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## Channel Capacity and Outage Probability

The notion of channel capacity is established in the pioneering work of Shannon [Sha48]. The channel capacity defines the maximum amount of information that one can reliably transmit over the channel. It is measured as the maximum of the mutual information between the input and output of the channel, with respect to the input distribution. C = max I(X; Y ) (1.11) p(x).

The capacity of a coherent MIMO system where the CSI is known only at the re-ceiver is achieved by an input distributed as a zero-mean circularly symmetric complex Gaussian random variable [Tel99, FG98]. The capacity of the random MIMO channel is given by.

C = EH [I(X; Y )] C = EH log2 det INr + ρ HH H (1.12).

Nt = EH log2 det INt + ρ H H H Nt.

where ρ is the average signal-to-noise ratio per receive antenna, the superscript H stands for conjugate transpose. The expectation is evaluated over the statistics of the matrix H. The capacity can be computed by either analytical expression [Tel99, RV05] or Monte Carlo simulations. The capacity of a MIMO system with uncorrelated Rayleigh fading channel achieves almost r = min(Nt, Nr) bps/Hz for every 3 dB increase in SNR while it achieves only one additional bps/Hz for every 3 dB increase for a single-input single-output (SISO) system [Tel99, FG98]. However, the correlation between antennas alter this capacity and may result to a substantial degradation of the MIMO capacity especially for high correlations [Loy01, RV05]. The outage probability is an important measure for the performance evaluation of communication systems over quasi-static fading channels. It is defined as the probability that a given realization of the channel cannot support a required data rate R. It can then be written as Pout = Pr [I < R] (1.13)ρ H = Pr log2 det INr + HH < R Nt.

In this case, we say that the channel is in outage. The computation of outage probabil-ities can be evaluated either by analytical expressions or by Monte Carlo simulations. However, exact analytical form of outage probabilities for any Nt × Nr are not easily tractable. Upper-bounds or Monte Carlo simulations are often used to compute these outage probabilities [Nar05, Nar06, OC07, RHGA10].

### Diversity Techniques

Diversity techniques refer to methods of improving the reliability of a message by eﬀectively transmitting diﬀerent replicas of the same information over diﬀerent indepen-dent branches, and thereby combating channel fading. Each replica experiences diﬀerent channel conditions (fading, interference, etc.), in the hope that at least one of these repli-cas will be correctly received. The receiver wisely combines these replicas and try to recover the transmitted information. The diversity order d is defined as the number of independent received replicas. At high SNRs, the average error probability decays inversely with the d-th power of the SNR i.e., SNR−d [ZT03]. Thus, it is important to maximize this diversity order while maintaining, if possible, the same transmission rate. Several diversity methods can be distinguished: Time diversity The same message is transmitted several times in diﬀerent time in-tervals. The separation of these time intervals has to be suﬃciently large, i.e., greater than the coherence time of the channel, so that the fading channel coeﬃcients change, and diﬀerent channel gains are observed.

Frequency diversity Diﬀerent replicas of the message are transmitted over diﬀerent frequency bands. To ensure that the channel coeﬃcients seen are diﬀerent, the separation of these frequency bands has to be greater than the coherence bandwidth of the channel.

When using simple repetition coding, the price to pay for time diversity and frequency diversity methods is an increased bandwidth occupation or a reduced data rate. Another method to combat the eﬀects of fading without wasting available resources is space diversity, also known as antenna diversity. Space diversity The signal is transmitted via several transmit antennas and received via diﬀerent receive antennas. These antennas should be placed at least half a wavelength of the carrier frequency apart to ensure that the signal undergoes diﬀerent independent channel fades [WSG94]. The maximal achievable diversity order of a MIMO system, assuming the path gains between individual pairs are independent and identically dis-tributed Rayleigh faded, is d = NtNr. One way to exploit the diversity with the order d is to use repetition coding with a rate 1/d. Coding techniques can be considered as more sophisticated forms to exploit the diversity. This coding can be done at several levels in the transmission chain. When considering space diversity, one can distinguish two major coding categories: Channel coding which consists of the addition of redundancy bits to the trans-mitted message, in order to detect and/or correct erroneous bits at the reception.

Space-time coding for MIMO systems. It consists of the transmission on multiple antennas of a redundant and/or correlated version of the mapped symbols during several channel use periods, and thereby can exploit only space diversity or both space and time diversities. Space-time coding for MIMO systems not only increases the diversity order, but also increases the spectral eﬃciency or the multiplexing gain of the channel. Indeed, it is possible to achieve simultaneously both diversity and multiplexing gains with a fundamental tradeoﬀ known as diversity-multiplexing tradeoﬀ (DMT) [ZT03].

#### 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(ρ) bits per channel use with a packet error probability Pe(ρ). The asymptotic multiplexing gain rasymptotic is defined as the ratio of the achievable rate to the logarithm of ρ where ρ → ∞. Asymptotically, an increase of 3 dB in ρ allows a data rate increase of rasymptotic bits. rasymptotic = lim R(ρ) . (1.14) ρ→∞ log ρ.

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 ρ in the log-log scale.

**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**