多天線-正交分頻多工存取無線網路之資源配置

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國立交通大學

電信工程學系

碩士論文

多天線-正交分頻多工存取無線網路之資源

配置

Resource Allocation for MIMO-OFDMA

based Wireless Network

研究生：翁志倫

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多天線-正交分頻多工存取無線網路之資源配置

Resource Allocation for MIMO-OFDMA based Wireless Network

研究生：翁志倫 Student：Chih-Lun Weng

指導教授：蘇育德博士 Advisor：Dr. Yu T. Su

國立交通大學

電信工程學系

碩士論文

A Thesis

Submitted to Department 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

August 2009

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i

多天線-正交分頻多工存取無線網路之資源配置

學生：翁志倫

指導教授

：

蘇育德博士

國立交通大學電信工程學系碩士班

摘

要

在傳收兩端同時使用多根天線，配合預先編碼技術(precoding)，我們可以在每個子載波上面獲得多個不同的空間通道(spatial channel)。另一方面，正交分頻多工存取(OFDMA)將一個寬頻帶切成許多窄頻子通道，以便平行傳送的訊號只受到非頻率選擇性衰退(frequency nonselective fading)，不但可以簡化接收複雜度，且因子通道會隨著時間以及使用者的位置而變化，傳送端可利用這些變化所造成的分集(diversity)，視各子通道之增益適當地調整其傳輸功率與調變訊號之階數，進而大大提高其頻寬使用效率。因此多天線-正交分頻多工存取系統可以分配的資源就包含了空間、頻率以及使用者這三個範疇。在本論文中我們將會探討如何有效的配置空間、頻率以及使用者這三個維度的資源將用戶的傳輸功率與平均的位元錯誤率降到最低。我們首先提出了兩種以奇異值分解(singular value decomposition)為基礎的預先編碼技術。我們首先利用奇異值向量的線性組合來合成預先編碼向量以建立多個正交通道。我們提出的第二個預先編碼技術則把通道矩陣的秩(rank)限制移除，以增加頻譜使用效率。雖然如此一來將會讓空間通道間產生干擾，而讓資源配置的問題因需滿足某些訊號干擾比之要求而更形複雜，但透過適當的設計，仍可將用戶間的干擾控制在一定的範圍下。針對這兩個預先編碼技術我們分別提出不同的動態資源分配演算法讓用戶的總傳輸功率降到最低。除此之外，我們還針對實用的碼書(codebook)預先編碼技術之資源配置進行探討，我們提出了動態的子載波分配及傳送功率調整等兩種基本方式及其組合來讓用戶的平均位元錯誤率降到最低。

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Resource Allocation for MIMO-OFDMA based

Wireless Networks

Student : Chih-Lun Weng Advisor : Yu T. Su

Department of Communications Engineering National Chiao Tung University

Abstract

By using multiple antennas at both the transmit and receive sides, one obtains mul-tiple eigenmodes (eigenchannels) on the same carrier through beamforming (precoding). With each eigenchannel represents an equivalent SISO channel, array gain is obtained by using only the strongest eigenmode but the capacity is maximized by allocate the transmit power across subchannels according to the water-filling result. The orthogonal frequency division multiplexing (OFDM) scheme divide a wideband channel into parallel narrowband subchannels so that signals propagate through each subchannel suffer from frequency nonselective fading. The OFDM-based multiple access (OFDMA) scheme has been shown to be capable of achieving the maximum spectral efficiency with extremely high probability. A key ingredient of an OFDMA system is that it can exploit the diver-sity offered by the time-varying and user(location)-dependent nature of the subchannels via proper scheduling and power/subchannel allocation. Hence a MIMO-OFDMA sys-tem is expected to offer high capacity through an efficient use of the spectral, spatial and user domain resources.

The main design issue we try to solve in this thesis is the following. Given the users’ rate requirements of a MIMO-OFDMA system, how to apportion the transmission re-sources in space, frequency, and user domains so that the total transmit power and each

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user’s average bit error rate are minimized. We first consider two orthogonal precoding schemes based on singular value decomposition (SVD). In the first design we construct orthogonal eigenchannels by performing linear operations on the channel matrix’s sin-gular vectors under the channel rank constraint. To improve spectral efficiency, we then remove the rank constraint on the number of users allowed on a subcarrier. Although the resulting co-channel interference may cause performance degradation, it is more than compensated for by the increased capacity through a proper RA plan that ensure the associated signal-to-interference ratios are within the tolerable limits. The proposed RA algorithms for both precoding scenarios are designed to minimize the total transmit power while satisfying the users’ QoS constraints.

Finally, we examine the resource allocation (RA) issue for MIMO systems with lim-ited feedback. More speciﬁcally, we consider the codebook based precoding scheme and suggest subcarrier assignment scheme based on the Lagrange multiplier method. For a given subcarrier assignment, we then present a power allocation method which minimizes the average bit error rate performance.

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誌謝

時間過得好快，在交大的六年(大學四年、研究所兩年)眼看著即將進入尾聲，感謝學校和系上對我的栽培，讓我在這六年的求學生涯中過得充實而有意義，也讓我在電信這個領域裡學到了許多知識，我要特別感謝我的指導教授蘇育德博士，沒有他的指導，我的研究及論文不會那麼順利地完成，除了學業方面，也要感謝蘇老師在做人處世上教導我們的一切，讓我在研究所的這兩年成熟了不少。除此之外，還要感謝實驗室的林淵斌學長和林坤昌學長，謝謝你們在這兩年在研究方面指導我，給我很多建議與經驗分享，幫助我論文的完成；還有實驗室的同學們，謝謝大家這兩年的陪伴，感謝盧彥碩和廖俊傑(我的室友)這兩年陪我經歷了很多事，希望未來大家都能有很好的發展。還要特別感謝實驗室的助理(淑琪姊、昱岑姊)，謝謝妳們幫實驗室處理好多大大小小的事情，幫我們解決好多問題；還要感謝 6 號家族的林政翰學長與郭政錦學長，謝謝你們在我進大學之後就一直照顧我，像我的哥哥一樣，謝謝你們。感謝我的家人，謝謝我的爸爸媽媽，因為你們不求回報的付出，讓我可以沒有後顧之憂的在求學路上朝自己的目標邁進，如果有一天能讓你們感到驕傲，我的努力才有價值；感謝疼愛我的姊姊，對我的問題與要求總是有求必應，常常給我很多關於生活上的意見與經驗；感謝我的好朋友徐志寧，在求學的過程中受到你太多的幫助與指導，謝謝你。最後要謝謝我的女朋友，泳妍，謝謝妳一直陪在我身邊，在我遇到挫折的時候鼓勵我，在我開心的時候與我一起分享，謝謝妳這些日子和我一起度過、一起成長。最後，對於每一個幫助過我的人，謹以此論文獻上我最深的的敬意，感謝你們！翁志倫謹誌于新竹國立交通大學

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List of Figures

2.1 Flow Chart Description of Algorithm I. . . 14 2.2 A search tree representing the multi-stage bit-loading procedure

(Algo-rithm II). . . 20 2.3 Average power ratio per user for a MIMO-OFDM uplink; 32 subcarriers,

64 bits per OFDM symbol, 4 users. . . 24 2.4 Average power ratio per user for a MIMO-OFDM downlink; 32

subcarri-ers, 64 bits per OFDM symbol, 4 users. . . 25 2.5 Average power ratio per user for a MIMO-OFDM downlink; 8 subcarriers,

16 bits per OFDM symbol, 2 users. . . 26 2.6 Average power ratio per user for a MIMO-OFDM downlink; 32

subcarri-ers, 64 bits per OFDM symbol, 2 users. . . 28 2.7 Average power ratio per user for a MIMO-OFDM downlink; 32

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3.1 Average power ratio per user for a MIMO-OFDM downlink; 32 subcarri-ers, 128 bits per OFDM symbol, 4 ussubcarri-ers, rank=3. . . 41 3.2 Average power ratio per user for a MIMO-OFDM downlink; 32

subcarri-ers, 192 bits per OFDM symbol, 4 ussubcarri-ers, rank=4. . . 41 3.3 Average power ratio per user for a MIMO-OFDM downlink; 32

subcarri-ers, 192 bits per OFDM symbol, 4 ussubcarri-ers, rank=4. . . 43

4.1 Average BER performance for the ZF receiver ; 128 subcarriers, 8 users, 2 substreams. . . 53 4.2 Average BER performance for the ZF receiver ; 128 subcarriers, 8 users,

3 substreams. . . 54 4.3 Average BER performance for the MMSE receiver ; 128 subcarriers, 8

users, 2 substreams. . . 54 4.4 Average BER performance for the MMSE receiver ; 128 subcarriers, 8

users, 3 substreams. . . 55 4.5 Average BER performance for the ZF receiver; ﬁxed subcarrier

assign-ment without codebook precoding ; 128 subcarriers, 16 users, 2 substreams. 55 4.6 Average BER performance for the ZF receiver ; ﬁxed subcarrier

assign-ment with codebook precoding ; 128 subcarriers, 16 users, 2 substreams. 56 4.7 Average BER performance for the ZF receiver ; dynamic subcarrier

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List of Tables

1.1 A brief review and comments of previous works I . . . 5

1.3 A brief review and comments of previous works II . . . 6

2.1 Algorithm for computing the required eigenchannel number. . . 16

2.2 The eigenchannel assignment algorithm. . . 18

2.3 The conventional bit-loading algorithm. . . 19

2.4 η for the 2 users case . . . . 27

2.5 η for the 4 users case . . . . 28

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Chapter 1 Introduction

Notwithstanding the recent worldwide financial crisis, the demand for high data rate multiuser multimedia wireless communications continues to grow. Future wireless communication systems are expected to provide even higher rate multimedia services with more varieties of QoS requirements. It has been shown that, for a given frequency band, the Orthogonal Frequency-Division Multiple Access (OFDMA) is the optimal multiple access (MA) scheme that provides the highest capacity in almost all cases if the channel information is perfectly known. The OFDMA refers to an Orthogonal Frequency-Division Multiplexing (OFDM) based MA scheme that assigns disjoint sub-sets of subcarriers to different users. Besides having anti-interference and anti-fading capabilities, OFDM offers another practical advantage for multimedia transmission due to its flexibility in allotting transmission resources to meet various media’s bandwidth and performance requirements.

The conventional wireless capacity, however, is limited to make the most of time and frequency (or equivalent code) degrees of freedom only. The capacity can be greatly enhanced by exploiting the space domain through the use of multiple antennas at either or both ends of a wireless link. The the multiple antenna systems, now commonly known as multiple-input multiple-output (MIMO) systems, provide not only spatial and interference diversities and multiplexing gain but also makes possible space-division multiple access (SDMA). Hence it is only natural to incorporate the MIMO technique

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in an OFDMA system to achieve the maximum capacity.

Adaptive resource allocation (RA) is not only important for efficient resource usage but also mandatory if the theoretical channel capacity is to be achieved or approached. It is an effective means to mitigate interference and reduce the outage probability in a interference-limited system. With the spatial channels as part of the radio resource, a MIMO-OFDMA system enjoys larger degrees of freedom in distribute its radio resources (power, subcarriers, time-slots, and spatial channels). Adaptive RA methods for maxi-mizing the capacity or throughput have been proposed [1]-[5]. Taking users’ quality of service (QoS) requirements into account, [6]– [10] proposed adaptive RA algorithms that minimize the total transmit power. These earlier results either have the single-user-per-subcarrier constraint or the single (strongest)-eigenmode-per-single-user-per-subcarrier constraint. In contrast, this thesis presents new RA schemes for MIMO-OFDM systems without the above constraints. Our schemes differ from these earlier results in that we propose two different precoding schemes that permit assigning eigenchannels on the same subcarrier to different users. We improve the “resolution” of the radio resources so that diversity gain and greater flexibility are obtained. It is also noted that the proposed precoding and the associated RA schemes can be used for uplink and downlink transmissions. A brief review of and comments on earlier works are given in Table 1.1 and Table 1.2. Comparison with our work can also be found in the same table.

In order that multiple user signals over the same subcarrier can be decoupled at the receiver, we follow the conventional approach invoking the Singular Value Decomposition (SVD) to obtain the pre-processing and the post-processing vectors for diﬀerent users through proper linear combinations of the singular vectors. Based on the above concept, we propose an orthogonal precoding scheme for the MIMO-OFDMA systems. The users transmitting data on the same subcarrier will cause no interference to each other. This feature will make resource allocation process more easily because once the eigen-channel assignment is done, users can do their own bit and power management individually

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without consider other users’ eﬀects.

However, the user number on the same subcarrier of the orthogonal precoding scheme will be bounded by the rank of the MIMO channel matrix such that the spectrum ef-ficiency may be constrained. In order to improve the efficiency of the radio resource utilization, we bring up another precoding scheme which not only provides orthogo-nal but also non-orthogoorthogo-nal eigenchannels for transmission. In this scheme, we allow more users to transmit data on the same subcarrier than the first scheme to improve spectrum efficiency. Nevertheless, The co-channel interference is no longer avoidable in this scheme. In this situation, the resource optimization problem will become more complicated.

For the ﬁrst scheme, we propose two adaptive RA algorithms that take care of subcar-rier assignment, pre-processing and post-processing vectors selection and bit allocation. They are designed to minimize the total power and meet each user’s QoS requirement. It should be noted that although the power consumption problem is always considered as a critical issue especially for the uplink transmission due to the power-limited feature of the mobile devices, we still take both the downlink and uplink transmission into ac-count in this thesis for generality. Similarly, we propose an adaptive RA algorithm for the non-orthogonal scheme according to its system structure. The adaptive algorithm also aims to minimize the total consuming power while each user’s QoS requirement is guaranteed.

In addition to the two SVD based precoding schemes we design, we also take the precoder with limited feedback into account [11]-[12]. For such the codebook based precoding scheme, we perform subcarrier assignment and dynamic power loading in order to minimize the average bit error rate (BER).

The rest of this thesis is organized as follows. The ensuing chapter describes the orthogonal precoding scheme for the MIMO-OFDMA systems and the two proposed adaptive RA algorithms. In Chapter 3, we discuss the design of the non-orthogonal

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precoding scheme and the corresponding adaptive RA algorithm. In Chapter 4, the resource allocation for codebook based precoding scheme is investigated. The numerical simulation results are all given in the end of these chapters. Finally, we give a conclusion for the resource allocation we have done for the MIMO-OFDMA systems in the last chapter.

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T able 1.1: A brief review and commen ts of previous w orks I Source Assumptions Preco ding Criterion Constrain t Prop osed algorithm Commen ts [1] Multiuser, do wnlink, one user p er sub car-rier SVD, all eigenmo des are used. Maximize total throughput B E R a n d p owe r constrain t Sub carrier assign-men t and p o w er/bit allo cation Lo w complexit y, but p o w er/bit allo cation is not optimal. [2] Multiuser, do wnlink, one user p er sub car-rier SVD, all eigenmo des are used. Maximize total throughput p owe r co n st ra in t and pac k et dela y Sub carrier assign-men t and p o w er/bit allo cation Lo w complexit y, bit allo cation is not op-timal. [3] Multiuser, do wnlink, one user p er sub car-rier SVD, all eigenmo des are used. Maximize total throughput B E R a n d p owe r constrain t Sub carrier assign-men t and p o w er/bit allo cation Lo w complexit y, but the assignmen t for su b ca rr ie rsi sn o t clearly describ ed. [4] Multiuser, do wnlink, one user p er sub car-rier VBLAST Maximize total throughput Powe r co n st ra int and prop ortional data rate fair-ness Sub carrier assign-men t and p o w er/bit allo cation Resource allo cation for VBLAST system is considered. [5] Multiuser, do wnlink, one user p er sub car-rier SVD, all eigenmo des are used. Maximize total throughput BER constrain t and user fairness Sub carrier as-signmen t and bit allo cation Lo w complexit y, and the deﬁnition of the user priorit y con-siders user through-put, buﬀer size, and pac k et dela y.

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T able 1.3: A brief review and commen ts of previous w orks II [6] Multiuser, do wnlink, one user/sub carrier Minimize total transmit p o w er Data rate con-strain t Sub carrier assign-men t and p o w er/bit allo cation Lo w complexit y, but the bit-loading algo-rithm is not optimal. [7] Multiuser, uplink, one user/sub carrier SVD, strongest eigen-mo de is used, ZF re-ceiv er Minimize total transmit p o w er Data rate con-strain t Lagrange m ultiplier metho d is used to get loading bits The ZF receiv er is restricted b y the rank of the channel matrix. [8] Multiuser, do wnlink, one user/sub carrier Alamouti sc heme, Minimize total transmit p o w er Data rate and BER constrain t Sub carrier assign-men t Adaptiv e mo du-lation and p o w er allo cation is not discussed. [9] Multiuser, do wnlink, one user/sub carrier, one an tenna at the receiv er Orthogonal b eam-forming, Minimize total transmit p o w er Data rate and BER constrain t User selection Only consider the case with one an-tenna at the re-ceiv er. [10] Single user, do wnlink, one user/sub carrier SVD, strongest eigen-mo de is used. Minimize total transmit p o w er Data rate and BER constrain t B ita n d p o w era ll o -cation Only consider the single user case. [a],[b] Multiuser, do wn-link/uplink, m ultiuser p er sub carrier SVD, all eigenmo des can b e used. Minimize total transmit p o w er Data rate and BER constrain t Eigenc hannel assign-men t and bit allo ca-tion Higher radio re-source resolution, aﬀordable complex-it y. [c] Multiuser, uplink, one user p er sub carrier Co deb o ok based pre-co ding Minimize a v er-age BER Powe r co n st ra int and prop ortional data rate fair-ness Sub carrier assign-m ent a n d p owe r allo cation Lo w complexit y, p o w er allo cation is optimal. [a]:Orthogonal preco ding sc heme and the corresp onding resource allo cation algorithms prop osed in this thesis. [b]:Non-orthogonal preco ding sc heme and the corresp onding resource allo cation algorithm prop osed in this thesis. [c]:Resource allo cation for co deb o ok based preco ding prop osed in this thesis.

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Chapter 2 Resource Allocation for

Orthogonally Precoded MIMO

Systems

2.1 Background

In recent years, the MIMO technology has drawn much attention since it promises a capacity that is proportional to the smallest number of antennas used at the transmit and the receive sites [13]. Many novel MIMO transceiver designs have been proposed and veriﬁed in past decade. In [14], the BLAST (Bell Labs Layered Space-Time) ar-chitectures proposed exploits the capacity advantage of multiple antenna systems for multiplexing. A simple but ingenious transmit diversity technique-the Alamouti scheme [15], is designed to achieve diversity gain and has been adopted in the standards of many communication systems. In addition, SVD also can be used in MIMO systems as the beam patterns of the beamforming technology to improve the system performance. Some SVD based orthogonalization schemes have been proposed [16]-[17] for MIMO precoding such that the CCI can be minimized. A basic assumption used is that there exits enough orthogonal spatial channels that each user will have access to at least one of them, which, unfortunately, may not always be valid. Besides SVD based precoding,

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there are also other precoding schemes such as lattice-reduction based precoding [18] or codebook based precoding [11]-[12]. In the design of our precoding scheme, we use the SVD to obtain the basis of the precoding vectors.

2.2 System Parameters and Transceiver Model

Consider a MIMO-OFDMA system with a single base station (BS) equipped with Tx

antennas and K mobile station (MS) users, each equipped with Rx antennas. The

frequency band used contains M subcarriers which are to be allocated to the K MS’. Besides orthogonal subcarriers, such a system provides additional spatial channels for transmission.

Let the kth MS’ channel matrix for subcarrier m be denoted by the Rx× Tx matrix

Hmk. Applying SVD to Hmk gives

Hmk = UmkΛmkV†mk (2.1)

where Umk contains the left singular vectors of Hmkand Um,k contains the right singular

vectors of Hmk. Λmk is the diagonal matrix with diagonal entries being the singular

values (SVs). In order to separate the signals from diﬀerent user perfectly the proposed scheme provides at most R eigen-channels on the same subcarrier where R is the rank of the MIMO channel matrix. (Here we assume that the channel matrices of all users are all full rank. Although there are still the case that the channel matrix may be rank-deﬁcient due to the spatial correlation, we can still suppose the channel matrix be full rank with some neglectable eigenmode magnitudes.) It is well known that the right and left singular vectors can be used as the pre-processing and post-processing vectors such that the receiver can easily extract the data symbol without interference.

Deﬁne the eigenchannel coeﬃcient Armk by Armk = 1 if user k is to use the mth

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corre-sponds to subcarrier m of user k can be expressed as ymk = Hmk R i=1 K j=1

Aimj√pimjtimjximj+ nmk, (2.2)

where ximj denotes the data symbol of user j carried by the ith eigenchannel of subcarrier

m. Timj and pimj are the pre-processing vector and transmit power, respectively. The

entries of the Rx× 1 noise vector nmk are i.i.d. zero-mean complex Gaussian random

variables with variance σ2. xrmk, the kth user’s data symbol transmitted over the rth

eigenchannel of subcarrier m, is pre-multiplied by the pre-processing vector Trmkto form

the Tx data symbols which are then transmitted with power prmk. Pre-multiplying the

received signal ymk by the post-processing vector Wrmk, we obtain

rrmk = w†rmkymk = w†_rmkHmk R i=1 k j=1

Aimj√pimjtimjximj+ w†_rmknmk

where † denotes conjugate transpose.

2.3 Spatial Channel Assignment and Related Signal

Processing

We use a simple example to illustrate the basic idea of the proposed scheme. Assume

Tx = Rx = 2, K = 2 and the subcarrier m channel matrices for the two users, Hm1 and

Hm2, are of full rank and have the SVDs

Hm1 = Um1Λm1V†m1 = [um₁₁um₁₂] sm 11 0 0 sm 12 vm₁₁† vm₁₂† (2.3) and Hm2 = Um2Λm2V†m2 = [um₂₁um₂₂] sm 21 0 0 sm 22 vm₂₁† vm₂₂† (2.4)

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where sm₁₁ = max{sm_ij : i, j = 1, 2}. We assign the strongest eigenchannel to user 1 and allow user 2 to use one that is orthogonal to the strongest one. In other words, we use

vm₁₁ and um₁₁ as the pre-processing vectors and assume those for user 2 are of the form ¯

vm

2 = α1v21m + α2vm22 and ¯um2 = β1um21+ β2um22, where α and β are weighting coeﬃcients

to be determined. The corresponding received signal from user 1 is given by

ym1 = Hm1(√pm1v₁₁mxm1+√pm2v¯m₂ xm2) + nm1, (2.5)

which, after post-processing, becomes

r_1m1 = um₁₁†ym1

= um₁₁†Hm1(√pm1vm₁₁xm1+√pm2v¯m₂ xm2) + nm1

= √pm1sm₁₁xm1+ sm₁₁(α1v₁₁m†v₂₁m + α2vm₁₁†v₂₂m)um₁₁+ um₁₁†nm1. (2.6)

To eliminate cochannel interference from user 2, we need

α₁ =−v m† 11v22m vm₁₁†vm 21 α₂. (2.7)

Similarly, to completely suppress interference into user 2’s received waveform, we need

β₁ =−s m 22vm22†vm11 sm 21vm21†vm11 β₂. (2.8)

The resulting α’s and β’s should then be normalized such that the norm of the processing vectors are all equal to unity.

The received signal-to-noise ratio (SNR) for the two eigenchannels are given by

SN Rm1(1)= (sm₁₁)2pm1 σ2 , (2.9) SN Rm2(2) = (sm₂₁α₁β₁+ s₂₂mα₂β₂)2pm2 σ2 (2.10)

where the numbers in the subscript brackets denote the indices of the users who have the access to the corresponding eigenchannels.

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To be more speciﬁc, if the BS wants to transmit the data to user k through the rth eigenchannel on subcarrier m, we should have

w_imj† _iHmjitrmk √ prmkxrmk = 0 i = 1,· · · , (r − 1) (2.11) and w†_rmkHmktimji √_p imjiximji = 0 i = 1,· · · , (r − 1) (2.12)

where j_i denotes the index of the user to whom the ith eigenchannel is assigned. Since

trmk and wrmk can be written as

trmk = r l₌₁ αlvmkl, (2.13) wrmk = r l=1 βlumkl (2.14)

(2.9) and (2.10) are equivalent to

w†_imj iHmji _r l=1 αlvklm √ prmkxrmk = 0 i = 1,· · · , (r − 1) (2.15) and _r l=1 βlumkl _† Hmktimji √_p imjiximji = 0 i = 1,· · · , (r − 1). (2.16)

The above equations and the condition that the the precoding vectors should be nor-malized to render unity norm imply that the corresponding gain to noise ratio (GNR) is given by GN Rmr(k) = _i₌₁r αiβiski 2 σ2 . (2.17)

Similarly, if user k wants to transmit the data to the BS through the rth eigenchannel on subcarrier m, the following identities should be satisﬁed.

_r l=1 βlumkl _† Hmktimji √ prmkxrmk = 0 i = 1,· · · , (r − 1) (2.18)

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and w†_imj_iHmji _r l=1 αlvmkl √ pimjiximji = 0 i = 1,· · · , (r − 1). (2.19)

Our design philosophy is force the user whose candidate transmit channels have weaker eigenmodes to “ﬁt” the user(s) with stronger eigenmodes by transmitting over an eigen-channel which lies within the dual space of the space spanned by all previously se-lected eigenchannels. Each new eigenchannel is obtained by using proper processing vectors which are linear combinations of known eigenvectors. The process is similar to a Gram-Schmidt orthonormalization process except that the process follows the descend-ing eigen-magnitude order. Hence, a precoder based on the above design procedure is henceforth referred to as a Gram-Schmidt (GS) precoder.

Once the assignment and the orthogonalizing weighting coeﬃcients of the ﬁrst r eigenchannels are determined, the corresponding GNR can be computed accordingly.

2.4 Problem Formulation

Now we are ready to recast in mathematical form the RA problem of assigning subcar-riers, and the corresponding power and the number of bits loaded to users such that the total transmit power of the system is minimized while the QoS of each user is satisﬁed. Let Rkbe the rate requirement for user k (bits/per OFDM symbol) and brmk the number

of bits transmitted over the mth subcarrier using the rth eigenchannel. bmax denotes

the maximum bit number (the highest modulation order) allowed to be carried by an eigenchannel. The RA problem can then be stated as

arg min Armk,prmk M m=1 R r=1 K k=1 Armkprmk (2.20)

subject to the constraints:

R r=1 M m=1 brmk = Rk ∀ k (2.20a)

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R r=1 K k=1 Armk = R ∀ m (2.20b) Armk ∈ {0, 1} ∀ r, m, k (2.20c) prmk ≥ 0 ∀ r, m, k (2.20d) bmax ≥ brmk ≥ 0 ∀ r, m, k (2.20e)

where prmk = f (BERk, brmk, GN Rmr(k)) if Armk = 1 and prmk = 0, otherwise. BERk

represents user k’s target BER and f (·, ·, ·) usually has a closed-form expression. If an

M -ary quadrature amplitude modulation (M-QAM) is employed, then f (·, ·, ·) or prmk

is given by [19] prmk = 1 GN Rmr(k) ln 1 5BERk 2brmk − 1 1.5 (2.21)

It is noted that we only consider the case thatK_i₌₁Ri ≤ RMbbmax since that if we

haveK_i₌₁Ri > RM bbmax, the optimization problem will have no feasible solutions. The

above optimization problem is a mixed-integer problem which is NP-hard. To ﬁnd the optimal solution all transmission resources–subcarriers, eigenchannels, bits and power– should be jointly allocated, which, unfortunately requires very high computational com-plexity. Suboptimal but aﬀordable-complexity solutions are perhaps more practical and desirable.

2.5 Resource Allocation Algorithms

In this section, we present two adaptive resource allocation algorithms for MIMO systems with the orthogonal precoding scheme described in previous sections. The first algorithm is an efficient space/frequency allocation algorithm (Algorithm I). It first determines the eigenchannel number for each user and then assigns eigenchannels to the users based on their eigenmode strengths. Once the eigenchannel assignment is done, we

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use convention bit-loading algorithm to load bits on users’ eigenchannel set and compute the corresponding transmitting power. The second algorithm is a constraint relaxation based greedy search algorithm (algorithm II). The algorithm contains total M stage and each stage, we will check all possible combinations of the eigenchannel assignment for certain subcarrier to decide which combination results in smallest increment of the transmitting power and repeat the process until all eigenchannels on all subcarriers are allocated.

2.5.1 An Space/Frequency Allocation Algorithm

We ﬁrst determine the required eigenchannel number for each user and assign the eigen-channels to the users using a modiﬁed version of the two-phase algorithm of [20]-[21]. Then we use the conventional bit-loading algorithm to allocate bits over each user’s eigenchannel subset and compute the required transmit power.

Determine the required

eigenchannel number for each user.

Assign eigenchannels to users.

Use the conventional bit-loading algorithm to

allocate bits over each user's eigenchannel subset.

Compute the required transmit power.

Figure 2.1: Flow Chart Description of Algorithm I.

In the ﬁrst phase we compute the required eigenchannel number for each user accord-ing to the QoS and the average channel condition. For each subcarrier, say, the mth, we

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sort the maximum eigenmodes λmax(k, m) of the channel matrices Hmk, k = 1, 2,· · · , K

in descending order, i.e.,

λmax(k1, m) > λmax(k2, m) >· · · > λmax(kK, m),

where

ki = arg max k∈IK\{k1,k₂,ki−1}

λmax(k, m), and IK ={1, 2, · · · , K}

and set Armk = 1 if k = kr. If R < K we set ARmk = 1 for those k = ki, i > R.

The computing of the weighting coeﬃcients and the corresponding GNR follow that described in Section II-B. Deﬁne the average channel condition for user k by

Tk= 1 M M i=1 R j=1 AjikGN Rij(k). (2.22)

The minimum required eigenchannel number for user k is Rk/bmax. The actual

eigen-channel number ck is determined by iteratively verifying the relative reduction of the

total transmit power after the allocation of an additional subcarrier. The detailed algo-rithm is given in Table 2.1.

Step 1: (assume Tk has been computed for all k

according to (3.1) ) (initialization) cmax

k =Rk/bmin;

ck = cmink =Rk/bmax for each k

Step 2: while K k=1 ck < RM and ck < cmaxk ∀k for k = 1 : K if ck < cmaxk ¯ Pk = ck· f(BERk,R_ck k, Tk) ¯ P_knew= (ck+ 1)· f(BERk,_(cRk k+1), Tk) ΔPk= ¯Pk− ¯Pknew end end w = arg maxkΔPk cw = cw+ 1 end

After determining ck’s, we then assign the eigenchannels on all subcarriers to each user

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Table 2.1: Algorithm for computing the required eigenchannel number.

Section II-B. Let the kth eigenchannel of the mth subcarrier be represented by the two-tuple (k, m). The channel assignment follows the order (1, 1) → (1, 2) → · · · → (1, M ) → (2, 1) → (2, 2) → · · · → (2, M) → (3, 1) → · · · . That is, we assign the first eigenchannel on all subcarriers first, and then assign the second eigenchannel on all subcarriers, and so on. The ordering of subcarriers in the channel assignment process is important as once the eigenchannels of a subcarrier are assigned, no re-assignment is allowed. When assign rth eigenchannel on all subcarriers, we first sort the user on each subcarrier according to their GN Rmr(k) in descending order and denote the largest

GN Rmr(k) on the mth subcarrier as Qm and then sort subcarriers according to Qm in

descending order. Once the order of the subcarrier is determined, we assign the eigen-channel to the user with largest GN R; see Table 2.2 for details. After ﬁnishing eigen-channel assignment, we use the conventional bit-loading algorithm to allocate bits and compute the corresponding required transmit power for each user. This algorithm initially allo-cates zero bit to all subcarriers and then alloallo-cates bit by bit to the subcarrier which requires the least additional transmit power. The allocation process repeats until all data rate requirements are satisﬁed. The details of the bit-loading algorithm is given in

Table 2.3. For a given set of assigned eigenchannels, the proposed bit-loading algorithm

is optimal which we summarize below.

Lemma 2.5.1. For a fixed eigenchannel assignment, the bit allocation algorithm

de-scribed by Table 2.3 is optimal, i.e., it results in minimum power consumption.

Proof. For the given BER and the GNR of the eigenchannel assigned to user k, deﬁne

Δf (brmk) as

Δf (brmk) =

f (BERk, brmk, GN Rmr(k))− f(BERk, brmk− 1, GNRmr(k)), if brmk ≥ 1

f (BERk, brmk, GN Rmr(k))− f(BERk, 0, GN Rmr(k)), if brmk < 1

The author of [22] introduce necessary and suﬃcient conditions for a discrete bit alloca-tion to be optimal:

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1. Δf (brmk)≤ Δf(brmk+ 1) ∀r, r = 1, 2, . . . , R , ∀m, m = 1, 2, . . . , M (eﬃcient) 2. R r=1 M m=1 brmk = Rk (B-tight)

Any bit distribution that satisfies the above conditions will be the optimal solution. The second condition is clearly satisfied since the bit-loading algorithm will not stop loading bits until the loaded bits achieve the user data rate. As for the first condition, we first show that Δf (brmk + 1) > Δf (brmk) for all r, m and k. Since we use a closed form

expression to estimate the require power when QAM modulation is used (2.21), we have

Δf (brmk+ 1) = f (BERk, brmk+ 1, GN Rmr(k))− f(BERk, brmk, GN Rmr(k)) = 1 GN Rmr(k) ln 1 5BERk 2brmk+1− 2brmk 1.5 = 1 GN Rmr(k) ln 1 5BERk 2brmk 1.5 > 1 GN Rmr(k) ln 1 5BERk 2brmk−1 1.5 = 1 GN Rmr(k) ln 1 5BERk 2brmk − 2brmk−1 1.5 = Δf (brmk) for brmk > 0 and Δf (brmk+ 1) = f BERk, brmk+ 1, GN Rmr(k))− f(BERk, brmk, GN Rmr(k) = 1 GN Rmr(k) ln 1 5BERk 2brmk+1− 2brmk 1.5 = 1 GN Rmr(k) ln 1 5BERk 2brmk 1.5 > 0 = 1 GN Rmr(k) ln 1 5BERk 2brmk − 0 1.5 = Δf (brmk) for brmk = 0

If there exist a brmk and a brmk in the result of the bit-loading algorithm such that

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when deciding to increase the bits of the rth eigenchannel on the mth subcarrier from

brmk− 1 to brmk, the power increment of loading a bit to the rth eigenchannel on the

mth subcarrier is less than the power increment of loading a bit to the rth eigenchannel on the mth subcarrier. Therefore, the result of the bit-loading process must satisfy the ﬁrst condition.

Step 1: (initialization) Set all Armk = 0.

Step 2: while ck > 0 ∀k

for r = 1 : R

Qm = maxkGN Rmr(k)

Arrange all subcarriers by decreasing order of Qm such that Q1 ≥ Q2 ≥ ... ≥ QM.

for m = 1 : M

Compute GN Rmr(k)∀k according to the

previous 1∼(r-1) channel assignment process. Let Dm ={GNRmr(1), GN Rmr(2)

..., GN Rmr(K)}. (If r = 1, then let

GN Rmr(k)= λmax(k, m) ∀k. ) while K k=1 Armk = 0 w = arg maxkGN Rmr(k) ∈ Dm if cw > 0 Armw = 1, cw = cw− 1 else Dm = Dm− {GNRmr(w)} end end end end end

Table 2.2: The eigenchannel assignment algorithm.

One of the advantages of this algorithm is its low computational complexity. We need only to perform bit-loading for each user once; the complexity analysis is discussed later. Another advantage of this algorithm is that it considers not only the fairness but also the eﬃciency of the resource utilization. In step 1, we insure that every user is assigned enough eigenchannels to transmit data so that outage will not occur. In step 2,

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we assign eigenchannels to the user who has the highest eigenmode magnitude, making the most of the available spatial resources.

Step 1: (initialization) Set all brmk = 0 and prmk = 0 for all r, m, k.

Step 2: for k = 1 : K while M m=1 R r=1 Armkbrmk < Rk

set pmin =∞, mindex = 0, and rindex= 0

for m = 1 : M for r = 1 : R if Armk = 1 ptemp = f (BERk, brmk+ 1, GN Rmr(k))− f (BERk, brmk+ 1, GN Rmr(k)) if ptemp< pmin

mindex = m,rindex = r,pmin = ptemp

end end end end

brindexmindexk = brindexmindexk+ 1

pr_indexm_indexk = pr_indexm_indexk+ pmin

end end

Table 2.3: The conventional bit-loading algorithm.

2.5.2 A Constraint-Relaxation based Greedy Search Algorithm

The second algorithm begins with a fair initial condition that gives all users the oppor-tunity to access all its eigenchannels over all subcarriers. Each user uses a bit-loading algorithm to obtain the local power minimization solution based on the allocated eigen-channel subset. The proposed eigen-channel allocation process consists of a series of M -stage deletion decisions. At each stage, the eigenchannels associated with a subcarrier is given to the users who have the desired spatial channel condition. These eigenchannels are then removed from the serving channel subsets of all other users. The order of subcar-riers assignment is the same as Algorithm I and the assignment process is carried out

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on a search tree whose root node has KR outgoing branches to represent all possible assignments of subcarrier 1.

Stage i=1~M

i=1

i=2

i=3

i=M-1

i=M

1

2 K

R

Figure 2.2: A search tree representing the multi-stage bit-loading procedure (Algorithm II).

Similarly, every node at any given level of the search tree, say the tth level, has

KR _{outgoing branches (to K}R _{child nodes), each represents a possible eigenchannel}

assignment (removal) decision and a tentative eigenchannels allocation. The resource allocation is tentative because only the eigenchannels for the ﬁrst t subcarriers are as-signed and those for the remaining M − t subcarriers are still unassigned. Given the initial fair channel allocation and the ultimate object of minimizing the required total power, the cost for a decision at any level should be the minimum required power for the corresponding tentative eigenchannel allocation. Repeating such a channel assignment and power allocation process for M times, we complete the search over the M -level tree and ﬁnish allocating all the eigenchannels and subcarriers; the corresponding power/rate allocations are accomplished simultaneously.

We ﬁrst set Armk = 1 for all r, m, k and re-index subcarriers by the same order

de-scribed in Algorithm I. The channel-deletion process begins at the 1st subcarrier and continues until the last one. Let F_i(m)= [f₁ f₂ · · · fR]T be the ith eigenchannel

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assign-ment vector for the mth subcarrier that assign the ith eigenchannel to user fi. There

are Rk candidate eigenchannel assignment vectors in total, each represents a possible eigenchannel assignment. Reset Armk based on Fi(m). As for the eigenchannels on the

M − 1 unassigned subcarriers, each user treat these eigenchannels without considering

other users’ interference. That is, when an user loads bits to the rth eigenchannel on the unassigned subcarriers, the channel gain is exactly equal to the rth singular value. Use the bit-loading algorithm to allocate bits and compute the corresponding power. After we check all RK _{candidate assignment options, we choose the assignment vector with}

the minimal transmit power and re-set Armk accordingly. The same procedure continues

for the (m + 1)th subcarrier and repeat the deletion process until all eigenchannels for all subcarriers are assigned.

Note that a node of the, say, mth stage, represents a particular assignment of the eigenchannels associated with the mth subcarrier. The corresponding optimal bit-loading scheme can be found by reconsidering only bit-bit-loading on these eigenchannels with bit-loadings on all other eigenchannels remain intact. This fact is summarized as

Lemma 2.5.2. For the constraint-relaxation based greedy search algorithm, the required

power associated with a node of the mth stage remains unchanged no matter one re-loads only the eigenchannels of the mth subcarrier or all spatial channels.

Proof. For user k, let brnk (r = 1 ∼ R, n = 1 ∼ M, n = (m − 1)) denote the bit

distribution of eigenchannels on all subcarriers except subcarrier m− 1 on the m − 1 stage and B = R r=1 M n=1 brnk− R r=1

br(m−1)k denote the number of bits of user k which are

allocated to the eigenchannels on all subcarriers except subcarrier m− 1. If the result of only re-allocating the bits on the eigenchannels which are assigned on the (m− 1)th stage is not the same as re-allocating all the bits on all the eigenchannels, this means that the result of only re-allocating the bits on the eigenchannels which are assigned on the (m− 1)th stage is not optimal. In other words, there must exist another bit distribution b_rnk (r = 1 ∼ R, n = 1 ∼ M, n = (m − 1)) for these B bits with lower

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power consumption than bit distribution brnk. But this will contradict the fact that the

bit distribution of stage M − 1 is brnk since the power consumption of bit distribution

brnk should be minimal. Therefore, it is impossible to have another bit distribution for

these B bits with lower power consumption than bit distribution brnk. So we only need

to re-allocate the bits on the eigenchannels which are assigned on the (m− 1)th stage instead of re-allocating all the bits on all the eigenchannels.

2.6 Complexity Analysis and Numerical Results

2.6.1 Computational Complexity Analysis

In this subsection, we analyze the complexity of the proposed resource allocation algo-rithms for the orthogonal precoding scheme.

First we check the complexity of the efficient space/frequency resource allocation algorithm. The efficient space/frequency resource allocation algorithm consists of three parts: compute eigenchannel number for each user, assign eigenchannels to the users and finally the bit-loading algorithm. The complexity of computing eigenchannel number for each user and assigning eigenchannels to the users are O(RM − RkK/bmax)≤ O(RM)

and O(R(Klog₂K + M log₂M ) + RM ), respectively. The R(Klog₂K + M log₂M ) term

in eigenchannel assignment is the complexity of the sorting before assigning the rth eigenchannel on all subcarriers. And the complexity of the bit-loading algorithm is

O(

K

k=1

RkM R)≤ O(KRmaxM R) where Rmax is the maximum date rate of the users. So

the overall complexity of the eﬃcient space/frequency resource allocation algorithm is

O(RM + R(Klog₂K + M log₂M ) + RM + KRmaxM R)≈ O(KRmaxM R).

Now we examine the complexity of the constraint-relaxation based greedy search algorithm. The the constraint-relaxation based greedy search algorithm contains total

M stage and in each stage, we have to check KR _{possible choices.} _{Therefore, the}

computational complexity of the constraint-relaxation based greedy search algorithm is

O(KR_R

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we have to perform complete bit-loading algorithm for each user. But for the later stage, The bit-loading algorithm only need to reload the bits on the eigenchannels which are assigned on the last stage (second term). The complexity can be further approximated by O(M2KR+1_R2_b

max).

2.6.2 Numerical Results

Selected simulated performance of the proposed RA algorithms are presented in this subsection. First we evaluate the performance of the proposed RA algorithms for the orthogonal precoding scheme in Fig. 2.2 ∼ 2.4. For Fig. 2.5 ∼ 2.10, we compare the performance of the proposed RA algorithm for the orthogonal precoding scheme and BD [16].

The performance of the proposed algorithms for uplink and downlink transmissions is shown in Fig. 2.2 and Fig. 2.3 respectively. In Fig. 2.4 we compare the performance of the proposed algorithms with that of the optimal solution. The average power is normalized by that of the single-user case, i.e., when a single user has access to all eigenchannels and all subcarriers. We deﬁne the average power ratio at BER=B as :

PB = 10 log₁₀ Pavg,B P_avg,₁₀−5_,single (2.23)

where Pavg,B represents the average transmit power for a given modulation scheme at

BER=B and Pavg,10−5,singlerepresents the average transmit power for the single user case

at BER=10−5.

We assume each Hmk is a 2× 4 (4 antennas at the BS and 2 antennas at each MS)

matrix with i.i.d. zero-mean, unit-variance complex Gaussian entries. The system has six diﬀerent modulation modes, BPSK, QPSK, 8QAM, 16QAM, 32QAM, and 64 QAM, respectively. For simplicity, we assume that the required data rate and BER are the same for all users.

In Fig. 2.2, we compare the performance of the proposed algorithms with that of the adaptive zero-forcing MIMO-OFDM receiver proposed in [23] and its non-adaptive

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counterpart. Algorithm I (the eﬃcient space/frequency resource allocation algorithm) is superior to the adaptive MIMO-OFDM ZF approach by a 2.5 dB margin and Algorithm II (the constraint-relaxation based greedy search algorithm) oﬀers additional 0.5 dB performance gain. Both algorithms achieve more than 12 dB performance gain against the non-adaptive MIMO-OFDM ZF receiver.

−10 −5 0 5 10 15 20 10−5 10−4 10−3 10−2 10−1

Average Power Ratio (dB)

BER

non−adaptive MIMO−ZF adaptvie MIMO−ZF algorithm I algorithm II

Figure 2.3: Average power ratio per user for a MIMO-OFDM uplink; 32 subcarriers, 64 bits per OFDM symbol, 4 users.

In Fig. 2.3, we consider downlink transmission and compare the performance of the FDMA scheme and our algorithms. For the FDMA scheme (without bit-loading), the subcarriers are allocated to the users like the classical FDMA scheme according to their data rate requirements. Each user has access to all the eigenchannels on the corresponding subcarrier subset. Each required data rate (bits/transmission) is equally distributed among all eigenchannels available to a user. We also consider the FDMA scheme with bit-loading. Similar to the uplink case, Algorithm I outperforms the FDMA scheme with bit-loading by 2.5 dB and Algorithm II provides additional 0.5 dB gain.

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Both algorithms have more than 10 dB advantage over the FDMA scheme without bit-loading. −10 −5 0 5 10 15 10−5 10−4 10−3 10−2 10−1

BER

FDMA with bit−loading FDMA without bit−loading algorithm I

algorithm II

Figure 2.4: Average power ratio per user for a MIMO-OFDM downlink; 32 subcarriers, 64 bits per OFDM symbol, 4 users.

Finally, Fig. 2.4 plots the performance of both the optimal solution and the proposed algorithms. It is found that Algorithm II yields performance almost identical to that of the optimal solution while Algorithm II suﬀers only minor degradation.

In Figs. 2.5–2.9, we compare the proposed orthogonal precoding scheme (using Algo-rithm I) and BD in downlink transmission. BD is a popular linear precoding technique for the multiple antenna multicast channel that involves transmission of multiple data streams to each receiver such that no multiuser interference is experienced at any of the receivers. In BD, each user data vector is multiplied by a precoding matrix to project the transmitted signal to the null space of the space spanned by all other users’ spatial channels.

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−12 −10 −8 −6 −4 −2 0 2 10−5 10−4 10−3 10−2 10−1

BER

optimal solution algorithm I algorithm II

assume that Tx = 4 and Rx = 2 and for the 4-user case, Tx = 8 and Rx = 2. As for the

required user data rate, we consider three cases: 64 bits per OFDM symbol, 128 bits per OFDM symbol, and 256 bits per OFDM symbol, respectively. The system oﬀers eight diﬀerent modulation options ranging from from BPSK to 256 QAM.

First we examine the performance of the 2-user case in Figs. 2.5 and 2.6. It is found that the proposed precoding scheme outperforms the BD approach when desired data rate is 64 or 128 bits per OFDM symbol. However, Fig. 2.7 indicates that BD outperforms the proposed precoding scheme by almost 0.5 dB. The same behavior can be found in the 4-user case. In Figs. 2.8 and 2.9, the performance of the proposed precoding scheme is better than that of the BD precoder when desired data rate is 64 bits or 128 bits per OFDM symbol. However, when the data rate increases to 256 bits per OFDM symbol, BD is superior to our precoder by a 1.5 dB margin.

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trans-mit power of an eigenchannel is a function of both GNR (GN Rmr(k)) and the

num-ber of bits loaded (brmk). When the eigenchannel’s GNR is large (strong eigenmode)

or the number of loaded bits is small, the corresponding required transmit power is low. For the BD precoder, the number of eigenchannels per subcarrier for each user is

Tx − (Rx∗ (K − 1)) while that for the proposed precoder is R = min(Tx, Rx). For the

2-user case, the numbers of eigenchannel per subcarrier for the BD and GS precoders are 4 and 2. For the 4-user case, the number of eigenchannel per subcarrier for the BD precoder are 8 while that for the GS precoder remains to be 2. Moreover, both simu-lation and theoretical analysis show that the eigenmode magnitude suffer degradation by using BD (this will be proved in the last of this section). Define the channel loading coefficient η as η = K i=1 Rk RM bmax . (2.24)

When η is more close to 1, this means that each eigenchannel must be loaded more bits since the user’s data rate is close to the system limit such that the total consuming power is higher. When η is more close to 0, this means that the bits on each eigenchannel is less and the total consuming power is also less relatively. When the channel loading is light (η is close to 1), the bit number on each eigenchannel (brmk) is small and the eigenmode

magnitude (GN Rmr(k)) will dominate the system performance. However, when we

in-crease the user’s data rate such that the channel loading is heavy, the exponential term

brmk in (2.21) becomes an important factor that will aﬀect the system performance.

The channel loading coeﬃcient for BD and the proposed precoding scheme is listed as follows:

Table 2.4: η for the 2 users case

Precoding scheme/Users’ data rate 64 bits 128 bits 256 bits BD 0.125 0.25 0.5 Orthogoal precoding scheme 0.25 0.5 1

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Table 2.5: η for the 4 users case

Precoding scheme/Users’ data rate 64 bits 128 bits 256 bits BD 0.0625 0.125 0.25 Orthogoal precoding scheme 0.25 0.5 1

Numerical results given in the above two tables indicate that the BD precoder has lower channel loading coeﬃcients (≤ 0.5) as it oﬀers more eigenchannels. Therefore, when the data rate requirement is low, the performance of BD precoder is inferior to that of GS precoder due to the fact that BD precoding results in weaker eigenmodes; see

Lemma 2.6.2 below. But for high data rate requirements, the per eigenchannel loading

for the BD precoder remains relatively low which then leads to better performance.

−14 −12 −10 −8 −6 −4 −2 0 2 4 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

To prove the eigenmode degradation suﬀered in the BD scheme, we need

Lemma 2.6.1. (Weak Majorization Lemma) Let x₁, . . . , xn, y1, . . . , yn be 2n given real

numbers such that x₁ ≥ . . . ≥ xn, y1 ≥ . . . ≥ yn and k i=1 yi ≥ k i=1 xi, k = 1, . . . , n.

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−12 −10 −8 −6 −4 −2 0 2 4 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

Then for any real-valued function f (·) which is increasing and convex on the interval

[min{xn, yn}, y1],f (x1)≥ . . . ≥ f(xn), f (y1)≥ . . . ≥ f(yn), and k i=1 f (yi)≥ k i=1 f (xi), k = 1, . . . , n.

Using the above lemma we can prove that the sum magnitude of the eigenmodes associated with the augmented matrix XY is always less than the sum of magnitude product for the component matrixes X and Y.

Lemma 2.6.2. Consider the m× p and p × n matrices, X and Y and let σi(X) be the

ith singular value of X. Then

k i=1 [σi(X)σi(Y)] ≥ k i=1

[σi(XY)] for k = 1, . . . , q, where

q = min{m, p, n}.

Proof. We ﬁrst show that k

i=1

[σi(X)σi(Y)] ≥ k

i=1

[σi(XY)] for k = 1, . . . , q, where q =

min{m, p, n}. Performing SVD on XY gives XY = UΣ V† = [Uk U×]Σ[Vk V×]†

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−8 −6 −4 −2 0 2 4 6 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

of YVk yields YVk = U Σ V † = [ Uk U×] Σk 0

V† where Uk consists of the ﬁrst k

columns of U. Then we have

σ₁(XY)· · · σk(XY) = | det(U†kXYVk)|

= | det(U†_kX U Σ V†)| = | det(U†_kX UkV † V ΣkV † )| = | det(U†_kX U)|| det( V ΣkV † )| ≤ [σ1(X)· · · σk(X)][σ1( Σk)· · · σk( Σk)] = [σ₁(X)· · · σk(X)][ σ₁(V†_kY†YVk)· · · σk(V†_kY†YVk) ] ≤ [σ1(X)· · · σk(X)][ σ₁(Y†Y)· · · σk(Y†Y))] = [σ₁(X)· · · σk(X)][ σ₁2(Y)· · · σ_k2(Y)] = k i=1 [σi(X)σi(Y)].

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−10 −8 −6 −4 −2 0 2 4 6 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

It is noted that the inequality becomes equality when m = p = n and k = n. If we take log for both sides, we get

k i=1 ln(σi(X)σi(Y)) ≥ k i=1

ln(σi(XY)). Let xi = ln(σi(XY)),

yi = ln(σi(X)σi(Y)) and f (·) is to take the exponential of the argument matrix. The

Weak Majorization Lemma then implies

k i=1 [σi(X)σi(Y)]≥ k i=1 [σi(XY)] for k = 1,· · · , q, where q = min{m, p, n}.

Let X = H and Y be the precoder matrix. For the BD precoder, σi(Y) = 1, ∀ i.

Hence, the above lemma tells us that BD precoding decreases the sum strength of all eigenchannels. Moreover, our simulation results show that not only the sum of the eigenmode magnitudes but also the individual eigenmode magnitude degrades after BD precoding.

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−8 −6 −4 −2 0 2 4 6 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

−6 −4 −2 0 2 4 6 8 10 12 10−5 10−4 10−3 10−2 10−1 BER

BD

Proposed scheme

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Chapter 3 Resource Allocation for MIMO

Systems with Non-Orthogonal

Precoding

3.1 System and Transceiver Models

In the previous chapter, we consider MIMO systems that use a orthogonal precoding scheme so that system users can transmit through distinct eigenchannels on the same subcarrier without causing interference to each other. For such a scheme, however, the maximum eigenchannel number is bounded by the rank of the MIMO channel matrix (R) and thus the spectrum eﬃciency may be constrained. To increase the spectrum eﬃciency, we allow more than R users to transmit over the eigenchannels on the same subcarrier. In this situation, the co-channel interference among users is no longer avoidable. Therefore, the associated optimization problem becomes more complicated due to the constraints on the tolerable inter-channel interference (ICI).

Similar to the previous system setup, we consider a MIMO-OFDMA system with a single base station (BS) equipped with Tx antennas and K mobile station (MS) users,

each equipped with Rxantennas. The frequency band used contains M subcarriers which

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orthogonal eigenchannels for users with no interference and additional Q (R∼ R+Q−1) eigenchannels with various tolerable interference levels.

For the R− 1 orthogonal eigenchannels, the way to choose the pre-processing and post-processing vectors is the same as that described in Chapter 2. That is, for the user to whom the rth eigenchannel is given, the pre-processing and post-processing vectors are the linear combinations of ﬁrst r left and right singular vectors, respectively. In order that the Q non-orthogonal eigenchannels do not induce interference to the R− 1 orthogonal eigenchannels, we require that the users who are allocated non-orthogonal eigenchannels to transmit over an eigenchannel which lies in the null space spanned by all R− 1 orthogonal eigenchannels. More speciﬁcally, they use linear combinations of R singular vectors as the processing vectors to project the transmitting signal to the null space of the R− 1 dimensional space spanned by orthogonal eigenchannels.

Although the non-orthoganal eigenchannels will not interfere with the R− 1 orthog-onal eigenchannels, the co-channel interference among the non-orthogorthog-onal eigenchannels is unavoidable. Here we deﬁne B_mk = 1 if user k is to transmit on the mth subcarrier’s non-orthogonal eigenchannel and B_mk = 0, otherwise. The GINR (gain to interference and noise ratio) for users k who is allowed to transmit data on the non-orthogonal eigen-channels can be expressed as:

GIN Rmk = _R i=1 αkiβkiski 2 σ2+ K i=1,i=k w†mkHmkBmitmi 2 p_mi (3.1) If we deﬁne R i₌₁

αkiβkiski as gmk (the channel gain of user k) and w†mkHmkBmitmi(the

correlation between user k and user i) as ρmki, then (1) can be simpliﬁed as:

GIN Rmk = |gmk| 2 σ2+ K i=1,i=k |ρmki|2pmi (3.2)

多天線-正交分頻多工存取無線網路之資源配置

國 立 交 通 大 學

電信工程學系

碩 士 論 文

多天線-正交分頻多工存取無線網路之資源

配置

Resource Allocation for MIMO-OFDMA

based Wireless Network

研 究 生：翁志倫

多天線-正交分頻多工存取無線網路之資源配置

Resource Allocation for MIMO-OFDMA based Wireless Network

研究生：翁志倫 Student：Chih-Lun Weng

指導教授：蘇育德博士 Advisor：Dr. Yu T. Su

國 立 交 通 大 學

電 信 工 程 學 系

碩 士 論 文

多天線-正交分頻多工存取無線網路之資源配置

學生：翁志倫

指導教授

蘇育德 博士

國立交通大學電信工程學系碩士班

摘

要

Resource Allocation for MIMO-OFDMA based

Wireless Networks

誌 謝

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Resource Allocation for

Orthogonally Precoded MIMO

Systems

2.1

Background

2.2

System Parameters and Transceiver Model

2.3

Spatial Channel Assignment and Related Signal

Processing

2.4

Problem Formulation

2.5

Resource Allocation Algorithms

2.5.1

An Space/Frequency Allocation Algorithm

Determine the required

eigenchannel number for each user.

Assign eigenchannels to users.

Use the conventional bit-loading algorithm to

allocate bits over each user's eigenchannel subset.

Compute the required transmit power.

2.5.2

A Constraint-Relaxation based Greedy Search Algorithm

Stage i=1~M

i=1

i=2

i=3

i=M-1

i=M

1

2

K

2.6

Complexity Analysis and Numerical Results

2.6.1

Computational Complexity Analysis

2.6.2

Numerical Results

Chapter 3

Resource Allocation for MIMO

Systems with Non-Orthogonal

Precoding

3.1

System and Transceiver Models

國立交通大學

碩士論文

研究生：翁志倫

國立交通大學

電信工程學系

碩士論文

蘇育德博士

誌謝