Transmitter Strategies for Closed-Loop MIMO-OFDM. Joon Hyun Sung

Transcription

1 Trasmitter Strategies for Closed-Loop MIMO-OFDM A Thesis Preseted to The Academic Faculty by Joo Hyu Sug I Partial Fulfillmet of the Requiremets for the Degree Doctor of Philosophy School of Electrical ad Computer Egieerig Georgia Istitute of Techology July 2004

2 Trasmitter Strategies for Closed-Loop MIMO-OFDM Approved by: Professor Joh R. Barry, Adviser Professor D Professor Ye (Geoffrey) Li Professor E Professor Steve W. McLaughli Date Approved: 9 July 2004

3 To my parets. iii

4 ACKNOWLEDGEMENTS iv

5 TABLE OF CONTENTS DEDICATION iii ACKNOWLEDGEMENTS iv LIST OF TABLES viii LIST OF FIGURES ix SUMMARY xiii 1 INTRODUCTION MIMO Wireless Commuicatios Closed-Loop MIMO Objective ad Cotributios Notatio Summary MIMO CHANNELS AND PROPERTIES MIMO Wireless Chaels Spatially Ucorrelated Chael Spatial Correlated Chael MIMO-OFDM Fuctios for Eigevalues Arithmetic ad Geometric Meas of Eigevalues Empirical Distributio Joit Distributio of Eigevalues CLOSED-LOOP MIMO-OFDM Trasmit Beamformig Pricipal Eigemode Eigebeamformig Availability of CSI at the Trasmitter Feedback of CSI v

6 3.2.2 Time-Divisio Duplex Adaptive Eigebeamformig INFORMATION THEORY FOR MIMO Defiitio of Average ad Outage Capacity Spatial Multiplexig ad Diversity Orders CSI Ukow at Trasmitter Average Capacity Outage Performace CSI Kow at Trasmitter Power Allocatio Problem Average Capacity Zero-Outage Capacity Miimum Outage Probability Optimal Outage Regio i MIMO ASYMPTOTIC BEHAVIORS AT HIGH SNR Diversity Order CSI Ukow to Trasmitter Eigebeamformig with Short-Term Costrait Capacity at High SNR CSI Ukow to Trasmitter CSI Kow at Trasmitter Proof of Propositio Proof of Propositio Proof of Propositio Proof of Propositio Proof of Propositio APPROACHING ZERO-OUTAGE CAPACITY WITHOUT WATER- FILLING vi

7 6.1 Problem Statemet FUSE Allocatio Fixed-Rate Allocatio Asymptotic Behaviors FUSE Allocatio FIX Allocatio Numerical Results Spatially Ucorrelated Chaels Spatially Correlated Chaels Proof of Propositio RATE ALLOCATION WITH GRANULARITY CONSIDERATIONS Bit-Allocatio Problem ad Full-Search Allocatio Proposed Bit-Allocatio Strategies Robust Allocatio Strategy Simple Allocatio Algorithms for DMT Bit Allocatio for MIMO-OFDM Derivatio of (240) BIT-ERROR RATE PERFORMANCE Short-Term Eergy Costrait Ucoded Performace of Flat-Fadig MIMO MIMO-OFDM with a Outer Code Log-Term Eergy Costrait CONCLUSIONS Summary Directios for Future Research REFERENCES VITA vii

8 LIST OF TABLES Table 1 θl used i Example Table 2 Table 3 Table 4 Table 5 E[1/s (1) ] for a M M spatially ucorrelated Rayleigh fadig with M {1, 2, 3, 4, 6, 8, 10} /E[1/G Nu ] of M M Rayleigh-fadig chaels with chael memory L by Mote-Carlo simulatios Domiat allocatio cadidates for M = 6 ad R = 9, which are deoted i the form of [r (1) /β,..., r (M) ]/β Samples of Biary Search ad Fixed Allocatio strategies optimized to 2 2 Rayleigh fadig i the form of [r (1) /β,..., r (M) /β] viii

9 LIST OF FIGURES Figure 1 Block diagram for MIMO-OFDM ad its equivalet chael model. 15 Figure 2 Block diagram for trasmit beamformig Figure 3 OFDM ad eigebeamformig trasform a frequecy-selective MIMO chael ito a bak of scalar chaels over space ad frequecy Figure 4 Reciprocity of TDD Figure 5 The average capacity of a M M spatially ucorrelated Rayleighflat-fadig chael whe M {1, 2, 4, 6, 8, 10}. Also plotted are the high-snr asymptotes of the average capacity Figure 6 The ratio of C RCSI / log 2 (SNR) for M {1, 2, 4, 6} Figure 7 Figure 8 Figure 9 Average capacity for L {0, 1, 2, 3, 4, 5} whe the chael is spatially correlated with σ θ = Average capacity for L {0, 1, 2, 3, 4, 5} whe the chael is spatially correlated with σ θ = Outage probability o a spatially ucorrelated flat-fadig chael with M T = 1 ad M R 1 i (76) at R = 4 bits per sigalig iterval. 45 Figure 10 Outage probability o a spatially ucorrelated chael with M T = M R = M for M {1, 2, 4, 6} at R = 4 bits per sigalig iterval.. 46 Figure 11 The outage probability of a spatially ucorrelated 4 4 Rayleighfadig chael with memory L {0, 1, 2, 3, 4, 5} at R = 10 bits per sigalig iterval Figure 12 The outage capacity for 1 1 spatially ucorrelated Rayleigh flat fadig with P OUT {10 1, 10 2, 10 5 } Figure 13 The outage capacity C ɛ=0.01 o a M M spatially ucorrelated Rayleigh-flat-fadig chael with M {1, 2, 4, 6} Figure 14 Spatial multiplexig order o a M M spatially ucorrelated Rayleighflat-fadig chael with M {1, 2, 4, 6} Figure 15 1% outage capacity o a 4 4 spatially ucorrelated Rayleigh-fadig chael with memory L {0, 1, 2, 3, 4, 5} Figure 16 The PDF of mutual iformatio of a 4 4 spatially ucorrelated Rayleigh-fadig chael with L {0, 1, 2, 3, 4, 5} at ρ = 8 db ix

10 Figure 17 Compariso betwee C TRCSI ad C RCSI o a spatially ucorrelated Rayleigh fadig whe M T = M R = M with M {1, 2, 4, 6}. Note that both are idepedet of L Figure 18 C 0 ad C TRCSI o a M M spatially ucorrelated Rayleigh-flat-fadig chael for M {1, 2, 4, 6} Figure 19 Peak-power probability for M T = 1 ad M R > 1 o a spatially ucorrelated Rayleigh-flat-fadig chael for M R {2, 3,..., 10}.. 63 Figure 20 Peak-power probability o a M M spatially ucorrelated Rayleighflat-fadig chael for M {2, 4, 6} Figure 21 Outage probability for M T = 1 ad M R 1 spatially ucorrelated Rayleigh fadig whe CSI is kow at the trasmitter at R = 4 bits per sigalig iterval. As a bechmark, outage probability whe CSI is ukow at the trasmitter is also plotted Figure 22 Outage probability o a M M spatially ucorrelated Rayleigh fadig for M {2, 4, 6} whe CSI is kow at the trasmitter [7].. 66 Figure 23 Compariso betwee C OR ad C 0 o a 2 1 spatially ucorrelated chael with L = 0 memory. Also plotted is R(1 P OUT ) with P OUT miimized accordig to Theorem 4.7 for R {4, 6, 8} Figure 24 Compariso of C TRCSI, C OR, ad C 0 for M M Rayleigh flat fadig with M {1, 2, 4, 6} Figure 25 The lower boud i (136) of the outage probability o a 4 4 spatially ucorrelated Rayleigh-fadig chael with memory L {0, 1, 2, 3, 4, 10} Figure 26 CDF of G Nu o a 4 4 spatially ucorrelated chael for L {0, 1, 2, 3, 4, 5} Figure 27 CDF of A Nu o a 4 4 spatially ucorrelated chael for L {0, 1, 2, 5, 10, 100, 1000} Figure 28 SNR pealty i (150) for M {2, 3,..., 10} o a M M spatially ucorrelated Rayleigh flat fadig Figure 29 SNR pealty of the FUSE allocatio ad its approximatio as M grows at high SNR Figure 30 Approximate SNR pealty of the FUSE allocatio for L {0, 1, 2, 5, 10, 100, 1000}.105 Figure 31 Approximate SNR pealty of the FUSE allocatio for (M T = 1, M R = k), (M T = k, M R = k), ad (M T = 2k, M R = k) with k {2, 3,..., 10}.105 Figure 32 A plot of 1 M 1 M versus M o a M M MIMO chael x

11 Figure 33 Zero-outage capacities of the FUSE ad FIX allocatios o a M M spatially ucorrelated Rayleigh-fadig chael with L = 4 memory. 109 Figure 34 SNR pealty of the FUSE allocatio o a M M spatially ucorrelated Rayleigh-fadig chael with L = 4 memory for M {2, 4, 6}.110 Figure 35 SNR pealty of the FIX allocatio o a M M spatially ucorrelated Rayleigh-fadig chael with L = 4 memory for M {2, 4, 6}. 111 Figure 36 SNR pealty of the FUSE Allocatio o a 4 4 spatially ucorrelated Rayleigh-fadig chael with memory L {1, 2, 3, 4, 5} grows Figure 37 Zero-outage capacities of the FUSE ad FIX allocatios o spatially ucorrelated 2 2, 4 2, ad 6 2 chaels with L = 4 memory Figure 38 SNR pealty of FUSE costrait o a 4 4 spatially correlated chael with L = 4 memory whe σ θl = 0 ad σ θl = Figure 39 Size of full-search set for M {2, 4, 6} Figure 40 Probability that a certai allocatio is used for M = 6 ad R = Figure 41 Number of domiat cadidates (P j > 0.05) for 2 2, 4 4, ad 6 6 MIMO chaels, respectively Figure 42 SNR pealty as we delete allocatios oe by oe from a full-search set for M = 6 ad R = Figure 43 The partial SNR requiremet whe restrictig search to B 1 (ɛ fix j ) i compariso with the partial SNR requiremet of full search (ɛ j ) Figure 44 Achievable rates of Biary Search ad Fixed Allocatio Figure 45 SNR pealty of Biary Search relative to the full-search strategy Figure 46 SNR pealty of Biary Search relative to the full-search strategy Figure 47 Fadig mismatch of Biary Search optimized to Rayleigh fadig ad performace of a robust allocatio o a 4 4 Ricea fadig chael with K = Figure 48 Performace of Biary-Search ad Fixed Allocatio combied with the FUSE costrait (deoted as FUSE-BINARY ad FUSE-FIXED, respectively) o a M M spatially ucorrelated Rayleigh fadig chael with M {2, 4, 6} whe L = 4 ad N = Figure 49 Block diagram for flat-fadig MIMO Figure 50 Ucoded BER performace of eigebeamformig ad orthogoal STBC o 2 2 Rayleigh fadig xi

12 Figure 51 Ucoded BER performace of eigebeamformig ad orthogoal STBC o 4 4 Rayleigh fadig Figure 52 SNR per bit requiremet with the short-term eergy costrait Figure 53 Block diagram for a MIMO-OFDM system with a LDPC outer code at the trasmitter Figure 54 Block diagram for a MIMO-OFDM system with a LDPC outer code at the receiver Figure 55 BER results for eigebeamformig (Fixed Allocatio) ad STBC i 2 2 MIMO-OFDM with LDPC codes Figure 56 Block diagram for MIMO-OFDM Figure 57 BER results of 4 4 MIMO-OFDM with N = 128 ad L = 0 at R = 6 bits per sigalig iterval, averaged over 10 5 idepedet chael realizatios Figure 58 BER results of 4 4 MIMO-OFDM with N = 128 ad L = 4 at R = 6 bits per sigalig iterval, averaged over 10 5 idepedet chael realizatios xii

13 SUMMARY This thesis cocers commuicatio across chaels with multiple iputs ad multiple outputs. Specifically, we cosider the closed-loop sceario i which kowledge of the state of the multiple-iput multiple-output (MIMO) chael is available at the trasmitter. We show how this kowledge ca be exploited to optimize performace, as measured by the zero-outage capacity, which is the capacity correspodig to zero outage probability. O flat-fadig chaels, a closed-loop trasmitter allocates differet powers ad rates to the multiple chael iputs so as to maximize zero-outage capacity. Frequecy-selective fadig chaels call for a combiatio of orthogoalfrequecy-divisio multiplexig (OFDM) ad MIMO kow as MIMO-OFDM. This exacerbates the allocatio problem because it multiplies the umber of allocatio dimesios by the umber of OFDM toes. Fortuately, this thesis demostrates that simple allocatios are sufficiet to approach the zero-outage capacity. These simple strategies exploit the tedecy for radom MIMO chaels to behave determiistically as the umber of iputs becomes large. We propose two simplified allocatio strategies: the frequecy-uiform-spectralefficiecy (FUSE) allocatio ad the fixed-rate (FIX) allocatio. The FUSE allocatio simplifies the power allocatio by forcig each OFDM toe to have the same spectral frequecy, so that the scope of the power allocatio reduces to spatial chaels of each toe. I the FIX allocatio, the achievable rate of each scalar chael is fixed irrespective of chael, ad the fixed rates are predetermied to match the fadig statistics. As the umber of ateas teds to ifiity, we aalytically show that the proposed allocatios approach the chael s zero-outage capacity. We also show that xiii

14 the covergece is fast so that the FIX ad FUSE allocatios closely approach the chael capacity for a fiite umber of ateas. Experimetal results are provided to support the theoretical aalysis. We also cosider the bit-allocatio problem for the case where graularity cosideratios require that the rate be draw from a discrete ad fiite set. The best allocatio is based o the exhaustive search over all possible cadidates for the bit allocatio satisfyig the graularity costrait. However, a exhaustive search is uecessary. I fact, we demostrate that the search ca be restricted to a small set cotaiig oly a few well-chose cadidates, without sigificatly affectig the optimality of the search. I particular, o a flat-fadig chael, a biary search (oly two cadidates) ad a fixed allocatio (oly oe cadidate) perform very close to the optimal allocatio, as show by simulatio. The biary-search ad fixed-allocatio strategies exted to a MIMO-OFDM system by applyig them o a toe-by-toe basis. We provide the bit-error rate results o a MIMO-OFDM system with the proposed bit-allocatio strategies to show that the performace promised by theoretical aalysis is actually achieved. xiv

15 CHAPTER 1 INTRODUCTION 1.1 MIMO Wireless Commuicatios Recetly, there has bee a dramatic ad rapid growth i wireless commuicatios from cellular phoe service to high-defiitio televisio broadcastig. More ad more iformatio is set through wireless chaels, ad the demad for data rate is gettig higher ad higher. Wireless chaels are ope to everybody, but this opeess puts a strict limit o the badwidth ad trasmit power. To support high data rate for limited badwidth, we desire higher spectral efficiecy. I a multiple-iput multiple-output (MIMO) chael, created by employig multiple ateas at the trasmitter ad receiver, the spectral efficiecy dramatically icreases. Pioeerig work by Foschii [24] ad Telatar [60] showed the capacity of M T - iput M R -output MIMO ca be mi(m T, M R ) times larger tha the sigle-atea capacity. I other words, the multiple trasmit ateas are used to multiplex data i space, where the gai by spatial multiplexig ca be as large as mi(m T, M R ). A simplest form of spatial multiplexig is kow as V-BLAST (vertical Bell Labs layered space-time) trasmissio [27], where idepedet layers of data are trasmitted form each atea. For reliable commuicatios, high spectral efficiecy must be accompaied with low error rate. A major obstacle to reliability o wireless chaels is fadig, which refers to deep atteuatio of chael amplitude due to the mobility of users ad surroudig obstacles [53]. Traditioally, to mitigate the effects of fadig, the receiver uses multiple ateas, a techology kow as atea diversity. I a MIMO system, we ca obtai diversity from the M T M R liks betwee trasmitter ad receiver. If 1

16 the liks are statistically idepedet, the diversity gai ca be as large as M T M R compared to a sigle-iput sigle-output (SISO) chael. However, achievig the diversity is ot quite straightforward with multiple trasmit ateas. Alamouti itroduced a clever way to achieve the maximum (M T M R ) diversity gai for two trasmit ateas [1]. Sice the, may space-time codes have bee proposed, which effectively provide the spatial diversity o a MIMO chael [56, 57]. Aother source for higher spectral efficiecy ad stroger reliability is frequecy selectivity. I widebad trasmissio, the chael respose is frequecy selective, ad multiple copies of a trasmitted symbol arrive at the receiver over several sigalig itervals. Aalogous to multiple ateas, we ca harvest the diversity from frequecy selectivity [8]. Hece, the diversity gai is huge i a frequecy-selective MIMO chael. Ufortuately, all the advatages of MIMO are ot free. A dearest pealty would be the iterferece betwee sigals simultaeously emitted from the multiple trasmit ateas [6]. This iterferece cosiderably icreases the detectio complexity. For example, a maximum-likelihood (ML) detector suffers a sigificat icrease i complexity, which expoetially grows with the umber of trasmit ateas [59]. To avoid the expoetial growth i complexity, simpler solutios have bee explored, such as spatial equalizers or space-time codes specially desiged to simplify the detectio process. Resolvig iterferece i MIMO is aalogous to a traditioal problem of equalizig the effects of iter-symbol iterferece (ISI) o a frequecy-selective chael. We ca hece exted well-kow solutios for the equalizer to MIMO [6]. Particularly, a decisio-feedback equalizer is widely used i MIMO detectio. For example, BLAST uses zero-forcig decisio-feedback detector, a special case of successive iterferece cacellatio [64]. However, spatial equalizer ca cost a decrease i diversity. If the zero-forcig decisio-feedback detectio is used without ay help from a outer 2

17 chael code, the diversity gai is oly at most M R. Aother example is orthogoal space-time block codes (STBC) [57], a geeralizatio of Alamouti s code to ay umber of trasmit ateas. Orthogoal codes reduce the detectio complexity sice a symbol ca be detected without iterferece from others owig to the orthogoality. To maitai orthogoality, however, orthogoal STBC sacrifices spatial multiplexig gai, which is as large as uity (whe M T = 2) or less (whe M T > 2) rather tha the full multiplexig gai mi(m T, M R ). Geerally speakig, there is a tradeoff amog spatial multiplexig (icreasig spectral efficiecy), diversity (mitigatig fadig), ad complexity (mitigatig iterferece). For example, if a simple decisio-feedback detector is used, the diversity gai is o greater tha M R, smaller tha maximum gai M T M R, while it achieves full spatial multiplexig gai. Orthogoal STBC sacrifices multiplexig gai for simple detectio ad full diversity gai. I both cases, either multiplexig gai or diversity gai is abadoed to reduce the detectio complexity. O the cotrary, if a ML detector is employed, the complexity is highest, but we ca achieve full gais for diversity ad spatial multiplexig [63]. 1.2 Closed-Loop MIMO We cosider a closed-loop MIMO system where the trasmitter has chael state iformatio (CSI). By exploitig CSI at the trasmitter, eigebeamformig coverts a MIMO chael ito a bak of scalar chaels, with o crosstalk from oe scalar chael to ext [7, 12, 38]. Thus, the complexity for detectio oly liearly icreases with the umber of ateas. Eigebeamformig is a optimal space-time processig i the sese that it achieves the capacity of a MIMO chael, attaiig the full multiplexig gai. Furthermore, the diversity is also fully achieved, as will be discussed i depth later. Surprisigly, the trasmitter CSI magically makes the above-metioed tradeoff ieffective ad three goals of MIMO commuicatios are attaied at the same 3

18 time. Clearly, a ew problem is the availability of CSI at the trasmitter. A ituitive way to furish the trasmitter with CSI is to sed the estimated CSI to the trasmitter. However, the feedback is redudat iformatio, ad the feedback delay might cause a mismatch problem. O a slowly varyig chael, the delay is ot a big issue, but the additioal trasmissio load could be a problem. Thus, we have a ew tradeoff betwee the availability of CSI feedback ad the optimality of MIMO. I this work, we do ot directly ivestigate the tradeoff problem, but try to aswer i part by showig that the advatage of kowig CSI at the trasmitter is sigificat to allow the redudacy by the feedback. We base our research o iformatio theory for fadig chaels [8]. O fadig chaels, the mutual iformatio betwee trasmitter ad receiver is a radom variable, ad the capacity of a fadig chael is either the expected value of mutual iformatio (average capacity) or the rate that achieves a target outage probability (outage capacity), depedig o the chael geeratio process [60]. CSI at the trasmitter does ot dramatically impact the average capacity, as reported for a sigle-atea system [28]. The same is true for a MIMO chael, where there is a distict advatage of kowig CSI at low sigal-to-oise ratio (SNR), but the advatage almost disappears at high SNR [18, 24, 30, 60]. Meawhile, the trasmitter CSI helps improve the outage performace sigificatly, ot oly for the sigle-atea case [8, 13] but also for MIMO [7]. We use the zero-outage capacity, also kow as the delay-limited capacity [8, 32], to measure the outage performace, which is the maximum achievable rate while maitaiig zero outage probability. If CSI is ukow to the trasmitter, the outage probability caot be made zero ad thus the zero-outage capacity is zero [8]. I stark cotrast, if the trasmitter kows CSI, it is possible to achieve a positive zero-outage capacity by cotrollig power at the trasmitter to avoid ay outage [13]. For some fadig 4

19 statistics, the zero-outage capacity is zero eve whe CSI is kow at the trasmitter, such as o Rayleigh-flat-fadig SISO chael [28]. However, whe there is diversity, such as from frequecy selectivity or from multiple receive ateas, we ca achieve a positive zero-outage capacity [2, 8]. 1.3 Objective ad Cotributios We use the zero-outage capacity as a performace criterio to develop efficiet ad low-complexity trasmitter strategies for a closed-loop frequecy-selective MIMO system with orthogoal-frequecy-divisio multiplexig (OFDM) [47]. Cotributios ad orgaizatio of this thesis are as follows. We begi by describig a MIMO-OFDM system i Chapter 2. OFDM trasforms a MIMO chael with memory ito a set of memoryless chaels {H 1, H 2,..., H N } over N toes for both the cases of spatially correlated ad ucorrelated fadig. The eigevalues of H H are crucial i the iformatio-theoretical aalysis. We summarize the properties of eigevalues. I Chapter 3, we cosider a closed-loop MIMO system. A combiatio of OFDM ad eigebeamformig creates a bak of M N scalar chaels. Each toe has M spatial chaels with the eigevalues of H H as squared chael gais, where M is the rak of each memoryless MIMO chael. We also address efficiet implemetatio of eigebeamformig i a time-divisio duplex (TDD) system, ad develop adaptive algorithms for updatig receive filters. Chapter 4 reviews previous results o average ad outage capacity. We have two goals i this chapter: (1) to cofirm the icreased capacity ad improved outage performace o a MIMO chael; (2) to show the advatage of kowig CSI at the trasmitter. We explicitly show that CSI at the trasmitter does ot dramatically icrease the average capacity. I terms of the outage capacity, however, we ca achieve a substatial gai from the trasmitter CSI. At the ed of Chapter 4, we also 5

20 propose the procedures for optimizig outage regio i a closed-loop MIMO system. Chapter 5 summarizes our cotributios to the high-snr aalysis o average ad outage capacity. More specifically, we prove that a closed-loop MIMO system achieves full diversity ad multiplexig orders. We also rigorously derive the asymptotes of capacity i terms of the geometric mea of the eigevalues of H H. The geometricmea represetatio eables simple ad isightful aalysis o average ad outage capacity. Most importatly, we aalyze the zero-outage capacity with respect to chael memory L, showig that it is a odecreasig fuctio of L ad quatifyig the icrease i terms of M T, M R, ad L. We cosider the power-allocatio problem i Chapter 6. To achieve the zerooutage capacity, the trasmitter must fid the power allocatio, distributig power to scalar chaels. The optimal power allocatio is based o water-fillig over M N scalar chaels, but requires high complexity. I this work, we propose two simpler allocatio strategies: the frequecy-uiform-spectral-efficiecy (FUSE) allocatio ad the fixed-rate (FIX) allocatio. The FUSE allocatio reduces the complexity by restrictig the scope of water-fillig over M spatial chaels for each toe, such that each toe achieves the same spectral efficiecy. I the FIX allocatio, we furthermore abado water-fillig by fixig the achievable rate for each scalar chael. Thus, power allocatio ca be calculated by a simple closed-form formula. The proposed allocatio strategies sigificatly reduce the allocatio complexity, but ievitably icur capacity pealties. However, we prove that the pealties of both allocatios coverge to zero as the umber of atea array size teds to ifiity. We also quatify the pealties for a fiite atea array by high-snr aalysis ad show that the pealties quickly coverge to zero. The aalysis i Chapter 6 provides useful isight to solvig the bit-allocatio problem with a graularity costrait o supportable rates. The best bit allocatio is based o a full search over all possible cadidates for the allocatio, which is usually 6

21 itractable for the complexity reaso. I Chapter 7, we propose simple bit-allocatio strategies for flat-fadig MIMO chaels, Biary Search ad Fixed Allocatio, by exploitig the properties of MIMO chaels. Compared to the full-search allocatio, the proposed strategies exhibit early optimal performace, while the complexity reductio is remarkable. We also exted the proposed strategies to MIMO-OFDM by equally treatig each toe similar to the FUSE allocatio i Chapter 6. Fially, i Chapter 8, we evaluate the bit-error rate (BER) performace with the proposed bit-allocatio strategies. We cofirm that theoretical results i Chapter 6 ad Chapter 7 also hold with quadrature amplitude modulatio (QAM) costellatios ad practical outer chael codes. From the BER results, we will see that CSI at the trasmitter ideed improves the performace critically. To summarize, the mai cotributios of this work are as follows. We derive high-snr asymptotes of capacity i terms of the geometric mea of the eigevalues of the MIMO chael, ad aalyze the zero-outage capacity by usig the properties of the geometric mea. (Chapter 5) We propose simple power-allocatio strategies: the FUSE ad Fixed allocatios. We prove that the proposed allocatios are optimal i terms of the zero-outage capacity as the umber of ateas teds to ifiity. We also show that the proposed allocatios perform well at a moderate umber of ateas by high- SNR aalysis. (Chapter 6) We propose practical bit-allocatio strategies, Biary Search ad Fixed Allocatio, for MIMO flat-fadig chaels, which have remarkably low complexity. We show that both strategies are early optimal as the umber of ateas grows. Extesio to MIMO-OFDM is also cosidered. (Chapter 7) 7

22 1.4 Notatio Summary We summarize acroyms ad mathematical otatios. Notatio BER CDF CSI DMT FUSE i.i.d. ISI MIMO MISO MRC OFDM PDF RCSI SIMO SISO SNR STBC SVD TDD TRCSI A A F [A] p,q Descriptio Bit-error rate Cumulative distributio fuctio Chael state iformatio Discrete multitoe Frequecy uiform spectral efficiecy Idepedet ad idetically distributed Iter-symbol iterferece Multiple iput multiple output Multiple iput sigle output Maximum-ratio combiig Orthogoal-frequecy-divisio multiplexig Probability desity fuctio Receiver chael state iformatio Sigle iput multiple output Sigle iput sigle output Sigal-to-oise ratio Space-time block code Sigular-value decompositio Time-divisio duplex Trasmitter ad receiver chael state iformatio Cardiality of a set A Frobeius orm of a matrix A Elemet of a matrix A at the pth row ad qth colum 8

23 1{ } Idicator fuctio A L CN (µ, σ 2 ) det(a) E[X] E (x) I M G L L M Arithmetic mea over idex set L Complex Gaussia radom variable with i.i.d. N (µ, σ 2 /2) etries Determiat of a square matrix A Expectatio of a radom variable X Geeralized expoetial itegral M M idetity matrix Geometric mea over idex set L Number of chael memory Rak of MIMO chael matrix M Number of available chaels i FIX Allocatio (Chapter 6) M T M R N s Number of trasmit ateas Number of receive ateas Spatial idex set at ay toe i MIMO-OFDM N u Uiverse idex set for all scalar chaels of MIMO-OFDM {s (m) } A set of squared sigular values of chael matrix tr(a) Sum of diagoal elemets of a square matrix A γ Euler costat γ Γ Γ(x) Γ(x, y) ρ Ψ L (x) SNR gap for bit allocatio Gamma fuctio Icomplete complemetary Gamma fuctio SNR per receive atea Empirical distributio over idex set L 9

24 CHAPTER 2 MIMO CHANNELS AND PROPERTIES This chapter describes a chael model for MIMO wireless chaels. Amog may factors that characterize wireless chaels, we maily cosider two factors: fadig ad frequecy selectivity. Fadig refers to the severe atteuatio i the chael amplitude caused by the combiatio of multipath propagatio ad receiver movemet [53]. Ulike the time-ivariat chael for wired commuicatios, the chael is timevaryig ad its amplitude is ofte too small to deliver iformatio reliably. Frequecyselectivity is a typical pheomeo for widebad trasmissio. If the chael is frequecy selective, the received sigals are impaired by iter-symbol iterferece (ISI) [6]. Orthogoal-frequecy-divisio multiplexig (OFDM) is a efficiet solutio to removig the distortio by ISI [9]. We first describe a MIMO chael model with memory, which reflects fadig ad frequecy selectivity. The, we show how OFDM removes ISI ad coverts a frequecy-selective chael ito a bak of flat-fadig (memoryless) MIMO chaels. Fially, we preset importat properties of MIMO fadig chaels. 2.1 MIMO Wireless Chaels We cosider a discrete-time basebad model. Suppose that there are M T trasmit ateas ad M R receive ateas, which create a M R M T multiple-iput multipleoutput (MIMO) chael. Let x k = [x (1) k,..., x(m T ) k ] be the trasmit sigal vector at the kth sigalig iterval. If the chael has memory L, the received sigal vector, 10

25 y k, is a liear combiatio of {x k, x k 1,... x k L } [46], such that y k = L G l x k l + k. (1) l=0 The M R M T matrix G l represets the MIMO chael at the lth delay. We assume white Gaussia oise, such that the elemets of k i (1) are circularly symmetric complex Gaussia radom variable with zero mea ad E[ k k ] = N 0δ k k I MR, where ( ) deotes the Hermitia traspose, I M is a M M idetity matrix, ad the Kroecker delta fuctio δ k is uity whe k = 0 ad zero otherwise. The frequecy respose of (1) is G(e jθ ) = L G l e jlθ π < θ π, (2) l=0 where j = 1. The respose i (2) is frequecy selective whe L > 0. Whe L = 0, G(e jθ ) is flat i the frequecy domai (flat fadig), ad the chael i (1) reduces to a memoryless chael. We assume Rayleigh fadig, uless specified otherwise, throughout the thesis, so that each elemet of G l is a complex Gaussia radom variable [53]. Elemets of each G l ca be spatially ucorrelated or correlated, but we assume for aalytical simplicity that there is o correlatio betwee chael taps, that is, elemets of G l ad elemets of G l are ucorrelated if l l Spatially Ucorrelated Chael If the chael is spatially ucorrelated, the chael matrix is give by G l = σ l W l, where {σ 2 l } deote the power profile [53]. The M R M T matrix W l deotes a spatially ucorrelated matrix, whose elemets are i.i.d. CN (0, 1), where CN (µ, σ 2 ) deotes a circularly symmetric complex Gaussia radom variable, whose real ad imagiary parts are i.i.d. with mea µ ad variace σ 2 /2 for each. Without loss of geerality, we ormalize the chael, such that L l=0 σ2 l = 1. 11

26 Defiitio 2.1. The power profile {σ 2 l } is said to be uiform whe σ2 0 = σ 2 1 =... = σ 2 L = 1/(L + 1). The spatially ucorrelated model is useful for mathematical aalysis. Particularly, the radom matrix theory for i.i.d. Gaussia elemets provides powerful tools for aalysis [23] Spatial Correlated Chael For a spatially correlated chael, we itroduce correlatio matrices R Tl (M T M T ) ad R Rl (M R M R ), which represet correlatio at the trasmitter ad receiver, respectively, for the lth delay, such that E [ [H l ] p,j [H l ] q,k] = [RTl ] j,k [R Rl ] p,q [18], where [A] p,q deotes the elemet of a matrix A at the pth row ad the qth colum. We assume that the correlatio betwee the fadig from trasmit ateas j ad k to a particular atea is [R Tl ] j,k ad does ot deped o the receive atea. The same is true for [R Rl ] p,q. The, G l ca be factored i the form: G l = (R Rl ) 1 2 Wl (R Tl ) 1 2, (3) where ( ) 1 2 deotes the matrix square root, ad W l is a M R M T spatially ucorrelated matrix with i.i.d. CN (0, 1) etries. We ormalize the chael, such that L l=0 tr{r T l } = M T ad L l=0 tr{r R l } = M R, where tr{ } deotes the trace of diagoal elemets of a square matrix [33]. If the chael is ucorrelated, the correlatio matrices reduce to R Tl = I MT ad R Rl = I MR The degree of correlatio is measured by σ 2 θ l, the variace of the agle spread for the lth path with its departure or arrival agle θ l [10]. For the lth path, θ Tl ad θ Rl deote average departure ad arrival agles, respectively, ad a is the array respose vector, defied as a(θ) = [1, e j2π cos(θ),..., e j2π(k 1) cos(θ) ] T, (4) 12

27 where is the atea spacig relative to wavelegth ad K is the umber of ateas (K = M T for the trasmitter ad K = M R for the receiver). If there is o agle spread (σθ 2 l = 0) [10], the correlatio matrices the become R Tl = σl 2a( θ Tl )a T ( θ Tl ) ad R Rl = σl 2a( θ Rl )a T ( θ Rl ), such that, each tap H l ca be writte as [47] H l = σ 2 l α l a( θ Rl )a T ( θ Tl ), (5) where α l is i.i.d. CN (0, 1), represetig Rayleigh fadig, ad where {σl 2 } is the power profile. For ozero agle spread, we cosider correlatio oly at the receiver, that is, R Tl = I MT [10]. The receive correlatio matrix ca be approximated as [3] [R Rl ] p,q σ 2 l e j2π p q cos(θ R l ) e 0.5(2π p q si(θ R l )σ θl ) 2. (6) I fact, the approximatio i (6) is accurate oly for small agle spread, but it provides the correct tred for large spread [10]. Note that each R Rl collapses to a rak-1 matrix, R Rl = σ 2 l a( θ Rl )a T ( θ Rl ), whe σ θl = MIMO-OFDM If the chael is frequecy selective, the received sigals are distorted by ISI, which makes detectio difficult [6]. OFDM has emerged as oe of most efficiet ways to remove such ISI [9]. I this sectio, we briefly review how a MIMO chael with memory i (1) becomes a set of memoryless (o ISI) MIMO chaels by OFDM. Figure 1 illustrates the block diagram for MIMO-OFDM. Suppose that the trasmitter collects N symbol vectors: {u 1,..., u N }, where u = [u (1),..., u (M T ) ] T. We N regroup N sigals {u (m) 1,..., u (m) } for m = 1, 2,..., M T by the NM T NM T permutatio matrix ˆP T i Figure 1. Each group of N sigals is fed ito the iverse discrete Fourier trasform (IDFT) block, which produces {x (m) 1,..., x (m) }. The, we add a N 13

28 cyclic prefix of legth L, such that {x (m) N L+1,..., x(m) N, x (m) 1,..., x (m) N }{{} } (7) cyclic prefix is the set of sigals trasmitted from the mth atea after digital-to-aalog (D/A) coversio ad upcoversio. At the receiver, we remove the cyclic prefix from the received sigals after dowcoversio ad aalog-to-digital coversio (A/D), producig a group of N sigals: {y (m) N 1,..., y (m) } for m = 1, 2,..., M R. Discrete Fourier trasform (DFT) block trasforms each group ito {v (m) 1,..., v (m) }. After the permutatio by ˆP R, the received N sigals are recollected such that v = [v (1),..., v (M R) ] T for = 1, 2,..., N. as Equivaletly, as show i Figure 1, the received sigals after DFT ca be writte v 1 = H 1 u 1 + ñ 1..., (8) v N = H N u N + ñ N where {ñ } are additive oise. Sice DFT/IDFT ad permutatios are uitary processes, {ñ } are statistically idetical to { k } i (1). The M R M T matrix H represets the memoryless chael at the th toe, where H = G(e j2π/n ) = L G l e j2πl/n. (9) l=0 Note that H is idetical for all whe the chael is memoryless (L = 0). For the rest of the thesis, we will cosider the MIMO-OFDM model of (8) istead of the uderlyig chael of (1). Lemma 2.1. Give our assumptio that the chael taps are ucorrelated, all {H } are statistically idetical to each other. If fadig is spatially ucorrelated, each etry of H is i.i.d. CN (0, 1), that is, all {H } is statistically idetical to spatially ucorrelated Rayleigh flat fadig. 14

29 (1) u 1 u 1 v { (M u T ) 1 (1) u N u N v { (M u T ) N ^ P T (1) u 1 (1) u N (M u T ) 1 (M u T) N IDFT IDFT PREFIX & P/S PREFIX & P/S (1) x k (M x T ) k D/A D/A NOISE UP DOWN y k = l G l x k l + k UP DOWN NOISE A/D A/D (1) y k (M y R ) k S/P & DIS- CARD S/P & DIS- CARD DFT DFT (1) v 1 (1) v N (M v R ) 1 (M v R) N ^ P R (1) v 1 (M v R) 1 (1) v N (M v R ) N { { 1 N AWGN u 1 H 1 v 1 AWGN u N H N v N Figure 1: Block diagram for MIMO-OFDM ad its equivalet chael model. Proof. From (3), the chael matrix at each tap is G l = (R Rl ) 1 2 W l, where R 1/2 R l = [r l,1,, r l,mr ] T is the receiver correlatio matrix ad W l = [w l,1,, w l,mt ] is a M R M T matrix with i.i.d. CN (0, 1) elemets, with r l,i ad q l,i are M R 1 vectors. The, the elemet of H at the p-th row ad q-th colum is [H ] p,q = L r T l,pw l,q e j2πl/n, (10) l=0 where superscript T deotes the traspose of a vector. Sice [H ] p,q is a liear combiatio of zero-mea uit-variace complex Gaussia radom variables i w l,p, [H ] p,q is also a complex Gaussia radom variable. Clearly, its mea ad variace E [[H ] p,q ] = 0 ad E [ [H ] p,q ] 2 = L l=0 r l,i 2 are idepedet of, where is the Euclidea orm [33]. Sice a Gaussia radom variable is fully described by its mea ad variace, it is sufficiet to say that H has idetical statistics for all. For spatially ucorrelated fadig, the correlatio matrix is R Rl = σ 2 l I M R. Thus, each elemet of H = L l=0 σ2 l W le j2πl/n is a Gaussia radom variable with zero mea ad variace E [ [H ] p,q 2] = L l=0 σ2 l = 1. Also H iherits the spatially ucorrelated property from {W l }. Sice a Gaussia radom variable is fully described 15

30 by its mea ad variace, each H is statistically idetical to a flat-fadig MIMO chael (L = 0). Corollary 2.1. Let M be the rak of H, such that M mi(m T, M R ). The, M is costat for all with probability oe. If the chael is spatially ucorrelated, M = mi(m T, M R ) with probability oe. Proof. The proof is straightforward from Lemma Fuctios for Eigevalues Eigevalues of H H are crucial i the iformatio-theoretical aalysis. As H is a radom matrix, the eigevalues are also radom variables. This sectio itroduces some useful fuctios of eigevalues for the use i aalysis. Eigevalues of H H are closely related to sigular-value decompositio (SVD) of H. Theorem 2.1 (SVD). For ay complex M R M T matrix A with rak M, there exists a SVD of the form: A = UDV, (11) where U (M R M R ) ad V (M T M T ) are uitary. The M R M T matrix D = [d i,j ] has d i,j = 0 for all i j, ad d 1,1 d 2,2... d M,M > d M+1,M+1 =... = d a,a = 0, where a = mi(m T, M R ). The, the sigular values of A, {d i,i ; i = 1, 2,..., M}, are the ozero eigevalues of AA, ad hece are uiquely determied. Proof. See [33]. The uitary matrices U ad V are ot uique. If M R M T ad if AA has distict eigevalues, the V is determied up to a right diagoal factor T = diag(e jθ 1,..., e jθ ) with real-valued θ i ; that is, if A = U 1 DV 1 = U 2 DV 2, the V 2 = V 1 T. If M R < M T, the V is ever uiquely determied; if M R = M T = M 16

31 ad V is give, the U is uiquely determied. If M T M R, the uiqueess of U ad V is determied by cosiderig A i a similar way of the case where M R M T. Corollary 2.2. Let {d m,m } be M ozero sigular values of H. The, {s (m) are M ozero eigevalues of either H H or H H. = d 2 m,m} Proof. See [33] Arithmetic ad Geometric Meas of Eigevalues We begi with well-kow arithmetic ad geometric meas of {s (m) }. Defiitio 2.2. For a give idex set L for (m, ), we defie the arithmetic ad geometric meas of {s (m) ; (m, ) L} as: A L = 1 L (m,) L s (m), (12) ad respectively. G L = s (m) (m,) L 1/ L, (13) A well-kow iequality is that A L G L with equality if ad oly if s (m) is idepedet of m ad for ay idex set L of m ad. Both A L ad G L are radom variables sice they are just sum or product of {s (m) }. We ca relate A L with the chael taps {G l } of (1), as follows. Lemma 2.2. If N u is the uiverse idex set: N u = {(m, ); m = 1,..., M, ad = 1,..., N}, (14) which ecompasses all m ad, the we have A Nu = 1 M L G l 2 F = l=0 1 M(L + 1) L W l 2 F, (15) l=0 17

32 where W l is a M R M T matrix with i.i.d. CN (0, 1) elemets, ad where A F deotes the Frobeius orm of A defied as Proof. From the defiitio of arithmetic mea, A Nu = 1 N M s (m) = 1 MN MN A F = tr{aa }. (16) =1 m=1 Sice H 2 F = M R MT p=1 q=1 [H ] p,q 2, we have N H 2 F = =1 = = N M R M T [H ] p,q 2 =1 p=1 q=1 M R M T p=1 q=1 =1 M R M T p=1 q=1 =1 + l l 2Re N H 2 F. (17) =1 N L 2 [G l ] p,q e j2πl/n l=0 N { L [G l ] p,q 2 l=0 [ [G l ] p,q [G l ] p,qe j2π(l l )/N ] }. (18) Sice N =1 e j2π(l l )/N = 0, (18) reduces to N H 2 F = =1 L M R M T l=0 p=1 q=1 =1 Substitutig (19) ito (17), we obtai (15). N [G l ] p,q 2 = N L G l 2 F. (19) l=0 Corollary 2.3. If the chael is spatially ucorrelated ad the power profile is uiform, M(L + 1)A Nu is a chi-square radom variable with a degree of freedom M T M R (L + 1) [39]. The probability desity fuctio (PDF) is f M(L+1)ANu (x) = xm T M R (L+1) 1 e x Γ(M T M R (L + 1)) x 0, (20) where is the Gamma fuctio [29]. 1 Γ(x) = 0 t x 1 e t dt, (21) 1 Whe x is a iteger, Γ(x) = (x 1)!, where! deotes the factorial. 18

33 Proof. With the uiform power profile, A Nu = 1 M(L+1) L l=0 W l 2 F, where W l has i.i.d. CN (0, 1) etries. From the defiitio of the Frobeius orm, A Nu is a chi-square radom variable with a degree of freedom M T M R (L + 1) Empirical Distributio Aother importat fuctio for {s (m) } is the empirical distributio fuctio [30]. Defiitio 2.3. Let θ (m) a idex set L is give by = s (m) /M. The, the empirical distributio of {θ (m) } over Ψ L (x) = 1 L (m,) L 1{θ (m) x}, (22) where 1{ } is a idicator fuctio, such that 1{A} is uity if the coditio A is satisfied or zero otherwise, ad where L deotes the cardiality of L. Empirical distributio is a radom variable whe M is fiite, but it is kow to coverge to a o-radom limit as M. For a spatially ucorrelated chael with L = 0 memory, we ca explicitly evaluate the limit of Ψ Nu (x), where N u is the uiverse idex set i (14), as follows. Theorem 2.2. We cosider a spatially ucorrelated chael with L = 0 memory. Suppose that M teds to ifiity such that mi(m T,M R ) max(m T,M R ) θ (1) a (1 + β) 2 β 1. The, θ (M) b (1 β) 2, (23) with probability oe, for ay. I fact, the etire empirical distributio of a radomly selected eigevalue coverges. Also, the empirical distributio coverges to a oradom limit, such that dψ Nu (x) = 1 2πβx (x a)(b x) a x b 0 otherwise (24) 19

34 Proof. For a tutorial, see [30]. The proof for this theorem ca be foud i [5] for the smallest eigevalue ad [4] for the largest eigevalue, respectively. For the last statemet, see for example [35]. From Lemma 2.1, we deduce that Ψ Nu (x) also coverges to a o-radom limit whe L > Joit Distributio of Eigevalues Whe the chael is spatially ucorrelated, Lemma 2.1 states that each H is statistically idetical to W, a radom matrix with i.i.d. CN (0, 1) i its etries. I mathematics, WW is called a Wishart matrix [23, 30]. The eigevalues of a Wishart matrix has bee extesively studied. Particularly, the joit PDF of {s (m) } is kow [23, 30]. For otatioal simplicity, we discard the toe idex i this sectio, sice {s (m) ; m = 1, 2,..., M} are statistically idetical for ay. Theorem 2.3 (Ordered Eigevalue Distributio). The joit distributio of s = [s (1),..., s (M) ] is f s (x 1, x 2,..., x M ) = ( π M(M 1) M ) Γ MT (M R )Γ MR (M T ) exp M x u u=1 u=1 x D u (x u x v ) 2, (25) u<v where D = max(m T, M R ) mi(m T, M R ), ad where Γ m (x) = π m(m 1)/x m i=1 Γ(x i + 1). (26) Proof. See [30]. Corollary 2.4. Let A ad B be a b ad b a matrices, respectively, with CN (0, 1) i their etries. The eigevalues of AA ad BB have idetical joit PDF. Proof. The proof is straightforward from the symmetry of (25) with respect to M T ad M R. If we radomize the order of {s (m) }, we have a simpler distributio fuctio. 20

35 Theorem 2.4 (Uordered Eigevalue Distributio). Let s be a radomly selected eigevalue from {s (m) }. The, its PDF is give by f s (x) = 1 M M 1 k=0 k!x D e x { L D (k + D)! k (x) }, (27) where L m k (x) is the Laguerre polyomial of order k [29], defied as L m k (x) = ex x m k! d k dx k { e x x k+m} = k i=0 ( 1) i ( k + m k i ) x i i!. (28) Proof. See [52]. Theoretically, it is possible to derive the margial distributio by evaluatig (M 1)-fold itegratio of (25). The required itegrals, however, quickly become itractable. To our kowledge, margial PDF is kow oly for small M T ad M R. Example 2.1. Whe M T = M R = 2 ad M T = M R = 3, margial distributio fuctios are: f s (1)(x 1 ) = (2 2x 1 + x 2 1)e x 1 f s (2)(x 2 ) = 2e 2x 2, (29) ad, f s (1)(x 1 ) = 1 4 (12 24x x 2 1 8x x 4 1)e x (12 12x 1 + 6x x x 4 1)e 2x 1 + 3e 3x 1 f s (2)(x 2 ) = 1 2 (12 12x 2 + 6x x x 4 2)e 2x 2 6e 3x 2 f s (3)(x 3 ) = 3e 3x 3, (30) respectively [65]. I commuicatios, the largest eigevalue s (1) is of particular iterest sice it carries the largest amout of iformatio. A geeral form for the margial CDF of s (1) is kow [30]. 21

36 Theorem 2.5. The margial CDF of the largest eigevalue (s 1 ) is give by Prob[s (1) < x] = Γ MR (M R ) Γ MR (M T + M R ) xm T M R 1 F 1 (M T ; M T + M R ; xi), (31) where 1 F 1 (; ; ) is the hypergeometric fuctio of matrix argumet [29]. Proof. See [30]. The hypergeometric fuctio of matrix argumet is extremely difficult to compute as it is represeted by slowly-covergig Zoal polyomials [30]. Istead of (31), we ca directly derive the margial distributio fuctios for some special cases. Example 2.2. Whe M T = 1 ad M R 1, amely a sigle-iput multiple-iput (SIMO) chael ad fadig is spatially ucorrelated, there is oly oe ozero eigevalue (M = 1), ad (31) reduces to f s (1)(x) = xm R 1 e x Γ(M R ). (32) Whe M T 1 ad M R = 1, amely a multiple-iput sigle-output (MISO) chael, (32) is valid with M T replacig M R. Example 2.3. Whe M T = M R = M = 4, the margial PDF of the largest eigevalue is where f s (1)(x) = M ϕ k (x)e kx, (33) k=1 ϕ 1 (x) = 4 12x + 18x x x4 1 2 x x6 ϕ 2 (x) = x 24x x3 1 2 ϕ 3 (x) = 12 12x + 6x x x x x6 ϕ 4 (x) = 4. (34) 22

37 O the other had, the margial distributio of the smallest eigevalue is kow for M T = M R. Example 2.4. Whe M T = M R = M, the margial PDF of the smallest eigevalue s (M) is [23] f s (M)(x) = Me Mx. (35) Note that the smallest eigevalue is expoetially distributed, which meas that E[1/s (M) ]. 23

38 CHAPTER 3 CLOSED-LOOP MIMO-OFDM This chapter cosiders a closed-loop MIMO-OFDM system, where the trasmitter has perfect CSI. We address the trasmit beamformig techique for a MIMO chael by exploitig CSI. If the trasmitter uses the optimal beamformig, kow as eigebeamformig, to maximize the achievable rate, a MIMO chael is trasformed ito a bak of scalar chaels with o crosstalk from oe scalar chael to ext. Next, we briefly metio the feasibility of havig CSI at the trasmitter ad propose a filter-reuse scheme for a time-divisio duplex (TDD) system. Fially, we survey adaptive algorithms for the receive filter of eigebeamformig. 3.1 Trasmit Beamformig We cosider the effective MIMO-OFDM chael i Figure 1: v 1 = H 1 u 1 + ñ 1... (36) v N = H N u N + ñ N For each H, we set a trasmit filter ad a receive filter at the receiver, as illustrated i Figure 2. Joit optimizatio of the trasmit ad receiver filters for MIMO is already a familiar topic i sigal processig [45, 49, 51, 67]. I this thesis, we focus o maximizig the achievable rate Pricipal Eigemode First, we cosider the trasmit beamformig with traditioal weight vectors [26], where we set v as a trasmit filter ad u as a receive filter for each H i Figure 2. 24

39 AWGN u a trasmit v receive filter H filter z Figure 2: Block diagram for trasmit beamformig. The, the output of the receive filter is z = u H v a + u ñ = 1, 2,..., N, (37) where a is the data sigal ad ñ is the oise vector. The receive SNR is maximized by choosig v ad u as the pricipal (correspodig to the largest eigevalue) eigevectors to H H ad H H, respectively, uder the uit-orm costrait [66]. The, (37) becomes: z = s (1) a + w = 1, 2,..., N, (38) where w = u ñ ad where s (1) is the largest eigevalue of H H. For this reaso, we call the trasmit beamformig scheme i (37) the pricipal-eigemode trasmissio. We will later see that the pricipal eigemode achieves a full diversity o a spatially ucorrelated chael. However, sice there is oly oe scalar chael, the spatial multiplexig gai of MIMO disappears ad the pricipal eigemode ca suffer a sigificat loss i rate. Practically, the pricipal eigemode is cosidered to be competitive sice its implemetatio is quite simple. Especially, it is suitable to the outdoor eviromet, where the largest eigevalue is ofte domiatly larger tha others [25]. A extreme case is whe there is oly oe path (L = 0) i the model of (5). The, the rak drops to M = 1, meaig that the largest eigevalue is the oly ozero eigevalue. I such a case, the pricipal eigemode is optimal. 25