFC and SSFC estimator versus num-ber of BS antennas with different user location, hence different received

SNR at BS (indicated in the legend), where AS= 15◦, and full modeling

4.1 An illustration of the σ2

` distribution with respect to ` for M = 100. . . . 47

4.2 An illustration of the [ ˜B]`` distribution with respect to ` when M = 100. 51

4.3 MSE performance of the proposed SSFC estimator versus received SNR

and modeling order with estimated and perfect LSFC, where AS= 7.2◦,

and polynomial basis is used. . . 61

4.4 MSE performance of the proposed RR SSFC estimator versus received

SNR and modeling order with estimated and perfect LSFC, where AS=

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4.5 MSE performance of the proposed SSFC estimator versus received SNR

and modeling order with estimated and perfect LSFC, where AS= 7.2◦,

and DCT-II basis is used. . . 62

4.6 MSE performance of the proposed RR SSFC estimator versus received

SNR and modeling order with estimated and perfect LSFC, where AS=

15◦, and DCT-II basis is used. . . 63

4.7 MSE performance of the proposed SSFC estimators, ˆh(I) _{and ˆ}_h(II)_{, versus}

modeling order when using respectively DCT-II and polynomial basis,

where AS= 7.2◦, SNR= 10 dB, and imperfect LSFC is used. . . 63

4.8 MSE performance of the proposed SSFC estimators, ˆh(I) and ˆh(II), versus

modeling order when using respectively DCT-II and polynomial basis,

where AS= 15◦, SNR= 10 dB, and imperfect LSFC is used. . . 64

4.9 Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I) _and

ˆ

h(II)_{, compared with true (exact) spatial waveform when DCT-II basis}

being chosen, where AS= 7.2◦, modeling order=5, SNR= 10 dB, mean

AoA= ₂₁π, and imperfect LSFC is used. . . 64

4.10 Spatial waveform (real part) of the proposed SSFC estimators, ˆh(I) and

ˆ

h(II), compared with true (exact) spatial waveform when DCT-II basis

AoA= ₂₁π, and imperfect LSFC is used. . . 65

ˆ

h(II)_{, compared with true (exact) spatial waveform when DCT-II basis}

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ˆ

h(II), compared with true (exact) spatial waveform when DCT-II basis

AoA= π

21, and imperfect LSFC is used. . . 66

4.13 Diagonal distribution of the bias matrix, B, with respect to the SSFC

estimators, ˆh(I) _{and ˆ}_h(II)_{, where AS= 7.2}◦_{, mean AoA=} π

21, SNR= 10 dB

and KLT basis is used. . . 66

estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= ₂₁π, SNR= 10 dB

and KLT basis is used. . . 67

estimators, ˆh(I) _{and ˆ}_h(II)_{, where AS= 7.2}◦_{, mean AoA=} π

21, SNR= 10 dB

estimators, ˆh(I) _{and ˆ}_h(II)_{, where AS= 15}◦_{, mean AoA=} π

21, SNR= 10 dB

estimators, ˆh(I) and ˆh(II), where AS= 7.2◦, mean AoA= ₂₁π, SNR= 10 dB

estimators, ˆh(I) _{and ˆ}_h(II)_{, where AS= 15}◦_{, mean AoA=} π

21, SNR= 10 dB

estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 3π₂₁, SNR= 10 dB

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estimators, ˆh(I) and ˆh(II), where AS= 15◦, mean AoA= 5π₂₁, SNR= 10 dB

estimators, ˆh(I) _{and ˆ}_h(II)_{, where AS= 15}◦_{, mean AoA=} 7π

21, SNR= 10 dB

4.22 Convergence speed of proposed IMOD algorithm. Iteration number “Full ” represents the full modeling order and “Initialized ” means the

initializa-tion. Assume DCT-II basis is used, SNR= 10 dB and η = 1. . . 71

4.23 Diagonal distribution of the bias matrix, B, with respect to KLT,

DCT-II, and polynomial basis, where AS= 7.2◦, mean AoA= ₂₁π, SNR= 10 dB

and number of subcarriers is 100. Assume imperfect spatial correlation

matrix estimated by ML or shrinkage [20] method is used here. . . 72

4.24 Diagonal distribution of the bias matrix, B, with respect to KLT, DCT-II,

and polynomial basis, where AS= 7.2◦, mean AoA= ₂₁π, SNR= 10 dB and

number of subcarriers is 50. Assume imperfect spatial correlation matrix

estimated by ML or shrinkage [20] method is used here. . . 72

and polynomial basis, where AS= 7.2◦, mean AoA= π

21, SNR= 10 dB and

and polynomial basis, where AS= 7.2◦, mean AoA= ₂₁π, SNR= 10 dB and

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4.27 Diagonal distribution of the bias matrix, B, with respect to KLT,

DCT-II, and polynomial basis, where AS= 15◦, mean AoA= ₂₁π, SNR= 10 dB

and number of subcarriers is 100. Assume imperfect spatial correlation

matrix estimated by ML or shrinkage [20] method is used here. . . 74

and polynomial basis, where AS= 15◦, mean AoA= ₂₁π, SNR= 10 dB and

4.31 Performance of the estimated mean AoA (in radians) with different ASs

versus modeling order, where SNR= 10 dB, and polynomial basis is used. 77

5.1 Illustration of Distributed Massive MIMO System. . . 78

5.2 The flow chart of closed-loop transceiver design in TDD mode. Colored

blocks are done by MSs while others done by BS. . . 82

5.3 The flow chart of closed-loop transceiver design in FDD mode. Colored

blocks are done by MSs while others done by BS. . . 84

5.4 Sum rate performance of antenna selection algorithms (CS and GNS)

ver-sus MT provided that ZF and MRT precoding are used, where MT = 3MF

represents the number of RRH antennas, MF is the number of selected

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5.5 Sum rate performance of antenna selection algorithms (CS and GNS)

versus MF provided that ZF and MRT precoding are used, where MT =

200 represents the number of RRH antennas, MF is the number of selected

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Chapter 1 Introduction

A cellular mobile network in which each base station (BS) is equipped with an M -antenna array, is referred to as a large-scale multiple input, multiple output (MIMO) system or a massive MIMO system for short if M 1 and M K, where K is the number of active user antennas within its serving area. A massive MIMO system has the potentiality of achieving transmission rate much higher than those offered by cur-rent cellular systems with enhanced reliability and drastically improved power efficiency. It takes advantage of the so-called channel-hardening effect [1] which implies that the channel vectors seen by different users tend to be mutually orthogonal and frequency-independent [2]. As a result, linear receiver is almost optimal in the uplink and simple multiuser precoder are sufficient to guarantee satisfactory downlink performance. Al-though most investigation consider the co-located BS antenna array scenario [1], the use of a more general setting of massive distributed antennas has been suggested recently [4]. The Kronecker model [9], which assumes separable transmit and receive spatial statis-tics, is often used in the study of massive MIMO systems [16]. The spatial channel model (SCM) [7], which is adopted as the 3GPP standard, degenerates to the Kronecker model [8] when the number of subpaths approaches infinity. This model also implies that the distributions of angle of arrival (AoA) and angle of departure (AoD) are in-dependent. In general such an assumption is valid if the antenna number is small and large cellular system is in question. But if one side of a MIMO link consists of

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multi-ple single-antenna terminals, only the spatial correlation of the array side needs to be taken into account and thus the reduced Kronecker model and other spatial correlated channel models become equivalent. Throughout this paper our investigation focuses on this practical scenario, i.e., we consider a massive MIMO system where K is equal to the number of active mobile users.

We assume that the mobile users transmit orthogonal uplink pilots for the serving BS to estimate CSI that includes both small-scale fading coefficients (SSFCs) and large-scale fading coefficients (LSFCs). Besides data detection, CSI is needed for a variety of link adaptation applications such as precoder, modulation and coding scheme selec-tion. The LSFCs, which summarize the pathloss and shadowing effect, are proportional to the average received signal strength (RSS) and are useful in power control, location estimation, hand-over protocol and other applications. While most existing works focus on the estimation of the channel matrix which ignores the LSFC [18] [29], it is desirable to know SSFCs and LSFCs separately. LSFCs are long-term statistics whose estima-tion is often more time-consuming than SSFCs estimaestima-tion. Convenestima-tional MIMO CSI estimators usually assume perfect LSFC information and deal solely with SSFCs [4–6]. For co-located MIMO systems, it is reasonable to assume that the corresponding LSFCs remain constant across all spatial subchannels and the SSFC estimation can sometime be obtained without the LSFC information. Such an assumption is no longer valid in a multi-user MIMO system where the user-BS distances spread over a large range and SSFCs cannot be derived without the knowledge of LSFCs.

The estimation of LSFC has been largely neglected, assuming somehow perfectly known prior to SSFC estimation. When one needs to obtain a joint LSFC and SSFC estimate, the minimum mean square error (MMSE) or least squares (LS) criterion is not directly applicable. The expectation-maximization (EM) approach is a feasible al-ternate [32, Ch. 7] but it requires high computational complexity and cannot guarantee convergence. We propose an efficient algorithm for estimating LSFCs with no aid of

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SSFCs by taking advantage of the channel hardening effect and large spatial samples available to a massive MIMO BS. Our LSFC estimator is of low computational com-plexity, requires relatively small training overhead and yields performance far superior to that of an EM-based estimator. Our analysis show that it is unbiased and asymptotically optimal.

Estimation of SSFCs, on the other hand, is more difficult as the associated spatial correlation is not as high as that among LSFCs. Nevertheless, given an accurate LSFC estimator, we manage to derive a reliable SSFC estimator which exploits the spatial correlation induced channel rank reduction and calls for estimation of much less channel parameters than that required by conventional method [29] when the angle spread (AS) of the uplink signals is small. The proposed SSFC estimator provides excellent perfor-mance and offer additional information about the average angle of arrival (AoA) which is very useful in designing a downlink precoder.

In this thesis we present a method to obtain estimates for both LS- and SSFCs. We first propose a uplink-pilot-based LSFC estimator for a massive MU-MIMO TDD system that requires a small training overhead. Based on the facts that i) the user channels tend to be mutually orthogonal and ii) the resolution of massive MIMO antenna array is high, thus the AoA spread (AS) at the BS from each uplink channel is small, we use the esti-mated LSFCs obtained in the first step to derive an efficient estimator incorporating the SSFC and mean AoA estimation through a rank-reduced (RR) channel model similar to that proposed in [12]. When considering SSFC estimator with RR channel modeling, [12] suggest the use of polynomial basis. Nevertheless, by connecting the basis selection issue

here with that of image signal processing, we show that the Karhunen-Lo`eve

transfor-mation (KLT) basis is optimal in terms of its outstanding energy compaction property; furthermore, type-2 discrete cosine transform (DCT-II) basis is the best approximation of KLT basis whereas has low computational complexity. Simulation results are con-ducted to show the superiority of our estimators in massive MU-MIMO system. Finally,

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an low complexity iterative modeling order decision algorithm is investigated.

The rest of this thesis is organized as follows. In Chapter 2, we describe a massive MU-MIMO channel model that takes into account spatial correlations and large-scale fading. In Chapter 3, a novel uplink-pilot-based LSFC estimator is proposed and in Chapter 4, we devise an SSFC estimator by using the estimated LSFCs. After that, we analysis the effect of modeling order on SSFC estimation, and based on the analysis results, we propose a rank determination algorithm. Finally, we introduce the closed-loop transceiver design including the uplink pilot design issue for both TDD and FDD mode in Chapter 5. Our main contributions are summarized in Chapter 6.

The following notations are used throughout the thesis: upper case bold symbols

denote matrices and lower case bold symbols denote vectors. (·)T, (·)H, and (·)∗represent

the transpose, conjugate transpose, and conjugate of the enclosed items, respectively. vec(·) is the operator that forms one tall vector by stacking columns of the enclosed matrix, whereas Diag(·) translate a vector into a diagonal matrix with the vector entries

being the diagonal terms. While E{·}, k·k, k·k2, and k·kF denote the expectation, vector

`2-norm, matrix spectral norm and Frobenius norm of the enclosed items, respectively,

⊗ and respectively denote the Kronecker and Hadamard product operator. Denote

by IL, 1L, and 0L respectively the L × L identity matrix and L-dimensional all-one

and all-zero column vectors, whereas 1L×S, and 0L×S are the matrix counterparts of the

latter two. ei and Eij are all-zero vector and matrix except for their ith and (i, j)th

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Chapter 2 Preliminaries

2.1 Conventional MIMO System Model

Multiple input and multiple output antenna systems, or commonly referred as MIMO system, is a system with spatial separated antennas. Under suitable channel fading con-ditions, the MIMO channel provides an additional spatial dimension for communication and yields a degree-of-freedom gain [38]. These additional degree of freedom can be utilized by spatially multiplexing several data streams onto the MIMO channel, and increase the capacity.

2.1.1 Single-user MIMO System

For conventional single-user (SU) MIMO, the channel between transmitter and re-ceiver at time k can be modeled as

H(k, τ ) =

L

X

m=1

Hm(k)δ(t − τm), (2.1)

where L is the total number of paths between one antenna pair. τm is the delay of

mth path, and δ is the Dirac delta function. This representation reduce to a NR× NT

single-tap fading channel matrix as we consider a narrowband flat-fading MIMO channel

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signal vector is then given by

y = Hx + n, (2.2)

where x is the data vector transmitted, and H is the Rayleigh-fading channel. The elements of H are independent and identically distributed (i.i.d.), zero-mean circularly symmetric complex Gaussian random variables. n is the additive white Gaussian noise

(AWGN) vector, whose entries are of zero-mean and with σ2

z variance.

2.1.2 Multiuser MIMO System

Now we cast into a single-cell uplink (UL) multiuser MIMO (MU-MIMO) system having an M -antenna BS and K single-antenna mobile stations (MSs) as illustrated in Fig. 2.1. When referring to one of the K MSs, we focus on the “one-ring” model shown in Fig. 2.2 while the other channel models introduced in [12] are also considered. In such a MIMO setup, MS is surrounded by local scatterers and waveforms impending from the MS are richly scattered. On the other hand, BS is often unobstructed by local scatterers and has a mean angle of arrival (AoA) and small angle spread (AS) with respect to the transmitter. The clustered channel setup is typical in urban environments, and has been validated through field measurements [12].

The most apparent difference between SU-MIMO and MU-MIMO is that in addition to small-scale fading, the transmitted signals (from MSs) also suffer from different large-scale fading caused by pathloss and shadowing effect. The M × 1 Rx signal vector can be written as y = K X k=1 p βkhkxk+ n = HD 1 2 βx + n (2.3)

where H = [h1, · · · , hK] ∈ CM ×Kand Dβ = Diag(β) contain respectively the SSFCs and

LSFCs that characterize the K uplink channels, and n is the noise vector whose entries

are distributed according to CN (0, 1). The vector β = [β1, · · · , βK]T whose elements

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distributed (i.i.d.) sk’s with 10 log10(sk) ∼ N (0, σs2), and the pathloss which depends on

the distance between the BS and MS k, dk, with α > 2. We called HD

1 2

β in (2.3) the

composite fading channel matrix.

ܯ

ߚ_ଵࢎ_૚

ߚ_ଶࢎ_૛ ߚଷࢎ૜

Figure 2.1: A co-located MU-MIMO system with an M -antenna BS and K single-antenna MSs.

2.2 Effect of Massive MIMO System

Massive MIMO, very-large MIMO, large-scale MIMO all refer to a system where the BS has an enormous number of antennas larger than the number of Rx antennas,

that is, NT NR in SU-MIMO system and M K in MU-MIMO system. [1] and [16]

say that M is about the magnitude of several hundreds but within a thousand.

Several observations on the effect of massive MIMO are given in the following sub-sections.

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broadside

AOA

mean AOA

AS

Figure 2.2: “One-ring” model with M transmit antennas and a single-antenna MS. The angle spread (AS) and mean angle of arrival (AoA) is depicted.

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2.2.1 Channel Hardening Effect

Massive MIMO takes advantage of the channel hardening effect which says if the UL channel matrix representing the SSFCs between M transmit antennas and K

single-antenna MSs, M K, is H = [hij], whose entries are i.i.d. zero-mean circularly

symmetric complex Gaussian random variables with unit variance, hij ∼ CN (0, 1), then

1 MHH H a.s. −→ I and K X `=1 λ2_` = ||H||2_F −→ M K as M −→ ∞a.s.

where λ` is the `th singular value of H. This effect tell us that the channel gain is

inde-pendent of frequency, thus, we can assign full bandwidth for all users, and no frequency domain scheduling is needed.

Moreover, the corresponding achievable sum rate is given by

R = E log₂det I + P MHH H (2.4) = K X l=1 log₂ 1 + P Mλ 2 ` (2.5) a.s. −→ K log₂ 1 + P M K (2.6)

which means ideally, thermal noise, interference all vanish asymptotically and to

main-tain a fixed sum rate, BS power can be scaled as _M1 . It also implies that instead of

reducing the cell size one can increase the system capacity by simply putting more an-tennas on the existing base stations [3]. Marzetta further showed that [3] the channel vectors seen by different users become mutually orthogonal and the simplest precoders– detectors, i.e., eigen beamforming (BF) and matched filter (MF), are asymptotically optimal. Furthermore, a simple regularized zero-forcing (RZF) precoding scheme can achieve the same performance as BF with one order of magnitude fewer antennas in both uncorrelated and correlated fading channels.

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2.2.2 TDD and FDD system

The pilot sequences are needed for channel estimation and other synchronization purposes. Nevertheless, the time needed to transmit the DL pilot sequences to MSs equals to M , which may exceed the channel coherent interval, and result in an inaccurate estimation. Furthermore, as the CSI must be obtained through channel estimation, an FDD system would require the downlink receivers to estimate multitude of channels and feedback to the transmitter, costing immense overhead in bandwidth and power. A TDD system can largely bypasses such a need assuming reciprocity does hold and proper calibrations are readily available at the transmit side [19].

It is worth noting that the FDD option cannot become a serious contender unless we can greatly reduce the CSI requirement and pilot transmission time. In this the-sis, we propose some approach to reduce the pilot dimension and feedback overhead simultaneously in FDD mode.

2.2.3 Large Antenna Aperture and Small Antenna Spacing

Owing to the large number of antennas, the beam resolution of massive antenna array is high [1], thus, high energy efficiency and power efficiency is achieved. More precisely, we have array power gain of M , hence the power per BS antenna can be scaled

as _M12 when the total power is fixed [1]. Also, due to the high beam resolution, AoA

spread at BS from each MS is small enough to helps us adopt the rank-reduced (RR) channel representation [12] of SSFCs; measurement results in [2] also suggest that the ASs from different users are small.

Due to the limited space that is available for BS installation, the implementation of a large amount of antennas on a BS forces them to be packed tightly (small antenna spacings). As a result, serious mutual coupling and spatial correlation may exist. Spatial correlation among BS antennas also enables the use of an RR model [12] that effectively decreases the number of SSFC parameters needed to be estimated.

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2.3 Spatial-Correlated Small-Scale Fading Channel

Models

2.3.1 Conventional Spatial-Correlated Channel Model

Consider a single-cell massive SU-MIMO system with an NT-antenna BS and an

NR-antenna UE, where NT NR. The signal received by the UE can be expressed as

y = Hx + n (2.7)

where H = [hij] is the NR× NT SSFC channel matrix with complex Gaussian entries,

hij’s, x is the transmitted signal, and n ∼ CN (0, INR) represents the white noise.

Let Φ, ΦT, and ΦR be the spatial correlation matrices of vec(H)

Φdef_{= E}vec(H)vec(H)H , (2.8)

and those of the Tx and Rx antennas, respectively.

In general, a spatial-correlated Rayleigh fading MIMO channel can be modeled as

vec(H) = Φ12vec(H_w), (2.9)

where Hw is NR× NT with i.i.d., zero-mean, unit-variance complex Gaussian entries.

For a massive MU-MIMO channel where the receiving end consists of many single-antenna receivers at various locations, the above model must be modified in an user-by-user sense. The small-scale fading channel seen by user k can in general be written as

hk = Φ

1 2

kh˜k, (2.10)

where Φk is the transmit spatial correlation matrix with respect to the kth user and

˜

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2.3.2 Kronecker Model

Kronecker model [9] is commonly used which assumes that the spatial correlations among Tx antennas and those among Rx antennas are separable so that

Φ = ΦT⊗ ΦR = Φ

1

2(Φ

1

2)H, (2.11)

where the square root matrix Φ12 has a similar decomposition

Φ12 = Φ 1 2 T⊗ Φ 1 2 R (2.12)

and therefore yields

H = Φ 1 2 RHw Φ 1 2 T H . (2.13)

This model is reasonable accurate only when the main scattering is locally rich at the transmitter and receiver sides, respectively [?]. There are many field measurements and experiments that report inconsistencies with this model.

For a massive MU-MIMO channel where the receiving end consists of K single-antenna MSs, the Kronecker model is equivalent to conventional spatial-correlated chan-nel model in 2.10.

2.3.3 Virtual Channel Representation

To solve the deficiencies of the Kronecker model, Sayeed [?] suggested a so-called virtual channel representation that takes the Tx-Rx cross-correlation into account. This

model expands the spatial correlations by unitary matrices, ΦT, ΦR, relate the Tx and

Rx spatial modes by a coupling matrix. Using the Fourier basis, one obtains

H = FR(Ω Hw)FHT, (2.14)

where FR and FT are DFT matrices of dimension NR and NT and Ω is the coupling

matrix. However, this model is only applicable for single-polarized ULAs and the Fourier basis is far from the optimal choice.

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2.3.4 Weichselberger Model

An obvious optimal choice of basis that incurs no approximation error can be readily obtained by performing eigen-decomposition on the spatial correlation matrices. Weich-selberger et al. therefore suggested the model [?]

H = UR ˜ Ω Hw UH_T, (2.15)

where UT and UR are the eigenbases of ΦT, and ΦR, respectively, i.e.,

ΦR = URΛRUHR, ΦT = UTΛTUHT. (2.16)

with ΛR and ΛT being the diagonal matrices consist of the (nonnegative) eigenvalues

of ΦR and ΦT. ˜Ω is the element-wise square root of the coupling matrix Ω = [ωij] in

which each entry specifies the average energy coupled from a transmit eigenmode to a receiver eigenmode [?].

Let uR,i and uT,j be the ith and jth column vector of UR and UT, respectively. Then

ωij = E n_u_H R,iHuT,j 2o , i = 1, · · · , NR, j = 1, · · · , NT (2.17)

The Weichselberger model is perhaps more convenient to generate the SSFC matrix H and for evaluating the channel capacity of correlated MIMO channels as the coefficient matrix has independent entries. It is also useful to analyze MIMO system performance. However, it is not suitable for channel estimation applications because the number of parameters, including the unknown eigenbases, is even larger than that of H.

2.3.5 Rank-reduced Channel Representation

The rank-reduced (RR) model introduced in [12] is reviewed in this section. Singular value decomposition (SVD) of H gives

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Let Q1 and Q2 be two predefined unitary matrices. Then

Q1 = UP1, Q2 = VP2, (2.19)

where Λ is a NR× NT rectangular diagonal matrix with nonnegative entries and both

P1 and P2 are also unitary. This unitary transform leads to

H = Q1P−11 Λ(P

−1 2 )

H_QH

2 = Q1CQH2 , (2.20)

with a random matrix C characterizing the cross-coupling effect. It can be verified that all the aforementioned models are special cases of (2.20) which is valid for all slow-varying narrowband MIMO channels without any attached pre-assumptions needed.

It is equivalent to the Kronecker model if C satisfies

vec(C) = (ΞT⊗ ΞR)vec(Hω), (2.21)

where ΞTand ΞR are obtained via Gram-Schmidt orthonormalization with Φ

1 2 T = Q2ΞT and Φ 1 2 R = Q1ΞR.

If Q1and Q2are chosen to be composed of columns of DFT matrices, this generalized

model is compatible with the virtual channel representation [?]. Finally, the RR model is related to the Weichselberger model via

UT= Q2PHT, UR = Q1PHR (2.22)

with PT and PR being the eigenbasis matrices of E{CCH} and E{CTC∗} whose

eigen-values are the same as those of E{HHH_{} and E{H}T_H∗_}.

The use of pre-determined basis matrices avoids the need for the channel estimator to perform the extra work of determining the eigenbasis (matrices) and provides a con-venient way for RR representation. Furthermore, when the AoD spread is small, we can show that (2.20) can be expressed as

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where W = Diag[w1, w2, . . . , wM] and wi = exp(−j2π (i−1)ξ

λ sinφ), ξ being the distance

between neighboring elements at the BS linear array, and φ is the mean AoD.

As UL MU-MIMO introduced in Section 2.1.2 being considered, we can summarize the SSFC matrix (2.20) and (2.23) into the lemma below:

Lemma 2.3.1 (RR representations). The channel vector seen by kth user can be repre-sented by

h = Q(I)_mc(I) (2.24)

or alternately by

h = W(φ)Q(II)_m c(II) (2.25)

where Q(I)m, Q(II)m ∈ RM ×m are predetermined (unitary) basis matrices and c(I), c(II) ∈

Cm×1 are the channel vectors with respect to bases Q(I)m and Q(II)m for the user k-BS

link and W(φ) is diagonal with unit magnitude entries. The two equalities hold only if m = M and become approximations if m < M . Furthermore, if the AS is small,

[W(φ)]ii = exp

−j2π(i−1)ξ_λ sin φ with ξ and λ being the antenna spacing and signal wavelength and φ is the mean AoA.

That is, Lemma 2.3.1 suggests that the mean AoA (which is close to the incident angle of the strongest path) of each user is extractable if its AS is small. In addition, due to the large aperture massive MIMO antenna array has offered, good AoA resolution and thus accurate mean value extraction are guaranteed [1].

2.4 Antenna Selection

Antenna selection (ANS) is useful in lessening the RF and baseband implementation complexity. For conventional MIMO systems, the number of antennas is usually relative small and ANS is not a particular concern although proper ANS does save hardware cost

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and power consumption. For massive, however, it becomes an important design consid-eration, as the associated number of RF chains reduction can be significant. Moreover, it not always a good policy to use more antennas if the number of antennas used is already large enough; employing more antennas may lead to degraded performance.

An outstanding low-complexity ANS scheme is especially important for massive MIMO systems as it can significantly reduce the hardware requirement of a BS without compromising performance. The solution, however, is challenging to say the least. This

can be easily seen by simply counting the number of possible combinations 100₅₀≈ 1029

in selecting 50 out of 100 antennas that maximizes the system sum rate.

We will consider first the basic setting of an SU-MIMO system with large-scale BS antenna array and then extend the investigation to an MU-MIMO scenario. Further-more, we focus the ANS study on the DL case, where a massive MIMO BS acts as the transmitter. The UL scenario can be similarly treated with some minor modifications.

2.4.1 ANS for Massive SU-MIMO systems

Consider a DL massive SU-MIMO system with NT Tx antennas and NRRx antennas

whose received vector is given by (2.38) and NT NR.

In such a system, H is known by the receiver but unknown to the transmitter. The ergodic capacity is given by [21]

R(H) = log₂det INT + P NT HHH ≈ NR X j=1 log₂ 1 + P NT khjk2 . (2.26)

To facilitate subsequent discussion, we need the following definition [23].

Definition 2.4.1. Let f be a function defined as f : U → R+_{. Then f is called monotone}

if f (S ∪ {a}) − f (S) ≥ 0, ∀a ∈ U , S ⊆ U , a /∈ S, and is called a sub-modular function

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A nesting property of R(H) has been derived in [21] when one tries to select one more Rx antenna by a incremental capacity-based selection (CS) algorithm:

R(Hn+1) = R(Hn) + log2 1 + P MαJ,n (2.27) ≈ R(Hn) + log2 1 + P NT khJ,nk2 (2.28)

where Hnis the channel matrix after selecting n Rx antennas and αj,n = hHj

I + _NP TH H nHn −1 hj

with hj being the channel seen by the jth user antenna. (2.28) implies that in an

SU-MIMO system, the channel capacity is sub-modular over Rx antenna set {1, 2, . . . , NR}.

Hence, the main design criterion of SU-MIMO Rx ANS is to reduce the hardware com-plexity which is dominated by the number of radio-frequency (RF) chains. Given the set

of selected antennas, one selects, in each step, the J th antenna that maximize khJ,nk [21]

J = arg max

j khj,nk, (2.29)

which gives an important observation that the CS criterion is asymptotically equivalent to the norm-based selection (NS) criterion. Thus, low-complexity NS criterion will be good enough to retain the performance CS can achieve.

On the other hand, [22] and [23] reported that Hn is not sub-modular over the Tx

antenna set {1, 2, . . . , NT}. This means, for a massive MIMO downlink the use of all Tx

antennas may not offer better rate. In general, we have two Tx ANS selection criteria, i.e., capacity maximization and feedback overhead reduction for FDD mode.

2.4.2 ANS for Massive MU-MIMO systems

Owing to the results of [24], we know that when M K, user selection is unneces-sary. Hence, we focus more on the Tx ANS in DL massive MU-MIMO system provided that TDD mode is used.

For an MU-MIMO system with M BS antennas and K single-antenna MSs, M K, the composite forward link (downlink) channel matrix consists of SSFCs and LSFCs [1]

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is given by

G = D

1 2

βH. (2.30)

On the other hand, due to the channel reciprocity [19] in TDD mode, i.e., the forward link and reverse link channel are symmetric, the reverse link (uplink) composite channel

matrix is simply the M × K matrix GT_{. Therefore, the DL system model is given by [42]}

y = Gs + n = GWPx + n (2.31)

where s is the transmitted signal, x ∼ CN (0, IK) the uncoded data, W the M × K

precoding matrix, n ∼ CN (0, IK) the received noise and K × K matrix P diagonal with

its kth entry being the square root of power allocated to user k. Thus, the achievable rate

R(H) = log₂det IK + WHGHGWP2

, (2.32)

subject to a total power constraint P

tr WP2WH= kWPk2_F ≤ P, (2.33)

can be obtained. It has been proved that, as opposed to SU-MIMO, MU-MIMO using linear precoding techniques, e.g., zero-forcing (ZF) and minimum mean square error (MMSE) precoding, has sub-modular property R(H) over the Tx antenna set [42]. This is because in MU-MIMO, MSs cannot cooperate and no post-detection signal processing is allowed.

Assume decremental transmit antenna selection (TAS) algorithm [42] is used, the capacity loss when getting rid of one more Tx antenna, indexed α, can be computed by using (14) of [42]:

Lemma 2.4.2. Let S and S0 be two Tx antenna sets in MU-MIMO system and S =

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rate (throughput loss) between these two sets is given by RD(r) ≡ RD(S0) − RD(S) = K log₂  1 + SNR kuH r (GSGHS)−1k22 1−uH r (GSGHS)−1ur [tr(TS0)]2+ tr(T_S0) SNR + kuHr (GSGHS)−1k22 1−uH r (GSGHS)−1ur   (2.34)

using a ZF precoder, where TS0 = G_S0GH_S0

−1

, GS is the composite fading channel given

the transmit antenna set S, u` is the channel vector seen by `th Tx antenna, SNR = _σP2,

P is the total power constraint, and σ2 _{= 1 is the noise power.}

Because RD(r) ≥ 0, we know that TAS in MU-MIMO system indeed has

sub-modularity [23], hence the purpose of TAS is to reduce the hardware complexity and

reduce the amount of feedback in FDD mode. Suppose that we want to select MF K

out of M Tx antennas, the capacity-based selection (CS) algorithm can be summarized in Algorithm 1.

Algorithm 1 Decremental CS algorithm

1: Let S = 1, 2, · · · , M`; 2: while |S| > MF do 3: α = arg max r∈S RD(S − {r}); 4: S = S − {α}; 5: end while

6: The resulting set S is the desired transmit antenna set.

From Lemma 4 of [42], we obtain

α = arg max r∈S RD(S \ {r}) = arg min r∈S RD(r) = arg min r∈S kuH r (GSGHS)−1k22 1 − uH r (GSGHS)−1ur (2.35)

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Applying Lemma 3.3.1 to (2.35) and substituting GS = D1/2_β HS yields α = arg min r∈S kuH r (GSGHS)−1k22 1 − uH r (GSGHS)−1ur a.s. = arg min r∈S 1 |S|ku H r D −1 β k 2 2 1 − _|S|1 uH r D −1 β ur = arg min r∈S kD−1_β urk22 |S| − kD−12 β urk22 (2.36)

We call (2.36) generalized norm-based selection (GNS), since the form of its selection metric is similar to norm-based selection (NS).

When the serving MSs are not far away from each other, or equivalently, Dβ ≈ βIK,

which is in general the case because each RRH only serves the nearby users, α = arg min r∈S kD−1_β urk22 |S| − kD− 1 2 β urk22 ≈ arg min r∈S kurk22 |S|β2 _{− βku} rk22 = arg min r∈S kurk 2 2 = arg min r∈S kurk2 (2.37)

which is just the norm-based selection (NS).

Besides, [1] has shown that in massive MIMO system, ZF precoding is asymptoti-cally equivalent to MF precoding, hence we can get the same result for the case of MF precoding.

2.5 System Model

Throughout this thesis, we consider a singlecell MUMIMO system having an M -antenna BS and K single--antenna MSs, where M K. For a muti-cell system, pilot contamination [13] may become a serious design concern in the worst case when the same pilot sequences (i.e., the same pilot symbols are placed at the same time-frequency locations) happen to be used simultaneously in several neighboring cells and are per-fectly synchronized in both carrier and time. In practice, there are frequency, phase and

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timing offsets between any pair of pilot signals and the number of orthogonal pilots is often sufficient to serve mobile users in neighboring cells. Moreover, neighboring cells may use the same pilot sequence but the pilot symbols are located in non-overlapping time-frequency units [15], hence a pilot sequence is more likely be interfered by uncor-related asynchronous data sequences whose average effect is not as serious as the worst case and can be mitigated by proper inter-cell coordination, frequency planning and some interference suppression techniques [14]. We will, however, focus on the single-cell narrowband scenario throughout this thesis.

We assume a narrowband communication environment in which a transmitted signal suffers from both large- and small-scale fading. The K uplink packets place their pilot of length T at the same time-frequency locations so that, without loss of generality, the

corresponding received samples, arranged in matrix form, Y = [yij] at the BS can be

expressed as Y = K X k=1 p βkhkpHk + N = HD 1 2 βP + N (2.38)

where N is the noise matrix whose entries are distributed according to CN (0, 1). The

K ×T matrix is defined by P = [p1, · · · , pK]H, where T ≥ K and pk is the pilot sequence

sent by MS k and pH

j pk = 0, ∀ j 6= k. The optimality of using orthogonal pilots has

been shown in [29].

We invoke the assumption that channels linking different users are independent as they are relatively far (with respect to the wavelength) apart and the kth uplink channel vector is

hk = Φ

1 2

kh˜k, (2.39)

where Φk is the transmit spatial correlation matrix with respect to the kth user and

˜

hk ∼ CN (0M, IM). We assume that ˜hk’s are i.i.d. and the SSFC H remains constant

during a pilot sequence period, i.e., the channel’s coherence time is greater than T , while the LSFC β varies much slower.

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Chapter 3 Large-Scale Fading Coefficient

Estimation

Unlike previous works on MIMO channel matrix estimation which either ignore

LSFCs [18, 29] or assume perfect known LSFCs [4–6], we try to estimate H and Dβ

jointly. We first introduce an efficient LSFC estimator without SSFCs information in this Chapter. We treat separately channels with and without spatial correlation at the BS side and show that both cases lead to same estimators when the BS is equipped with a large-scale linear antenna array.

3.1 Uncorrelated BS Antennas

It is known that if the BS antenna spacings are large enough, say greater than 5λ, where λ is the signal wavelength, spatial mode correlation can be neglected and thus

Φk= IM, ∀k [9]. A statistic based on the received sample matrix Y and is asymptotically

independent of the SSFCs is derivable from the following property [31, Ch. 3]

Lemma 3.1.1. Let p, q ∈ CM ×1 _{be two independent M -dimensional random vectors}

whose elements are independent identically distributed (i.i.d.) according to CN (0, 1). Then by the law of large number,

1

Mp

H_p_{−→ 1 and}a.s. 1

Mp

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For a massive MIMO system with M T ≥ K, we have, as M → ∞, _M1 HH_H_−→a.s. IK, _M1 NHN a.s. −→ IT, _M1 HHN a.s. −→ 0K×T, and thus 1 MY H_{Y − I} T = PHDβP + 1 MN H_{N − I} T + PHD 1 2 β 1 MH H H − IK D 1 2 βP + 2 MR n PHD 1 2 βH H_No a.s. −→ PHDβP (3.1)

(3.1) indicates that the additive noise effect is reduced and the estimation of LSFCs can be decoupled from that of the SSFCs. Using the identity, vec(A · Diag(c) · F) =

(1S⊗ A) (FT ⊗ 1T)

c with A ∈ CT ×K, F ∈ CK×S, and c ∈ CK×1, we simplify

[12](3.1) as vec 1 MY H_{Y − I} T a.s. −→ 1T ⊗ PH PT _{⊗ 1} T β

This equation suggests that we solve the following unconstrained convex problem min β vec 1 MY H_{Y − I} T − 1T ⊗ PH PT _{⊗ 1} T β 2 , (3.2)

to obtain the LSFC estimate ˆ β = Diag kp1k−4, · · · , kpKk−4 · (1T T ⊗ P) (P ∗_{⊗ 1}T T) vec 1 MY H_{Y − I} T . (3.3) This LSFC estimator is of low complexity as no matrix inversion is needed when or-thogonal pilots are used and does not require any knowledge of SSFCs. Furthermore, the configuration of massive MIMO makes the estimator robust against noise, which is verified numerically later in Section 3.4.

3.2 Correlated BS Antennas

In practice, the spatial correlations are non-zero and Y is of the form

Y = ˜Φ    ˜ h1 · · · 0 .. . . .. ... 0 · · · h˜K   D 1 2 βP + N def = ˜Φ ˜HD 1 2 βP + N

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where ˜Φ = [Φ 1 2 1, · · · , Φ 1 2

K]. Following [16, 17], we assume that

Assumption 1. The spatial correlation at BS antennas seen by a user satisfies

lim sup M →∞ kΦ12 kk2 < ∞, ∀k; or equivalently, lim sup M →∞ kΦkk2 < ∞, ∀k. Therefore, (3.1) becomes 1 MY H_{Y − I} T a.s. −→ PH_D βP + 2 MR n PHD 1 2 βH˜ H_Φ_˜H_No + PHD 1 2 β 1 M ˜ HHΦ˜HΦ ˜˜H − IK D 1 2 βP def = PHDβP + N0

where N0 is zero-mean with seemingly non-diminishing variance due to the spatial

cor-relation. Nonetheless, we proved in Appendix A that Theorem 3.2.1. If lim sup

M →∞ sup 1≤k≤K kΦ 1 2 kk2 < ∞, then 1 M ˜ HHΦ˜HΦ ˜˜H −→ Ia.s. K, (3.4) 1 M ˜ HHΦ˜HN −→ 0a.s. K×T (3.5) as M → ∞.

This theorem implies that although the non-zero spatial correlation does cause the

increase of variance of N0, the channel hardening effect still exist and N0 is asymptotically

diminishing provided that Assumption 1 holds. In this case, LS criterion also mandates the same estimator as (3.3). Several remarks are worth mentioning.

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Remark 1. If J consecutive coherence blocks in which the LSFCs remain constant are available, our estimator can be easily extended to

ˆ β = Diag kp1k−4, · · · , kpKk−4 1T_T ⊗ P P∗ ⊗ 1T_T· vec 1 M J J X i=1 YH_i Yi − 1 JIT ! (3.6)

where Yi is the ith received block. Moreover, the noise reduction effect becomes more

evident as more received samples become available.

Remark 2. The proposed LSFC estimators (3.3) and (3.6) render element-wise expres-sions as ˆ βk = pH_k YHYpk− M kpkk2 M kpkk4 , ∀k, (3.7) ˆ βk = PJ i=1p H kYiHYipk− M Jkpkk2 M J kpkk4 , ∀k. (3.8)

Although these new expressions imply the same computational complexity, they are shown to be useful in Section 5.2 when designing uplink pilots.

Remark 3. After the LSFC estimates are got, we can use the conventional least squares (LS) estimator presented in [29] to estimate the SSFCs:

ˆ H = YPHDiag 1 ˆ γ1 , · · · , 1 ˆ γK , (3.9) where ˆγ` = q ˆ

β`kp`k2, or more desirable, adopt the SSFC estimator introduced later in

Chapter 4.

3.3 Performance Analysis

As LSFC estimator (3.7) is unbiased because

E n ˆ βk o = p H k(M PHDβP + M IK)pk− M kpkk2 M kpkk4 = M βkkpkk 4_{+ M kp} kk2− M kpkk2 M kpkk4 = βk, ∀k, (3.10)

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the mean squared error (MSE) of ˆβk is thus E ˆβk− βk 2 = Var n ˆ βk o . (3.11)

Lemma 3.3.1 (Lemma 4 of [16]). Let A ∈ CM ×M _{and p and q be two vectors with}

i.i.d. elements drawn from CN (0, 1). If lim sup

M →∞

kAk2 < ∞, then

pHAp −→ tr(A) anda.s. 1

Mp

H_Aq_{−→ 0 as M → ∞.}a.s.

Remark 4. Using [39, Lemma B.26], we can prove that the convergence rates in the

aforementioned asymptotic formulae follow O(kAkF/M ). More precisely,

Ep H_{Ap − tr(A)} M = O(kAkF/M ) (3.12) Ep H_Aq M = O(kAkF/M ). (3.13) By reformulating (3.7) as ˆ βk = βk+ pH k(NHN − M IK)pk M kpkk4 | {z } r1 +βk(h H khk− M ) M | {z } r2 + √ βk 2R hH kNpk M kpkk2 | {z } r3 ,

and invoking Assumption 1, Lemmas 3.1.1 and 3.3.1, and the fact that hk = Φ

1 2 kh˜k, we conclude that r1, r2, r3 a.s. −→ 0 as M → ∞, and thus Varnβˆk o = E|r1+ r2+ r3|2 a.s. −→ 0. (3.14)

As pk, N and hk are uncorrelated, we have

E|r1+ r2+ r3|2 ≈ E|r1|2 + E|r2|2 + E|r3|2 (3.15)

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Lemma 3.3.2. When the pilot length is T = K, the MSE convergence rates for E {|r1|2}, E {|r2|2} and E {|r3|2} follow O _β2 k T ·SNR2_k 1 M , O β_k2kΦkk2F M2 and O _4β2 k T ·SNRk 1 M , respec-tively, where SNRk def = βkkpkk2

T . Thus the convergence rate of Var{ ˆβk} is dominated by

E {|r2|2}. Proof. E|r1|2 = 1 kpkk8E ( tr pkpHk NH_{N − M I} K M 2) ≤ 1 kpkk8E ( tr pkpHk 2 tr NHN − M IK M 2) = 1 kpkk4E ( tr NH_{N − M I} K M 2) = 1 kpkk4E    K X i=1 nH i ni− M M !2_  = 1 kpkk4 K X i=1 E ( nH i ni− M M 2) = O K kpkk4 1 M = O β2 k T · SNR2_k 1 M (3.16) E|r2|2 = β_k2_E    ˜ hH kΦkh˜k− M M 2_  = O β_k2kΦkk 2 F M2 O β_k2 M M2 = O β_k2 1 M (3.17)

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E|r3|2 = 4βk kpkk4E    K X i=1 pkiR{hHkni} M 2   = 4βk kpkk4 K X i=1 p2_ki_E ( R{hH kni} M 2) = 4βk kpkk4 K X i=1 p2_ki_E    R{˜hH kΦ 1 2 kni} M 2   = 4βk kpkk2 O kΦ 1 2 kk 2 F M2 ! = O 4βk kpkk2 1 M = O 4β2 k T · SNRk 1 M , (3.18)

Lemma 3.3.3. The LSFC estimators (3.3) and (3.6) approach the minimum mean square error (MMSE) estimator with asymptotically diminishing MSE as M → ∞.

Remark 5 (Fisher Information Matrix). Denote θT =√β_k hT

k

, the Fisher informa-tion matrix for estimating θ is given by

I(θ) =      2kpkk2khkk2 √ β_kkpkk2hk1 · · · √ β_kkpkk2hkM √ β_kkpkk2hk1? βkkpkk2 · · · 0 .. . ... . .. ... √ β_kkpkk2hkM? 0 · · · βkkpkk2     . (3.19)

Moreover, the Cram´er-Rao lower bound (CRLB) of estimating βk is given by

Var q ˆ β k ≥ [I(θ)]₁₁, (3.20)

thus, (3.14) means that our LSFC estimator asymptotically achieves the CRLB.

Remark 6. From Lemma 3.3.2, the only term related to spatial correlation in (3.15)

is E {|r2|2}, and thus, for cases with finite M , the MSE-minimizing spatial correlation

matrix Φ? k is the solution of min A E ˜hHkA˜hk 2 − tr(A) s.t. [A]ii = 1, ∀ i. (3.21)

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Following the method of Lagrange multiplier, we obtain Φ?

k = IM. The convexity of (3.21)

implies that Var{ ˆβk} is an increasing function of kΦk− IMkF, i.e., the variance of the

LSFC estimator decreases as the channel becomes less correlated; meanwhile, Lemma 3.3.2 says the error convergence rate also improves.

Remark 7 (Finite M scenarios). Low normalized MSE, in the order of 10−5 to 10−4, is

obtainable with not-so-large BS antenna numbers (e.g., 50). The above MSE performance analysis is validated via simulation in Section 3.4.

3.4 Numerical Results and Discussion

Simulation results reported here using the channel generated by [40] whose spatial correlation at the BS is related to AoA distribution and antenna spacings. In addi-tion, the environment surrounding a user is of rich scattering with AoD’s uniformly distributed in [−π, π) making spatial correlation at MSs negligible. This setting accu-rate describes the environment where the BS with large-scale antenna array are mounted on an elevated tower or building. Throughout this section, we assume that there are 8 uniformly distributed users in a circular cell of radius R with the mean AoAs

equis-paced within [−60◦, 60◦]. The other simulation parameters are listed in Table 3.1 unless

explicitly stated otherwise. We define average received signal-to-noise power ratio as

SNRdef= βkkpkk2/T with k the index of farthest MS from the BS, and normalized mean

squared error (NMSE) as the MSE between the real and estimated vectors normalized by the former’s dimension and entry variance.

First, in Fig. 3.1 we compare the performance of the proposed LSFC estimator (3.3) with that of a conventional LS estimator [32, Ch. 8]

ˆ

β = (AHA)−1AHvec(Y)2 (3.22)

where A = (1T ⊗ H) PT ⊗ 1M

. As opposed to (3.3), the conventional estimator needs to know SSFCs beforehand, hence a full knowledge of SSFCs is assumed for the

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Table 3.1: Simulation parameters

Parameters Values

Operating frequency 2.6 GHz [2]

Cell radius R 100 meters

Pathloss exponent α 3

Shadow fading variance σs 10 dB

Number of BS antennas M 100

BS antenna spacing ξ 0.5λ

Number of MSs K 8

latter. Figure 3.1 shows that our proposed estimator outperforms the conventional sig-nificantly even if the latter has full knowledge of SSFCs. It is because the former takes advantage of the noise reduction effect that massive MIMO systems have offered. More-over, as antenna spacing increases, the channel decorrelates and thus the estimation error due to spatial correlation decreases, which verifies Theorem 3.2.1. Figure 3.2 illus-trates the effect of massive antennas to MSE. Owing to the fact that we have assumed perfect SSFC knowledge for the conventional LSFC estimator, MSE decreases with in-creasing sample amount as M increases. Unlike the conventional, the amount of known information does not grow with M for the proposed LSFC estimator. However, it still can be utilized to improve estimation and thus enables the proposed to outperform the conventional. In addition, the proposed is robust to SNR degradation due to noise or user-BS distance increment.

When comparing our proposed estimators with EM-based estimators, Fig. 3.3-3.7 shows the superiority of the proposed estimators. The details of EM approach is as follows: (i)set the initial value of bβ;

(ii)evaluate the LMMSE estimator of SSFC,

\ vec(H) = Diag (Φ1+ kp1k2βb1IM)−1· · · (ΦK+ kpKk2βbKIM)−1 D 1 2 βP ?_{⊗ I} M vec(Y);

(ii)evaluate the LMMSE estimator of LSFC, d p β = Enpβo+C−1√ β√β+ A H_A−1_AH vec(Y) − AEnpβo

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where A = 1T ⊗ bH PT _{⊗ 1} M ;

(iv)recursively compute until convergence occur.

Moreover, the modified EM (MEM) approach is obtained by replacing AHA = (HHH) (P?PT)−→ Diag M kpa.s. 1k2· · · M kpKk2

in the EM approach.

The results in Figs. 3.8 and 3.9 demonstrate the large-system performance of the proposed LSFC and full-order SSFC estimator. As can be seen, similar to the results in Fig. 3.2, the accurate LSFC estimates due to large received samples make its compensa-tion prior to the SSFC estimacompensa-tion reliable. Furthermore, such large sample size clearly improves the performance of the SSFC estimators directly. In addition, the proposed LSFC estimator has noise-robustness in the sense that the MSE performance is nearly the same when SNR = 0 dB to 15 dB, which gives similar result as Fig. 3.2.

0 5 10 15 10−6 10−5 10−4 10−3 BS antenna spacing (λ) NMSE 0 dB 10 dB 20 dB Proposed Conventional

Figure 3.1: MSE performance of the conventional and proposed LSFC estimator with

perfect SSFC knowledge assumed for the former, AS= 15◦.

After discussing some results about the average MSE performance of MSs, we recast to compare the MSE performance of MSs at different locations, or equivalently, different

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0 50 100 150 200 10−5 10−4 10−3 10−2 Number of antennas NMSE 0dB 10dB 20dB Conventional Proposed

Figure 3.2: MSE performance of the conventional and proposed LSFC estimator with

perfect SSFC knowledge assumed for the former, AS= 15◦.

2 4 6 8 10 12 14 16 18 20 10−5 10−4 10−3 10−2 10−1 100 101 102 AS=7.2o, SNR=10dB Iteration NMSE EM MEM proposed SSFC LSFC

Figure 3.3: MSE performance comparison between the proposed estimators (LSFC, SSFC) and the based estimators (LSFC, SSFC) versus iteration number of

EM-based estimators, where AS= 7.2◦, SNR=10dB, and full modeling order is used. Initial

b

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2 4 6 8 10 12 14 16 18 20 10−5 100 105 AS=7.2o, SNR=10dB Iteration NMSE EM MEM proposed LSFC SSFC

b β is chosen as 1₂1K. 2 4 6 8 10 12 14 16 18 20 10−6 10−4 10−2 100 102 104 AS=7.2o, SNR=0dB Iteration NMSE EM MEM proposed LSFC SSFC

b

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2 4 6 8 10 12 14 16 18 20 10−5 100 105 AS=15o, SNR=10dB Iteration NMSE EM MEM proposed LSFC SSFC

EM-based estimators, where AS= 15◦, SNR=10dB, and full modeling order is used. Initial

b β is chosen as 1₂1K. 2 4 6 8 10 12 14 16 18 20 10−6 10−4 10−2 100 102 104 AS=15o, SNR=0dB Iteration NMSE EM MEM proposed LSFC SSFC

EM-based estimators, where AS= 15◦, SNR=0dB, and full modeling order is used. Initial bβ

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0 20 40 60 80 100 120 140 160 180 200 10−5 10−4 10−3 10−2 10−1 100 AS=7.2o Number of antennas NMSE 0dB 5dB 10dB 15dB SSFC LSFC

Figure 3.8: MSE performance of the proposed LSFC and SSFC estimator versus number

of BS antennas and received SNR, where AS= 7.2◦, and full modeling order is used.

0 20 40 60 80 100 120 140 160 180 200 10−5 10−4 10−3 10−2 10−1 100 AS=15o Number of antennas NMSE 0dB 5dB 10dB 15dB SSFC LSFC

Figure 3.9: MSE performance of the proposed LSFC and SSFC estimator versus number

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LSFC. Consider Fig. 3.10, the red lines with × represents the nearest MS while the blue lines with ◦ represents the farthest MS from the BS. Assume the pilot power is such that the received SNR from farthest MS at BS equals to 10.22dB for left figure and 20.22dB for right figure. It can be seen clearly that the pilot power has little effect on the MSE performance of LSFC estimator due to the robustness of our estimator to noise. Nevertheless, when considering the full-order SSFC estimator, the nearest MS indeed has better MSE performance.

0 50 100 150 200 10−5 10−4 10−3 10−2 10−1 100 Number of antennas MSE/ β Nearest (36.71dB) Farest (10.22dB) 0 50 100 150 200 10−4 10−3 10−2 10−1 100 Number of antennas MSE/ β Nearest (SNR=46.71dB) Farest (SNR=20.22dB) SSFC LSFC SSFC LSFC

Figure 3.10: MSE performance of the proposed LSFC and SSFC estimator versus number of BS antennas with different user location, hence different received SNR at BS (indicated

大型天線陣列系統的秩值與複合通道估計及閉迴路傳收機之設計

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

大型天線陣列系統的秩值與複合通道估計

及閉迴路傳收機之設計

Rank Determination, Composite Channel Estimation and

Closed-loop Transceiver Design for Massive MIMO Systems

研 究 生：陳科夆

指導教授：蘇育德 教授

大型天線陣列系統的秩值與複合通道估計

及閉迴路傳收機之設計

Rank Determination, Composite Channel Estimation and

Closed-loop Transceiver Design for Massive MIMO Systems

研 究 生：陳科夆 Student：Ko-Feng Chen

指導教授：蘇育德 Advisor：Dr. Yu T. Su

國 立 交 通 大 學

電信工程研究所

碩 士 論 文

大型天線陣列系統的秩值與複合通道估計及閉迴路傳收機之設計

學生：陳科夆

指導教授：蘇育德

國立交通大學

電信工程研究所碩士班

摘

要

我們探討一個裝有大型天線陣列基地台(BS)來服務多個單天線用戶(UE)的

多輸入多輸出(MIMO)分時多工系統。此系統藉由上行(UL)的領航信號(pilot)來估

計包含大尺度衰減係數(LSFC)及小尺度衰減係數(SSFC)的通道狀態資訊(CSI)。

雖然在MU-MIMO或是分散式MIMO系統之運作，LSFC資訊是不可或缺的，然而

有關MIMO通道估計之研究往往被假設為已知或是被忽略。我們利用大型天線陣

列之通道硬化(channel-hardening)並能同時收到大量空間樣本的特性，在不需

SSFC資訊的前提下能有效的利用相對少量的領航符元精準地估計LSFC。

至於SSFC的估計，我們利用降秩(rank-reduced, RR)通道模型來完成。由於這

種方法之降秩效應需選擇適當的基底並有準確的秩值估計，後者又需先知道通道

的空間相關矩陣。針對這三項議題我們首先分析了最佳先設(predetermined)基底

的選擇，證明兩種常用的基底之近優性(near-optimality)。接著我們探討秩值對

SSFC估計的性能影響、設計一套秩值決定的演算法，最後並發展了估計空間相

關矩陣的演算法。這些成果乃是以我們對SSFC估計法的均方誤差(mean squared

error, MSE)性能的詳細分析為基礎。我們結合了大、小尺度衰減係數的估計並證

明在接收信號之入射角度擴散(AS)不大時，還可利用適當的RR模型來一併估計

平均的接收角度(AoA)。比起使用不含AoA資訊的模型之通道估計法，這種方法

不但可降低MSE而且所估得的角度資訊可用來形成下傳鏈路的波束。

最後，我們推導了上行領航信號的設計方式，並且在含括前述的複合通道估

計器後，提出了分別適用於分時多工和分頻多工模式的閉迴路傳收機(closed-loop

transceiver)設計流程與細部演算法。由電腦實驗結果可以看出我們的複合通道估

計器、秩值決定及空間相關矩陣估計等演算法在大型天線陣列系統中的優異表

現。

Rank Determination, Composite Channel

Estimation and Closed-loop Transceiver Design

for Massive MIMO Systems

誌

謝

對於論文得以順利完成，首先感謝指導教授 蘇育德博士。老師的諄諄教誨

使我對於通訊領域的研究有更深入的了解，也教導我們許多書上學不到的知識與

人生道理，讓我受益良多。並感謝口試委員蘇賜麟教授、祁忠勇教授、趙啟超教

授及林大衛教授給予的許多寶貴意見，以補足這份論文的缺失與不足之處。

也由衷感謝實驗室的劉彥成學長，在我研究上有問題時，能給予我建議以及

討論，使我的研究能夠順利完成，從中學習到的經驗是很珍貴的。另外也感謝實

驗的學長姐、同學、及學弟妹們，在這兩年內的互相支持與鼓勵。

最後，要感謝我的家人及朋友，他們總是在背後默默的關心與支持，使我有

動力可以努力往前進，在此僅獻上此論文，以代表我最深的敬意。

Contents

List of Tables

List of Figures

Chapter 1

Introduction

Chapter 2

Preliminaries

2.1

Conventional MIMO System Model

2.1.1

Single-user MIMO System

2.1.2

Multiuser MIMO System

ܯ

2.2

Effect of Massive MIMO System

broadside

mean AOA

國立交通大學

碩士論文

研究生：陳科夆

指導教授：蘇育德教授

研究生：陳科夆 Student：Ko-Feng Chen

國立交通大學

碩士論文

對於論文得以順利完成，首先感謝指導教授蘇育德博士。老師的諄諄教誨