DoF and Feedback Tradeoff over Two-User MISO BC 

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Contributions and outline of the thesis

As stated, the main work of thesis seeks to address the simple yet elusive and fundamental question of “How much quality of feedback, and when, must one send to achieve a certain performance in specific settings of multiuser communications”.
In Chapter 2, the work considers two-user MISO BC with imperfect and delayed CSIT, and explores the tradeoff between performance, and feedback timeliness and quality. The work considers a broad setting where communication takes place in the presence of a random fading process, and in the presence of a feedback process that, at any point in time, may or may not provide CSIT estimates – of some arbitrary quality – for any past, current or future channel realization. Under standard assumptions, the work derives the DoF region, which is optimal for a large regime of CSIT quality. This region concisely captures the effect of channel correlations, the quality of predicted, current, and delayed-CSIT, and generally captures the effect of the quality of CSIT offered at any time, about any channel. The bounds are met with novel schemes which – in the context of imperfect and delayed CSIT – introduce here for the first time, encoding and decoding with a phase-
Markov structure. The results hold for a large class of block and non-block 8 Chapter 1 Introduction fading channel models, and they unify and extend many prior attempts to capture the effect of imperfect and delayed feedback. This generality also allows for consideration of novel pertinent settings, such as the new periodically evolving feedback setting, where a gradual accumulation of feedback bits progressively improves CSIT as time progresses across a finite coherence period. The results were published in part at
– Jinyuan Chen and Petros Elia, “Can Imperfect Delayed CSIT be as Useful as Perfect Delayed CSIT ? DoF Analysis and Constructions for the BC”, in Proc. of 50th Annual Allerton Conf. Communication, Control and Computing (Allerton’12), October 2012.
– Jinyuan Chen and Petros Elia, “Degrees-of-Freedom Region of the MISO Broadcast Channel with General Mixed-CSIT”, in Proc. Information Theory and Applications Workshop (ITA’13), February 2013.
– Jinyuan Chen and Petros Elia, “MISO Broadcast Channel with Delayed and Evolving CSIT”, in Proc. IEEE Int. Symp. Information Theory (ISIT’13), July 2013. and will be published in part at
– Jinyuan Chen and Petros Elia, “Toward the Performance vs. Feedback Tradeoff for the Two-User MISO Broadcast Channel”, to appear in IEEE Trans. Inf. Theory, available on arXiv :1306.1751.
– Jinyuan Chen and Petros Elia, “Optimal DoF Region of the Two- User MISO-BC with General Alternating CSIT”, to appear in Proc. 47th Asilomar Conference on Signals, Systems and Computers (Asilomar’13), 2013, available on arXiv :1303.4352. In Chapter 3, extending the results of two-user MISO BC setting, the work explores the performance of the two user multiple-input multiple-output (MIMO) BC and the two user MIMO interference channel (MIMO IC), in the presence of feedback with evolving quality and timeliness. Under standard assumptions, and in the presence of M antennas per transmitter and N antennas per receiver, the work derives the DoF region, which is optimal for a large regime of CSIT quality. This region concisely captures the effect
of having predicted, current and delayed-CSIT, as well as concisely captures the effect of the quality of CSIT offered at any time, about any channel. In addition to the progress towards describing the limits of using such imperfect and delayed feedback in MIMO settings, the work offers different insights that include the fact that, an increasing number of receive antennas can allow for reduced quality feedback, as well as that no CSIT is needed for the direct links in the IC. The results were published in part at
– Jinyuan Chen and Petros Elia, “Symmetric Two-User MIMO BC and IC with Evolving Feedback”, June 2013, available on arXiv : 1306.3710.

DoF region of the MISO BC

We proceed with the main DoF results, which are proved in Section 2.4 (inner bound) and Section 2.6 (outer bound). We here remind the reader of the sequences f(1) t gnt=1, f(2)t gnt=1, f(1) t gnt =1,f(2) t gnt =1 of quality exponents, as these were defined in (2.4)-(2.7), as well as of the corresponding averages (1); (2); (1); (2) from (2.9). We also remind the reader that we consider communication over an asymptotically large time duration n. We henceforth label the users so that (2) (1). We start with the following proposition, the proof of which can be found in Section 2.4 which describes the scheme that achieves the corresponding DoF corner points.

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Periodically evolving CSIT

We here focus on the block fading setting with a finite coherence period of Tc channel uses, during which the channel remains fixed, and during which a gradual accumulation of feedback provides a progressively increasing CSIT quality, as time progresses across the coherence period (partially delayed current CSIT), or at any time after the end of the coherence period (delayed and potentially obsolete CSIT) 4. Such gradual improvement could be sought in FDD (frequency division duplex) settings with limited-capacity feedback links that can be used more than once during the coherence period to progressively refine CSIT, as well as in TDD (time division duplex) settings that use reciprocity-based estimation that progressively improves over time.

Table of contents :

Acknowledgements
Abstract
Résumé
Contents
List of Figures
List of Tables
Acronyms
Notations
1 Introduction 
1.1 Channel model
1.1.1 MISO BC channel model
1.1.2 MIMO BC channel model
1.1.3 MIMO IC channel model
1.2 Degrees-of-freedom
1.3 Delay-and-quality effects of feedback
1.4 Channel and CSIT feedback process
1.5 Early, current, and delayed CSIT
1.6 Examples
1.7 Diversity
1.8 Global CSIR
1.9 Contributions and outline of the thesis
2 DoF and Feedback Tradeoff over Two-User MISO BC 
2.1 Introduction
2.1.1 Channel model
2.1.2 Delay-and-quality effects of feedback
2.1.3 Channel and feedback process
2.1.4 Notation, conventions and assumptions
2.1.5 Prior work
2.1.6 Structure
2.2 DoF region of the MISO BC
2.3 Periodically evolving CSIT
2.4 Universal encoding-decoding scheme
2.4.1 Scheme : Encoding
2.4.2 Scheme : Decoding
2.4.3 Scheme : Calculating the achieved DoF
2.4.4 Scheme : Examples
2.5 Conclusions
2.6 Appendix – Proof of outer bound Lemma
2.7 Appendix – Further details on the scheme
2.7.1 Explicit power allocation solutions
2.7.2 Encoding and decoding details for equations (2.64),(2.66)
2.8 Appendix – Discussion on estimates and errors assumption
2.9 Appendix – Another Outer Bound Proof
3 DoF and Feedback Tradeoff over MIMO BC and IC 
3.1 Introduction
3.1.1 MIMO BC and MIMO IC channel models
3.1.2 Degrees-of-freedom as a function of feedback quality
3.1.3 Predicted, current and delayed CSIT
3.1.4 Notation, conventions and assumptions
3.1.5 Existing results directly relating to the current work
3.2 DoF region of the MIMO BC and MIMO IC
3.2.1 Imperfect current CSIT vs. perfect current CSIT
3.2.2 Imperfect delayed CSIT vs. perfect delayed CSIT
3.3 Outer bound proof
3.3.1 Outer bound proof for the BC
3.3.2 Outer bound proof for the IC
3.4 Phase-Markov transceiver for imperfect and delayed feedback
3.4.1 Encoding
3.4.2 Decoding
3.4.3 Calibrating the scheme to achieve DoF corner points
3.4.4 Modifications for the IC
3.5 Conclusions
4 DoF and Feedback Tradeoff over K-User MISO BC 
4.1 Introduction
4.1.1 CSIT quantification and feedback model
4.1.2 Structure and summary of contributions
4.2 Main results
4.2.1 Outer bounds
4.2.2 Optimal cases of DoF characterizations
4.2.3 Inner bounds
4.3 Converse proof of Theorem 3
4.4 Details of achievability proofs
4.4.1 Achievability proof of Theorem 6
4.4.2 Achievability proof of Theorem 5
4.4.3 Proof of Proposition 3
4.4.4 Proof of Proposition 4
4.4.5 Proof of Proposition 5
4.5 Conclusions
4.6 Appendix – Proof details of Proposition
4.6.1 Proof of Lemma 4
4.6.2 Proof of Lemma 5
4.6.3 Proof of Lemma 6
4.6.4 Proof of Proposition 6
5 On the Imperfect Global CSIR and Diversity Aspects 
5.1 On the Imperfect Global CSIR Aspect
5.1.1 Introduction
5.1.2 Related work
5.1.3 Quantification of CSI and CSIR quality
5.1.4 Conventions and structure
5.1.5 Main results
5.1.6 Scheme
5.1.7 Conclusions
5.2 Diversity
5.2.1 Introduction
5.2.2 Outline
5.2.3 System model
5.2.4 Original MAT scheme
5.2.5 Interference alignment scheme
5.2.6 Diversity analysis of the proposed scheme
5.2.7 Conclusions
5.2.8 Appendix – Proof of Proposition
6 Conclusions and Future Work 
7 French Summary 
7.1 Modèle Canal
7.1.1 MISO BC
7.1.2 MIMO BC
7.1.3 MIMO IC
7.2 Degrés de liberté
7.3 Effets et de qualité de retard de rétroaction
7.4 Manche et processus de rétroaction
7.5 Début, le courant et différé CSIT
7.6 Exemples
7.7 Diversité
7.8 Global CSIR
7.9 Les contributions et les grandes lignes de la thèse
7.10 Résumé du chapitre
7.10.1 Modèle canal
7.10.2 Processus de canal et de feedback
7.10.3 Notation, conventions et hypothèses
7.10.4 Région DoF des deux-utilisateur MISO BC
7.10.5 CSIT evoluant periodiquement
7.10.6 Généralisation des paramètres existants .

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