Semi-Blind Channel Estimation for LTE DownLink

Transcription

1 TECHNICAL UNIVERSITY OF DENMARK Semi-Blid Chael Estimatio for LTE DowLik by Nicolò Michelusi A thesis submitted i partial fulfillmet for the degree of Master of Sciece MSc i Telecommuicatio Egieerig Supervisors: Lars Christese Nokia Ole Wither DTU Jue 2009

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3 TECHNICAL UNIVERSITY OF DENMARK Abstract Telecommuicatio Egieerig Departmet of Iformatics ad Mathematical Modelig IMM Master of Sciece Nicolò Michelusi I a MIMO system the umber of chael parameters is much larger tha i a typical SISO sceario, makig the chael estimatio task particularly critical. I fact, this icrease i the umber of chael parameters traslates ito a smaller estimatio accuracy, which is couteracted by trasmittig a loger pilot sequece. This i tur egatively impacts the badwidth efficiecy of the system, makig pilot based approaches less attractive. I this thesis we ivestigate the Semi-Blid approach to chael estimatio i MIMO- OFDM systems, ad i particular for LTE dowlik. This techique, by exploitig the observatios associated to the ukow symbols other the the pilot sequece to perform the chael estimate, potetially leads to a improvemet i the estimatio accuracy compared to the typical pilot based estimatio approach, without requirig a log pilot sequece, despite the large umber of parameters typical of a MIMO sceario. Through simulatios performed o the LTE system we show that the proposed Semi- Blid approaches lead to sigificat improvemets i the estimatio accuracy, both from a MSE ad BER perspective, compared to the typical pilot based techique. However, exploitig the true discrete distributio of the ukow symbols is computatioally demadig, therefore we propose the use of two approximatios o the ukow symbols: the Gaussia ad the Costat Modulus assumptios. These, though sub-optimal from a poit of view of the estimatio accuracy, still lead to sigificat improvemets with respect to the pilot based approach, while reducig the computatioal overhead icurred whe usig true discrete distributio of the ukow symbols.

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5 Cotets Abstract iii List of Figures vii 1 Itroductio Chael Estimatio i MIMO systems MIMO-OFDM priciples ad system model MIMO model MIMO-OFDM model Model Assumptios Problem Formulatio Traiig sequece chael estimatio of MIMO-OFDM FIR chaels Maximum-Likelihood chael estimatio of a MIMO-OFDM FIR chael Chael Idetifiability Coditios Properties of ML chael estimator Bias of Maximum Likelihood chael estimator Variace of Maximum Likelihood chael estimator White Gaussia Noise at the receiver Semi-Blid chael estimatio Geeral formulatio of Semi-Blid ML estimatio of MIMO-OFDM FIR chaels Brief itroductio to the EM-Algorithm ML solutio through EM-algorithm Semi-Blid ML estimatio: true discrete distributio of the ukow symbols ML solutio through EM-algorithm Semi-Blid ML estimatio: Gaussia approximatio for the ukow symbols ML estimate through EM Algorithm Semi-Blid ML estimatio: Costat Modulus approximatio for the ukow symbols ML solutio through EM-algorithm Joit Semi-Blid Estimatio of chael ad oise covariace matrix Noise Model v

6 vi CONTENTS 4.2 Noise Covariace matrix Estimatio Joit Semi-Blid Estimatio of chael ad oise covariace matrix Pilot based approach Semi-Blid approach Simulatio Results ad Discussio LTE frame structure Simulatio setup Compariso of Semi-Blid ad pilot based approaches for differet atea setups T 1R MIMO T 2R MIMO T 1R MIMO T 2R MIMO, trasmissio rak S = T 2R MIMO, trasmissio rak S = Estimatio accuracy as a fuctio of the sub-carriers Estimatio accuracy as a fuctio of the costellatio order Covergece of the EM-Algorithm, Gaussia approximatio Joit Estimatio of Chael ad oise covariace matrix Coclusio 97 A Complex derivatives 101 B Computatio of the posterior mea of costat modulus symbols 103 C Cramér Rao lower boud 107 C.1 Ubiased Cramér Rao lower boud for Complex parameters C.2 Ubiased CRLB for pilot based estimator of MIMO-FIR chaels C.2.1 The Fisher Iformatio Matrix for the estimatio of h C.3 Ubiased CRLB for Semi-Blid estimatio of MIMO-OFDM FIR Chaels113 Bibliography 119

7 List of Figures 3.1 g N x for differet values of N Plot of fuctio gx ad its approximatio 1 e x Gaussia approximatio versus CM with uiform phase approximatio, stadard deviatio o the posterior expectatio; N = L = 1,R = T = LTE frame structure Pilot allocatio o oe resource block 12 sub-carriers times 7 OFDM symbols for the cases 1,2 ad 4 trasmittig ateas Compariso of pilot based ad Semi-Blid approaches MSE, 1T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches BER, 1T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches MSE, 1T 2R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches BER, 1T 2R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches MSE, 2T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches MSE, equivalet chael, 2T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches BER, 2T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches MSE, 2T 2R MIMO-OFDM, trasmissio rak 1, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches BER, 2T 2R MIMO-OFDM, trasmissio rak 1, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches MSE, 2T 2R MIMO-OFDM, trasmissio rak 2, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches BER, 2T 2R MIMO-OFDM, trasmissio rak 2, 4-QAM, 72 sub-carriers Compariso of pilot based ad Semi-Blid approaches for differet umber of sub-carriers MSE, 1T 2R MIMO-OFDM, 4-QAM Compariso of pilot based ad Semi-Blid approaches for differet umber of sub-carriers BER, 1T 2R MIMO-OFDM, 4-QAM Compariso of pilot based ad Semi-Blid approaches for differet costellatio orders MSE, 1T 2R MIMO-OFDM, 72 sub-carriers Evolutio of MSE ad BER over the iteratios of the EM-algorithm, 1T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers vii

8 viii LIST OF FIGURES 5.18 Evolutio of MSE ad BER over the iteratios of the EM-algorithm, 2T 2R MIMO-OFDM, trasmissio rak S = 2, 4-QAM, 72 sub-carriers Evolutio of MSE ad BER over the iteratios of the EM-algorithm, 1T 2R MIMO-OFDM, 4-QAM, 72 sub-carriers Joit Estimatio of chael ad oise covariace matrix, MSE of chael estimator, 2T 2R MIMO-OFDM, trasmissio rak 2, 4-QAM, 72 subcarriers Joit Estimatio of chael ad oise covariace matrix BER, 2T 1R MIMO-OFDM, 4-QAM, 72 sub-carriers

9 Chapter 1 Itroductio Durig the last few decades we have experieced a extraordiary growth of wireless commuicatios, which lead to the defiitio of ew mobile commuicatio stadards, with the aim of providig broadbad ubiquitous access to the Iteret. I this cotext, LTE Log Term Evolutio is a 3GPP project uder stadardizatio, promisig dowlik data rates of up to 300Mbps. This is accomplished by employig advaced techologies at the physical layer, such as Orthogoal Frequecy Divisio Multiplexig OFDM ad Multiple-Iput Multiple-Output MIMO to icrease the capacity of the wireless chael. Typically, the badwidth available to wireless commuicatio systems is limited by a series of factors, the most importat of which is the ature of the wireless chael. A defiig characteristic of the wireless chael is multipath fadig, which cosists i the variatio of the chael stregth over time ad frequecy, due to costructive ad destructive superpositio of multiple paths travelig from the trasmitter to the receiver through the wireless medium. The frequecy variatio of the chael is due to the fact that the sigal propagates through distict paths to the receiver, thus arrivig at distict times, causig a spreadig of the chael impulse respose over time, which is equivalet to frequecy selectivity i the frequecy domai. The time variatio of the chael is due to the fact that distict paths ecouter movig obstacles while propagatig through the wireless medium. Moreover, trasmitter ad receiver might be movig etities. These effects cause the chael impulse respose to vary over time. The time widow durig which the chael is assumed to be timeivariat is called time coherece, ad is approximately iversely proportioal to the speed of the receiver, typically of the order of magitude of a few ms. 1

10 2 Chapter 1 Itroductio Typically, wireless commuicatio is established betwee oe trasmittig ad oe receivig atea SISO, Sigle-Iput Sigle-Output systems. However, the capacity achievable i such systems is severely limited by fadig, sice the sigal is severely atteuated whe the chael is i a deep fade. I recet years, MIMO Multiple-Iput Multiple-Output has emerged as the ew atea techology. This has bee proposed as a techique to icrease the capacity ad the reliability of wireless chaels through the adoptio of multiple ateas at the trasmitter ad receiver sides. By adoptig multiple ateas at the receiver, multiple copies of the same sigal propagates through idepedet chaels. Globally, the probability that all the chaels are i a deep fade is reduced, thus improvig chael reliability. This techique is called Receive Diversity. A similar effect is achieved by adoptig multiple ateas at the trasmitter, a techique called Trasmit Diversity. By adoptig multiple ateas at both the trasmitter ad the receiver sides, multiple iformatio streams ca be multiplexed through the trasmittig atea array, a techique called Spatial Multiplexig. Compared to a SISO system, this techique allows a icrease i the capacity of the overall chael by a factor proportioal to the miimum betwee the umber of receivig ad the umber of trasmittig ateas actually to the chael rak. We suggest the iterested reader to read [1] for a thorough treatmet of MIMO systems ad the derivatio of this result. Although MIMO represet a solutio to icrease the capacity ad the reliability of wireless chaels, it is particularly challegig from a chael estimatio perspective. This is explaied i the followig sectio. 1.1 Chael Estimatio i MIMO systems Typically, chael estimatio is performed by isertig a sequece of symbols kow at the receiver termed pilot symbols i the trasmitted frame. At the receiver side the, by observig the output i correspodece of the pilot symbols, it is possible to estimate the chael. This approach is the most commoly used i commuicatio systems, for its low computatioal complexity ad robustess. Its drawback cosists i the fact that the pilot symbols do t carry useful iformatio, therefore they represet a badwidth waste. Moreover, most of the observatios those related to the ukow symbols are discarded i the estimatio process, thus represetig a missed opportuity to ehace the accuracy of the chael estimate. I a MIMO system, chael estimatio is eve more critical tha i a SISO system. I fact, a T R MIMO system where T ad R represet the umber of trasmittig

11 Chapter 1 Itroductio 3 ad receivig ateas respectively ca be represeted as a set of RT idepedet SISO chaels, oe betwee each trasmittig-receivig atea pair. It is clear that the umber of chael parameters to estimate i a MIMO system icreases with the product RT. Uder this coditio, the pilot based chael estimatio approach has a severe limitatio: as we will also demostrate i the course of the thesis, a larger umber of parameters require the trasmissio of a loger pilot sequece. However, the trasmissio of a loger pilot sequece is ot desirable i a commuicatio system, sice they do t carry useful iformatio ad represet a badwidth waste. I this cotext, it becomes importat to develop a ew estimatio approach capable of improvig the chael estimatio accuracy without the eed to trasmit a loger pilot sequece. I this thesis the solutio proposed is Semi-Blid chael estimatio, which cosists i exploitig also the ukow iformatio other tha the pilot sequece to estimate the chael. The potetial advatage, compared to the pilot based approach, cosists i the fact that all the iformatio available at the receiver is exploited i the estimatio process, therefore there is a potetial improvemet i the achievable estimatio accuracy. However, this comes at the cost of a icreased receiver complexity with respect to a pilot based approach, as we will demostrate i the course of the thesis. We start the treatmet by modelig i sectio 1.2 the MIMO-OFDM system, ad itroducig the model assumptios used throughout the thesis. I sectio 1.3 we briefly formalize the chael estimatio problem i MIMO-OFDM systems. The, i chapter 2 we derive a Maximum-Likelihood estimator of MIMO-OFDM chaels usig the typical pilot based approach. We derive i particular a relatio betwee the estimatio accuracy ad the order of the MIMO-OFDM system that is the umber of trasmittig-receivig ateas, highlightig the weakesses of this approach for MIMO systems. I chapter 3 we treat i detail Semi-Blid chael estimatio of MIMO-OFDM chaels, studyig i particular three cases, depedig o the assumptios used for the ukow symbols i the estimatio process: i the first case we exploit the true discrete distributio, i the secod we approximate the distributio of the ukow symbols with a circular Gaussia distributio, i the third case we assume the symbols have costat amplitude ad phase uiformly distributed i [0, 2π valid oly for Costat Modulus costellatios. As we will show i the simulatio results i chapter 5, these three assumptios represet a trade-off betwee estimatio accuracy ad complexity: while usig the true discrete distributio of the ukow symbols is optimal from the poit of view of the estimatio accuracy, from the perspective of the computatioal complexity it is far too demadig; therefore the use of approximatios represet a solutio to reduce the computatioal overhead, i spite of a reduced estimatio accuracy. I geeral, sice i the Semi-Blid approach the Maximum Likelihood solutio caot be determied

12 4 Chapter 1 Itroductio i closed form, the use of iterative algorithms to coverge to a local maximum of the likelihood fuctio is required. We propose the Expectatio-Maximizatio algorithm as a geeral framework to solve this maximizatio problem, where the ukow symbols are treated as hidde variables. So far, we have assumed that the statistical properties of the oise are kow at the receiver. However, this is ot the case i a real commuicatio system. Moreover, the wireless chael is a shared medium. Cosequetly, the receiver udergoes iterferece from other users, whose statistical properties have to be estimated at the receiver. I chapter 4 we derive a algorithm for the joit estimatio of the oise covariace matrix ad of the chael usig the Semi-Blid approach. Fially, i chapter 5 we preset some simulatio results performed o the LTE system, comparig the performace achievable with the Semi-Blid approaches ad the pilotbased approach described i the thesis. 1.2 MIMO-OFDM priciples ad system model MIMO model MIMO Multiple-Iput Multiple-Output is the use of multiple ateas at the trasmitter ad receiver sides, with the purpose of combatig fadig ad icreasig the capacity of wireless commuicatio systems. Let T ad R be the umber of trasmittig ad receivig ateas, respectively. This MIMO system is labeled as T R MIMO, ad ca be represeted as a set of RT SISO chaels, oe betwee each trasmittig-receivig atea pair. Now, let s cosider the sigal at the receiver. Usig, ow ad i the rest of the thesis, the equivalet discrete basebad model, ad assumig that, durig the time spa we observe the evolutio of the model, the chael is time-ivariat block-fadig chael, each SISO chael, modeled as a Fiite Impulse Respose FIR filter of legth L, is described by meas of L complex taps. Therefore, the sigal received at atea r is give by the superpositio of the sigals trasmitted by each atea t = 0... T 1, filtered through the SISO chael betwee atea pairs r, t, plus the oise. This ca be writte as T 1 L 1 y r k = h l r, tx t k l + η r k 1.1 t=0 l=0

13 Chapter 1 Itroductio 5 where y r k is the sigal received o atea r at time k, h l r, t is the lth tap of the FIR SISO chael betwee atea pairs r, t, x t k is the sigal trasmitted through atea t at time k ad η r k is the oise o receivig atea r at time k. Now, stackig the observatios, the trasmitted sigal ad the oise at time k o the colum vectors yk, ηk ad xk respectively, ad lettig h l be a R T matrix with etries give by the lth tap betwee each atea pairs r, t, we ca rewrite 1.1 i matrix form as L 1 yk = h l xk l + ηk 1.2 l=0 which is the Iput-Output relatio of a MIMO system MIMO-OFDM model Now, we go oe step further, ad we defie the iput-output relatio of a MIMO-OFDM system. Orthogoal Frequecy Divisio Multiplexig OFDM is a modulatio techique which cosists i subdividig the available spectrum ito multiple sub-carriers orthogoal to each other. Each sub-carrier is the idepedetly modulated with a low-rate data stream, ad trasmitted through the chael. However, by combiig i the time domai the streams associated to each sub-carrier, the overall data rate achieved is much higher tha the data rates associated to the sigle streams. The advatage of this approach cosists i the fact that the frequecy-selective chael is trasformed ito a set of flat-fadig chaels. This is possible because the badwidth occupied by each subcarrier is much smaller tha the overall badwidth, therefore each sub-carrier udergoes approximately a flat-fadig chael. I this thesis, we treat the implemetatio of OFDM usig the DFT Discrete Fourier Trasform ad Cyclic Prefix, which is the actual implemetatio of OFDM i LTE. Let s cosider a MIMO-OFDM system with N sub-carriers, T trasmittig ad R receivig ateas T R MIMO. Let X k be the MIMO sigal trasmitted o sub-carrier at time k this is a T 1 vector. With OFDM, the time domai sigal is obtaied with the Iverse DFT trasformatio, through the relatio x k p = 1 X ke i2π p N p = CP... N N

14 6 Chapter 1 Itroductio Here x k p is the pth sample of the kth MIMO-OFDM symbol, where this latter term refers to the ordered set of the symbols trasmitted o all the sub-carriers, that is {X k, = 0... N 1}. These samples are the trasmitted i sequece through the chael across the ateas array. Observe that the time-domai sigal is composed of two parts: x k p, p = 0... N 1 is a whole period of the Iverse DFT, whereas x k p, p = CP 1 is the Cyclic Prefix of legth CP, which is added at the begiig of the time-domai stream to make the chael appear cyclic, as we show ow. Notice that, sice the DFT is a periodic sigal with period N, we have x k p = x k N + p, p = CP 1, therefore the isertio of the Cyclic Prefix correspods to the isertio of the last samples of the Iverse DFT at the begiig of the stream. Now, observe the Iput-Output relatio of a MIMO system give by 1.2. Sice the chael is FIR of legth L, the output of the model at time k depeds oly o the trasmitted symbols at times k L k. Therefore, assumig that the Cyclic Prefix satisfies the coditio CP L 1, the output i correspodece of the kth OFDM symbol, cosiderig oly the output samples p = 0... N 1, depeds solely o the symbols trasmitted o the kth symbol. I fact L 1 y k p = h l x p p l + η k p l=0 = 1 L 1 p l i2π h l e N X k + η k p p = 0... N N l=0 It is thus clear that Iter-Symbol Iterferece from previous OFDM symbols is elimiated by settig CP L 1. The, usig this assumptio o the legth of the Cyclic Prefix, ad lettig H be the frequecy domai chael, defied as N times the DFT of the time domai chael h l, we obtai y k p = 1 H X ke i2π p N + η k p p = 0... N N

15 Chapter 1 Itroductio 7 At the receiver, the time-domai sigal is processed usig the N-poits DFT. O subcarrier m we have Y m k = 1 N 1 y k pe i2π pm N N = 1 N = p=0 N 1 p m i2π H X k e N + 1 N 1 η k pm i2π pe N N p=0 p=0 H X kδ m + η m k = H m X m k + η m k 1.6 where η m is the oise vector o sub-carrier m at time k. From this relatio we see that the isertio of the Cyclic Prefix of legth CP L 1 has trasformed a frequecy selective chael ito a set of N flat-fadig chaels. Fially, assumig K OFDM symbols are trasmitted, ad collectig the received ad the trasmitted sigals ad the oise at time k o a matrix, we have the followig Iput-Output relatio for a MIMO-OFDM system: Y = H X + η for = 0... N Here, the subscript represets the sub-carrier idex, Y is the R K observatio matrix with etries Y r, k C represetig the sigal received o sub-carrier at time k o receivig atea r, X is the T K matrix of the trasmitted symbols with elemets X t, k C represetig the symbol trasmitted o sub-carrier at time k from trasmittig atea t, H is the R T chael matrix with etries H r, t C represetig the chael coefficiet betwee atea pair r, t, ad η is the R K oise matrix. Now, let s assume that the matrix of the trasmitted symbols X is a collectio of both pilot symbols, used at the receiver for performig the chael estimate, ad iformatio symbols. We assume also that, i order to suppress multi-atea iterferece durig the estimatio process, at a geeric time k o sub-carrier, either all the ateas are trasmittig pilots or oe of them i this case they are all trasmittig iformatio symbols. With these assumptios, we ca split the matrix of the trasmitted symbols ito the sum of two matrices, the former oe carryig the cotributio from the pilot symbols X tr, with ull etries i correspodece of the ukow symbols, the latter carryig the cotributio from the ukow symbols X bl, with ull etries i correspodece of the pilot symbols. Similarly, we ca split the observatio ad oise matrices ito the observatio ad oise matrices associated to pilot symbols Y tr ad η tr ad ukow symbols respectively Y bl ad η bl. Therefore, o each sub-carrier

16 8 Chapter 1 Itroductio we have the followig decompositio of the observatio, symbol ad oise matrices: X = X tr Y = Y tr η = η tr + X bl + Y bl + η bl Usig this otatio, we ca split the Iput-Output relatio 1.7 as 1.8 { tr Y = H X tr + η tr for = 0... N 1 Y bl = H X bl + η bl for = 0... N The first relatio describes the iput-output model associated to the pilot symbols, the secod istead describes the iput-output model associated to the iformatio symbols. Notice that, i the pilot based approach to chael estimatio, oly the first iputoutput relatio is cosidered, sice oly the pilot observatios are used for the estimate. Coversely, i the Semi-Blid approach all the iformatio is cosidered at the receiver, both Y tr ad Y bl Model Assumptios Based o the MIMO-OFDM model described i the previous sectio, we ow defie the geeral assumptios used throughout the thesis. I particular, we defie the assumptios o the ukow symbols ad o the oise at the receiver. As regards the ukow symbols, we assume that they are obtaied by ecodig across the trasmittig atea array a set of S idepedet streams. The model used is the followig: X bl = CV bl 1.10 where C is a T S precodig matrix, which ecodes S idepedet streams of symbols ito the T trasmittig ateas array, ad V bl is the S K matrix of the iformatio symbols. The etries of this matrix are assumed to be draw uiformly from a discrete costellatio C, idepedetly ad idetically distributed, with zero mea ad mea power σ 2 s. Therefore we have E[V kv k H ] = σ 2 si S. Notice that matrix C ecodes the symbols oly across the trasmittig ateas, ot across time. Its colums represet a set of Hadamard vectors, with the property that C H C = I S, where I S is the S S idetity matrix. Therefore, also the trasmitted symbols are idepedet across time ad across sub-carriers, but they are ot ecessarily idepedet across the trasmittig ateas.

17 Chapter 1 Itroductio 9 I our treatmet we assume S mi {R, T }, sice detector performace would be severely reduced i the case S > mi {R, T } ad good approximate detector desig is sigificatly harder for this case. This assumptio is coheret with the fact that the capacity of a MIMO system liearly icreases with the miimum betwee the umber of receivig ad the umber of trasmittig ateas, which correspods to the rak of the chael matrix assumig there is eough diversity i the wireless medium to make the chael matrix full-rak. As regards the oise, we assume it is a zero mea multivariate Gaussia process, statistically idepedet across time ad across sub-carriers, with covariace matrix o each sub-carrier E[η kη k H ] = Covη or equivaletly precisio matrix B η = Covη 1. Fially, observe that a OFDM system is desiged to support a chael of legth up to the legth of the Cyclic Prefix, i order to suppress Iter-Symbol Iterferece at the receiver. Therefore, i the course of our treatmet, we always assume that the coditio CP L 1 is fulfilled. Moreover, i the study of the chael estimators carried o i the followig chapters, we always assume that the chael legth L is kow at the receiver. 1.3 Problem Formulatio Now that we have defied the system model ad the assumptios used throughout the thesis, before proceedig with the treatmet it is importat to formulate the problem of chael estimatio i MIMO-OFDM systems. Problem Statemet 1.1 Chael Estimatio i MIMO-OFDM systems. Based o a set of observatios correspodig to pilot symbols Y tr ad to the ukow symbols Y bl, ad based o the sequece of trasmitted pilots X tr, the chael estimator attempts to approximate the ukow chael taps {H, = 0... N 1}. I our case, the chael is assumed to be a FIR filter of legth L, therefore there is a fuctioal depedecy of the chael taps i the frequecy domai, give by the Discrete Fourier Trasform. We ca i fact write L 1 H = h l e i2π l N = f h = 0... N l=0 where f. is a fuctio expressig the depedecy of H o the time domai chael h.

18 10 Chapter 1 Itroductio This fact has to be take ito accout i the estimatio process, as we will do i the course of the thesis. I order to read ad uderstad the followig chapters, the reader should be cofidet with the basics of estimatio theory, i particular with Maximum-Likelihood estimatio ad its properties. The iterested reader is suggested to read [2] or [3] for a geeral itroductio to estimatio theory.

19 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels I this chapter we derive a Maximum Likelihood estimator of MIMO-OFDM FIR chaels based solely o the trasmissio of a pilot sequece ad o the observatio of the correspodig output. We study the case of Gaussia oise at the receiver, idepedet across sub-carriers ad across time, with covariace matrix Covη o each sub-carrier. The we apply the results to the simpler case of white Gaussia oise at the receiver, with variace σw 2 o each receivig atea, i order to better uderstad the limits of applicability of the pilot based approach to MIMO systems. I this chapter ad i the followig, where we treat the Semi-Blid approach, we assume to have perfect kowledge of the statistics of the oise at the receiver. However, observe that this assumptio does ot hold true i practice, therefore the oise covariace matrix eeds to be estimated at the receiver as well. This issue is aalyzed i chapter 4 i detail. This chapter is orgaized as follows: based o the system model ad o the assumptios described i the previous chapter i sectio 1.2, i sectio 2.1 we derive a pilot based Maximum Likelihood ML estimator of MIMO-OFDM FIR chaels. We also aalyze the properties of such estimator, i terms of its mea ad its Mea Square Error ad we compare it with the Cramér Rao lower boud which is derived i sectio C.2 of the Appedix. I particular, we aalyze the case of white Gaussia oise at the receiver, sice this gives a deeper isight o the limits of the pilot approach whe applied to a MIMO system. 11

20 12 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels We also derive the ecessary coditio for the idetifiability of the MIMO-OFDM FIR chael, ad we determie the miimal pilot structure which satisfies these idetifiability coditios. 2.1 Maximum-Likelihood chael estimatio of a MIMO- OFDM FIR chael I this sectio, we derive a ML estimator of a MIMO-OFDM FIR chael usig the pilot based approach. As such, oly the observatios correspodig to pilot symbols are cosidered for the estimate Y tr, therefore the blid observatios Y bl are discarded i this chapter. Sice the chael is FIR of legth L, there is a fuctioal depedecy of the chael taps i the frequecy domai, expressed through the DFT L 1 H = h l e i2π l N = 0... N l=0 Therefore the Maximum-Likelihood solutio is determied with respect to the chael taps i the time-domai collected o the parameter matrix h, sice these represet a ucostraied set of parameters, from which the frequecy domai chael is determied through the liear trasformatio give above. Sice the oise at the receiver is statistically idepedet across sub-carriers ad across time, with covariace matrix Covη or equivaletly precisio matrix B η = Covη 1, the likelihood of the observatios, coditioed o the trasmitted pilots ad o the timedomai chael h, is give by p Y tr h, X tr = N 1 =0 { exp trace N 1 =0 [B η Y tr tr 1 K π R B η 2.2 H X tr Y tr ]} H H X tr where K tr is the umber of pilot symbols trasmitted o sub-carrier.

21 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 13 The maximizatio of the likelihood fuctio 2.2 with respect to its argumets is equivalet to the miimizatio of the egative log-likelihood, give by N 1 l p Y tr h, X tr 1 = K tr l π R B η + N 1 + =0 trace [B η Y tr =0 H X tr Y tr ] H H X tr 2.3 I order to eforce the chael legth costrait, the miimizatio of 2.3 is performed with respect to the time-domai chael matrix h. Keepig oly the terms depedig o h, the ML estimate, ĥ, is solutio to the followig miimizatio problem: ĥ = mi h = mi h { N 1 =0 { N 1 =0 trace [B η Y tr trace H H B η H X tr H X tr Y tr N 1 X trh 2real ] } H H X tr 2.4 =0 trace H H B η Y tr X trh Sice this problem will be ecoutered ofte i the course of this thesis, we express the above equatio i a more geeral form, by defiig the two matrices Λ xx ad Λ yx } as { Λ xx = X tr yx = Y tr Λ X trh X trh = 0 N The, we ca rewrite 2.4 as ĥ = mi h { N 1 =0 trace H H B η H Λ xx N 1 2real =0 trace H H B η Λ H yx = mi h {f h} 2.6 where we have defied the cost fuctio } f h = N 1 =0 trace H H B η H Λ xx N 1 2real =0 trace H H B η Λ H yx 2.7 The miimizatio is carried out by computig the derivative of 2.7 with respect to the chael etries {h l r, t, l = 0... L 1, r = 0... R 1, t = 0... T 1}, ad equalig this derivative to zero. The complex derivative defied i Appedix A with respect to

22 14 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels etry h l r, t of the time-domai chael is give by N 1 f h h l r, t = [ ] trace δt, rb η H Λ xx δt, rb η Λ yx e i2π l N = =0 N 1 =0 [B η H Λ xx Λ l yx ei2π N = ]rt where δt, r is a matrix whose etries are equal to zero except for the etry at positio t, r which is equal to 1. Rewritig the above equatio i matrix form we have N 1 =0 B η Λ yx e i2π l N N 1 = =0 B η H Λ xx e i2π l N 2.9 Now, expressig H as the Fourier trasform of the time-domai chael, we obtai N 1 =0 B η Λ yx e i2π l N = L 1 N 1 p=0 =0 B η h p Λ l p i2π xx e N 2.10 I order to determie a solutio to this problem, let s cosider the etry at positio r, t of the above system of equatios, ad let s make explicit the matrix product operatio i the followig way N 1 =0 L 1 = B η Λ yx e i2π l N N 1 p=0 =0 L 1 = p=0 r 1 t 1 rt L 1 = N 1 p=0 =0 B η h p Λ xx e B η r, r 1 h p r 1, t 1 Λ xx t 1, te r 1 t 1 B η r, r 1 Λ l p i2π xx t 1, te N l p i2π N l p i2π N rt h p r 1, t where for coveiece we dropped the extrema of the sum over the sub-carrier umber. Now, let Γ tr xx be a LRT LRT matrix with elemets Γ tr xx RT l + T r + t; RT p + T r 1 + t 1 = = B η r, r 1 Λ l p i2π xx t 1, te N B η Λ xx l p i2π e N T r+t,t r 1 +t

23 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 15 where the otatio A B represets the Kroecker product betwee A ad B, that is, assumig A is a M N matrix A0, 0 B A0, 1 B... A0, N 1 B. A1, 0 B A1, 1 B... A B = AM 1, 0 B AM 1, 1 B... AM 1, N 1 B The, let s redefie h as a LRT 1 colum vector with etries hrt l + T r + t = h l r, t 2.14 ad similarly let Γ tr yx be a colum vector with etries Γ tr yx RT l + T r + t = N 1 =0 B η Λ yx e i2π l N rt 2.15 The, we ca rewrite 2.10 i matrix form as Fially, assumig that Γ tr xx Γ tr yx = Γ tr xx h 2.16 is full rak, ad therefore ivertible we will discuss about the ecessary coditios i sectio 2.1.1, the Maximum Likelihood estimate of the time-domai chael is give by ĥ tr = Γ tr 1 xx Γ tr yx 2.17 The, lettig Ĥtr be a NRT dimesioal colum vector with elemets Ĥ tr RT + T r + t = H r, t 2.18 ad ŨN a N L matrix obtaied by takig the first L colums of the N N Fourier matrix U N with elemets U N, l = 1 N e i2π l N, we ca write the frequecy domai chael estimate as Ĥ tr = N ŨN I RT ĥtr 2.19 where I K is the K K idetity matrix. Sice this estimator will be used ofte i the course of our treatmet of Semi-Blid estimators, it is coveiet to iclude all the operatios ivolved i the estimatio of the time-domai chael ito a Black-Box, that is a fuctio H, takig as iput the

24 16 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels symbol autocorrelatio Λ xx, the correlatio betwee the observatios ad the trasmitted symbols Λ yx ad the oise precisio matrix B η o each sub-carrier the chael legth L, the umber of receivig ad trasmittig ateas R ad T are dropped for coveiece, ad returig the time-domai chael estimate. That is ĥ tr = H Λ xx, Λ yx, B η, = 0... N With these iputs we ca easily compute 2.12 ad 2.15, which ca the be used to estimate the chael usig We ow preset a result o the ecessary coditios for the idetifiability of the chael said i aother way, the chael is idetifiable if there are o ambiguities o the determiatio of the ML estimate Chael Idetifiability Coditios Theorem 2.1 Necessary idetifiability coditio of the chael for the pilot based approach through ML estimate. The ecessary but ot sufficiet coditio for the idetifiability of the chael is N 1 =0 rak X tr X trh LT 2.21 Proof. From equatio 2.17 we see that the chael matrix h is idetifiable if ad oly if Γ tr xx is ivertible, or equivaletly, if ad oly if it is full rak. Observe that Γ tr xx ca be rewritte i the followig form: Γ tr xx H = N ŨN I RT Λ ŨN I RT 2.22 where Λ is a block diagoal NRT NRT matrix obtaied by stackig the matrices B η Λ xx alog the diagoal. The for the rak of Γ tr xx, usig the product rule we have rak Γ tr xx { } mi rak ŨN I RT, rak Λ 2.23 Now, matrix ŨN is full rak, sice its colums belog to a set of orthoormal vectors the colums of the Fourier matrix U N represet a set of orthoormal vectors, therefore we have: rak ŨN I RT = LRT 2.24

25 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 17 Λ is a block diagoal matrix, therefore its rak is equal to the sum of the raks of its diagoal blocks: rak Λ = N 1 =0 { rak B η Λ xx } 2.25 ad sice B η is a full-rak square matrix with rak R we ca rewrite, usig the fact that rak A B = rak A rak B: N 1 rak Λ = R =0 rak Λ xx 2.26 Fially, usig 2.23 we have: rak Γ tr xx { mi LT, N 1 =0 } rak Λ xx 2.27 Therefore, substitutig Λ xx rak is: = X tr X trh the ecessary coditio for Γ tr xx to be full N 1 =0 rak X tr X trh LT 2.28 which completes the proof. We ow preset aother broader ecessary coditio for the chael idetifiability, determiig the miimum umber of pilot symbols ecessary for the chael to be idetifiable. Lemma 2.2. The miimum umber of pilots ecessary for the chael to be idetifiable is { mi K tr }, T LT 2.29 Proof. I fact from chael idetifiability coditio 2.21 we have N 1 =0 rak X tr X trh LT 2.30

26 18 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels Now, for the pilot correlatio terms we have X tr X trh = k X tr kx trh k 2.31 where X tr k is the vector of the pilots trasmitted at time k, or zero if o pilots are trasmitted at time k. Observe that each matrix X tr kx trh k has rak oe if a pilot is trasmitted at time k, or rak zero otherwise, therefore, sice o sub-carrier K tr pilots are trasmitted, the rak of the correlatio matrices is give by rak X tr Fially we obtai X trh N 1 =0 = rak rak X tr k X tr X trh kx trh From the above iequality we see that, if mi { k { mi K tr { mi K tr }, T }, T K tr }, T < LT, the ecessarily coditio 2.21 is ot satisfied. Therefore a ecessary coditio o the umber of pilots is which proves the lemma. { mi K tr }, T LT 2.34 However observe that, eve if coditio 2.29 of the lemma is satisfied, the ecessary coditio 2.21 may still ot be satisfied. This is a cosequece of the iequality used i Assumig K tr T o all the N tr N sub-carriers carryig pilots, the above lemma reduces to the coditio N tr L Properties of ML chael estimator I this sectio we study the properties of the Maximum Likelihood chael estimator give by equatio 2.19, i terms of its Bias ad Variace. However, oly for the calculatio of the bias, we assume that the oise precisio matrix B η used for estimatig the chael is ot ecessarily equal to the true oise precisio matrix. This result will be used later durig the thesis. Therefore, let B η be the true oise precisio matrix o sub-carrier, ad B η the oe actually used to estimate the chael.

27 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels Bias of Maximum Likelihood chael estimator Calculatig the expectatio of 2.19 with respect to the observatios we obtai [Ĥtr ] E = N ŨN I RT [ Γ xx tr 1 E Γ tr yx ] 2.35 Now, usig B η istead of B η for the expressio of Γ tr xx [ ] E from 2.15 we have Γ tr yx ad Γ tr yx, for the etries of [ ] N 1 E Γ tr yx RT l + T r + t = p=0 =0 L 1 N 1 = B η H X tr =0 B η h p X tr X trh e i2π l N X trh rt l p i2π e N rt 2.36 where i the last equality we expressed the frequecy domai chael as the Fourier trasform of the time domai chael. The, makig explicit the matrix products we obtai [ ] E Γ tr yx RT l + T r + t = L 1 N 1 p=0 r 1 t 1 =0 B η r, r 1 X tr X trh t 1 t ei2π l p N hp r 1, t Recogizig ad substitutig i the above expressio the etries of Γ tr xx give by 2.12, we ca rewrite the above expressio as [ E Γ tr yx Fially, substitutig ito the bias of the estimator ] = Γ tr xx h 2.38 [Ĥtr ] E = N ŨN I RT Γ tr 1 xx Γ tr xx h = N ŨN I RT h = H 2.39 which demostrates that the ML estimator is ubiased. This results shows also that the ML chael estimator is ubiased eve if we do t use the true oise covariace matrix for the estimate. This result will be used i chapter 4 for the joit estimatio of the chael ad of the oise covariace matrix o each sub-carrier.

28 20 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels Variace of Maximum Likelihood chael estimator Now, we defie the Mea Square Error of the estimator as the sum of the Mea Square Error for the estimatio of each etry of the chael matrix, divided by the umber of etries. This ca be thought as the Mea Square Error for the estimatio of the chael matrix etries, averaged over all the etries. For the case uder cosideratio, the Mea Square Error correspods to the variace of the estimator sice the estimator is ubiased, as demostrated i Therefore for the variace we have = 1 RT E { [ Ĥtr trace H Ĥtr ]} H H Ĥtr Var = 1 NRT E = 1 { [ RT E trace ŨN ĥtr I RT h ĥtr H ]} H h ŨN I RT { [ ĥtr H trace h ĥtr h ]} 2.40 where i the last equality we used the fact that traceab = traceba, ad Ũ H N ŨN = I L. The, substitutig ito ĥtr the ML solutio give by 2.17 we obtai Ĥtr Var = 1 { RT trace Γ tr 2 xx E [ Γ tr yx [ E Γ tr yx ] [ Γ tr yx E Γ tr yx ] H ]} 2.41 where we used the fact that h = Γ tr 1 xx For the etries of the term E E = { [ Γ tr yx N 1 =0 [ E E { [ B η [B η Y tr [ Γ tr yx Γ tr yx Y tr E[Γ tr [ E ] Γ tr yx H X tr yx ] from 2.38 ad Γ tr xx is Hermitia. ] [ ] ] H E we have Γ tr yx [ E H X tr Γ tr yx Γ tr yx ] X trh ] H ] r 1 t 1 Γ tr yx l,r 1,t 1 ;p,r 2,t 2 ] } X trh l p i2π e N 2.42 r 2 t 2 }

29 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 21 Now, makig explicit the matrix products we obtai E = = { [ Γ tr yx N 1 =0 s 1 s 2 [ E k Γ tr yx [ E Y tr B η r 1, s 1 X tr N 1 =0 ] [ Γ tr yx E Γ tr yx ] H ] H X tr Y tr s 1 k t 1, kb η s 2, r 2 X tr t 2, ke B η r 1, s 1 Cov η s1 s 2 B η s 2, r 2 s 1 s 2 l,r 1,t 1 ;p,r 2,t 2 } H X tr s 2 k l p i2π N k X tr = 2.43 ] t 1, kx tr t 2, ke l p i2π N where i the last equality we used the defiitio of oise covariace matrix Cov η. Fially, observig that s 1 s 2 B η r 1, s 1 Cov η s1 s 2 B η s 2, r 2 = B η Cov η B η r1 r 2 = B η r 1, r we obtai E = { [ Γ tr yx N 1 =0 [ E Γ tr yx B η r 1, r 2 ] X tr Γ tr yx [ E Γ tr yx ] H ] l,r 1,t 1 ;p,r 2,t 2 X trh e i2π l p N 2.45 t 2 t 1 } = ad comparig the above expressio with the etries of Γ tr xx {[ [ E Γ tr yx E Γ tr yx ] Γ tr yx [ E Γ tr yx ] H ]} i 2.12 we ca rewrite = Γ tr xx 2.46 Fially, substitutig this expressio ito the expressio for the variace of the estimator i 2.41 we obtai the followig result: Ĥtr Var = 1 RT trace Γ xx tr 1 which represets the variace MSE of the Maximum Likelihood estimator I coclusio, the Maximum Likelihood estimator derived i the previous sectio is Ĥtr ubiased with variace Var = 1 RT trace Γ tr 1 xx. I the Appedix, i sectio C.2 we derive the Cramér Rao lower boud for the pilot based approach, showig that the ML estimator achieves the CRLB for ay cofiguratio of the pilot grid.

30 22 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels White Gaussia Noise at the receiver I the previous sectio we derived the expressio for the ML FIR chael estimator for Gaussia oise at the receiver with covariace matrix Covη o each sub-carrier, ad we derived its properties i terms of bias ad variace. I order to better uderstad the estimatio accuracy achievable with the pilot based approach, it is iterestig to study the particular case of white Gaussia oise at the receiver, with variace σw 2 o all sub-carriers ad o all receivig ateas. I this case the covariace matrix is give by Covη = σ 2 wi R precisio matrix is give by B η = 1 σ 2 w I R., or equivaletly the Moreover, we also assume a typical sceario where the pilots are allocated o sub-carrier 0 < S tr ad the o the followig sub-carriers spaced by S tr, where S tr is the pilot sub-carriers spacig, a divisor of N. We also assume that o all these sub-carriers ad o all the trasmittig ateas the total power ρ assiged to the pilots is the same, ad that the pilot sequece is orthogoal across the trasmittig ateas array. This ca be mathematically writte as { tr X X trh = ρi T = 0 + ks tr, k = 0... N S tr 1 X tr X trh = 0 otherwise 2.48 Sice oly N S tr sub-carriers over N are used for the allocatio of pilots, ad the rak of X tr X trh is either 0 o pilots allocated o sub-carrier or T sub-carrier is used for allocatig pilots, the ecessary idetifiability coditio becomes: N 1 =0 rak X tr X trh = NT S tr LT 2.49 or equivaletly N S tr L, which is the same result obtaied i lemma 2.2, assumig that K tr T o the sub-carriers carryig pilots. I order to eforce the orthogoality of the pilot sequece across the trasmittig ateas, oe solutio is to trasmit a set of orthogoal vectors of symbol. For example, o the sub-carriers carryig pilots we may trasmit T pilots i T distict MIMO-OFDM symbols, where oe oly atea trasmits at a time with a power equal to ρ, while the others are silet, ad each atea

31 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 23 trasmits a pilot o oe of the T time-slots. This ca be writte mathematically as ρe iθ X tr 0 ρe iθ = 0 0 ρe iθ ρe iθ This is the solutio used also to allocate the pilots i the LTE slots ad i the course of our simulatios. Substitutig 2.48 ito the expressio for Γ tr xx, we obtai Γ tr xx = Nρ S tr σw 2 I LRT 2.51 Notice that i this case the coditio N S tr L represets ot oly a ecessary but also a sufficiet coditio for the idetifiability of the chael. It ca be show that this pilot allocatio method is optimal i the case of white Gaussia oise, sice it miimizes the variace of the estimator. I fact, let s assume we have a pilot power costrait, that is trace X tr X trh = P 2.52 This traslates ito a costrait o the trace of matrix Γ tr xx, i fact trace Γ tr xx = 1 σw 2 LR trace X tr X trh = 1 LRT 1 σw 2 LRP = p=0 λ p 2.53 where i the last equality we used the fact that the trace of Γ xx is equivalet to the sum of its eigevalues {λ p }. The optimizatio of the pilot structure is performed by miimizig the variace of the estimator, give by 2.47, uder the costrait Usig the Lagrage multipliers i order to eforce the costrait we have the followig cost fuctio: f = 1 RT trace = 1 RT LRT 1 p=0 Γ xx tr µ λ p + µ LRT 1 p=0 LRT 1 p=0 λ p 1 σ 2 w LRP = λ p 1 σw 2 LRP = 2.54

32 24 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels The, calculatig the derivative of the cost fuctio with respect to the eigevalue λ r ad equalig it to zero we have λ r = 1 RT µ 2.55 Fially, eforcig the power costrait we obtai λ r = P σ 2 wt r 2.56 which demostrates that the optimal Γ xx miimizig the variace of the estimator is give by Γ tr xx = P σ 2 wt I LRT 2.57 This is achieved with the pilot allocatio method 2.48, by settig P = NT ρ S tr. For the variace of the estimator, substitutig Γ tr xx ito 2.47 we have Var Ĥtr = LS trσw 2 Nρ 2.58 Notice that i a commo system the average trasmissio power per sub-carrier σ 2 T x is the same o all sub-carriers, ad is equally distributed across the trasmittig ateas. The, usig this assumptio, sice ρ is the total power assiged to the pilots o each trasmittig atea ad o each sub-carrier, ρ NT S tr = σ 2 T x N T OT, where N T OT is the total umber of pilot symbols allocated o the OFDM grid. Therefore we ca rewrite the variace as Var Ĥtr = σ2 wlt σt 2 x N T OT = LT SNR N T OT 2.59 where SNR = σ2 T x σ 2 w is the sigal to oise ratio of the system. The above expressio highlights some importat limitatios of the pilot based approach. Observe that the variace of the estimator grows proportioally to the umber of trasmittig ateas, ad iversely to the umber of pilots N T OT. This implies that a larger umber of trasmittig ateas has to be compesated by a loger pilot sequece, i order to achieve a give estimatio accuracy while keepig fixed the other parameters. This behavior ca be easily uderstood by ispectig the pilot allocatio structure 2.50, which we showed to be optimal sice it miimizes the variace of the estimator: oly oe atea trasmits at a time, sice i this way the iterferece from the other ateas is suppressed, ad each receivig atea is able to effectively estimate the SISO chael betwee itself ad the atea trasmittig the pilot symbol. Therefore T pilot symbols

33 Chapter 2 Traiig sequece chael estimatio of MIMO-OFDM FIR chaels 25 are eeded for all the ateas to trasmit oe pilot, i other words the umber of pilot symbols ecessary to achieve a give estimatio accuracy is proportioal to the umber of trasmittig ateas. It is clear that, as the order of the MIMO system icreases, while keepig fixed the other parameters, i order to achieve a acceptable estimatio accuracy more pilots have to be collected at the receiver. This i tur is achieved either elargig the observatio time, or allocatig more pilots o the OFDM grid. However, the first approach larger observatio time compromises the ability of the receiver to track fast-varyig chaels, which is ot acceptable i a Mobile Commuicatio System. O the other had, the secod approach more pilots o the OFDM grid compromises the badwidth efficiecy of the system, sice the pilots represet a waste of badwidth. Therefore, it becomes importat to exploit also other iformatio at the receiver tha relyig solely o pilots. The approach studied i this thesis for improvig the estimatio accuracy cosists i exploitig also the ukow symbols at the receiver semi-blid chael estimatio. I the ext chapter we study differet Semi-Blid approaches ad algorithms, ad we compare them with the pilot based approach studied i this chapter.

34

35 Chapter 3 Semi-Blid chael estimatio I chapter 2 we have derived a Maximum Likelihood estimator of a MIMO-OFDM FIR chael based exclusively o pilot symbols, assumig Gaussia oise at the receiver with covariace matrix Covη o sub-carrier. We have also show that the estimatio accuracy of such a estimator equals the correspodig Cramér Rao lower boud ad, i the case of white Gaussia oise at the receiver, ad orthogoal pilots, equally spaced across the sub-carriers, the variace of the estimator is give by equatio 2.59 ad reported here Var Ĥtr = σ2 wlt σt 2 x N T OT = LT SNR N T OT 3.1 where N T OT is the total umber of pilots used for the estimate. From this result it is clear that, i order to improve the estimatio accuracy, for a give sigal to oise ratio ad umber of trasmittig ateas, a larger umber of pilot observatios have to be collected at the receiver. I a MIMO-OFDM system it is required to estimate a larger umber of parameters with respect to a simple SISO system. This egatively impacts the accuracy of the estimator. I fact, observig the expressio give above, we see that the variace for the estimatio of the etries of the chael matrix icreases liearly with the umber of trasmittig ateas. Moreover, otice that the oe give above represets the average variace per etry of the chael matrix. If we sum the variace for the estimatio of each chael etry, istead of averagig it over the umber of etries, the overall variace is a quadratic fuctio of the umber of trasmittig ateas ad a liear fuctio of the umber of receivig ateas. This depedecy o the dimesio of the MIMO- OFDM system, i the case of estimatio techiques based o pilots aloe, traslates ito the eed for a loger traiig sequece with respect to a simple SISO i order to achieve a give performace. This is achieved either by elargig the observatio time, 27