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Challenges in MIMO synchronization
An important challenge in MIMO synchronization is that there exists a performance-complexity trade-off; when the number of antennas increases, so does the complexity of the algorithms. From a complexity point of view, it is thus appropriate to ask the question whether we should add extra antennas for the specific system and application under study. Complexity is of course also a problem from the hardware cost and compatibility point of view. Increasing the number of antennas in a system is an important system decision, and the advantages and possible disadvantages should be carefully weighed. Another important consideration in MIMO systems is the assumptions on interference of the system. Point-to-point systems can be a reasonable assumption in certain applications, for example the IEEE 802.11n standard, since it is designed to ensure that its short range links do not suffer from interference. But in general, MIMO cellular systems are interference limited, suffering from internal and other cell interference. As the number of interfering streams increases, the interference cannot be suppressed by spatial signal processing, and is treated as noise. Therefore, as opposed to what one may believe, adding transmit antennas in the system can actually decrease the throughput at low SINR. An additional important system feature is the channel characteristics. A common, often simplified, assumption is that of uncorrelated antenna arrays and Rayleigh fading channels. To fully cover the problem, there is a need for investigating the performance of MIMO systems even for deterministic channels, and understanding which system parameters to choose according to a suitable channel model.
Considering the above challenges, the following section outlines the state of the art in MIMO synchronization, with the goal of validating the need for our study.
State of the art in MIMO synchronization
Time and frequency synchronization of MIMO systems have been strongly studied in the last fifteen years, mainly in the context of direct-sequence coded division multiple access (DS-CDMA) and orthogonal frequency division multiplex (OFDM) links. Both coarse and fine time synchronization jointly with frequency offset estimation and compensation have been analyzed, and many techniques have been proposed either for time-frequency synchronization [1–4] or time synchronization only [5–9]. Nevertheless, most of these techniques assume an absence of interference. The scarce papers of the literature dealing with MIMO synchronization in the presence of interference correspond to [2, 7–9]. However, [2] and [7] only consider the problem of MIMO synchronization in the presence of multi-user interference (MUI), and [8] seems to be the only paper dealing with MIMO synchronization in the presence of interference of any kind, such as hostile jammers. In [8], several statistics are proposed for time synchronization for both flat fading and frequency selective fading channels. Despite the numerous existing algorithms for MIMO time synchronization, many important questions about their optimality, performance and complexity have arisen. First of all, none of the receivers developed for MIMO synchronization in the absence of interference [1, 3–6, 10–27] has been developed through a GLRT approach in the general case of an SC non-DS-CDMA link and arbitrary potentially non-orthogonal sequences. It is wellknown [28], however, that contrary to likelihood ratio test (LRT) statistics, GLRT statistics may be suboptimal from a detection point of view. With this in mind, one may wonder if a non-GLRT statistics can be equal or better than existing GLRT statistics.
New low-complexity statistics
The direct computation of the determinant (2.3) is not so straightforward for K > 2 while the MMSE statistics (2.5) has been shown in [8] to become sub-optimal for non-orthogonal synchronization sequences. In this context, a second way of decreasing the complexity of GLRT for arbitrary values of K while trying to keep its performance is to develop alternative statistics. To this aim, it seems natural to think that non-GLRT statistics corresponding to good estimates GLRT,kn, the GLRT statistics in known total noise, have good chances of approaching the performance of GLRT. For this reason, in this section, we introduce GLRT,kn and propose two new low-complexity statistics corresponding to two different estimates of this test.
Table of contents :
1 Introduction
1.1 MIMO in synchronization: Friend or foe?
1.1.1 Introduction: Evolution and advantages of MIMO
1.1.2 Challenges in MIMO synchronization
1.1.3 State of the art in MIMO synchronization
1.2 Model and problem formulation
1.2.1 Hypotheses
1.2.2 Synchronization as hypothesis testing
1.2.3 Generalized Likelihood Ratio Test (GLRT)
1.2.4 Performance of a binary hypothesis test
1.3 Objective
1.4 Parameter optimization and complexity reduction for time synchronization
1.4.1 Introduction to the problem
1.4.2 Synchronization statistics studied in the thesis
1.4.3 Complexity reduction and parameter optimization of MIMO synchronization
1.5 Large system analysis using random matrix methods
1.5.1 Introduction to the problem
1.5.2 Why large system approximation?
1.5.3 Preliminaries
1.5.4 Example of asymptotic analysis: Expected value of N under H0
1.5.5 Limit distributions
2 Parameter optimization for time synchronization of multi-antenna systems
2.1 Introduction
2.2 Statistics optimized for the presence of interference
2.2.1 GLRT: GLRT synchronization statistics
2.2.2 MMSE: MMSE synchronization statistics
2.3 Optimization of the synchronization for fixed K,M
2.3.1 Direct expression of GLRT for K=2
2.3.2 New low-complexity statistics
2.3.3 Computation rate decrease of ˆRyy
2.3.4 Complexity analysis
2.4 Optimization of K for a fixed M and statistics
2.4.1 Deterministic channel
2.4.2 Random channel
2.5 Conclusion
2.6 Appendix: Derivation of GLRT,we
2.7 Appendix: Derivation of GLRT,wu
2.8 Appendix: Derivation of GLRT
2.9 Appendix: Derivation of GLRT,kn
2.10 Appendix: Complexity of common matrix operations
3 Large system analysis of a GLRT for detection with large sensor arrays in temporally white noise
3.1 Introduction
3.2 Presentation of the problem.
3.3 Standard asymptotic analysis of N.
3.3.1 Hypothesis H0.
3.3.2 Hypothesis H1.
3.4 Main results.
3.4.1 Asymptotic behaviour of N when the number of paths L remains fixed when M and N increase.
3.4.2 Asymptotic behaviour of N when the number of paths L converges towards
1 at the same rate as M and N.
3.5 Numerical results.
3.5.1 Influence of cN = M N on the asymptotic means and variances.
3.5.2 Comparison of the asymptotic means and variances of the approximations of N under H0
3.5.3 Validation of asymptotic distribution under H0
3.5.4 Comparison of the asymptotic means and variances of the approximations of N under H1.
3.5.5 Validation of asymptotic distribution under H1
3.6 Conclusion.
3.7 Appendix: Useful technical results.
3.8 Appendix: Proofs of Theorems 1 and 2
3.9 Appendix: Proof of (3.87)
4 Conclusion
Résumé long
4.1 Synchronisation MIMO: ami ou ennemi?
4.1.1 Introduction: Évolution et avantages de MIMO
4.1.2 Défis de la synchronisation MIMO
4.1.3 Etat de l’art en synchronisation MIMO
4.2 Modèle et problème
4.2.1 Hypothèses
4.2.2 Synchronisation comme test d’hypothèse
4.2.3 Test du rapport de vraisemblance généralisé (GLRT)
4.2.4 Performance d’un test d’hypothèse binaire
4.3 Objectif
4.4 Optimisation des paramètres et réduction de la complexité de la synchronisation temporelle
4.4.1 Introduction au problème
4.4.2 Les statistiques de synchronisation étudiées dans la thèse
4.4.3 Réduction de la complexité et optimisation des paramètres de synchronisation MIMO
4.5 Analyse du grand système avec matrices aléatoires
4.5.1 Introduction au problème
4.5.2 Pourquoi l’approximation du grand système?
4.5.3 Préliminaires
4.5.4 Exemple d’analyse asymptotique: l’espérance de N sous de H0
4.5.5 Distributions limites
