分散式放大前送多輸入多輸出中繼站網路之設計

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國

立交通大學

電子工程學系電子研究所

博士論文

分散式放大前送多輸入多輸出中繼站網路之設計

Design of Distributed Amplify-Forwarding MIMO Relay

Networks

研究生：吳俊榮

指導教授：林大衛

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分散式放大前送多輸入多輸出中繼站網路之設計

Design of Distributed Amplify-Forwarding MIMO Relay Networks

研究生：吳俊榮 Student：Chun-Jung Wu

指導教授：林大衛 Advisor：David W. Lin

國立交通大學

電子工程學系電子研究所

博士論文

A Dissertation

Submitted to Department of Electronics Engineering and Institute of Electronics

College of Electrical and Computer Engineering National Chiao Tung University

in partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in

Electronics Engineering

July 2011

Hsinchu, Taiwan, Republic of China

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分散式放大前送多輸入多輸出中繼站網路之設計

研究生：吳俊榮指導教授：林大衛博士國立交通大學電子工程學系電子研究所

摘要

多進多出系統(MIMO)搭配中繼站之應用可提升系統涵蓋率或提高系統容量。採用分散式平行單天線中繼站網路則額外享有建置彈性、多集性與成本之優勢，然而亦面臨因結構上缺乏中繼站相互連通而衍生的設計議題。本論文針對分散式放大 -前送(amplify-forwarding)中繼站型態，首先考慮提升中繼站容量之中繼前送增益(forwarding)設計。在設計中數學結構的限制下，目前並無實用有效之快速設計方式，因此我們採用矩陣低階更新(matrix low-rank updating)的方式，設計遞迴式最佳化解法。此解法雖可快速計算並提供提高系統容量之設計，對於分散式中繼站之特性研究並無實質貢獻，因此我們設計雜訊優勢模型以簡化系統，並據此提除對應之增益設計方法。此設計方法搭配中繼站部分啟用之模式，我們發現此設計方法可類比於單站多天線選擇系統 (single-hop MIMO antenna selection)。因此我們可採用多天線選擇系統之已知特性，類比分析我們針對中繼站所提出之設計方法。由模擬結果可知，我們所設計之設計方法可有效提升系統容量，且印證類比於多天線選擇系統之容量分佈特性。當中繼站的數量增加時，基於中繼站選擇之設計方法將面臨運算量大幅上升之問題。同樣還是由優勢雜訊模型出發，我們設計新的增益最佳化運算並避免中繼站選擇。此演算法更進一步簡化並轉換設計問題，並使用修正之解題空間(solution

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模擬結果可知，免除中繼站選擇之演算法，相較於使用中繼站選擇方法，其效能相若或更好。

在第三部分我們討論針對高度散佈(highly dispersive)通道之分頻正交多工系統(OFDM)之通道估計，以及其應用於使用 OFDM 之中繼站網路。對於 OFDM 通道估計問題，時域 (time-domain) 通道估計方法可利用有限的領航訊號 (pilot symbols)提供精確之通道估計。然而此類方法須掌握通道多路徑之延遲位置 (multipath delay positions)。已知針對延遲位置估計的方法需要特定格式之領航訊號，且在通過慢速衰落(slow fading)通道時可能會降低估計效能。我們所設計之延遲位置估計，採用追求最大程度相合(matching pursuit)之演算法，考慮相鄰 OFDM 訊框中領航訊號，且不限制領航訊號的位置。經由模擬結果可知，我們所設計之延遲位置估計方法，可大幅提升通道估計之準確率。另外我們討論採用 OFDM 之放大前送中繼站其對應之傳輸通道模型，經由分析得知可對應於一高度散佈之傳輸通道，因此適用於本文所提出之估計方法。

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Design of Distributed Amplify-Forwarding

MIMO Relay Networks

Student: Chun Jung Wu Advisor: Dr. David-W. Lin

Department of Electronics Engineering & Institute of Electronics National Chiao Tung University

Abstract

Relays can potentially enhance the transmission performance of multi-input multi-output (MIMO) systems. A parallel single-antenna relay network has additional advantages in flexibility, diversity, and cost, but also poses significant design problems because the absence of inter-antenna connections over different relays makes the underlying mathematical problems much more difficult to solve. In this thesis, we first consider the design of parallel amplify-and-forward relay networks. More specifically, we focus on the design of relay gains to maximize the system capacity. As no closed-form analytic solution can be found, we first develop an iterative algorithm based on low-rank updating to efficiently find a locally optimal solution. Since iterative optimization provides little insight into the analytical properties of the solution, we attempt analytical solutions based on asymptotic noise conditions and relay selection. It is discovered that the proposed algorithms could be modeled as single-hop MIMO antenna selection systems and could be analyzed in the similar way. We examine the resulting capacity outage diversity orders and confirm the analysis and equivalent modeling with simulation results.

As the algorithms based on relay selection may encounter expensive computation burden as number of relays increases, we develop algorithms based on noise dominant models but avoid computational intensive relay selection. The proposed algorithms are made possible by modified criterion and specifically constrained solution space. Low-leakage beamforming used for multiuser communication is applied in our algorithm. Numerical results demonstrate the proposed algorithms exhibit comparable

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In the third part of this thesis, we consider OFDM channel estimation with highly dispersive channel and application in distributed relay network. Time-domain channel estimation techniques have been proposed for OFDM systems for their ability to yield relatively accurate estimates with only a few pilots. Key information needed in such techniques is the multipath delays of the channel. Prior approaches to estimation of multipath delays require regular pilot structures and may not work in slow fading. We propose a group matching pursuit technique for channel estimation. The technique is an extension of the orthogonal matching pursuit technique. It employs the pilots in several OFDM symbols to estimate the multipath delays in a sequential manner, where the pilots can have an arbitrary structure. Simulation results show that the proposed algorithm has superior performance. We then demonstrate that distributed AF relay network with OFDM transmission could be modeled as single-hop OFDM system with equivalently high dispersive delay profile. Thus we apply and examine the proposed channel estimation schemes for the relay network of interest.

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誌

謝

這份論文與漫長的修業得以完成，首先要感謝我的指導教授林大衛博士。林教授在這一路艱辛的過程中，提供我全方位的支持，在我的研究進度陷入困頓之時，還是不斷的鼓勵與協助。林教授長期的栽培與指導，對於我的研究與學位是最關鍵的因素，對於我個人則是永誌不忘的師恩。電子系桑梓賢教授亦提供諸多寶貴意見，並協助提升研究的品質與成果。在兩位老師的提攜之下，我才得以在專業的道路上徐緩學步前行。我的家人與玉如在我成長與學業的過程中，以永恆無盡的包容與關懷，陪伴我走過幽暗，支持我奮力向前。親情的力量與偉大無以名狀，如果個人幸得尺寸之功，所有的成果皆榮歸於我的親人。

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List of Figures

2.1 MIMO system with distributed relays. . . 9

2.2 Progression of average capacity with number of iterations. . . 20

2.3 Average of normalized capacity with respect to varying β values . . . 20

3.1 Comparison of MIMO antenna selection and full MIMO systems in terms of CDF of capacity (top) and biased capacity (bottom). . . 24

3.2 Capacity CDF of distributed relay system designed under destination noise-dominant assumption. . . 29

3.3 Horizontally “biased” capacity CDF curves of distributed relay systems for diversity comparison. . . 30

3.4 Capacity CDF of distributed relay system designed under relay noise-dominant assumption. . . 30

3.5 Comparison of diversity orders of capacity CDFs of distributed relay sys-tems of different sizes designed under relay noise-dominant assumption. . . 31

4.1 Downlink signal and interference flows for user 1 . . . 34

4.2 Downlink signal and leakage flows from user 1 . . . 35

4.3 Simulate under relay-noise dominating condition. . . 45

4.4 Simulate under destination-noise dominating condition. . . 45

4.5 Simulation with two noise terms having equal power level . . . 46

5.1 In the pth iteration,projection and selection based on projection residual . 51 5.2 Relay operations for OFDM transmission . . . 53

5.3 Normalized mean-square channel estimation errors of different channel es-timation methods . . . 55

5.4 Average symbol error rates (SERs) at data subcarriers with different chan-nel estimation methods. . . 56

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5.5 Average SERs at data subcarriers with different channel estimation meth-ods when multipath delay are spaced apart. . . 56 5.6 Average SERs of OFDM distributed relay network at data subcarriers with

different channel estimation methods when multipath delay are randomly spaced. . . 57

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Chapter 1 Introduction

1.1 Research Background, Motivation and Issues

1.1.1 On the Capacity of Distributed AF MIMO Relay Network

Relays have been considered a useful means for coverage extension and capacity enhance-ment of wireless systems [33]. Among all conceivable relaying strategies, two have received the most attention: amplify-and-forward (AF) [30] and decode-and-forward (DF) [31]. In AF systems the relays amplify or beamform the received signals without further process-ing, while in DF systems they decode (or demodulate if there is no channel coding) the received signals and transmit the re-encoded (or remodulated) signals to the destination. Besides the forwarding strategies, an important subject in relay system design is the over-all wireless system architecture. In this, due to the capacity advantage of multi-input multi-output (MIMO) transmission over single-input single-output (SISO) transmission, many have sought to incorporate some MIMO concepts one way or another. The present work is concerned with AF-based distributed relay networks, whose architecture will be described further later.

The simplest relay-aided transmission system consists of three nodes: source, relay (cooperator) and destination [44]. To facilitate MIMO transmission, an intuitive approach is to install multiple antennas on one or more of the nodes. For simplicity, consider the situation where the source and the destination have an equal number of antennas. A case with a single-antenna source (SAS) and a single relay (SR) equipped with multiple antennas (MAR) is considered in [48]. A natural extension to have a multiple-antenna source (MAS) and an SR-MAR to enable spatial multiplexing [25, 6]. In studies of MAS-SR-MAR systems, the multiple antennas on each terminal are usually assumed to be fully connected and may have arbitrary interconnection weights. In this case, known matrix theory can be used to decompose a MIMO transmission channel into parallel SISO links (via, for example, the singular value decomposition (SVD) or the QR decomposition).

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Each spatially multiplexed signal stream can then be transported over one parallel link, and the matrix decomposition can be viewed as simultaneous beamforming for these streams. Typical performance measures, such as the signal-to-interference-plus-noise ratio (SINR) or the mean-square error (MSE) in received signal values, can be expressed in

terms of the parameters of the decomposed channel. System optimization may then

become essentially a problem of power allocation among the individual streams [25, 6]. On the other hand, use of multiple, parallel relays (PR) has also been considered by many researchers and shown to be potentially beneficial in various aspects [5, 21, 40, 22, 13, 9, 12, 24, 26, 11, 15, 47, 20, 8, 3]. For example, it is found that an increased number of relays can benefit the system capacity [5]. In fact, the parallel relays can function as virtual transmitter antennas and effect transmitter diversity either in the form of distributed space-time coding [21, 40] or in the form of distributed beamforming [22, 13, 9]. The corresponding diversity order has been examined in [22] and [12], respectively. Moreover, parallel relaying has also been studied in the contexts of sensor networks [24], two-way relaying [26], and secrecy communication [11]. However, despite the potential benefits, the fact that the relays are not connected but stand in parallel raises a cooperation problem which, if not dealt with, could severely limit the realizable benefit.

To see why, let L be the number of parallel relays and Ni (1 ≤ i ≤ L) the number

of antennas on relay i. Let M denote the number of antennas on the source terminal as well as that on the destination terminal. Consider first the simplest case where each

terminal has only one antenna, i.e., SAS-PR-SAR where M = Ni = 1 ∀i [13, 9]. In

this case, the relays effectively constitute a distributed beamformer for the single signal stream. Applying the same design philosophy to an MAS-PR-MAR system with M > 1

and Ni > 1 ∀i, there can be MS = min{M, Ni ∀i} concurrent signal streams. The

beamforming techniques used in MAS-SR-MAR systems can be extended to this scenario with a twist [15, 47]. That is, the available antennas on the relays can be used to provide MS parallel subchannels between the source and the destination. Systems operating in

the above ways have been considered in some works [48, 13, 9, 15, 47, 25, 6]. In terms of capacity, however, such systems suffer from two consequences. First, the number of supported subchannels (i.e., the number of concurrent spatially multiplexed streams) does not grow with the relay number L, but is upper-bounded by MS. Secondly, to increase the

number of streams we need to ensure that all relays are equipped with sufficient antennas. Designs that can obviate the above limits are of interest and importance.

In this work we consider the design of MAS-PR-SAR systems (where Ni = 1 and

P Ni = L) to support multiple signal streams. More specifically, we consider the design

of AF relay forwarding gains for maximization of system capacity. Previously, Jin et al. [20] considered the case where the relays had equal gain and analyzed the statistics of the resulting ergodic capacity. Chen et al. [8] considered the minimization of transmission power subject to per-stream SINR targets. The problem is related to system capacity, but somewhat indirectly. Bae and Lee [3] proposed algorithms for capacity optimization under the condition that the product of the source-to-relay and the relay-to-destination

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channel matrices was asymptotically diagonal in the limit of a large number of relays. But in sum, there is as yet no extensive work on the design of distributed parallel relay networks for capacity maximization. Actually, the relationship between number of relay terminals and system capacity also needs to be further clarified. The present work is motivated by these observations.

We consider two approaches to maximizing the capacity of a distributed relay network with presence of perfect channel state information (CSI). The first is algorithmic, as

so far no closed-form solution to the problem exists. However, although algorithmic

optimization can yield good results, it provides little insight into the analytical properties of the solutions. We thus also attempt an analytical approach. Because no closed-form solution can be obtained for the general situation, we consider two asymptotic situations which are more amenable to analysis. In one of them the relay noise dominates the overall noise in the received signal at the destination and in the other the destination terminal noise dominates. Alternatively, these two situations can also be viewed as providing two upper bounds to the system capacity.

Given the simplification lead by noise-dominant models, it is still a challenging task to optimize the relay network performance. Instead of considering all relays simultaneously, we discover that closed form optimization is possible if only partial of relays are active. In consequence, we develop algorithms for capacity improvement which work with relay selection. In addition, we then observe the proposed selection-based algorithms could be modeled and analyzed with equivalent single-hop MIMO antenna selection systems [16, 17]. Given the results and analysis of existing works in the area of MIMO antenna selection, we gain more in depth understanding about the application of proposed algo-rithms for distributed relay networks.

As the number of relays increases, it is found that both capacity and outage diversity order increases. However, the proposed algorithms based on relay selection may become intractable as the number of relay terminals is large. To counteract the situation, we design further simplifications by modified criterion and shrunken solution space. Though being suboptimal in nature, the proposed algorithms avoid computational intensive oper-ation and could fully utilize the whole relay network. Simuloper-ations confirm the proposed approaches could result in slightly inferior or better performance than selection-based schemes, with considerably smaller computation cost as number of relays is large.

1.1.2 On the OFDM Channel Estimation and Applications in

Relay Networks

Coherent demodulation of orthogonal frequency-division multiplexing (OFDM) signals critically depends on proper channel estimation. Since OFDM systems usually reserves some subcarriers as pilots, most channel estimation methods are pilot-aided. A com-mon approach is to estimate the channel frequency response at pilot locations first, and

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then “extend” the estimate to other subcarrier locations. One frequently considered way of “extension” is low-order polynomial interpolation, which can take the form of one-dimensional interpolation in the frequency domain (within the boundary of one OFDM symbol) or two-dimensional interpolation over frequency and time (across several OFDM symbols) [23], [7]. The performance of this sort of methods is limited by the pilot density and the channel characteristics. For example, if the channel has small coherence band-width (i.e., long delay spread) and low coherence time (e.g., due to fast motion) and the pilots are widely spaced in frequency, then they would have difficulty obtaining accurate channel estimates.

Another way of “extension” is based on exploiting the time-domain characteristics of the channel [29]. Since, in many cases, only a few multipaths contribute significantly to the channel response (in other words, the channel is “sparse”), the unknowns in time-domain estimation (which consist of the path coefficients of the significant multipaths if their delays are known) are usually much fewer than that in frequency-domain-based interpolation (which consist of the frequency response at all subcarriers). Hence the few pilots can be put to better use and result in more accurate channel estimates. This is especially the case when the pilots are very few and very widely spaced (as, for example, in the case of the IEEE 802.16-2004 OFDMA uplink [19]).

Evidently, a fundamental issue in time-domain channel estimation is to find the delays of the significant multipaths. In [46], an effective delay acquisition technique is developed, but the pilots need to be equally spaced. In [32], the MUSIC algorithm widely used for spectrum analysis is employed for channel estimation, but again assuming equally spaced pilots. The algorithm can be easily extended to deal with irregular pilot spacings, but the pilot locations in the multiple OFDM symbols used in channel estimation should be identical. To the best of our knowledge, there is no time-domain channel estimation technique proposed so far that makes use of arbitrarily organized pilots in multiple OFDM symbols in the presence of channel fading.

In this work, we propose an effective technique for time-domain sparse channel esti-mation based on the matching pursuit (MP) approach. MP algorithms have been used in audio and video signal processing to select bases of subspaces [27], [1]. We extend the prior MP method for multipath delay estimation by jointly considering a group of OFDM symbols; thus we term the proposed algorithm a group MP (GMP) algorithm. And we design the algorithm such that it can deal with arbitrary pilot assignment that may vary from OFDM symbol to OFDM symbol. In [43], [34] the similar MP based processing for multiple measurement vectors are discussed. Note that the dictionary, or the range of basis searching, in these works is unique. In the scenario of this contribution, however, there are multiple reference dictionaries due to arbitrary pilot assignment.

The proposed subspace-based approaches for OFDM channel estimation is effective to highly dispersive frequency-selective channels with limited resource of pilots, which is also an potential threat to distributed relays with OFDM transmission. For most works on relay systems, flat-fading channels are the presumed channel models. However in

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practice frequency selective channels would pose substantial effects to relay networks We demonstrate that for AF distributed relays, the end-to-end OFDM transmission could be modeled as equivalent single-hop OFDM with a delay profile composed of summation of multiple convolution, and consequently exhibits severe channel dispersion. We would apply the proposed schemes for the relay transmission and examine the performance.

1.2 Outline of This Thesis

In Chap. 2, we first briefly describe the system model of amplify-forwarding distributed relay network used in this work, then present capacity (i.e. mutual information) of this model and the associated optimization problem. To simplify the optimization and gain some insights into system design, we discuss the effects of relay transmission power scaling which ultimately leads to an upper bound of capacity. As our first approach to capacity optimization, in the third part of this chapter we present the alternating optimization. Specifically, this approach applies successive greedy optimization, low-rank updating of matrix computation, and quadratic approximation. Finally some numerical results are shown to confirm the performance of proposed approach.

While alternating optimization based on low-rank updating discussed in Chap. 2 can yield good results, it provides little insight into the analytical properties of the solutions. We thus consider an analytical approach in Chap. 3. In particular, note that in AF sys-tems the receiver noise arises from two sources: the relay noise nR and the destination

terminal noise nD. The design problem becomes mathematically more tractable when

one of the two dominates in the overall receiver noise so that the other may be ignored. We term the simplification as noise-dominant models. Algorithms to design relay gains corresponding to the models are discussed in this chapter. We discover that the proposed algorithm could be linked to equivalent single-hop MIMO antenna selection system. Thus we would first briefly review the important results and findings on single-hop MIMO an-tenna selection systems. For analytical insights we show how to model equivalent MIMO antenna selection systems. Finally some numerical results are shown for performance eval-uation and validating the links between equivalent MIMO antenna selection and proposed relay selection schemes.

We discuss system simplification and approximation of distributed relay network based on noise-dominant models in Chap. 3, wherein relay selection algorithms and associated relay gains designs are also presented. Though the algorithms based on relay selection is conceptually concise, the computation burden would growth exponentially as the number of relays increases. To circumvent the problem, in Chap. 4 we discuss algorithms based on noise-dominant models but avoid relay selection. For relay-noise dominant model we cast the problem into projections with two subspace and minimizing the ratio of projected vector norms. In Sec. 4.1.1 the idea and algorithm for multiuser low-leakage beamform-ing [35] are briefly reviewed. We would apply the algorithm to solve minimization of

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norms in Sec. 4.1.3. As for destination-noise dominant model, in Sec. 4.2.2 we simplify the problem by making the end-to-end MIMO channel H in (2.3) an upper-triangular matrix so that the matrix determinant maximization in (2.4) could be approximated as product of diagonal terms of H. Then in Sec. 4.2.3 we transform the design problem with specifically constrained solution space and propose iterative algorithm to reach local optimizer. Finally we summarize all the proposed algorithms (with and without relay se-lection), and compare the order of computation complexity for selection-based algorithms and proposed efficient designs. Numerical results of respective algorithms are shown and compared in Sec. 4.3.

In Chap. 5 we discuss subspace-based algorithms for OFDM transmission with highly dispersive delay profile and limited pilot resources. We propose MP based algorithms to reconstruct the delay subspace which enables efficient and accurate time-domain channel estimation. Then we present the channel modeling of distributed AF relay network with OFDM transmission. It turns out the relay network of interest actually suffer equivalent highly dispersive channel. Thus we propose the apply subspace-based algorithm to handle the channel estimation. Finally in Chap. 6 we provide some concluding remarks on this thesis work.

1.3 Contribution of This Thesis

We briefly outline the contribution of this work.

1. We propose iterative alternating optimization to improve the capacity of distributed AF MIMO relay network. The proposed algorithm is based on matrix low-rank updating and thus could efficiently generate optimized results.

2. Noise-dominant models is proposed to simplify system design on distributed AF MIMO relay network. We show the models serve as upper bounds of system capacity and lead to several efficient design algorithms.

3. Based on noise-dominant models and relay selection, we develop two corresponding relay gains designs. We also discover the performance of proposed algorithm could be modeled as equivalent MIMO antenna selection systems. Thus we could reply on the well-studied characteristics of MIMO antenna selection to approximate and analyze the proposed algorithms.

4. To circumvent the potential expensive computation burden of selection-based schemes, we further transform and simplify the design problem in order to consider and uti-lize all the relays simultaneously. For the relay noise dominant model, we develop algorithms based on low-leakage beamforming designs. As to destination noise dom-inant model, we propose to simplify the problem by matrix triangularization. An

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algorithms with specifically constrained solution space is developed to iteratively improve the results.

5. To perform OFDM channel estimation with highly dispersive delay profile but with few pilot or training symbols, we apply subspace-based algorithm to efficiently utilize limited pilot resource. Since the unknown subspace is critical to this approach, we apply OMP and invent GMP to iterative reconstruct the subspace. The proposed MP based algorithms could bypass some limitations of other algorithms and utilize the information brought by consecutive OFDM symbols. Also we discover that AF distributed relay network with OFDM transmission could be modeled as single-hop OFDM with equivalent highly dispersive channel. Thus the proposed subspace-based algorithms is suitable to handle the channel estimation for the relay network of interest.

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Chapter 2 Capacity of Relay Network and

Alternating Optimization

In this chapter, we first briefly describe the system model of amplify-forwarding dis-tributed relay network used in this work, then present capacity (i.e. mutual information) of this model and the associated optimization problem. To simplify the optimization and gain some insights into system design, we discuss the effects of relay transmission power scaling which ultimately leads to an upper bound of capacity. As our first approach to capacity optimization, in the third part of this chapter we present the alternating op-timization. Specifically, this approach applies successive greedy optimization, low-rank updating of matrix computation, and quadratic approximation. Finally some numerical results are shown to confirm the performance of proposed approach.

2.1 System Model and Capacity of Relay Networks

2.1.1 System Model and Power Limits

We consider a distributed MIMO relay system consisting of one source terminal, one destination terminal and L single-antenna relay terminals. Both source and destination are equipped with M antennas. To preserve the degree of freedom in the end-to-end transmission, it is reasonable to assume L ≥ M . Fig. 2.1 demonstrates a conceptual representation for the system of interest.

Let x and y (x, y ∈ CM_{) denote the signal vector transmitted from the source and}

received at the destination, respectively. Let r = [r(1) r(2) · · · r(L)]T represent the for-warding gain vector of relay network, wherein the ith relay performs amplify-forfor-warding by multiplying the received signal with the complex gain r(i) and then transmitting the result. Signals in the system pass through two MIMO channels, F and GH, to go from the source to the destination, where F ∈ CM ×L _{and G}H

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des-Fig. 2.1: MIMO system with distributed relays.

tination and relays and channel between relays and source, respectively. The superscript ()H _{defines matrix Hermitian transpose. The received signals at the relays are assumed to}

be subject to additive complex circular white Gaussian noise nR ∼ CN (0, σR2IL), where

IL denotes an identity matrix of size L × L. Likewise, the received signals at the

des-tination are assumed to be subject to additive complex circular white Gaussian noise nD ∼ CN (0, σD2IM). Assume that there is no direct propagation path from the source to

the destination. The received signal vector y at the destination can be described as

y = F diag(r)GHx + F diag(r)nR+ nD

, F RGHx + F RnR+ nD (2.1)

where R is a diagonal matrix defined from r, and diag(r) denotes a diagonal matrix whose diagonal terms are equal to r.

To optimize the system capacity fairly, transmission power limits are imposed on the source and the relays. Specifically, we assume that the source transmits independent signal streams over its M antennas with equal power σ2

x, subject to a total power limit of

PS. Further, the relays are subject to a total power limit of PR. Mathematically, these

constraints can be expressed as

PS ≥ tr(E{xxH}) = M σ2x, PR ≥ tr(E{(RGHx + RnR)(RGHx + RnR)H}) = L X i=1 (σ_R2 + σ_x2|G(i)|2)|r(i)|2 _, L X i=1 p(i)|r(i)|2,

where |G(i)|2 _{= (G}(i)

)HG(i), with G(i) representing the ith column of G and being asso-ciated with the channel vector between the source and the ith relay.

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2.1.2 Noise Whitening and Capacity

Note that the noise vector (F RnR+nD) in (2.1) as received by the destination is spatially

correlated. To find the system capacity, a noise whitening filter W−1/2 at the destination is used, where W is given by

W = E{(F RnR+ nD)(F RnR+ nD)H}

= σ_D2(I + σ2(F R)(F R)H), (2.2)

with σ , σR/σD. Let H denote the noiseless end-to-end equivalent MIMO channel, i.e.,

H ,pσ2

xF RG

H_{. Then we may derive the system capacity as a function of R as [41]}

log₂_{C(R) , log}₂det(IN + HHW−1H)

= log₂det(W + HHH) − log₂det(W ) (2.3)

where det(·) denotes matrix determinant. The optimization problem can be stated as Ropt = arg max

R C(R) (2.4) subject to PS ≥ M σx2, PR ≥ L X i=1 p(i)|r(i)|2. (2.5)

By observing (2.3) it is clear increasing source transmitting power or σx is always

beneficial to capacity regardless of channel condition or the design of R. Thus we may readily set σ2

x = M PS. Further, the combination of σ2x and G matrix in (2.3) could be

replaced by a scaled version of G without lose any generality, so we ignore σx hereafter

in this work.

2.2 Power Scaling and Upper bounds of Capacity

The inequality about PS in (2.5) is discussed in last section. The remaining inequality

power constraint naturally prompts one to think: is it possible to simplify the constraint by considering only the equality therein without impacting the optimality of the solution? Or, alternatively, given a certain r that satisfies (2.5) with inequality, will the system capacity be increased by scaling r to reach equality in (2.5)? Intuitively, the answer may seem to be a no-brainer as increasing the transmission power should be beneficial to the signal-to-noise ratio (SNR) and thus the capacity. However, because R affects both H and W , it is not easy to intuitively determine in (2.3) the consequence of scaling R up. Thus a solid proof nevertheless requires some works. We state the result as a theorem and present the proof as follows.

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Theorem 2.2.1 (Capacity scaling). When the (complex) relay gains R are scaled by s ∈ C with |s| > 1, C(sR) > C(R).

Proof. First, it is clear in (2.3) that C(sR) = C(|s|R). Without loss of generality we assume s ∈ R+ _{(the set of positive real numbers) hereafter.}

Consider a singular value decomposition of F R given by

F R = U ΛVH (2.6)

where for convenience we let Λ be M × M . Thus U ∈ CM ×M _{is the matrix of left singular}

vectors as usual, but the matrix of right singular vectors V becomes L × M , that is,

V ∈ CL×M. Further, let the singular values along the diagonal of Λ be arranged in

descending numerical order. Let λi denote the ith diagonal element in Λ. Substituting

the above into (2.2) and (2.3), we get

W = U ΣUH,

C(R) = log det{IM + G[(F R)HW−1(F R)]GH}

= log det{IM + G[V ΣVH]GH}, (2.7)

where Σ and Σ are diagonal matrices with their ith diagonal terms given by

Σ(i, i) = σ_D2 + (σR|λi|)2, (2.8) Σ(i, i) = |λi| 2 σ2 D+ (σR|λi|)2 = |λi| 2 Σ(i, i). (2.9)

By scaling R to sR and using Ws to denote the resulting noise correlation matrix at

the destination (in place of W ), we find

Ws = σ2DI + (sσR)2(F R)(F R)H = U ΣsUH,

C(sR) = log det{IM + G[s2(F R)HW−1s (F R)]G H_}

= log det{IM + G[V ΣsVH]GH}, (2.10)

where Σs and Σs are diagonal matrices with their ith diagonal terms given by

Σs(i, i) = σ2D+ (sσR|λi|)2, Σs(i, i) = s|λi|2 σ2 D+ (sσR|λi|)2 , a iΣ(i, i), (2.11) with ai defined as ai = s2[σ_D2 + (σR|λi|)]2 σ2 D + s2(σR2|λi|)2 . Note that ai > 1 and ai ≤ aj for i ≤ j.

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From (2.10) and (2.11), Σs can be expressed as the sum of two diagonal matrices as

Σs= a1Σ + Σ∆,

where Σ∆is some nonnegative diagonal matrix. Then, based on the eigenvalue inequalities

concerning the sum of two nonnegative-definite matrices [37, Sec. 6.4], we have

C(sR) = log det{IM + G[V ΣsVH]GH}

≥ log det{IM + a1G[V ΣVH]GH}

> log det{IM + G[V ΣVH]GH} = C(R).

Therefore, we confirm that scaling up of the relay gains can increase system capacity. Hence we may simplify the optimization constraint to

PR= L X i=1 (σ2_R+ |G(i)|2_)|r(i)|2 , L X i=1 p(i)|r(i)|2. (2.12)

That is, the relays should transmit at the maximum allowed total power.

Next, one may wonder if the capacity could increase without bound if the total relay transmission power tends to infinity. Intuitively, the answer may appear to be another no-brainer because, from (2.1), the quality of the source-to-relay links should place a cap on the amount of information rate that the system can support no matter how much the relay transmission power can be. But again, a solid mathematical proof requires a few lines of reasoning. Again we state the result as a theorem and present the proof as follows. Theorem 2.2.2 (Asymptotic capacity with high relay power). As |s| → ∞, C(sR) is upper-bounded by

log det[IM + σR−2GG H_]

and it approaches the upper bound if and only if G and F R span the same row space. Proof. To complete the proof, we will need the following inequality of eigenvalues about matrix product [37, Sec. 6.6]. Assume A and B are N × N Hermitian non-negative definite matrices, and let ρi(A) be the ith largest eigenvalue of A. Then we have

ρi(A)ρN(B) ≤ ρi(AB) ≤ ρi(A)ρ1(B)

From (2.11), as |s| → ∞ the significance of σD vanishes, so that Σs(i, i) . σ−2R and

C(sR) ≈ log det[IM + σR−2GV V

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where V is the matrix of right singular vectors of F R as given in (2.6). To obtain the result, the key is to grasp the eigenvalue structure of GV VHGH, or equivalently that of GHGV VH. For this, let ρi(M ) denote the ith largest eigenvalue of a matrix M that

has real eigenvalues. We have

ρ1(GHG) ≥ ρ2(GHG) ≥ · · · ≥ ρM(GHG) > 0,

ρi(V VH) = 1, 1 ≤ i ≤ M,

ρi(GHG) = ρi(V VH) = 0, M + 1 ≤ i ≤ L.

Therefore, based on the eigenvalue properties concerning matrix products [37, Sec. 6.6], we have

ρi(GHGV VH) ≤ ρi(GHG)ρ1(V VH)

= ρi(GHG). (2.14)

The equality in the first line of the above equation holds if and only if G and V span the same row space, or equivalently, if and only if G and F R span the same row space. In conclusion, as |s| → ∞,

C(sR) ≤ log det[IM + σ−2_R GGH], (2.15)

where the equality holds if and only if F R and G span the same row space.

With Theorem 2.2.2, it is verified that C(R) is upper-bounded irrespective of the power level of the relays.

2.3 Iterative Alternating Optimization

In Sec. 1.1.1 we briefly review the state of the art of relay network designs and the challenges of distributed relay networks. In this section we take a closer look at the optimization problem in (2.4) and present the efficient suboptimal successive alternating algorithms based on low-rank updating and quadratic approximation.

Observing the criteria in (2.3), it is clear the computation for matrix determinant is critical for optimization. Given multiple-antenna relays, the forwarding gain matrix R is not diagonal, which mean we could rely on matrix decomposition (such as SVD or QR) to transform cascaded MIMO channels into parallel subchannels then simplify determinant computation [6]. Now to deal with diagonal R and non-convex problem, intuitively we have two options to design the optimization algorithms:

1. Brute-force or systematic global search: since (2.4) could not apply convex program-ming, we could always utilize systematic global optimization (e.g. genetic algorithm

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[10]) or even brute-force search to find the optimal R. However, this approach would cost huge computation when L is large.

2. Replace determinant computation with high-order multivariate polynomial function: the matrix determinant of a N × N matrix H is defined to be the scalar [28]

det(H) =X

p

s(p)H(1, p1)H(2, p2) · · · H(N, pN), (2.16)

where the sum is taken over the N ! permutations p = (p1, p2· · · pN) of (1, 2 · · · N ),

H(i, j) denotes the item at ith row and jth column of H, and the sign scalar s(p) ∈ {+1, −1} is defined by permutation p. Following (2.3) and (2.16), we may transform the optimization cost function in (2.4) into a high-order multivariate polynomial of r(i) then perform derivative-based algorithms (e.g. Newton’s method [10]) to approach local optimizer. Such algorithms would simplify the original multivariate problem as successive one-dimensional line searches and guide the searches to most favorable directions. However, even heading to favorable directions, line searches with repeated calculation of high-order multivariate polynomial function would still cost considerable computations and thus may not suit practical purpose.

Given the above two observations, we should design the optimization which not only simplifies the multivariate problem but does effectively suppress extensive computation. Alternating optimization [4] is one approach that meets the requirements. Instead of si-multaneously considering all variables, by definition alternating optimization focus on one variable at one time and alternates between variables. By optimizing one variable (and fix the others) in one iteration and considering another variable in next round, alternating optimization again transform multivariate problem into successive one-dimensional opti-mization. Since we deal with cost function with single variable throughout line searches, we may apply low-rank updating methods [28, Sec. 6.2] to avoid extensive matrix compu-tation. Further, we could approximate polynomial function based on low-rank updating as second-order function. Thus the approximated optimizer could be derived without line search.

2.3.1 Low-rank Updating for Capacity Computation

Some matrix computation, such as decomposition, inversion or determinant, require con-siderable computing resource. When the variation of a matrix comes from one or two vectors, it is possible to compute some quantities associated with the matrix efficiently via an updating procedure rather than via a full-blown computational procedure. Since the variations are frequently of rank one or two, the efficient updating methods are often termed low-rank updating. In what follows we describe some updating methods used in our work.

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Assume square matrices A, B and C with updating equalities as follows B , A + u1vH1 , C , B + u2vH2

where A is assumed to be full-rank Hermitian matrix. Given det(A) and A−1, we have rank-one updating for efficient computation about B as

B−1 = A−1− A −1 u1vH1 A −1 1 + vH 1 A −1 u1 , (2.17) det(B) = det(A)(1 + vH₁ A−1u1). (2.18)

Using (2.18) and (2.17) we could save a great deal of calculation and extend to rank-two updating for det(C) as

det(C) = det(B)(1 + vH₂ B−1u2) = det(A)(1 + vH₁ A−1u1) 1 + vH₂ A−1−A −1 u1vH1 A −1 1 + vH 1 A −1 u1 u2 = det(A){(1 + vH₁ A−1u1)(1 + vH2 A −1 u2) − vH2 A −1 u1vH1 A −1 u2}. (2.19)

Now we describe how the above-mentioned low-rank updating is applied in our alter-nating optimization. In the proposed iterative relay gain adjustment, we adjust only one relay gain at a time. Specifically, in each iteration, we replace one relay gain by some value α ∈ C. The other relay gains are multiplied by a factor β ∈ R+ (where R+ stands

for the set of positive real numbers) such that the power constraint (2.12) is satisfied. The factors α and β are chosen to maximize C(R).

In more detail, let r = [r(1), . . . , r(L)]T _{denote the gain vector of the relay network}

where superscript T denotes transpose. Let the ith term of r, or r(i), be the relay to be optimized in certain iteration, and ro be the same as r except with r(i) replaced by zero.

Also let ru denote the gain vector after the above-described update. Then

ro = (IL− Si)r, ru = βro+ αSi1, (2.20)

where Si denotes the “selection matrix” whose elements are all zero except for a 1 at the

ith diagonal position and 1 represents an all-ones vector. Clearly, following (2.12) α is subject to the constraint

0 ≤ |α| <pPR/p(i), (2.21)

and for given α, we have

β = s PR− |α|2p(i) P ip(i)|ro(i)|2 . (2.22)

Let Ro = diag(ro) and Ru = diag(ru). Then the noise-free equivalent end-to-end

channel matrix after gain updating is given by

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where Ho = F RoGH and Hu = F RuGH. And the autocorrelation matrix of the received

noise vector at the destination becomes

Wu = σD2[IM + (σβ)2(F Ro)(F Ro)H] + |σRα|2F(i)(F(i))H

, σ2DWo+ |σRα|2F(i)(F(i))H. (2.24)

We see that Hu is different from βHo by a rank-one matrix and that Wu is different

from σ2

DWo also by a rank-one matrix. Further, we could express Wu + HuHHu with

rank-two updating as Wu+ HuHHu = (σ 2 DWo+ β2HoHHo ) + (α H β)(HoG(i))(F(i))H

+F(i)[(αβ)(HoG(i))H + (|αG(i)|2+ |σRα|2)(F(i))H]. (2.25)

Given these matrix updating forms, we proceed the design of alternating optimization by decomposing the problem into three parts: 1) how to express C(Ru) in terms of α and

β; 2) how to optimize the values of α and β; and 3) how to iterate. We address these subproblems in order below.

First, consider the term det(Wu). From (2.24) it can readily be seen to be a polynomial

in |α|2 _{and β}2_{. Applying the rank-one determinant update formula to it results in}

det(Wu) = σD2Mdet(Wo)(1 + |σα|2(F(i))HW−1o F

(i)_). _(2.26)

Its dependence on |α|2 _{and β}2 _{can be expressed more concretely in terms of an eigenvalue}

decomposition of (F Ro)(F Ro)H:

(F Ro)(F Ro)H = V1Σ1VH1 . (2.27)

Then, letting e1(i) denote the ith eigenvalue (i.e., the ith diagonal element of Σ1), we

have det(Wu) = σ2MD M Y i=1 {1 + (σβ)2_e

1(i)}{1 + |σα|2(F(i))HV1[IM + (σβ)2Σ1]−1VHF(i)},

(2.28) where a leading product in (2.18) is canceled by the common denominator of the braced quantity.

Next, consider the term det(Wu+HuHHu ), for which we make use of the rank-2 update

formula for matrix determinants. Employing (2.19) with the following identifications of variables:

C ↔ Wu+ HuHHu,

A ↔ σ2_D[IM + (σβ)2F Ro(F Ro)H] + β2HoHHo ,

u1 ↔ F(i), v1 ↔ (αHβ)(HoG(i)) + (|αG(i)|2+ |σRα|2)F(i),

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we get det(Wu+ HuHHu) = det(A){(1 + vH₁ A−1u1)(1 + vH2 A −1 u2) − vH2 A −1 u1vH1 A −1 u2}. (2.29)

As in the case of det(Wu), its polynomial functional dependence on α and β can be

brought out more concretely with an eigenvalue decomposition of a constituent factor of A:

F Ro(F Ro)H + σ−2R HoHHo = V2Σ2VH2 . (2.30)

Then, letting e2(i) denote the ith eigenvalue, we have

det(A) = σ_D2M M Y i=1 [1 + (σβ)2e2(i)] det(Wu+ HuHHu) (2.31) = det(A)[(1 + l1pH1 p1+ l2pH2 p1)(1 + l H 2 p H 1 p2) − (l H 2 p H 1 p1)(l1pH1 p2+ l2pH2 p2)] = σ_D2M M Y i=1 [1 + (σβ)2e2(i)] × [1 + l1|p1| 2_{+ 2<(l} 2pH2 p1) + |l2|2(|pH1 p2| 2_{− |p} 1| 2_|p 2| 2_)],

where <(·) denotes the real part of a quantity and we have made the following definitions to simplify the notation:

l1 , (|αG(i)|2+ |σRα|2)σ−2D , l2 , αβ2σD−2, (2.32) p₁ _{, [I}M + (σβ)2Σ2]− 1 2VH 2 F (i) , p₂ _{, [I}M + (σβ)2Σ2]− 1 2VH 2 (HoG(i)). (2.33)

In summary, C(Ru) can be expressed as the difference between (2.31) and (2.28). We

now turn to the problem of finding α and β that maximize it.

2.3.2 Optimization of α and β

To start, note that none of the terms constituting det(Wu) and det(Wu +2 HuHHu)

depend on the phase of α except <(l2pH2 p1) that appears in (2.31). As a result, for any

given |α| and β, C(Ru) can be maximized by choosing the phase of α such that <(l2pH2 p1)

is maximized. _{This can be achieved by letting ∠α = ∠p}H₁ p₂, so that <(l2pH2 p1) =

|l2pH2 p1|. The problem thus reduces to one of finding the best |α| and β. But since there

is a one-to-one relation between |α| and β (see (2.22)), we only need to solve for β. After some straightforward algebra based on (2.3), (2.28), (2.31), we can show that the optimal β is one that maximizes the following function:

q(β) , 1 + l1|p1| 2 _{+ 2|l} 2pH2 p1| + |l2|2(|pH1 p2|2− |p1|2|p2|2) 1 + |σα|2_(F(i)₎H_V 1[IM + (σβ)2Σ1]−1VHF(i) M Y i=1 1 + (σβ)2e2(i) 1 + (σβ)2_e 1(i) , (2.34)

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where |α|, l1, l2, p1 and p2 are all functions of β.

Due to the complicated nature of (2.34), there is in general no closed-form solution for the optimal β. We need to resort to a search technique, and such techniques are innu-merable. A simplest one is non-iterative line search, in which one examines a sufficiently dense subset of all admissible values of β to find the one maximizing q(β). From (2.21) and (2.22), the set of admissible values of β are given by

0 < β ≤ s

PR

PL

i=1p(i)|ro(i)|2

. (2.35)

A second method is to iteratively update a trial solution to β by solving a low-order

polynomial approximation to q(β) in each iteration. For example, one may, in each

iteration, use a quadratic approximation obtained by taking the second-order Taylor series expansions of q(β) around some β value and take the β value that maximizes the quadratic approximation as the updated trial solution. If this value should fall outside the admissible range given in (2.35), we may replace it by the nearest boundary value of the range. In addition, if the resulting q(β) value should decrease in some iteration, then we may stop the iteration and revert to an earlier solution.

In fact, to find the optimal β one need not work with q(β) directly. Any monotone increasing function of q(β) can be used in its stead. For example, since, from (2.34), q(β) is a product of multiple factors, it may be easier to consider maximizing a logarithm of q(β) than q(β) itself, for then products become sums. This approach is taken in our implementation of the quadratic approximation method. Moreover, in implementing the quadratic approximation method we have chosen to take the Taylor series expansion at β = 1. The reason is that, since we adjust one relay gain at a time, the overall opti-mization process belong to the category of alternating optiopti-mization which is guaranteed to converge to a local optimum [4]. Upon convergence, the values in ru will change little

from one iteration to the next. In other words, the optimal β values will approach unity upon convergence of the overall algorithm. Hence a series expansion around β = 1 should provide a good approximation to the performance surface in the later stages of algorithm progression and benefit its convergence behavior there. In summary, in our implementa-tion of the quadratic approximaimplementa-tion method we seek to maximize ql(β) , log q(β). And

for it we define qa(β) , ql(1) + ql0(1)(β − 1) + q_l00(1) 2 (β − 1) 2 , (2.36)

where q_l0(1) and q00_l(1) are the first and second derivatives of ql(β) evaluated at β = 1. The

solution to the equation q_a0(β) = 0 is then taken to be the current trial solution for β. We summaries the proposed successive optimization algorithm as follows. It finds a locally optimal solution.

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1. Select some relay i for gain adjustment, where i can be chosen in round-robin fashion, for example.

2. Perform eigenvalue decomposition of (F Ro)(F Ro)H and F Ro(F Ro)H+σR−2HoHHo

as in (2.27) and (2.30) to find Σi and Vi, i = 1, 2.

3. Solve for the β that maximizes q(β) as given in (2.34) by a search method, such as the line search or the quadratic approximation method described in the last subsection. Obtain the corresponding |α| using (2.22) and let ∠α = ∠pH

1 p2.

4. Update the relay gains by setting the gain of the ith relay to α and multiplying the gains of all other relays by β.

5. Exit if some stopping criteria are satisfied, or go to step 1 otherwise.

2.3.3 Numerical Results

In presenting the simulation results, we arbitrarily let M = 4, PR = 10, and σR2 = σD2 =

0.1. And we consider two relay network sizes: L = 6 and L = 12. The channel matri-ces F and G are generated by letting all their elements be independent and identically distributed (i.i.d.) complex Gaussian random variables. The relays are initialized to an identical gain that satisfies the power constraint (2.12). Thus their initial performance also serves as a benchmark to compare algorithm results with.

Fig. 2.2 illustrates the progression of average capacity with number of iterations under two methods of solving for the relay gains adjustment factor β: line search and quadratic approximation, where the former has a much higher computational complexity than the latter. The results show that line search performs better than quadratic approximation, but both show a qualitatively similar convergence behavior and the final results after convergence are quite close. Also note that the converged capacity, for both approaches, are much better than the initial capacity, especially when L is large. Such significant im-provement of capacity suggests that the proposed algorithms could contribute substantial performance upgrading when compared to simple identical gain design.

In Fig. 2.3 we show the average behavior of capacity variation when performing line search of optimal β value in step 3 of procedures described in Sec. 2.3.2. Note the capacity curve is first normalized with maximal q(β) in step 3, then is averaged over random channel conditions as set for Fig. 2.2. With this simulation, we could confirm that the optimal solution to line search is around β = 1 in average. Thus applying second-order Taylor Series expansion at β = 1 not only lead to efficient suboptimal q(β) solution, but also provides a good approximation to actual q(β) behavior around optimal β.

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0 20 40 60 80 100 25 26 27 28 29 30 31 32 33

Average capacity for L=6 and L=12, M=4

iteration index

Average capacity (bps/Hz)

L=12

L=6

solid: line search

dashed: quadratic approximation

Fig. 2.2: Progression of average capacity with number of iterations.

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 β Normalized C(R) P R=10, σR 2₌_σ D 2_{=0.1, average of normalized C(R) vs.}_β

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Chapter 3 Capacity Improvement based on

Noise-Dominant Models

While alternating optimization based on low-rank updating discussed in Chap. 2 can yield good results, it provides little insight into the analytical properties of the solutions. We thus consider an analytical approach in this chapter. Since no closed-form solution can be obtained for the general situation, we consider several simplified situations which are more amenable to analysis. In particular, note that in AF systems the receiver noise arises from two sources: the relay noise nR and the destination terminal noise nD. The design

problem becomes mathematically more tractable when one of the two dominates in the overall receiver noise so that the other may be ignored. The results obtained from ignoring one noise source may be viewed as upper bounds on system capacity or as asymptotic performance of the system. For convenience, we term the two simplified conditions the relay noise-dominant condition and the destination noise-dominant condition, respectively. Interestingly, closed-form analytical solutions are not available for arbitrary L even in these simplified conditions. But such solutions can be found if L is restricted to some specific values depending on M . It thus prompts a (suboptimal) relay selection approach wherein a judiciously selected subset of the relays is used to participate in signal transmission and the subset size is such that an analytical solution exists. This approach also helps us to study the resulting capacity outage diversity and compare it to that of single-hop MIMO systems with or without antenna selection [17, 14].

Since MIMO antenna selection systems lay the foundation for the analytical inter-pretation of the proposed relay selection schemes, In this chapter we would first briefly review the important results of the studies on single-hop MIMO antenna selection systems. Then we present noise-dominant models for relay network design and propose correspond-ing algorithms for individual models. For analytical insights we design equivalent MIMO antenna selection systems. Finally some numerical results are shown for performance eval-uation and validating the links between equivalent MIMO antenna selection and proposed relay selection schemes.

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3.1 Preliminary: MIMO Antenna-Selection Systems

Diversity techniques based on antenna selection or maximal ratio combining are mature and popular approach to utilize multiple antennas for performance improvement. When only single stream is transmitted with NtTx antennas and Nr Rx antennas, the diversity

order of NtNr could be achieved [2]. However, this principle could not be applied directly

to spatial multiplexing transmission with multiple concurrent streams. To this end, MIMO antenna selection systems are developed [16] [17] and shown to be an efficient way to provide additional diversity for spatial multiplexing systems.

Consider a point-to-point MIMO system with M transmitter antennas and N receiver

antennas, where N ≥ M . Let H ∈ CN ×M _{be the channel matrix and let the}

transmit-ted signal-to-received noise power ratio (transmit-to-receive SNR) ρ2. Then the system capacity is given by

log₂C(H) = log₂det(IM + ρ2HHH). (3.1)

For a flat-fading H, a statistical lower bound is [14]

log₂C(H) ≥ M X i=1 log₂(1 + ρ2γ_{N −i+1}2 ) (3.2) where γ2

N −i+1 denotes a gamma-distributed random variable with N − i + 1 degrees of

freedom. This lower bound indicates that the capacity of an M × N MIMO system is statistically equivalent to or better than that of a system composed of M parallel independent single-input multi-output (SIMO) subsystems wherein the ith subsystem performs maximal-ratio combining (MRC) on N − M + i receiver antennas. In other words, the overall capacity outage diversity of an M × N MIMO system is bounded between N − M + 1 and N .

Consider a system where the receiver selects M out of its N antennas for use in signal

detection. Let HS be the M × M channel matrix of the resulting MIMO channel. This

matrix contains the M rows in H that correspond to the selected receiver antennas. There are _MN possible antenna choices. Let (M, N; M )S denote a system wherein the antennas

are chosen to maximize the capacity. Then the system capacity can be described as log₂CS = max

HS

log₂det(I + ρ2HH_SHS). (3.3)

It is shown in [17] that the capacity of such a system is again statistically equivalent to or better than a MIMO system composed of M parallel independent SIMO subsystems wherein the ith subsystem performs antenna selection by choosing one out of N − i + 1 receiver antennas.

In a nutshell, both the full system (M, N ; N )S and the receiver antenna selection

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thus share a similar capacity outage diversity order. We show the cumulative distribution function (CDF) curves of capacity for both systems. Given M = 3, in Fig. 3.1 the top part presents CDF curves of both systems with N = 5 and N = 12, respectively. We could observe that increasing N would improve capacity, and full systems outperform antenna selection systems. To gain more insights into the capacity distribution, in the bottom part of Fig. 3.1 we show biased CDF where curves are overlapped and shifted horizontally with the 50% points collocated. With the biased CDF it is clear that antenna selection systems share similar capacity distribution characteristic with corresponding full systems.

3.2 Designs for Destination-Noise Dominant

Condi-tions

Following Sec. 2.1.2, the noise in system could be represented as

W = E{(F RnR+ nD)(F RnR+ nD)H}

= σ_D2(I + σ2(F R)(F R)H). (3.4)

Thus, when the destination noise dominates in the overall noise we have W & σ2

DI. Then

we have capacity approximation as

C(R) = log₂det(I + HHW−1H) . det[I + σD−2H

H

H]. (3.5)

Hence the capacity is approximately that of an M × M single-hop point-to-point MIMO system with channel matrix H and transmitted signal-to-received noise power ratio (transmit-to-receive SNR) 1/σ_D2. However, even in this rather simplified condition, no general solution is available to the optimization problem (2.4) for arbitrary L > M . But an analytical solution can be obtained for L = M . Thus we consider a relay selection approach wherein M relays are selected to perform the relaying. Let the total of _ML selections be indexed from 1 to _ML. For the kth selection define the corresponding optimization target based on (3.5) as

CD(k, RD) , det[IM + σD−2(FkRDGHk )(FkRDGHk)H] (3.6)

where Fk ∈ CM ×M and Gk ∈ CM ×M, respectively, denote the submatrices of F and

G constructed by collecting the columns corresponding to the active relays in the kth selection, and RD denotes the diagonal matrix of relay gains of the active relays. Let

rD(i) be the ith diagonal term in RD. In high SNR,

CD(k, RD) . det[σ−2D (FkRDGHk)(FkRDGHk) H_] = σ_D−2M det(FkFHk) det(GkGHk) Y i |rD(i)|2. (3.7)

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42 44 46 48 50 52 54 56 10−2 10−1 100 capacity (bps/Hz) CDF

dashed: antenna selection solid: full MIMO system

N = 5 N = 12 −4 −3 −2 −1 0 1 2 3 4 5 10−2 10−1 100 biased capacity (bps/Hz) CDF N = 5

dashed: antenna selection solid: full MIMO system

N = 12

Fig. 3.1: Comparison of MIMO antenna selection and full MIMO systems in terms of CDF of capacity (top) and biased capacity (bottom).

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To maximize CD(k, RD) subject to the power constraint (2.12) given Fk and Gk, we

equivalently find the optimum rD such that

rD = arg max rD Y i |rD(i)| (3.8) subject to M X i=1 (σ_R2 + kG(i)_k k2)|rD(i)|2 = PR, (3.9) where rD = [rD(1), . . . , rD(M )]T and G (i)

k denotes the ith column of Gk. Employing the

Lagrange multiplier technique leads to the optimum relay power allocation as |rD(i)| = s PR M (σ2 R+ kG (i) k k2) . (3.10)

Denote the resulting CD(k, RD) by CDO(k). The final solution is then given by the optimal

selection

¯

k , arg max

k CDO(k) (3.11)

together with its corresponding optimum relay power allocation.

To analyze its performance, substitute (3.10) into (3.7) and assume kG(i)_k k2 _σ2 R

(i.e., consider the high SNR limit). Then we get an upper bound for any C_DO(k)(RD) as

CDO(k) . σD−2Mdet(FkFHk) det(GkGHk) Q ikG (i) k k2 (PR M ) M_. _(3.12)

A simpler upper bound can be obtained by considering a QR decomposition of GH_k as

GH_k = QT , where Q is a unitary matrix and T is an upper triangular matrix. Denote the ith column of T by T(i) and the ith diagonal term of T by T (i, i). Then kT(i)k2 _{= kG}(i)

k k2

because T(i) and G(i)_k are related by a unitary transform Q, and |T (i, i)|2 ≤ kT(i)k2_.

Consequently, det(GkGHk) Q ikG (i) k k2 = | det(Q)| 2_{| det(T )|}2 Q ikG (i) k k2 = M Y i=1 |T (i, i)|2 kG(i)_k k2 ≤ M Y i=1 kT(i)_k2 kG(i)_k k2 = 1, (3.13)

where equality holds only when Gk has orthogonal columns. Substituting into (3.12)

yields the desired upper bound

CDO(k) < det(FkFHk)(

PR

σ2 DM

)M. (3.14)

Thus we obtain an upper bound CDU on the capacity measure for the suboptimal solution

as CDO(¯k) < det(F_k¯FH¯_k)( PR σ2 DM )M ≤ max k det(FkF H k)( PR σ2 DM )M _{, C}DU. (3.15)

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Note that log₂CDU is actually the asymptotic capacity of an (M, L; M )S system at

transmit-to-receive SNR PM

R /(σD2M )M, where (X, Y ; Z)S denotes an X × Y single-hop

point-to-point MIMO system wherein the receiver selects, out of the total Y received an-tenna signals, the Z that yields the maximum capacity for receiver processing. (See the review in Sec. 3.1)

By (3.12) we may also obtain a lower bound for CDO(¯k): Letting

k = arg max k det(GkGHk) Q ikG (i) k k2 , (3.16)

we have the lower bound CDL as

CDL , CDO(k) ≤ max

k CDO(k). (3.17)

When L M , it becomes more likely to find a set of M nearly orthogonal columns in G. In this case, we will have det(GkGHk)/

Q ikG (i) k k2 . 1 and thus CDL = CDO(k) . det(FkFHk)( PR σ2 DM )M. (3.18)

Now since k is a selection based on G without taking F into consideration and since from (3.18) log₂CDL resembles the form of the capacity of an M × M single-hop point-to-point

MIMO system with channel matrix Fk at transmit-to-receive SNR PRM/(σD2M )M, we can

view log₂CDL as the capacity of an (M, M ; M )S system.

Therefore, from (3.15) and (3.17) we conclude that in the destination-noise dominant condition, the performance of relay selection with optimal power allocation is

asymptot-ically upper-bounded by that of the (M, L; M )S MIMO antenna selection system and

lower-bounded by that of (M, M ; M )S. The capacity outage diversity order is thus

sim-ilarly bounded by that of these two systems. Simulation results in Sec. 3.4 will show that, although the above derivation has been carried out mostly assuming asymptotic conditions, relay selection systems operating in practical conditions exhibit some similar performance characteristics.

3.3 Designs for Relay-Noise Dominant Conditions

We now turn to the relay noise-dominant situation. Again, no closed-form general solution can be found for arbitrary values of L and M , but a solution can be found if they are related in a specific way. We thus again propose a relay selection scheme.

To start, let N out of the L relays be selected to participate in the relaying, where N ≥ M but is otherwise undetermined for the moment. Altogether there are _NL selections. For the jth selection we let Fj ∈ CM ×N and Gj ∈ CM ×N denote the corresponding channel

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submatrices of F and G, respectively. From the derivation up to (2.9) in Theorem. 2.2.1 we may infer that, in the relay noise-dominant situation,

CR(j, RR) . det(IM + σR−2GjVjVHj G H

j ) (3.19)

where RRis the diagonal matrix of relay gains, Vj ∈ CN ×M is the matrix of right singular

vectors of FjRR with its jth column corresponding to the jth largest singular value of

FjRR. Comparing with the situation addressed in Theorem 2.2.1 (in particular, see

(2.13)) we find that relay noise-dominant systems behave similarly to systems with very high relay transmission power. Hence by Theorem 2.2.2, C_R(j)(RR) is upper-bounded as

CR(j, RR) ≤ det(IM + σ−2_R GjGHj ), (3.20)

where equality holds and CR(j, RR) is maximized if the rows of FjRR span the same

space as that of Gj. Note that, contrary to the destination noise-dominant case, in the

present case the total relay transmission power does not affect the performance at all, only the row space of FjRR matters. And the relay network should try to align the row

space of FjRR with that of Gj, which is a beamforming problem.

To proceed, let F<i>_j denote the ith row of Fj. Let Oj ∈ CN ×(N −M ) be a matrix

of basis vectors for the orthogonal complement of the row space of Gj; that is, Oj is

such that GjOj = 0 where 0 denotes a zero matrix. Also, let Φij , diag(F<i>j )Oj.

Immediately we have rT_RΦij = rTRdiag(F <i> j )Oj = (F<i>j ) H RROj (3.21)

where rR ∈ CN is the vector formed of the diagonal elements of RR. To make the row

space of FjRR equal to that of Gj, we may equivalently find rRsuch that rTRΦij = 0 ∀i.

For this, define

Φj , [Φ1j Φ2j · · · ΦM j] ∈ CN ×N (N −M ). (3.22)

Then the optimal solution or beamformer rRshould be such that rTRΦj = 0. The existence

of such a solution would require Φj to have a non-empty null column space. Therefore let

M (N − M ) < N . Combined with the earlier assumption that N ≥ M , the only choice is N = M + 1 for any M ≥ 2.

In conclusion, the final solution is given by the selection ¯

j = arg max

1≤j≤(_{M +1}L )

CR(j, RR) (3.23)

where for each j, RR is given by diag(rR) with rR being the solution to the equation

rT

RΦj = 0 and “normalized” such that

PM +1 i=1 (σ 2 R+ kG (i) j k2)|rR(i)|2 = PR (where G (i) j is

the ith column of Gj and rR(i) is the ith element of rR).

Regarding its performance, from (3.20) we see that the resulting capacity measure is approximately given by CRO as follows:

max

j CR(j, RR) ≈ maxj det(IM + σ −2

分散式放大前送多輸入多輸出中繼站網路之設計

國

立 交 通 大 學

電子工程學系 電子研究所

博 士 論 文

分散式放大前送多輸入多輸出中繼站網路之設計

Design of Distributed Amplify-Forwarding MIMO Relay

Networks

研 究 生：吳俊榮

指導教授：林大衛

分散式放大前送多輸入多輸出中繼站網路之設計

Design of Distributed Amplify-Forwarding MIMO Relay Networks

研 究 生：吳俊榮 Student：Chun-Jung Wu

指導教授：林大衛 Advisor：David W. Lin

國 立 交 通 大 學

電子工程學系 電子研究所

博 士 論 文

分散式放大前送多輸入多輸出中繼站網路之設計

分散式放大前送多輸入多輸出中繼站網路之設計

分散式放大前送多輸入多輸出中繼站網路之設計

分散式放大前送多輸入多輸出中繼站網路之設計

摘要

摘要

摘要

摘要

Design of Distributed Amplify-Forwarding

MIMO Relay Networks

Abstract

誌

誌

誌

誌

謝

謝

謝

謝

Contents

List of Figures

Chapter 1

Introduction

1.1

Research Background, Motivation and Issues

1.1.1

On the Capacity of Distributed AF MIMO Relay Network

1.1.2

On the OFDM Channel Estimation and Applications in

Relay Networks

1.2

Outline of This Thesis

1.3

Contribution of This Thesis

Chapter 2

Capacity of Relay Network and

Alternating Optimization

2.1

System Model and Capacity of Relay Networks

2.1.1

System Model and Power Limits

2.1.2

Noise Whitening and Capacity

2.2

Power Scaling and Upper bounds of Capacity

2.3

Iterative Alternating Optimization

2.3.1

Low-rank Updating for Capacity Computation

2.3.2

Optimization of α and β

2.3.3

Numerical Results

Chapter 3

Capacity Improvement based on

Noise-Dominant Models

3.1

Preliminary: MIMO Antenna-Selection Systems

3.2

Designs for Destination-Noise Dominant

Condi-tions

3.3

Designs for Relay-Noise Dominant Conditions

立交通大學

電子工程學系電子研究所

博士論文

研究生：吳俊榮

研究生：吳俊榮 Student：Chun-Jung Wu

國立交通大學

電子工程學系電子研究所

博士論文