Effect of Spatial Correlation and Antenna Selection on Coded MIMO Systems 

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Diversity-Multiplexing Tradeoff for MIMO systems

A MIMO system can provide two types of gains: diversity gain and multiplexing gain. In [ZT03], Zheng and Tse have proposed the DMT framework to characterize the interplay between reliability and rate at infinite SNR (SNR → ∞). They have proved that both gains can be simultaneously obtained, but there is a fundamental tradeoff between them: a higher multiplexing gain comes at the price of sacrificing diversity. The DMT curve defines the upper-bound achievable by any space-time coding scheme over a Nt × Nr MIMO channel.
Let us consider a space-time code X[Nt×T] transmitting at rate R() bits per channel use with a packet error probability Pe(). The asymptotic multiplexing gain rasymptotic is defined as the ratio of the achievable rate to the logarithm of where → ∞. Asymptotically, an increase of 3 dB in allows a data rate increase of rasymptotic bits. rasymptotic = lim →∞ R() log . (1.14) The multiplexing gain is always less than or equal to min(Nt,Nr). Indeed, in the high SNR regime, the MIMO channel can be viewed as min(Nt,Nr) parallel spatial channels since min(Nt,Nr) is the total number of degrees of freedom available for communication [Fos96].
The asymptotic diversity gain dasymptotic is defined as the negative (asymptotic) slope of the packet error probability curve as a function of in the log-log scale. dasymptotic = − lim →∞ log Pe() log .

Space-Time Coding for MIMO Systems

At the transmitter, MIMO coding techniques are employed for data rate increase and/or better reliability. Spatial multiplexing schemes [WFGV98] aim at increasing the channel spectral efficiency while space-time codes are designed to exploit the channel diversity. The main idea of space-time coding is to introduce redundancy or correlation between transmitted symbols on spatial and temporal dimensions. A space-time code is characterized by its rate, diversity gain and coding gain. The rate of a space-time code is equal to the average number of transmitted symbols per a channel use period. The diversity order is equal to the number of independent received replicas of transmitted symbols and the coding gain is the gain provided by coded system with respect to a non-coded one. A space-time code is said to be full-rate (FR) when the rate is equal to the total number of degrees of freedom available for communication i.e., min(Nt,Nr). Besides, it is said to be full-diversity (FD) when it exploits all the available MIMO channel diversity NtNr.
One can distinguish two main types of space-time codes: space-time trellis codes (STTCs) and space-time block codes (STBCs). Space-time trellis coding consists of the coding on a trellis of transmitted symbols over multiple antennas and multiple channel uses. STTCs can provide both coding gain and diversity gain [TSC98]. The coding optimization is done for each targeted spectral efficiency value. The detection is done using the Viterbi algorithm by minimizing a cumulative likelihood metric then selecting the most likely path in the trellis. Due to their lack of flexibility and high maximum likelihood (ML)-detection complexity, they are excluded from practical communication system standards [SOZ11]. Therefore, they are beyond the scope of this thesis. On the other hand, STBCs enjoy relatively lower ML-detection complexity and are often incorporated in communication system standards. In the sequel, we introduce space-time block codes and their conventional design criteria. Then we focus on some well-known STBCs considered throughout this thesis. More details on coding for MIMO systems can be found in [DG07, SOZ11].

Space-Time Block Codes

Space-time block codes are usually represented by a Nt × T matrix where each row represents a transmit antenna and each column represents a channel use period. They were first introduced by Alamouti in [Ala98]. The scheme proposed by Alamouti is a low ML-detection STBC achieving full transmit diversity for 2 × 1 MISO systems with rate 1, equal to the rate of a SISO system. Thanks to its orthogonality property, the Alamouti code can be ML-detected by a simple linear processing on the set of received signals. Tarokh et al. in [TJC99] generalize the work of Alamouti and introduce orthogonal space-time block codes (OSTBCs). OSTBCs achieve full-diversity but, in order to maintain the orthogonality property, are not full-rate. To mitigate this problem, Hassibi et al. introduce in [HH02] linear dispersion STBCs (LD-STBCs) where they remove the orthogonality constraint. LD-STBCs are designed to maximize the symbolwise mutual information. However, LD-STBCs in [HH02] do not guarantee the FR-FD properties, requisite to achieve lower error probabilities at high SNRs. The well-known rank-determinant criteria formulated in [TSC98] aim to design such space-time codes.

Conventional Rank-Determinant Design Criteria

The rank-determinant criteria are conventionally used while designing FR-FD STBCs for quasi-static flat Rayleigh fading MIMO channels. They aim to minimize the pairwise error probability (PEP) Pr(X, ˆX), defined as the probability that the detector estimates an erroneous codeword ˆX instead of the transmitted codeword X, between all possible transmitted codewords. The PEP is upper-bounded by [TSC98]

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Table of contents :

Abstract
R´esum´e
Acknowledgments
R´esum´e de la th`ese
List of Figures
List of Tables
Abbreviations
Notations
Introduction
1 Introduction to Coded MIMO Systems 
1.1 Coded MIMO System Model
1.1.1 Noise Model
1.1.2 Fading Model
1.1.3 MIMO Channel Model
1.2 MIMO Channel Capacity
1.2.1 Entropy and Mutual Information
1.2.2 Channel Capacity and Outage Probability
1.3 Diversity Techniques
1.3.1 Diversity-Multiplexing Tradeoff for MIMO systems
1.4 Space-Time Coding for MIMO Systems
1.4.1 Space-Time Block Codes
1.4.2 Conventional Rank-Determinant Design Criteria
1.4.3 Conventional MIMO Codes
1.4.3.1 ML-Detection of STBCs
1.5 Background on Capacity-Approaching Channel Coding
1.5.1 Convolutional Codes
1.5.2 The WiMAX 8-State Double Binary Turbo Code
1.5.3 Log-Likelihood Ratios
1.5.4 Iterative (Turbo) Decoding
1.5.5 EXtrinsic Information Transfer Charts
1.5.6 FlexiCodes
1.6 An Example of Coded MIMO System: WiMAX
1.6.1 Bit Error Rate of Uncoded Transmission
1.6.2 Bit Error Rate of Coded Transmission
1.7 Chapter Summary
2 Improving Standardized STBCs for Coded MIMO Systems: Towards Adaptive STBCs 
2.1 Matrix D Space-Time Block Code
2.1.1 Original Low Complexity Detector
2.1.2 Soft Detection for MD STBC
2.1.2.1 Original Detector Impairments
2.1.2.2 Proposed Soft Detector for MD STBC
2.1.2.3 Comparison Between Detectors
2.1.3 Robustness of the Detector with respect to Erasure Events
2.2 Matrix D STBC for Coded MIMO Communication Systems
2.2.1 Trace Criterion
2.2.2 MD STBC Optimization according to the Trace Criterion
2.2.3 Performance Evaluation
2.2.3.1 BER Curves of the Uncoded MIMO System
2.2.3.2 BER Curves of the Coded MIMO System
2.2.3.3 Erasure Effects on the BER of the Coded MIMO System
2.3 On the Complexity of the WiMAX Receiver
2.3.1 ML-Detection Complexity Assessment
2.3.2 Turbo Decoding Complexity Assessment
2.3.3 WiMAX Receiver Complexity Assessment
2.4 Bitwise Mutual Information Criterion
2.4.1 Why Bitwise Mutual Information?
2.4.2 Criterion Definition
2.4.3 Criterion Validation
2.5 Proposal of Adaptive Space-Time Block Codes
2.5.1 Adaptive Matrix D STBC
2.5.2 Adaptive Trace-Orthonormal STBC
2.5.2.1 Presentation of Adaptive Trace-Orthonormal STBC
2.5.2.2 Analysis of the Transmitted Constellations
2.5.2.3 Complexity Reduction of ML-Detection
2.6 Performance Evaluation of the Proposed Adaptive TO STBC for aWiMAX System
2.6.1 Adaptive TO STBC Parameter Computation for Coded Systems .
2.6.2 BMI Study of the Adaptive TO STBC
2.6.3 Closed-Loop System
2.6.3.1 4-QAM Modulation
2.6.3.2 16-QAM Modulation
2.6.3.3 Higher Order Modulations
2.6.3.4 Conclusion
2.6.4 Broadcast Transmission: Open-Loop System
2.7 Performance Evaluation with FlexiCode
2.8 Chapter Summary
3 Effect of Spatial Correlation and Antenna Selection on Coded MIMO Systems 
3.1 Effect of Spatial Correlation on Coded MIMO Systems
3.1.1 Modeling Spatially Correlated Channels
3.1.2 BMI Study of the Effect of Spatial Correlation
3.1.2.1 Correlation between Transmit Antennas
3.1.2.2 Correlation between Receive Antennas
3.1.2.3 Correlation between both Transmit and Receive Antennas
3.1.2.4 Discussion on the Obtained Results
3.1.3 Adaptive TO STBC Design for Spatially Correlated Systems .
3.1.4 Enhanced Spatial Multiplexing Scheme
3.1.5 BER Performance of the Proposed Adaptive STBCs
3.1.5.1 Low Transmit Correlation
3.1.5.2 High Transmit Correlation
3.1.6 BER Performance of the Enhanced Spatial Multiplexing Scheme .
3.1.6.1 High Transmit Correlation
3.1.6.2 Full Transmit Correlation
3.1.7 Conclusions on the Effect of Spatial Correlation
3.2 Antenna Selection for Coded MIMO Systems
3.2.1 Transmit Antenna Selection
3.2.1.1 Antenna Selection Algorithm
3.2.2 BER Curves of the Uncoded MIMO System with Transmit AS .
3.2.3 Transmit AS for Coded MIMO systems: Adaptive STBCs design .
3.2.3.1 Adaptive Matrix D STBC
3.2.3.2 Adaptive Trace-Orthonormal STBC
3.2.4 BER Performance of the Coded MIMO System with Transmit AS
3.3 Chapter Summary
4 Finite-SNR Diversity-Multiplexing Tradeoff for RayleighMIMO Channels 
4.1 Background on Finite-SNR DMT
4.1.1 Finite-SNR DMT for Orthogonal Space-time Block Codes
4.1.2 Finite-SNR DMT for 2 × 2 MIMO Channels
4.2 Finite-SNR DMT for MIMO Channels with Dual Antennas
4.2.1 Mutual Information for Nt × 2 and 2 × Nr MIMO Systems
4.2.2 Finite-SNR DMT for Uncorrelated Flat Rayleigh Channels
4.2.2.1 Mutual Information pdf Derivation
4.2.2.2 Outage Probability Derivation
4.2.2.3 Analytical Expression of Finite-SNR DMT
4.2.3 Finite-SNR DMT for Correlated Flat Rayleigh Channel
4.2.3.1 Mutual Information pdf Derivation
4.2.3.2 Outage Probability Derivation
4.2.3.3 Analytical Expression of Finite-SNR DMT
4.3 Numerical Results
4.3.1 2 × 2 MIMO System
4.3.2 MIMO System with n ≥ 3
4.4 Chapter Summary
Conclusions and perspectives
A Optimization of Matrix D STBC Parameter according to the BMI Criterion
B Optimization of Trace-Orthonormal STBC Parameter according to the BMI Criterion
C Transmitted Constellation for Trace-Orthonormal STBC
D Derivation of Finite-SNR DMT for uncorrelated RayleighMIMO Channels
Bibliography

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