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Communications over the broadcast channel with limited and delayed feedback : Fundamental limits and novel encoders and decoders Jinyuan Chen
To cite this version: Jinyuan Chen. Communications over the broadcast channel with limited and delayed feedback : Fundamental limits and novel encoders and decoders. Networking and Internet Architecture [cs.NI]. T´el´ecom ParisTech, 2013. English. .
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2012-ENST-0035
EDITE - ED 130
Doctor of Philosophy ParisTech DISSERTATION In Partial Fulfi llment of the Requirements for the Degree of Doctor of Philosophy from
TELECOM ParisTech Specialization « Electronics and Communications » publicly presented and defended by
Jinyuan Chen on 21 June 2013
Communications over the Broadcast Channel with Limited and Delayed Feedback : Fundamental Limits and Novel Encoders and Decoders Thesis Advisors :
Petros ELIA Raymond KNOPP
Jury Dirk SLOCK, Professor, EURECOM, France Daniela TUNINETTI, Professor, University of Illinois in Chicago, US Mari KOBAYASHI, Assistant Prof., SUPELEC, France Syed Ali JAFAR, Professor, University of California, Irvine, US Sheng YANG, Assistant Prof., SUPELEC, France David GESBERT, Professor, EURECOM, France TELECOM ParisTech An Institute Telecom School - Member of ParisTech
President Reviewer Reviewer Examiner Examiner Examiner
Acknowledgement s It is my pleasure t o express my t hanks t o all t he people who cont ribut ed in many ways t o t he success of t his dissert at ion. Many t hanks must begin wit h my advisers, Prof. Pet ros Elia and Prof. Raymond K nopp. T his dissert at ion would not have been possible wit hout t heir guidance, support and encouragement . My first and sincere appreciat ion goes t o Prof. Pet ros Elia. I appreciat e his cont inuing int eract ions and helps, in all st ages of t his t hesis, which made my Ph.D. experience product ive and st imulat ing. I remember t hat he used t o work on a paper wit h me overnight in t he offi ce, and t hat he used t o discuss on a paper wit h me in a McDonald’s st ore during t he weekend. His at t it ude and ent husiasm t o research inspired me t o become a bet t er researcher. I am also t hankful t o Prof. Raymond K nopp, whose advices and support s were invaluable t o me. In addit ion, I would like t o t hank Prof. Sheng Yang and Prof. Syed Ali Jafar for t he insight ful t echnical discussions on my t opic, as well as my t hesis jury members Prof. Daniela Tuninet t i, Prof. Mari Kobayashi, Prof. Dirk Slock and Prof. David Gesbert for t heir valuable comment s and encouragement . My warm and sincere t hanks now goes t o all of my friends. I would like t o t hank Arun Singh and Erhan Yilmaz for t heir helps and suggest ions. I would also like t o t hank my offi ce-mat es Paul de K erret and Amélie Gyrard ; my lunch-mat es Fidan Mehmet i, Rajeev Gangula, Robin T homas, NgocDuy Nguyen, T ien-T hinh Nguyen, Ankit Bhamri and Rui Pedro FerreiraDa-Cost a ; as well as Tania Villa, Imran Latif, Siouar Bensaid, Milt iades Filippou, Fat ma Hrizi, Maha Alodeh, Haiyong Jiang, Xueliang Liu, Jinbang Chen, Rui Min, Xuran Zhao, Lusheng Wang, K aijie Zhou, Shengyun Liu, Xinping Yi, Xiaohu Wu and ot her friends. During t he last t hree years, I have had lot s of good memories t oget her wit h t hem. Last but not least , I wish t o express my deepest grat it ude t o my family for t heir love and support t hroughout my life. I would like t o give a special t hanks t o my wife, Haiwen Wang, who has been at my side and has support ed me in so many ways. Also, I would like t o t hank my lovely son, Lucas Chen, who is an excit ing and special result for us, and who is bringing us happiness and hopes.
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A bst ract In many mult iuser wireless communicat ions scenarios, good feedback is a crucial ingredient t hat facilit at es improved performance. While being useful, perfect feedback is also hard and t ime-consuming t o obt ain. Wit h t his challenge as a st art ing point , t he main work of t he t hesis seeks t o address t he simple yet elusive and fundament al quest ion of “HOW MUCH QUALITY of feedback, AND WHEN, must one send t o achieve a cert ain degreesof-freedom (DoF) performance in specific set t ings of mult iuser communicat ions”. Emphasis is first placed on communicat ions over t he two-user mult ipleinput single-out put (MISO) broadcast channel (BC) wit h imperfect and delayed channel st at e informat ion at t he t ransmit t er (CSIT ) ; a set t ing for which t he work explores t he t radeoff between performance on t he one hand, and CSIT timeliness and accuracy on t he ot her hand. T he work considers a broad set t ing where communicat ion t akes place in t he presence of a random fading process, and in t he presence of a feedback process t hat , at any point in time, may or may not provide CSIT est imates - of some arbit rary quality for any past , current or fut ure channel realizat ion. T his feedback quality may fluct uat e in t ime across all ranges of CSIT accuracy and t imeliness, ranging from perfect ly accurat e and inst ant aneously available est imat es, t o delayed est imat es of minimal accuracy. Under st andard assumpt ions, t he work derives t he DoF region, which is t ight for a large range of CSIT quality. T his derived DoF region concisely capt ures t he eff ect of channel correlat ions, t he accuracy of predict ed, current , and delayed-CSIT , and generally capt ures t he eff ect of t he quality of CSIT off ered at any t ime, about any channel. T he work also int roduces novel schemes which - in t he cont ext of imperfect and delayed CSIT - employ encoding and decoding wit h a phase-Markov st ruct ure. T he result s hold for a large class of block and non-block fading channel models, and t hey unify and ext end many prior at t empt s t o capt ure t he eff ect of imperfect and delayed feedback. T his generality also allows for considerat ion of novel pert inent set t ings, such as t he new periodically evolving feedback set t ing, where a gradual accumulat ion of feedback bit s progressively improves CSIT as t ime progresses across a finit e coherence period. T he above result s are achieved for t he two-user MISO-BC, and are t hen immediat ely ext ended t o t he two-user mult iple-input mult iple-out put (MIMO) iii
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BC and MIMO int erference channel (MIMO IC) set t ings, again in t he presence of a random fading process, and in t he presence of a feedback process t hat , at any point in t ime, may or may not provide CSIT est imat es - of some arbit rary quality - for any past , current or fut ure channel realizat ion. Under st andard assumpt ions, and in t he presence of M ant ennas per t ransmit t er and N ant ennas per receiver, t he work derives t he DoF region, which is opt imal for a large regime of CSIT quality. In addit ion t o t he progress t owards describing t he limit s of using such imperfect and delayed feedback in MIMO set t ings, t he work off ers diff erent insight s t hat include t he fact t hat , an increasing number of receive ant ennas can allow for reduced quality feedback, as well as t hat no CSIT is needed for t he direct links in t he IC. T hen, t he work considers t he more general set t ing of t he K -user MISO broadcast channel, where a t ransmitt er wit h M ant ennas t ransmit s informat ion t o K single-ant enna users, and where again, t he quality and t imeliness of CSIT is imperfect . In t his mult iuser set t ing, t he work est ablishes bounds on t he t radeoff between DoF performance and CSIT feedback quality. Specifically, t his work provides a novel DoF region out er bound for t he general K -user M ⇥ 1 MISO BC wit h imperfect quality current -CSIT , which nat urally bridges t he gap between t he case of having no current CSIT (only delayed CSIT , or no CSIT ) and t he case wit h full CSIT . In t his set t ing, t he work t hen charact erizes t he minimum current CSIT feedback t hat is necessary t o achieve any sum DoF point . T his charact erizat ion is opt imal for t he case where M K , and t he case where M = 2, K = 3. Moving t owards a diff erent direct ion, t he work also considers anot her aspect of communicat ing wit h imperfect and delayed feedback, i.e., t he aspect of having addit ional imperfect ion on t he receiver est imat es of t he channel of t he ot her receiver (global CSIR), in addit ion t o t he imperfect ion on t he CSIT . T he work focuses on a MIMO broadcast channel wit h fixed-quality imperfect delayed CSIT and imperfect delayed global CSIR, and proceeds t o present schemes and DoF bounds t hat are oft en t ight , and t o const ruct ively reveal how even subst ant ially imperfect delayed-CSIT and subst ant ially imperfect delayed-global CSIR, are in fact suffi cient t o achieve t he opt imal DoF performance previously associat ed t o perfect delayed CSIT and perfect global CSIR. Moving t owards one anot her diff erent direct ion, t he work also considers t he diversity aspect of communicat ing - over t he two-user MISO BC - wit h delayed CSIT . In t his set t ing, t he work proposes a novel broadcast scheme which employs delayed CSIT and a form of int erference alignment t o achieve bot h t he maximum possible DoF (2/ 3) as well as full diversity. In addit ion t o t he t heoret ical limit s and novel encoders and decoders, t he work applies t owards gaining insight s on pract ical quest ions on t opics relat ing t o how much feedback quality (delayed, current or predict ed) allows for a cert ain DoF performance, relating t o t he usefulness of delayed feedback, t he usefulness of predict ed CSIT , t he impact of imperfect ions in t he quality
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of current and delayed CSIT , t he impact of feedback t imeliness and t he eff ect of feedback delays, t he benefit of having feedback symmet ry by employing comparable feedback links across users, t he impact of imperfect ions in t he quality of global CSIR, and relat ing t o how t o achieve bot h full DoF and full diversity.
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Résumé Dans des nombreux scénarios de communicat ion sans fil mult i-ut ilisat eurs, une bonne rét roact ion est un ingrédient essent iel qui facilit e l’ameliorat ion des performances. Bien qu’ét ant ut ile, une rétroact ion parfait e rest e diffi cile et fast idieuse à obt enir. En considérant ce défi comme point de départ , les présent s t ravaux cherchent à adresser la quest ion simple et pourt ant insaisissable et fondament ale suivant e : “ Quel niveau de qualit é de la rét roact ion doit -on rechercher, et à quel moment faut -il envoyer pour at t eindre une cert aine performance en degrés de libert é (DoF en anglais) avec des paramèt res spécifiques de communicat ions mult i-ut ilisat eurs”. L’accent est t out d’abord mis sur les communicat ions à t ravers un canal de diff usion (BC en anglais) à deux ut ilisat eurs, mult i-ent rées, unique sort ie (MISO en anglais) avec informat ions imparfait es et ret ardées à l’émet t eur de l’ét at du canal (CSIT en anglais), un paramèt re pour lequel la présent e t hèse explore le compromis ent re la performance et la rapidit é et la qualit é de la rét roact ion. La présent e ét ude considère un cadre général dans lequel la communicat ion a lieu en présence d’un processus d’at t enuat ion aléat oire, et en présence d’un processus de rét roact ion qui, à t out moment , peut ou non fournir des est imat ions CSIT - d’une qualit é arbit raire - pour t out e réalisat ion passée, act uelle ou fut ure du canal. Sous des hypot hèses st andard, dans cet t e t hèse est dérivée la région DoF qui est opt imale pour un large régime de qualit é CSIT . Cet t e région capt ure de manière concise l’eff et des corrélat ions de canaux, la qualit é de la valeur prédit e, la valeur courant e et ret ardée du CSIT , et capt ure généralement l’eff et de la qualit é du CSIT fourni à n’import e quel moment , sur n’importe quel canal. Les encadrement s sont obt enus à l’aide de nouveaux schémas qui - dans le cont ext e de CSIT imparfait et ret ardé - sont présent és ici pour la première fois, avec encodage et décodage sur st ruct ure de phase Markovienne. Les résultat s sont validés pour une grande classe de modèles de canaux d’at t énuat ion en bloc et non-bloc, et ils unifient et ét endent de nombreuses t ent at ives ant érieures de capt ure de l’eff et de rét roact ion imparfait e et ret ardée. Cet t e généralit é permet également d’examiner de nouveaux paramèt res pert inent s, t els que la nouvelle t echnique de rét roact ion à évolut ion périodique, où une accumulat ion progressive de bit s de rét roact ion améliore progressivement le CSIT avec le t emps, ce dernier progressant à t ravers une période de cohérence vii
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finie. Les résult at s ci-dessus sont at teint s dans le cas du MISO-BC à deux ut ilisat eurs, et sont ensuit e immédiat ement ét endus aux cas ent rées mult iples sort ies mult iples (MIMO) BC à deux ut ilisat eurs et de canaux d’int erférence MIMO (MIMO IC), encore une fois en présence de processus d’at t énuat ion aléat oire, et en présence d’un processus de rét roact ion qui, à t out moment , peut ou non fournir des est imat ions CSIT - d’une qualit é arbit raire - pour t out e réalisat ion passée, act uelle ou fut ure du canal. Sous les hypot hèses st andard, et en présence de M ant ennes par émet t eur et N ant ennes par récept eur, la région DoF, qui est opt imale pour un large régime de qualit é CSIT , est dérivée. En plus de présent er une avancée dans la descript ion des limit es d’ut ilisat ion de t elles informat ions imparfait es et ret ardées dans les milieux MIMO, la present e ét ude propose diff érent es idées dont le fait qu’un nombre croissant d’ant ennes de récept ion peuvent causer la réduct ion de la qualit é de la rét roact ion, ainsi que le fait qu’aucun CSIT n’est requis pour les liens direct s dans l’IC. Dans un deuxieme t emps, la t hèse considère le cadre plus général de la chaîne de diff usion K -ut ilisat eurs MISO, où un émet t eur avec M ant ennes t ransmet des informat ions aux K ut ilisat eurs à une seule ant enne, et où, une nouvelle fois, la qualit é et la rapidit é du CSIT est imparfait e. Dans ce cont ext e mult i-ut ilisat eurs, la t hèse ét ablit des limit es sur le compromis ent re la performance du DoF et la qualit é de la rét roact ion CSIT . Plus précisément , elle fournit une nouvelle région de la borne ext erne DoF dans le cas general à K - ut ilisat eurs MISO BC avec un CSIT à qualit é imparfait e courant e, ce qui nat urellement comble le lien entre le cas sans CSIT courant (ou le CSIT est seulement ret ardé, ou sans CSIT ) et le cas où l’on dispose d’un CSIT complet . Dans ce cont ext e, la présent e ét ude caract érise alors la rét roact ion CSIT à courant minimum nécessaire pour at t eindre n’import e quel ét at de somme DoF. Cet t e caract érisat ion est opt imale dans le cas où M K , et le cas M = 2, K = 3. Dans une aut re perspect ive, l’ét ude considère également un aut re aspect de la communicat ion à rét roact ion imparfait e et ret ardée : celui où une imperfect ion supplément aire sur les est imat ions du récept eur sur le canal d’un aut re récept eur (CSIR global) est présent e, en plus de l’ imperfect ion du CSIT . L’ét ude se concent re sur un canal de diff usion MIMO avec un CSIT de qualit é donnée, imparfait et ret ardé, et un CSIR global imparfait et ret ardé. Et des schémas, ainsi que des encadrement s des DoFs, souvent précises, sont présent és. L’ét ude cont inue ensuit e en révélant de manière const ruct ive comment même des CSIT sensiblement imparfait s ret ardés et des CSIR globaux essent iellement imparfait s ret ardés sont en fait suffi sant s pour at t eindre la performance opt imale en DoF qui ét ait précédemment associée au CSIT parfait ret ardé et au CSIR parfait global. En s’avent urant plus loin encore, l’ét ude t ient également compt e de l’aspect diversit é de la communicat ion - ent re deux ut ilisat eurs MISO BC -
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avec CSIT ret ardé. Dans ce cadre, l’ouvrage propose un nouveau syst ème de diff usion qui fait appel au CSIT ret ardé et à une forme d’ alignement d’int erférence pour obt enir à la fois le maximum DoF possible (2/ 3), ainsi qu’une diversit é complèt e. En plus de fournir des limit es t héoriques et des nouveaux encodeurs et décodeurs, l’ét ude s’applique à obt enir une meilleure comprehension sur des quest ions prat iques relat ives à combien la qualit é de rét roact ion (diff érée, en cours ou prévue) influe sur les performances DoF. Sur des quest ions relat ives à l’ut ilit é de la rét roact ion ret ardée, l’ut ilit é du CSIT prédit , à l’impact des imperfect ions s
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Table of Cont ent s Acknowledgement s Abst ract . . . . . . Résumé . . . . . . Cont ent s . . . . . . List of Figures . . List of Tables . . . Acronyms . . . . . Not at ions . . . . .
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1 I nt r oduct ion 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 eff ect s 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 Cont ribut ions and out line of t he t hesis
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2 D oF and Feedback Tr adeoff over Two-U ser M I SO 2.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Channel model . . . . . . . . . . . . . . . . 2.1.2 Delay-and-quality eff ect s of feedback . . . . 2.1.3 Channel and feedback process . . . . . . . . 2.1.4 Not at ion, convent ions and assumpt ions . . . 2.1.5 Prior work . . . . . . . . . . . . . . . . . . . 2.1.6 St ruct ure . . . . . . . . . . . . . . . . . . . 2.2 DoF region of t he MISO BC . . . . . . . . . . . . . 2.3 Periodically evolving CSIT . . . . . . . . . . . . . . 2.4 Universal encoding-decoding scheme . . . . . . . . xi
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Table of C ont ent s
2.5 2.6 2.7
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2.4.1 Scheme : Encoding . . . . . . . . . . . . . . . . . . . . 2.4.2 Scheme : Decoding . . . . . . . . . . . . . . . . . . . . 2.4.3 Scheme : Calculat ing t he achieved DoF . . . . . . . . . 2.4.4 Scheme : Examples . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix - Proof of out er bound Lemma . . . . . . . . . . . Appendix - Furt her det ails on t he scheme . . . . . . . . . . . 2.7.1 Explicit power allocat ion solut ions . . . . . . . . . . . 2.7.2 Encoding and decoding det ails for equat ions (2.64),(2.66) Appendix - Discussion on est imat es and errors assumpt ion . . Appendix - Anot her Out er Bound Proof . . . . . . . . . . . .
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3 D oF and Feedback Tr adeoff over M I M O B C and I C 67 3.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.1 MIMO BC and MIMO IC channel models . . . . . . . 67 3.1.2 Degrees-of-freedom as a funct ion of feedback quality . 68 3.1.3 Predict ed, current and delayed CSIT . . . . . . . . . . 69 3.1.4 Not at ion, convent ions and assumpt ions . . . . . . . . . 70 3.1.5 Exist ing result s direct ly relat ing t o t he current work . 71 3.2 DoF region of t he MIMO BC and MIMO IC . . . . . . . . . . 72 3.2.1 Imperfect current CSIT vs. perfect current CSIT . . . 75 3.2.2 Imperfect delayed CSIT vs. perfect delayed CSIT . . . 75 3.3 Out er bound proof . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.1 Out er bound proof for t he BC . . . . . . . . . . . . . . 76 3.3.2 Out er bound proof for t he IC . . . . . . . . . . . . . . 78 3.4 Phase-Markov t ransceiver for imperfect and delayed feedback 78 3.4.1 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.2 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4.3 Calibrat ing t he scheme t o achieve DoF corner point s . 84 3.4.4 Modificat ions for t he IC . . . . . . . . . . . . . . . . . 89 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 D oF and Feedback Tr adeoff over K -U ser M I SO B C 4.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 CSIT quant ificat ion and feedback model . . . 4.1.2 St ruct ure and summary of cont ribut ions . . . 4.2 Main result s . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Out er bounds . . . . . . . . . . . . . . . . . . 4.2.2 Opt imal cases of DoF charact erizat ions . . . 4.2.3 Inner bounds . . . . . . . . . . . . . . . . . . 4.3 Converse proof of T heorem 3 . . . . . . . . . . . . . 4.4 Det ails of achievability proofs . . . . . . . . . . . . . 4.4.1 Achievability proof of T heorem 6 . . . . . . . 4.4.2 Achievability proof of T heorem 5 . . . . . . .
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T abl e of C ont ent s
4.4.3 Proof of Proposit ion 3 . . . . . . . 4.4.4 Proof of Proposit ion 4 . . . . . . . 4.4.5 Proof of Proposit ion 5 . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . 4.6 Appendix - Proof det ails of Proposit ion 6 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 Proposit ion 6 . . . . . . .
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5 On t he I m p er fect G lobal C SI R and D iver sit y A sp ect s 5.1 On t he Imperfect Global CSIR Aspect . . . . . . . . . . 5.1.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . 5.1.2 Relat ed work . . . . . . . . . . . . . . . . . . . . 5.1.3 Quant ificat ion of CSI and CSIR quality . . . . . 5.1.4 Convent ions and st ruct ure . . . . . . . . . . . . . 5.1.5 Main result s . . . . . . . . . . . . . . . . . . . . . 5.1.6 Scheme . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . 5.2 Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . 5.2.2 Out line . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Syst em model . . . . . . . . . . . . . . . . . . . . 5.2.4 Original MAT scheme . . . . . . . . . . . . . . . 5.2.5 Int erference alignment scheme . . . . . . . . . . . 5.2.6 Diversity analysis of t he proposed scheme . . . . 5.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . 5.2.8 Appendix - Proof of Proposit ion 7 . . . . . . . .
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108 109 110 112 113 113 116 118 118
. . . . . . . . . . . . . . . . .
123 124 124 124 125 126 126 128 132 133 133 133 134 135 136 138 140 140
6 C onclusions and Fut ur e W or k
143
7 Fr ench Sum m ar y 7.1 Modèle Canal . . . . . . . . . . . . . . . . . . . . 7.1.1 MISO BC . . . . . . . . . . . . . . . . . . 7.1.2 MIMO BC . . . . . . . . . . . . . . . . . 7.1.3 MIMO IC . . . . . . . . . . . . . . . . . . 7.2 Degrés de libert é . . . . . . . . . . . . . . . . . . 7.3 Eff et s et de qualit é de ret ard de rét roact ion . . . 7.4 Manche et processus de rét roact ion . . . . . . . . 7.5 Début , le courant et diff éré CSIT . . . . . . . . . 7.6 Exemples . . . . . . . . . . . . . . . . . . . . . . 7.7 Diversit é . . . . . . . . . . . . . . . . . . . . . . . 7.8 Global CSIR . . . . . . . . . . . . . . . . . . . . 7.9 Les cont ribut ions et les grandes lignes de la t hèse
145 147 147 149 149 149 150 152 152 152 153 153 153
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
x iv
Table of C ont ent s
7.10 Résumé du chapit re 2 . . . . . . . . . . . . . . . . 7.10.1 Modèle canal . . . . . . . . . . . . . . . . . 7.10.2 Processus de canal et de feedback . . . . . . 7.10.3 Not at ion, convent ions et hypot hèses . . . . 7.10.4 Région DoF des deux-ut ilisat eur MISO BC 7.10.5 CSIT evoluant periodiquement . . . . . . . 7.10.6 Généralisat ion des paramèt res exist ant s . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
157 157 157 157 159 164 168
List of Figures 1.1 Syst em model of K -user MISO BC wit h CSIT feedback. . . . 1.2 Syst em model of two-user int erference channel. . . . . . . . . 2.1 2.2 2.3 2.4 2.5 2.6
. . . . . .
22 22 23 36 36 41
3.1 Opt imal DoF regions for two-user MIMO BC and MIMO IC .
74
4.1 4.2 4.3 4.4 4.5 4.6
DoF region inner bound for BC. . . . . . . . . . . . . . . . Opt imal DoF region for BC. . . . . . . . . . . . . . . . . . DoF region of two-user MISO BC wit h symmet ric feedback Illust rat ion of coding across phases. . . . . . . . . . . . . . . Illust rat ion of coding over a single phase. . . . . . . . . . . . Illust rat ion of decoding st eps. . . . . . . . . . . . . . . . . .
2 2
Syst em model of K -user MISO BC wit h CSIT feedback. . . . 92 Opt imal sum DoF d⌃ vs. P for t he MISO BC wit h M K . 98 Opt imal sum DoF d⌃ vs. CP for 3-user 2 ⇥1 MISO BC. . . . 98 Achievable sum DoF d⌃ vs. P for K ( 3)-user 2⇥1 MISO BC.101 Achievable sum DoF d⌃ vs. P for t he MISO BC wit h M < K . 101 3, M = 2, P = 0 . . . . . . 102 d⌃ vs. D for MISO BC wit h K
5.1 DoF region of MIMO BC wit h imperfect CSIT and CSIR . . 127 5.2 Int erference alignment scheme illust rat ion. . . . . . . . . . . . 138 5.3 DMT upper bound for t he MAT scheme. . . . . . . . . . . . . 142 7.1 7.2 7.3 7.4 7.5
Modèle de syst ème de K -ut ilisat eur MISO BC. . Modèle de syst ème de canal d’int erfŕence de deux Région DoF opt imale pour BC. . . . . . . . . . . DoF région int érieure liée à BC. . . . . . . . . . Région DoF MISO BC avec feedback symét rique
xv
. . . . . . . ut ilisat eurs. . . . . . . . . . . . . . . . . . . . . .
148 148 161 161 162
xvi
L ist of F i gur es
List of Tables 2.1 Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s, s = 1, 2, · · · , S 1. . . . . 2.2 Opt imal corner point s summary, for suffi cient ly good delayed ( 1) ( 2) ( 2) CSIT such t hat min{ ¯ (1) , ¯ (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . . . 2.3 DoF inner bound corner point s, for delayed CSIT such t hat ( 1) ( 2) ( 2) min{ ¯ (1) , ¯ (2) } < min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . . . . . . . . . . . . 2.4 Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s = 1, 2, · · · , S 1. . . . . . 2.5 Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s, s = 1, 2, · · · , S 1, of t he alt ernat ing CSIT scheme. . . . . . . . . . . . . . . . . . . . . 3.1 Number of bit s carried by privat e and common symbols, and by t he quant ized int erference (phase s). . . . . . . . . . . . . 3.2 Out er bound corner point s. . . . . . . . . . . . . . . . . . . .
39 46 46 48
50 82 85
P = 2, C? = 2, M = 2, K = 3 . . . . . 107 4.1 Scheme summary for d⇤ P 2 4.2 Scheme summary for d⌃ = 43 , D = 3K . . . . . . . . . . . . . 110 3 9 4.3 Scheme summary for d⌃ = 2 , D = 8K . . . . . . . . . . . . . . 111
5.1 Number of bit s carried by privat e and common symbols, and by t he quant ized int erference (t ime t). . . . . . . . . . . . . . 130
xvii
x v i ii
L i st of T abl es
Acronyms We provide here t he main acronyms used in t his dissert at ion. T he meaning of an acronym is usually indicat ed once, when it first occurs in t he t ext . T he English acronyms are also used for t he French summary. AWGN BC CSI CSIR CSIT cf. DMT DoF e.g. et al. FDD FDMA QAM IC i.i.d. i.e. MAC ML MMSE MIMO MISO pdf resp. SNR SISO T DD ZF
Addit ive Whit e Gaussian Noise Broadcast Channel Channel St at e Informat ion Channel St at e Informat ion at t he Receiver Channel St at e Informat ion at t he Transmit t er Confer (compare t o, see also) Diversity-Mult iplexing Tradeoff Degrees-of-Freedom exempli grat ia (for t he sake of example) et alii, et alia (and ot hers) Frequency Division Duplex Frequency Division Mult iple Access Quadrat ure Amplit ude Modulat ion Int erference Channel Independent and Ident ically Dist ribut ed id est (t hat is) Mult iple Access Channel Maximum Likelihood Minimum Mean Square Error Mult i-Input Mult i-Out put Mult i-Input Single-Out put probability density funct ion Respect ively Signal-t o-Noise Rat io Single-Input Single-Out put T ime Division Duplex Zero-Forcing
xix
xx
A cr ony m s
Not at ions We summarize here t he symbols and not at ions t hat are commonly used in t his dissert at ion, wit h t he rest not at ions defined in t he t ext where t hey occur. Specifically, we use t he bold uppercase lett ers, e.g., H , t o refer t o mat rices, while bold lowercase let t ers, e.g., h , t o refer t o column vect ors. We . . use = t o denot e exponential equality, i.e., we writ e f (P ) = P B t o denot e . . log f (P ) lim = B . Similarly and denot e exponent ial inequalit ies. P! 1 log P o(• ) and O(• ) come from t he st andard Landau not at ion, where f (x) = o(g(x)) implies limx ! 1 f (x)/ g(x) = 0, wit h f (x) = O(g(x)) implying t hat lim supx ! 1 |f (x)/ g(x)| < 1 . We also use A ⌫0 t o denot e t hat A is posit ive semidefinit e, and use A B t o mean t hat B A ⌫0. e? denot es a unit -norm vect or ort hogonal t o e. Logarit hms are of base 2. Ot her not at ional convent ions are summarized as follows : R, C R+ |• | (• ) + ||x || x? X? diag(x ) CN(µ,
2)
EX [• ] X⇤ XH XT ||X ||F rank(X ) t r(X ) det (X )
T he set s of real and complex numbers, respect ively T he set of posit ive real numbers Eit her t he magnit ude of a scalar or t he cardinality of a set T he operat ion max(0, • ) T he Euclidian (l 2 ) norm of a vect or A unit -norm vect or ort hogonal to vect or x A unit -norm mat rix ort hogonal t o mat rix X T he diagonal mat rix whose diagonal ent ries are represent ed by t he element s of a vect or x in order T he circularly symmet ric complex Gaussian dist ribut ion wit h mean µ and variance 2 T he expect at ion operat or over the random variable X T he conjugat e operat ion on X T he complex conjugat e (Hermitian) operat ion on X T he t ranspose operat ion on X T he Frobenius norm of X T he rank of X T he t race of X T he det erminant of X xxi
xxii
X Im 0
N ot at i ons 1
T he inverse of X T he m ⇥m ident ity mat rix T he all-zeros mat rix of appropriat e dimensions
Chapt er 1
I nt roduct ion
In many mult iuser wireless communicat ions scenarios, t he capacity performance depends heavily on t he t imeliness and quality of channel st at e informat ion at t he t ransmit t er (CSIT ). T his timeliness and quality of CSIT t hough may be reduced, due t o t he t ime-varying nat ure of wireless fading channel, as well as limit ed-capacity feedback links. Wit h t his challenge as a st art ing point , t he main work of t he t hesis seeks t o address t he simple yet elusive and fundament al quest ion of “How much quality of feedback, and when, must one send t o achieve a cert ain performance in specific set t ings of mult iuser communicat ions”.
1.1
Channel model
Wit h t he rapid growt h of wireless connect ivity, mult iuser communicat ions have received a t remendous amount of int erest in t he lit erat ure. In t he cont ext of mult iuser communicat ions scenarios (see for example in Fig 1.1 and in Fig 1.2 ), we will consider mult iple-input single-out put broadcast channel (MISO BC), mult iple-input mult iple-out put BC (MIMO BC), as well as MIMO int erference channel (MIMO IC).
1.1.1
M I SO B C channel m odel
We first focus on t he mult iuser broadcast channel wit h a t ransmit t er communicat ing t o K receiving users, in t he presence of imperfect and delayed CSIT feedback. We begin wit h two user MISO BC where an M -t ransmit ant enna t ransmitt er communicat es t o two (K = 2) single-ant enna users. For h t and gt 1
2
C hapt er 1
I nt r oduct ion
Feedback
Channel 1 User 1
Tx
Channel K Feedback
User K
Figur e 1.1 – Syst em model of K -user MISO BC wit h CSIT feedback. Channel 11
l 12 nne Ch a
Tx 1
Cha nne l
Rx 1
21
Channel 22 Tx 2
Rx 2
Figur e 1.2 – Syst em model of two-user int erference channel. denot ing t his channel at t ime t for t he first and second user respect ively, and for x t denot ing t he t ransmit t ed vect or at t ime t, t he corresponding received signals at t he first and second user t ake t he form (1)
= h Tt x t + zt
(2)
= gTt x t + zt
yt yt (1)
(1)
(1.1)
(2)
(1.2)
(2)
(t = 1, 2, · · · ), where zt , zt denot e t he unit power AWGN noise at t he receivers. T he above t ransmit vect ors accept a power const raint E[||x t ||2] P = ⇢, for some power P (or ⇢) which also here t akes t he role of t he signalt o-noise rat io (SNR).
1.1.2
M I M O B C channel model
For t he set t ing of t he MIMO BC, we consider t he case where an M ant enna t ransmit t er, sends information t o two receivers wit h N receive an-
1.2
D egr ees-of-fr eedom
3
t ennas each. In t his set t ing, t he received signals at t he two receivers t ake t he form (1)
yt
(2) yt (1)
= H = H
(1) t xt (2) t xt
(1)
(1.3)
(2) zt
(1.4)
+ zt +
(2)
where H t 2 CN ⇥M , H t 2 CN ⇥M respect ively represent t he first and se(1) (2) cond receiver channels at t ime t, where z t , z t represent unit power AWGN noise at t he two receivers, where x t 2 CM ⇥1 is t he input signal wit h power const raint E[||x t ||2 ] P .
1.1.3
M I M O I C channel m odel
For t he set t ing of t he MIMO IC, we consider a case where two t ransmit t ers, each wit h M t ransmit ant ennas, send informat ion t o t heir respect ive receivers, each having N receive ant ennas. In t his set t ing, t he received signals at the two receivers t ake t he form (1)
yt
(2) yt (11)
= H = H
(11) (1) xt t (21) (1) xt t
+ H + H
(12) (2) xt t (22) (2) xt t
(1)
(1.5)
(2) zt
(1.6)
+ zt +
(22)
where H t 2 CN ⇥M , H t 2 CN ⇥M represent t he fading mat rices of t he (12) (21) direct links of t he two pairs, while H t 2 CN ⇥M , H t 2 CN ⇥M , represent t he fading mat rices of t he cross links at t ime t.
1.2
D egr ees-of-fr eedom
T he main work of t hesis will focus on t he degrees-of-freedom (DoF) performance. In t he high-SNR set t ing of int erest, for an achievable rat e t uple (R 1, R 2, · · · , R K ), t he corresponding degrees-of-freedom t uple (d1, d2, · · · , dK ) is given by Ri di = lim , i = 1, 2, · · · , K . P ! 1 log P T he corresponding DoF region D is t hen t he set of all achievable DoF t uples (d1, d2, · · · , dK ). Alt hough t he main work will focus on t he high-SNR regime, t here is subst ant ial evidence t hat , t he high-SNR analysis off ers good insight on t he performance at moderat e SNR regime.
1.3
D elay-and-qualit y eff ect s of feedback
As in many mult iuser wireless communicat ions scenarios, t he performance of t he broadcast channel depends on t he t imeliness and quality of
4
C hapt er 1
I nt r oduct ion
CSIT . T his t imeliness and quality t hough may be reduced by limit ed-capacity feedback links, which may off er consist ent ly low feedback quality, or may offer good quality feedback which t hough comes lat e in t he communicat ion process and can t hus be used for only a fract ion of t he communicat ion durat ion. T he corresponding performance degradat ion, as compared t o t he case of having perfect feedback wit hout delay, forces t he delay-and-quality quest ion of how much feedback is necessary, and when, in order t o achieve a cert ain performance. T hese delay-and-quality eff ect s of feedback, nat urally fall between t he two ext reme cases of no CSIT and of full CSIT (immediat ely available and perfect CSIT ), for t he two-user MISO BC set t ing, wit h full CSIT allowing for t he opt imal 1 degrees-of-freedom per user (cf., [1]), while t he absence of any CSIT reduces t his t o just 1/ 2 DoF per user (cf., [2, 3]). A valuable t ool t owards bridging t his gap and furt her underst anding t he delay-and-quality eff ect s of feedback, came wit h [4] showing t hat arbit rarily delayed feedback can st ill allow for performance improvement over t he noCSIT case. In a set t ing t hat diff erentiat ed between current and delayed CSIT - delayed CSIT being t hat which is available aft er t he channel elapses, i.e., aft er t he end of t he coherence period corresponding t o t he channel described by t his delayed feedback, while current CSIT corresponded t o feedback received during t he channel’s coherence period - t he work in [4] showed t hat perfect delayed CSIT , even wit hout any current CSIT , allows for an improved 2/ 3 DoF per user. Wit hin t he same cont ext of delayed vs. current CSIT , t he work in [5–8] int roduced feedback quality considerat ions, and managed t o quant ify t he usefulness of combining perfect delayed CSIT wit h immediat ely available imperfect CSIT of a cert ain quality t hat remained unchanged t hroughout t he ent ire coherence period. In t his set t ing t he above work showed a furt her bridging of t he gap from 2/ 3 t o 1 DoF (two-user MISO BC case.), as a funct ion of t his current CSIT quality. Furt her progress came wit h t he work in [9] which, in addit ion t o exploring t he eff ect s of t he quality of current CSIT , also considered t he eff ect s of t he quality of delayed CSIT , t hus allowing for considerat ion of t he possibility t hat t he overall number of feedback bit s (corresponding t o delayed plus current CSIT ) may be reduced. Focusing again on t he specific set t ing where t he current CSIT quality remained unchanged for t he ent irety of t he coherence period, t his work revealed among ot her t hings t hat imperfect delayed CSIT can achieve t he same opt imality t hat was previously at t ribut ed t o perfect delayed CSIT , t hus equivalent ly showing how t he amount of delayed feedback required, is proport ional t o t he amount of current feedback. A useful generalizat ion of t he delayed vs. current CSIT paradigm, came wit h t he work in [10] which deviat ed from t he assumpt ion of having invariant CSIT quality t hroughout t he coherence period, and allowed for t he possibility t hat current CSIT may be available only aft er some delay, and
1.4
C hannel and C SI T feedback pr ocess
5
specifically only aft er a cert ain fract ion of t he coherence period. Under t hese assumpt ions, in t he presence of more t han two users, and in t he presence of perfect delayed CSIT , t he above work showed t hat for up t o a cert ain delay, one can achieve t he opt imal performance corresponding t o full (and immediat e) CSIT . Anot her int erest ing generalizat ion came wit h t he work in [11] which, for t he set t ing of t he t ime-select ive two-user MISO BC, t he CSIT for t he channel of user 1 and of user 2, alt ernat e between t he t hree ext reme st at es of perfect current CSIT , perfect delayed CSIT , and no CSIT . T he above set t ings, and many ot her set t ings wit h imperfect and delayed CSIT , such as t hose in [12–32], addressed diff erent inst ances of t he more general problem of communicat ing in t he presence of feedback wit h diff erent delay-and-quality propert ies, wit h each of t hese set t ings being mot ivat ed by t he fact t hat perfect CSIT may be generally hard and t ime-consuming t o obtain, t hat CSIT precision may be improved over t ime, and t hat feedback delays and imperfect ions generally cost in t erms of performance. T he generalizat ion here t o t he set t ing wit h general imperfect , delayed and limit ed CSIT , incorporat es t he above considerat ions and mot ivat ions, and allows for insight on pert inent quest ions such as : – How much CSIT quality (delayed, current or predict ed) allows for a cert ain DoF performance? – Can imperfect delayed CSIT achieve the same opt imality t hat was previously at t ribut ed t o perfect delayed CSIT ? – When is delayed feedback unnecessary ? – Is predict ed CSIT useful in t erms of t he DoF performance? – Can symmet ric feedback off er DoF benefit over t he asymmet ric feedback ? – What is t he impact of imperfect global CSIR (imperfect receiver est imat es of t he channel of t he ot her receiver) ? – Can a communicat ion scheme achieve bot h full diversity and full DoF for t he set t ing wit h only delayed CSIT ?
1.4
Channel and CSI T feedback pr ocess
We will first focus on t he two-user MISO BC, and will consider communicat ion of an infinit e durat ion n (unless wit h specific argument ), a random channel fading process { h t , gt } nt= 1 drawn from a st at ist ical dist ribut ion, and ˆ t ,t 0} nt,t 0= 1 (of channel a feedback process t hat provides CSIT est imates { hˆ t ,t 0, g h t , gt ) at any t ime t 0 - before, during, or after mat erializat ion of h t , gt at t ime t - and does so wit h a cert ain quality { E[||h t
hˆ t ,t 0||2], E[||gt
ˆ t ,t 0||2 ]} nt,t 0= 1 . g
6
1.5
C hapt er 1
I nt r oduct ion
Ear ly, cur r ent , and delayed CSI T
ˆ t ,t } , est imat es For t he channel h t , gt at t ime t, t he est imat es { hˆ t ,t , g ˆ t ,t 0} t 0> t , and est imat es { hˆ t ,t 0, g ˆ t ,t 0} t 0< t form what can be described { hˆ t ,t 0, g as current CSIT est imat es, delayed CSIT est imat es and early CSIT est imat es, respect ively. Unlike t he current CSIT est imat es available at t ime t, t he delayed CSIT est imat es are available at t ime t 0 > t due t o t he delay, while t he early CSIT est imat es are available at t ime t 0 < t at t ribut ed t o predict ion.
1.6
Examples
Here let us consider some examples t hat are incorporat ed and considered in our generalizat ion. Exam ple 1 (Delayed CSIT ). One of the incorporated setting is the delayed CSI T (without any current CSI T ) setting in [4]. Extended from the delayed CSI T setting in [4], our generalization follows and reveals that imperfect delayed CSI T can be as useful as perfect delayed CSI T. Exam ple 2 (Imperfect current and delayed CSIT ). Another incorporated setting is the imperfect current and delayed CSI T setting in [6, 7]. Exam ple 3 (Asymmet ric CSIT ). One another incorporated setting is the asymmetric CSI T setting in [33], where both users off ered perfect delayed CSI T, but where only one user off ered perfect current CSI T. Such asymmetry could reflect feedback links with diff erent capacity or diff erent delays. Exam ple 4 (Not -so-delayed CSIT ). Not-so-delayed CSI T setting in [10] is one of the incorporated setting, which corresponds to the block fading channel with periodic feedback. Exam ple 5 (Alt ernat ing CSIT ). Alternating CSI T setting in [11] is one of the incorporated setting, where CSI T alternates between perfect, delayed and no CSI T states. Exam ple 6 (Evolving CSIT ). One of the novel setting considered is the periodically evolving feedback setting over the quasi-static block fading channel, where a gradual accumulation of feedback bits results in a progressively increasing CSI T quality as time progresses across a finite coherence period. T his powerful setting captures many of the engineering options relating to feedback, as well as captured many interesting settings previously considered.
1.7
1.7
D i ver sit y
7
D iver sit y
In addit on t o t he DoF aspect , t he t hesis also considers t he diversity aspect of communicat ing wit h imperfect and delayed CSIT feedback. For Pe denot ing t he probability t hat at least one user has decoded erroneously, we recall t he not ion of diversity t o be d=
lim
P! 1
log Pe log P
(cf. [34]).
1.8
Global CSI R
In addit ion t o t he challenge of communicating CSIT over feedback channels wit h limit ed capacity and limit ed reliability, anot her known bot t leneck is the non-negligible cost of dist ribut ing global CSIR (receiver est imat es of t he channel of t he ot her receiver) across t he diff erent receiving nodes (see [35], [36]). For t his reason, we explore the case where, in addit ion t o limit ed and imperfect CSIT , we also have t he addit ional imperfect ion of t he global CSIR, which means t hat each user has imperfect est imat es of t he ot her user’s channel, as well as, in t his case, no access t o t he est imat es of t he t ransmit t er.
1.9
Cont r ibut ions and out line of t he t hesis
As st at ed, t he main work of t hesis seeks t o address t he simple yet elusive and fundament al quest ion of “How much quality of feedback, and when, must one send t o achieve a cert ain performance in specific set t ings of mult iuser communicat ions”. In Chapt er 2, t he work considers two-user MISO BC wit h imperfect and delayed CSIT , and explores t he t radeoff between performance, and feedback t imeliness and quality. T he work considers a broad set t ing where communicat ion t akes place in t he presence of a random fading process, and in t he presence of a feedback process t hat , at any point in t ime, may or may not provide CSIT est imat es - of some arbit rary quality - for any past , current or fut ure channel realizat ion. Under st andard assumpt ions, t he work derives t he DoF region, which is opt imal for a large regime of CSIT quality. T his region concisely capt ures t he eff ect of channel correlat ions, t he quality of predict ed, current , and delayed-CSIT , and generally capt ures t he eff ect of t he quality of CSIT off ered at any t ime, about any channel. T he bounds are met wit h novel schemes which - in t he cont ext of imperfect and delayed CSIT - int roduce here for t he first t ime, encoding and decoding wit h a phaseMarkov st ruct ure. T he result s hold for a large class of block and non-block
8
C hapt er 1
I nt r oduct ion
fading channel models, and t hey unify and ext end many prior at t empt s t o capt ure t he eff ect of imperfect and delayed feedback. T his generality also allows for considerat ion of novel pert inent set t ings, such as t he new periodically evolving feedback set t ing, where a gradual accumulat ion of feedback bit s progressively improves CSIT as t ime progresses across a finit e coherence period. T he result s were published in part at – Jinyuan Chen and Pet ros Elia, “C an I m p er fect D elayed C SI T b e as U seful as Per fect D elayed C SI T ? D oF A naly sis and C onst r uct ions for t he B C ”, in Proc. of 50th Annual Allerton Conf. Communication, Control and Computing (Allerton’12), Oct ober 2012. – Jinyuan Chen and Pet ros Elia, “D egr ees-of-Fr eedom R egion of t he M I SO B r oadcast C hannel w it h G ener al M ix ed-C SI T ”, in Proc. I nformation Theory and Applications Workshop (ITA’13), February 2013. – Jinyuan Chen and Pet ros Elia, “M I SO B r oadcast C hannel w it h D elayed and Evolving C SI T ”, in Proc. IEEE I nt. Symp. I nformation T heory (I SI T’13), July 2013. and will be published in part at – Jinyuan Chen and Pet ros Elia, “Towar d t he Per for m ance v s. Feedback Tr adeoff for t he Two-U ser M I SO B r oadcast C hannel”, t o appear in I EEE Trans. I nf. Theory, available on arXiv :1306.1751. – Jinyuan Chen and Pet ros Elia, “Opt im al D oF R egion of t he TwoU ser M I SO-B C w it h G ener al A lt er nat ing C SI T ”, t o appear in Proc. 47th Asilomar Conference on Signals, Systems and Computers (Asilomar’13), 2013, available on arXiv :1303.4352. In Chapt er 3, ext ending t he result s of two-user MISO BC set t ing, t he work explores t he performance of t hetwo user mult iple-input mult iple-out put (MIMO) BC and t he two user MIMO int erference channel (MIMO IC), in t he presence of feedback wit h evolving quality and t imeliness. Under st andard assumpt ions, and in t he presence of M ant ennas per t ransmit t er and N ant ennas per receiver, t he work derives t he DoF region, which is opt imal for a large regime of CSIT quality. T his region concisely capt ures t he eff ect of having predict ed, current and delayed-CSIT , as well as concisely capt ures t he eff ect of t he quality of CSIT off ered at any t ime, about any channel. In addit ion t o t he progress t owards describing t he limit s of using such imperfect and delayed feedback in MIMO set t ings, t he work off ers diff erent insight s t hat include t he fact t hat , an increasing number of receive ant ennas can allow for reduced quality feedback, as well as t hat no CSIT is needed for t he direct links in t he IC. T he result s were published in part at – Jinyuan Chen and Pet ros Elia, “Sym m et r ic Two-U ser M I M O B C and I C w it h Evolving Feedback ”, June 2013, available on arXiv : 1306.3710. – Jinyuan Chen and Pet ros Elia, “M I M O B C w it h I m p er fect and D elayed C hannel St at e I nfor m at ion at t he Tr ansm it t er and
1.9
C ont r ibut ions and out l ine of t he t hesis
9
R eceiver s”, in Proc. I EEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC’13), June 2013. In Chapt er 4, t he work considers t he K -user MISO BC, and est ablishes bounds on t he t radeoff between DoF performance and CSIT feedback quality. Specifically, for t he general MISO BC wit h imperfect current CSIT , a novel DoF region out er bound is provided, which naturally bridges t he gap between t he case of having no current CSIT and t he case wit h full CSIT . T he work t hen charact erizes t he minimum CSIT feedback t hat is necessary for any point of t he sum DoF, which is opt imal for many cases. T he result s will be published in part at – Jinyuan Chen, Sheng Yang, and Pet ros Elia, “On t he Fundam ent al Feedback-v s-Per for m ance Tr adeoff over t he M I SO-B C w it h I m p er fect and D elayed C SI T ”, in Proc. I EEE I nt. Symp. I nformation T heory (I SI T’13), July 2013. In Chapt er 5, t he work furt her considers t he ot her fundament al aspect s on t he communicat ions wit h imperfect and delayed feedback. One furt her work focuses on a MIMO broadcast channel wit h fixed-quality imperfect delayed CSIT and imperfect delayed global CSIR (t he receiver est imat es of t he channel of t he ot her receiver), and proceeds t o present schemes and DoF bounds t hat are oft en t ight , and t o const ruct ively reveal how even subst ant ially imperfect delayed-CSIT and subst ant ially imperfect delayed-global CSIR, are in fact suffi cient t o achieve t he opt imal DoF performance previously associat ed t o perfect delayed CSIT and perfect global CSIR. Anot her furt her work st udies t he diversity aspect of t he communicat ion wit h delayed CSIT . T he work proposes a novel broadcast scheme which, over broadcast channel wit h delayed CSIT , employs a form of int erference alignment t o achieve bot h full DoF as well as full diversity. T he result s were published in part at – Jinyuan Chen, Raymond K nopp, and Pet ros Elia, “I nt er fer ence A lignm ent for A chiev ing b ot h Full D OF and Full D iver sit y in t he B r oadcast C hannel w it h D elayed C SI T ”, in Proc. I EEE I nt. Symp. I nformation T heory (I SI T’12), July 2012. – Jinyuan Chen and Pet ros Elia, “M I M O B C w it h I m p er fect and D elayed C hannel St at e I nfor m at ion at t he Tr ansm it t er and R eceiver s”, in Proc. I EEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC’13), June 2013. Chapt er 6 finally describes t he conclusions and fut ure work. In addit ion t o t he t heoret ical limit s and novel encoders and decoders, t he work applies t owards gaining insight s on pract ical quest ions on t opics relat ing t o how much feedback quality (delayed, current or predict ed) allows for a cert ain DoF performance, relat ing t o t he usefulness of delayed feedback, t he usefulness of predict ed CSIT , t he impact of imperfect ions in t he quality of current and delayed CSIT , t he impact of feedback t imeliness and t he eff ect of feedback delays, t he benefit of having feedback symmet ry by employing comparable feedback links across users, t he impact of imperfect ions in t he
10
C hapt er 1
I nt r oduct ion
quality of global CSIR, and relat ing t o how t o achieve bot h full DoF and full diversity. Furt hermore, in addit on t o t he above result s, some ot her result s achieved in my PhD st udy were published in part at – Jinyuan Chen, Pet ros Elia, and Raymond K nopp, “R elay -A ided I nt er fer ence N eut r alizat ion for t he M ult iuser U plink -D ow nlink A sy m m et r ic Set t ing”, in Proc. I EEE I nt. Symp. I nformation Theory (I SI T’12), July 2011. – Jinyuan Chen, Arun Singh, Pet ros Elia and Raymond K nopp, “I nt er fer ence N eut r alizat ion for Separ at ed M ult iuser U plink D ow nlink w it h D ist r ibut ed R elay s”, in Proc. I nformation Theory and Applications Workshop (ITA), February 2011.
Chapt er 2
Fundament al Performance and Feedback Tradeoff over t he Two-User M I SO BC
For t he two-user MISO broadcast channel wit h imperfect and delayed channel st at e informat ion at t he t ransmit t er (CSIT ), t he work explores t he t radeoff between performance on t he one hand, and CSI T timeliness and accuracy on t he ot her hand. T he work considers a broad set t ing where communicat ion t akes place in t he presence of a random fading process, and in t he presence of a feedback process t hat , at any point in t ime, may provide CSIT est imat es - of some arbit rary accuracy - for any past , current or fut ure channel realizat ion. T his feedback quality may fluct uat e in t ime across all ranges of CSIT accuracy and t imeliness, ranging from perfect ly accurat e and instant aneously available est imat es, t o delayed est imat es of minimal accuracy. Under st andard assumpt ions, t he work derives t he degrees-of-freedom (DoF) region, which is t ight for a large range of CSIT quality. T his derived DoF region concisely capt ures t he eff ect of channel correlat ions, t he accuracy of predict ed, current , and delayed-CSIT , and generally capt ures t he eff ect of t he quality of CSIT off ered at any t ime, about any channel. T he work also int roduces novel schemes which - in t he cont ext of imperfect and delayed CSIT - employ encoding and decoding wit h a phase-Markov st ruct ure. T he result s hold for a large class of block and non-block fading channel models, and t hey unify and ext end many prior at t empt s t o capt ure t he eff ect of imperfect and delayed feedback. T his generality also allows for considerat ion of novel pert inent set t ings, such as t he new periodically evolving feedback set t ing, where a gradual accumulat ion of feedback bit s progressively 11
12
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D oF and Feedback Tr adeoff over Two-U ser M I SO B C
improves CSIT as t ime progresses across a finit e coherence period.
2.1 2.1.1
I nt r oduct ion Channel m odel
We consider t he mult iple-input single-out put broadcast channel (MISO BC) wit h an M -t ransmit ant enna (M 2) t ransmit t er communicat ing t o two receiving users wit h a single receiving ant enna each. Let h t , gt denot e t he channel of t he first and second user respect ively at t ime t, and let x t denot e t he t ransmit t ed vect or at t ime t, satisfying a power const raint E[||x t ||2 ] P , for some power P which also here takes t he role of t he signal-t o-noise rat io (SNR). Here h t and gt are drawn from a random dist ribut ion, such t hat each has zero mean and ident ity covariance (spat ially - but not necessarily t emporally - uncorrelat ed), and such t hat h t is linearly independent of gt wit h probability 1. In t his set t ing, t he corresponding received signals at t he first and second user t ake t he form (1)
yt
(2) yt (1)
(1)
(2.1)
(2) zt
(2.2)
= h Tt x t + zt = gTt x t +
(2)
(t = 1, 2, · · · ), where zt , zt denot e t he unit power AWGN noise at t he receivers. In t he high-SNR set t ing of int erest , for an achievable rat e pair (R 1 , R 2 ) for t he first and second user respect ively, t he corresponding degrees-of-freedom (DoF) pair (d1 , d2 ) is given by di = lim
P! 1
Ri , i = 1, 2 log P
and t he corresponding DoF region is t hen t he set of all achievable DoF pairs.
2.1.2
D elay-and-qualit y eff ect s of feedback
As in many mult iuser wireless communicat ions scenarios, t he performance of t he broadcast channel depends on t he t imeliness and precision of channel st at e informat ion at t he t ransmit t er (CSIT ). T his t imeliness and precision t hough may be reduced by limit ed-capacity feedback links, which may off er CSIT wit h consist ent ly low precision and high delays, i.e., feedback t hat off ers an inaccurat e represent at ion of t he t rue st at e of t he channel, as well feedback t hat can only be used for an insuffi cient fract ion of t he communicat ion durat ion. T he corresponding performance degradat ion, as compared t o t he case of having perfect feedback wit hout delay, forces t he delay-andquality quest ion of how much CSIT precision is necessary, and when, in order t o achieve a cert ain performance.
2.1
I nt r oduct ion
2.1.3
13
Channel pr ocess and feedback pr ocess wit h pr edict ed, cur r ent , and delayed CSI T
We here consider communicat ion of an infinit e durat ion n, a channel fading process { h t , gt } nt= 1 drawn from a st at ist ical dist ribut ion, and a feedback process t hat provides CSIT est imat es ˆ t ,t 0} nt,t 0= 1 { hˆ t ,t 0, g (of channel h t , gt ) at any t ime t 0 - before, during, or aft er mat erializat ion of h t , gt at t ime t - and does so wit h precision/ quality defined by t he st at ist ics of ˆ t ,t 0)} nt,t 0= 1 { (h t hˆ t ,t 0), (gt g (2.3) where we consider t hese est imat ion errors to have zero-mean circularlysymmet ric complex Gaussian ent ries. Nat urally any at t empt t o capt ure and meet t he t radeoff between performance, and feedback t imeliness and quality, must consider t he full eff ect of t he st at ist ics of t he channel and of CSIT preˆ t ,t 0)} nt,t 0= 1 at any point in t ime, about any channel. cision { (h t hˆ t ,t 0), (gt g P r edict ed, cur r ent , and delayed C SI T ˆ t ,t 0} nt0= 1 For t he channel h t , gt at t ime t, t he set of all est imat es { hˆ t ,t 0, g is formed by what can be described as t he set of predicted estimates { hˆ t ,t 0, ˆ t ,t 0} t 0< t , by t he current estimates hˆ t ,t , g ˆ t ,t at time t, and by t he set of delayed g ˆ 0 0 ˆ t ,t 0} t > t comprising of est imat es t hat are not available at t ime CSI T { h t ,t , g t. Predict ed CSIT may pot ent ially allow for reduct ion of t he eff ect of fut ure int erference, current CSIT may be used t o ‘separat e’ t he current signals of t he users, while delayed CSIT may facilit at e ret rospect ive compensat ion for t he lack of perfect quality feedback ( [4]).
2.1.4
N ot at ion, convent ions and assumpt ions
We will use t he not at ion (1)
↵t
,
(2)
↵t
,
lim
log E[||h t hˆ t ,t ||2] log P
(2.4)
lim
ˆ t ,t ||2] log E[||gt g log P
(2.5)
P! 1
P! 1
t o describe t he current quality exponent for t he current est imat e of t he chan(1) nel of each user at t ime t (↵ t is for user 1), while we will use (1) t
,
lim
P! 1
log E[||h t hˆ t ,t + ⌘||2] log P
(2.6)
14
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
ˆ t ,t + ⌘||2] log E[||gt g (2.7) P! 1 log P - for any suffi cient ly large but finite int eger ⌘ > 0 - t o denot e t he delayed quality exponent for each user. To clarify, wit h delayed CSIT consist ing of all channel est imat es t hat arrive aft er t he channel mat erializes, t he above use of a finit e ⌘, reflect s t he fact t hat we here only consider delayed CSIT t hat arrives up t o a finit e t ime of ⌘ channel uses from t he moment t he (1) channel mat erializes. In words, ↵ t measures t he precision/ quality of t he (1) CSIT (about h t ) t hat is available at t ime t, while t measures t he (best ) quality of t he CSIT (again about h t ) which arrives st rict ly aft er t he channel (2) (2) appears, i.e., st rict ly aft er t ime t (similarly ↵ t , t for t he channel gt of t he second user). It is easy t o see t hat wit hout loss of generality, in t he DoF set t ing of int erest , we can rest rict our at t ent ion t o t he range 1 (2) t
lim
,
0 (1)
(i )
↵t
(i ) t
1
(2.8)
(2)
where t = t = 1 corresponds t o being able t o event ually gat her (asymp(1) (2) t ot ically) perfect delayed CSIT for h t , gt , while ↵ t = ↵ t = 1, simply corresponds t o having inst ant aneously available CSIT of asympt ot ically perfect precision. Furt hermore we will use t he notat ion n
↵¯
(i )
1X (i ) , lim ↵t , n! 1 n t= 1
n
X ¯ (i ) , lim 1 n! 1 n
(i ) t ,
i = 1, 2
(2.9)
t= 1
t o denot e t he average of t he quality exponent s. A ssum pt ions Our result s, specifically t he achievability part , will hold under t he soft as(1) + T (2) ⌧ +T sumpt ion t hat any suffi cient ly long subsequence { ↵ t } t⌧ = ⌧ (resp. { ↵ t } t = ⌧, (1) + T (2) ⌧ +T { t } t⌧ = ⌧, { t } t = ⌧) has an average t hat approaches t he long t erm average ↵¯ (1) (resp. ↵¯ (2) , ¯ (1) , ¯ (2) ), for some finit e T t hat can be chosen t o be suffi cient ly large t o allow for t he above convergence. Such an assumpt ion which has also been employed in works like [11] - essent ially imply t hat t he long t erm st at ist ics of t he feedback process, remains t he same in t ime, i.e., t hat t he average feedback behavior - averaged over large amount s of t ime remains t he same t hroughout t he communicat ion process. 1. To see t his, we recall from [16, 17] t hat under a peak-power const raint of P , having CSIT est imat ion error in t he order of P 1 causes no DoF reduct ion as compared t o t he perfect CSIT case. In our DoF high-SNR set t ing of int erest where P > > n, t his same (i ) observat ion also holds under an average power const raint of P . T he fact t hat ↵ t (i ) comes nat urally from t he fact t hat one can recall, at a lat er t ime, st at ist ically good t est imat es.
2.1
I nt r oduct ion
15
We also adhere t o t he common convent ion (see [4, 6, 7, 33]) of assuming perfect and global knowledge of channel st ate informat ion at t he receivers (perfect global CSIR), where t he receivers know all channel st at es and all est imat es. We furt her adopt t he common assumpt ion (see [5–7,15]) t hat t he current estimation error is st at ist ically independent of current and past estimates, and consequent ly t hat t he input signal is a funct ion of t he message and of t he CSIT . T his assumpt ion fit s well with many channel models spanning from t he fast fading channel (i.i.d. in t ime), t o t he correlat ed channel model as t his is considered in [5], t o t he quasi-st at ic block fading model where t he CSIT est imat es are successively refined while t he channel remains st at ic (see [16], see also t he discussion in t he appendix in Sect ion 2.8). Addit ionally we consider t he ent ries of each est imat ion error vect or h t hˆ t ,t 0 ˆ t ,t 0) t o be i.i.d. Gaussian, clarifying t hough t hat we are (similarly of gt g just referring t o t he M ent ries in each such specific vect or h t hˆ t ,t 0, and t hat we do not suggest t hat t he error ent ries are i.i.d. in t ime or across users. T he appendix in Sect ion 2.8 off ers furt her det ails and just ificat ion on t he above assumpt ions and convent ions. . Finally we safely assume t hat E[||h t hˆ t ,t 0||2] E[||h t hˆ t ,t 00||2] (similarly . ˆ t ,t 0||2 ] ˆ t ,t 00||2]), for any t 0 > t 00. T his assumpt ion - which E[||gt g E[||gt g simply suggest s t hat one can revert back t o past est imat es of st at ist ically bett er quality - is used here for simplicity of not at ion, and can be removed, aft er a small change in t he definit ion of t he quality exponent s, wit hout an eff ect t o t he main result .
2.1.5
P r ior wor k
T he delay-and-quality eff ect s of feedback, nat urally fall between t he two ext reme cases of no CSIT and of full CSIT (immediat ely available, perfect quality CSIT ), wit h full CSIT allowing for t he opt imal 1 DoF per user (cf. [1]), while t he absence of any CSIT reduces t his t o just 1/ 2 DoF per user (cf. [2, 3]). Toward bridging t his gap, diff erent works have considered t he use of imperfect and delayed feedback. For example, t he work by Lapidot h, Shamai and Wigger in [15] considered t he case where t he amount of feedback is limit ed t o t he ext ent t hat t he channel-est imat ion error power does not vanish with increasing SNR, in t he sense t hat limP ! 1 (log E[||h t hˆ t ,t ||2])/ log P = ˆ t ,t ||2 ])/ log P = 0. In t his set t ing - which corresponds limP ! 1 (log E[||gt g (1)
(2)
(1)
(2)
t o t he case here where ↵ t = ↵ t = t = t = 0, 8t - t he work in [15] showed t hat t he symmet ric DoF is upper bounded by 2/ 3 DoF per user, again under t he assumpt ion placed here t hat t he input signaling is independent of t he est imat ion error. It is wort h not ing t hat finding t he exact DoF in t his zero-exponent set t ing, current ly remains an open problem. At t he ot her ext reme, t he work by Caire et al. [17] (see also t he work
16
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
of Jindal [16], as well as of Lapidot h and Shamai [37]) showed t hat having immediat ely available CSIT est imat es wit h est imat ion error power t hat is in t he order of P 1 - i.e., having limP ! 1 (log E[||h t hˆ t ,t ||2])/ log P = ˆ t ,t ||2])/ log P = 1, 8t, corresponding here t o having limP ! 1 (log E[||gt g (1)
(2)
↵ t = ↵ t = 1, 8t - causes no DoF reduct ion as compared t o t he perfect CSIT case, and can t hus achieve t he opt imal 1 DoF per user. A valuable t ool t oward bridging t his gap and furt her underst anding t he delay-and-quality eff ect s of feedback, came wit h t he work by Maddah-Ali and Tse in [4] which showed t hat arbit rarily delayed feedback can st ill allow for performance improvement over the no-CSIT case. In a fast -fading blockfading set t ing, t he work diff erent iated between current and delayed CSIT wit h delayed CSIT defined in [4] as t he CSIT which is available aft er t he channel’s coherence period - and showed t hat delayed and complet ely obsolet e CSIT , even wit hout any current CSIT , allows for an improved opt imal 2/ 3 DoF per user. T his set t ing - which in principle corresponded t o perfect delayed CSIT - is here represent ed by current -CSIT exponent s of t he form (1) (2) ↵ t = ↵ t = 0, 8t. Wit hin t he same block-fading cont ext of delayed vs. current CSIT , t he work by Kobayashi et al., Yang et al., and Gou and Jafar [5–7], quant ified t he usefulness of combining delayed and complet ely obsolet e CSIT wit h immediat ely available but imperfect CSIT of a cert ain quality ↵ = ˆ t ,t ||2])/ log P limP ! 1 (log E[||h t hˆ t ,t ||2])/ log P = limP ! 1 (log E[||gt g t hat remained unchanged t hroughout t he communicat ion process. T his work - which again in principle assumed perfect delayed CSIT , and which is here (1) (2) represent ed by current -CSIT exponent s of t he form ↵ t = ↵ t = ↵, 8t derived t he opt imal DoF region t o be t hat wit h a symmet ric DoF of (2+ ↵)/ 3 DoF per user. Int erest ingly, despit e t he fact t hat in principle, t he above set t ings in [4–7] corresponded t o perfect delayed CSIT , t he act ual schemes in t hese works in fact achieved t he opt imal DoF, by using delayed CSIT for only a fract ion of t he channels. T his possibility t hat imperfect and sparse delayed CSIT may be as good as perfect and omnipresent delayed CSIT (cf. [9]), is one of t he many facet s t hat are explored in det ail in Sect ions 2.2-2.3. Anot her int erest ing approach was int roduced by Tandon et al. in [11] who considered t he fast -fading two-user MISO BC set t ing, where each user’s CSIT changes every coherence period by alt ernat ing between t he t hree ext reme st at es of perfect current CSIT , perfect delayed CSIT , and no CSIT . Addit ionally, Lee and Heat h in [10] considered, in t he set t ing of t he quasist at ic block-fading channel, t he possibility t hat current CSIT may be available only aft er a cert ain fract ion of a finit e-durat ion coherence period Tc. Ot her work such as t hat by Maleki et al. in [33] considered, again in t he MISO BC cont ext , an asymmet ric set t ing where bot h users off ered perfect delayed CSIT , but where only one user off ered perfect current CSIT while
2.1
I nt r oduct ion
17
t he ot her user off ered no current CSIT . In t his set t ing, t he opt imal DoF corner point was calculat ed t o be (1, 1/ 2) (sum-DoF d1 + d2 = 3/ 2). Anot her asymmet ric-feedback set t ing was considered in [8]. In addit ion t o t he above works t hat are immediat ely relat ed t o our own result , many ot her works t hat have provided int erest ing result s in t he cont ext of delayed or imperfect feedback, include [12, 13, 18–32, 38].
2.1.6
St r uct ur e
Sect ion 2.2 will give t he main result of t his work by describing, under t he aforement ioned common assumpt ions, t he DoF off ered by a CSIT process ˆ t ,t 0} nt= 1,t 0= 1 of a cert ain quality { (h t hˆ t ,t 0), (gt g ˆ t ,t 0)} nt= 1,t 0= 1. Spe{ hˆ t,t 0, g cifically Proposit ion 1 and Lemma 1 lower and upper bound t he DoF region, and t he result ing T heorem 1 provides t he optimal DoF for a large range of ‘suffi cient ly good’ delayed CSIT . T he result s capt ure specific exist ing cases of int erest , such as t he Maddah-Ali and Tse set t ing in [4], t he Yang et al. and Gou and Jafar set t ing in [6,7], t he Lee and Heat h ‘not -so-delayed CSIT ’ set ting in [10] for two users, t he Maleki et al. asymmet ric set t ing in [33], and in the range of suffi cient ly good delayed CSIT , also capt ure t he result s in t he Tandon et al. set t ing of alt ernat ing CSIT [11] . Towards gaining furt her insight , we t hen proceed t o provide diff erent corollaries for specific cases of int erest . Again in Sect ion 2.2, Corollary 1a dist ills t he main result down t o t he symmet ric feedback case where ↵¯ (1) = ↵¯ (2) and ¯ (1) = ¯ (2) , and immediat ely aft er t hat , Corollary 1b explores t he benefit s of such feedback symmet ry, by quant ifying t he ext ent t o which having similar feedback quality for t he two users, off ers a gain over t he asymmet ric case where one user has generally more feedback t han t he ot her. One of t he out comes here is t hat such ‘symmet ry gains’ are oft en nonexist ent . Corollary 1c generalizes t he pert inent result in t he set t ing in [33] corresponding t o feedback asymmet ry ; a set t ing which we consider t o be import ant as it capt ures t he inherent non-homogeneity of feedback quality of diff erent users. Corollary 1d off ers insight on t he need for delayed CSIT , and shows how, reducing ↵¯ (1) , ↵¯ (2) allows - t o a cert ain ext ent - for furt her reducing of ¯ (1) , ¯ (2) , wit hout an addit ional DoF penalty. It will be surprising t o not e t hat t he expressions from Corollary 1d, mat ch t he amount of delayed CSIT used by diff erent previous schemes which were designed for set t ings t hat in principle off ered perfect delayed CSIT , and which were t hus designed wit hout an expressed purpose of reducing t he amount of delayed CSIT . At t he ot her ext reme, Corollary 1e off ers insight on t he need for using predict ed channel est imat es (forecast ing channel st at es in advance), by showing t hat at least in t he range of suffi cient ly good delayed CSIT - employing predict ed CSIT is unnecessary. Sect ion 2.3 highlight s t he newly considered periodically evolving feedback set ting over t he quasi-st at ic block fading channel, where a gradual accu-
18
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
mulat ion of feedback, result s in a progressively increasing CSIT quality as t ime progresses across a finit e coherence period. T his set t ing is powerful as it capt ures t he many feedback opt ions t hat one may have in a block-fading environment where t he statistical nat ure of feedback may remain largely unchanged across coherence periods. To off er furt her underst anding, we provide examples which - under very clearly specified assumpt ions - describe how many feedback bit s t o int roduce, and when, in order t o achieve a cert ain DoF performance. In t he same sect ion, smaller result s and examples off er furt her insight - again in t he cont ext of periodically evolving feedback over a quasist at ic channel - like for example t he result in Corollary 1g which bounds t he quality of current and of delayed CSIT needed t o achieve a cert ain t arget symmet ric DoF, and in t he process off ers int uit ion on when delayed feedback is ent irely unnecessary, in t he sense t hat t here is no need t o wait for feedback t hat arrives aft er t he end of t he coherence period of t he channel. Similarly Corollary 1h provides insight on t he feedback delays t hat allow for a given t arget symmet ric DoF in t he presence of const raint s on current and delayed CSIT qualit ies. T his quant ifies t o a cert ain ext ent t he int uit ive argument t hat , wit h a t arget DoF in mind, feedback delays must be compensat ed for, wit h high quality feedback est imat es. Sect ion 2.4 corresponds t o t he achievability part of t he proof of t he main result , and present s t he general communicat ion scheme t hat ut ilizes t he avaiˆ t ,t 0} nt= 1,t 0= 1 , t o achieve t he corlable informat ion of a CSIT process { hˆ t ,t 0, g responding DoF corner point s. T his is done - by properly employing diff erent combinat ions of zero forcing, superposit ion coding, int erference compressing and broadcast ing, as well as specifically t ailored power and rat e allocat ion - in order t o t ransmit privat e informat ion, using current ly available CSIT est imat es t o reduce int erference, and using delayed CSIT est imat es t o alleviat e t he eff ect of past int erference. T he scheme has a forward-backward phase-Markov st ruct ure which, in t he cont ext of imperfect and delayed CSIT , was first int roduced in [8, 9] t o consist of four main ingredient s t hat include, block-Markov encoding, spat ial precoding, int erference quant izat ion, and backward decoding. Aft er t he descript ion of t he scheme in it s general form, and t he explicit descript ion of how t he scheme achieves t he diff erent DoF corner point s, Sect ion 2.4.4 provides example schemes - dist illed from t he general scheme - for specific set t ings such as t he imperfect -delayed CSIT set t ing, t he (ext ended) alt ernat ing CSIT set t ing of Tandon et al. [11], as well as discusses schemes wit h small delay. Sect ion 2.5 off ers concluding remarks, t he appendix in Sect ion 2.6 provides t he det ails of t he out er bound, t he appendix in Sect ion 2.7 off ers det ails on t he proofs, while t he appendix in Sect ion 2.8 off ers a discussion on some of t he assumpt ions employed in t his work. In t he end, t he above result s provide insight on pert inent quest ions such
2.1
I nt r oduct ion
19
as : – What CSIT feedback precision should be provided, and when, in order t o achieve a cert ain t arget DoF performance? (T heorem 1) – When is delayed feedback unnecessary ? (Corollary 1g) – Is t here any gain in early predict ion of fut ure channels? (Corollary 1e) – What current -CSIT and delayed-CSIT qualit ies suffi ce t o achieve a cert ain performance? (Corollary 1g) – Can delayed CSIT t hat is sparse and of imperfect -quality, achieve t he same DoF performance t hat was previously at t ribut ed t o sending perfect delayed CSIT ? (Corollary 1d) – How much more valuable are feedback bit s t hat are sent early, t han t hose sent lat e? (Sect ion 2.3) – In t he quasi-st at ic block-fading case, is it bet t er t o send less feedback early, or more feedback lat er ? (Sect ion 2.3) – What is t he eff ect of having asymmet ric feedback links, and when can we have a ‘symmet ry gain’ ? (Corollary 1b)
20
2.2
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
D oF r egion of t he M I SO B C
We proceed wit h t he main DoF result s, which are proved in Sect ion 2.4 (inner bound) and Sect ion 2.6 (out er bound). (1) (2) (1) We here remind t he reader of t hesequences { ↵ t } nt= 1 , { ↵ t } nt= 1, { t } nt= 1 , (2) { t } nt= 1 of quality exponent s, as t hese were defined in (2.4)-(2.7), as well as of t he corresponding averages ↵¯ (1) , ↵¯ (2) , ¯ (1) , ¯ (2) from (2.9). We also remind t he reader t hat we consider communicat ion over an asympt ot ically large t ime durat ion n. We hencefort h label t he users so t hat ↵¯ (2) ↵¯ (1) . We st art wit h t he following proposit ion, t he proof of which can be found in Sect ion 2.4 which describes t he scheme t hat achieves t he corresponding DoF corner point s. P r op osit ion 1. The DoF region of the two-user MI SO BC with a CSI T ˆ t ,t 0} nt= 1,t 0= 1 of quality { (h t hˆ t ,t 0), (gt g ˆ t ,t 0)} nt= 1,t 0= 1 , is inner process { hˆ t ,t 0, g bounded by the polygon described by d1
1,
d2
1
(2.10)
(1)
2d1 + d2
2 + ↵¯
2d2 + d1
2 + ↵¯ (2)
d1 + d2
1 + min{
(2.11) (2.12) ¯ (1)
,
¯ (2)
}.
(2.13)
Figure 2.1 corresponds t o t he result in Proposit ion 1. Towards t ight ening t he above inner bound, we here draw from t he DoF out er bound in [6] t hat focused on CSIT wit h invariant and symmet ric quality and non-st at ic channels, and employ t echniques t hat allow for t he new bound t o hold for a broad range of channels including t he st at ic-channel case t hat is of part icular int erest here. T he proof of t he new bound can be found in Sect ion 2.6. L em m a 1. The DoF region of the two-user MI SO BC with a CSI T process ˆ t ,t 0} nt= 1,t 0= 1 of quality { (h t hˆ t ,t 0), (gt g ˆ t ,t 0)} nt= 1,t 0= 1 , is upper bounded { hˆ t ,t 0, g as d1
1,
d2
1
(2.14)
(1)
(2.15)
2d1 + d2
2 + ↵¯
2d2 + d1
2 + ↵¯ (2) .
(2.16)
Comparing t he above inner and out er bounds, and observing t hat t he last bound in Proposit ion 1 becomes inact ive in t he range of suffi cient ly ( 1) ( 2) ( 2) good delayed-CSIT where min{ ¯ (1) , ¯ (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } , gives t he main result of t his work in t he form of t he following t heorem t hat provides t he opt imal DoF for t his large range of ‘suffi cient ly good’ delayed CSIT .
2.2
D oF r egi on of t he M I SO B C
21
T heor em 1. T he optimal DoF region of the two-user MISO BC with a CSI T ˆ t ,t 0} nt= 1,t 0= 1 of quality { (h t hˆ t ,t 0), (gt g ˆ t ,t 0)} nt= 1,t 0= 1 is given process { hˆ t ,t 0, g by d1
1,
d2
1
(2.17)
2d1 + d2
2 + ↵¯
(1)
(2.18)
2d2 + d1
2 + ↵¯ (2)
(2.19)
for any suffi ciently good delayed-CSI T process such that min{ ¯ (1) , ¯ (2) } ( 1) ( 2) ( 2) min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . As ment ioned, t he achievability part of t he proof can be found in Sect ion 2.4. Figure 2.2 corresponds t o t he main result in t he t heorem. Before proceeding t o specific corollaries that off er furt her insight , it is wort h making a comment on t he fact t hat t he ent ire complexity of t he problem is capt ured by t he quality exponent s. R em ar k 1. T he results suggest that the quality exponents capture - in the DoF setting of interest, and under our assumptions - the eff ect of the staˆ t ,t 0)} nt,t 0= 1. This is indeed tistics of the CSI T precision { (h t hˆ t ,t 0), (gt g the case since the following two hold. Firstly, given the Gaussianity of the ˆ t ,t 0)} nt,t 0= 1 are captured estimation errors, the statistics of { (h t hˆ t ,t 0), (gt g by the 2n 2 ⇥ 2n 2 covariance matrix 2 of the 2n 2 -length vector consisting of ˆ t ,t 0)} nt,t 0= 1 . The diagonal entries of this cothe elements { (h t hˆ t ,t 0), (gt g ˆ t ,t 0||2]} nt,t 0= 1. With variance matrix are simply { M1 E[||h t hˆ t ,t 0||2 ], M1 E[||gt g the above in mind, we also note that the outer bound has kept open the possibility of having arbitrary off -diagonal elements in this covariance matrix (this is specifically seen in the steps in (2.95), (2.96)), thus allowing for the outer bound to hold irrespective of the off -diagonal elements of this covariance matrix. Consequently, under our assumptions, the essence of the statistics is ˆ t ,t 0||2]} nt,t 0= 1, and its eff ect is captured captured by { E[||h t hˆ t ,t 0||2], E[||gt g - in the high-SNR DoF regime - by the quality exponents.
Sy m m et r ic v s. asy m m et r ic feedback We proceed t o explore t he special case of symmet ric feedback where t he long-t erm accumulat ed feedback quality at t he two users is similar, in t he sense t hat t he feedback links of user 1 and user 2 share t he same long-t erm 2. T his size of t he covariance mat rix reflect s t he fact t hat t he M ent ries of each h t hˆ t , t 0 ˆ t , t 0 ). Please not e t hat we refer t o independence across t he spatial are i.i.d. (similarly of g t g dimensions of t he channel of one user, and cert ainly do not refer t o independence across t ime or across users.
22
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
d2
d2
B 1
B
1
E
E A
C
G
F D
d1
0
(a) Case 1:
2
(1)
(2)
d1
0
1
1
(b) Case 2:
1
(1)
2
(2)
1
Figur e 2.1 – DoF region inner bound for t he two-user MISO BC. T he corner point s t ake t he following values : E = (2¯ ↵¯ (2) , 1 + ↵¯ (2) ¯ ), F = (1 + ↵¯ (1) ¯ , 2¯ ↵¯ (1) ), and G = (1, ¯ ), where ¯ , min{ ¯ (1) , ¯ (2) , 1+ ↵¯ ( 1) + ↵¯ ( 2) , 1+ ↵¯ ( 2) } . 3 2
d2
d2
(1)
d2 2d1 2
d2 2d1
B
B
1
1
(1)
2
A
C
d1 2d2
2
C
( 2)
d1 2d2
2
(2)
D
0
1
d1
(1)
2
0
2
2
(a) Case 1:
2
(1)
d1
(1)
2
1
(2)
(b) Case 2:
1
2
(1)
( 2)
1
Figur e 2.2 – Opt imal DoF region for t he two-user MISO BC, for t he case ( 1) ( 2) ( 2) of min{ ¯ (1) , ¯ (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . T he corner point s t ake t he ( 2) following values : A = (1, 1+ ↵2¯ ), B = ( ↵¯ (2) , 1), C = ( 2+ 2¯↵
( 1)
3
↵¯ ( 2)
, 2+ 2¯↵
( 2)
3
↵¯ ( 1)
), and D = (1, ↵¯ (1) ).
2.2
D oF r egi on of t he M I SO B C
23 No CSI T Delayed CSI T [MAT]
d2 B=
1
Current + delayed CSI T
(1 2 ) / 3
Current + delayed CSI T
(1 2 ) / 3
,1
E
C= 2
3
,2
3
d1 d2 1
2/3
F D = 1,
0
2/3
d1
1
Figur e 2.3 – DoF region of two-user MISO BC wit h symmet ric feedback, ↵¯ (1) = ↵¯ (2) = ↵¯ , ¯ (1) = ¯ (2) = ¯ . T he opt imal region t akes t he form of a polygon wit h corner point s { (0, 0), (0, 1), ( ↵¯ , 1), ( 2+3 ↵¯ , 2+3 ↵¯ ), (1, ↵¯ ), (1, 0)} for 1+ 2¯↵ ¯ < 1+ 2¯↵ , t he derived region t akes t he form of a polygon ¯ 3 . For 3 wit h corner points ¯ ¯ ), (1 + ↵¯ ¯ , 2 ¯ ↵¯ ), (1, ↵¯ ), (1, 0)} . { (0, 0), (0, 1), ( ↵¯ , 1), (2 ↵¯ , 1 + ↵¯ exponent averages ↵¯ (1) = ↵¯ (2) = : ↵¯ and ¯ (1) = ¯ (2) = : ¯ . Most exist ing works, wit h an except ion in [33] and [8], fall under t his symmet ric feedback set ting. T he following holds direct ly from T heorem 1 and Proposit ion 1. C or ollar y 1a (DoF wit h symmet ric feedback). T he optimal DoF region for the symmetric feedback case, takes the form d1
1,
d2
1,
2d1 + d2
1+ 2¯↵ ¯ < when ¯ 3 , while when achievable region
d1
1+ 2¯↵ 3
1,
2 + ↵¯ ,
2d2 + d1
2 + ↵¯
this region is inner bounded by the 1
(2.20)
2d1 + d2
2 + ↵¯
(2.21)
2d2 + d1
2 + ↵¯ 1+ ¯.
(2.22)
d2 + d1
d2
(2.23)
Figure 2.3 depict s t he DoF region of t he two-user MISO BC in t he presence of CSIT feedback wit h long-t erm symmet ry. We now quant ify t he ext ent t o which having symmet ric feedback off ers a benefit over t he asymmet ric case where one user accumulat es - in t he long
24
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
t erm - bet t er feedback t han t he other. Such ‘symmet ry gains’ have been recorded in diff erent inst ances (cf. [11], [33]). T he following broad comparison focuses on t he case of perfect delayed CSIT ( ¯ = 1), and cont rast s t he symmet ric feedback case ↵¯ (1) = ↵¯ (2) , t o t he asymmet ric case ↵¯ (1) 6 = ↵¯ (2) . Nat urally such comparison is performed under an overall feedback const raint , which - reflect ing t he spirit of previous works t hat have ident ified symmet ry gains - is here chosen t o be in t he form of a fixed sum ↵¯ (1) + ↵¯ (2) . T he comparison is in t erms of t he opt imal sum DoF d1 + d2, where again we recall t hat t he users are labeled so t hat ↵¯ (1) ↵¯ (2) . To clarify, t he symmet ry gain will be t he diff erence in t he sum-DoF performance of two cases; t he symmet ric case where t he two exponent averages are t he same and are equal t o 12 ( ↵¯ (1) + ↵¯ (2) ), and t he asymmet ric case where t he two dist inct exponent averages are ↵¯ (1) and ↵¯ (2) . T he proof is direct from T heorem 1 and Corollary 1a. C or ollar y 1b (Symmet ric vs. asymmet ric feedback). T he symmetry sumDoF gain is equal to 16 (2¯↵ (1) ↵¯ (2) 1) + , i.e., if 2¯↵ (1) ↵¯ (2) 1 > 0, the ( 1) ( 2) 1 symmetric sum-DoF gain is 2¯↵ 6↵¯ > 0, else there is no symmetry gain. Exam ple 7. For example, consider the asymmetric feedback option ↵¯ (1) = 1, ↵¯ (2) = 0 which corresponds to an optimal sum-DoF of d1 + d2 = 3/ 2 (see T heorem 1, and consider perfect delayed CSI T ), and compare this with the symmetric option where both exponent averages are equal to 1/ 2. T he symmetric option provides a sum-DoF of d1 + d2 = 5/ 3, and a symmetry gain of 5/ 3 3/ 2 = 1/ 6. As expected, the gain is positive since 2¯↵ (1) ↵¯ (2) 1 = 2 0 1 = 1 > 0. On the other hand, an asymmetric option ↵¯ (1) = 3/ 5, ↵¯ (2) = 2/ 5 corresponds to an optimal sum DoF of d1 + d2 = 5/ 3, which matches the aforementioned DoF performance of the symmetric option. The symmetry gain here is zero, since 2¯↵ (1) ↵¯ (2) 1 = 6/ 5 2/ 5 1 = 1/ 5 < 0. Finally, before concluding our discussion on feedback symmet ry/ asymmet ry, it is wort h not ing t hat t he asymmetric set t ing here - where ↵¯ (1) 6 = ↵¯ (2) and (1) (2) ¯ ¯ where and need not be equal - yields a nat ural generalizat ion for t he asymmet ric set t ing of Maleki et al. in [33] which, as we have ment ioned, in t he presence of abundant delayed CSIT , had an opt imal DoF corresponding t o DoF corner point (1, 1/ 2) (and a sum-DoF d1 + d2 = 3/ 2). T he following corollary - which again corresponds t o t he range of suffi cient ly good delayed ( 1) ( 2) ( 2) CSIT where min{ (1) , (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } - off ers a broad generalizat ion of t he corresponding result in [33]. T he proof is direct from t he main result . C or ollar y 1c (Asymmet ric and periodic CSIT ). In the range of suffi ciently good delayed CSI T , the optimal DoF region is defined by corner points B =
2.2
D oF r egi on of t he M I SO B C ( 1)
( 2)
( 2)
25 ( 1)
( ↵¯ (2) , 1), C = ( 2+ 2¯↵ 3 ↵¯ , 2+ 2¯↵ 3 ↵¯ ) and D = (1, ↵¯ (1) ) whenever 2¯↵ (1) ( 2) ↵¯ (2) < 1, else by corner points A = (1, 1+ ↵2¯ ) and B . As an example we can see t hat t he same DoF corner point A = (1, 1/ 2) derived in [33] under t he general principle of perfect delayed CSIT for bot h users, and perfect current CSIT for t he first user - can in fact be achieved with a plet hora of opt ions wit h lesser current and delayed CSIT , such as (1)
↵t
(2)
= 1/ 2, ↵ t
= 0,
(1) t
=
(2) t
= 1/ 2, 8t.
N eed for delayed feedback : I m p er fect vs. p er fect delayed C SI T We now shift emphasis t o explore t he fact t hat imperfect delayed CSIT ¯ ( < 1) can - in some cases - be as useful as (asympt ot ically) perfect delayed CSIT ( ¯ = 1), and t o provide insight on t he overall feedback quality (t imely and delayed) t hat is necessary t o achieve a cert ain DoF performance. Before proceeding wit h t he result , we briefly mot ivat e our int erest in imperfect and sparse delayed CSIT . Towards t his we recall t hat ↵¯ (1) , ↵¯ (2) are more represent at ive of t he quality (and inevit ably of t he amount ) of timely feedback, while ¯ (1) , ¯ (2) are more represent at ive of t he quality of t he entirety of feedback (t imely plus delayed). In this sense, any at t empt t o limit t he t ot al amount and quality of feedback - t hat is communicat ed during a cert ain communicat ion process - must include reducing ¯ (1) , ¯ (2) , rat her t han just focusing on reducing ↵¯ (1) , ↵¯ (2) . For example, even if we removed ent irely (1) (2) all current CSIT (↵ t = ↵ t = 0, 8t), but insist ed on always sending perfect (1) (2) delayed CSIT ( t = t = 1, 8t), we would achieve lit t le t owards reducing t he t ot al amount of feedback, and we would mainly shift t he t ime-frame of t he problem, again irrespect ive of t he drast ic reduct ion in ↵¯ (1) , ↵¯ (2) . As we will see t hough, having reduced ↵¯ (1) , ↵¯ (2) can in fact t ranslat e t o having overall reduced feedback because, int erest ingly, having reduced ↵¯ (1) , ↵¯ (2) , can t ranslat e - t o a cert ain ext ent - t o needing lesser quality delayed feedback, i.e., can t ranslat e t o furt her reduct ions in ¯ (1) , ¯ (2) . T his is quant ified in t he following, t he proof of which is direct , because it simply restat es part of what is in t he t heorem. C or ollar y 1d (Imperfect vs. perfect delayed CSIT ). A CSI T process { hˆ t ,t 0, ˆ t ,t 0} nt= 1,t 0= 1 that off ers g 1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , } (2.24) 3 2 gives the same DoF as a CSI T process that off ers perfect delayed CSIT for (1) (2) each channel realization ( t = t = 1, 8t, i.e., ¯ (1) = ¯ (2) = 1). For the symmetric case, having min{ ¯ (1) , ¯ (2) }
min{
¯
1 + 2¯↵ 3
(2.25)
26
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
guarantees the same. It is int erest ing t o observe t hat t he expressions in t he above corollary mat ch t he amount of delayed CSIT used by schemes in t he past , even t hough such schemes were not designed wit h t he expressed purpose of reducing t he amount of delayed CSIT . For example, t he Maddah-Ali and Tse scheme in [4] (1) (2) (feedback wit h ↵ t = ↵ t = 0, 8t, over an i.i.d fast -fading channel), while in principle corresponding t o abundant delayed CSIT , in fact was based on a precoding design t hat only needed delayed CSIT only for every t hird channel realizat ion, corresponding t o ( 1 if t = i ( mod 3) (i ) , user i = 1, 2 (2.26) t = 0 ot herwise and 3 t hus corresponding t o ¯ (i ) = 1/ 3, i = 1, 2, which happens t o mat ch t he above expression in (2.25) ( ↵¯ = 0). T his same general expression in (2.25) addit ionally t ells us t hat , in t he Maddah-Ali and Tse set t ing, any combinat ion of CSIT quality exponent s t hat allows for ¯ (1) = ¯ (2) 1/ 3, will allow for t he same opt imal DoF region in [4]. For example, one such choice would (1) (2) be t o use t = t = 1/ 3, 8t. (1) A similar observat ion holds for t he opt imal schemes in [6, 7] (↵ t = (2) ↵ t = ↵, 8t) which again operat ed in a set t ing t hat in principle allowed for unlimit ed delayed CSIT , but which in fact asked for delayed CSIT only for every t hird channel realizat ion ( 1 if t = i ( mod 3) (i ) (2.27) t = ↵ ot herwise corresponding t o ¯ (i ) = (1 + 2↵)/ 3, i = 1, 2. T his again can be seen as a special inst ance of t he general expression in (2.25), which is powerful enough t o reveal t hat any combinat ion of CSIT quality exponent s t hat allows for ¯ (1) = ¯ (2) (1 + 2¯↵ )/ 3, will achieve t he same opt imal DoF in [6, 7]. One (1) (2) such choice would be t o have t = t = 1+32↵ , 8t. Along t he same lines, t he opt imal asymmet ric scheme in [33] which operat ed under t he general principle of perfect delayed CSIT for bot h users, and perfect current CSIT for t he first user, in fact employed a scheme t hat used lesser feedback. In t his scheme, which had durat ion of two channel (1) (1) (2) uses, t he act ual required CSIT corresponded t o ↵ 1 = 1 = 1 = 1, (2) (1) (2) (1) (2) and ↵ 1 = ↵ 2 = ↵ 2 = 2 = 2 = 0, t hus corresponding t o ↵¯ (1) = (2) (1) (2) 1/ 2, ↵¯ = 0, ¯ = 1/ 2, ¯ = 1/ 2, which mat ches t he expression in (2.24) ( 1) ( 2) ( 1) ( 2) since min{ ¯ (1) , ¯ (2) } = min{ 1/ 2, 1/ 2} = min{ 1+ ↵¯ + ↵¯ , 1+ min{ ↵¯ ,↵¯ } } = 3
2
3. Here when we say t = i (mod 3), we refer t o t he modulo operat ion, i.e., we mean t hat t = 3k + i for some int eger k.
2.2
D oF r egi on of t he M I SO B C
27
min{ 1+ 1/32+ 0 , 1+2 0 } = 1/ 2. T his same expression in (2.24) furt her reveals ot her CSIT opt ions t hat allow for t he same opt imal DoF. N eed for pr edict ed C SI T We now shift emphasis from delayed CSIT t o t he ot her ext reme of predict ed CSIT . As we recall, we considered a channel process { h t , gt } t and a ˆ t ,t 0} t ,t 0, consist ing of estimat es hˆ t ,t 0 - available at any CSIT process { hˆ t ,t 0, g 0 t ime t - of t he channel h t t hat mat erializes at any t ime t. We also advocated t hat we can safely assume t hat E[||h t hˆ t ,t 0||2 ] E[||h t hˆ t ,t 00||2] ˆ t ,t 0||2] ˆ t ,t 00||2 ]), for any t 0 > t 00, simply be(similarly E[||gt g E[||gt g cause one can revert back t o past est imat es of st at ist ically bet t er quality. T his assumpt ion t hough does not preclude t he possible usefulness of early (predict ed) est imat es, even if such est imat es are generally of lesser quality (st at ist ically) t han current est imat es (i.e., of lesser quality t han est imat es t hat appear during or aft er t he channel mat erializes). It is st ill conceivable t hat t ransmission at a cert ain t ime t ⇤, can benefit from being a funct ion of an est imat e hˆ t ,t 0 of a fut ure channel t > t ⇤, where t his est imat e became available - nat urally by predict ion - at any t ime t 0 t ⇤ < t. T he following addresses t his, in t he range of suffi ciently good delayed CSIT where ( 1) ( 2) ( 2) min{ ¯ (1) , ¯ (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . C or ollar y 1e (Need for predict ed CSIT ). I n the range of suffi ciently good delayed CSI T, transmission need not consider predicted estimates of future channels, to achieve the optimal DoF. Proof. T he proof is by const ruct ion ; t he designed schemes do not use predict ed est imat es, while t he t ight out er bound does not preclude t he use of such predict ed est imat es.
28
2.3
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Per iodically evolving CSI T
We here focus on t he block fading set t ing wit h a finit e coherence period of Tc channel uses, during which t he channel remains fixed, and during which a gradual accumulat ion of feedback provides a progressively increasing CSIT quality, as t ime progresses across the coherence period (part ially delayed current CSIT ), or at any t ime aft er t he end of t he coherence period (delayed and pot ent ially obsolet e CSIT ) 4. Such gradual improvement could be sought in FDD (frequency division duplex) set t ings wit h limit ed-capacity feedback links t hat can be used more t han once during t he coherence period t o progressively refine CSIT , as well as in T DD (t ime division duplex) set t ings t hat use reciprocity-based est imat ion t hat progressively improves over t ime. In t his set t ing, where t he channel remains t he same for a finit e durat ion of Tc channel uses, t he t ime index is arranged so t hat h ` Tc + 1 = h ` Tc + 2 = · · · = h (` + 1)Tc g` Tc + 1 = g` Tc + 2 = · · · = g(` + 1)Tc for a non-negat ive int eger `. As a result , in t he presence of a periodic feedback process which repeat s wit h period Tc, we are present ed wit h a periodic sequence of current -CSIT quality exponent s (i )
(i )
↵ t = ↵ ` Tc + t , 8` = 0, 1, 2, · · · , i = 1, 2.
(2.28)
We focus here - simply for t he sake of clarity of exposit ion - on t he symmet ric feedback case ( ↵¯ (1) = ↵¯ (2) = : ↵¯ ). In t his set t ing - and aft er adopt ing a periodic t ime index corresponding t o having ` = 0 (cf. (2.28)) - t he t ime horizon of int erest spans t = 1, 2, · · · , Tc, and t he feedback quality is now represent ed by t he Tc current CSIT quality exponent s { ↵ t } Tt =c 1 and by t he delayed CSIT exponent . Specifically each ↵ t describes t he high SNR precision of t he current CSIT est imat es at t ime t Tc, whereas capt ures t he precision of t he best CSIT est imat e received aft er t he channel has elapsed, i.e., aft er t he coherence period of t he channel. In t his set t ing we have t hat 0
↵1
···
↵ Tc
1
(2.29)
where - since t he channel remains fixed during t he coherence period - any diff erence between two consecut ive exponent s is at t ribut ed t o feedback t hat was received during t hat t ime slot . One of t he ut ilit ies of t his set t ing is t hat it concisely capt ures pract ical t iming issues, capt uring t he eff ect s of feedback t hat off ers an inaccurat e represent at ion of t he t rue st at e of t he channel, as well t he eff ect s of feedback 4. T his definit ion of current vs. delayed CSIT , originat es from [4], and is t he st andard definit ion adopt ed by most exist ing works on t he t opic.
2.3
Per i odi cal ly evol v i ng C SI T
29
t hat can only be used for a small fract ion of t he communicat ion durat ion. Having for example ↵ 1 = 1 simply refers t o t he case of asympt ot ically perfect and immediat ely available (full) CSIT , whereas having ↵ Tc = 0 simply means t hat no (or very limit ed) current feedback is sent during t he coherence period of the channel. Similarly having ↵ Tc = 0, for some 2 [0, 1], simply means t hat no (or very limit ed) current feedback is sent during t he first fract ion of the coherence period 5 . Exam ple 8. Having a periodic feedback process that sends refining feedback, let’s say, two times per coherence period, at times t = 1Tc + 1, t = 2 Tc + 1 and never again about that same channel, will result in having Before feedback
z }| 0 = ↵1 = · · · = ↵
{ 1 Tc
z ↵
A fter fi r st feedback 1 Tc +
1
}| = ··· = ↵
↵ |
{ 2 Tc 2 Tc +
1
= · · · = ↵ Tc = {z }
(2.30)
A fter second feedback
whereas if the same feedback system is modified to further add some delayed feedback after the channel elapses, may allow for > ↵ Tc . One can not e t hat reducing ↵ Tc , implies a reduced amount of feedback - about a specific channel - t hat is sent during t he coherence period of t hat same channel. On t he ot her hand, reducing implies a reduced amount of feedback, during and aft er t he channel’s coherence period. Along t hese lines, reducing ( ↵ Tc ) implies a reduced amount of feedback, about a specific fading coeffi cient , t hat is sent aft er t he coherence period of t he channel. T he result s here hold direct ly from t he previous result s in t his work, where direct ly from (2.9), we now simply have t hat T
↵¯ =
1 Xc ↵t . Tc
(2.31)
t= 1
T he following - which is placed here for complet eness - holds direct ly from Corollary 1a, for t he case of a periodically evolving feedback process over a quasi-st at ic channel. C or ollar y 1f (Periodically evolving feedback). For a periodic feedback process with { ↵ t } Tt =c 1 and perfect delayed CSI T (received at any time after the end of the coherence period), the optimal DoF region over a block-fading channel is the polygon with corner points { (0, 0), (0, 1), ( ↵¯ , 1), (
2 + ↵¯ 2 + ↵¯ , ), (1, ↵¯ ), (1, 0)} . 3 3
(2.32)
5. Our ignoring here of int eger rounding is an abuse of not at ion t hat is only done for t he sake of clarity of not at ion, and it carries no real eff ect on t he result .
30
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
T his same optimal region can in fact be achieved even with imperfect-quality 1+ 2¯↵ delayed CSI T, as long as 3 . R em ar k 2 (Feedback quality vs. quant ity). While all the results here are in terms of feedback quality rather than in terms of feedback quant ity, there are distinct cases where the relationship between the two is well defined. Such is the case when CSI T estimates are derived using basic - and not necessarily optimal - scalar quantization techniques [39]. In such cases, which we mention here simply to off er some insight 6 - and remaining in the high SNR regime - dedicating ↵ log P quantization bits, per scalar, to quantize h into an estimate hˆ , allows for a mean squared error [39] . hˆ k2 = P
Ekh
↵
.
Drawing from t his, and going back t o our previous example, let us consider a similar example. Exam ple 9. Consider a periodic feedback process that sends refining feedback two times per coherence period, by first sending ↵ 0log P bits of feedback per scalar at time t = 1 Tc + 1, then by sending extra ↵ 00log P bits of feedback per scalar at time t = 2Tc + 1, and where it finally sends ( (↵ 0 + ↵ 00)) log P extra bits of refining feedback per scalar, at some fixed point in time after the end of the coherence period of the channel. This would result in having Before feedback
z }| 0 = ↵1 = · · · = ↵
{ 1 Tc 00
A fter fi r st feedback
z ↵0= ↵
↵0+ ↵ = ↵ |
2 Tc +
{z
}| = ··· = ↵
1 Tc + 1
1
= · · · = ↵ Tc }
A fter second feedback, before Tc
{ 2 Tc
| { z}
. (2.33)
A fter coherence per i od
For instance, if this periodic feedback process sends 49 log P feedback bits per scalar, at time t = 13 Tc + 1, and then sends extra 19 log P bits of feedback at time t = 23 Tc + 1, it will allow for A fter fi r st feedback
Before feedback
z }| { 0 = ↵ 1 = · · · = ↵ 1 Tc 3
z }| { 4 = ↵ 1 Tc + 1 = · · · = ↵ 2 Tc 3 3 9 5 = ↵ 2 Tc + 1 = · · · = ↵ Tc 3 |9 {z }
(2.34)
A fter second feedback, before Tc
6. We clarify t hat t his relat ionship between CSIT quality and feedback quant ity, plays no role in t he development of t he result s, and is simply ment ioned in t he form of comment s t hat off er int uit ion. Our focus is on quality exponent s, and we make no opt imality claim regarding t he number of quant izat ion bit s.
2.3
Per i odi cal ly evol v i ng C SI T
31
which gives ↵¯ = (0 + 4/ 9 + 5/ 9)/ 3 = 1/ 3, which in turn gives (Corollary 1f) an optimal DoF region which is defined by the polygon with corner points { (0, 0), (0, 1), (1/ 3, 1), (7/ 9, 7/ 9), (1, 1/ 3), (1, 0)} .
(2.35)
Not e t hat in t his example, t here is no need for ext ra bit s of (delayed) feedback aft er t he end of t he coherence period, because t he exist ing amount and t iming of feedback bit s - again under scalar quant izat ion - guarant ees t hat 1 + 2¯↵ 1 + 2/ 3 = ↵ Tc = 5/ 9 = = 3 3 which we have seen (Corollary 1f) t o already be as good as perfect delayed feedback ( = 1). Placing our focus back on feedback quality, and remaining on t he set t ing of periodically evolving feedback, we proceed wit h a corollary t hat off ers insight on t he quest ion of what CSIT quality and t iming, suffi ce t o achieve a cert ain DoF performance. For ease of exposit ion, we focus on t he hardest t o-achieve DoF point d1 = d2 = d. T he proof is again direct . C or ollar y 1g (Suffi cient feedback for t arget DoF). Having ↵¯ 3d 2 with 2d 1, or having ↵¯ 3d 2 with ↵ Tc 2d 1 (and no extra delayed feedback), suffi ces to achieve a symmetric target DoF d1 = d2 = d. One can see t hat having ↵¯ 3d 2 wit h ↵ Tc 2d 1 simply means t hat t here is no need t o send delayed feedback, i.e., t here is no need t o send feedback aft er t he end of t he coherence period. Anot her pract ical aspect t hat is addressed here - again in t he cont ext of periodically evolving feedback - has t o do wit h feedback delays. Such delays might cause performance degradat ion, which might be mit igat ed if t he feedback - albeit wit h delays - has higher precision. T he following corollary provides some insight on t hese aspect s, by describing t he feedback delays t hat allow a given t arget symmet ric DoF d in t he presence of const raint s on current and delayed CSIT qualit ies. We will be specifically int erest ed in t he allowable fract ional delay of feedback (cf. [10]) , arg max {↵ 0
0T
c
= 0}
(2.36)
i.e., t he fract ion 1 for which ↵ 1 = · · · = ↵ Tc = 0, ↵ Tc + 1 > 0. We are also int erest ed t o see how t his allowable delay reduces in t he presence of a const raint ↵ t ↵ max 8t on t imely feedback, or in t he presence of a const raint on . C or ollar y 1h (Allowable feedback delay). Under a current CSI T quality constraint ↵ t ↵ max 8t, a symmetric target DoF d can be achieved with any
32
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
fractional delay (
3d 2 ↵ m ax
1 1 while under a constraint ( 1 1 1 2 2d 1
if d 2 [2/ 3, (2 + ↵ max )/ 3] if d 2 [0, 2/ 3] max ,
it can be achieved with any
if d 2 [0, 1+ min{ 2m ax ,1/ 3} ] else if d 2 [ 1+ min{ 2m ax ,1/ 3} , 1+
1
m ax
2
]
1, the above reveals that under no specific Finally since ↵ max max constraint on CSI T quality, d can be achieved with ( 3(1 d) if d 2 [2/ 3, 1] 1 if d 2 [0, 2/ 3]. To see t he above, we first not e t hat in t he first case (↵ t ↵ max ), d 2 [0, 2/ 3] can be achieved by using perfect but delayed feedback sent at any point in t ime aft er t = Tc ↵ 1 = · · · = ↵ Tc = 0, {z } |
= 1 | {z }
No feedback
delayed CSI T at t > Tc
(cf. [4]), while d 2 (2/ 3, (2 + ↵ max )/ 3] can be achieved by set t ing ↵ 1 = 2 2d 1 · · · = ↵ Tc = 0, ↵ Tc + 1 = · · · = ↵ Tc = ↵ max , = 1 3d ↵ m ax , (cf. Corollary 1a). 1+ min{ m ax ,1/ 3} In t he second case ( ], t hen d can be max ), if d 2 [0, 2 achieved by using imperfect and delayed feedback sent at any point in t ime aft er t = Tc ↵ 1 = · · · = ↵ Tc = 0, = | { zmax} | {z } No feedback
delayed CSI T at t > Tc
(cf. Corollary 1a), else if d 2 [ 23 , 1+ 2m ax ] and max 1/ 3, t hen d can be achieved by set t ing ↵ 1 = · · · = ↵ Tc = 0, ↵ Tc + 1 = · · · = ↵ Tc = = 2d 1, = 12 2d1 1 1 . Finally, in t he unconst rained case, d 2 [0, 2/ 3] can be achieved by set t ing ↵ 1 = · · · = ↵ Tc = 0 and = 1, while d 2 [2/ 3, 1] can be achieved by using perfect (but part ially delayed) feedback sent at t = Tc + 1 ↵1 = · · · = ↵ | {z
Tc
No feedback
= 0, ↵ } |
Tc + 1
= · · · ↵ Tc = {z
= 1. }
Perfect qualit y CSI T
Exam ple 10. Consider a symmetric target DoF of d1 = d2 = d = 79 . T his can be achieved with = 3(1 d) = 2/ 3 if there is no bound on the quality exponents, and with = 1 (3d 2)/ ↵ max = 1/ 3 if the feedback
2.3
Per i odi cal ly evol v i ng C SI T
33
link only allows for ↵ t ↵ max = 1/ 2, 8t. If on the other hand, feedback timeliness is easily obtained, we can substantially reduce the amount of CSI T and achieve the same d = 79 with ↵ 1 = · · · = ↵ Tc = ↵¯ = 3d 2 = 1/ 3 ( = 0, = 1+32¯↵ = 2d 1 = 5/ 9).
34
2.4
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
U niver sal encoding-decoding scheme
We proceed t o describe t he universal scheme t hat achieves t he aforement ioned DoF corner point s. T he challenge ent ails designing a scheme of an ˆ t ,t 0} nt= 1,t 0= 1 asympt ot ically large durat ion n, t hat ut ilizes a CSIT process { hˆ t ,t 0, g ˆ 0)} n 0 . T his of quality defined by t he st at ist ics of { (h t hˆ t ,t 0), (g g t
t ,t
t = 1,t = 1
will be achieved by focusing on t he corresponding quality-exponent sequences (1) (2) (1) (2) { ↵ t } nt= 1, { ↵ t } nt= 1 , { t } nt= 1 , { t } nt= 1 , as t hese were defined in (2.4)-(2.7). T he opt imal DoF region in T heorem 1 and t he addit ional corner point s in Proposit ion 1, will be achieved by properly ut ilizing diff erent combinat ions of zero forcing, superposit ion coding, int erference compressing and broadcast ing, as well as proper power and rat e allocat ion. P hase-M ar kov for war d-back war d schem e T he scheme has a forwardbackward phase-Markov st ruct ure which, in t he cont ext of imperfect and delayed CSIT , was first int roduced in [8,9] t o consist of four main ingredient s t hat include – block-Markov encoding – spat ial precoding – int erference quant izat ion – backward decoding. T he scheme asks t hat t he accumulat ed quant ized int erference bit s of a cert ain (current ) phase, be broadcast ed t o bot h users inside t he common informat ion symbols of t he next phase, while also a cert ain amount of common informat ion can be t ransmit t ed t o bot h users during t he current phase, which will t hen help resolve t he accumulat ed int erference of t he previous phase. As previously suggest ed, t his causal scheme does not require knowledge of fut ure quality exponent s, nor does it use predict ed CSIT est imat es of fut ure channels. T he t ransmit t er must know t hough t he long t erm averages ↵¯ (1) , ↵¯ (2) , ¯ (1) , ¯ (2) , which - as is commonly assumed of long t erm st at ist ics - can be derived. By ‘feeding’ t his universal scheme wit h t he proper paramet ers, we can get schemes t hat are t ailored t o t he diff erent specific set t ings we have discussed. We will see such examples lat er in this sect ion. We remind t he reader t hat t he users are labeled so t hat ↵¯ (2) ↵¯ (1) . We also remind t he reader of t he soft assumpt ion t hat any suffi cient ly long subse(1) + T (2) ⌧ (1) ⌧ (2) ⌧ +T +T +T quence { ↵ t } t⌧ = ⌧ (resp. { ↵ t } t = ⌧, { t } t = ⌧, { t } t = ⌧) is assumed t o have (1) an average t hat converges t o t he long t erm average ↵¯ (resp. ↵¯ (2) , ¯ (1) , ¯ (2) ), for a finit e T t hat can be suffi cient ly large t o allow for t his convergence. We briefly not e t hat , as we will see lat er, in periodic set t ings such as t hose described in Sect ion 2.3, T need not be large. We proceed t o describe in Sect ion 2.4.1 t he encoding part , and in Sect ion 2.4.2 t he decoding part . In Sect ion 2.4.3 we show how t he scheme
2.4
U niver sal encoding-decoding schem e
35
achieves t he diff erent DoF corner point s of interest . Finally in Sect ion 2.4.4 we provide example inst ances of our general scheme, for specific cases of part icular int erest . For not at ional convenience, we will use hˆ t , hˆ t ,t , hˇ t , hˆ t ,t + ⌘,
ˆt , g ˆ t ,t g ˇt , g ˆ t ,t + ⌘ g
t o denot e t he current and delayed est imat es of h t and gt , respect ively 7 , wit h corresponding est imat ion errors being
2.4.1
h˜ t , h t
hˆ t ,
˜ t , gt g
ˆt g
(2.37)
h¨ t , h t
hˇ t ,
¨ t , gt g
ˇ t. g
(2.38)
Scheme X : encoding
Scheme X is designed t o have S phases, where each phase has a durat ion of T channel uses, and where T is finit e but - unless st at ed ot herwise suffi cient ly large. Specifically each phase s (s = 1, 2, · · · , S) will t ake place over all t ime slot s t belonging t o t he set B s = { B s,` , (s 1)2T + `} T`= 1,
s = 1, · · · , S.
(2.39)
As st at ed, T is suffi cient ly large so t hat 1 X T
t2 Bs
(i )
↵ t ! ↵¯ (i ) ,
1 X T
(i ) t
! ¯ (i ) , s = 1, · · · , S
(2.40)
t2 Bs
i = 1, 2. T he above allocat ion in (2.39) guarant ees t hat t here are T channel uses in between any two neighboring phases. Having T being suffi cient ly large allows for t he delayed CSIT corresponding t o t he channels appearing during phase s, t o be available before t he beginning of t he phase t hat we label as phase s + 1. T his implies t hat T > ⌘(cf. (2.6),(2.7)), alt hough t his assumpt ion can be readily removed 8. Nat urally t here is no silent t ime, and over t he remaining channel uses t 2 { (2s
,s= S 1)T + `} `` == T1,s= 1
7. Recall t hat ⌘is a suffi cient ly large but finit e int eger, corresponding t o t he maximum delay allowed for wait ing for delayed CSIT . 8. T he assumpt ion can be removed because we can, inst ead of split t ing t ime int o two int erleaved halves and ident ifying each half t o a message, t o inst ead split t ime int o more part s, each corresponding t o a diff erent message. For a suffi cient ly large number of part s, t his would allow for t he removal of t he assumpt ion t hat T ⌘, and t he only assumpt ion t hat would remain would be t hat T is large enough so t hat (2.40) is sat isfied. In periodic set t ings, such T can be small.
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Map
Used
Residual
Common information
Common information User 1 private information
Quantized Interference
Map
Used
User 2 private information
Phase s
Residual
36
User 1 private information
Quantized Interference
Map
User 2 private information
Phase (s+1)
Figur e 2.4 – Illust rat ion of coding across phases.
Used
Residual
Common information User 1 private information Help
Quantized Interference
User 2 private information
Recover
One phase
Figur e 2.5 – Illust rat ion of coding over a single phase. we simply repeat scheme X wit h a diff erent message. Wit h n being generally infinit e, S is also infinit e (except for specific inst ances, some of which are highlight ed in Sect ion 2.4.4). We proceed t o give t he general descript ion t hat holds for all phases s = 1, 2, · · · , S 1, except for t he last phase S, which we describe separat ely aft erwards. A brief corresponding illust rat ion can be found in Figure 2.4 and Figure 2.5. P hase s, for s = 1, 2, · · · , S
1
We proceed t o describe t he way t he scheme, in each phase s 2 [1, S 1], combines zero forcing and superposit ion coding, power and rat e allocat ion, and int erference compressing and broadcast ing, in order t o t ransmit privat e informat ion, using current ly available CSIT est imat es t o reduce int erference, and using delayed CSIT est imat es t o alleviat e t he eff ect of past int erference. Zer o for cing and sup er p osit ion coding t ransmit t er sends
During phase s, t 2 B s , t he
0 ? 0 ˆ ?t at + hˆ t at + hˆ t bt + g ˆ t bt x t = w t ct + g
(2.41)
2.4
U niver sal encoding-decoding schem e
37
0
0
where at , at are t he symbols meant for user 1, bt , bt for user 2, where ct is a common symbol, where e? denot es a unit -norm vect or ort hogonal t o e, and where w t is a predet ermined randomly-generat ed vect or known by all t he nodes. Power and r at e allocat ion p olicy In describing t he power and rat es of t he symbols in (2.41), we use t he not at ion (x )
Pt
, E|x t |2
(2.42) (x )
t o denot e t he power of x t corresponding t o time-slot t, and we use r t t o (x ) denot e t he prelog fact or of t he number of bits r t log P o(log P ) carried by symbol x t at t ime t. When in phase s, during t ime-slot t, t he powers and (normalized) rat es are set as (c) . Pt = P, ( 2) (a) . (a) (2) Pt = P t , rt = t ( 1) (b) . (b) (1) (2.43) Pt = P t , rt = t 0) ( 2) ( 2) (a0) . (a (2) (2) Pt = P t ↵t , r t = ( t ↵ t )+ 0) ( 1) ( 1) (b0) . (b (1) (1) Pt = P t ↵ t , r t = ( t ↵ t )+ where (• ) + , max{ • , 0} . We design t he scheme so t hat t he ent irety of common informat ion symbols { cB s, t } Tt= 1 , carry ¯ ) log P
T (1
o(log P )
bit s, and design t he power paramet ers {
1 X T
(i ) t (1) t
t2 Bs
1 X ( T
(i ) t
(2.44)
(1) (2) t , t } t2 Bs
(i ) t
1 X = T
t o sat isfy
i = 1, 2, t 2 B s (2) t
= ¯
(2.45) (2.46)
t2 Bs
(i ) ↵ t )+
= (¯
↵¯ (i ) ) +
i = 1, 2,
(2.47)
t2 Bs
for some ¯ t hat will be bounded by ¯
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) min{ ¯ (1) , ¯ (2) , , }. 3 2 (1)
(2)
(2.48)
T here indeed exist solut ions { t , t } t 2 B s that sat isfy t he above, and an explicit solut ion is shown in Appendix 2.7.1. Our solut ion for power and rate allocat ion allows t hat , at t ime t, t he t ransmit t er needs only acquire
38
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
(1)
(1)
(2)
(2)
knowledge of { ↵ t , t ; ↵ t , t } , in addit ion t o t he derived long-t erm averages ↵¯ (1) , ↵¯ (2) , ¯ (1) , ¯ (2) . T his nat ure of t he derived solut ions is crucial for (1) (2) (1) (2) handling asymmet ry (↵ t 6 = ↵t , t 6 = t ). Aft er t ransmission, t he received signals t ake t he form (1)
yt
0 (1) ˆ ? at + h Tt hˆ t at + zt = h Tt w t ct + h Tt g | { z } | { zt } | {z } | { z}
P
P
( 2) t
( 2) t
P
( 1)
( 1)
ˇ◆
z }t | { ? 0 T ˆ ˇ ˆ b)+ + h t ( h t bt + g | { z t t} P (2)
yt
( 1) t
( 1) ↵t
P0
( 2) ↵t
◆
( 1)
ˇ◆
t z }| t { ? 0 T ¨h ( hˆ bt + g ˆ b) | t t { z t t} ( 1) t
P
( 1) t
(2.49)
P0
? 0 (2) ˆ b + zt = gTt w t ct + gTt hˆ t bt + gTt g | {z } | {z } | { zt }t | { z} P
P
( 1) t
P
( 1) t
( 2)
( 2)
ˇ◆
P0
( 1) ↵t
◆
( 2)
ˇ◆
t z }t | { z }| t { 0 0 T ? T ? ˇ t (ˆgt at + hˆ t at ) + g ¨ t (ˆgt at + hˆ t at ) + g | {z } | {z } ( 2) t
P
( 2) ↵t
P
( 2) t
( 2) t
(2.50)
P0
where 0 0 (1) (2) T ˆ ? T ˆ t bt ), ◆ ◆ g?t at + hˆ t at ) t , h t ( h t bt + g t , g t (ˆ
(2.51)
denot e t he int erference at user 1 and user 2 respect ively, and where 0 0 (1) (2) ˇT ˆ? ˆ t bt ), ˇ◆ ˇ Tt (ˆg?t at + hˆ t at ) ˇ◆ t , h t ( h t bt + g t , g
(2.52)
denot e t he t ransmit t er’s delayed est imat es of t he scalar int erference t erms (1) (2) ◆t , ◆ t . In t he above - where under each t erm we not ed t he order of t he summand’s average power - we considered t hat (1) 2 ˇ Tˆ ? 2 ˇ T ˆ b0|2 E|ˇ◆ t t t | = E|h t h t bt | + E|h t g T T = E|( hˆ t + h˜ t T
= E|( h˜ t ( 1) . =P t
? 0 T T ˆ t bt |2 h¨ t ) hˆ t bt |2 + E|hˇ t g
? 0 T T ˆ t bt |2 h¨ t ) hˆ t bt |2 + E|hˇ t g ( 1)
↵t
0 (2) 2 ¨ Tt )ˆg?t at |2 + E|ˇgTt hˆ t at |2 E|ˇ◆ gTt g t | = E|(˜ ( 2) ( 2) . = P t ↵t .
(2.53)
Quant izing and br oadcast ing t he accumulat ed int er fer ence Aft er t he end of phase s and before t he beginning of t he next phase - which st art s
2.4
U niver sal encoding-decoding schem e
39
Tabl e 2.1 – Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s, s = 1, 2, · · · , S 1.
Privat e symbols for user 1 Privat e symbols for user 2 Common symbols Quant ized int erference
Total bit s (⇥log P ) T ( ¯ + ( ¯ ↵¯ (2) ) + ) T ( ¯ + ( ¯ ↵¯ (1) ) + ) T (1 ¯ ) (1) ¯ T (( ↵¯ ) + + ( ¯ ↵¯ (2) ) + )
T channel uses aft er t he end of phase s, i.e., aft er t he accumulat ion of all de(1) (2) layed CSIT - t he t ransmit t er reconst ruct s ˇ◆ t , ˇ◆ t , t 2 B s using it s knowledge of delayed CSIT , and quant izes t hese int o (1) (1) ¯ˇ◆ = ˇ◆t t (1)
(1)
(1)
(2) (2) ¯ˇ◆ = ˇ◆ t t
˜◆t ,
(2)
(2)
˜◆ t
(2.54)
(2)
↵ t ) + log P and ( t ↵ t ) + log P quant izat ion bit s respect ively, with ( t (1) (2) allowing for bounded power of quant izat ion noise ˜◆t , ˜◆t , i.e, allowing for (2) 2 . (1) 2 . E|˜◆ t | = E|˜◆ t | = 1 ( 2) (2) 2 . t since E|ˇ◆ t | = P t er evenly split s t he
X
( 2)
↵t
(1) 2 . , E|ˇ◆ t | = P
( 1) t
⇣ (
(1) t
(1)
↵ t )+ + (
( 1)
↵t
(2) t
(cf. [39]). T hen t he t ransmit ⌘ (2) ↵ t ) + log P
(2.55)
t2 Bs
quant izat ion bit s int o t he common symbols { ct } t 2 B s+ 1 t hat will be t ransmit t ed during t he next phase (phase s + 1), conveying t hese quant izat ion bit s t oget her wit h ot her new informat ion bit s for the users. T his t ransmission of { ct } t 2 B s+ 1 in t he next phase, will help each of t he users cancel t he dominant part of t he int erference, and it will also serve as an ext ra observat ion (see (2.67) lat er on) that allows for decoding of all privat e informat ion of t hat same user. Table 2.1 summarizes t he number of bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s, s = 1, 2, · · · , S 1. We now proceed wit h t he descript ion of encoding over t he last phase S. P hase S T he last phase, in addit ion t o communicat ing new privat e symbols, conveys t he remaining accumulat ed int erference from t he previous phase, and does so in a manner t hat allows for t erminat ion at t he end of t his phase.
40
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
During t his last phase, t he t ransmit t er sends ? ˆ ?t at + hˆ t bt x t = w t ct + g
(2.56)
t 2 B S , wit h power and rat es set as (c)
Pt
( 2) ( 1) . (a) . (b) . = P, Pt = P ↵ t , Pt = P t (a) (2) (b) (1) r t = ↵t , rt = t .
(2.57)
Wit h t he ent irety of common informat ion symbols { cB S, ` } T`= 1 now carrying 9 T (1 bit s, t he power paramet ers {
↵¯ (2) ) log P (1) t } t 2 BS (1)
↵t 1 X T
o(log P )
(2.58)
are designed such t hat
(1) t (1) t
8t
(2.59)
= ↵¯ (2) .
(2.60)
t2 BS
T he solut ion t o t he above problem is similar t o t hat in (2.45),(2.46),(2.47). T his concludes t he part of encoding. Aft er t ransmission, t he received signals are t hen of t he form (1)
yt
T ? (1) ˆ ?t at + h˜ t hˆ t bt + zt = h Tt w t ct + h Tt g | { z } | { z } | { z } | { z}
P (2)
yt
P0
( 2)
P↵ t
P0
? (2) ˜ Tt g ˆ ?t at + gTt hˆ t bt + zt . = gTt w t ct + g | { z } | { z } | { z } | { z} P
P0
P
( 1) t
(2.61)
(2.62)
P0
We now move t o describe decoding at bot h receivers, where t his decoding part has a phase Markov st ruct ure (see Figure 2.6), similar t o t he encoding part .
2.4.2
Scheme X : decoding
As it may be apparent (more det ails will be shown in Sect ion 2.4.3), t he power and rat e allocat ion in (2.45),(2.46),(2.47) guarant ees t hat t he quant ized int erference accumulat ed during phase s (s = 1, · · · , S 1) has fewer bit s t han t he load of t he common symbols t ransmit t ed during t he next phase (cf. (2.55)). Consequent ly decoding of t he common symbols during a cert ain phase, helps recover t he int erference accumulat ed during t he previous phase. As a result , decoding moves backwards, from t he last t o t he first phase. 9. We remind t he reader of t he definit ion of B s, ` (cf. (2.39)) which denot es t he `t h element of set B s consist ing of all t ime indexes of phase s. For example, saying t hat t = B 1, ` simply means t hat t = `.
2.4
U niver sal encoding-decoding schem e
Private information decoding
41
help
Step Step
Common information joint decoding
Step
Step Interference cancellation
Step Interference reconstruction
help
Phase (s-1)
Phase (s+1)
Phase s
Figur e 2.6 – Illust rat ion of decoding st eps. P hase S At t he end of phase S, we consider joint decoding of all common symbols [cB S, 1 , · · · , cB S, T ]T . Specifically user i , i = 1, 2, decodes t he corresponding (i )
(i )
(i )
common-informat ion vect or using it s received signal vect or [yB S, 1 , yB S, 2 , · · · , yB S, T ]T , and does so by t reat ing t he ot her signals as noise. We now not e t hat t he accumulat ed mut ual informat ion sat isfies (1)
(1)
I ([cB S, 1 , · · · , cB S, T ]T ; [yB S, 1 , · · · , yB S, T ]T ) Y ( 2) = log P1 ↵t o(log P ) t2 BS
= T (1
↵¯ (2) ) log P (2)
o(log P ) (2)
I ([cB S, 1 , · · · , cB S, T ]T ; [yB S, 1 , · · · , yB S, T ]T ) Y ( 1) = log P1 t o(log P ) t2 BS
= T (1
↵¯ (2) ) log P
o(log P )
(2.63)
(cf. (2.59),(2.60)), t o conclude t hat bot h users can reliably decode all T (1
↵¯ (2) ) log P
o(log P )
(2.64)
bit s in t he common informat ion vect or [cB S, 1 , · · · , cB S, T ]T . T his is proved in Lemma 2 in t he appendix of Sect ion 2.7.2, which in fact guarant ees t hat both users will be able t o decode t he amount of feedback bit s described in (2.64), even for finit e and small T . T his is done t o ensure t he validity of t he schemes also for finit e T , and is achieved by employing specific lat t ice codes t hat have good propert ies in t he finit e-durat ion high-SNR regime. T he details for t his st ep can be found in t he aforement ioned appendix.
42
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Aft er decoding [cB S, 1 , · · · , cB S, T ]T , user 1 removes h Tt w t ct from t he received signal in (2.61), t o decode at . Similarly user 2 removes gTt w t ct from it s received signal in (2.62), t o decode bt . Now we go back one phase and ut ilize knowledge of { ct } t 2 B S , t o decode t he corresponding symbols. P hase s, s = S
1, S
2, · · · , 1
We here describe, for phase s, t he act ions of int erference reconst ruct ion, int erference cancelat ion, joint decoding of common informat ion symbols, and decoding of privat e informat ion symbols, in t he order t hey happen. I nt er fer ence r econst r uct ion In t his phase (phase s), each user employs knowledge of { ct } t 2 B s+ 1 from phase s + 1, t o reconst ruct t he delayed est imat es of all t he int erference accumulat ed in phase s, i.e., t o reconst ruct (2) ¯ (1) { ¯ˇ◆ t , ˇ◆ t } t2 Bs . (2) (1) I nt er fer ence cancelat ion Now wit h knowledge of { ˇ¯◆t , ¯ˇ◆t } t 2 B s , each (i ) user can remove - up t o noise level - all t he int erference ◆ t , t 2 B s , by (i ) (i ) subt ract ing t he delayed int erference est imat es ¯ˇ◆ t from yt .
Joint decoding of com m on infor m at ion sy mb ols At t his point , user i decodes t he common informat ion vect or cs , [cB s , 1 , · · · , cB s, T ]T from it s (mo(i ) (i ) (i ) (i ) ¯ˇ◆ ¯ˇ◆ dified) received signal vect or [y ,··· ,y ]T by t reat ing t he B s, 1
B s, 1
B s, T
Bs,T
ot her signals as noise. T he accumulat ed mut ual informat ion t hen sat isfies (1) (1) (1) (1) T I (cs ; [yB s, 1 ¯ˇ◆B s, 1 , · · · , yB s, T ¯ˇ◆ B s, T ] ) Y ( 2) = log P1 t o(log P ) = T (1 ¯ ) log P
o(log P )
t2 Bs (2) (2) (2) (2) T I (cs ; [yB s, 1 ¯ˇ◆B s, 1 , · · · , yB s, T ¯ˇ◆ B s, T ] ) Y ( 1) = log P1 t o(log P ) = T (1 ¯ ) log P
o(log P )
(2.65)
t2 Bs
(cf. (2.45)-(2.50)), and we conclude t hat bot h users can reliably decode all T (1
¯ ) log P
o(log P )
(2.66)
bit s of t he common informat ion vector cs . T he det ails for t his st ep, can again be found in t he appendix of Sect ion 2.7.2. (1) (1) ¯ˇ◆ Aft er decoding cs , user 1 removes h Tt w t ct from yt t , while user 2 (2) (1) T ¯ removes gt w t ct from yt ˇ◆ t , t 2 B s.
2.4
U niver sal encoding-decoding schem e
43
D ecoding of pr ivat e infor m at ion sy mb ols Aft er removing t he int erference, and decoding and subt ract ing out t he common symbols, each user now decodes it s privat e informat ion symbols of phase s. Using knowledge of (2) ¯ (1) (2) ¯ (2) { ¯ˇ◆ (of ˇ◆ t , ˇ◆ t } t 2 B s , user 1 will use t he est imat e ˇ◆ t t ) as an ext ra observa(1) (1) T t ion which, t oget her wit h t he observat ion yt h t w t ct ¯ˇ◆t , will allow for 0 decoding of bot h at and at , t 2 B s . Specifically user 1, at each inst ance t, can ‘see’ a 2 ⇥2 MIMO channel of t he form " " # # (1) (1) (1) T ⇥ ⇤ a z ˜ yt h Tt w t ct ¯ˇ◆ h t t t t = (2.67) ˆ ?t hˆ t 0 + g T (2) (2) ˇ ¯ˇ◆ g a ˜ ◆ t t t t where (1)
z˜ t
? 0 T (1) (1) ˆ t bt ) + zt + ˜◆ = h¨ t ( hˆ t bt + g t .
0 . (1) T he fact t hat E|˜zt |2 = 1, allows for decoding of at and at , corresponding t o (a) (2) (a0) (2) (2) t he aforement ioned rat es r t = t , r t = ( t ↵ t ) + , t 2 B s . Similar 0 actions are t aken by user 2, allowing for decoding of bt and bt , again wit h 0 (b) (1) (a ) (1) (1) ↵ t ) + , t 2 B s. rt = t , rt = ( t At t his point , each user has decoded all t he informat ion symbols (common and privat e) corresponding t o phase s, goes back one phase (t o phase s 1) t o ut ilize it s knowledge of { ct } t 2 B s , and decodes t he common and privat e symbols of t hat phase. T he whole decoding eff ort nat urally t erminat es aft er decoding of t he symbols in t he first phase.
2.4.3
Scheme X : Calculat ing t he achieved D oF
In t he following DoF calculat ion we will consider two separat e cases. Case 1 will correspond t o 2¯↵ (1) which in t urn implies t hat ↵¯ (1) correspond t o 2¯↵ (1)
↵¯ (2) < 1 1+ ↵¯ ( 1) + ↵¯ ( 2) 3
↵¯ (2)
1+ ↵¯ which in t urn implies t hat ↵¯ (1) users are labeled so t hat ↵¯ (1) ↵¯ (2) .
( 1) +
3
(2.68) 1+ ↵¯ ( 2) 2
1 ↵¯ ( 2)
, while case 2 will
(2.69) 1+ ↵¯ ( 2) 2
. We recall t hat t he
G ener ic D oF p oint To calibrat e t he DoF performance, we first not e t hat for any fixed ¯ ( 1) ( 2) ( 2) min{ ¯ (1) , ¯ (2) , 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } (cf. (2.48)), t he rat e and power allocat ion in (2.45),(2.46),(2.47) (as t his policy is explicit ly described in t he appendix
44
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
of Sect ion 2.7.1) t ells us t hat , t he t ot al amount of informat ion, for user 1, in t he private symbols of a cert ain phase s < S, is equal t o ¯ + (¯
↵¯ (2) ) + T log P
(2.70)
↵¯ (1) ) + T log P
(2.71)
bit s, while for user 2 t his is ¯ + (¯
bit s. T he next st ep is t o see how much int erference t here is t o load ont o common symbols. Given t he power and rat e allocat ion in (2.45),(2.46),(2.47),(2.48), it is guarant eed t hat t he accumulat ed quant ized int erference in a phase s < S (cf. (2.55)) has ( ¯ ↵¯ (1) ) + + ( ¯ ↵¯ (2) ) + T log P bit s, which can be carried by t he common symbols of t he next phase (s+ 1) since t hey can carry a t ot al of 1 ¯ T log P bit s (cf. (2.44)). This leaves an ext ra space of com T log P bit s in t he common symbols, where com
,1
¯
(¯
(¯
↵¯ (1) ) +
is guarant eed t o be non-negat ive for any given ¯
↵¯ (2) ) + min{ ¯ (1) , ¯ (2) ,
1+ ↵¯ ( 2) 2
(2.72) 1+ ↵¯ ( 1) + ↵¯ ( 2) 3
,
} . T his ext ra space can be split between t he two users, by allocat ing ! com T log P bit s for t he message of user 1, and t he remaining (1 ! ) com T log P bit s for t he message of user 2, for some ! 2 [0, 1]. Consequent ly t he above, combined wit h t he informat ion st ored in privat e symbols (cf. (2.70),(2.71)), allows for d1 = ¯ + ( ¯ d2 = ¯ + ( ¯
↵¯ (2) ) + + ! ↵¯ (1) ) + + (1
(2.73)
com
!)
com .
(2.74)
T he above considers t hat S is large, and t hus removes t he eff ect of having a last phase t hat carries less new message informat ion. In t he following, we will achieve diff erent corner point s by accordingly set t ing t he value of ! 2 [0, 1] ( 1) ( 2) ( 2) and of ¯ min{ ¯ (1) , ¯ (2) , 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . D oF cor ner p oint s in T heor em 1 To achieved t he DoF region in Theorem 1, we will show how t o achieve t he following DoF corner point s (see also Table 2.2) 1 + ↵¯ (2) 2 (2) B = ↵¯ , 1 A = 1,
C=
2+
2¯↵ (1)
D = 1, ↵¯ (1)
3 .
(2.75) (2.76) ↵¯ (2)
,
2+
2¯↵ (2) 3
↵¯ (1)
(2.77) (2.78)
2.4
U niver sal encoding-decoding schem e
45
To achieve t he DoF region of T heorem 1 we need suffi cient ly good (but cert ainly not perfect ) delayed CSIT such t hat min{ ¯ (1) , ¯ (2) }
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , } 3 2
(2.79)
(cf. T heorem 1), which in t urn implies t hat (cf. (2.48)) ¯
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , }. 3 2
Under t he condit ion of (2.79), t he DoF corner point s are achievable by ( 1) ( 2) ( 2) set ting t he value of ! 2 [0, 1] and of ¯ min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } as in Table 2.2. Specifically when (2.79) and (2.68) hold, we achieve DoF point B by set ting ! = 0, ¯ = ↵¯ (2) which indeed gives (cf. (2.72),(2.73),(2.74)) d1 = ¯ + ( ¯ d2 = ¯ + ( ¯
↵¯ (2) ) + = ↵¯ (2) ↵¯ (1) ) + +
com
= ↵¯ (2) + 1
↵¯ (2) = 1.
To achieve DoF point D we set ! = 1 and ¯ = ↵¯ (1) and get d1 = ¯ + d2 = ¯ +
(¯ (¯
↵¯ (2) ) + +
com
= ↵¯ (1) + 1
↵¯ (1) ) + = ↵¯ (1)
while t o achieve DoF point C we set ! = 0 and ¯ = d1 = ¯ + ( ¯ d2 = ¯ + ( ¯
↵¯ (1) = 1
1+ ↵¯ ( 1) + ↵¯ ( 2) 3
2 + 2¯↵ (1) ↵¯ (2) 3 2 + 2¯↵ (2) ↵¯ (1) ) + + com = 3
and get
↵¯ (2) ) + =
↵¯ (1)
.
On t he ot her hand, when (2.69) (case 2) and (2.79) hold, t o achieve DoF point B we set ! = 0 and ¯ = ↵¯ (2) as before, while t o achieve DoF point A, ( 2) we set ! = 0 and ¯ = 1+ ↵2¯ . Finally t he ent ire DoF region of T heorem 1 is achieved using t ime sharing between t hese corner point s. D oF cor ner p oint s of P r op osit ion 1 Now we focus on t he DoF point s of Proposit ion 1 (see Table 2.3). T hese are t he point s we label as DoF point s B and D , as t hese were defined in (2.76) and (2.78), as well as t hree new DoF point s E = 2 min{ ¯ (1) , ¯ (2) }
↵¯ (2) , 1 + ↵¯ (2) min{ ¯ (1) , ¯ (2) } F = 1 + ↵¯ (1) min{ ¯ (1) , ¯ (2) } , 2 min{ ¯ (1) , ¯ (2) } ↵¯ (1) G = 1, min{ ¯ (1) , ¯ (2) } .
(2.80) (2.81) (2.82)
46
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Tabl e 2.2 – Opt imal corner point s summary, for suffi cient ly good delayed ( 1) ( 2) ( 2) CSIT such t hat min{ ¯ (1) , ¯ (2) } min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } .
Cases
Corner point s
¯
Case 1
C
1+ ↵¯ ( 1) + ↵¯ ( 2) 3
0
D
↵¯ (1)
1
B
↵¯ (2)
0
B
↵¯ (2)
0
A
1+ ↵¯ ( 2) 2
0
Case 2
!
Tabl e 2.3 – DoF inner bound corner point s, for delayed CSIT such t hat ( 1) ( 2) ( 2) min{ ¯ (1) , ¯ (2) } < min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } .
Cases Case 1 and case of min{ ¯ (1) , ¯ (2) }
Corner point s ↵¯ (1)
E , F, B , D
Case 1 and case of min{ ¯ (1) , ¯ (2) } < ↵¯ (1)
B, E, G
Case 2
B, E, G
2.4
U niver sal encoding-decoding schem e
47
As st at ed in t he proposit ion, we are int erest ed in t he range of reduced-quality delayed CSIT , as t his is defined by 1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) min{ ¯ (1) , ¯ (2) } < min{ , } 3 2
(2.83)
and which implies t hat ¯ min{ ¯ (1) , ¯ (2) } (cf. (2.48)). In addit ion t o t he two cases in (2.68),(2.69), we now addit ionally consider t he cases where ↵¯ (1) min{ ¯ (1) , ¯ (2) } min{ ¯ (1) , ¯ (2) } < ↵¯ (1) .
(2.84) (2.85)
When (2.68),(2.83) and (2.84) hold, we set ! = 0, ¯ = ↵¯ (2) as before t o achieve DoF point B . To achieve point D , we set ! = 1 and ¯ = ↵¯ (1) as before, whereas t o achieve point E , we set ! = 0, ¯ = min{ ¯ (1) , ¯ (2) } t o get (cf. (2.72), (2.73), (2.74)) d1 = ¯ + d2 = ¯ +
(¯ (¯
↵¯ (2) ) + = 2 min{ ¯ (1) , ¯ (2) }
↵¯ (2)
↵¯ (1) ) + +
min{ ¯ (1) , ¯ (2) } .
com
= 1 + ↵¯ (2)
Finally t o achieve DoF point F , we set ! = 1 and ¯ = min{ ¯ (1) , ¯ (2) } . When (2.68),(2.83) and (2.85) hold, we achieve point s B and E wit h t he same paramet ers as before, while t o achieve point G, we set ! = 1, ¯ = min{ ¯ (1) , ¯ (2) } . Similarly when (2.69) and (2.83) hold, we achieve point s B , E , G by set t ing ! and ¯ as above. Finally t he ent ire DoF region of Proposit ion 1 is achieved wit h t ime sharing between t he corner point s.
2.4.4
Scheme X : ex am ples
We proceed t o provide example inst ances of our general scheme, for specific cases of part icular int erest . F ixed and im p er fect qualit y delayed C SI T , no cur r ent C SI T (i )
We consider t he case of no current CSIT (↵ t = 0, 8t, i ) and of imperfect (1) (2) delayed CSIT of an unchanged quality t = t 1, 8t. We focus on t he (1) (2) case of t = t = 1/ 3, 8t. T he universal scheme - wit h t hese paramet ers achieves t he opt imal DoF by achieving t he opt imal DoF corner point (d1 = 2 2 3 , d2 = 3 ), as in t he case of [4] which assumed t hat t he delayed feedback of a channel could be sent wit h perfect quality. (i ) (i ) For t his case of t = 1/ 3, ↵ t = 0, we have ↵¯ (1) = ↵¯ (2) = 0, ¯ (1) = ¯ (2) = 1/ 3. Toward designing t he scheme, we set ¯ = 1/ 3 (cf. (2.48)). For t he case of block fading where we can rewrit e the t ime index t o reflect a unit
48
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Tabl e 2.4 – Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s = 1, 2, · · · , S 1.
Privat e symbols for user 1 Privat e symbols for user 2 Common symbols Quant ized int erference
Tot al bit s (log P ) 2/ 3 2/ 3 2/ 3 2/ 3
coherence period, delayed CSIT is simply t he CSIT t hat comes during t he next coherence period, i.e., during t he next t ime slot . Given t he i.i.d. fast fading assumpt ion ( [4]), we can set ⌘= 1 (cf. (2.6),(2.7)), which allows for a simpler variant of our scheme where now t he phases have durat ion T = 1. In t his simplified variant , t he t ransmit t ed signal (cf. (2.41)) t akes t he simple form x t = w t ct +
bt at 0 + 0 at bt
wit h t he power and rat es of t he symbols (cf. (2.43)) set as (c) . (c) Pt = P, r t = 1 1/ 3 0 (a) . (a ) . (b) . (b0) . Pt = Pt = Pt = Pt = P 1/ 3 (a) (a0) (b) (b0) r t = r t = r t = r t = 1/ 3.
(2.86)
During each phase, t he t ransmit t er quant izes - as inst ruct ed in (2.55) - t he int erference accumulat ed in t hat phase, wit h a quant izat ion rat e of 2/ 3 log P , which is mapped int o t he common symbol ct + 1 t hat will be t ransmit t ed in t he next phase (at t ime-slot t + 1). For large enough communicat ion lengt h, simple calculat ions can show t hat this can achieve t he opt imal DoF (d1 = 2 2 3 , d2 = 3 ), and can do so wit h imperfect quality CSIT . Table 2.4 summarizes t he rat es associat ed t o t he symbols in t his scheme. A lt er nat ing b et ween t wo cur r ent -C SI T st at es In t he cont ext of t he two-user MISO BC wit h spat ially and t emporally i.i.d. fading and M = 2, t he work in [11] considered t he alternating CSI T set t ing where CSIT for t he two users, alt ernat es between perfect current CSIT (labeled here as st at e P ), perfect delayed CSIT (D ), or no CSIT (N ). In t his set t ing where I i denot ed t he CSIT st at e for t he channel of user i at any given t ime (I 1 , I 2 2 { P, D , N } ), t he work in [11] considered communicat ion where, for a fract ion I 1 I 2 of t he t ime, t he CSIT st at es are equal t o I 1, I 2 (st at e I 1 for t he first user, st at e I 2 for t he second user).
2.4
U niver sal encoding-decoding schem e
49
T he same work focused on t he symmet ric case where I 1 I 2 = I 2 I 1 . For P , P I 2 2 { P,D ,N } P I 2 being t he fract ion of t he t ime where one user has perP fect CSIT , and D , I 2 2 { P,D ,N } D I 2 being t he fract ion of t he t ime where one user had delayed CSIT , t he work in [11] charact erized t he opt imal DoF region t o t ake t he form d1
1, d1
1,
d1 + 2d2
2+
P
d2 + 2d1
2+
P
d1 + d2
1+
P
+
D. (1)
T he above set t ing corresponds t o our symmet ric set t ing where ↵ t , (2) 2 { 0, 1} , 8t, and where t P D
= =
↵¯ (1) = ↵¯ (2) ¯ (1) ↵¯ (1) = ¯ (2)
(1) (2) t , ↵t ,
(2.87) ↵¯
(2)
(2.88)
in which case our DoF inner bound mat ches the above, and as a result , for 1+ 2¯↵ any ¯ 3 , T heorem 1 generalizes [11] t o any set of quality exponent s, avoiding t he symmet ry assumpt ion, as well as easing on t he i.i.d. block-fading assumpt ion. T he universal scheme described in t his sect ion, can be direct ly applied t o opt imally implement more general alt ernat ing CSIT set t ings. We here off er an example where, in t he presence of suffi cient ly good delayed CSIT , t he current CSIT of t he two users alt ernat es between two quality exponent s equal t o 12 and 34 , i.e., (1) ↵t (2) ↵t
= =
t = 1 t = 2 t = 3 t = 4 ··· 1 3 1 3 ··· 2 4 2 4 3 1 3 1 ··· 4 2 4 2
In t his case, which corresponds t o having ↵¯ (1) = ↵¯ (2) = 5/ 8, we can choose any delayed CSIT process t hat gives ¯ (1) = ¯ (2) = 3/ 4 which suffi ces (see Corollary 1d) t o achieve t he opt imal DoF region by achieving t he opt imal DoF point (d1 = 78 , d2 = 78 ). Toward designing t he scheme, we set ¯ = 3/ 4. For t his example, and again considering a block-fading fast -fading set t ing (unit -lengt h coherence period), t he scheme can have phases wit h durat ion T = 2. T he t ransmit t ed signal (cf. (2.41)) now t akes t he form 0 ? 0 ˆ ?t at + hˆ t at + hˆ t bt + g ˆ t bt x t = w t ct + g
with power and rat es of t he symbols being set as inst ruct ed in (2.43). Again as inst ruct ed by t he general descript ion of t he scheme, at t he end of phase s =
50
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Tabl e 2.5 – Bit s carried by privat e symbols, common symbols, and by t he quant ized int erference, for phase s, s = 1, 2, · · · , S 1, of t he alt ernat ing CSIT scheme.
Privat e symbols for user 1 Privat e symbols for user 2 Common symbols Quant ized int erference
Tot al bit s (⇥log P ) (7 ⇥2)/ 8 (7 ⇥2)/ 8 (1 ⇥2)/ 4 (1 ⇥2)/ 4
1, 2, · · · , S 1, t he t ransmit t er quantizes t he int erference accumulat ed during t hat phase, and does so using a t ot al of 2(1/ 8 + 1/ 8) log P quant izat ion bit s (cf. (2.55)). T hese bit s are t hen mapped int o t he common symbols t hat will be t ransmit t ed in t he next phase. For a large number of phases, t he proposed scheme achieves t he opt imal DoF point (d1 = 78 , d2 = 78 ). Table 2.5 summarizes t he rat es associat ed t o t he symbols in t his scheme. Schem es w it h shor t dur at ion We recall t hat t he Maddah-Ali and Tse scheme [4] uses (under t he employed assumpt ion in [4] of a unit coherence period) T = 3 channel uses, (1) (2) during which it employs 10 1 = 1, 1 = 1 (t he rest of t he exponent s are zero). T he scheme manages t o have t he informat ion bit s of t he quant ized int erference, ‘fit ’ inside t he common symbols in t he above t hree t ime slot s. A similar set t ing where again t he informat ion bit s of t he quant ized int erference, can fit in t he common symbols of a single, short phase, would be if t = 1 t = 2 t = 3 t = 4 ··· (1) 1 ↵t = 0 0 0 ··· 4 (1) 1 1 = 1 0 ··· t 4 4 (2) 1 ↵t = 0 0 0 ··· 4 (2) 1 1 = 1 0 ··· t 4 4 where t he corresponding single-phase (T = 4 t ime-slot s) scheme, can achieve 11 t he opt imal DoF corner point (d1 = 11 16 , d2 = 16 ).
10. We here refer t o an equivalent M AT scheme t hat can be seen as a special case of t he scheme in [6] for ↵ = 0.
2.5
2.5
C oncl usions
51
Conclusions
T he work made progress t oward est ablishing and meet ing t he limit s of using imperfect and delayed feedback. Considering a general CSIT process and a primit ive measure of CSIT quality, t he work provided DoF expressions t hat are simple and insight ful funct ions of easy t o calculat e paramet ers which concisely capt ure t he problem complexity. T he derived insight addresses pract ical quest ions on t opics relat ing t o t he usefulness of predict ed, current and delayed CSIT , t he impact of estimat e precision, t he eff ect of feedback delays, and t he benefit of having feedback symmet ry by employing comparable feedback links across users. Furt her insight was derived from t he int roduced periodically evolving feedback set t ing, which capt ures many of t he engineering opt ions in pract ical feedback set t ings. In t erms of t he applicability of t he DoF high-SNR asympt ot ic approach, for our chosen set t ing of a small number of users (two in t his case), we expect t he high-SNR insight s t o hold for SNR values of operat ional int erest . T he nat ure of t he improved bounds and novel const ruct ions, allows for t his same insight t o hold for a broad family of block fading and non-block fading channel models. We believe t hat t he adopt ed approach is fundament al, in t he sense t hat it considers a general fading process, a general CSIT process, and a primit ive measure of feedback quality in t he form of t he precision of est imat es at any t ime about any channel, i.e., in t he form of t he ent ire set of est imat ion errors ˆ t ,t 0)} nt,t 0= 1 at any t ime about any channel. As we have { (h t hˆ t ,t 0), (gt g seen, t his set of errors nat urally fluct uat es depending on t he inst ance of t he problem, and as expect ed, t he overall opt imal performance is defined by t he st at ist ics of t his error set . T hese st at ist ics are mildly const rained t o t he case of having Gaussian est imat ion errors which are independent of t he prior and current channel est imat es 11. Under t hese assumpt ions, t he result s capt ure t he performance eff ect of t he st at istics of feedback. Int erest ingly t his eff ect - at least for suffi cient ly good delayed CSIT , and for high SNR is capt ured by t he averages of t he quality exponent s. As not ed, t his can be t raced back t o t he assumpt ion t hat t he est imation errors are Gaussian, which ˆ t ,t 0)} nt,t 0= 1 are capt ured by a means t hat t he st at ist ics of { (h t hˆ t ,t 0), (gt g covariance mat rix t hat has diagonal (block) ent ries of t he form { M1 E[||h t ˆ t ,t 0||2F ]} nt,t 0= 1 , and whose off -diagonal ent ries are not used hˆ t ,t 0||2F ], M1 E[||gt g by t he scheme, but where t his scheme t hough meet s an out er bound t hat has kept open t he possibility of any off -diagonal element s. Hence, as st at ed, under our assumpt ions, t he essence of t he CSIT error st at ist ics is capt ured by t he diagonal block element s (of t he aforement ioned covariance mat rix) whose eff ect s are in t urn capt ured - in t he high-SNR regime - by t he quality 11. Again we caut ion t he reader t hat t his is not an assumpt ion about independence between errors, but rat her between errors and est imat es.
52
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
exponent s. T his general approach allows for considerat ion of many facet s of t he performance-vs-feedback quest ion in t he two-user MISO BC set t ing, accent uat ing some import ant facet s while revealing t he reduced role of ot her facet s. For example, while t he approach allows for considerat ion of predict ed CSIT - i.e., of est imat es for fut ure channels - t he result at t he end reveals t hat such est imat es do not provide DoF gains, again under our assumpt ions. In a similar manner, t he result leaves open t he possibility of a role in t he off -diagonal element s of t he aforement ioned covariance mat rix of est imat ion errors, but in t he end again reveals t hat t hese can be neglect ed wit hout a DoF eff ect . Similarly, t he approach allows for any ‘typical’ sequence of quality exponent s - t hus avoiding t he need t o assume periodic or st at ic feedback processes or a block-fading st ruct ure - but despit e t his generality in t he range of t he considered exponent s, in t he end t he result reveals t hat what really mat t ers is t he long-t erm average of each of t hese sequences of current and delayed CSIT exponent s. Finally we believe t he main assumpt ions here t o be mild. Regarding t he high SNR assumpt ion, t here is subst ant ial evidence t hat for primit ive net works (such as t he BC and t he IC) wit h a reasonably small number of users, DoF analysis off ers good insight on t he performance at moderat e SNR. Any possible ext ensions t hough t o t he set t ing of larger cellular networks, may need t o consider sat urat ion eff ect s on t he high-SNR spect ral effi ciency, as t hese were recent ly revealed in [29] t o hold for set t ings where communicat ion involves clust ers of large size. Furt hermore t he assumpt ion of having global CSIR, allowed us t o focus on t he quest ion of feedback t o t he t ransmit t ers, which is a fundament al quest ion on it s own. While t he overhead of gat hering global CSIR must not be neglect ed, it has been repeat edly shown (cf. [35,36]) t hat t his overhead is manageable in t he presence of a reduced number of users. When considering ext ensions t o ot her mult iuser networks wit h pot ent ially more users, such analysis may have t o be combined wit h finding ways t o disseminat e imperfect global CSIR (cf. [35, 36, 38], see also [30, 40]) whose eff ect increases as t he number of users increases. Addit ionally asking t hat current est imat ion errors are independent of current est imat es, is a widely accept ed assumpt ion. Similarly accept ed is t he assumpt ion t hat t he est imat ion error is independent of t he past est imat es, as t his assumpt ion suggest s good feedback processes t hat ut ilize possible correlat ions t o improve current channel est imat es. Finally t he requirement t hat t he running average of t he quality exponent s of a single user, converges t o a fixed value aft er a suffi cient ly long t ime, is also believed t o be reasonable, as it would hold even if t hese exponent s were t hemselves t reat ed as random variables from an ergodic process.
2.6
2.6
A pp endi x - P r oof of out er b ound L em m a
53
A ppendix - Pr oof of out er bound L emma
Proof. Let W1 , W2 respect ively denot e t he messages for t he first and second user, and let R 1 , R 2 denot e t he two users’ rat es. Each user sends t heir message over n channel uses, where n is large. For ease of exposit ion we int roduce t he following not at ion.
ˇT ˇS t , h t , ˇ Tt g
h Tt , gTt
St , (i )
(i )
y[n] , { yt } nt= 1,
ˆT ˆS t , h t , ˆ Tt g
"
(1)
#
z z t , t(2) zt
i = 1, 2
ˇ ˆ n ⌦ [n] , { S t , S t , S t } t = 1 .
T he first st ep is t o const ruct a degraded BC by providing t he first user with complet e and immediat ely available informat ion on t he second user’s received signal. In t his improved scenario, t he following bounds hold.
nR 1 = H (W1 ) = H (W1 |⌦ [n] ) (1)
(2)
I (W1; y[n] , y[n] |⌦ n [n] ) + n✏ I
(1) (2) (W1; W2 , y[n] , y[n] |⌦ [n] ) (1)
(2.89)
+ n✏ n
(2)
= I (W1; y[n] , y[n] |W2, ⌦ n [n] ) + n✏ (1)
(2)
= h(y[n] , y[n] |W2 , ⌦ [n] )
(1)
(2)
h(y[n] , y[n] |W1, W2, ⌦ n [n] ) + n✏ | {z } no(log P )
Xn =
(1)
(2)
(1)
h(yt , yt |y[t
(2)
1]
, y[t
1]
, W2 , ⌦ n [n] ) + no(log P ) + n✏
(2.90)
t= 1
(i )
where (2.89) result s from Fano’s inequality, where y0 was set t o zero by convent ion, and where t he last equality follows from t he ent ropy chain rule and t he fact t hat t he knowledge of { W1 , W2 , ⌦ [n] } implies knowledge of (1)
(2)
{ y[n] , y[n] } up t o noise level.
54
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Similarly
nR 2 = H (W2) (2)
I (W2 ; y[n] |⌦ n [n] ) + n✏ =
(2)
h(y[n] |⌦ [n] ) | {z }
(2.91)
(2)
h(y[n] |W2, ⌦ n [n] ) + n✏
n log P + no(log P ) Xn (2) (2) h(yt |y[t 1] , W2, ⌦ [n] ) + n log P + t= 1 Xn (2) (1) (2) h(yt |y[t 1] , y[t 1] , W2, ⌦ [n] ) t= 1
no(log P ) + n✏ n
+ n log P + no(log P ) + n✏ n
(2.92)
(2.93)
(2.94)
where (2.93) follows from t he ent ropy chain rule and from t he fact t hat received signals are scalars, while t he last st ep is due t o t he fact t hat condit ioning reduces ent ropy. Now given (2.90) and (2.94), we upper bound R 1 + 2R 2 as
n(R 1 + 2R 2 ) 2n log P + no(log P ) + 3n✏ n ⌘ Xn ⇣ (1) (2) (2) ˆt) + h(yt , yt |U, St , Sˆ t ) 2h(yt |U, St , S
(2.95)
t= 1
where
(1)
U , { y[t
(2) [n] } 1] , y[t 1] , W2 , ⌦
(1) (2) ˆt) and where each t erm h(yt , yt |U, St , S
\ St , Sˆ t
(2) ˆ t ) in t he summa2h(yt |U, St , S
2.6
A pp endi x - P r oof of out er b ound L em m a
55
t ion, can be upper bounded as (1)
(2)
(2)
ˆ t ) 2h(y |U, St , S ˆt) h(yt , yt |U, St , S t h i (1) (2) ˆ t ) 2h(y(2) |U, St , S ˆt) max h(yt , yt |U, St , S t PX
t
E[t r(X t X tH )] P
ESˆ t
h (1) (2) ˆt = S ˆ t) ESt |Sˆ t h(yt , yt |U, St = S t , S
max PX
t
E[t r(X t X tH )] P (2) ˆt = S ˆ t) 2h(yt |U, St = S t , S
= ESˆ t
max PX
h ES˜ t h(S t x t + z t |U)
(2)
i i
2h(gTt x t + zt |U)
t
E[t r(X t X tH )] P
= ESˆ t ESˆ t
i ⇥ ES˜ t log det (I + S t S Ht ) 2 log (1+ gHt gt ) ⌫0:t r( ) P i ⇥ max ES˜ t log (1 + h Ht h t ) log (1 + gHt gt ) . max
⌫0:t r( ) P
(2.96) (2.97)
In t he above, (2.96) uses t he result s in [41, Corollary 4] t hat t ell us t hat Gaussian input maximizes t he weight ed diff erence of two diff erent ial ent ro(2) (1) (2) pies 12 , as long as : 1) yt is a degraded version of { yt , yt } ; 2) U is (1) (2) independent of zt , zt ; 3) t he input maximizat ion is done given a fixed faˆ t , and is independent of S ˜ t 13. Furt hermore, in t he above, ding realizat ion S (2.97) comes from Fischer’s inequality which gives t hat det (I + S t S Ht ) (1 + h Ht h t )(1 + gHt gt ). At t his point we follow t he st eps involving equat ion (25) in [6], t o upper bound t he right hand side of (2.97) as i ⇥ max ES˜ t log (1 + h Ht h t ) log (1 + gHt gt ) ESˆ t ⌫0:t r( ) P
(2)
↵ t log P + o(log P ).
(2.98)
Combining (2.95) and (2.97), gives t hat Xn ⇣ n(R 1 + 2R 2)
⌘ (2) (2 + ↵ t ) log P + o(log P ) + 3✏ n
t= 1
12. We not e t hat t he result s in [41, Corollary 4] are described for t he non-fading channel model, however, as argued in t he same work in [41, Sect ion V ], t he result s can be readily ext ended t o t he fading channel model by linearly t ransforming t he fading channel int o an equivalent non-fading channel, wit h t he new channel act ually maint aining t he same capacity and t he same degradedness order. 13. We recall t hat x t is only a funct ion of t he messages and of t he CSIT (current and delayed) est imat es up t o t ime t, and t hat t hese CSIT est imat es are assumed t o be independent of t he current est imat e errors at t ime t.
56
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
and consequent ly t hat d1 + 2d2
2 + ↵¯ (2) .
Similarly, int erchanging t he roles of t he two users, allows for d2 + 2d1
2 + ↵¯ (1) .
Finally t he fact t hat each user has a single receive ant enna, gives t hat d1 1, d2 1.
2.7
A pp endi x - Fur t her det ai ls on t he schem e
2.7 2.7.1
57
A ppendix - Fur t her det ails on t he scheme Explicit p ower allocat ion solut ions under const r aint s in equat ions (2.45),(2.46),(2.47)
We remind t he reader t hat , in designing the power allocat ion policy of (1) (2) t he scheme, we must design t he power paramet ers { t , t } t 2 B s t o sat isfy equat ions (2.45),(2.46),(2.47) which asked t hat (i ) t
1 X T
(1) t
t2 Bs
1 X ( T
(i ) t
(i ) t
i = 1, 2, t 2 B s
1 X = T
(2) t
= ¯
t2 Bs
(i )
↵ t )+ = ( ¯
↵¯ (i ) ) +
i = 1, 2
t2 Bs
for a given ¯ 2 [0, 1]. For each phase s, we here explicit ly describe such (1) (2) sequence { t , t } t 2 B s , which is const ruct ed using a wat erfilling-like approach. We first consider t he case where ¯ ↵¯ (i ) . At any given t ime t = B s,1 , B s,2, · · · , B s,T , we set ( (i ) (i ) (i ) T ( ¯ ↵¯ (i ) ) T ( ¯ ↵¯ (i ) ) (i ) ,t + ↵ t if t ,t + ↵ t = t (i ) (i ) (i ) if t < T ( ¯ ↵¯ (i ) ) ,t + ↵ t t where
,t
is init ialized t o zero (
t hat t he calculat ion of
(i ) t + 1,
,B s, 1
= 0), and is updat ed each t ime, so
uses
,t + 1
=
,t
+
In t he end, t he solut ion t akes t he form 8 (i ) > t , < (i ) ¯ ↵¯ (i ) ) + ↵ (i ) P ⌧0 1 ( t = T( t `= 1 > : (i ) ↵t ,
(i ) t
(i )
↵t .
t = B s,1, · · · , B s,⌧0 1 (i ) ↵ B s, ` ), t = B s,⌧0
(i ) B s, `
t = B s,⌧0+ 1 , · · · , B s,T
0 is a funct ion 14 of t he quality exponent s during phase s. T his design where ⌧ (i ) of { t } t 2 B s sat isfies (2.45),(2.46), as well as (2.47), since, for t he case where ¯ ↵¯ (i ) 0, we deliberat ely force (i ) ↵ (i ) 0, t 2 B s . t t Similarly for ¯ ↵¯ (i ) , we set ( (i ) (i ) ↵t if ↵ t T¯ (i ) ,t = t (i ) ¯ ¯ T if ↵ t > T ,t ,t 0 14. Not e t hat t here is no need t o explicit ly describe ⌧ , because t he schemes are explicit ly (i ) 0 described as a funct ion of t he above t , which - aft er calculat ion - also reveal ⌧ which by design - falls wit hin t he proper range.
58
C hapt er 2
where
,t
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
is init ialized t o zero, and is updat ed as ,t + 1
=
,t
+
(i ) t .
In t he end, in t his case, t he solut ion t akes t he form 8 (i ) > t = B s,1, · · · , B s,⌧0 1 < ↵t , P ⌧0 1 (i ) (i ) ¯ = T t = B s,⌧0 t ` = 1 ↵ B s, ` , > : 0, t = B s,⌧0+ 1 , · · · , B s,T 0 where again ⌧ is a funct ion of t he quality exponent s during phase s. T his sat isfies (2.45),(2.46), as well as (2.47) since, for t he case where ¯ ↵¯ (i ) 0, (i ) (i ) we again have t ↵t 0, t 2 B s .
2.7.2
Encoding and decoding det ails for equat ions (2.64),(2.66)
We here elaborat e on how t he users will be able t o decode t he amount of feedback bit s described in equat ions (2.64) and (2.66). We first provide t he following lemma, which holds for any T . L em m a 2. Let (1) y¯ t (2)
y¯ t
= ct + P = ct + P
( 2) t 2 ( 1) t 2
(1)
z¯ t , (2)
z¯ t ,
(2.99) t = 1, 2, · · · ,T
(2.100)
P . (i ) (i ) ¯ ⇤ for a given where E[|ct |2] P , P r (|¯zt |2 > P ✏) = 0, and T1 Tt= 1 t ¯ ⇤ 2 [0, 1], i = 1, 2. Also let r , 1 ¯ ⇤ ✏for a vanishingly small but positive ✏> 0, and consider communication over T channel uses. T hen for any rate up to R = r log P o(log P ) (bits/ channel use), the probability of error can be made to vanish with asymptotically increasing SNR. Proof. We will draw each T -lengt h codevect or c , [c1, · · · , cT ]T from a lat t ice code of t he form { ✓M q | q 2 @}
(2.101)
where @⇢ CT is t he T -dimensional 2R -QAM const ellat ion, where M 2 CT ⇥T is a specifically const ruct ed unit ary mat rix of algebraic conjugat es t hat allows for t he non vanishing product distance property (t o be described lat er on see for example [42]), and where ✓= P
1
r 2
= P(
¯ ⇤+ ✏ )/ 2
(2.102)
2.7
A pp endi x - Fur t her det ai ls on t he schem e
59
. is designed t o guarant ee t hat E||c||2 = P (to derive t his value of ✓, just . . recall t he QAM property t hat E||q||2 = 2R = P r ). Specifically for any two T ? ? ? codevect ors c = [c1 , · · · , cT ] , c = [c1 , · · · , cT ]T , M is designed t o guarant ee t hat YT
c?t )|2 ˙ ✓2T .
|(ct
(2.103)
t= 1
T his can be readily done for all dimensions T by, for example, using t he proper root s of unity as ent ries of a circulant M (cf. [42]), which in t urn allows for t he above product - before normalizat ion wit h ✓- t o t ake non-zero int eger values. In t he post -whit ened channel model at user i = 1, 2, we have ( 2) 1 /
y¯ (1) , diag(P
( 2) 1 /
= diag(P
( 1) 1 /
y¯ (2) , diag(P
2
2
2
( 2) T /
,··· ,P
2
( 2) T /
,··· ,P
( 1) T /
,··· ,P
) y¯ (1)
2
2
)c + z¯ (1)
)c + z¯ (2)
where, as we have st at ed, t he noise z¯ (i ) has finit e power in t he sense t hat P r (||¯z (i ) ||2 > P ✏) ! 0.
(2.104)
At t he same t ime, aft er whit ening at each user, t he codeword dist ance for any two codewords c, c? , is lower bounded as (i ) 1 /
||diag(P XT = .
2
(i ) t /
|P t= 1 YT
(i ) t /
|P
,··· ,P 2
2
(i ) T /
2
)(c
c? )||2
c?t)|2
(ct
c?t )|2/ T
(ct
(2.105)
t= 1
= P
1 T
P
T t= 1
(i ) t
YT |(ct
c?t)|2/ T
t= 1 .
P P
1 T
P
¯⇤
T t= 1
P
(i ) t
¯ ⇤+ ✏
✓2
= P
(2.106) ✏
(2.107)
for i = 1, 2, where (2.105) result s from t he arithmet ic-mean geomet ric-mean inequality, (2.106) is due t o (2.103), and where (2.107) uses t he assumpt ion P (i ) ¯ ⇤. Set t ing ✏posit ive but vanishingly small, combined t hat T1 Tt= 1 t with (2.104), proves t he result .
60
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
At t his point , we use t he lat t ice code of t he above lemma, t o design t he T -lengt h vect or c t ransmit t ed during phase s. T his encoding guarant ees successful decoding of t his vect or, at bot h users, at a rat e R = r log P o(log P ), where r = 1 ↵¯ (2) for phase S, else r = 1 ¯ (✏is set posit ive but vanishingly small, recall (2.64), (2.66)). We not e t hat for phase S, user i = 1, 2 (i ) can linearly t ransform t heir signal observat ions { yt } t 2 B S (cf. (2.61),(2.62)) t o t ake t he form in (2.99),(2.100), while for phase s = 1, 2, · · · , S 1, (i ) (i ) ¯ˇ◆ user i = 1, 2 can linearly t ransform t heir signal observat ions { yt t } t2 Bs (i ) (aft er removing t he int erference ¯ˇ◆ t , cf. (2.65),(2.49),(2.50)), again t o t ake t he form in (2.99),(2.100). Finally we not e t hat t he achievable rat e is det ermined by t he exponent P (i ) (i ) average T1 Tt= 1 t and not by t he inst ant aneous exponent s t .
2.8
A pp endi x - D iscussi on on est im at es and er r or s assum pt i on
2.8
61
A ppendix - D iscussion on independence of est imat ion er r or and past est imat es
T he assumpt ion on independence of est imat ion error and past est imat es, is consist ent wit h a large family of channel models ranging from t he fast fading channel (i.i.d in t ime), t o t he correlat ed channel as t his was present ed in [6] 15 , and even t he quasi-st at ic slow fading model where t he CSIT est imat es are successively refined over t ime. Successive CSIT refinement - as t his is t reat ed in [16] - considers an increment al amount of quant izat ion bit s t hat progressively improve t he CSIT est imat es. For example, focusing on t he est imat es of channel h 1, t he quality of t his estimat e would improve in t ime, with a successive refinement t hat would ent ail h 1 = hˆ 1,1 + h˜ 1,1 ˆ˜ ˜ = hˆ 1,1 + h 1,1,2 + h 1,2 | {z } hˆ 1, 2
ˆ˜ ˆ˜ ˜ = hˆ 1,1 + h 1,1,2 + h 1,2,3 + h 1,3 | {z } hˆ 1, 3
.. . where
˜ˆ 1,t 0,t 00 + h˜ 1,t 00 h˜ 1,t 0 , h
ˆ˜ 0 00 and where h 1,t ,t denot es t he est imat e correct ion t hat happens between t ime 0 0 0 t and t . Generalizing t his t o t he est imat e of any channel h t , and accept ing t hat ˆ˜ 0 00 ˜ t he est imat e correct ion h t ,t ,t and est imat e error h t ,t 00 are st at ist ically independent , allows t hat t he est imat ion error h˜ t ,t 00 of h t is independent of t he previous and current est imat es { hˆ t ,⌧} ⌧ t 00, which in t urn allows for t he aforement ioned assumpt ion t o hold even for t he block fading channel model. As a side not e, even t hough we consider t he quant ificat ion of CSIT quality as in (2.37), we not e t hat our result s can be readily ext ended t o t he case where we est imat e channel direct ions (phases), in which case we would simply ˆt + g ˜ t (cf. [23]). consider ||h1t || h t = hˆ t + h˜ t , ||g1 || gt = g t
15. Not e t hat our assumpt ion is soft er t han t he assumpt ion in [6] where ⇤ ˆ t , t 0 } tt 0= 1 , h t , g t } tt = 1 1 $ { hˆ t ⇤, t ⇤, g ˆ t ⇤, t ⇤} $ { h t ⇤, g t ⇤} was assumed t o be a M ar{ { hˆ t ,t 0 , g kov chain ; an assumpt ion which may not direct ly hold in block fading set t ings where for example, having h t ⇤ 1 = h t ⇤ (resp. g t ⇤ 1 = g t ⇤), breaks t he chain { h t ⇤ 1 , g t ⇤ 1 } $ ˆ t ⇤, t ⇤} $ { h t ⇤, g t ⇤} because, given h t ⇤ 1 = h t ⇤, t he following condit ional pro{ hˆ t ⇤, t ⇤, g bability density funct ions hold f h t ⇤ 1 , h t ⇤| hˆ t ⇤ = f h t ⇤| hˆ t ⇤ 6 = f h t ⇤| hˆ t ⇤ f h t ⇤ 1 | hˆ t ⇤ as long ˆ as h t ⇤ 6 = h t ⇤ (nat urally for two random variables X 1 and X 2 , t hen X 1 |X 2 cannot be independent of X 1 |X 2 no mat t er what X 1 and X 2 are, unless X 1 = X 2 ).
62
C hapt er 2
2.9
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
A ppendix - A not her Out er B ound Pr oof
We here provide anot her out er bound proof (cf. Lemma 1), remaining in (1) (2) t he general set t ing of having general CSIT feedback qualit ies (↵ t 6 = ↵ t ), except t hat now t he channel coeffi cient s are assumed t o be i.i.d. t emporal variat ions, and t hat M = 2. Adopt ing t he out er bound approach in [7], we first linearly convert t he original BC in (2.1),(2.2) t o an equivalent BC (see (2.108a),(2.108b)) having t he same DoF region as t he original BC (cf. [7]), and we t hen consider t he degraded version of t he equivalent BC in t he absence of delayed feedback, which mat ches in capacity t he degraded BC wit h feedback (for t he memoryless case), and which exceeds t he capacity of t he equivalent BC. T he final st ep considers t he compound and degraded version of t he equivalent BC wit hout delayed feedback, whose DoF region will serve as an out er bound on t he DoF region of t he original BC.
T he equivalent degr aded com p ound B C direct ly from (2.1),(2.2) we have t hat
(1)
yt
(2)
yt
Towards t he equivalent BC,
(1)
= h Tt x t + zt p 1 (1) = h Tt P Q t p Q t 1x t + zt P p 0 (1) T = h t P Q t x t + zt p p T ? (1) ˆ ?t x 1t + P h˜ t hˆ t x 2t + zt = P h Tt g
(2.108a)
(2)
= gTt x t + zt p 0 (2) = gTt P Q t x t + zt p p ? (2) ˜ Tt g ˆ ?t x 1t + P gTt hˆ t x 2t + zt , = Pg
where
0
x t , [x 1t x 2t]T , p
1 Q t 1x t , P
(2.108b)
2.9
A pp endi x - A not her Out er B ound P r oof
63
? where Q t , [ˆg?t hˆ t ] 2 C2⇥2 is, wit h probability 1, an invert ible mat rix. Furt hermore each receiver normalizes t o get 0(1)
yt
(1)
= = 0(2)
yt
yt ˆ ?t h Tt g
=
p
P x 1t (2) yt ? gTt hˆ t
=
p p q
0(1)
( 1)
zt ˆ ?t h Tt g
0
+ q
p
=
=
p
P x 1t
P x 2t
P1
+
p + q
P x 2t
T ? (1) P h˜ t hˆ t x 2t zt + ˆ ?t ˆ ?t h Tt g h Tt g 0(1)
0
ht x 2t + zt
,
(2.109a)
,
(2.109b)
(2)
˜ Tt g ˆ ?t x 1t Pg z + t ? ? gTt hˆ t gTt hˆ t P1
+
( 1)
↵t
( 2)
↵t
0(2)
0
gt x 1t + zt
q ( 1) ↵t
˜
T
ˆ?
0(2)
z
( 2)
0
( 2)
P↵t
˜ Tg ˆ? g
t t , ht = P h T gˆ h? t h t , zt = Tt ˆ ? , gt = . Conse? gt h t g Tt hˆ t t t q q ( 1) ( 2) ˜ t have ident ity covariance mat rices, and t he quent ly P ↵ t h˜ t and P ↵ t g 0(1) 0(2) 0 0 average power of ht , gt , zt and zt does not scale wit h P , i.e., in t he high-SNR region t his power is of order P 0 . Wit h t he same CSIT knowledge mapped from t he original BC, it can be shown (see [7]) t hat t he DoF region of t he equivalent BC in (2.109a)(2.109b) mat ches t he DoF region of t he original BC in (2.1)(2.2). Towards designing t he degraded version of t he above equivalent BC, we 0(1) supply t he second user wit h knowledge of yt , and t owards designing t he compound version of t he above degraded equivalent BC, we add two ext ra users (user 3 and 4). In t his compound version, t he received signals for t he first two users are as in (2.109a)(2.109b), while t he received signals of t he added (virt ual) users are given by q p 00(1) 00(1) ( 1) 00 1 yt = P x t + P 1 ↵ t ht x 2t + zt , (2.110a) q p 00(2) 00(2) ( 2) 00 yt = P x 2t + P 1 ↵ t gt x 1t + zt . (2.110b)
where zt
=
00
00
We here not e t hat by definit ion, ht and gt are st at ist ically equivalent t o 00(1) 00(2) 0 0 t he original ht and gt respect ively, and t hat zt and zt are st at ist ically 0(1) 0(2) equivalent t o t he original zt and zt . Furthermore we not e t hat user 3 is int erest ed in t he same message as user 1, while user 4 is int erest ed in t he same message as user 2. Also we recall t hat in t he specific degraded 0(1) 0(2) 0(1) compound BC, user 1 knows yt , user 2 knows yt and yt , user 3 knows
64
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
00(1)
00(2)
00(1)
yt , and user 4 knows yt and yt . Finally we remove delayed feedback - a removal known t o not aff ect t he capacity of t he degraded BC wit hout memory [43]. We now proceed t o calculat e an out er bound on t he DoF region of t his degraded compound BC which at least mat ches t he DoF of t he previous degraded BC and which serves as an out er bound on t he DoF region of t he original BC. Out er b ound We consider communicat ion over t he described equivalent degraded compound BC, let t ing n be t he large number of fading realizat ions over which communicat ion t akes place, and let t ing R 1, R 2 be t he rat es of t he 0 0 ˆ ˆ t } n , y (i ) , { y (i ) } n first and second user. We also let ⌦ [n] , { h t , g t , h t , g t= 1 t t= 1 [n] 00(i )
00(i )
and y[n] , { yt } nt= 1 for i = 1, 2. Using Fano’s inequality, we have nR 1
0(1)
I (W1; y[n] |⌦ [n] ) + no(n) 0(1)
n logP + no(logP ) h(y[n] |W1 , ⌦ [n] ) + no(n),
(2.111)
as well as nR 1
00(1)
I (W1; y[n] |⌦ [n] ) + no(n) 00(1)
n logP + no(logP ) h(y[n] |W1, ⌦ [n] ) + no(n),
(2.112)
which is added t o (2.111) t o give 2nR 1
0(1)
h(y[n] |W1, ⌦ [n] )
2n log P + 2no(log P ) 00(1)
h(y[n] |W1 , ⌦ [n] ) + 2no(n) 2n log P + 2no(log P ) 0(1)
00(1)
h(y[n] , y[n] |W1 , ⌦ [n] ) + 2no(n).
(2.113)
Let q y¯ t , diag(1,
1
0
P
( 1) ↵t
2 p 1 P xt 6q = p 2 + 4 P xt
1 ht ) 00 1 ht 0( 1)
zt
P
00
00( 1)
h t zt 00 0 h t ht
( 1) ↵t
00( 1) zt 00 ht
p P x 1t z¯ 0 p = + t + 0 zt P x 2t
"
0(1)
#
yt 00(1) yt 3 0
ht
0( 1) zt 0 ht
7 5
(2.114)
2.9
A pp endi x - A not her Out er B ound P r oof 0( 1)
where z¯ t =
zt
Consequent ly
00
00( 1)
h t zt 00 0 ht h t
q
0
ht
( 1)
P↵t
, zt =
00( 1)
zt
00
ht
65 0( 1)
zt 0 ht
, and let z[n] , { zt } nt= 1.
nR 1 + nR 2 = H (W1 , W2) 0(1)
00(1)
= I (W1 , W2 ; y[n] , y[n] , z[n] |⌦ [n] ) 0(1)
00(1)
+ H (W1, W2 |y[n] , y[n] , z[n] , ⌦ [n] ) 0(1)
00(1)
= I (W1 , W2 ; y[n] , y[n] , z[n] |⌦ [n] ) + no(log P ) + no(n) 0(1)
(2.115)
00(1)
= I (W1 ; y[n] , y[n] , z[n] |⌦ [n] ) 0(1)
00(1)
+ I (W2; y[n] , y[n] , z[n] |⌦ [n] , W1 ) + no(log P ) + no(n),
(2.116)
where t he t ransit ion t o (2.115) uses t he fact t hat t he high SNR variance ( 1)
of z¯ t and zt scales as P 0 and P ↵ t respect ively, which in t urn means t hat 0(1) 00(1) n knowledge of { yt , yt , zt , ⌦ [n] } t = 1 , implies knowledge of W1 , W2 and of 1 2 n { x t , x t } t = 1 , up t o bounded noise level. Furt hermore
nR 1= H (W1 ) 0(1)
00(1)
0(1)
00(1)
0(1)
00(1)
= I (W1 ; y[n] , y[n] ,z[n] |⌦ [n])+H (W1 |y[n] , y[n] ,z[n] , ⌦ [n]) = I (W1 ; y[n] , y[n] , z[n] |⌦ [n]) + no(logP ) + no(n),
0(1)
since again knowledge of { yt ded noise level.
00(1)
, yt
(2.117)
n , zt , ⌦ [n] } t = 1 provides for W1 up t o boun-
66
C hapt er 2
D oF and Feedback Tr adeoff over Two-U ser M I SO B C
Now combining (2.116) and (2.117), gives 0(1)
00(1)
nR 2 = I (W2; y[n] , y[n] , z[n] |⌦ [n] , W1 ) + no(log P ) + no(n) 0(1)
00(1)
= I (W2; y[n] , y[n] |⌦ [n] , W1 ) 0(1)
00(1)
+ I (W2; z[n] |y[n] , y[n] , ⌦ [n] , W1 ) + no(log P ) + no(n) 0(1)
00(1)
0(1)
00(1)
= h(y[n] , y[n] |⌦ [n] , W1) h(y[n] , y[n] |⌦ [n] , W1 , W2 ) | {z } no(log P ) 0(1)
00(1)
h(z[n] |y[n] , y[n] , ⌦ [n] , W1 , W2 ) | {z } no(log P ) 0(1)
00(1)
+ h(z[n] |y[n] , y[n] , ⌦ [n] , W1 ) + no(log P ) + no(n) | {z } h(z[n ] ) 0(1)
00(1)
0(1)
00(1)
h(y[n] , y[n] |⌦ [n] , W1) + h(z[n] ) + no(log P ) + no(n) h(y[n] , y[n] |W1 , ⌦ ¯ (1) log P [n] ) + n ↵ + no(log P ) + no(n), which is combined wit h (2.113) t o give 2nR 1 + nR 2
2n log P + n ↵¯ (1) log P + no(log P ) + no(n),
(2.118)
which in t urn proves t he out er bound 2d1 + d2
2 + ↵¯ (1) .
(2.119) (1)
(2)
Finally int erchanging t he roles of t he two users and of ↵ t , ↵ t , gives d1 + 2d2
2 + ↵¯ (2) .
Nat urally t he single ant enna const raint gives t hat d1
(2.120) 1, d2
1.
2
Chapt er 3
Fundament al Performance and Feedback Tradeoff over t he M I M O BC and I C
Ext ending recent findings on t he two-user MISO broadcast channel (BC) with imperfect and delayed channel st at e informat ion at t he t ransmit t er (CSIT ), t he work here explores t he performance of t he two user MIMO BC and t he two user MIMO int erference channel (MIMO IC), in t he presence of feedback wit h evolving quality and t imeliness. Under st andard assumpt ions, and in t he presence of M ant ennas per t ransmit t er and N ant ennas per receiver, t he work derives t he DoF region, which is opt imal for a large regime of suffi cient ly good (but pot ent ially imperfect ) delayed CSIT . T his region concisely capt ures t he eff ect of having predict ed, current and delayed-CSIT , as well as concisely capt ures t he eff ect of t he quality of CSIT off ered at any t ime, about any channel. In addit ion t o t he progress t owards describing t he limit s of using such imperfect and delayed feedback in MIMO set t ings, t he work off ers diff erent insight s t hat include t he fact t hat , an increasing number of receive ant ennas can allow for reduced quality feedback, as well as t hat no CSIT is needed for t he direct links in t he IC.
3.1 3.1.1
I nt r oduct ion M I M O B C and M I M O I C channel models
For t he set t ing of t he mult iple-input multiple-out put broadcast channel (MIMO BC), we consider t he case where an M -ant enna t ransmit t er, sends 67
68
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
informat ion t o two receivers wit h N receive ant ennas each. In t his set t ing, t he received signals at t he two receivers t ake t he form (1)
yt
(2) yt (1)
= H = H
(1) t xt (2) t xt
(1)
(3.1)
(2) zt
(3.2)
+ zt +
(2)
where H t 2 CN ⇥M , H t 2 CN ⇥M respect ively represent t he first and se(1) (2) cond receiver channels at t ime t, where z t , z t represent unit power AWGN noise at t he two receivers, where x t 2 CM ⇥1 is t he input signal wit h power const raint E[||x t ||2 ] P . For t he set t ing of t he MIMO interference channel (MIMO IC), we consider a case where two t ransmit t ers, each wit h M t ransmit ant ennas, send informat ion t o t heir respect ive receivers, each having N receive ant ennas. In t his set t ing, t he received signals at t he two receivers t ake t he form (1)
= H
(2)
= H
yt yt (11)
(11) (1) xt t (21) (1) xt t
+ H + H
(12) (2) xt t (22) (2) xt t
(1)
(3.3)
(2)
(3.4)
+ zt + zt
(22)
where H t 2 CN ⇥M , H t 2 CN ⇥M represent t he fading mat rices of t he (12) (21) 2 CN ⇥M , H t 2 CN ⇥M , represent direct links of t he two pairs, while H t t he fading mat rices of t he cross links at t ime t.
3.1.2
D egr ees-of-fr eedom as a funct ion of feedback qualit y
In t he presence of perfect channel st at e informat ion at t he t ransmit t er (CSIT ), t he degrees-of-freedom (DoF) performance 1 for t he case of t he MIMO BC, is given by (cf. [1]) { d1
min{ M , N } , d2
min{ M , N } , d1 + d2
min{ M , 2N } }
(3.5)
whereas for t he MIMO IC, t his DoF region wit h perfect CSIT , is given by (cf. [44]) { d1
min{ M , N } , d2
min{ M , N } , d1 + d2
min{ 2M , 2N , max{ M , N } } } . (3.6)
In t he absence of any CSIT t hough, t he BC performance reduces, from t hat in (3.5), t o t he DoF region { d1 + d2
min{ M , N } }
(3.7)
1. We remind t he reader t hat in t he high-SNR set t ing of int erest , for an achievable rat e pair (R 1 , R 2 ) for t he first and second receiver respect ively, t he corresponding DoF Ri pair (d1 , d2 ) is given by di = limP ! 1 l og P , i = 1, 2 and t he corresponding DoF region is t hen t he set of all achievable DoF pairs.
3.1
I nt r oduct ion
69
corresponding t o a symmet ric DoF corner point (d1 = d2 = min{ M , N } / 2) (cf. [3, 45]). Similarly t he performance of t he MIMO IC wit hout any CSIT , reduces from t he DoF region in (3.6), t o t he DoF region { d1
min{ M , N } , d2
min{ M , N } , d1 + d2
min{ N , 2M } }
(3.8)
corresponding t o a symmet ric DoF corner point (d1 = d2 = min{ N , 2M } / 2) (cf. [3, 45]). T his gap necessit at es t he use of imperfect and delayed feedback, as t his was st udied in works like [12, 13, 19–30, 38] for specific inst ances. T he work here makes progress t owards describing t he limit s of t his use of imperfect and delayed feedback.
3.1.3
P r edict ed, cur r ent and delayed CSI T
As in [14], we consider communicat ion of an infinit e durat ion n. For t he case of t he BC, we consider a random fading process { H (2) n H t } t = 1, and a feedback process t hat provides CSIT est imat es { Hˆ
(1) t ,
(1) ˆ (2) n t ,t 0, H t ,t 0} t ,t 0= 1
(1) (2) t ,H t )
(1)
(2)
at any t ime t 0 = 1, · · · , n. For t he channel H t , H t (1) (2) at a specific t ime t, t he set of all available estimat es { Hˆ t ,t 0, Hˆ t ,t 0} t 0, can be (1) (2) naturally split in t he predicted estimates { Hˆ 0, Hˆ 0} t 0< t t hat are off ered (of channel H
t ,t
t ,t
(1) (2) before t he channel mat erializes, t he current estimate Hˆ t ,t , Hˆ t ,t at t ime t, (1) (2) and t he delayed estimates { Hˆ 0, Hˆ 0} t 0> t t hat may allow for ret rospect ive t ,t
t ,t
compensat ion for t he lack of perfect quality feedback. Nat urally t he fundament al measure of feedback quality is given by t he precision of est imat es at any t ime about any channel, i.e., is given by { (H
(1) t
Hˆ
(1) (2) t ,t 0), (H t
Hˆ
(2) n t ,t 0)} t ,t 0= 1 .
(3.9)
T hese est imat ion-error set s of course fluct uat e depending on t he inst ance of t he problem, and as expect ed, t he overall opt imal performance is defined by t he st at ist ics of t he above est imat ion errors. We here only assume t hat t hese errors have zero-mean circularly-symmet ric complex Gaussian ent ries, t hat are spatially uncorrelat ed, and t hat at any time t, t he current est imat ion error is independent of t he channel est imat es up t o t hat t ime. (11) Similarly for t he case of t he IC, we consider a fading process { H t , (12) (21) (22) H t , H t ,H t } nt= 1, a set of CSIT est imat es { Hˆ
(11) ˆ (12) ˆ (21) ˆ (22) n t ,t 0 , H t ,t 0 , H t ,t 0 , H t,t 0 } t ,t 0= 1
70
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
and an overall feedback quality which, at any inst ance, is defined by { (H
(i j ) t
Hˆ
(i j ) n t ,t 0 )} t ,t 0= 1 ,
i , j = 1, 2
(3.10)
where again, t he st at ist ics of t he above error set s define t he opt imal performance. We will here seek t o capt ure t his relat ionship between performance and feedback.
3.1.4
N ot at ion, convent ions and assumpt ions
We will generally follow t he notat ions and assumpt ions in [14], and will adapt t hem t o t he MIMO and IC set t ings. When addressing t he BC, we will use t he not at ion (i )
↵t ,
lim
log E[||H
(i ) t
log P
P! 1
Hˆ
(i ) 2 t ,t ||F ]
(i ) t
,
,
lim
log E[||H
P! 1
(i ) t
Hˆ log P
(i ) 2 t ,t + ⌘||F ]
(3.11) where is used t o describe t he current quality exponent for t he CSIT for (i ) (i ) channel H t of receiver i , i = 1, 2, while t is used t o describe t he delayed quality exponents for each user. In the above, ⌘can be as large as necessary, but it must be finit e, as we here consider delayed CSIT t hat arrives aft er a finit e delay from t he channel it describes. T he above used || • ||F t o denot e t he Frobenius norm of a mat rix. Similarly when considering t he MIMO IC, we will use t he same not at ion, except t hat now (i ) ↵t
(i ) ↵t (i ) t (1)
(1)
, ,
lim
log E[||H
lim
Hˆ
(i j ) 2 t ,t || F ]
Hˆ
(i j ) 2 t ,t + ⌘|| F ]
log P
P! 1
P! 1
(i j ) t
log E[||H
(i j ) t
log P
, ,
i 6 = j
(3.12) (12)
where ↵ t , t will correspond t o the CSIT quality for t he cross link H t (2) (2) where t his CSIT is available at t ransmit t er 2, and where ↵ t , t will cor(21) respond t o t he CSIT quality for t he cross link H t where t his CSIT is 2 available at t ransmit t er 1 As argued in [14], t he result s in [16, 17] easily show t hat wit hout loss of generality, in t he DoF set t ing of int erest , we can rest rict our at t ent ion t o t he range (i ) (i ) 0 ↵t 1. (3.13) t 2. When t reat ing t he IC case, emphasis is placed on t he CSIT of t he cross links because, as it will t urn out , t he DoF region will be achieved wit hout any knowledge of t he direct links. T his is a small improvement over [25] where bot h t ransmit t ers were assumed t o have ( 21) ( 21) ( 11) ( 22) st at ic-quality CSIT for all t he channels (H t , H t , H t , H t ).
3.1
I nt r oduct ion (1)
71 (2)
Here having ↵ t = ↵ t = 1, corresponds t o the highest quality CSIT wit h perfect t iming (full CSIT ) for t he specific channel at t ime t, while having (i ) t = 1 corresponds t o having perfect delayed CSIT for t he same channel, i.e., it corresponds t o t he case where at some point t 0 > t, t he t ransmit t er has perfect est imat es of t he channel t hat materialized at t ime t. Furt hermore we will use t he not at ion n
↵¯
(i )
1X (i ) , lim ↵t , n! 1 n t= 1
n
X ¯ (i ) , lim 1 n! 1 n t= 1
(i ) t ,
i = 1, 2
(3.14)
t o denot e t he average of t he quality exponent s. As in [14] we will adopt (1) + T t he mild assumpt ion t hat any suffi cient ly long subsequence { ↵ t } t⌧ = ⌧ (resp. (2) ⌧ (1) ⌧ (2) ⌧ +T +T +T { ↵ t } t = ⌧, { t } t = ⌧, { t } t = ⌧) has an average t hat converges t o t he long t erm average ↵¯ (1) (resp. ↵¯ (2) , ¯ (1) , ¯ (2) ), for any ⌧and for some finit e T t hat can be chosen t o be suffi cient ly large t o allow for t he above convergence. Implicit in our definit ion of t he quality exponent s, is our assumpt ion t hat (1) (1) (2) (1) (1) (2) (2) E[||H t Hˆ t ,t 0||2F ] E[||H t Hˆ t ,t 00||2F ], E[||H t Hˆ t ,t 0||2F ] E[||H t (2) Hˆ t ,t 00||2F ], for any t 0 > t 00, (similarly for t he IC case) which simply reflect s t he fact t hat one can revert back t o past est imat es of st at ist ically bet t er quality. T his assumpt ion can be removed - aft er a small change in t he definit ion of t he quality exponent s - wit hout an eff ect t o t he main result . We adhere t o t he common convent ion (see [4, 6, 7, 25, 33]) of assuming perfect and global knowledge of channel st ate informat ion at t he receivers (perfect global CSIR), where t he receivers know all channel st at es and all est imat es. We will also adopt t he common convent ion (see [5–7, 15]) of assuming t hat t he current est imat ion error is st at ist ically independent of current and past est imat es. A discussion on t his can be found in [14] which argues t hat t his assumpt ion fit s well wit h many channel models, spanning from t he fast fading channel (i.i.d. in t ime), t o t he correlat ed channel model as t his is considered in [5], t o t he quasi-st at ic block fading model where t he CSIT est imat es are successively refined while t he channel remains st at ic. Addit io(i ) (i ) nally we consider t he ent ries of each est imat ion error mat rix H t Hˆ t ,t 0 t o be i.i.d. Gaussian 3. Finally we will refer t o a CSIT process wit h ‘sufficient ly good delayed CSIT ’, t o be a process for which min{ ¯ (1) , ¯ (2) } min{ 1, M
3.1.5
( 1)
( 2)
( 1)
(1+ ↵¯ + ↵¯ ) N (1+ min{ ↵¯ + ↵¯ min{ M , N } , Nmin{ M ,2N } + N , min{ M ,2N }
( 2) } )
}.
Exist ing r esult s dir ect ly r elat ing t o t he cur r ent wor k
T he work here builds on t he ideas of [4] on using delayed CSIT t o ret rospect ively compensat e for int erference due t o lack of current CSIT , on 3. We here make it clear t hat we are simply referring t o t he M N ent ries in each such (i ) (i ) specific mat rix H t Hˆ t ,t 0 , and t hat we cert ainly do not suggest t hat t he error ent ries are i.i.d. in t ime or across users.
72
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
t he ideas in [5] and lat er in [6, 7] on exploit ing perfect delayed and imperfect current CSIT , as well as t he work in [8, 9] which - in t he cont ext of imperfect and delayed CSIT - int roduced encoding and decoding wit h a phase-Markov st ruct ure t hat will be used lat er on. T he work here is also mot ivat ed by t he work in [19] which considered t he use of delayed feedback in diff erent MIMO BC set t ings, as well as by recent progress in [25] t hat considered MIMO BC and MIMO IC set t ings t hat enjoyed perfect delayed feedback as well as imperfect current feedback of a quality t hat remained unchanged t hroughout t he communicat ion process (↵ (1) = log E[||H
( 1)
Hˆ
( 1)
|| 2 ]
log E[||H
( 2)
Hˆ
( 2)
||2 ]
t ,t F t ,t F t t limP ! 1 , ↵ (2) = limP ! 1 , 8t). T he log P log P work is finally mot ivat ed by t he recent approach in [14] t hat employed sequences of evolving quality exponent s t o address a more fundament al problem of deriving t he performance limit s given a general CSIT process of a cert ain quality.
3.2
D oF r egion of t he M I M O B C and M I M O I C
We proceed wit h t he main DoF result s, which are proved in Sect ion 3.3 t hat describes t he out er bound, and in Sect ion 3.4 t hat describes an inner bound by ext ending t he schemes from [14] t o t he symmet ric MIMO BC and MIMO IC cases of int erest . We recall t hat we consider communicat ion (1) (2) of large durat ion n, a possibly correlat ed channel process { H t , H t } nt= 1 (11) (12) (21) (22) ({ H t , H t , H t , H t } nt= 1 for t he IC), and a feedback process of qua(1) (2) (1) (2) (i j ) lity defined by t he st at ist ics of { (H Hˆ 0), (H Hˆ 0)} n 0 ({ (H t
t ,t
t
t ,t
t = 1,t = 1
t
(i j ) n t ,t 0 )} t ,t 0= 1 ,
Hˆ i , j = 1, 2 for t he IC). We hencefort h, wit hout loss of generality, label t he users so t hat ↵¯ (2) ↵¯ (1) . We proceed wit h t he DoF region for any CSIT process wit h suffi cient ly good delayed CSIT .
T heor em 2. T he optimal DoF region of the two-user (M ⇥(N , N )) MI MO (1) (2) (1) (1) (2) BC with a CSI T process { Hˆ 0, Hˆ 0} n 0 of quality { (H Hˆ 0), (H t ,t
t ,t
t = 1,t = 1
t
t ,t
t
3.2
Hˆ
D oF r egi on of t he M I M O B C and M I M O I C
(2) n t ,t 0)} t = 1,t 0= 1
73
that has suffi ciently good delayed CSI T, is given by d1
min{ M , N }
(3.15)
d2
min{ M , N }
(3.16)
d1 + d2 d2 d1 + min{ M , N } min{ M , 2N } d1 d2 + min{ M , 2N } min{ M , N }
min{ M , 2N } (3.17) min{ M , 2N } min{ M , N } (1) 1+ ↵¯ min{ M , 2N } (3.18) min{ M , 2N } min{ M , N } (2) 1+ ↵¯ min{ M , 2N } (3.19) (i j )
while for the (M , M ) ⇥ (N , N ) MI MO I C with feedback quality { (H t (i j ) Hˆ t ,t 0 )} nt,t 0= 1, i , j = 1, 2, the above holds after substituting (3.17) with d1 + d2
min{ 2M , 2N , max{ M , N } } .
(3.20)
T he following proposit ion provides t he DoF region inner bound for t he regime of low-quality delayed CSIT . T he proof is shown in Sect ion 3.4. P r op osit ion 2. The DoF region of the two-user (M ⇥(N , N )) MI MO BC (1) (2) (1) (2) Hˆ t ,t 0), (H Hˆ t ,t 0)} n 0 such with a CSIT process of quality { (H t
t
t = 1,t = 1
(1+ ↵¯ ( 1) + ↵¯ ( 2) ) N (1+ ↵¯ ( 2) ) that min{ ¯ (1) , ¯ (2) } < min{ 1, M min{ M , N } , Nmin{ M ,2N } + N , min{ M ,2N } } , is inner bounded by the polygon described by
d1
min{ M , N }
(3.21)
d2
min{ M , N }
(3.22)
d1 + d2
min{ M , 2N }
(3.23) d1 + d2 min{ M , N } + (min{ M , 2N } min{ M , N } ) min{ ¯ (1) , ¯ (2) } (3.24) d1 d2 min{ M , 2N } min{ M , N } (1) + 1+ ↵¯ min{ M , N } min{ M , 2N } min{ M , 2N } (3.25) d1 d2 min{ M , 2N } min{ M , N } (2) + 1+ ↵¯ . min{ M , 2N } min{ M , N } min{ M , 2N } (3.26) (i j )
while for the (M , M ) ⇥ (N , N ) MI MO I C with feedback quality { (H t (i j ) Hˆ t ,t 0 )} nt,t 0= 1, i , j = 1, 2, the above holds after substituting (3.23) with d1 + d2
min{ 2M , 2N , max{ M , N } } .
(3.27)
74
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
d2
d2 L2 L2
L1
N
L0
L0 B*
L1
N E*
C*
(1)
D*
(1)
(2)
when
M/ N
A*
E*
C* F*
M/ N
d1
d1
0 (a) Case: N (1
(2)
when
F*
B*
0
N ( 2)
)/M
(1)
and M
N
(b) Case: N(1
N ( 2)
)/ M
(1)
and M N
Figur e 3.1 – Opt imal DoF regions for two diff erent cases for t he two-user MIMO BC and MIMO IC, wit h M > N and (1+ ↵¯ ( 1) + ↵¯ ( 2) ) N (1+ ↵¯ ( 2) ) (1) ¯ (2) ¯ min{ , } min{ 1, M min{ M , N } , Nmin{ M ,2N } + N , min{ M ,2N } } . T he ( 2)
corner point s t ake t he following values : A ⇤ = N , (M N )NM(1+ ↵¯ ) , B ⇤ = (M N ) ↵¯ (2) , N , MN N (2) N (1) ⇤ C = M + N (1 + ↵¯ (1) M ↵¯ ), MM+NN (1 + ↵¯ (2) M ↵¯ ) , ⇤ = N , (M (1) ⇤ (2) N ) ↵¯ D , E = M N ↵¯ , N ↵¯ (2) , F ⇤ = N ↵¯ (1) , M N ↵¯ (1) . Line L 0 corresponds t o t he bound in (3.17), Line L 1 corresponds t o t he bound in (3.19), while line L 2 corresponds t o t he bound in (3.18).
3.3
Out er b ound pr oof
75
R em ar k 3. As a small comment, and to place the above proposition in the context of previous work, we briefly note that deriving the DoF for even the simplest instance in the setting of low-quality delayed CSI T - corresponding to the case of (1) = (2) = ↵ (1) = ↵ (2) = 0 - has been a long lasting open problem. I n this simple setting of all-zero quality exponents, the conjectured DoF of d1 = d2 = 1/ 2 in [15], matches the above inner bound.
3.2.1
I mp er fect cur r ent CSI T can be as useful as p er fect cur r ent CSI T
T he above result s allow for direct conclusions on t he amount of CSIT t hat is necessary t o achieve t he opt imal DoF performance associat ed t o perfect and immediat ely available CSIT . T he following corollary holds for t he BC and t he IC case, for which we also remember that t here is no need for CSIT when M N (cf. [3, 45]). T he proofs for t he following corollary, and of t he corollary immediat ely aft er t hat , are direct from t he above t heorems. C or ollar y 2a. Having a CSI T process that off ers ↵¯ (1) + ↵¯ (2) min{ M , 2N } / N , allows for the optimal sum-DoF associated to having perfect and immediately (full) CSI T ( ↵¯ (1) = ↵¯ (2) = 1). T he above suggest s a reduct ion in t he required feedback quality ↵¯ (1) , ↵¯ (2) , as t he number of receive ant ennas increases. Furt hermore as st at ed before, when applied t o t he IC case, t he above also reveals t hat no CSIT is needed for t he direct links.
3.2.2
I mp er fect delayed CSI T can be as useful as p er fect delayed CSI T
Along t he same lines, t he following describes t he amount of delayed CSIT t hat suffi ces t o achieve t he DoF associat ed t o perfect delayed CSIT . C or ollar y 2b. Any CSI T process that off ers min{ ¯ (1) , ¯ (2) } min{ 1, M N (1+ ↵¯ ( 1) + ↵¯ ( 2) ) N (1+ ↵¯ ( 2) ) min{ M , N } , min{ M ,2N } + N , min{ M ,2N } } , can achieve the same DoF region as a CSI T process that off ers perfect delayed CSIT ( ¯ (1) = ¯ (2) = 1).
3.3
Out er bound for t he M I M O B C and M I M O I C wit h evolving feedback
We proceed t o first describe t he out er bound for t he BC case. T he bound, present ed in t he following lemma, draws from [13] and [14], and for t his we here mainly focus on t he proof st eps t hat are import ant in t he MIMO case.
76
C hapt er 3
3.3.1
D oF and Feedback T r adeoff over M I M O B C and I C
Out er bound pr oof for t he B C
L em m a 3. The DoF region of the two-user MI MO BC with a CSI T process (1) (2) (1) (2) (1) (2) { Hˆ t ,t 0, Hˆ t ,t 0} nt= 1,t 0= 1 of quality (H t Hˆ t ,t 0), (H t Hˆ t ,t 0)} nt= 1,t 0= 1, is upper bounded as
d1
min{ M , N }
(3.28)
d2
min{ M , N }
(3.29)
d1 + d2 d2 d1 + min{ M , N } min{ M , 2N } d1 d2 + min{ M , 2N } min{ M , N }
min{ M , 2N } (3.30) min{ M , 2N } min{ M , N } (1) 1+ ↵¯ min{ M , 2N } (3.31) min{ M , 2N } min{ M , N } (2) 1+ ↵¯ . min{ M , 2N } (3.32)
Proof. For not at ional convenience we define h• i 0, min { • , M } , ⌦ [n] , { H (1) (2) (2) (1) (1) (2) (2) H t , Hˆ t ,t 0, Hˆ t ,t 0, } nt = 1 nt 0= 1 , y [n] , { y t } nt= 1 and y [n] , { y t } nt= 1.
(1) t ,
We first design a degraded version of t he BC by giving t he observat ions and messages of receiver 1 t o receiver 2. T his allows for
(1)
nR 1
I (W1; y [n] |⌦ [n] ) + n✏
nR 2
I (W2; y [n] , y [n] |W1 , ⌦ [n] ) + n✏
(1)
(2)
(3.33) (3.34)
due t o Fano’s inequality, due t o t he basic chain-rule of mut ual informat ion, and due t o t he fact t hat messages from diff erent users are independent . T his now gives t hat
(1)
(1)
nR 1
h(y [n] | ⌦ [n] )
nR 2
h(y [n] , y [n] | W1, ⌦ [n] )
(1)
(2)
h(y [n] | W1 , ⌦ [n] ) + n✏ (1)
(2)
h(y [n] , y [n] | W1 , W2, ⌦ [n] ) + n✏
(3.35) (3.36)
3.3
Out er b ound pr oof
77
and t hat nR1 nR 2 n n ✏ 0+ 0 0+ hN i h2N i hN i h2N i 0 1 1 1 (1) (1) (2) (1) [n] ) + [n] ) [n] ) 0h(y [n] | ⌦ 0h(y [n] , y [n] | W1 , ⌦ 0h(y [n] | W1 , ⌦ hN i h2N i hN i 1 (1) (2) [n] ) 0h(y [n] , y [n] | W1 , W2 , ⌦ h2N i 1 1 1 (1) (1) (2) (1) = h(y [n] | W1 , ⌦ [n] ) + [n] ) [n] ) 0h(y [n] | ⌦ 0h(y [n] , y [n] | W1 , ⌦ hN i h2N i hN i 0 + no(log P ) (3.37) 1 1 (1) (2) (1) n log P + no(log P ) + h(y [n] | W1 , ⌦ [n] ) [n] ) 0h(y [n] , y [n] | W1 , ⌦ h2N i hN i 0 + no(log P ) (3.38) 0 1 (1) (2) (1) (2) (1) (1) Xn h(y t , y t | y [t 1] , y [t 1] , W1 , ⌦ h(y t | y [t 1] , W1 , ⌦ [n] ) [n] ) @ A = 0 0 h2N i hN i t= 1 + n log P + no(log P ) 0 (1) (2) (1) (2) Xn h(y t , y t | y [t 1] , y [t 1] , W1, ⌦ [n] ) @ h2N i 0 t= 1
(3.39) 1 (1) (1) (2) ) h(y t | y [t 1] , y [t 1] , W1 , ⌦ [n] A 0 hN i
+ n log P + no(log P ) (3.40) ⌘ Xn ⇣ 1 0 0 0 (1) hN i ) hN i ↵ log P + o(log P ) + n log P + no(log P ) (h2N i t h2N i 0hN i 0t = 1 (3.41) ⇣ ⌘ n = (h2N i 0 hN i 0) hN i 0↵¯ (1) log P + o(log P ) + n log P + no(log P ) h2N i 0hN i 0 n(h2N i 0 hN i 0) (1) = ↵¯ log P + n log P + no(log P ). (3.42) h2N i 0 In the above, (3.37) is due t o t he fact t hat knowledge of { W1, W2, ⌦ [n] } allows (1)
(2)
for reconst ruct ion of y [n] , y [n] up t o noise level, while (3.38) is due t o t he fact (1)
t hat h(y [n] | ⌦ hN i 0log P + o(log P ). Addit ionally (3.39) is due t o t he [n] ) chain rule of diff erent ial ent ropy, (3.40) is due t o t he fact t hat condit ioning reduces diff erent ial ent ropy, and (3.41) is directly from Proposit ion 6 of Chap(1) (2) (1) (2) (1) (2) t er 4, aft er set t ing U = { y ,y , W1 , ⌦ , H , Hˆ t ,t , Hˆ t ,t } 4. [n] } \ { H [t 1]
[t 1]
t
t
T he above gives t he bound in (3.31). Int erchanging t he roles of t he users gives t he bound in (3.32). T he bounds in (3.28),(3.29) are basic single-user 4. We not e t hat t he result in Proposit ion 6 of Chapt er 4, holds for a large family of channel models, under t he assumpt ion t hat t he CSIT est imat es up t o t ime t are independent of t he current est imat e errors at t ime t.
78
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
const raint s, while t he bound in (3.30) corresponds t o an assumpt ion of user cooperat ion.
3.3.2
Out er bound pr oof for t he I C
T he t ask is t o show t hat t he above bounds (3.28),(3.29),(3.31),(3.32) hold (1) (2) for t he case of t he IC. First let us set y t , y t t o t ake t he form in (3.3),(3.4), and let us denot e ⌦ [n] , { H
(12) (21) (11) (22) (11) (12) (22) (21) , H t , H t , H t , Hˆ t ,t 0 , Hˆ t ,t 0 , Hˆ t ,t 0 , Hˆ t ,t 0 } nt = 1 nt 0= 1. t
Focusing on t he bound in (3.31), we not e t hat t he t ransit ion from (3.40) t o (3.41) holds in t he IC set t ing, because knowledge of { W1 , ⌦ [n] } implies knowledge of H (1)
(2)
h(y t , y t
(11) (1) xt t (1)
| y [t
and H
(21) (1) xt , t
(2) [n] ) 1] , y [t 1] , W1 , ⌦ 0
which in t urn implies t hat (1)
(1)
h(y t | y [t
h2N i h(H =
(12) (2) xt t
+
(1) (22) (2) zt , H t x t
+
h2N i h(H
(12) (2) xt t
(1)
+ zt
(1)
| y [t
(2) [n] ) 1] , y [t 1] , W1 , ⌦ 0
hN i
(2) zt 0
(1) (2) | y [t 1] , y [t 1] , W1, ⌦ [n] )
(2) [n] ) 1] , y [t 1] , W1 , ⌦
hN i 0 which has t he form of t he diff erence of t he diff erent ial ent ropies corresponding t o a BC set t ing, where t he role of t he BC t ransmit t er is replaced now by t he second IC t ransmit t er. T his implies t hat (3.31) holds for t he IC. Bounds (3.28),(3.29),(3.32) follow easily. Finally, t he bound d1 + d2 min{ 2M , 2N , max{ M , N } , max{ M , N } } is direct ly from [44].
3.4
Phase-M ar kov t r ansceiver for imper fect and delayed feedback
We proceed t o ext end t he MISO BC scheme in [14], t o t he current set t ing of t he MIMO BC and MIMO IC. While part of t he ext ension of t he schemes in [14] involves keeping t rack of t he dimensionality changes t hat come wit h MIMO, t here are here modificat ions t hat are not t rivial. T hese include changes in t he way t he scheme performs int erference quant izat ion as well as power and rat e allocat ion, differences in decoding, as well as diff erences in t he way t he informat ion is aggregat ed t o achieve t he corresponding DoF corner point s. A part icular ext ra challenge corresponding t o t he MIMO IC, has t o do wit h t he fact t hat now t he signals must be sent by two independent t ransmit t ers. T his will reflect on t he power and rat e allocation at each t ransmit t er, and on t he way common and privat e informat ion is decoded at each receiver.
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback
79
Before proceeding wit h t he schemes, we again not e t hat for t he achievability proof of bot h BC and IC, we only focus on t he case where N < M 2N simply because, for t he case of having M N , t he opt imal DoF can be achieved wit hout any CSIT , while for t he case of having M 2N , we will consider it as having M = 2N , again just for t he achievability proof. We also note t hat , for t he case wit h suffi cient ly good delayed CSIT , having M 2N can be shown t o be equivalent , in t erms of DoF, wit h t he case of having M = 2N (cf. T heorem 2). We first begin wit h t he scheme descript ion for t he BC set t ing, while at t he end we will describe t he modificat ions required t o achieve t he result for t he IC case. Sect ion 3.4.1 will describe t he encoding part , Sect ion 3.4.2 t he decoding part , and Sect ion 3.4.3 will describe how we calibrat e t he paramet ers of t his universal scheme t o achieve t he diff erent DoF corner point s. T he challenge here will be t o design a scheme of large durat ion n, t hat (1) (2) ut ilizes t he CSIT process { Hˆ t ,t 0, Hˆ t ,t 0} nt= 1,t 0= 1. As in [14], t he causal scheme will not require knowledge of fut ure quality exponent s, nor of predict ed CSIT est imat es of fut ure channels. We remind t he reader t hat t he users are labeled so that ↵¯ (2) ↵¯ (1) . For not at ional convenience, we will use Hˆ Hˇ
(1) t
(1) t
(1) t ,t ,
Hˆ
(2) t
, Hˆ
(2) t ,t
(3.43)
(1) t ,t + ⌘,
Hˇ
(2) t
, Hˆ
(2) t ,t + ⌘
(3.44)
, Hˆ
, Hˆ
t o denot e t he current and delayed est imat es of H ponding est imat ion errors being
(1) (2) t ,H t ,
wit h t he corres-
H˜
(1) t
,H
(1) t
Hˆ
(1) t ,
H˜
(2) t
,H
(2) t
Hˆ
(2) t
(3.45)
H¨
(1) t
,H
(1) t
Hˇ
(1) t ,
H¨
(2) t
,H
(2) t
Hˇ
(2) t .
(3.46)
We will also use t he not at ion (e)
Pt
, E|et |2
(3.47)
t o denot e t he power of a symbol et corresponding t o t ime-slot t, and we will (e) (e) use r t t o denot e t he prelog fact or of t he number of bit s r t log P o(log P ) carried by symbol et at t ime t.
3.4.1
Encoding
As in [14], we subdivide t he overall t ime durat ion n, int o S phases, each of durat ion of T , such t hat each phase s (s = 1, 2, · · · , S) t akes place over t he t ime slot s t 2 B s B s = { B s,` , (s 1)2T + `} T`= 1,
s = 1, · · · , S.
(3.48)
80
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
Nat urally in t he gap of what we define here t o be consecut ive phases, anot her message is sent , using t he same exact scheme. Going back t o t he aforement ioned assumpt ion, T is suffi cient ly large so t hat 1 X 1 X (i ) (i ) ¯ (i ) , s = 1, · · · , S ↵ t ! ↵¯ (i ) , (3.49) t ! T T t2 Bs
t2 Bs
i = 1, 2. For not at ional convenience we will also assume t hat T > ⌘(cf. (3.11)), alt hough t his assumpt ion can be readily removed, as t his was argued in [14]. Finally wit h n being infinit e, S is also infinit e. Adhering t o a phase-Markov st ruct ure which - in t he cont ext of imperfect and delayed CSIT , was first int roduced in [8,9] - t he scheme will quant ize t he accumulat ed int erference of a cert ain phase s, broadcast it t o bot h receivers over phase (s + 1), while at t he same t ime it will send ext ra informat ion t o bot h receivers in phase s, which will help recover t he int erference accumulat ed in phase (s 1). We first describe t he encoding for all phases except t he last phase which will be addressed separat ely due t o it s diff erent st ruct ure. P hase s, for s = 1, 2, · · · , S
1
In each phase, t he scheme combines zero forcing and superposit ion coding, power and rat e allocat ion, and int erference quant izing and broadcast ing. We proceed t o describe t hese st eps. Zer o for cing and sup er p osit ion coding t he t ransmit t er sends 0
0
At t ime t 2 B s (of phase s), 0
0
x t = W t ct + U t a t + U t a t + V t bt + V t bt
(3.50)
0
where a t 2 C(M N )⇥1, a t 2 CN ⇥1 are t he vect ors of symbols meant for re0 ceiver 1, bt 2 C(M N )⇥1 , bt 2 CN ⇥1 are t hose meant for receiver 2, where (2) ct 2 CM ⇥1 is a common symbol vect or, where U t = ( Hˆ ) ? 2 CM ⇥(M N ) t
(2) (1) is a unit -norm mat rix t hat is ort hogonal t o Hˆ t , where V t = ( Hˆ t ) ? 2 (1) 0 0 CM ⇥(M N ) is ort hogonal t o Hˆ , and where W t 2 CM ⇥M , U 2 CM ⇥N , V 2 t
t
t
CM ⇥N are predet ermined randomly-generat ed mat rices known by all nodes.
Power and r at e allocat ion T he powers and (normalized) rat es during phase s t ime-slot t, are (c) . Pt = P, ( 2) (a ) . (a ) (2) Pt = P t , r t = (M N ) t ( 1) (b) . (b) (1) (3.51) Pt = P t , r t = (M N ) t 0) ( 2) ( 2) (a 0) . (a (2) (2) Pt = P t ↵t , rt = N ( t ↵ t )+ 0 ( 1) ( 1) (b0) . (b ) (1) (1) Pt = P t ↵t , rt = N ( t ↵ t )+ .
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback (1) (2) t , t } t2 Bs
where {
81
are designed such t hat (i ) t
(i ) t
1 X T
t2 Bs
1 X ( T
(i ) t
i = 1, 2, t 2 B s 1 X (1) (2) = = ¯ t t T
(3.52) (3.53)
t2 Bs
(i ) ↵ t )+
= (¯
↵¯ (i ) ) +
i = 1, 2
(3.54)
t2 Bs
for some ¯ t hat will be bounded by ¯
N (1 + ↵¯ (1) + ↵¯ (2) ) N (1 + ↵¯ (2) ) , } min{ 1, ¯ (1) , ¯ (2) , M + N M
(3.55)
and which will be set t o specific values lat er on, depending on t he DoF corner point we wish t o achieve. (1) (2) T he exact solut ions for { t , t } t 2 B s sat isfied (3.52),(3.53),(3.54) are shown in [14], and t he rat es of t he common symbols { cB s, t } Tt= 1 are designed t o joint ly carry T (N
N ) ¯ ) log P
(M
o(log P )
(3.56)
bit s. To put t he above allocat ion in perspect ive, we show t he received signals, and describe under each t erm t he order of t he summand’s average power. T hese signals t ake t he form (1)
yt
= H |
(1) t W t ct
(1)
(1)
0
{z P
P
( 2) t
P
( 2) t
( 2) ↵t ( 1)
( 1)
z + Hˇ |
ˇ◆t
}| { 0 0 (1) t (V t bt + V t bt ) + {z } P
(2) yt
= H |
( 1) t
(1)
P
P
( 1) t
}| { z 0 0 (2) ¨ t (U t a t + U t a t ) + H {z } | ( 2) t
( 1) t
( 2) ↵t
◆t
(3.57)
P0
}
(2)
z | {tz} P0
( 1) ↵t ( 2)
( 2)
ˇ◆t
P
( 1) t
+H } | {z
( 1) t
( 1)
ˇ◆
(2) 0 0 t V t bt +
(2) t V t bt
P
P0
}| t { 0 0 (1) t (V t bt + V t bt ) {z } P
+H } | {z
{z
◆t
z H¨ |
( 1) ↵t
(2) t W t ct
z + Hˇ |
0
+ H U t at + H U a + z } | t { z } | t { z t }t | {tz}
( 2)
ˇ◆
}| t { 0 0 (2) t (U t a t + U t a t ) {z } P
( 2) t
( 2) t
(3.58)
P0
where (1)
◆ t , H
(1) t (V t bt
0 0
(2)
+ V t bt ), ◆ t , H
(2) t (U t a t
0
0
+ U t at )
(3.59)
82
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
Tabl e 3.1 – Number of bit s carried by privat e and common symbols, and by t he quant ized int erference (phase s).
Privat e symbols for user 1 Privat e symbols for user 2 Common symbols Quant ized int erference
Tot al bit s (⇥log P ) T ((M N ) ¯ + N ( ¯ ↵¯ (2) ) + ) T ((M N ) ¯ + N ( ¯ ↵¯ (1) ) + ) T (N (M N ) ¯ ) ¯ T N (( ↵¯ (1) ) + + ( ¯ ↵¯ (2) ) + )
denot e t he int erference at receiver 1 and receiver 2 respect ively, and where (1) ˇ ˇ◆ t , H
(1) t (V t bt
0 0 (2) ˇ + V t bt ), ˇ◆ t , H
(2) t (U t a t (1)
0
0
+ U t at )
(3.60)
(2)
denot e t he t ransmit t er’s delayed estimat es of ◆t , ◆t . Quant izing and br oadcast ing t he accumulat ed int er fer ence Before (1) (2) t he beginning of phase (s + 1), t he t ransmit t er reconst ruct s ˇ◆ for all t , ˇ◆ t t 2 B s , using it s knowledge of delayed CSIT , and quant izes t hese int o (1) ¯ˇ◆(1) = ˇ◆t t (1)
(1)
˜◆ t ,
(2) (2) ¯ˇ◆ = ˇ◆ t t
(1)
(2)
˜◆ t
(2)
(3.61) (2)
using a t ot al of N ( t ↵ t ) + log P and N ( t ↵ t ) + log P quant izat ion bit s respect ively. T his allows for bounded power of quant izat ion noise ( 2) ( 2) . (1) (2) (2) (1) 2 . (2) 2 . ↵t , t ˜◆t , ˜◆t , i.e, allows for E|˜◆t |2 = E|˜◆ t | = 1, since E|ˇ◆ t | = P ( 1) ( 1) . (1) E|ˇ◆t |2 = P t ↵ t (cf. [39]). T hen t he t ransmit t er evenly split s t he X
⇣ N(
(1) t
(1)
↵ t )+ + N (
t2 Bs
⇣ = TN (¯
↵¯ (1) ) + + ( ¯
(2) t
⌘ (2) ↵ t ) + log P
⌘ ↵¯ (2) ) + log P
(3.62)
(cf. (3.54)) quant izat ion bit s int o t he common symbols { ct } t 2 B s+ 1 t hat will be t ransmit t ed during t he next phase (phase s+ 1), and which will convey t hese quant izat ion bit s t oget her wit h ot her new informat ion bit s for t he receivers. T hese { ct } t 2 B s+ 1 will help t he receivers cancel int erference, as well as will serve as ext ra observat ions (see (3.63) lat er on) t hat will allow for decoding of all privat e informat ion (see Table 3.1). Finally, for t he last phase S, t he main t arget will be t o recover t he informat ion on t he int erference accumulat ed in phase (S 1). For large S, t his last phase can focus ent irely on t ransmit t ing common symbols. T his concludes t he part of encoding, and we now move t o decoding.
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback
3.4.2
83
D ecoding
In accordance t o t he phase-Markov st ruct ure, we consider decoding t hat moves backwards, from t he last t o t he first phase. T he last phase was specifically designed t o allow decoding of t he common symbols { ct } t 2 B S . Hence we focus on t he rest of t he phases, t o see how - wit h knowledge of common symbols from t he next phase - we can go back one phase and decode it s symbols. During phase s, each receiver uses { ct } t 2 B s + 1 t o reconst ruct t he delayed (2) (1) est imat es { ˇ¯◆t , ˇ¯◆ t } t 2 B s , t o remove - up t o noise level - all t he int erference (i ) (i ) (i ) ◆t , t 2 B s , by subt ract ing t he delayed int erference est imat es ˇ¯◆ t from y t . (2) (1) (1) ¯(2) Now given { ˇ¯◆t , ˇ¯◆ t } t 2 B s , receiver 1 combines { ˇ◆ t } t 2 B s wit h { y t 0 (1) ˇ¯◆t } t 2 B s t o decode { ct , a t , a t } t 2 B s of phase s. T his is achieved by decoding over all accumulat ed MIMO mult iple-access channel (MIMO MAC) of t he general form 2 (1) (1) 3 2 (1) 3 y B s, 1 ¯ˇ◆ B s, 1 H B s, 1 W B s ,1 7 6 6 3 (2) 72 ¯ˇ◆ 7 6 6 0 7 cB s ,1 B s, 1 7 6 6 76 . 7 7 6 6 .. .. 74 .. 5 7= 6 6 . . 7 7 6 6 7 (1) 7 (1) 6 y (1) ¯ 4 H B s, T W B s, T5 cB s , T 4 B s, T ˇ◆ B s, T 5 ¯ˇ◆(2) 0 B s, T 32 2 " (1) #h 3 2 (1) 3 i z˜ B s , 1 H B s, 1 0 a 76 B0 s, 1 7 6 (2) 7 6 ˇ (2) U B s, 1 U B s ,1 6 H B s, 1 76 a B s, 1 7 6 ˜◆ 7 B s ,1 7 6 76 7 6 7 6 6 7 . . . .. 7 .. + 6 .. 7 + 6 76 6 7 7 6 6 " (1) # 7 6 (1) 7 6 7 i 76 h a 4 B H s, T 0 5 0 5 4 z˜ B s, T 5 4 B s, T U U B s, T B s, T (2) (2) a B s, T ˜◆ Hˇ B s, T B s, T (3.63) where
(1)
z˜ t
= H¨
(1) t (V t bt
0 0
(1)
+ V t bt ) + z t
(1)
+ ˜◆ t
. (1) and where E|˜z t |2 = 1. It can be readily shown (cf. [39]) t hat opt imal decoding in such a MIMO MAC set t ing, allows user 1 t o achieve t he aforement ioned rat es in (3.51),(3.53),(3.54),(3.56) i.e., allows for r ⇤(a ) log P = T (M N ) ¯ + N ( ¯ ↵¯ (2) ) + log P bit s t o be reliably carried by h iT 0 0 a B s, 1 a B s, 1 · · · a B s, T a B s, T as well as allows for r ⇤(c) log P = T N (M N ) ¯ log P bit s t o be carried ⇥ ⇤T by cB s, 1 · · · cB s, T . Similarly receiver 2 can again accumulat e enough received signals t o const ruct a similar MIMO MAC, which will again allow
84
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
for decoding of it s own privat e and common symbols at t he aforement ioned rat es in (3.51),(3.56). Now t he decoders shift t o phase s 1 and use { ct } t 2 B s t o decode t he common and privat e symbols of t hat phase. Decoding st ops aft er decoding of t he symbols in phase 1.
3.4.3
Calibr at ing t he scheme t o achieve D oF cor ner point s
We now describe how t o regulat e t he scheme’s paramet ers t o achieve t he diff erent DoF point s of int erest . As previously discussed, we can focus wit hout an eff ect t o our result 5 - on t he case where N < M 2N . Focusing first on t he DoF region of t he out er bound in Lemma 3, we not e t hat t he DoF corner point s t hat define t his region, vary from case t o case, as t hese cases are each defined by each of t he following inequalit ies min{ ¯ (1) , ¯ (2) }
min{ 1, M
min{ M , N } ,
min{ ¯ (1) , ¯ (2) } < min{ 1, M
min{ M , N } ,
N (1 + ↵¯ (1) + ↵¯ (2) ) N (1 + ↵¯ (2) ) , } min{ M , 2N } + N min{ M , 2N } (3.64) N (1 + ↵¯ (1) + ↵¯ (2) ) N (1 + ↵¯ (2) ) , } min{ M , 2N } + N min{ M , 2N } (3.65)
N (1 + ↵¯ (2) ) M N (1 + ↵¯ (2) ) M M > N M . N
↵¯ (1)
2N .
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback
85
Tabl e 3.2 – Out er bound corner point s. Cases (3.66) and (3.68) (3.66) and (3.69) (3.67)
Corner point s D ⇤, B ⇤, E ⇤, F ⇤ D ⇤, B ⇤, C ⇤ B ⇤, A ⇤
(3.68) hold t hen t he ‘act ive’ out er bound corner point s are D ⇤, B ⇤, E ⇤, F ⇤, whereas if (3.66) and (3.69) hold, t hen t he act ive corner point s are D ⇤, B ⇤, C ⇤, while if (3.67) holds t hen t he act ive out er bound corner point s are B ⇤, A ⇤ (see Table 3.2). We proceed t o show how t he designed scheme achieves t he above point s. To do so, we will show how t he scheme, in it s general form, achieves a range of DoF point s (see (3.80),(3.81) lat er on), which can be shift ed t o t he DoF corner point s of int erest by properly adapt ing t he power allocat ion and t he rate split t ing of t he new informat ion carried by t he common symbols. G ener al D oF p oint Remaining on t he case where (3.64) holds, we see t hat t he bound in (3.55) now implies t hat ¯
min{ 1,
N (1 + ↵¯ (1) + ↵¯ (2) ) N (1 + ↵¯ (2) ) , }. M + N M
(3.76)
Changing ¯ - wit hin t he bounds of (3.76) - will achieve t he diff erent DoF point s. Such changing of ¯ , amount s t o changing t he power allocat ion (cf. (3.51)) (1) (2) by changing { t , t } t 2 B s which are a funct ion of ¯ (cf. (3.53),(3.54)). T he first st ep is t o see t hat for any fixed ¯ , t he rat e allocat ion in (3.51) t ells us t hat , t he t ot al amount of informat ion, for user 1, in t he private symbols of a cert ain phase s < S, is equal t o (M
N )¯ + N (¯
↵¯ (2) ) + T log P
(3.77)
↵¯ (1) ) + T log P
(3.78)
bit s, while for user 2 t his is (M
N )¯ + N (¯
bit s. T he next st ep is t o see t he amount of int erference informat ion bit s accumulat ed in one phase. Given t he power and rat e allocat ion in (3.52), (3.53), (3.54), (3.55), it is guarant eed t hat t he accumulat ed quant ized int erference in a phase s < S has N ( ¯ ↵¯ (1) ) + + N ( ¯ ↵¯ (2) ) + T log P bit s (cf. (3.62)), which ‘fit ’ int o t he common symbols of t he next phase (s + 1) t hat can carry
86
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C
a t ot al of N (M N ) ¯ T log P bit s (cf. (3.56)). T his leaves an ext ra space of com T log P bit s in t he common symbols, where , N
com
(M
N )¯
N (¯
↵¯ (1) ) +
N (¯
↵¯ (2) ) +
(3.79)
is guarant eed t o be non-negat ive due t o (3.52),(3.53),(3.54),(3.55). T his ext ra space can be split between t he two users, by allocat ing ! com T log P bit s for t he message of user 1, and t he remaining (1 ! ) com T log P bit s for t he message of user 2, for some ! 2 [0, 1]. Consequent ly, considering (3.77),(3.78), and given (3.64), t he scheme allows for DoF performance in it s general form 6 d1 = (M d2 = (M
N )¯ + N (¯ N )¯ + N (¯
↵¯ (2) ) + + ! ↵¯ (1) ) + + (1
(3.80)
com
!)
com .
(3.81)
Again under t he set t ing of (3.64), we can now move t o t he diff erent cases, and set ! and ¯ (and t hus com ) t o achieve t he diff erent DoF corner point s (cf. Table 3.2). Case 1 - (3.64) and (3.66) and (3.68) (points D ⇤, B ⇤, E ⇤, F ⇤) : We first consider t he case where (3.66) and (3.68) hold, and show how t o achieve DoF corner point s D ⇤, B ⇤, E ⇤, F ⇤. In t his set t ing, (3.76) gives t hat N (1+ ↵¯ ( 1) + ↵¯ ( 2) ) N (1+ ↵¯ ( 2) ) ¯ 1 because (3.66) implies t hat ↵¯ (2) ↵¯ (1) M+N M while at t he same t ime (3.68) implies t hat To achieve E ⇤ = M which gives N )¯ + N (¯
d1 = (M = M
N ↵¯
= (M = N ↵¯
N ) + N (1 ↵¯ (2)
= 1.
N ↵¯ (2) , N ↵¯ (2) , we set ¯ = 1, ! = 0 (cf. (3.80)),
↵¯ (2) ) +
(3.82)
(2)
N )¯ + N (¯
d2 = (M
N (1+ M ) N M+N
N (1+ ↵¯ ( 1) + ↵¯ ( 2) ) M+N
(3.83) ↵¯ (1) ) + + (1) +
) + N
(3.84)
com
(M
N ) N (1 ↵¯
(1) +
)
N (1 ↵¯
(2) +
)
(3.85)
where (3.82) and (3.84) are direct ly from (3.80),(3.81) aft er set t ing ! = 0, and where (3.83) and (3.85) consider t he value of com in (3.79) and t he fact t hat ¯ = 1. To achieve F ⇤ = N ↵¯ (1) , M N ↵¯ (1) , we set ¯ = 1 and we set ! = 6. T his expression considers t hat S is large, and t hus removes t he eff ect of having a last phase t hat carries no new message informat ion.
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback
N (1+ ↵¯ ( 1) + ↵¯ ( 2) ) (M + N ) ¯ com
87
= 1, t o get N )¯ + N (¯
d1 = (M
↵¯ (2) ) + + !
(3.86)
com
(2) +
(1)
N ) + N (1 ↵¯ ) + com = N ↵¯ N ) ¯ + N ( ¯ ↵¯ (1) ) + + (1 ! ) com
= (M d2 = (M = (M
↵¯ (1) ) + = M
N ) + N (1
(3.87) (3.88)
N ↵¯ (1)
(3.89)
where again (3.86) and (3.88) are direct ly from (3.80),(3.81), and where (3.87) and (3.89) consider t he value of com in (3.79), t oget her wit h t he fact t hat ¯ = 1. To achieve B ⇤ = (M N ) ↵¯ (2) , N , we set ! = 0 and ¯ = ↵¯ (2) , which aft er recalling t hat we label t he receivers so t hat ↵¯ (1) ↵¯ (2) - gives com = N (M N ) ↵¯ (2) , which in t urn gives (cf. (3.80),(3.81)) N )¯ + N (¯ N )¯ + N (¯
d1 = (M d2 = (M
↵¯ (2) ) + = (M ↵¯
(1) +
) +
(3.90) (3.91)
com
N ) ↵¯ (2) + N ( ↵¯ (2) ↵¯ (1) ) + + N
= (M
N ) ↵¯ (2) (M
N ) ↵¯ (2) N ( ↵¯ (2) ↵¯ (1) ) +
= N.
(3.92) ( 1)
( 2)
¯
To achieve D ⇤ = N , (M N ) ↵¯ (1) , we set ! = N (1+ ↵¯ + ↵¯ com) (M + N ) = 1 and ¯ = ↵¯ (1) , which gives com = N (M N ) ↵¯ (1) N ( ↵¯ (1) ↵¯ (2) ) + = N M ↵¯ (1) + N ↵¯ (2) , which in t urn gives (cf. (3.80),(3.81)) d1 = (M = (M d2 = (M
N )¯ + N (¯ (1)
↵¯ (2) ) + + ! (1)
com
(2) +
N ) ↵¯ + N ( ↵¯ ↵¯ ) + com = N N ) ¯ + N ( ¯ ↵¯ (1) ) + + (1 ! ) com = (M
(3.93) N ) ↵¯ (1) .
(3.94)
Case 2 - (3.64) and (3.66) and (3.69) (points D ⇤, B ⇤, C ⇤) : Again under t he condit ion of (3.64), we now consider t he case where (3.66) and (3.69) hold, and seek t o achieve point s D ⇤, B ⇤, C ⇤. For point s D ⇤, B ⇤, we can use t he same paramet ers ! , ¯ that we used before t o achieve t hese same point s (for B ⇤ we set ¯ = ↵¯ (2) , ! = 0, and for D ⇤ we set ¯ = N (1+ ↵¯ ( 1) + ↵¯ ( 2) ) (M + N ) ¯
= 1). To achieve point C ⇤, we need t o set ( 2) ! = 0 and ¯ = N (1+M + N ↵¯ ) which, as before, gives d1 = (M N ) ¯ + N ( ¯ ↵¯ (2) ) + = M N (1+ ↵¯ (1) N ↵¯ (2) ) and d2 = (M N ) ¯ + N ( ¯ ↵¯ (1) ) + + com = ↵¯ (1) , ! =
com ↵¯ ( 1) +
M+N
MN M+N
(1 + ↵¯ (2)
M
N M
↵¯ (1) ).
Case 3 - (3.64) and (3.67) (points B ⇤, A ⇤) : Again given (3.64), we now move t o t he case where (3.67) holds, and seek t o achieve point s B ⇤ and A ⇤. To achieve B ⇤ we can use t he same paramet ers ( 2) as before, and t hus set ¯ = ↵¯ (2) , ! = 0. To achieve A ⇤ = (N , (M N )NM(1+ ↵¯ ) )
88
C hapt er 3
we simply set ¯ =
D oF and Feedback T r adeoff over M I M O B C and I C N (1+ ↵¯ ( 2) ) M
and ! =
N (1+ ↵¯ ( 2) ) M ¯ com
= 1.
We now focus on t he DoF point s in t he inner bound of Proposit ion 2, corresponding t o t he set t ing where (3.65) holds, rat her t han (3.64). In addit ion t o t he aforement ioned point s D ⇤ and B ⇤, we will seek t o achieve t he new point s E = M min{ ¯ (1) , ¯ (2) } F = N ↵¯
(1)
+ N (1
G = N , (M
N ↵¯ (2) , N ↵¯ (2) + N (1 min{ ¯ (1) , ¯ (2) } ) min{ ¯ (1) , ¯ (2) } ), M min{ ¯ (1) , ¯ (2) } N ↵¯ (1)
N ) min{ ¯ (1) , ¯ (2) } .
(3.95) (3.96) (3.97)
Before proceeding, we not e t hat under (3.65), t he bound on ¯ in (3.55) now becomes ¯
min{ ¯ (1) , ¯ (2) } .
(3.98)
We proceed wit h t he diff erent cases, and now addit ionally consider t he cases where ↵¯ (1) min{ ¯ (1) , ¯ (2) } min{ ¯ (1) , ¯ (2) } < ↵¯ (1) .
(3.99) (3.100)
Case 4a - (3.65) and (3.66) and (3.99) (points D ⇤, B ⇤, E , F ) : To achieve D ⇤, B ⇤ we use t he same paramet ers as before, where for B ⇤ we set ! = 0, ¯ = ↵¯ (2) , and for D ⇤ we set ¯ = ↵¯ (1) , ! = 1, all of which sat isfy t he condit ions in (3.98) and (3.99). To get point E , we set ! = 0 and ¯ = min{ ¯ (1) , ¯ (2) } , and calculat e t hat d1 = (M d2 = (M = (M + N = N
N )¯ + N (¯ N )¯ + N (¯ N )¯ + N (¯
↵¯ (2) ) + = M min{ ¯ (1) , ¯ (2) } ↵¯
(1) +
) +
N ↵¯ (2)
(3.102)
com
↵¯ (1) ) + N ) ¯ N ( ¯ ↵¯ (1) ) +
(M ¯ N( ↵¯ (2) ) + = N ↵¯ (2) + N (1
(3.101)
N (¯
↵¯ (2) ) + min{ ¯ (1) , ¯ (2) } ).
(3.103) (3.104)
To get point F , we set ¯ = min{ ¯ (1) , ¯ (2) } and ! = 1, and calculat e t hat N ) ¯ + N ( ¯ ↵¯ (2) ) + + ! com N ) min{ ¯ (1) , ¯ (2) } + N (min{ ¯ (1) , ¯ (2) }
d1 = (M = (M
(3.105) ↵¯
(2) +
) +
com
(3.106) = N ↵¯ d2 = (M
(1)
¯ (1)
¯ (2)
+ N (1 min{ , }) N ) ¯ + N ( ¯ ↵¯ (1) ) + + (1
(3.107) !)
com
= M min{ ¯ (1) , ¯ (2) }
N ↵¯ (1) . (3.108)
3.4
P hase-M ar kov t r anscei ver for i m p er fect and del ayed feedback
89
Case 4b - (3.65) and (3.66) and (3.100) (points B ⇤, E , G) : Again under (3.65), we now consider t he case where (3.66) and (3.100) hold, and seek t o achieve point s B ⇤, E and G. To achieve B ⇤ we set as before ! = 0, ¯ = ↵¯ (2) , and for E we set as before ! = 0 and ¯ = min{ ¯ (1) , ¯ (2) } , both in accordance wit h t he condit ions in (3.98) and (3.100). To get point G = N , (M N ) min{ ¯ (1) , ¯ (2) } , we simply set ¯ = min{ ¯ (1) , ¯ (2) } and ! = 1, and t he calculat ions follow immediat ely. Case 4c - (3.65) and (3.67) (points B ⇤, E , G) : In t he last case where (3.65) and (3.67) hold, we can achieve point s B ⇤, E , G using t he same paramet ers as above. Finally t he DoF regions in t he t heorem and proposit ion are achieved by t ime sharing between t he proper DoF corner point s.
3.4.4
M odifi cat ions for t he I C
We here briefly describe t he modificat ions t hat adapt our scheme t o (1) (1) t he IC set t ing. In t erms of not at ion, t he role of Hˆ t , Hˆ t ,t is t aken by (12) (12) (2) (2) (21) (21) (1) (1) (12) Hˆ , Hˆ , of Hˆ , Hˆ by Hˆ , Hˆ , of Hˇ , Hˆ by Hˇ t
t ,t
t
t ,t (2) Hˇ t
(12) t ,t + ⌘,
t (2) t ,t + ⌘ is
t,t
t
(21) t
t ,t + ⌘ (21) Hˆ t ,t + ⌘.
t
, Hˆ and t he role of , Hˆ t aken by Hˇ , Many of t he st eps follow from t he BC, wit h t he main diff erence being t hat now t he common symbols must be t ransmit t ed by two independent t ransmit t ers. For t hat we change t he st ruct ure of t he signaling (cf. (3.50)) and now consider t hat at t ime t 2 B s (phase s), t ransmit t er 1 sends (1)
xt
0
0
= W
(1) (1) t ct
+ U t at + U t at
= W
(2) (2) t ct
+ V t bt + V t bt
(3.109)
and t ransmit t er 2 sends (2)
xt
0 0
(3.110)
(i )
2 CM ⇥1 is t he common informat ion vect or sent by t ransmit t er i (21) (12) (i = 1, 2), where U t is ort hogonal t o Hˆ t , V t is ort hogonal t o Hˆ t , and 0 0 (1) (2) where W t , U t , W t , V t are randomly picked precoding mat rices. Finally 0 0 a t , a t , bt , bt accept t he same rat e and power allocat ion previously described in (3.51). (1) In t he above, t he common symbol vect ors { cB s, t } Tt= 1 convey informat ion where ct
(2)
(21)
0
0
on t he (quant ized version of t he) int erference ◆ (U t a t + U t a t ), t , H t (cf. (3.59)) accumulat ed during phase (s 1). T hese symbols will carry !T
com
+ TN (¯
↵¯ (2) ) + log P
T o(log P )
(3.111)
bit s, where com is defined in (3.79), and where ! 2 [0, 1] will be set depen(2) ding on t he t arget DoF point . Similarly { cB s, t } Tt= 1 will carry t he (1
! )T
com
+ TN (¯
↵¯ (1) ) + log P
T o(log P )
(3.112)
90
C hapt er 3
D oF and Feedback T r adeoff over M I M O B C and I C (1)
bit s of informat ion corresponding t o { ◆ t } t2 Bs 0 0
(12)
(1)
1
(1)
where we recall t hat ◆ = t (2)
H t (V t bt + V t bt ) (cf. (3.59)). Joint ly { cB s, t , cB s, t } Tt= 1 will carry T (N (M N ) ¯ ) log P o(log P ) bit s, which mat ches t he amount in t he BC set t ing (cf. (3.56)). Decoding is similar t o t he case of t he BC, except t hat now t he corresponding MIMO MAC (for receiver 1) t akes t he form 2
(1) (1) 3 2 y B s, 1 ¯ˇ◆ B s, 1 H 6 7 6 (2) 7 6 ˇ¯◆ B s, 1 7 6 6 6 7 6 .. 6 7= 6 . 6 7 6 6 (1) 7 6 y (1) ¯ 4 ˇ ◆ 4 B s, T B s, T 5 (2) ¯ˇ◆ B s, T
2 6 6 6 + 6 6 6 4
H
2 " 6 6 6 6 + 6 6 6 4
3
(11) (1) B s, 1 W B s, 1
0 ..
. H
0 ..
. H
(11) B s, 1 (21) B s, 1
#
W 0
(1) B s, T
2
3
(12) (2) B s, 1 W B s, 1
H Hˇ
(11) B s, T
(12) B s ,T
(2)
W B s, T 0
h i 0 U B s ,1 U B s, 1 ..
.
"
H Hˇ
2 3 7 c(1) 7 Bs,1 76 . 7 7 6 .. 7 74 5 7 (1) 5 cB s, T 0(1)
3
z˜ B s ,1
3 6 (2) 7 2 6 ˜◆ 7 7 c(2) 6 B 7 7 7 6 B.s, 1 7 6 . s, 1 7 7 6 .. 7 + 6 .. 7 74 7 5 6 6 0(1) 7 7 (2) 6 z˜ B 7 5 cB s, T 4 s, T 5 (2) ˜◆ B s, T 32
(11) B s, T (21) B s, T
# h U B s, T
3 a B s, 1 76 0 7 76 a B s, 1 7 76 76 .. 7 76 . 7 7 76 74 a B 7 i 0 5 0 s, T 5 U B s, T a B s, T (3.113)
where t he eff ect ive noise t erm at t he end can be shown t o have bounded power. As before, t he receivers recover t he signals at t he rat es described in (3.51), (3.111), (3.112) (cf. [39]).
3.5
Conclusions
T he work, ext ending on recent work on t he MISO BC, considered t he symmet ric MIMO BC and MIMO IC, and made progress t owards est ablishing and meet ing t he t radeoff between performance, and feedback t imeliness and quality. Considering a general CSIT process, t he work provided simple DoF expressions t hat reveal t he role of t he number of ant ennas in est ablishing t he feedback quality associat ed t o a cert ain DoF performance.
Chapt er 4
Fundament al Performance and Feedback Tradeoff over t he K -User M I SO BC
T his chapt er considers t he mult iuser mult iple-input single-out put (MISO) broadcast channel (BC), where a t ransmit t er wit h M ant ennas t ransmit s informat ion t o K single-ant enna users, and where - as expect ed - t he quality and t imeliness of channel st at e informat ion at t he t ransmit t er (CSIT ) is imperfect . Mot ivat ed by t he fundament al question of how much feedback is necessary t o achieve a cert ain performance, we seeks t o est ablish bounds on t he t radeoff between degrees-of-freedom (DoF) performance and CSIT feedback quality. Specifically, t he work provides a novel DoF region out er bound for t he general K -user M ⇥ 1 MISO BC wit h part ial current CSIT , which naturally bridges t he gap between t he case of having no current CSIT (only delayed CSIT , or no CSIT ) and t he case wit h full CSIT . T he work t hen charact erizes t he minimum CSIT feedback t hat is necessary for any point of t he sum DoF, which is opt imal for t he case wit h M K , and t he case wit h M = 2, K = 3.
4.1
I nt r oduct ion
We consider t he mult iuser mult iple-input single-out put (MISO) broadcast channel (BC), where a t ransmit t er wit h M ant ennas, t ransmit s informat ion t o K single-ant enna users. In t his set t ing, t he received signal at t ime 91
92
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Feedback
h
(1)
y (1) User 1
Tx
(K)
h Feedback
y(K) User K
Figur e 4.1 – Syst em model of K -user MISO BC wit h CSIT feedback. t, is of t he form (k)
yt (k)
(k) T
= ht
(k)
x t + zt ,
k = 1, · · · , K
(4.1) (k)
where h t denot es t he M ⇥ 1 channel vect or for user k, zt denot es t he unit power AWGN noise, and where x t denot es t he t ransmit t ed signal vect or adhering t o a power const raint E[||x t ||2 ] P , for P t aking t he role of t he signal-t o-noise rat io (SNR or snr). We here consider t hat t he fading coeffi (k) cient s h t , k = 1, · · · , K , are independent and ident ically dist ribut ed (i.i.d.) complex Gaussian random variables wit h zero mean and unit variance, and are i.i.d. over t ime. It is well known t hat t he performance of t he BC is great ly aff ect ed by t he t imeliness and quality of feedback ; having full CSIT allows for t he opt imal min{ M , K } sum degrees-of-freedom (DoF) (cf. [1]) 1, while t he absence of any CSIT reduces t his t o just 1 sum DoF (cf. [2, 3]). T his gap has spurred a plet hora of works t hat seek t o analyze and opt imize BC communicat ions in t he presence of delayed and imperfect feedback. One of t he works t hat st ands out is t he work by Maddah-Ali and Tse [4] which recent ly revealed t he benefit s of employing delayed CSIT over t he BC, even if t his CSIT is complet ely obsolet e. Several int erest ing generalizat ions followed, including t he work in [10] which showed t hat in t he BC set t ing wit h K = M + 1, combining delayed CSIT wit h perfect (current ) CSIT (over t he last KK 1 fract ion of communicat ion period) allows for t he opt imal sum DoF M corresponding t o full CSIT . A similar approach was exploit ed in [46] which revealed t hat , t o achieve t he maximum sum DoF min{ M , K } , each user has t o symmet rically feed back perfect CSIT over a min{KM ,K } fract ion of t he communicat ion t ime, 1. We remind t he reader t hat for an achievable rat e t uple (R 1 , R 2 , · · · , R K ), where R i is Ri for user i , t he corresponding DoF t uple (d1 , d2 , · · · , dK ) is given by di = limP ! 1 l og , i = P 1, 2, · · · , K . T he corresponding DoF region D is t hen t he set of all achievable DoF t uples (d1 , d2 , · · · , dK ).
4.1
I nt r oduct ion
93
and t hat t his fract ion is opt imal. Ot her int erest ing works in t he cont ext of ut ilizing delayed and current CSIT , can be found in [5–8] which explored t he set t ing of combining perfect delayed CSIT wit h immediat ely available imperfect CSIT , t he work in [9, 47] which addit ionally considered t he eff ect s of the quality of delayed CSIT , t he work in [11] which considered alt ernat ing CSIT feedback, t he work in [48] which considered delayed and progressively evolving (progressively improving) current CSIT , and t he works in [19–24,49] and many ot her publicat ions. Our work here generalizes many of t he above set t ings, and seeks t o est ablish fundament al t radeoff between DoF performance and CSIT feedback quality, over t he general K -user M ⇥1 MISO BC.
4.1.1
CSI T quant ifi cat ion and feedback m odel
We proceed t o describe t he quality and t imeliness measure of CSIT feed(k) back, and how t his measure relat es t o exist ing work. We here use hˆ to (k)
t
denot e t he current channel est imat e (for channel h t ) at t he t ransmit t er at t imeslot t, and use (k) (k) h˜ t = h t
(k) hˆ t
(k) t o denot e t he est imat e error assumed t o be mut ually independent of hˆ t and assumed t o have i.i.d. circularly symmet ric complex Gaussian ent ries wit h zero mean and power ⇥ (k) ⇤ . (k ) E kh˜ t k2 = P ↵ t , (k)
for some CSI quality exponent ↵ t 2 [0, 1] describing t he quality of t his (k) est imat e. We not e t hat ↵ t = 0 implies very lit t le current CSIT knowledge, (k) and t hat ↵ t = 1 implies perfect CSIT in t erms of t he DoF performance 2 . T he approach ext ends over non-alt ernat ing CSIT set t ings in [4] and [5– 8], as well as over an alt ernat ing CSIT set ting (cf. [11, 46]) where CSIT (k) knowledge alt ernat es between perfect CSIT (↵ t = 1), and delayed or no (k) CSIT (↵ t = 0). In a set t ing where communicat ion t akes place over n such coherence periods (assuming t hat a single channel use per such coherence period, t = 1, 2, · · · , n), t his approach off ers a nat ural measure of a per-user average feedback cost , in t he form of n
↵¯ (k) ,
1X (k) ↵t , n
k = 1, 2, · · · , K ,
t= 1
2. T his can be readily derived, using for example t he work in [17].
94
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
as well as a measure of current CSIT feedback cost XK CC ,
↵¯ (k) ,
(4.2)
k= 1
accumulat ed over all users. A lt er nat ing C SI T set t ing In a set t ing where delayed CSIT is always available, t he above model (k) capt ures t he alt ernat ing CSIT set t ing where t he exponent s are binary (↵ t = 0, 1), in which case (k) ↵¯ (k) = P simply describes t he fract ion of t ime during which user k feeds back perfect CSIT , wit h XK (k) CC = CP , P k= 1
describing t he total perfect CSI T feedback cost. Sy m m et r ic and asy m m et r ic C SI T feedback Mot ivat ed by t he fact t hat diff erent users might have diff erent feedback capabilit ies due t o t he feedback channels wit h diff erent capacit ies and different reliabilit ies, symmet ric CSIT feedback ( ↵¯ (1) = · · · = ↵¯ (K ) ) and asym0 0 met ric CSIT feedback ( ↵¯ (k) 6 = ↵¯ (k ) 8 k 6 = k ) are considered in t his work.
4.1.2
St r uct ur e and sum mar y of cont r ibut ions
Sect ion 4.2 provides t he main result s of t his work : – In T heorem 3 we first provide a novel out er bound on t he DoF region, for t he K -user M ⇥ 1 MISO BC wit h part ial current CSIT quant i(k) zed wit h { ↵ t } k,t , which bridges t he case wit h no current CSIT (only delayed CSIT , or no CSIT ) and t he case wit h full CSIT . T his result ma(k) nages t o generalize t he result s by Maddah-Ali and Tse (↵ t = 0, 8t, k), (k) Yang et al. and Gou and Jafar (K = 2, ↵ t = ↵, 8t, k), Maleki (1) (2) et al. (K = 2, ↵ t = 1, ↵ t = 0, 8t), Chen and Elia (K = 2, (1) (2) (k) ↵t 6 = ↵ t , 8t), Lee and Heath (M = K + 1, ↵ t 2 { 0, 1} , 8t, k), and (k) Tandon et al. (↵ t 2 { 0, 1} , 8t, k). – From T heorem 3, we t hen provide t he upper bound on t he sum DoF as a funct ion of t he current CSIT feedback cost , which is t ight for t he case wit h M K (cf. T heorem 4) and t he case wit h M = 2, K = 3 (cf. T heorem 5, Corollary 5a).
4.1
I nt r oduct ion
95
– Furt hermore, T heorem 6 charact erizes t he minimum t ot al current CSIT feedback cost C?P t o achieve t he maximum sum DoF, where t he t ot al feedback cost C?P can be dist ribut ed among all t he users wit h any (k) (asymmet ric and symmet ric) combinat ions { P } k . – In addit ion, t he work considers some ot her general set t ings of BC and provides t he DoF inner bound as a funct ion of t he CSIT feedback cost . T he main converse proof, t hat is for T heorem 3, is shown in t he Sect ion 4.3 and appendix. Most of t he achievability proofs are shown in t he Sect ion 4.4. Finally Sect ion 4.5 concludes t his work. T hroughout t his work, we will consider communicat ion over n coherence periods where, for clarity of not at ion, we will focus on t he case where we employ a single channel use per such coherence period (unit coherence period). Furt hermore, unless st at ed ot herwise, we assume perfect delayed CSIT , as well as adhere t o t he common convent ion (see [4,6,7,11,33,46]), and assume perfect and global knowledge of channel st at e informat ion at t he receivers.
96
C hapt er 4
4.2
M ain r esult s
4.2.1
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Out er bounds
We first present t he DoF region out er bound for t he general K -user M ⇥1 MISO BC. T heor em 3 (DoF region out er bound). T he DoF region of the K -user M ⇥1 MI SO BC, is outer bounded as XK k= 1
KX 1✓
d⇡ (k) min{ k, M }
1+
dk
1,
k= 1
◆ 1 min{ k, M }
1 min{ K , M }
↵¯ (⇡ (k))
k = 1, 2, · · · , K
(4.3) (4.4)
where ⇡ denotes a permutation of the ordered set { 1, 2, · · · , K } , and ⇡ (k) denotes the k th element of set ⇡ .
Proof. T he proof is shown in Sect ion 4.3. (k)
R em ar k 4. I t is noted that the bound captures the results in [4] (↵ t = (k) 0, 8t, k), in [6, 7] (K = 2, ↵ t = ↵, 8t, k), in [33] (M = K = 2, (1) (2) (1) (2) ↵ t = 1, ↵ t = 0, 8 t), in [8] (K = 2, ↵ t 6 = ↵ t , 8 t), in [11, 46] (k) (k) (↵ t 2 { 0, 1} , 8t, k), as well as in [26] (M K , ↵ t = ↵, 8t, k). 3 , we direct ly have Summing up from t he above t he K diff erent bounds PK t he following upper bound on t he sum DoF d⌃ , k= 1 dk . In t he following, we will use some not at ions given as
dM AT ,
K 1+
, P
1 min{ 2,M }
+
1 min{ 3,M }
+ ··· +
M K M 1 M 1 i 1 i= 1 i( M )
+
( MM 1 ) K
(4.5)
1 min{ K ,M }
M
(
P
K i= K
1 M+1 i )
.
(4.6)
C or ollar y 3a (Sum DoF out er bound). For the K -user M ⇥ 1 MI SO BC, the sum DoF is outer bounded as d⌃
dM AT
✓ + 1
dM AT min{ K , M }
◆ XK
↵¯ (k) .
(4.7)
k= 1
3. For bound k, k = 1, 2, · · · , K , we have ⇡ = { ⇡ (i ) = mod(k + i mod(x) K is t he modulo operat or.
2) K + 1} Ki= 1 , where
4.2
M ai n r esult s
97
T he above t hen readily t ranslat es ont o a lower on t he minimum P K bound (k) possible t ot al current CSIT feedback cost CC = ↵ ¯ needed t o achieve k= 1 t he maximum sum DoF 4 d⌃ = min{ K , M } . C or ollar y 3b (Bound on CSIT cost for maximum DoF). T he minimum CC required to achieve the maximum sum DoF min{ K , M } of the K -user M ⇥1 MISO BC, is lower bounded as C?C
min{ K , M } .
(4.8) (k)
Transit ioning t o t he alt ernat ing CSIT sett ing where ↵ t 2 { 0, 1} , we have t he following sum-DoF out er bound as a funct ion of t he perfect -CSIT (k) durat ion ↵¯ (k) = P = P , 8 k. We not e t hat t he bound holds irrespect ive of whet her, in t he remaining fract ion of t he t ime 1 P , t he CSIT is delayed or non exist ent . C or ollar y 3c (Out er bound, alt ernat ing CSIT ). For the K -user M ⇥ 1 MISO BC, and given symmetric perfect-CSI T feedback cost P , the sum DoF is outer bounded as ✓ ◆ ⇢ K dM AT min{ K , M } d⌃ dM AT + K min P , . (4.9) min{ K , M } K
4.2.2
Opt imal cases of D oF char act er izat ions
We now provide t he opt imal cases of DoF charact erizat ions as a funct ion of t he current CSIT feedback cost . T he case wit h M K is first considered in the following. T heor em 4 (Opt imal case, M K ). For the K -user M ⇥1 MI SO BC with M K , and given symmetric perfect-CSI T feedback cost P , the optimal sum DoF is characterized as d⌃ = (K
dM AT ) min{
P , 1}
+ dM AT .
(4.10)
Proof. T he converse and achievability proofs are derived from Corollary 3c and Proposit ion 4 (shown in t he next subsect ion), respect ively. R em ar k 5. I t is noted that, for the special case with M = K = 2, the above characterization captures the result in [11]. Moving t o t he case where M < K , we have t he following opt imal sum DoF charact erizat ions for t he case wit h M = 2, K = 3. T he first int erest is placed on t he minimum C?P (d⌃ ) t o achieve a sum DoF d⌃ , recalling t hat PK (k) C?P = k= 1 P describes t he t ot al perfect CSIT feedback cost . 4. Nat urally t he result is limit ed t o t he case where min{ K , M } > 1.
98
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
d∑ K
d
dMAT
(K dMAT ) P dMAT
0
1
Figur e 4.2 – Opt imal sum DoF d⌃ vs.
P
P
for t he MISO BC wit h M
K .
d∑ 2 d
3/ 2
3 2
1 CP 4
0
2
CP
Figur e 4.3 – Opt imal sum DoF (d⌃ ) vs. t ot al perfect CSIT feedback cost (CP ) for t hree-user 2 ⇥1 MISO BC. T heor em 5 (Opt imal case, M = 2, K = 3). For the three-user 2 ⇥1 MISO BC, the minimum total perfect CSIT feedback cost is characterized as C?P (d⌃ ) = (4d⌃
6) + ,
8 d⌃ 2 [0, 2]
(4.11)
where the total feedback cost C?P (d⌃ ) can be distributed among all the users (k) (k) with some combinations { P } k such that P C?P (d⌃ )/ 2 for any k. Proof. T he converse proof is direct ly from Corollary 3a, while t he achievability proof is shown in Sect ion 4.4.2. T heorem 5 reveals t he fundament al t radeoff between sum DoF and t ot al perfect CSIT feedback cost (see Fig 4.3). T he following examples are provided t o off er some insight s corresponding t o T heorem 5.
4.2
M ai n r esult s
99
Exam ple 11. For the target sum DoF d⌃ = 3/ 2, 7/ 4, 2, the minimum total perfect CSI T feedback cost is C?P = 0, 1, 2, respectively. Exam ple 12. T he target d⌃ = 7/ 4 is achievable with asymmetric feedback P = [1/ 6 1/ 3 1/ 2], and symmetric feedback P = [1/ 3 1/ 3 1/ 3], and some other feedback such that C?P (7/ 4) = 1. Exam ple 13. The target d⌃ = 2 is achievable with asymmetric feedback P = [1/ 3 2/ 3 1], and symmetric feedback P = [2/ 3 2/ 3 2/ 3], and some other feedback such that C?P (2) = 2. (k)
Transit ioning t o t he symmet ric set t ing where P = P 8 k, from T heorem 5 we have t he fundament al t radeoff between opt imal sum DoF and CSIT feedback cost P . C or ollar y 5a (Opt imal case, M = 2, K = 3, P ). For the three-user 2 ⇥ 1 MISO BC with symmetrically alternating CSIT feedback, and given P , the optimal sum DoF is characterized as ⇢ 3(2 + P ) d⌃ = min ,2 . (4.12) 4 Now we address t he quest ions of what is t he minimum C?P t o achieve t he maximum sum DoF min{ M , K } for t he general BC, and how t o dist ribut e C?P among all t he users, recalling again t hat C?P is t he t ot al perfect CSIT feedback cost . T heor em 6 (Minimum cost for maximum DoF). For the K -user M ⇥ 1 MISO BC, the minimum total perfect CSIT feedback cost to achieve the maximum DoF is characterized as ⇢ 0, if min{ M , K } = 1 ? CP (min{ M , K } ) = min{ M , K } , if min{ M , K } > 1 where the total feedback cost C?P can be distributed among all the users with (k) any combinations { P } k . Proof. For t he case wit h min{ M , K } = 1, simple T DMA is opt imal in t erms of the DoF performance. For t he case wit h min{ M , K } > 1, t he converse proof is direct ly derived from Corollary 3b, while t he achievability proof is shown in Sect ion 4.4.1. It is not ed t hat T heorem 6 is a generalization of t he result in [46] where only symmet ric feedback was considered. T he following examples are provided t o off er some insight s corresponding t o T heorem 6.
100
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Exam ple 14. For the case where M = 2, K = 4, the optimal 2 sum DoF performance is achievable, with asymmetric feedback P = [1/ 5 2/ 5 3/ 5 4/ 5], and symmetric feedback P = [1/ 2 1/ 2 1/ 2 1/ 2], and any other feedback such that C?P = 2. Exam ple 15. For the case where M = 3, K = 5, the optimal 3 sum DoF performance is achievable, with asymmetric feedback P = [1/ 5 2/ 5 3/ 5 4/ 5 1], and symmetric feedback P = [3/ 5 3/ 5 3/ 5 3/ 5 3/ 5], and any other feedback such that C?P = 3. T he following corollary is derived from T heorem 6, where t he case wit h min{ M , K } > 1 is considered. C or ollar y 6a (Minimum cost for maximum DoF). For the K -user M ⇥ 1 MI SO BC, where J users instantaneously feed back perfect (current) CSI T, with the other users feeding back delayed CSI T, then the minimum number J is min{ M , K } , in order to achieve the maximum sum DoF min{ M , K } .
4.2.3
I nner bounds
In t his subsect ion, we provide t he following inner bounds on t he sum DoF as a funct ion of t he CSIT cost , which are t ight for many cases as st at ed. P r op osit ion 3 (Inner bound, M = 2, K MI SO BC, the sum DoF is bounded as d⌃
3 K + min{ 2 4
3). For the K (
P,
2 }. K
3)-user 2 ⇥ 1
(4.13)
Proof. T he proof is shown in Sect ion 4.4.3.
P r op osit ion 4 (Inner bound, M K and M < K ). For the K -user M ⇥1 MI SO BC, the sum DoF for the case with M K is bounded as d⌃
(K
dM AT ) min{
P , 1}
+ dM AT ,
(4.14)
while for the case with M < K , the sum DoF is bounded as d⌃
(K
K ) min{ M
Proof. T he proof is shown in Sect ion 4.4.4.
P,
M }+ K
.
(4.15)
4.2
M ai n r esult s
101
d∑ 2
3 2
d 3/ 2
K 4
0
P
P
2/ K
Figur e 4.4 – Achievable sum DoF d⌃ vs. MISO BC.
P
for t he K (
3)-user 2 ⇥1
d∑ M
d
0
(K
K ) M
P
P
M/ K
Figur e 4.5 – Achievable sum DoF d⌃ vs. M < K.
P
for t he MISO BC wit h
102
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
d∑ d
4( K
3) / 11
D
3/ 2 4/ 3
d
1
0
1 K
2/ (3K)
D
/2
9/ (8K)
Figur e 4.6 – Achievable sum DoF d⌃ vs. K 3, M = 2, where
D
for t he MISO BC wit h = 0.
D P
Finally, we consider a case of BC wit h delayed CSIT feedback only, where (k) t o denot e t he fract ion of t ime during P = 0. In t his case, we use D which CSIT fed back from user k is delayed, and focus on t he case wit h (k) D = D , 8k. P r op osit ion 5 (Inner bound on DoF wit h delayed CSIT ). For the K ( 3)user 2 ⇥ 1 MISO BC, and for the case of P = 0, the sum DoF is bounded as ⇢ K 12 4K 3 d⌃ min 1 + + . (4.16) D, D, 2 11 11 2 Proof. T he proof is shown in Sect ion 4.4.5. R em ar k 6. For the K -user MI SO BC with current and delayed CSI T feedback, by increasing the number of users, the same DoF performance can be achievable with decreasing feedback cost per user. For example, for the K user MI SO BC with M = 2, by increasing K we can achieve any fixed DoF 2 9 within the range of (1, 2], with decreasing P K , and D 8K , both of which approach to 0 as K is large.
4.3
4.3
C onver se pr oof of T heor em 3
103
Conver se pr oof of T heor em 3
We first provide t he Proposit ion 6 t o be used, where we drop t he t ime index for simplicity. P r op osit ion 6. Let T
y(k) = h (k) x + z(k) , y k , [y
(1)
y
(2)
··· y
k = 1, 2 · · · , K (k)
]T
z k , [z(1) z(2) · · · z(k) ]T H
k
, [h (1) h (2) · · · h (k) ]T
H , [h (1) h (2) · · · h (K ) ]T H = Hˆ + H˜ (k)
where h˜ 2 CM ⇥1 has i.i.d. N C (0, k2 ) entries. T hen, for any U such that pX |U Hˆ H˜ = pX |U Hˆ and K m l, we have l h(y m |U, Hˆ , H˜ ) 0
m h(y l |U, Hˆ , H˜ ) 0
(m
0
0
Xl
l )
log
2 k
+ o(log snr)
k= 1
(4.17) where we define l 0 , min { l, M } and m 0 , min { m, M } .
Proof. T he proof is shown in t he Sect ion 4.6. Now giving t he observat ions and messages of users 1, . . . , k 1 t o user k, we est ablish t he following genie-aided upper bounds on t he achievable rat es (1)
nR 1
I (W1; y[n] | ⌦ [n] ) + n✏
nR 2
I (W2; y[n] , y[n] | W1 , ⌦ [n] ) + n✏
(1)
(4.18)
(2)
(4.19)
.. . nR K
(1)
(2)
(k)
I (WK ; y[n] , y[n] , . . . , y[n] | W1, . . . , WK
1, ⌦ [n] )
+ n✏
(4.20)
where we apply Fano’s inequality and some basic chain rules of mut ual informat ion using t he fact t hat messages from diff erent users are independent , where we define h iT h i (1) (K ) T (K ) ˆ ˆ ˆ S t , h (1) S , · · · ht t ht · · · ht t ˆ n ⌦ [n] , { S t , S t } t = 1
(k)
(k)
y[n] , { yt } nt= 1.
104
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Alt ernat ively, we have (1)
(1)
nR 1
h(y[n] | ⌦ [n] )
nR 2
h(y[n] , y[n] | W1 , ⌦ [n] )
(1)
h(y[n] | W1, ⌦ [n] ) + n✏
(2)
(1)
(2)
h(y[n] , y[n] | W1 , W2 , ⌦ [n] ) + n✏
(4.21) (4.22)
.. . (1)
nR K
(K )
h(y[n] , . . . , y[n] | W1, . . . , WK (1) (K ) h(y[n] , . . . , y[n]
1, ⌦ [n] )
| W1 , . . . , WK , ⌦ . [n] ) + n✏
(4.23)
T herefore, it follows t hat XK n (R k k0 k= 1 KX 1✓
✏ )
1 (1) (k+ 1) h(y[n] , . . . , y[n] | W1, . . . , Wk , ⌦ [n] ) 0 (k + 1) k= 1 ◆ 1 (1) (k) h(y , . . . , y[n] | W1 , . . . , Wk , ⌦ [n] ) k 0 [n] 1 (1) (1) (K ) + h(y[n] | ⌦ h(y , . . . , y[n] | W1, . . . , WK , ⌦ (4.24) [n] ) [n] ) K 0 [n] KX 1 Xn ✓ 1 (1) (k+ 1) (1) (k) h(y , . . . , yt | y[t 1] , . . . , y[t 1] , W1, . . . , Wk , ⌦ [n] ) (k + 1) 0 t k= 1 t = 1 ◆ 1 (1) (k) (1) (k) h(y , . . . , yt | y[t 1] , . . . , y[t 1] , W1 , . . . , Wk , ⌦ [n] ) k0 t + n log P + n o(log P ) (4.25) KX 1 Xn
k
log P k= 1 t = 1 KX 1
= n log P
(4.26)
i= 1
k
(k + 1) 0 k 0 X ↵¯ (i ) + n log P + n o(log P ) k 0(k + 1) 0
(4.27)
1 ⌘ (k) ↵¯ + n log P + n o(log P ) K0
(4.28)
k 0 , min { k, M } ;
(4.29)
k= 1 KX 1⇣
= n log P k= 1 5
(k + 1) 0 k 0 X (i ) ↵ t + n log P + n o(log P ) k 0(k + 1) 0
i= 1
1 k0
where we define
t he inequality (4.25) is due t o 1) t he chain rule of diff erent ial ent ropy, 2) t he fact t hat removing condit ion does not decrease diff erent ial ent ropy, (i )
5. We denot e y0
as an empty t erm, for all i .
4.3
C onver se pr oof of T heor em 3
105
(1)
3) h(yt | ⌦ log P + o(log P ), i.e., Gaussian dist ribut ion maximizes dif[n] ) ferent ial ent ropy under covariance const raint , and 4) (1)
(K )
(k)
K
n h(y[n] , . . . , y[n] | W1, . . . , WK , ⌦ [n] ) = h({ zt } t = 1 k= 1 ) > 0;
(4.26) is from Proposit ion 6 by set t ing (1) (k) ˆ U = { y[n] , . . . , y[n] , W1 , . . . , Wk , ⌦ [n] } \ { S t , S t } ,
H = St ,
Hˆ = Sˆ t ;
t he last equality is obt ained aft er put t ing t he summat ion over k inside t he summat ion over i and some basic manipulations. Similarly, we can int erchange t he roles of t he users and obt ain t he same genie-aided bounds. Finally, t he single ant enna const raint gives t hat dk 1, k = 1, · · · , K . Wit h t his, we complet e t he proof.
106
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
4.4
D et ails of achievabilit y pr oofs
In t his sect ion, we provide t he det ails of t he achievability proofs. Specifically, t he achievability proof of T heorem 6 is first described in Sect ion 4.4.1, which can be applied in part s for the achievability proof of T heorem 4.4.2 shown in Sect ion 4.4.2, wit h t he proposit ion proofs shown in t he rest of t his sect ion.
4.4.1
A chievabilit y pr oof of T heor em 6
We will prove t hat , t he opt imal sum DoF d⌃ = min{ M , K } is achievable (1) (2) (K ) K wit h any CSIT feedback cost P , [ P ··· P P ] 2 R such t hat PK (k) CP = = min{ M , K } . First of all, we not e t hat t here exist s a k= 1 P minimum number n such t hat P
0
,[
(1) 0 P
(2) 0 P
···
(K ) 0 ], P
n
P
= [n
(1) P
n
(2) P
··· n
(K ) P ]
2 ZK
is an int eger vect or. T he explicit communicat ion wit h n channel uses is given as follows : – St ep 1 : Init ially set t ime index t = 1. (U(1)) 0 – St ep 2 : Permut e user indices orderly int o a set U such t hat P (U(2)) 0 P
(U(K )) 0
··· , where U(k) denot es t he k t h element of set P U, and where U(k) 2 { 1, 2, · · · , K } . – St ep 3 : Select min{ M , K } users t o communicat e : users U(K ), U(K 1), · · · , U(K min{ M , K } + 1). – St ep 4 : Let select ed users feed back perfect CSIT at t ime t, keeping t he rest users silent . – St ep 5 : T he t ransmit t er sends min{ M , K } independent symbols t o t hose select ed users respect ively, which can be done wit h simple zeroforcing. (U(k)) 0 (U(k)) 0 – St ep 6 : Set P = P 1, k = K min{ M , K } + 1, · · · , K 1, K . – St ep 7 : Set t = t + 1. If renewed t > n t hen t erminat e, else go back t o st ep 2. In t he above communicat ion wit h n channel uses, t he algorit hm guarant ees (k) 0 (k) t hat user i is select ed by P = n P t imes t ot ally, and t hat min{ M , K } diff erent users are select ed in each channel use. As a result , t he opt imal sum DoF d⌃ = min{ M , K } is achievable. Now we consider an example wit h M = 2, K = 3, and P = [1/ 3 2/ 3 1], and show t hat t he opt imal sum DoF d⌃ = 2 is achievable wit h t he following communicat ion : (1) 0 (1) (2) 0 (2) (3) 0 (3) – Let n = 3. Init ially P = n P = 1, P = n P = 2, P = n P = 3. – For t = 1, we have U = { 1, 2, 3} , and (U(3)) 0 = P
(U(1)) 0 P
= 1,
3. Users 3 and 2 are select ed t o communicat e.
(U(2)) 0 P
= 2,
4.4
D et ail s of achievabil it y pr oofs
107
P = 2 wit h C? = 2, Tabl e 4.1 – Summary of t he scheme for achieving d⇤ P
where M = 2, K = 3,
(1) P
t ime t U {
(U(1)) 0 , P
(2) P
= 1/ 3,
1 { 1, 2, 3}
(U(2)) 0 , P
= 2/ 3, 2 { 1, 2, 3}
(3) P
= 1. 3 { 2, 1, 3}
(U(3)) 0 } P
{ 1, 2, 3} { 1, 1, 2} { 0, 1, 1} user 2, 3 user 2, 3 user 1, 3 user 3 : yes user 3 : yes user 3 : yes user 2 : yes user 2 : yes user 2 : no user 1 : no user 1 : no user 1 : yes No. of t ransmit t ed symbols 2 2 2 Act ive users Perfect CSIT feedback
– For t = 2, we updat e t he paramet ers as U = { 1, 2, 3} , and (U(2)) 0 P
(U(3)) 0 P
(U(2)) 0 = P
(U(3)) 0
(U(1)) 0 = P
1,
= 1, = 2. At t his t ime, again user 3 and user 2 are select ed t o communicat e. (U(1)) 0 – For t = 3, we updat e t he paramet ers as U = { 2, 1, 3} , and P = 0, 1, P = 1. At t his t ime, user 3 and user 1 are select ed t o communicat e. Aft er t hat t he communicat ion t erminat es. In t he above communicat ion wit h t hree channel uses, t he t ransmit t er sends two symbols in each channel use, which allows for t he opt imal sum DoF d⌃ = 2 (see Table 4.1).
4.4.2
A chievabilit y pr oof of T heor em 5
We proceed t o show t hat , any sum DoF d⌃ 2 [3/ 2, 2] is achievable wit h t he feedback (k) P
X3
CP , k = 1, 2, 3, 2
such t hat
CP =
(k) P
= 4d⌃
6.
k= 1
First of all, we not e t hat t here exist s a minimum number n such t hat [2n
(1) P / CP
2n
(2) P /
CP
n2
(3) P /
CP ] 2 Z 3 ,
and 2n/ CP 2 Z.
T he scheme has two blocks, wit h t he first block consist ing of n channel uses, and t he second block consist ing of 0
n = 2n/ CP
n
channel uses. In t he first block, we use t he algorit hm shown in t he Sect ion 4.4.1 t o achieve t he full sum DoF in t hose n channel uses, during which (k) user k feeds back perfect CSIT in 2n P / CP channel uses, for k = 1, 2, 3. In
108
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
t he second block, we use t he Maddah-Ali and Tse scheme in [4] t o achieve 0 3/ 2 sum DoF in t hose n channel uses, during which each user feeds back delayed CSIT only. T he communicat ion wit h n channel uses for t he first block is given as follows : (k) 0 (k) – St ep 1 : Let P = 2n P / CP for all k. Init ially, set t = 1. – T he st eps 2, 3, 4, 5, 6 are t he same as t hose in t he algorit hm shown in Sect ion 4.4.1, for M = 2, K = 3. – St ep 7 : Set t = t + 1. If renewed t > n t hen t erminat e, else go back t o st ep 2. In t he above communicat ion wit h n channel uses, t he algorit hm guarant ees (k) 0 (k) t hat user k, k = 1, 2, 3, is select ed by P = 2n P / CP t imes. We not e t hat P (k) 0 (k) (k) 0 n under t he const raint P CP / 2 for any k, and t hat Kk= 1 P = P 2n, t o suggest t hat in each t imeslot two diff erent users are select ed, which allows for t he opt imal 2 sum DoF in t his block. As st at ed, in t he second block, we use t he MAT scheme t o achieve t he 3/ 2 0 sum DoF in t hose n channel uses, during which each user feeds back delayed 0 CSIT only. As a result , in t he t otal n + n channel uses communicat ion, 0 (k) (k) user k = 1, 2, 3 feeds back perfect CSIT in 2n P / (CP (n+ n )) = P fract ion of communicat ion period, wit h achievable sum DoF given as 0
d⌃ =
2n 3n 3 1 + CP . 0 + 0 = (n + n ) 2(n + n ) 2 4
We not e t hat t he achievability scheme applies t o t he case of having some (1) (2) (3) CP / 2 such t hat CP = 4d⌃ 6, and allows t o achieve any sum P , P , P DoF d⌃ 2 [3/ 2, 2]. Apparent ly, CP = 0 allows for any sum DoF d⌃ 2 [0, 3/ 2],
which complet es t he proof.
4.4.3
P r oof of P r oposit ion 3
T he achievability scheme is based on t ime sharing between two st rat egies of CSIT feedback, i.e., delayed CSIT feedback wit h P 0 = 0 and alt ernat ing CSIT feedback wit h P 00 = K2 , where t he first st rat egy achieves d0⌃ = 3/ 2 by applying Maddah-Ali and Tse (MAT ) scheme (see in [4]), wit h t he second st rat egy achieving d0⌃0 = 2 by using alt ernat ing CSIT feedback manner (see in [46]). Let 2 [0, 1] (resp. 1 ) be t he fract ion of t ime during which t he first (resp. second) CSIT feedback st rat egy is used in t he communicat ion. As a result , t he final feedback cost (per user) is given as P
=
P
0
+
P
00
(1
),
(4.30)
4.4
D et ail s of achievabil it y pr oofs
109
implying t hat 00 P 00 P
=
P , 0
(4.31)
P
with final sum DoF given as d⌃ = d0⌃ =
+ d0⌃0(1
d0⌃0 +
(d0⌃
= d0⌃0 + (d0⌃ =
3 K + 2 4
) d0⌃0)
d0⌃0)
00 P 00 P
P 0 P
(4.32)
P
which complet es t he proof.
4.4.4
P r oof of P r oposit ion 4
For t he case wit h M K , t he proposed scheme is based on t ime sharing between delayed CSIT feedback wit h P 0 = 0 and full CSIT feedback wit h 0 00 P = 1, where t he first feedback st rat egy achieves dP = dM AT by applying 00
MAT scheme, wit h t he second one achieving dP = K . As a result , following t he st eps in (4.30), (4.31), (4.32), t he final sum DoF is calculat ed as d⌃ = d0⌃0 + (d0⌃ = (K
d0⌃0)
dM AT )
P
00 P 00 P
P 0
P
+ dM AT
where P 2 [0, 1] is t he final feedback cost (per user) for t his case. Similar approach is exploit ed for t he case wit h M < K . In t his case, we apply t ime sharing between delayed CSIT feedback wit h P 0 = 0 and alt ernat ing CSIT feedback wit h P 00 = M / K . In t his case, t he first feedback st rat egy achieves d0⌃ = by applying MAT scheme, wit h t he second st rat egy achieving d0⌃0 = M by using alt ernat ing CSIT feedback manner. As a result , for P 2 [0, M K ] being t he final feedback cost for t his case, t he final sum DoF is calculat ed as d⌃ = d0⌃0 + (d0⌃ = (K which complet es t he proof.
K ) M
d0⌃0) P
+
00 P 00 P
P 0 P
110
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Tabl e 4.2 – Summary of t he achievability scheme for achieving d⌃ = 2 wit h D = 3K . block index No. of channel uses Act ive users Delayed CSIT feedback fract ion in a block Sum DoF in a block
4.4.5
1 3 user 1, 2 user 1 : 1/ 3 user 2 : 1/ 3 t he rest : 0 4/ 3
2 3 user 2, 3 user 2 : 1/ 3 user 3 : 1/ 3 t he rest : 0 4/ 3
··· ··· ··· ···
···
4 3
K 3 user K , 1 user K : 1/ 3 user 1 : 1/ 3 t he rest : 0 4/ 3
P r oof of P r oposit ion 5
As shown in t he Fig 4.6, t he sum DoF performance has t hree regions : 8 2 < 1 + K2 D , D 2 [0, 3K ] 12 4K 2 9 d⌃ = + 11 D , D 2 [ 3K , 8K ] : 11 9 3/ 2, D 2 [ 8K , 1]. In t he following, we will prove t hat t he sum DoF d⌃ = 1, 43 , 32 are 2 9 achievable wit h D = 0, 3K , 8K , respect ively. At t he end, t he whole DoF performance declared can be achievable by t ime sharing between t hose performance point s. 2 T he proposed scheme achieving d⌃ = 43 wit h D = 3K , is a modified version of t he MAT scheme in [4]. The new scheme has K blocks, wit h each block consist ing of t hree channel uses. In each block, four independent symbols are sent t o two orderly select ed users, which can be done wit h MAT scheme wit h each of two chosen user feeding back delayed CSIT in one chan2 nel use. As a result , d⌃ = 43 is achievable wit h D = 3K , using t he fact t hat each of K users needs t o feed back delayed CSIT twice only in t he whole communicat ion (see Table 4.2). 9 Similarly, t he proposed scheme achieving d⌃ = 32 wit h D = 8K has K blocks, wit h each block consist ing of 8 channel uses. In each block, 3 out of K users are select ed t o communicate. In t his case, 12 independent symbols are sent t o t he chosen users during each block, which can be done wit h anot her MAT scheme wit h each of chosen users feeding back delayed CSIT 9 in 3 channel uses. As a result , d⌃ = 32 is achievable wit h D = 8K , using t he fact t hat each of K users needs t o feed back delayed CSIT 9 t imes only in t he whole communicat ion (see Table 4.3). Finally, d⌃ = 1 is achievable wit hout any CSIT . By now, we complet e t he proof.
4.4
D et ail s of achievabil it y pr oofs
111
Tabl e 4.3 – Summary of t he achievability scheme for achieving d⌃ = 9 wit h D = 8K . block index No. of channel uses Act ive users Delayed CSIT feedback fract ion in a block
Sum DoF in a block
1 8 user 1, 2, 3 user 1 : 3/ 8 user 2 : 3/ 8 user 3 : 3/ 8 t he rest : 0 3/ 2
2 8 user 2, 3, 4 user 2 : 3/ 8 user 3 : 3/ 8 user 4 : 3/ 8 t he rest : 0 3/ 2
··· ··· ··· ···
···
3 2
K 8 user K , 1, 2 user K : 3/ 8 user 1 : 3/ 8 user 2 : 3/ 8 t he rest : 0 3/ 2
112
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
4.5
Conclusions
T his work considered t he general mult iuser MISO BC, and est ablished inner and out er bounds on t he t radeoff between DoF performance and CSIT feedback quality, which are opt imal for many cases. T hose bounds, as well as some analysis, were provided wit h t he aim of giving insight s on how much CSIT feedback t o achieve a cert ain DoF performance.
4.6
A pp endi x - P r oof det ai ls of P r op osi t ion 6
4.6
113
A ppendix - Pr oof det ails of Pr oposit ion 6
In t he following, we will prove Proposit ion 6 used for t he converse proof, as well as t hree lemmas t o be used here. As st at ed, we will drop t he t ime index for simplicity. ˆ + G ˜ 2 Cm ⇥m where G ˜ has i.i.d. N c(0, 1) entries, L em m a 4. 6 Let G = G ˜ is independent of G ˆ . T hen, we have and G ⇥ ⇤ X⌧ H EG˜ log det (G G ) = log
ˆ ) + o(log snr) ˆ HG
i (G
(4.33)
i= 1
ˆ HG ˆ ) denotes the i th largest eigenvalue of G ˆ HG ˆ ; ⌧is the number of where i ( G H ˆ G ˆ that do not vanish with snr (SNR), i.e., i ( G ˆ HG ˆ ) = o(1) eigenvalues of G when snr is large, 8 i > ⌧ . L em m a 5. For P 2 Cm ⇥m a permutation matrix and A 2 Cm ⇥m , let A P = QR be the QR decomposition of the column permuted version of A . Then, there exist at least one permutation matrix P such that r i2i
m
1 H i (A A ), i+ 1
i = 1, . . . , m
(4.34)
where as stated i (A H A ) is the i th largest eigenvalue of A H A ; r i i is the i th diagonal elements of R . L em m a 6. For any matrix A 2 Cm ⇥m , there exists a column permuted version A¯ , such that H det ( A¯ I A¯ I )
m
|I |
Y i (A
H
A ),
8 I ✓ { 1, . . . , m}
(4.35)
i2I
where A¯ I = [A j i : j 2 { 1, . . . , m} , i 2 I ] 2 Cm ⇥|I | is the submatrix of A formed by the columns with indices in I .
4.6.1
P r oof of L emm a 4
ˆ, Let us perform ai singular value decomposit ion (SVD) on t he mat rix G h D H m ⇥m 1 ˆ = U i.e., G are unit ary mat rices and D 1 D 2 V where U , V 2 C 0 0 0 0 and D 2 are ⌧ ⇥⌧ and (m ⌧ ) ⇥(m ⌧ ) diagonal mat rices of t he singular ˆ . Wit hout loss of generality, we assume t hat t he i t h singluar values of G value, i = 1, . . . , m, scales wit h snr as snrbi , when snr is large. Moreover, t he
6. We not e t hat Lemma 4 is a slight ly more general version of t he result in [25, Lemma 6].
114
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
singular values in D 1 are such t hat bi > 0 and t hose in D 2 verify bi First , we have t he following lower bound ⇥ ⇤ EG˜ log det (G H G ) ⇣⇣h D1 = EM log det
⌘H ⇣ h
i D
2
+ M
D
⌘⌘
i 1
D
2
0.
+ M
(4.36)
⇣ ⇥ ⌘ ⇤ ⇤ H ⇥ D1 D1 EM log det + M + M 0 0 h = EM log det (D 1 + M 11) det M 22 M 21 (D 1 + M | {z
11 )
1
}
(4.37) i M
2
12
B
h 2 = log det (D 1 ) + EM 11 log det I + D 1 1 M ⇤ ⇥ H + EB EM˜ log det ( M˜ (I + B B H ) M˜ ) h 2 log det (D 1 ) + EM 11 log det I + D 1 1 M ⇤ ⇥ H + EM˜ log det ( M˜ M˜ ) | {z } (ln 2)
1
P
m ⌧0 1 l= 0
2
i
(4.38)
11
2
i
(4.39)
11
(4.40)
0 l )= O(1) (m ⌧
h i ˜ V = M 11 M 12 wit h M 11 2 C⌧0⇥⌧0, and remind where we define M , U H G M 21 M 22 t hat t he ent ries of M , t hus of M i j , i , j = 1, 2, are also i.i.d. N c(0, 1) ; (4.37) is from t he fact t hat expect at ion of t he log det erminant of a non-cent ral Wishart mat rix is non-decreasing wit h in t he “line-of-sight ” component [50] ; (4.38) is due t o t he ident ity ⇣h det
N N
11 21
N N
i⌘ 12 22
= det (N
11 ) det (N 22
N
1 21 N 11 N 12 )
whenever N 11 is square and invert ible; in (4.39), we not ice t hat , given t he mat rix B , M 21 (D 1 + M 11 ) 1 , t he columns of M 22 B M 12 are i.i.d. N c(0, I + B B H ), from which |det (M 22 B M 12 )|2 is equivalent in dist ribuH 0 0 t ion t o det ( M˜ (I + B B H ) M˜ ) where M˜ 2 C(m ⌧)⇥(m ⌧) has i.i.d. N c(0, 1) H H ent ries; t he last inequality is from M˜ (I + B B H ) M˜ ⌫M˜ M˜ and t hereH H fore det ( M˜ (I + B B H ) M˜ ) det( M˜ M˜ ), 8 B ; t he closed-form t erm in t he last inequality is due t o [51] with (·) being Euler’s digamma funct ion. ⇥ 2⇤ In t he following, we show t hat E log det I + D 1 1 M 11 O(1) as well. To t hat end, we use t he fact t hat t he dist ribut ion of M 11 is invariant t o rot at ion, and so for D 1 1M 11. Specifically, int roducing ✓⇠ Unif(0, 2⇡ ] t hat
4.6
A pp endi x - P r oof det ai ls of P r op osi t ion 6
115
is independent of t he rest of t he random variables, we have h i 2 log det I + D 1 1M 11 h ⇣ ⌘ i 2 = EM 11 ,✓ log det I + D 1 1 M 11 ej ✓ h ⇣ ⌘ i 2 = EM 11 ,✓ log det e j ✓I + D 1 1M 11
EM
11
0
X⌧ =
⇥ EJ E✓ log|e
j✓
+
i= 1 0 X⌧
=
|
i (D
1
M {z
11 ) |
2
(4.41) (4.42)
⇤ (4.43)
}
Ji
EJ E✓[log(1 + |J i |2 + 2|J i | cos(✓+
(J i )))]
(4.44)
i= 1 0
X⌧ =
EJ E✓[log(1 + |J i |2 + 2|J i | cos(✓))]
(4.45)
i= 1 0
X⌧ ⇥ EJ log(1 + |J i |2)
⇤ 1
(4.46)
i= 1 0
⌧
(4.47)
where t he first equality is from t he fact t hat M 11 is equivalent t o M 11ej ✓ as long as ✓ is independent of M 11 and t hat M 11 has independent circularly symmet ric Gaussian ent ries; (4.43) is due t o t he charact erist ic polynomial of the mat rix D 1M 11 ; in (4.44) we define (J i ) t he argument of J i t hat is independent of ✓; (4.45) is from t he fact t hat mod(✓+ ) 2⇡ ⇠ Unif(0, 2⇡ ] and is independent of , as long as ✓⇠ Unif(0, 2⇡ ] and is independent of , also known as t he Crypt o Lemma [52] ; (4.46) is from t he ident ity Z
1
log(a + bcos(2⇡ t)) dt = log 0
a+
p
a2 2
b2
log(a)
1, 8 a
b > 0.
Combining (4.40) and (4.47), we have t he lower bound ⇥ ⇤ EG˜ log det (G H G ) log det (D 1)
2
+ O(1)
(4.48)
when snr is large. In fact , it has been shown t hat t he O(1) t erm here, sum 0 in (4.47), does not depend on snr at all. of the O(1) t erm in (4.40) and ⌧ ⇥ ⇤ T he next st ep is t o derive an upper bound on E log det (G H G ) . Following
116
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
Jensen’s inequality, we have ⇥ ⇤ EG˜ log det (G H G ) log det EG˜ [G H G ] ⇣h 2 i ⌘ D1 H = log det + E[M M ] 2 D
(4.49) (4.50)
2
= log|det (D 1 )|2 + log det I + mD 1 2 + log det mI + D 22 | {z } | {z } o(1)
(4.51)
o(log snr)
2
= log|det (D 1 )| + o(log snr)
(4.52) ⇥ ⇤ Put t ing t he lower and upper bounds t oget her, we have E log det (G H G ) = ˆ ) =. snr0 , i = ˆ HG log|det (D 1 )|2 + o(log snr). Finally, not e t hat , since i ( G 0 + 1, . . . , ⌧ ⌧ , we have 0
log det (D 1 )
2
X⌧ =
ˆ) ˆ HG
log
i (G
log
ˆ Hˆ i (G G )
(4.53)
i= 1
X⌧ = i= 1
log i=
X⌧ =
X⌧ 0+ ⌧
ˆ HG ˆ)
i (G
1
ˆ ) + o(log snr) ˆ HG
log
(4.54)
(4.55)
i (G
i= 1
from which t he proof is complet e.
4.6.2
P r oof of L emm a 5
T he exist ence is proved by const ruct ion. Let a j , j = 1, . . . , m, be t he j t h column of A . We define j 1⇤ as t he index of t he column t hat has t he largest Euclidean norm, i.e., j 1⇤ = arg max ka j k.
(4.56)
j = 1,...,m
Swapping t he j 1⇤ and t he first column, and denot ing A 1 = A , we have B 1 , A 1 T 1,j 1⇤
(4.57)
where T i j 2 Cm ⇥m denot es t he permut at ion mat rix t hat swaps t he i t h and j t h columns. Now, let U 1 2 Cm ⇥m be any unit ary mat rix such t hat t he first aj ⇤ column is aligned wit h t he first column of B 1, i.e., equal t o ka 1⇤k . T hen, we j1
can const ruct a block-upper-t riangular mat rix R 1 = U H1 B 1 = U H1 A 1T 1,j 1⇤ wit h t he following form R1 =
r 11 ⇤ 0(m 1)⇥1 A 2
(4.58)
4.6
A pp endi x - P r oof det ai ls of P r op osi t ion 6
117
where it is readily shown t hat 2 r 11 = ka j 1⇤k2 1 ||A 1||2F m 1 H 1 (A 1 A 1). m
(4.59) (4.60) (4.61)
Repeat ing t he same procedure on A 2 , we will have R 2 = U H2 B 2 = U H2 A 2 T 2,j 2⇤ where all t he involved mat rices are similarly defined as above except for t he reduced dimension (m 1) ⇥(m 1) and R2 =
r 22 ⇤ 0(m 2)⇥1 A 3
(4.62)
where it is readily shown t hat 1
2 r 22
m
1 1
m
1
1 (A 2 A 2 )
H
(4.63)
H
(4.64)
2 (A 1 A 1 ).
Here, t he last inequality is from t he fact t hat , for any mat rix C and a submat rix C k by removing k rows or columns, we have [53, Corollary 3.1.3] H
i (C k C k )
i + k (C
H
C)
(4.65)
where we recall t hat i is t he i t h largest eigenvalue. Let us cont inue t he procedure on A 3 and so on. At t he end, we will have all t he { U i } and T i ,j i⇤ such t hat h |
Im
i 1
U
H m
h ···
ih
I2
{z QH
U
H 3
1
i UH 2
h U H1 A T 1,j 1⇤ } |
1
ih T 2, j ⇤ 2
i
I2 T 3, j ⇤
h ···
Im
i 1
T m ,j m ⇤
{z3
2 6 6 = 6 4 |
P
r 11
⇤
⇤ ⇤ .. .
r 22
{z
}
3 ⇤ ⇤7 7 .. 7 . 5 r mm
}
R
where it is obvious t hat P is a permut at ion mat rix and Q is unit ary. T he proof is t hus complet ed.
118
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
4.6.3
P r oof of L emm a 6
Let A¯ , A P = QR wit h P a permut at ion mat rix such t hat (4.34) holds. T hen, we have det ( A¯ I A¯ I ) = det (R HI Q H QR I ) H
(4.66)
H
(4.67)
H
(4.68)
= det (R I R I ) det (R I I R I I ) Y = r i2i i2I
m
|I |
(4.69)
Y i (A
H
A)
(4.70)
i2I
where t he first inequality result s from t he Cauchy-Binet formula, and t he last inequality is due t o Lemma 5.
4.6.4
P r oof of P r oposit ion 6
T he inequality (4.17) is t rivial when m t he chain rule
l
M , i.e., l 0 = m 0 = M . From
h(y m | U, Hˆ , H˜ ) = h(y l | U, Hˆ , H˜ ) + h(y(l + 1) , . . . , y(m ) | y l , Hˆ , H˜ ) = h(y l | U, Hˆ , H˜ ) + o(log snr)
(4.71) (4.72)
since wit h l M , t he observat ions y(l + 1) , . . . , y(m ) can be represent ed as a linear combinat ion of y l , up t o t he noise error. In t he following, we focus on t he case l M . First of all, let us writ e h(y m |U, Hˆ , H˜ ) h = EHˆ EH˜ [h(H
µ h(y l |U, Hˆ , H˜ ) mx
+ z m | U, Hˆ = Hˆ , H˜ = H˜ )]
µ EH˜ [h(H l x + z l | U, Hˆ = Hˆ , H˜ = H˜ )]
i (4.73)
In t he following, we focus on t he t erm inside t he expect ion over Hˆ in (4.73), i.e., for a given realizat ion of Hˆ . Since y l is a degraded version of y m , we can apply t he result s in [41, Corollary 4] and obt ain t he opt imality of Gaussian input , i.e., ⇥ ⇤ ⇥ ⇤ ˆ = Hˆ , H˜ = H˜ ) ˆ = Hˆ , H˜ = H˜ ) max E h(y |U, H µ E h(y |U, H ˜ ˜ m l H H p : X | U Hˆ
E[t r(X X H )] snr
=
max
⌫0:t r( ) snr
⇥ EH˜ log det (I + H
m
H
H
m)
⇤
⇥ µ EH˜ log det (I + H
l
H Hl )
⇤
(4.74)
4.6
A pp endi x - P r oof det ai ls of P r op osi t ion 6
for any µ of (4.74).
119
1. T he next st ep is t o upper bound t he right hand side (RHS)
Next , let = V ⇤V H be t he eigenvalue decomposit ion of t he covariance mat rix where ⇤ is a diagonal mat rix and V is unit ary. Not e t hat it is without loss of generality t o assume t hat all eigenvalues of are st rict ly posit ive, i.e., i ( ) c > 0, 8i , in t he sense that H H)
log det (I + H
log det (I + H (cI + log det (I + H
)H H )
H H ) + log det (I + cH H H ) .
(4.75)
In ot her words, a const ant lift of t he eigenvalues of does not have any impact on t he high SNR behavior. T his regularizat ion will however simplify t he analysis. T he following is an upper bound for t he first t erm in t he RHS of (4.74). h i EH˜ log det I + H m H Hm ⇣ ⌘ ⇥ 1 1 ⇤ = EH˜ log det I M + 2 H Hm H m 2 ⇣ h i ⌘ 1 1 2 EH˜ log det I M + 2 U H kH m k F I m 0 0 U 2 h i = EH˜ log det I m 0 + kH m k2F ˜ h i ⇥ ⇤ 1 = EH˜ log det ˜ + EH˜ log det ˜ + kH m k2F I Xm
(4.77) (4.78) (4.79)
0
log i= 1
i(
) + log det (c |
1
+ m + kHˆ {z
2 m kF )I
}
(4.80)
o(log snr)
⇤) + o(log snr) log det (⇤ 1
(4.76)
1
(4.81)
2
2 is such t hat 2 where = ; (4.77) is due t o fact t hat H H mH m h i 2 H kH m k F I m 0 U wit h U being t he mat rix of eigenvect ors of H H U mH 0
m
and kH m kF being t he Frobenius norm of H m , where m 0, min{ m, M } ; in (4.78), we define ˜ as t he m 0⇥m 0 upper left block of U U H ; t he first t erm Q 0 Q m0 Q m0 H ˜ in (4.80) is due t o det ( ˜ ) = m i= 1 i ( ) i = 1 i (U U ) = i= 1 i ( ) ; t he second t erm in (4.80) is from Jensen’s inequality and using fact P m t he 1 2I t hat c 1 I M by assumpt ion and t hat EH˜ (H H H ) = m m k= 1 k M + H H Hˆ m Hˆ m mI + Hˆ m Hˆ m ; t he last inequality is from t he assumpt ion t hat every eigenvalue of is lower-bounded by some const ant c > 0 independent of snr. Now, we need t o lower bound t he second expect at ion in t he RHS of
120
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
(4.74). To t his end, let us writ e det (I l + H
l
H Hl ) = det (I l + H l V ⇤V H H Hl ) = det (I M + ⇤V H H Hl H l V )
(4.82) (4.83)
2
= det I M + ⇤ ⌃ X ⇤I I ) det ( = 1+ det (⇤ H
(4.84) I⌃ H
2
I)
(4.85)
)
(4.86)
I✓ { 1, . . . , M } I6 =;
2
⌃ ) det (⌃
XM
⇤I j I j ) det ( det (⇤
H Ij
Ij
j=1
0
⇣
YM
⌃ 2) @ M det (⌃
⇤I j I j ) det ( det (⇤
H Ij
1 ⌘ A Ij )
1 M
(4.87)
j=1
0 l
⇤) M ⌃ 2 ) det (⇤ = M det (⌃
1
YM @ det (
H Ij
Ij
1 M
)A
(4.88)
j=1
where (4.83) is an applicat ion of t he ident ity det (I + A B ) = det (I + B A ) ; in (4.84), we define ⌃ , diag(
1, . . . ,
l ),
, ⌃
1
H l V , and ˆ , ⌃
1
Hˆ l V ;
in (4.85), we define I , [ j i : j = 1, . . . , l, i 2 I ] 2 Cl ⇥|I | as t he submat rix of wit h columns indexed in I and ⇤I I = [⇤j i : i , j 2 I ] 2 C|I |⇥|I | , wit h I denot ing a nonempty set ; t he equality (4.85) is an applicat ion of t he ident ity [54] X det (I + A ) = 1 + det (A I I ) I✓ { 1, . . . ,M } I6 =;
for any A 2 CM ⇥M ; in (4.86), we define I 1 , . . . , I M as t he so-called sliding window of indices I 1 , { 1, 2, · · · , l} , I 2 , { 2, 3, · · · , l, l + 1} , · · · , I M , { M , 1, 2, · · · , l 1} (4.89) i.e., I j , { mod(j + i
1) M + 1 : i = 0, 1, · · · , l
1} , j = 1, 2, · · · , M (4.90)
wit h mod(x) M being t he modulo operat or ; (4.87) is from t he fact t hat arit hmet icQmean is not smaller t han geomet ric mean ; in (4.88), we use t he fact ⇤I j I j ) = det (⇤ ⇤) l . t hat M j = 1 det (⇤ Wit hout loss of generality, we assume t hat t he M columns of H l V are ordered in such a way t hat 1) t he first l columns are linearly independent , i.e., ˆ I 1 has full rank, and 2) A = ˆ I 1 sat isfies Lemma 6. Not e t hat t he
4.6
A pp endi x - P r oof det ai ls of P r op osi t ion 6
121
former condit ion can almost always be sat isfied since rank( ˆ ) = l almost surely. Hence, we have ⇥ EH˜ log det (
⇤ Ij ) =
H Ij
rank( ˆ
Ij
X
)
log i= 1 rank( ˆ I
X
T
j
i(
ˆ H ˆ I ) + o(log snr) Ij j
(4.91)
I1 )
log
i(
ˆ H ˆ I ) + o(log snr) Ij j
(4.92)
i= 1 rank( ˆ
X Ij
T
I1 )
log
i(
ˆH
Ij
T
I1
ˆI
T j
I1)
+ o(log snr)
i= 1
(4.93) H
= log det ( ˆ I j T I 1 ˆ I j T I 1 ) + o(log snr) Y ˆ H ˆ ) + o(log snr) log i( i2Ij
T
(4.94) (4.95)
I1
where (4.91) is from Lemma 4 by not icing t hat I j = ˆ I j + ˜ I j wit h t he ent ries of ˜ I j , ⌃ 1H˜ l V being i.i.d. N c(0, 1) ; (4.92) is from t he fact t hat H ˆH T ˆI TI ) rank( ˆ I ) rank( ˆ I T I ) ; (4.93) is due t o i ( ˆ ˆ I ) i( j
j
Ij
1
j
Ij
I1
j
1
where we recall t hat i (A H A ) is defined as t he i t h largest eigenvalue of A H A ; and t he last inequality is due t o Lemma 6. Summing over all j , we have ! XM YM Y ⇥ ⇤ H ˆ ˆ ) + o(log snr) (4.96) E ˜ log det ( H I ) log i( Ij
H
j
j=1
j = 1 i2Ij
T
! l!
Y = log
i(
Y
I1 H
ˆ ˆ)
+ o(log snr)
(4.97)
i2I1 i(
l log
ˆ H ˆ I ) + o(log snr) I1 1
(4.98)
i2I1
⇣ H ⌘ = l log det ˆ I 1 ˆ I 1 + o(log snr)
(4.99)
l log det ⌃ 2 + o(log snr)
(4.100)
=
H ˆH ˆ where (4.98) is due t o i ( ˆ ˆ ) i ( I 1 I 1 ), 8 i = 1, . . . , l ; t he last equality is from t he fact t hat ˆ I 1 = ⌃ 1 Hˆ l V I 1 and t hat Hˆ l V I 1 has full rank by const ruct ion. From (4.88) and (4.100), we obtain
⇥ EH˜ log det (I l + H
l
H Hl )
⇤
l M l ⇤) + ⌃ 2 ) + o(log snr) log det (⇤ log det (⌃ M M (4.101)
122
C hapt er 4
D oF and Feedback Tr adeoff over K -U ser M I SO B C
and finally ⇥ EH˜ log det (I m + H
m
H
H
m)
⇤ M ⇤ ⇥ EH˜ log det (I l + H l H Hl ) l M l ⌃ 2) + o(log snr). log det (⌃ l
(4.102)
When m < M , t he above bound (4.102) is not t ight . However, we can show t hat , in t his case, (4.102) st ill holds when we replace M wit h m. To see t his, let us define ⇤0 , diag( 1 , . . . , m ). First , not e t hat when m < M , (4.81) holds if we replace ⇤ wit h ⇤0 on t he RHS. T hen, t he RHS of (4.82) becomes a lower bound if we replace ⇤ wit h ⇤0 and V wit h V 0 2 CM ⇥m , t he first m columns of V . From t hen on, every st ep holds wit h M replaced by m. (4.102) t hus follows wit h M replaced by m. By t aking t he expect at ion on bot h sides of (4.102) over Hˆ and plugging it int o (4.73), we complet e t he proof of (4.17).
Chapt er 5
On t he I mperfect Global CSI R and Diversity A spect s
In t his chapt er we consider furt her aspect s on t he communicat ions wit h imperfect , limit ed and delayed feedback. T he first focus in t his chapt er is placed on t he aspect of having imperfect global CSIR, i.e., wit h imperfect receiver est imat es of t he channel of t he ot her receiver, in t he communicat ion wit h limit ed feedback. It is mot ivat ed by t he challenge of dist ribut ing global CSIR across t he diff erent receiving nodes, in addit ion t o t he challenge of communicat ing CSIT , over t he resource links with limit ed capacity and limit ed reliability (cf. [35], [36]). Our work st udies t he mult iple-input mult iple-out put (MIMO) broadcast channel (BC) communicat ing wit h imperfect delayed CSIT and wit h imperfect global CSIR, and proceeds t o present schemes and degrees-of-freedom (DoF) bounds t hat are oft en t ight , and t o const ruct ively reveal t hat even subst ant ially imperfect delayed CSIT and imperfect global CSIR, are in fact suffi cient t o achieve t he optimal DoF performance previously associat ed t o perfect delayed CSIT and perfect global CSIR. T he second focus in t his chapt er is placed on t he diversity aspect of t he communicat ion wit h delayed CSIT . Not e t hat , most of t he works considering communicat ions wit h delayed CSIT , e.g., t he works in [4–7,10,11,19–23], just focus on t he DoF aspect . Besides t he DoF aspect , anot her aspect so called di ver si ty ( [34]), corresponding t o t he charact erizait on of t he communicat ion reliability, is also very import ant for t hose delayed CSIT communicat ion scenarios. Our work proposes a novel broadcast scheme which, over broadcast channel wit h delayed CSIT , employs a form of int erference alignment t o achieve bot h full DoF as well as full diversity. 123
124 C hapt er 5
5.1 5.1.1
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
On t he I mper fect Global CSI R A spect I nt r oduct ion
We here consider t he two-user (M , N ) mult iple-input mult iple-out put (MIMO) broadcast channel (BC), where a M -ant enna t ransmit t er communicat es informat ion t o two N -ant enna receivers. In t his set t ing, t he channel model t akes t he form (1)
= H
(2)
= H
yt yt
(1) t xt (2) t xt
(1)
(5.1)
(2)
(5.2)
+ zt + zt
(1)
where mat rices H t , H (2) 2 CN ⇥M represent t he t ransmit t er-t o-user 1 and (1) (2) t he t ransmit t er-t o-user 2 channels respect ively at t ime t, where z t , z t represent unit power AWGN noise vect ors at t he two receivers, where x t 2 CM ⇥1 is t he input signal vect or at t ime t wit h power const raint E[||x t ||2 ] P , and where in t his case, P also t akes the role of t he signal-t o-noise rat io (SNR). (1) In t his work we assume t hat t he element s of t he channel mat rices H t and (2) H t , are spat ially and t emporally i.i.d. Gaussian random variables, wit h zero mean and unit variance. Wit h channel st at e informat ion at t he t ransmit t er (CSIT ) being a crucial ingredient t hat facilit at es improved performance, and wit h CSIT oft en being limit ed, imperfect and delayed, we here explore t he eff ect s of t he quality of delayed CSI T, corresponding t o how well t he t ransmit t er knows t he chan(1) (2) nel H t (resp. H t ) aft er t his channel st at e has fully changed. Nat urally, reduced CSIT quality relat es t o limit at ions in t he capacity and reliability of t he feedback links. Similar issues, which addit ionally mot ivat e t his work, and which are addressed here, pert ain t o t he quality of delayed global CSIR, i.e., t o t he quality of t he est imat es, at a given receiver, of t he channel of t he ot her receiver (see for example the work of [35], [36] on t he challenge of obt aining such global CSIR).
5.1.2
R elat ed wor k
It is well known t hat in t he two-user (M , N ) MIMO BC set t ing of int erest , t he presence of perfect CSIT allows for t he opt imal sum degreesof-freedom (DoF) min{ M , 2N } (t his is wit h perfect global CSIR, cf. [1]), whereas t he complet e absence of CSIT causes a subst ant ial degradat ion t o just min{ M , N } (cf. [3]) 1 . An int erest ing scheme t hat mit igat es t his degradat ion by ut ilizing part ial CSIT knowledge, was recent ly present ed in [4] by Maddah-Ali and Tse, which 1. We remind t he reader t hat for an achievable rat e pair (R 1 , R 2 ), t he corresponding Ri DoF pair (d1 , d2 ) is given by di = limP ! 1 l og P , i = 1, 2. T he corresponding DoF region is t hen t he set of all achievable DoF pairs.
5.1
On t he I m p er fect G l obal C SI R A sp ect
125
showed t hat in t he absence of current CSIT , delayed CSIT knowledge can st ill be useful in improving t he DoF region of t he mult iple-input singleoutput (MISO) broadcast channel (N = 1). T his result was lat er generalized by Vaze and Varanasi in [19] t o t he MIMO case (again, t his is wit h perfect global CSIR). 2 Our work ext ends t he work in [19], and studies t he general case of communicat ing wit h imperfect delayed CSIT and imperfect global CSIR. Specifically t his work reveals t hat even subst ant ially imperfect delayed-CSIT and imperfect global CSIR, are in fact suffi cient t o achieve t he opt imal DoF performance previously associat ed t o perfect delayed CSIT and perfect global CSIR.
5.1.3
Quant ifi cat ion of CSI and CSI R qualit y
In t his work we will consider t he case wit hout any current CSIT , but with imperfect delayed CSIT . In t erms of delayed CSIT , we consider t he (1) (2) case where t he t ransmit t er’s (best ) delayed est imat es Hˇ t , Hˇ t of channels (1) (2) H t , H t come wit h est imat ion errors H¨
(1) t
= H
(1) t
Hˇ
(1) t ,
H¨
(2) t
= H
(2) t
Hˇ
(2) t
(5.3)
having independent and ident ically dist ribut ed (i.i.d.) Gaussian ent ries wit h power . ¨ t ||2 ] =. P E[||H¨ t ||2F ] = E[||G F for some CSI quality exponent describing t he general quality of t he delayed est imat es. Wit hout loss of generality, here we assume t hat t he delayed est i(1) (2) (1) (2) mat es Hˇ t , Hˇ t of channels H t , H t become known t o t he t ransmit t er (1) (2) with unit coherence delay, i.e., Hˇ t , Hˇ t are available at t ime (t + 1). In t his set t ing, an increasing exponent implies an improved delayed CSIT quality, wit h = 0 implying very lit t le delayed CSIT knowledge, and with = 1 implying perfect delayed CSIT . In addit ion t o t he challenge of communicating CSIT over feedback channels wit h limit ed capacity and limit ed reliability, anot her known bot t leneck is the non-negligible cost of dist ribut ing global CSIR across t he diff erent receiving nodes (see [35], [36]). For t his reason, we explore t he case where, in addit ion t o limit ed and imperfect CSIT , we also have t he addit ional imperfect ion of t he global CSIR, which means t hat each user has imperfect est imat es of t he ot her user’s channel, as well as, in t his case, no access t o 2. Ot her int erest ing works in t he cont ext of ut ilizing delayed and current CSIT , can be found in [6–8] which explored t he set t ing of combining perfect delayed CSIT wit h immediat ely available imperfect CSIT , t he work in [9] which addit ionally considered t he eff ect s of t he quality of delayed CSIT for t he M ISO BC, t he work in [48] which considered delayed and progressively evolving (progressively improving) current CSIT , and t he works in [11, 21, 23] and many ot her publicat ions.
126 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
t he est imat es of t he t ransmit t er. In t he spirit of communicat ing global CSIR across feedback links ( [35]) we also focus on t he associat ed case of having imperfect delayed global CSIR corresponding t o t he same quality exponent , and addit ionally having no receiver access t o t he CSIT est imat es of t he ˇ (1) denot ing t he delayed est imat e of H (1) at user 2, and t ransmit t er. Wit h H t t (2) (2) ˇ H t denot ing t he delayed est imat e of H t at user 1, we maint ain as before ˇ (1) , H ˇ (2) are available at t ime (t + 1), and t hat t he est imat ion errors t hat H t t ˇ (1) , H ˇ (2) ¨¨ (2) = H (2) H ¨¨ (1) = H (1) H (5.4) H t
t
t
have i.i.d. Gaussian ent ries wit h power ¨¨ (1) ||2 ] =. E[||H ¨¨ E[||H t
F
t
(2) 2 t ||F ]
t
t
. = P
again for t he same as before, now also describing t he quality of t he global CSIR delayed est imat es. R em ar k 7. We here note that without loss of generality, we can restrict our attention to the range 0 1 (cf. [17]), where again = 1 corresponds the case of perfect delayed CSI T .
5.1.4
Convent ions and st r uct ur e
In Sect ion 5.1.5, for t he aforement ioned two user MIMO BC, and for t he general case of imperfect delayed CSIT and imperfect global-CSIR, we derive a DoF region inner bound, which t urns out t o be t ight for any N min(M ,2N )+ N previously associat ed t o perfect delayed CSIT and perfect globalCSIR. Sect ion 5.1.6 t hen present s the novel mult i-phase precoding schemes associat ed t o t he aforement ioned DoF regions. Finally adhering t o t he common convent ion, we consider a unit coherence period, as well as assume t hat each receiver knows perfect ly it s own channel (perfect local CSIR).
5.1.5
D oF of t he M I M O B C wit h I m per fect D elayed CSI T and I m per fect global-CSI R
It is not ed t hat , for t he case wit h M N , t he DoF region is charact erized as d1 + d2 M , which is achievable by T DMA scheme wit hout any CSIT and wit hout any global-CSIR. T hus in t he following, we focus on t he case wit h M > N . T heor em 7. For the (M > N , N ) MI MO BC with imperfect delayed CSI T and imperfect global-CSIR, the optimal DoF region takes the form d1 d2 + 1 min{ M , N } min{ M , 2N } d2 d1 + 1 min{ M , N } min{ M , 2N }
5.1
On t he I m p er fect G l obal C SI R A sp ect
127
No CSIT
d2 N
Perfect Delayed CSIT or
N min{M ,2N} N
Imperfect Delayed CSIT
N min{ M ,2N}
min{ M ,2 N } , N (1
N
)
min{ M ,2N} N min{ M ,2N} N , min{ M ,2N} N min{ M ,2N} N
N (1
0
),
min{ M ,2N}
d1
N
Figur e 5.1 – DoF region of MIMO BC wit h imperfect delayed CSIT and imperfect global-CSIR (M > N ). N when < min(M N,2N )+ N this region is inner min(M ,2N )+ N , while when bounded by the achievable region d1 d2 + 1 min{ M , N } min{ M , 2N } d2 d1 + 1 min{ M , N } min{ M , 2N } d1 + d2 min{ M , N } + (min{ M , 2N } min{ M , N } )
which, for
0
, min{ , min(M N,2N )+ N } , takes the form of a polygon with corner
points { (0, 0), (0, N ), (min{ M , 2N } (N , 0)} .
0
, N (1
0
)), (N (1
0
), min{ M , 2N }
0
),
At t his point we can draw an int erest ing conclusion on t he amount of delayed CSIT and global CSIR needed t o achieve a cert ain symmet ric DoF performance d0. For all t he cases considered here, t he derived t hreshold value ⇤, arg min { d( ) = d0} , accept s t he simple form of ⇤
= (d0
d(0))/ (d(1)
d(0))
(5.5)
describing t he fract ion of t he DoF gap - between t he no-CSIT -and-no-globalCSIR case and t he perfect -delayed-CSIT -and-perfect -global-CSIR case - t hat is covered t o reach d0. As an example of t his derived t hreshold quality, we see t hat for t he case of t he MIMO case wit h N < M < 2N , t he t arget opt imal d0 = d⇤ 1 = M N/ (M + N ) (cf. [19]) corresponds t o t he aforement ioned ⇤
=
d0 d(0) M N / (M + N ) N / 2 N = = . d(1) d(0) M / 2 N/ 2 M + N
128 C hapt er 5
5.1.6
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
Schemes for M I M O B C wit h imp er fect delayed CSI T and imp er fect global CSI R
It is not ed t hat in Chapt er 3, a novel scheme has been proposed for t he set t ing wit h imperfect CSIT but with perfect global CSIR. Here we proceed t o describe a modified scheme t hat achieves t he corresponding DoF corner point s, and does so wit h imperfect delayed CSIT and imperfect global CSIR. In t his specific set t ing, t he encoding of t his modified scheme is similar t o t he encoding of t he previous general scheme in Chapt er 3, by set t ing specific paramet ers for t his set t ing, while t he decoding here is light ly diff erent from t he decoding of t he previous scheme, i.e., imperfect delayed global CSIR is suffi cient for t he decoding in t his modified scheme. For 0
, min{ ,
N } min(M , 2N ) + N
(5.6)
t his modified scheme will achieve DoF corner point s E = (min{ M , 2N }
0
, N (1
0
)),
F = (N (1
0
), min{ M , 2N }
0
) (5.7)
bot h of which converge t o t he opt imal DoF point C= (
min{ M , 2N } N min{ M , 2N } N , ) min{ M , 2N } + N min{ M , 2N } + N
N when min(M ,2N )+ N , recalling t hat we focus on t he case wit h M > N . It is not ed t hat , DoF corner point (min{ M , N } , 0) and point (0, min{ M , N } ) are easily achievable by single-user t ransmission scheme wit hout any CSIT and wit hout any global CSIR. As in Chapt er 3, t he scheme has a forward-backward phase-Markov st ruct ure which, in t he cont ext of imperfect and delayed CSIT , was first int roduced in [8, 9] t o consist of four main ingredient s t hat include – block-Markov encoding – spat ial precoding – int erference quant izat ion – backward decoding. In t he scheme, t he accumulat ed quant ized int erference bit s of phase s can be broadcast ed t o bot h users inside t hecommon informat ion symbols of t he next phase (phase (s + 1)), while also a cert ain amount of common informat ion can be t ransmit t ed t o bot h users during phase s, which will t hen help resolve t he accumulat ed int erference of phase (s 1). As in Chapt er 3, t he scheme has infinit e communicat ion durat ion n, and in t his set t ing each phase has one channel use.
5.1
On t he I m p er fect G l obal C SI R A sp ect
129
Encoding We first describe t he encoding for t = 1, 2, · · · , n 1 except t he last channel use (t = n) which will be addressed separat ely due t o it s diff erent st ruct ure. Zer o for cing and sup er p osit ion coding t ransmit t er sends
At t ime t = 1, 2, · · · , n
x t = ct + a t + bt
1, t he (5.8)
where a t 2 CM ⇥1 is a symbol vect or meant for receiver 1, bt 2 CM ⇥1 is meant for receiver 2, where ct 2 CM ⇥1 is a common symbol vect or. Power and r at e allocat ion For t his set t ing, in order t o achieve t he DoF corner point E and point F (cf. (5.7)), t he powers and (normalized) rat es at t ime-slot t are set as 0 (c) . (c) Pt = P, r t = (N (M N ) ) 0 0 (a ) . (a ) (5.9) Pt = P , rt = M 0 0 (b) . (b) Pt = P , rt = M . (e)
0
where is defined in (5.6), where Pt , E|et |2 denot es t he power of a symbol (e) vect or et corresponding t o t ime-slot t, and where r t denot es t he prelog (e) fact or of t he number of bit s r t log P o(log P ) carried by symbol vect or et at time t. To put t he above allocat ion in perspect ive, we show t he received signals, and describe under each t erm t he order of t he summand’s average power. T hese signals t ake t he form ( 1)
ˇ◆
(1)
yt
P
P
0
P
0
( 2)
ˇ◆
(2)
yt
( 1)
ˇ◆
P
0
P0
( 2)
◆
P (1)
◆ t , H
0
(1) t bt ,
P
0
(2)
◆ t , H
P
(5.10)
P0
( 2)
ˇ◆
z }t | { z t } | t { (2) (2) (2) (2) (2) = H t ct + H t bt + Hˇ t a t + H¨ t a t + z t | {z } | {z } | {z } | {z } | { z} P
where
( 1)
◆
z }t | { z t } | t { (1) (1) (1) (1) (1) = H t ct + H t a t + Hˇ t bt + H¨ t bt + z t | {z } | {z } | {z } | {z } | { z}
0
P0
(2) t at
(5.11)
P0
(5.12)
denot e t he int erference at receiver 1 and receiver 2 respect ively, and where (1) ˇ ˇ◆ t , H
(1) t bt ,
(2) ˇ ˇ◆ t , H
(2) t at (1)
(2)
denot e t he t ransmit t er’s delayed est imat es of ◆t , ◆ t .
(5.13)
130 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
Tabl e 5.1 – Number of bit s carried by privat e and common symbols, and by t he quant ized int erference (t ime t).
Privat e symbols for user 1 Privat e symbols for user 2 Common symbols Quant ized int erference
Tot al bit s (⇥log P ) 0 M 0 M 0 T (N (M N ) ) 0 2N
Quant izing and br oadcast ing t he accumulat ed int er fer ence Before t he beginning of next phase (t ime (t + 1)), t he t ransmit t er reconst ruct s (1) (2) ˇ◆t , ˇ◆t , using it s knowledge of delayed CSIT , and quant izes t hese int o (1) ¯ˇ◆(1) = ˇ◆t t 0
(1)
˜◆ t ,
(2) (2) ¯ˇ◆ = ˇ◆ t t
(2)
˜◆ t
(5.14)
0
using a t ot al of N log P and N log P quant izat ion bit s respect ively. T his (1) (2) (2) 2 . allows for bounded power of quant izat ion noise˜◆ t , ˜◆ t , i.e, allows for E|˜◆ t | = 0 0 (1) 2 . (2) 2 . (1) 2 . E|˜◆t | = 1, since E|ˇ◆ (cf. [39]). T hen t he t ranst | = P , E|ˇ◆ t | = P mit t er evenly split s t he 0 2N log P quant izat ion bit s int o t he common symbols ct + 1 t hat will be t ransmit t ed during t he next phase (at t ime (t + 1)), and which will convey t hese quant izat ion bit s t oget her wit h ot her new informat ion bit s for t he receivers. T hese ct + 1 will help t he receivers cancel int erference, as well as will serve as ext ra observat ions t hat will allow for decoding of all privat e informat ion (see Table 5.1). Finally, for t he last phase at t ime t = n, t he main t arget will be t o recover t he informat ion on t he int erference accumulat ed in phase (t 1). For large n, t his last phase can focus ent irely on t ransmit t ing common symbols. T his concludes t he part of encoding, and we now move t o decoding. D ecoding As in Chapt er 3, in accordance t o t he phase-Markov st ruct ure, we consider decoding t hat moves backwards, from t he last t o t he first phase. T he last phase (at t ime n) was specifically designed t o allow decoding of t he common symbols cn . Hence we focus on t he rest of t he phases, t o see how - wit h knowledge of common symbols from t he next phase - we can go back one phase and decode it s symbols. During phase t (at t ime t), each receiver uses ct + 1 t o reconst ruct t he (2) ¯ (1) delayed est imat es { ¯ˇ◆ t , ˇ◆ t } , t o remove - up t o noise level - all t he int erference
5.1
On t he I m p er fect G l obal C SI R A sp ect
131
(i ) (i ) (i ) ◆t , by subt ract ing t he delayed int erference est imat es ¯ˇ◆ t from y t , for i = 1, 2. (2) ¯ (1) (1) (1) ¯ (2) ¯ˇ◆ Now given ¯ˇ◆ wit h (y t t , ˇ◆ t , receiver 1 combines ˇ◆ t t ) t o decode ct and a t of phase t. T his is achieved by decoding over a MIMO mult iple-access channel (MIMO MAC) of t he general form " # " # " # " # (1) (1) (1) (1) ¯ˇ◆(1) ˜ yt H z H t t t = ct + ˇ t(2) a t + (5.15) 0(2) (2) ¯ˇ◆ 0 ˜ H z t t t
where
(1)
z˜ t and
= H¨
(1) t bt
(1)
+ zt
(1)
+ ˜◆ t
(2) ¨¨ (2) H¨ (2) )a = (H ˜◆ t t t t 0(2) (1) 2 . 2 . and where E|˜z t | = E|˜z t | = 1. It can be readily shown (cf. [39]) t hat optimal decoding in such a MIMO MAC set ting - doing so wit h imperfect ˇ (2) - allows user 1 t o achieve t he aforement ioned rat es in global CSIR H t 0 (a ) (c) (5.9), i.e., allows for decoding of a t wit h r t = M and of ct wit h r t = 0 (N (M N ) ) , Similarly receiver 2 can const ruct a similar MIMO MAC, which will again allow for decoding of it s own privat e and common symbols at the aforement ioned rat es in (5.9). Now t he decoders shift t o phase (t 1) and use ct t o decode t he common and privat e symbols of t hat phase. Decoding st ops aft er decoding of t he symbols in phase 1. It is not ed t hat , t he above const ruct ion in (5.15) is diff erent from t hat in Chapt er 3, and t he const ruct ion in (5.15) allows us t o handle t he issue of t he imperfect ion on global CSIR. 0(2)
z˜ t
D oF calculat ion We now describe, as in Chapt er 3, how to regulat e t he scheme’s paramet ers t o achieve t he DoF point E and point F (cf. (5.7)). T he rat e and power allocat ion in (5.9) t ells us t hat , t he t ot al amount of informat ion in 0 t he private symbols of a cert ain phase t < n, for user 1 is equal t o M log P 0 bit s, while for user 2 t his is also M log P bit s (as shown in t he t able 5.1). Given t he power and rat e allocat ion in (5.9), it is guarant eed t hat t he accu0 mulat ed quant ized int erference in a phase t < n has 2N log P bit s, which ‘fit ’ int o t he common symbols of t he next phase t hat can carry a t ot al of 0 N (M N) log P bit s. T his leaves an ext ra space of com log P bit s in the common symbols, where com
,N
(M
N)
0
2N
0
= N
(M + N )
0
(5.16)
is guarant eed t o be non-negat ive due t o (5.9) and (5.6). T his ext ra space can be split between t he two users, by allocat ing ! com log P bit s for t he
132 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
message of user 1, and t he remaining (1 ! ) com log P bit s for t he message of user 2, for some ! 2 [0, 1]. Consequent ly, for large n, t he scheme allows for DoF performance in t he form d1 = M d2 = M
0 0
+ ! + (1
com
!)
= M com
0
+ ! (N
= M
0
0
(M + N ) )
+ (1
! )(N
(5.17) 0
(M + N ) ).
(5.18)
As a result , as in Chapt er 3, set t ing ! = 0 allows us t o achieve t he DoF corner point E , while set t ing ! = 1 allows us t o achieve t he DoF corner point F , which complet es t he proof.
5.1.7
Conclusions
T he work provided analysis and novel communicat ion schemes for t he set t ing of t he two-user MIMO BC wit h imperfect delayed CSIT , as well as, in t he presence of addit ional imperfect ions in t he global CSIR. T he derived DoF region is oft en opt imal and, while corresponding t o imperfect delayed CSIT and imperfect global CSIR, oft en mat ches t he region previously associat ed t o perfect delayed CSIT and perfect global CSIR. In addit ion t o t he t heoret ical limit s and pract ical schemes, t he work provided insight on how much delayed CSIT feedback and global CSIR feedback are necessary t o achieve a cert ain t arget performance, off ering possible advant ages in t he presence of feedback links wit h limit ed capacity and limit ed reliability.
5.2
D i ver sit y
5.2
133
D iver sit y
Maddah-Ali and Tse have recent ly shown that delayed t ransmit t er channel st at e informat ion (CSIT ) can st ill be useful in increasing t he degreesof-freedom (DoF) over t he MIMO broadcast channel. T his was achieved by const ruct ing a scheme t hat , in t he presence of two t ransmit ant ennas, of two single-ant enna receivers, and of CSIT t hat is delayed by one coherence t ime, manages t o provide each user wit h 2/ 3 DoF, improving upon t he 1/ 2 DoF corresponding t o no CSIT . T his same scheme t hough, as well as all subsequent pert inent schemes, achieve DoF gains by suppressing t he inherent diversity of t he broadcast parallel channel. T he current work proposes a novel broadcast scheme which, over t he above described set t ing of t he delayed CSIT broadcast channel, employs a form of int erference alignment t o achieve both full DoF as well as full diversity.
5.2.1
I nt r oduct ion
Many mult iuser wireless communicat ions set t ings are known t o benefit great ly from t he use of CSIT feedback. It is t he case t hough t hat such feedback is oft en hard t o obt ain suffi cient ly fast , and as a result , eff ort s have been made t o find ways t o ut ilize delayed CSIT . One such case involves communicat ion over t he broadcast channel (BC), where recent advances by Maddah-Ali and Tse [4] have shown t hat st ale CSIT can st ill allow for improvement s in t he channel’s degrees-of-freedom (DoF) region. T his same work in [4] developed a novel scheme t hat , in t he specific set t ing of t he mult ipleinput single-out put (MISO) BC wit h 2 t ransmit ant ennas and 2 users each with a single receive ant enna, can off er 2/ 3 DoF t o each user, even when t he CSIT is delayed by a coherence period. While achieving t he opt imal DoF, t his novel scheme, as well as subsequent pert inent t echniques, neglect diversity considerat ions, t hus result ing in subst ant ial subopt imality wit h respect t o diversity. T his subopt imality nat urally cont ribut es t o subst ant ial performance degradat ion, and t hus brings t o the fore t he need for novel designs t hat can combine t he signal manipulat ions t hat allow for full diversity, wit h t he signal manipulat ions t hat ut ilize st ale CSIT t o give full DoF. We here propose a novel design which employs int erference alignment [55] t echniques t o provide, in t he aforement ioned twouser MISO BC wit h st ale CSIT , full DoF and full diversity.
5.2.2
Out line
Aft er describing t he syst em model and briefly recalling t he original MAT scheme, Sect ion 5.2.5 describes t he new DoF-opt imal design, and Sect ion 5.2.6 shows t hat t he scheme achieves full diversity. As a side result , t he int erest ed reader can find in Appendix 5.2.8 an upper bound on t he diversity-
134 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
mult iplexing t radeoff (DMT ) of t he MAT scheme.
5.2.3
Sy st em model
As st at ed, we focus on t he frequency-flat MISO BC wit h two t ransmit ant ennas and two users wit h a single receive ant enna. We consider a coherence period of Tc channel uses, during which t he channel remains t he same. Specifically we consider communication over L coherence periods, or equivalent ly L phases, where each phase coincides wit h a coherence period. During phase l (l = 1, · · · , L ), t he first user’s channel is denot ed as h l = [hl,1 hl ,2]T 2 C2 and t he second user’s as gl = [gl ,1 gl,2 ]T 2 C2 . For x l ,t denot ing t he t ransmit t ed signals during t imeslot t of phase l, t hen t he corresponding received signals at t he first and second user respect ively, t ake t he form (1)
(1)
(5.19)
(2)
(5.20)
yl ,t = h Tl x l ,t + zl,t , (2)
yl ,t = gTl x l ,t + zl ,t , (1)
(2)
(t = 1, 2, · · · , Tc), where zl ,t , zl ,t denot es t he AWGN noise. T he fading coeffi cient s are assumed t o be i.i.d. circularly symmet ric complex Gaussian CN(0, 1) dist ribut ed, and are assumed t o remain fixed during a phase and change independent ly from phase to phase. We let ⇢denot e t he signal-t onoise rat ion (SNR), and we consider a short -t erm power const raint where E||x ` (t)||2 ⇢. We also consider a communicat ion rat e of R bit s per channel use, which is here t aken t o be t he same for bot h users. We recall t hat t he corresponding mult iplexing gain (or equivalent ly DoF) t akes t he form r = lim ⇢! 1
Ri log ⇢
where r is also referred t o as t he mult iplexing gain. Finally, for Pe denot ing t he probability t hat at least one user has decoded erroneously, we recall t he not ion of diversity t o be log Pe d= lim ⇢! 1 log ⇢ (cf. [34]). Regarding knowledge of t he channel st at e, we assume perfect channel st at e informat ion at t he receivers (perfect CSIR), but we only allow t he t ransmit t er perfect knowledge of CSIT wit h a delay of a single phase (single coherence t ime), and provide no knowledge of current CSIT .
5.2
D i ver sit y
135
We consider t he minimum delay case where communicat ion, just as in [4], t akes place over L = 3 phases, and we not e t hat nat urally t he achieved optimal DoF performance cannot furt her benefit from using L > 3 phases. T he proposed design is present ed here for simplicity only for t he minimum phase-delay case (L = 3), for which it achieves t he opt imal diversity of 6 (3 coherence int ervals, 2 t ransmit ant ennas). We not e t hough t hat t his design can be readily ext ended t o t he case where L is a mult iple of 3, t o again achieve t he opt imal diversity order of 2L . We will hencefort h consider t hat L = 3.
5.2.4
Or iginal M AT scheme
T he MAT scheme [4] applies irrespect ive of t he coherence durat ion Tc, and it considers communicat ion over L = 3 phases. Wit hout considerat ion for t he t ime index, in describing t he scheme, we denot e by { a1 , a2 } t he two symbols int ended for t he first user, and by { b1, b2} t he symbols for t he second user. During t he first , second and t hird phase, t he t ransmit t er sequent ially sends x 1, x 2, x 3 2 C2 where x1 =
a1 a2
, x2 =
b1 b2
h T2 x 2 + gT1 x 1 0
, x3 =
.
(5.21)
Consequent ly t he result ing input -out put relationship seen by t he first user, t akes t he form 2 (1) 3 2 2 (1) 3 3 2 3 y1 z1 h T1 0 6 7 6 7 y (1) = 4 y2(1) 5 = 4 0 5 x 1 + 4 h T2 5 x 2 + 4 z2(1) 5 T (1) (1) h3,1 gT1 h3,1h 2 y z 3
3
which is convert ed t o t he equivalent form " # (1) y 1 y˘ (1) , = H˘ x 1 + z˘ (1) , (1) (1) y3 h3,1y2
(5.22)
where " H˘ =
h1,1 h1,2 h3,1g1,1 h3,1 g1,2
, z˘ (1) =
(1)
z3
# (1) z1 (1) . h3,1 z2
(5.23)
Not ing t hat H˘ is almost surely of full rank, allows us t o conclude t hat user 1 can achieve 2/ 3 DoF. Due t o symmet ry, t he same holds for t he second user. Regarding t he diversity of t he MAT scheme, we have t he following. P r op osit ion 7. T he MAT scheme gives diversity that is upper bounded by 3. T his is shown in Appendix 5.2.8.
136 C hapt er 5
5.2.5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
I nt er fer ence alignment for achieving bot h full D oF and full diver sit y
We briefly not e t hat t he maximum achievable diversity t hat we can hope for is 6, simply because t he t ransmitt er has 2 ant ennas and because communicat ion t akes place over L = 3 st at ist ically independent channel realizat ions (L = 3 phases). T he scheme applies t o t he case where t he coherence period Tc is no less t han 8. Wit hout loss of generality, we will assume t hat Tc = 8, and t hat t he ent ire coding durat ion is L Tc = 24 channel uses, spanning t hree diff erent phases of 8 channel uses each. For l = 1, · · · , L , t = 1, · · · , T2c = 4, we denot e by x l,2t 1 t he vect ors t ransmit t ed during odd t imeslot s of phase l (i.e., during t imeslot s 1, 3, 5, 7 of phase l), and we denot e by x l ,2t (l = 1, · · · , L , t = 1, · · · , 4) t he vect ors t ransmit t ed during even t imeslot s of phase l. We assign 16 informat ion symbols { a1,t , a2,t , a3,t , a4,t } 4t= 1 t o t he first user, and 16 symbols { b1,t , b2,t , b3,t , b4,t } 4t= 1 t o t he second user. Consequent ly for any given t = 1, 2, 3, 4, t he t ransmit ted signal vect ors are designed as follows x 1,2t
= [a1,t + b1,t a3,t + b3,t ]T P P ⌘3 2j= 1 g1,j a2j 1,t + ⌘1 1= 0 P2 P ⌘4 j = 1 g1,j a2j 1,t + ⌘2 1= 0
1
x 2,2t x 3,2t
2 j = 1 h 1,j b2j
1,t
2 j = 1 h 1,j b2j
1,t
x 1,2t =[a2,t + b2,t a4,t + b4,t ]T x 2,2t =
P ⌘3
2 j = 1 g1,j
0 P a2j ,t + ⌘1
2 j = 1 h 1,j b2j ,t
x 3,2t =
P ⌘4
2 j = 1 g1,j
0 P a2j ,t + ⌘2
2 j = 1 h 1,j b2j ,t
(5.24)
i = 1, 2
(5.25)
, i = 3, 4,
(5.26)
where ⌘i =
P
i+
and ⌘i =
i+
P
1
, 2 2 j = 1 |h1,j | 1 2 2 j = 1 |g1,j |
for some posit ive const ant s { 1 , 2, 3, 4} t hat are specifically designed lat er on. Due t o symmet ry, we can focus only on t he first user. T he received signals, accumulat ed at t he first receiver, can be rearranged t o t ake t he form (1)
y¯ t
= H
AA at +
H
A B bt
(1)
+ z¯ t , t = 1, · · · , Tc/ 2,
(5.27)
5.2
D i ver sit y
137
where
=
h (1) y1,2t h (1) z1,2t
at
=
[a1,t a3,t a2,t a4,t ]T ,
bt
=
[b1,t b3,t b2,t b4,t ]T ,
(1)
y¯ t
(1)
z¯ t
=
(1)
1
y2,2t
1
z2,2t
i (1) (1) (1) T y y y 1 1,2t 2,2t 3,2t i (1) (1) (1) T 1 z1,2t z2,2t z3,2t
(1)
1
y3,2t
1
z3,2t
(1)
(1)
(5.28)
and where 2
H
H
AA
AB
=
=
h1,1 h1,2 0 0 6h2,1 ⌘3g1,1 h2,1⌘3 g1,2 0 0 6 6h3,1 ⌘4g1,1 h3,1⌘4 g1,2 0 0 6 6 0 0 h1,1 h1,2 6 4 0 0 h2,2 ⌘3g1,1 h2,2⌘3g1,2 0 0 h3,2 ⌘4g1,1 h3,2⌘4g1,2 2 h1,1 h1,2 0 0 6h2,1 ⌘1h1,1 h2,1⌘1h1,2 0 0 6 6h3,1 ⌘2h1,1 h3,1⌘2h1,2 0 0 6 6 0 0 h h 1,1 1,2 6 4 0 0 h2,2⌘1h1,1 h2,2⌘1 h1,2 0 0 h3,2⌘2h1,1 h3,2⌘2 h1,2
3 7 7 7 7 7 7 5 3 7 7 7 7. 7 7 5
Rewrit ing (5.27) we now get (1)
y¯ t
= [H
AA
H
AB ]
at bt
(1)
+ z¯ t .
(5.29)
Not e t hat H A A 2 C6⇥4, H A B 2 C6⇥4 , and t hat t he rank of [H A A H A B ] 2 C6⇥8 can generally not support decoding of [a Tt bTt ]T 2 C8 . T his problem is bypassed by t he st ruct ure of t he designed scheme which allows for aligning some of t he int erference at t he first receiver (cf. Fig. 5.2), such t hat H
A B bt
= H¯
¯
A B bt
(5.30)
where 2 6 6 6 H¯ A B = 6 6 6 4
h1,1 0 h2,1⌘1h1,1 0 h3,1⌘2h1,1 0 0 h1,1 0 h2,2 ⌘1h1,1 0 h3,2 ⌘2h1,1
3 7 " 7 7 b1,t + ¯t = 7, b 7 b2,t + 7 5
h 1, 2 h 1, 1 b3,t h 1, 2 h 1, 1 b4,t
# .
Consequent ly we can rewrit e (5.27) as (1)
y¯ t
(1) = H¯ A x¯ t + z¯ t ,
(5.31)
138 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
h 2 a2,t
h 3 a3,t
h 5 b3,t
h1a1,t
h 5b1,t
h 6 b 2 ,t
h 6 b4 ,t
h 4 a 4 ,t
Figur e 5.2 – Received signal space at user 1 aft er int erference alignment . h h¯ i denot es t he i t h column of H¯ A , and ✓= h 1,1, 21 . where H¯
A
⇥ = H
AA
H¯
AB
⇤ , x¯ t =
at ¯t b
,
(5.32)
and where H¯ A 2 C6⇥6, x¯ A 2 C6. At t his point we randomly pick 1 , 2, 3, 4 from t he set of all possible numbers t hat guarant ee ⌘3 ⌘1 = 6 ⌘4 ⌘2 as well as guarant ee t hat . . 2 2 3 = 4 = max(|g1,1 | , |g1,2 | ), . . 2 2 (5.33) 1 = 2 = max(|h1,1 | , |h 1,2 | ). It is t hen easy t o show t hat t his random choice of 1, 2 , 3 , 4 , while sat isfying t he power const raint s, also guarant ees t hat t he rank of H¯ A is full wit h probability 1. Simple zero-forcing (ZF) decoding guarant ees t he 2/ 3 DoF for bot h users.
5.2.6
D iver sit y analysis of t he pr oposed scheme
We again focus, wit hout loss of generality, on t he first user, and consider joint ML decoding for t he MAC channel corresponding t o (5.31). We remind t he reader t hat we are int erest ed only in est ablishing t he diversity of t he scheme, i.e., we are int erest ed in t he case of r = 0 (R is fixed). Direct ly from [56] we know t hat t he probability of error in t his light ly loaded regime (cf. [56]) is dominat ed by t he out age event ⇢ ⌘ 1 ⇣ (1) ¯ ¯ O , H¯ A : I a t ; y¯ t |b , H (5.34) t A < R , 6
5.2
D i ver sit y
139
and as a result , t he corresponding probability of error t akes t he form . Pe = = . =
P r (O) ◆ ✓ 1 (1) ¯ ¯ Pr I (a t ; y¯ t |bt , H A ) < R 6 ✓ ◆ 1 H Pr log det (I + ⇢H A A H A A ) < R 6
where for t he above we considered opt imal Gaussian dist ribut ions for a t and bt . For 2 3 h1,2 h1,1 H A A ,j = 4 h2,j ⌘3 g1,1 h2,j ⌘3g1,2 5 , (5.35) h3,j ⌘4 g1,1 h3,j ⌘4g1,2 and ⌦ j = det I + ⇢H and aft er considering t hat H implies t hat
AA
. Pe =
A A ,j
H
H
A A ,j
, for j = 1, 2,
(5.36)
has a block-diagonal st ruct ure, we see t hat (5.35)
P r (log(⌦ 1⌦ 2 ) < 6R).
(5.37)
Using t he law of expansion of det erminant s by diagonal element s [54], and t he Cauchy-Binet equat ion, we t hen have ⌦ j = 1+
X2
⇢n
X
X det (([H
A A ,j ]J,S )
H
[H
A A ,j ]J,S)
n= 1 J ⇢{ 1,2,3} S⇢{ 1,2} |J |= n
|S|= n
2
= 1 + ⇢|h1,1 | + ⇢|h2,j |2 |⌘3 |2 |g1,1 |2 + ⇢|h3,j |2 |⌘4 |2|g1,1|2 + ⇢|h1,2 |2 + ⇢|h2,j |2 |⌘3|2|g1,2|2 + ⇢|h3,j |2|⌘4|2|g1,2|2 + ⇢2 |h2,j |2 |⌘3 |2 |h1,1 g1,2
h1,2 g1,1|2
+ ⇢2 |h3,j |2 |⌘4 |2 |h1,1 g1,2
h1,2 g1,1|2,
where in t he above, [E]J,S denot es t he submat rix of mat rix E t hat includes t he rows of E labeled by t he element s of set J , and t he columns labeled by t he element s of set S. Cont inuing we get t hat (a)
. ⌦ = j
1 + ⇢|h1,1 |2 + ⇢|h1,2 |2 + ⇢|h2,j |2 + ⇢|h3,j |2 + ⇢2(|h2,j |2 + |h3,j |2 )|⌘B |2|h1,1g1,2
h1,2g1,1 |2
1 + ⇢|h1,1 |2 + ⇢|h1,2 |2 + ⇢|h2,j |2 + ⇢|h3,j |2
(5.38)
140 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
where in (a) we used t hat ⌘A ,
1 |h1,1
|2+|h
1,2
|2
, ⌘B ,
1 |g1,1|2+|g1,2 |2
,
(5.39)
and t hat (cf. (5.25),(5.33)) . . . . ⌘A = ⌘1 = ⌘2 , ⌘B = ⌘3 = ⌘4. Consequent ly (5.38) direct ly gives t hat Pe ˙ P r (log(1 + ⇢|h1,1 |2 + ⇢|h1,2|2 + ⇢|h2,1|2 + ⇢|h3,1|2 + ⇢|h2,2 |2 + ⇢|h3,2 |2) < 6R), which direct ly shows t hat t he diversity of t he scheme is 6 (again for r = 0). At t his point we also not e t hat t he above design can be readily ext ended t o t he case where L is a mult iple of 3. T he proof for t his is simple and it is omit t ed. T he result is summarized in t he following. P r op osit ion 8. I n the setting of the described two-user MI SO BC with delayed CSI T, the proposed interference alignment based precoding scheme achieves full DoF and full diversity.
5.2.7
Conclusions
In t he set t ing of t he two-user MISO broadcast channel wit h delayed CSIT , we designed t he first scheme t o achieve full DoF as well as full diversity. T he scheme borrows from t he t echniques of int erference alignment , which allow for combining t he signal manipulat ions t hat increase t he DoF wit h t he signal manipulat ions t hat allow for full diversity. Fut ure work can ext end t he result by analyzing t he ent ire DMT behavior of t he scheme, as well as ext end t he result t o ot her BC set t ings.
5.2.8
A pp endix - Pr oof of Pr op osit ion 7
In deriving t he diversity achieved by t he MAT scheme, we again focus, wit hout loss of generality, on t he performance of t he first user. From (5.22) we first recall t hat decoding is based on y˘ (1) = H˘ x 1 + z˘ (1) . Consequent ly, in t he presence of Gaussian input x 1 = [a1 a2 ]T (cf. (5.21)), and not ing t hat t he noise t erm z˘ (1) in (5.22) has zero mean and covariance E[˘z (1) ( z˘ (1) ) H ] = diag( 1 , 1+ |h3,1 |2),
5.2
D i ver sit y
141
we calculat e t he probability of out age t o t ake t he form ⇣ ⌘ Pout (r ) = P r I (x 1 ; y˘ (1) |H˘ ) < 3R ⌘ ⇣ H . = P r log det (I + ⇢H˘ H˘ ) < 3R = where we used t hat
P r (log
(5.40)
< 3r log ⇢),
(5.41)
⇣ ⌘ H = det I + ⇢H˘ H˘ ,
and t hat
. ⇢✏) = exp( ⇢✏) = 0,
P r (|h3,1|2
for any posit ive ✏ . Consequent ly expanding the det erminant and applying t he Cauchy-Binet rule, gives ⇣ ⌘ X2 X X = 1+ ⇢n det [H˘ ]J,S ([H˘ ]J,S ) H n= 1 J ⇢{ 1,2} S⇢{ 1,2} |J |= n
|S|= n
2
= 1+ ⇢|h1,1 | + ⇢|h1,2 |2 + ⇢|h3,1|2|g1,1|2 + ⇢|h3,1|2 |g1,2|2 + ⇢2|h3,1|2|h1,1g1,2
h1,2g1,1|2 .
(5.42)
T he fact t hat |h1,1 g1,2
h1,2 g1,1|2
|h1,1 |2|g1,2 |2 + |h1,2 |2|g1,1 |2 + 2|h1,1 ||g1,2 ||h1,2||g1,1 |,
t oget her wit h a change of variables where hl ,j = ⇢ ↵ l , j , gl ,j = ⇢ and along wit h t he fact t hat
⇢
. P r (↵ l ,j ) =
l ,j
,
⇢ 1 , for ↵ l ,j < 0 , ⇢ ↵ l , j , for ↵ l ,j 0
(5.43)
gives t hat t he diversity d(r ) of t he MAT scheme is upper bounded as d(r )
dM (r ) , inf (↵ 1,1 + ↵ 1,2 +
where
OM (r )
8 > > > > > > > > < OM (r ) =
> > > > > > > > :
(2
1,1
+
1,2
+ ↵ 3,1 )
(1 ↵ 1,1) + 3r, + (1 ↵ 1,2) 3r, + (1 ↵ 3,1 3r, 1,1 ) + (1 ↵ 3,1 ) 3r, 1,2 + (2 ↵ 3,1 ↵ 1,1 3r, 1,2 ) + (2 ↵ 3,1 ↵ 1,2 ) 3r, 1,1 P2 ↵ 3,1 0.5 j = 1 (↵ 1,j + 1,j )) +
(5.44)
9 > > > > > > > > = > > > > > > > > ; 3r
.
At t his point it is easy t o see t hat t he diversity is upper bounded by 3 (see also Fig. 5.3). 2
142 C hapt er 5
On t he I m p er fect G l obal C SI R and D i ver si t y A sp ect s
Figur e 5.3 – DMT upper bound for t he MAT scheme.
Chapt er 6
Conclusions and Fut ure Work
Wit h a st art ing point t hat good feedback is crucial but hard and t imeconsuming t o obt ain, t he main work in t he t hesis, sought t o address t he simple yet elusive and fundament al quest ion of “HOW MUCH QUALIT Y of feedback, AND WHEN, must one send t o achieve a cert ain performance in specific set t ings of mult iuser communicat ions”. T he t hesis first considered t he two-user MISO BC, and made progress t owards est ablishing and meet ing t he t radeoff between performance, and feedback t imeliness and quality. Considering a general CSIT process, t he work provided DoF expressions t hat are simple and insight ful funct ions of easy t o calculat e paramet ers. T he result s - bounds and novel schemes- hold for a broad family of channel models spanning a large class of block fading and non-block fading channel models. It also int roduced t he novel periodically evolving feedback set t ing over t he quasi-st at ic block fading channel, where a gradual accumulat ion of feedback bit s result s in a progressively increasing CSIT quality as t ime progresses across a finit e coherence period. T his powerful set t ing capt ures many of t he engineering opt ions relat ing t o feedback, as well as capt ured many int erest ing set tings previously considered. T he derived DoF expressions are a result of novel schemes and improved outer bounds. T hese derived expressions off er insight on pract ical quest ions on t opics relat ing t o how much feedback quality (delayed, current or predict ed) allows for a cert ain DoF performance, relat ing t o t he usefulness of delayed feedback, t he usefulness of predict ed CSIT , t he impact of imperfect ions in t he quality of current and delayed CSIT , t he impact of feedback t imeliness and t he eff ect of feedback delays, t he int erplay between t imeliness and quality of feedback, and t he benefit of having feedback symmet ry by 143
144
C hapt er 6
C onclusi ons and Fut ur e W or k
employing comparable feedback links across users. Int erest ingly, t he above result s can be ext ended t o t he two user MIMO BC and t he two user MIMO IC, in t he presence of feedback wit h evolving quality and t imeliness. In addit ion t o t he progress t owards describing t he limit s of using such imperfect and delayed feedback in MIMO set t ings, t he work off ers diff erent insight s t hat include t he fact t hat , an increasing number of receive ant ennas can allow for reduced quality feedback, as well as t hat no CSIT is needed for t he direct links in t he IC. T hen, t he work considered t he general K -user MISO BC, and est ablished inner and out er bounds on t he t radeoff between DoF performance and CSIT feedback quality. For t he general K -user BC wit h imperfect current CSIT , t he work provided a novel out er bound on t he DoF region, which nat urally bridges t he gap between t he case of having no current CSIT and t he case wit h full CSIT . In addit ion, t he work charact erized t he minimum current CSIT feedback t hat is necessary for any point of t he sum DoF, which is opt imal for many cases. A furt her work focused on t he global CSI R aspect , and provided analysis and novel communicat ion schemes for t he two-user MIMO BC wit h imperfect delayed CSIT , as well as, in t he presence of addit ional imperfect ions in t he global CSIR. T he derived DoF region is oft en opt imal and, while corresponding t o imperfect delayed CSIT and imperfect global CSIR, oft en mat ches t he region previously associat ed t o perfect delayed CSIT and perfect global CSIR. Finally, t he t hesis focused on t he diversity aspect of t he communicat ion wit h limit ed and delayed feedback. In t he set t ing of two-user MISO BC wit h delayed CSIT , t he first scheme was provided t o achieve full DoF as well as full diversity. T he scheme borrows from t he t echniques of int erference alignment , which allow for combining t he signal manipulat ions t hat increase t he DoF gain and diversity gain. In addit ion t o t he t heoret ical limit s and novel encoders and decoders, t he work provided int erest ing insight s on many pract ical and fundament al quest ions. However, up t o now a number of fundament al quest ions are st ill open, regarding t o t he communicat ion scenarios wit h limit ed, imperfect and delayed feedback. For example, what is DoF region out er bound for t he set t ing wit h low-quality delayed CSIT ? What is DoF region out er bound for t he two-user MISO BC where, t he t ransmit t er has perfect and inst ant aneous CSIT for one user’s channel, but never has CSIT for t he ot her user’s channel ? Moving t o t he ot her set t ings, such as t he set t ing of X channel, and t he set t ings wit h diff erent t opologies, what are t he fundament al limit s of communicat ions wit h limit ed feedback ? T hose are t he int erest ing quest ions t hat will be explored in t he fut ure work.
Chapt er 7
French Summary
Dans des nombreux scénarios de communicat ion sans fil mult i-ut ilisat eurs, une bonne rét roact ion est un ingrédient essent iel qui facilit e l’ameliorat ion des performances. Bien qu’ét ant ut ile, une rétroact ion parfait e rest e diffi cile et fast idieuse à obt enir. En considérant ce défi comme point de départ , les présent s t ravaux cherchent à adresser la quest ion simple et pourt ant insaisissable et fondament ale suivant e : “ Quel niveau de qualit é de la rét roact ion doit -on rechercher, et à quel moment faut -il envoyer pour at t eindre une cert aine performance en degrés de libert é (DoF en anglais) avec des paramèt res spécifiques de communicat ions mult i-ut ilisat eurs”. L’accent est t out d’abord mis sur les communicat ions à t ravers un canal de diff usion (BC en anglais) à deux ut ilisat eurs, mult i-ent rées, unique sort ie (MISO en anglais) avec informat ions imparfait es et ret ardées à l’émet t eur de l’ét at du canal (CSIT en anglais), un paramèt re pour lequel la présent e t hèse explore le compromis ent re la performance et la rapidit é et la qualit é de la rét roact ion. La présent e ét ude considère un cadre général dans lequel la communicat ion a lieu en présence d’un processus d’at t enuat ion aléat oire, et en présence d’un processus de rét roact ion qui, à t out moment , peut ou non fournir des est imat ions CSIT - d’une qualit é arbit raire - pour t out e réalisat ion passée, act uelle ou fut ure du canal. Sous des hypot hèses st andard, dans cet t e t hèse est dérivée la région DoF qui est opt imale pour un large régime de qualit é CSIT . Cet t e région capt ure de manière concise l’eff et des corrélat ions de canaux, la qualit é de la valeur prédit e, la valeur courant e et ret ardée du CSIT , et capt ure généralement l’eff et de la qualit é du CSIT fourni à n’import e quel moment , sur n’importe quel canal. Les encadrement s sont obt enus à l’aide de nouveaux schémas qui - dans le cont ext e de CSIT imparfait et ret ardé - sont présent és ici pour la première 145
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fois, avec encodage et décodage sur st ruct ure de phase Markovienne. Les résult at s sont validés pour une grande classe de modèles de canaux d’at t énuat ion en bloc et non-bloc, et ils unifient et ét endent de nombreuses t ent at ives ant érieures de capt ure de l’eff et de rét roact ion imparfait e et ret ardée. Cet t e généralit é permet également d’examiner de nouveaux paramèt res pert inent s, t els que la nouvelle t echnique de rétroact ion à évolut ion périodique, où une accumulat ion progressive de bit s de rét roact ion améliore progressivement le CSIT avec le t emps, ce dernier progressant à t ravers une période de cohérence finie. Les résult at s ci-dessus sont at teint s dans le cas du MISO-BC à deux ut ilisat eurs, et sont ensuit e immédiat ement ét endus aux cas ent rées mult iples sort ies mult iples (MIMO) BC à deux ut ilisat eurs et de canaux d’int erférence MIMO (MIMO IC), encore une fois en présence de processus d’at t énuat ion aléat oire, et en présence d’un processus de rét roact ion qui, à t out moment , peut ou non fournir des est imat ions CSIT - d’une qualit é arbit raire - pour t out e réalisat ion passée, act uelle ou fut ure du canal. Sous les hypot hèses st andard, et en présence de M ant ennes par émet t eur et N ant ennes par récept eur, la région DoF, qui est opt imale pour un large régime de qualit é CSIT , est dérivée. En plus de présent er une avancée dans la descript ion des limit es d’ut ilisat ion de t elles informat ions imparfait es et ret ardées dans les milieux MIMO, la present e ét ude propose diff érent es idées dont le fait qu’un nombre croissant d’ant ennes de récept ion peuvent causer la réduct ion de la qualit é de la rét roact ion, ainsi que le fait qu’aucun CSIT n’est requis pour les liens direct s dans l’IC. Dans un deuxieme t emps, la t hèse considère le cadre plus général de la chaîne de diff usion K -ut ilisat eurs MISO, où un émet t eur avec M ant ennes t ransmet des informat ions aux K ut ilisat eurs à une seule ant enne, et où, une nouvelle fois, la qualit é et la rapidit é du CSIT est imparfait e. Dans ce cont ext e mult i-ut ilisat eurs, la t hèse ét ablit des limit es sur le compromis ent re la performance du DoF et la qualit é de la rét roact ion CSIT . Plus précisément , elle fournit une nouvelle région de la borne ext erne DoF dans le cas general à K - ut ilisat eurs MISO BC avec un CSIT à qualit é imparfait e courant e, ce qui nat urellement comble le lien entre le cas sans CSIT courant (ou le CSIT est seulement ret ardé, ou sans CSIT ) et le cas où l’on dispose d’un CSIT complet . Dans ce cont ext e, la présent e ét ude caract érise alors la rét roact ion CSIT à courant minimum nécessaire pour at t eindre n’import e quel ét at de somme DoF. Cet t e caract érisat ion est opt imale dans le cas où M K , et le cas M = 2, K = 3. Dans une aut re perspect ive, l’ét ude considère également un aut re aspect de la communicat ion à rét roact ion imparfait e et ret ardée : celui où une imperfect ion supplément aire sur les est imat ions du récept eur sur le canal d’un aut re récept eur (CSIR global) est présent e, en plus de l’ imperfect ion du CSIT . L’ét ude se concent re sur un canal de diff usion MIMO avec un CSIT de qualit é donnée, imparfait et ret ardé, et un CSIR global imparfait et re-
7.1
M odèl e C anal
147
t ardé. Et des schémas, ainsi que des encadrement s des DoFs, souvent précises, sont présent és. L’ét ude cont inue ensuit e en révélant de manière const ruct ive comment même des CSIT sensiblement imparfait s ret ardés et des CSIR globaux essent iellement imparfait s ret ardés sont en fait suffi sant s pour at t eindre la performance opt imale en DoF qui ét ait précédemment associée au CSIT parfait ret ardé et au CSIR parfait global. En s’avent urant plus loin encore, l’ét ude t ient également compt e de l’aspect diversit é de la communicat ion - ent re deux ut ilisat eurs MISO BC avec CSIT ret ardé. Dans ce cadre, l’ouvrage propose un nouveau syst ème de diff usion qui fait appel au CSIT ret ardé et à une forme d’ alignement d’int erférence pour obt enir à la fois le maximum DoF possible (2/ 3), ainsi qu’une diversit é complèt e. En plus de fournir des limit es t héoriques et des nouveaux encodeurs et décodeurs, l’ét ude s’applique à obt enir une meilleure comprehension sur des quest ions prat iques relat ives à combien la qualit é de rét roact ion (diff érée, en cours ou prévue) influe sur les performances DoF. Sur des quest ions relat ives à l’ut ilit é de la rét roact ion ret ardée, l’ut ilit é du CSIT prédit , à l’impact des imperfect ions sur la qualit é du CSIT courant et diff éré, l’impact de la rapidit é de la rét roact ion et l’eff et des ret ards de rét roact ion, l’avant age d’avoir de la symét rie dans la rét roact ion en ut ilisant des liens de rét roact ion comparables ent re les ut ilisat eurs, l’impact des imperfections sur la qualit é des CSIR globaux. Finalement sur des quest ions relat ives à la façon d’at t eindre à la fois un DoF et une diversit é complet s.
7.1
M odèle Canal
Dans le cadre de scénarios de communicat ion mult i-ut ilisat eurs (voir par exemple la figure 7.1 et la figure 7.2), nous allons examiner à ent rées mult iples et sort ie unique canal de diff usion (MISO BC), mult iple-input mult ipleoutput BC (MIMO BC), ainsi comme canal MIMO int erférence (MIMO IC).
7.1.1
M I SO B C
Nous nous concent rons d’abord sur le canal de diff usion mult i-ut ilisat eur, avec un émet t eur, et avec K ut ilisat eurs, en présence d’imparfait et ret ardée évaluat ions de la CSIT . Nous commençons avec deux-ut ilisat eurs MISO BC où un émet t eur communique à deux ut ilisat eurs. h t et gt représent ent ces canaux à l’inst ant t, pour la première et la deuxième ut ilisat eur. x t représent ent le vect eur émis au moment t. Les signaux reçus correspondant à la première et deuxième
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Feedback
Channel 1 User 1
Tx
Channel K Feedback
User K
Figur e 7.1 – Modèle de syst ème de K -ut ilisat eur MISO BC.
Channel 11
l nne Ch a
Tx 1
12 Rx 1
C ha nne l 21
Channel 22 Tx 2
Rx 2
Figur e 7.2 – Modèle de syst ème de canal d’int erfŕence de deux ut ilisat eurs.
ut ilisat eur prennent la forme (1)
= h Tt x t + zt
(2)
= gTt x t + zt
yt yt (1)
(1)
(2)
(7.1) t = 1, 2, · · ·
(2)
(7.2)
zt , zt sont le bruit AWGN, la puissance unit aire. Il exist e une cont raint e de puissance E[||x t ||2 ]
P= ⇢
et P (ou ⇢) prend le rôle du rapport signal-sur-bruit (SNR).
7.2
D egr és de l ib er t é
7.1.2
149
M IM O BC
Nous considérons également le MIMO BC. Dans ce cadre, le modèle canal prend la forme (1)
= H
(2)
= H
yt yt (1)
(1) t xt (2) t xt
(1)
(7.3)
(2)
(7.4)
+ zt + zt
(2)
H t 2 CN ⇥M , H t 2 CN ⇥M représent ent le premier et le deuxième canal (1) (2) du récept eur au t emps t ; z t , z t représent ent le AWGN bruit , la puissance de l’unit é; x t 2 CM ⇥1 est le signal d’entrée, la cont raint e de puissance E[||x t ||2 ] P .
7.1.3
M IM O IC
Nous considérons également le MIMO BC. Dans ce cadre, le modèle canal prend la forme (1)
= H
(2)
= H
yt yt (11)
(11) (1) xt t (21) (1) xt t
+ H + H
(12) (2) xt t (22) (2) xt t
(1)
(7.5)
(2)
(7.6)
+ zt + zt
(22)
Ht 2 CN ⇥M , H t 2 CN ⇥M représent ent les mat rices de canal des liens (12) (21) direct s, H t 2 CN ⇥M , H t 2 CN ⇥M représent ent les mat rices de canal des liaisons t ransversales, au t emps t.
7.2
D egr és de liber t é
La t hèse principale port era sur la performance degrés de libert é (DoF). Dans le régime de la haut e-SNR, ét ant donné les t aux réalisables (R 1 , R 2 , · · · , R K ) le correspondant DoF est donnée par di = lim
P! 1
Ri , i = 1, 2, · · · , K . log P
La région de la DoF D est l’ensemble de t ous les DoF réalisable (d1 , d2 , · · · , dK ). Lla analyse à haut SNR off re un bon aperçu de la performance au régime SNR modérée.
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7.3
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Eff et s et de qualit é de r et ar d de r ét r oact ion
Comme dans de nombreux scénarios de communicat ion sans fil mult iut ilisat eurs, la performance du canal de diff usion dépend de la rapidit é et la qualit é de la CSIT . Cet t e rapidit é et la qualit é peuvent êt re réduit es par des liens de ret our à capacit é limit ée, qui peuvent off rir une qualit é d’évaluat ions basse, ou peut off rir des informat ions de bonne qualit é qui vient si t ard dans le processus de communicat ion et peuvent donc êt re ut ilisés pour seulement une fract ion de la durée de communicat ion. La dégradat ion de performance correspondant e, par rapport au cas de la rét roact ion parfait e sans délai, la force de quest ions et de qualit é de ret ard de la quant it é d’informat ions qui est nécessaire, et lorsque, dans le but d’at t eindre un cert ain rendement . Ces eff et s et la qualit é de ret ard de réact ion, t ombent nat urellement ent re les deux cas ext rêmes de non CSIT et de plein CSIT (CSIT immédiat ement disponible et parfait ), avec CSIT complèt e permet t ant l’opt imal 1 degrés de libert é par ut ilisat eur (cf., [1]), t andis que l’absence de t out e CSIT réduit ce nombre à seulement 1/ 2 DoF par l’ut ilisat eur (cf., [2, 3]). Un out il précieux en vue de combler cet t e lacune et de comprendre davant age les eff et s et la qualit é de ret ard de réact ion, est venu avec [4] mont rant que les réact ions ret ardées arbit rairement peut encore permet t re l’améliorat ion des performances de l’aff aire sans CSIT . Dans un cadre qui ét ablit une dist inct ion ent re la CSIT courant et diff éré - CSIT t ardive ét ant celle qui est disponible après l’écoulement du canal, soit après la fin de la période de cohérence correspondant à la voie décrit e par cet t e réact ion t ardive, alors que CSIT act uelle correspondait aux comment aires reçus au cours de la période de la cohérence de la chaîne - le t ravail dans [4] a mont ré que parfait CSIT ret ardée, même sans CSIT act uel, permet une améliorat ion de 2/ 3 DoF par ut ilisat eur. Dans le même cont ext e de ret ard cont re courant CSIT , le t ravail dans [5–8] a int roduit des considérat ions de qualit é de rét roact ion, et a réussi à quant ifier l’ut ilit é de la combinaison parfait e CSIT ret ardé avec immédiat ement disponible CSIT imparfait d’une cert aine qualit é qui est rest é inchangé pendant t out e la durée de la cohérence. Dans ce cadre les t ravaux ci-dessus mont re un pont supplément aire de l’écart de 2/ 3 à 1 DoF, en fonct ion de cet t e qualit é CSIT act uel. De nouveaux progrès est venu avec le t ravail dans [9] qui, en plus d’explorer les eff et s de la qualit é de la CSIT act uel, également examiné les eff et s de la qualit é de la CSIT ret ardée, permet t ant ainsi l’examen de la possibilit é que l’ensemble nombre de bit s de rét roact ion (correspondant au ret ard en plus du courant CSIT ) peut êt re réduit e. Se concent rant à nouveau sur le réglage précis où la qualit é de la CSIT courant est rest é inchangé pour l’ensemble de la période de cohérence, ce t ravail a révélé ent re aut res que imparfait e CSIT ret ard peut at t eindre le même opt imalit é qui a ét é précédemment at t ribué à perfect ionner CSIT ret ard, donc équivalent e mont rant comment l’quant it é
7.3
Eff et s et de quali t é de r et ar d de r ét r oact ion
151
de feedback ret ard nécessaire, est proport ionnelle à la quant it é de ret our de courant . Une généralisat ion ut ile du paradigme de la CSIT ret ard par rapport courant , est venu avec le t ravail dans [10] qui s’écart e de l’hypot hèse d’avoir la qualit é de la CSIT invariant t out au long de la période de la cohérence, et a permis à la possibilit é que CSIT courant peut êt re disponible uniquement après un cert ain ret ard, et en part iculier seulement après une cert aine fract ion de la période de cohérence. Sous ces hypot hèses, en présence de plus de deux ut ilisat eurs, et en présence d’une parfait e CSIT ret ard, les t ravaux ci-dessus a mont ré que jusqu’à un cert ain ret ard, on peut at t eindre la performance opt imale correspondant à pleine CSIT (et immédiat ). Une aut re généralisat ion int éressant e est venu avec le t ravail dans [11] qui, pour le réglage de l’heure sélect if pour deux ut ilisat eurs MISO BC, le CSIT pour le canal de l’ut ilisat eur 1 et l’ut ilisat eur 2, alt erner ent re les t rois ét at s ext rêmes de la CSIT parfait e act uel, parfait CSIT ret ardé, et aucune CSIT . Les paramèt res ci-dessus, et de nombreux aut res paramèt res avec CSIT imparfait et ret ardée, comme ceux de [12, 13, 15–28, 48], a abordé diff érent es instances du problème plus général de la communicat ion en présence de rét roact ion avec des propriét és diff érent es et de qualit é de ret ard, avec chacun de ces paramèt res êt re mot ivé par le fait que CSIT parfait e peut êt re généralement diffi cile et fast idieux à obt enir, que la précision CSIT peut êt re améliorée au fil du t emps, et que les ret ards de rét roact ion et les imperfect ions coût e généralement en t ermes de performances. La généralisat ion ici pour la mise à CSIT imparfait , ret ardé et limit é générale, int ègre les considérat ions et mot ivat ions ci-dessus, et permet un aperçu sur les quest ions pert inent es t elles que : – Combien de qualit é CSIT (ret ard, act uel ou prévu) permet une cert aine performance DoF ? – Imparfait CSIT ret ardé pouvez obt enir le même opt imalit é qui a ét é précédemment at t ribué à parfait e CSIT ret ard ? – Quand est ret ardé comment aires inut iles? – On prévoit CSIT ut ile en t ermes de performances DoF ? – Comment aires symét rique off re DoF peut bénéficier au cours de la réact ion asymét rique? – Quel est l’impact du CSIR mondiale imparfait e (est imat ions de la récept ion imparfait es du canal de l’aut re récept eur) ? – Un syst ème de communicat ion permet tant à la fois t out e la diversit é et la pleine DoF pour la mise à CSIT seulement ret ardé?
152
7.4
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M anche et pr ocessus de r ét r oact ion
Ce t ravail se concent rera d’abord sur les deux - ut ilisat eur MISO BC, et examinera la communicat ion d’une durée infinie n (sauf avec l’argument spécifique), un processus de décolorat ion du canal aléat oire { h t , gt } nt= 1 t irée d’une dist ribut ion st at ist ique, et un processus de rét roact ion qui fournit des ˆ t ,t 0} nt,t 0= 1 (de canal h t , gt ) en t out t emps t 0 - avant , est imat ions CSIT { hˆ t ,t 0, g pendant ou après la mat érialisat ion de h t , gt au t emps t - et le fait avec une cert aine qualit é { E[||h t
7.5
hˆ t ,t 0||2], E[||gt
ˆ t ,t 0||2 ]} nt,t 0= 1 . g
D ébut , le cour ant et diff ér é CSI T
ˆ t ,t } , les est imaPour le canal h t , gt au t emps t, les est imat ions { hˆ t ,t , g ˆ ˆ ˆ t ,t 0} t 0> t , et les est imat ions { h t ,t 0, g ˆ t ,t 0} t 0< t forment ce que l’on t ions { h t ,t 0, g peut décrire comme des est imat ions de la CSIT , les est imat ions de la CSIT ret ardés et les est imat ions de la CSIT début , respect ivement . Cont rairement à la CSIT est imat ions act uelles disponibles au moment de t, les est imat ions de la CSIT ret ardées sont disponibles au t emps t 0 > t en raison du ret ard, alors que les est imat ions de la CSIT premières sont disponibles au t emps t 0 < t at t ribué à la prédict ion.
7.6
Exemples
Voici Prenons quelques exemples qui sont incorporés et pris en compt e dans not re généralisat ion. Exam ple 16 (Envoi diff éré CSIT ). Un des paramètre Incorporated est la CSI T retard (sans CSI T courant) de mise en [4]. étendue du réglage du CSI T retard dans [4], notre généralisation suit et révèle que imparfaite CSI T retard peut être aussi utile que parfait CSI T retardée. Exam ple 17 (CSIT courant et diff éré Imperfect ). Un des paramètre I ncorporated est le réglage de la CSI T courant et diff éré imparfaite dans [6, 7]. Exam ple 18 (Asymmet ric CSIT ). Un des paramètre Incorporated est le réglage asymétrique de la CSI T dans [33], où les utilisateurs off erts parfait CSI T retard, mais où un seul utilisateur off ert CSI T actuel parfait. Une telle asymétrie pourrait tenir compte des commentaires des liens avec des capacités diff érentes ou des retards diff érents. Exam ple 19 (Not -so-ret ardé CSIT ). Un des paramètre I ncorporated est le réglage de la CSI T pas si retardé dans [10] correspond au canal à évanouissements par blocs avec retour périodique.
7.7
D i ver sit é
153
Exam ple 20 (Alt ernat if CSIT ). Un des paramètre I ncorporated est le réglage de la CSI T alternance dans [11] où alterne entre la CSIT parfait, retardée et ne CSI T Unis. Exam ple 21 (Évolut ion CSIT ). L’un des paramètre de nouveau considéré est le paramètre de rétroaction évolue périodiquement sur le canal avec évanouissement bloc quasi-statique, où une accumulation progressive des bits de rétroaction des résultats dans une qualité CSI T augmentant progressivement à mesure que le temps progresse à travers une période de cohérence finie. Ce paramètre puissant capte de nombreuses options d’ingénierie relatifs à la rétroaction, ainsi que capturé de nombreux paramètres intéressants considérés précédemment.
7.7
D iver sit é
La t hèse examine également l’aspect de la diversit é de communiquer avec rét roact ion limit ée. Pe est la probabilit é d’erreur de la communicat ion. La diversit é est d=
lim
P! 1
log Pe log P
(cf. [34]).
7.8
Global CSI R
En plus limit ée CSIT , nous considérons également le Global CSIR limit ée, chaque ut ilisat eur dispose des est imat ions imparfait es de la chaîne de l’aut re ut ilisat eur.
7.9
L es cont r ibut ions et les gr andes lignes de la t hèse
Comme indiqué précédemment , le principal t ravail de t hèse vise à répondre à la quest ion simple et pourt ant insaisissable et fondament al de la “Combien qualit é des comment aires, et quand faut -il envoyer pour at t eindre un cert ain rendement dans des cont ext es spécifiques de communicat ions mult i-ut ilisat eurs”. Dans le chapit re 2, le t ravail considère deux ut ilisat eurs MISO BC avec CSIT imparfait et ret ardée, et explore le compromis ent re la performance et la rét roact ion rapidit é et la qualit é. Le t ravail est ime un cadre large où la communicat ion a lieu en présence d’un processus de décolorat ion aléat oire, et en présence d’un processus de rét roact ion qui, à t out moment , peuvent ou non fournir des est imat ions CSIT - d’une cert aine qualit é arbit raire - pour t out
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C hapt er 7
Fr ench Sum m ar y
passé, la réalisat ion act uelle ou fut ure canal. Sous les hypot hèses habit uelles, le t ravail vient de la région DoF, ce qui est opt imal pour un grand régime de qualit é CSIT . Cet t e région capt e de manière concise l’eff et des corrélat ions de canal, la qualit é de la valeur prédit e, le courant et diff éré CSIT , et capt ure généralement l’eff et de la qualit é de la CSIT off ert à n’import e quel moment , sur n’import e quel canal. Les limit es sont remplies avec de nouveaux régimes qui - dans le cont ext e de la CSIT imparfait et ret ardée - Int roduire ici pour la première fois, l’encodage et le décodage d’une st ruct ure de phase Markov. Les résult at s t iennent pour une large classe de bloc et modèles de canaux fading non - bloc, et qu’elles unifient et s’ét endent de nombreuses t ent at ives ant érieures pour capt urer l’eff et de rét roact ion imparfait et ret ardée. Cet t e généralit é permet également d’examiner les nouveaux paramèt res pert inent s, t els que le nouveau paramèt re de rétroact ion évoluer périodiquement , où une accumulat ion progressive de bit s de rét roact ion améliore progressivement CSIT que le t emps progresse à t ravers une période de cohérence finie. Les résult at s ont ét é publiés en part ie à – Jinyuan Chen and Pet ros Elia, “C an I m p er fect D elayed C SI T b e as U seful as Per fect D elayed C SI T ? D oF A naly sis and C onst r uct ions for t he B C ”, in Proc. of 50th Annual Allerton Conf. Communication, Control and Computing (Allerton’12), Oct ober 2012. – Jinyuan Chen and Pet ros Elia, “D egr ees-of-Fr eedom R egion of t he M I SO B r oadcast C hannel w it h G ener al M ix ed-C SI T ”, in Proc. I nformation Theory and Applications Workshop (ITA’13), February 2013. – Jinyuan Chen and Pet ros Elia, “M I SO B r oadcast C hannel w it h D elayed and Evolving C SI T ”, in Proc. IEEE I nt. Symp. I nformation T heory (I SI T’13), July 2013. et sera publié en part ie à – Jinyuan Chen and Pet ros Elia, “Towar d t he Per for m ance v s. Feedback Tr adeoff for t he Two-U ser M I SO B r oadcast C hannel”, t o appear in I EEE Trans. I nf. Theory, available on arXiv :1306.1751. – Jinyuan Chen and Pet ros Elia, “Opt im al D oF R egion of t he TwoU ser M I SO-B C w it h G ener al A lt er nat ing C SI T ”, t o appear in Proc. 47th Asilomar Conference on Signals, Systems and Computers (Asilomar’13), 2013, available on arXiv :1303.4352. Dans le chapit re 3, ét endant les résult at s de réglage BC MISO deux ut ilisat eurs, le t ravail explore la performance de l’ut ilisat eur deux ent rées mult iples sort ies mult iples (MIMO) BC et les deux canal MIMO int erférence de l’ut ilisat eur (MIMO IC), en présence de rét roact ion avec l’évolut ion qualit é et la rapidit é. Sous les hypot hèses habit uelles, et en présence de M ant ennes par émet t eur et N ant ennes par récept eur, le t ravail vient de la région DoF, ce qui est opt imal pour un grand régime de qualit é CSIT . Cet t e région capt e de manière concise l’eff et d’avoir prédit , le courant et diff éré CSIT , ainsi que de façon concise capt e l’eff et de la qualit é de la CSIT off ert à n’import e quel
7.9
L es cont r i but i ons et les gr andes li gnes de la t hèse
155
moment , sur n’import e quel canal. En plus de la progression vers décrivant les limit es de l’ut ilisat ion de t elles informat ions imparfait et ret ardée dans les milieux MIMO, l’ouvrage propose diff érent es idées qui incluent le fait que, un nombre croissant d’ant ennes de récept ion peuvent permet t re de réduire la rét roact ion de la qualit é, ainsi que qu’aucune CSIT est nécessaire pour les liens direct s dans l’IC. Les résult at s ont ét é publiés en part ie à – Jinyuan Chen and Pet ros Elia, “Sy m m et r ic Two-U ser M I M O B C and I C w it h Evolving Feedback”, June 2013, available on arXiv : 1306.3710. – Jinyuan Chen and Pet ros Elia, “M I M O B C w it h I m p er fect and D elayed C hannel St at e I nfor m at ion at t he Tr ansm it t er and R eceiver s”, in Proc. I EEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC’13), June 2013. Dans le chapit re 4, le t ravail considère les K - ut ilisat eur MISO BC, et ét ablit des limit es sur le compromis ent re la performance et la qualit é DoF de rét roact ion CSIT . Plus précisément , pour le MISO BC général avec CSIT act uel imparfait , un roman DoF région ext erne lié est fourni, qui relie naturellement l’écart ent re le cas de ne pas avoir CSIT act uelle et le cas de plein CSIT . Le t ravail caract érise alors le ret our de la CSIT minimum qui est nécessaire pour n’import e quel point de la somme DoF, ce qui est opt imal pour de nombreux cas. Les résult at s seront publiés en part ie à – Jinyuan Chen, Sheng Yang, and Pet ros Elia, “On t he Fundam ent al Feedback-v s-Per for m ance Tr adeoff over t he M I SO-B C w it h I m p er fect and D elayed C SI T ”, in Proc. I EEE I nt. Symp. I nformation T heory (I SI T’13), July 2013. Dans le chapit re 5, le t ravail considère en out re les aut res aspect s fondament aux sur les communicat ions avec les informat ions imparfait et ret ardée. Un aut re t ravail est axé sur un canal de diff usion MIMO avec fixe de qualit é imparfait e CSIT t ardive et imparfait e CSIR global ret ardée (les est imat ions du récept eur de la chaîne de l’aut re récept eur), et procède à des régimes act uels de géant DoF qui sont souvent serrés, et de manière const ruct ive révéler comment voire sensiblement imparfait e CSIR ret ardée global ret ardée CSIT et essent iellement imparfait , sont en fait suffi sant e pour at t eindre la performance opt imale DoF précédemment associé à perfect ionner CSIT ret ardée et CSIR global parfait . Encore d’aut res ét udes sur le t ravail de l’aspect de la diversit é de la communicat ion avec CSIT ret ardée. L’ouvrage propose un syst ème de diff usion roman qui, sur le canal de diff usion avec CSIT ret ardée, emploie une forme d’alignement d’int erférence pour obt enir à la fois DoF complet ainsi que t out e leur diversit é. Les résult at s ont ét é publiés en part ie à – Jinyuan Chen, Raymond K nopp, and Pet ros Elia, “I nt er fer ence A lignm ent for A chiev ing b ot h Full D oF and Full D iver sit y in t he B r oadcast C hannel w it h D elayed C SI T ”, in Proc. I EEE I nt. Symp. I nformation T heory (I SI T’12), July 2012.
156
C hapt er 7
Fr ench Sum m ar y
– Jinyuan Chen and Pet ros Elia, “M I M O B C w it h I m p er fect and D elayed C hannel St at e I nfor m at ion at t he Tr ansm it t er and R eceiver s”, in Proc. I EEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC’13), June 2013. Chapit re 6 décrit enfin les conclusions et les t ravaux fut urs. En plus des limit es t héoriques et de nouveaux encodeurs et décodeurs, le t ravail s’applique à gagner idées sur des quest ions prat iques sur des sujet s relat ifs à combien la qualit é de rétroact ion (diff érée, en cours ou prévus) permet une cert aine performance DoF, relat ive à l’ut ilit é de la rét roact ion ret ardée, l’ut ilit é de la CSIT prévu, l’impact des imperfect ions dans la qualit é de la CSIT courant et diff éré, l’impact des évaluat ions rapidit é et l’eff et des ret ards de rét roact ion, l’avant age d’avoir des comment aires symét rie en ut ilisant les liens de rét roact ion comparables ent re les ut ilisat eurs, l’impact des imperfect ions de la qualit é des CSIR global relat if à la façon d’at t eindre les deux DoF plein et la diversit é. En out re, dans addit on aux résult at s ci-dessus, cert ains aut res résult at s obt enus dans mon t ravail de t hèse ont ét é publiés en part ie à – Jinyuan Chen, Pet ros Elia, and Raymond K nopp, “R elay -A ided I nt er fer ence N eut r alizat ion for t he M ult iuser U plink -D ow nlink A sy m m et r ic Set t ing”, in Proc. I EEE I nt. Symp. I nformation Theory (I SI T’12), July 2011. – Jinyuan Chen, Arun Singh, Pet ros Elia and Raymond K nopp, “I nt er fer ence N eut r alizat ion for Separ at ed M ult iuser U plink D ow nlink w it h D ist r ibut ed R elay s”, in Proc. I nformation Theory and Applications Workshop (ITA), February 2011.
7.10
R ésum é du chapit r e 2
7.10 7.10.1
157
Résumé du chapit r e 2 M odèle canal
Dans ce chapit re, nous considérons les deux-ut ilisat eur MISO BC avec un M -ant enne d’émission (M 2) émet t eur communiqueant à deux ut ilisat eurs reçevant avec une seule ant enne de recept ion chacun. Le modèle de canal prend la forme de (2.1) et (2.2), ie, (1)
= h Tt x t + zt
(2)
= gTt x t + zt
yt yt
(1)
(2)
où h t , gt désigne le canal du premier et second ut ilisat eur respect ivement au t emps t, et où h t et gt sont t irés d’une dist ribut ion aléat oire, de t elle sort e que chacune a une moyenne nulle et de covariance d’ident it é (spat ialement décorrélée), et de t elle sort e que h t est linéairement indépendant de gt avec une probabilit é.
7.10.2
P r ocessus de canal et de feedback
Comme dans de nombreux scénarios de communicat ion sans fil mult iut ilisat eurs, la performance du canal de diff usion dépend de la rapidit é et de la qualit é de l’informat ion d’ét at de canal de l’émet t eur (CSIT ). Cet t e rapidit é et qualit é peuvent êt re réduit es par des liens de feedback à capacit é limit ée, ce qui peut off rir un feedback avec t oujours faible qualit é et d’import ant ret ards, c’est à dire, le feedback qui off re une représent at ion inexact e de l’ét at réel du canal, comme feedback qui ne peut êt re ut ilisé pour une fract ion suffi sant e de la durée de la communicat ion. La dégradat ion de performance correspondant e, par rapport au cas de feedback parfait sans délai, force la quest ion de delai et de qualit é de la quant it é nécessaire de la qualit é du feedback, dans le but d’at t eindre un cert ain rendement . Nous considérons ici la communicat ion d’une durée infinie n, un processus de décolorat ion du canal { h t , gt } nt= 1 t iré d’une dist ribut ion st at ist ique, et un ˆ t ,t 0} nt,t 0= 1 processus de feedback qui fournit des est imat ions de la CSIT { hˆ t ,t 0, g (du canal h t , gt ) en t out t emps t 0 - avant , pendant ou après la mat érialisat ion de h t , gt au t emps t - et ce, avec une qualit é définie par les st at ist iques de { (h t
hˆ t ,t 0), (gt
ˆ t ,t 0)} nt,t 0= 1 g
(7.7)
où l’on compt e de ces erreurs d’est imat ion d’avoir de moyenne nulle circulairement symét riques d’ent rées gaussiennes complexes.
7.10.3
N ot at ion, convent ions et hypot hèses
Nous allons ut iliser la not at ion (1)
↵t
,
lim
P! 1
log E[||h t hˆ t ,t ||2] log P
(7.8)
158
C hapt er 7 (2)
↵t ,
Fr ench Sum m ar y
ˆ t ,t ||2] log E[||gt g log P
lim
P! 1
(7.9) (1)
pour décrire l’exposant de la qualit é act uelle pour les deux ut ilisat eurs (↵ t est de ut ilisat eur 2), alors que nous allons ut iliser (1) t
,
(2) t
,
lim
log E[||h t hˆ t ,t + ⌘||2] log P
(7.10)
lim
ˆ t ,t + ⌘||2] log E[||gt g log P
(7.11)
P! 1
P! 1
- pour t out e suffi samment grand mais fini ent ier ⌘ > 0 - pour désigner la (1) qualit é des exposant s diff éré pour chaque ut ilisat eur. En d’aut res t ermes, ↵ t mesures de la qualit é de la CSIT (environ h t ) qui est disponible à l’inst ant (1) t, t andis que t mesure la qualit é (le meilleur) de la CSIT (environ h t ) qui arrive st rict ement après le canal apparaît , c’est st rict ement après le t emps t (2) (2) (même ↵ t , t pour le canal gt du second ut ilisat eur). Il est facile de voir que, sans pert e de généralit é, dans le cadre DoF d’int érêt , nous pouvons limit er not re at t ent ion à la plage 0
(i )
↵t
(i ) t
1
(i )
où t = 1 correspond à avoir (asympt ot iquement ) parfait CSIT ret ard pour (1) (2) h t , gt , et où ↵ t = ↵ t = 1, correspond au cas opt imal de la CSIT parfait e courant e (complet e). En out re, nous allons ut iliser la not at ion n
↵¯ (i ) , lim n! 1
1X (i ) ↵t , n t= 1
n
X ¯ (i ) , lim 1 n! 1 n
(i ) t ,
i = 1, 2
(7.12)
t= 1
pour désigner la moyenne des exposant s de qualit é. A ce st ade, nous const at ons que nos résult at s, en part iculier la part ie de faisabilit é, t iendront sous (1) + T l’hypot hèse que t out e suffi samment longue séquence { ↵ t } t⌧ = ⌧ (respect ive(2) ⌧ (1) ⌧ (2) ⌧ +T +T +T ment { ↵ t } t = ⌧, { t } t = ⌧, { t } t = ⌧) a une moyenne qui converge vers la moyenne à long t erme ↵¯ (1) (respect ivement ↵¯ (2) , ¯ (1) , ¯ (2) ), pour t out ⌧et , pour cert ains finie T qui peuvent êt re choisis pour êt re suffi samment large pour permet t re la convergence ci-dessus. Nous adhérons à la convent ion commune d’assumer une connaissance parfait e et globale de l’informat ion d’ét at de canal au niveau des récept eurs (CSIR global parfait ), où les récept eurs connaissent t ous les Et at s du canal et t out es les est imat ions. Nous adhérons également à la convent ion commune de supposer que l’erreur d’est imat ion act uelle est st at ist iquement indépendant e des est imat ions act uelles et passées, et par conséquent le signal d’ent rée est une fonct ion du message et de la CSIT . Cet t e hypot hèse s’accorde bien avec
7.10
R ésum é du chapit r e 2
159
de nombreux modèles de canaux s’ét endant du cqnql fast fading (i.i.d. dans le t emps), au modèle de canal corrélé car cela est considéré dans [5], le modèle fading bloc quasi-st at ique où les est imat ions de CSIT sont successivement raffi né t andis que le canal rest e st at ique. En out re, nous considérons que les ent rées de chaque vect eur d’erreur d’est imat ion h t hˆ t ,t 0 (de même de ˆ t ,t 0) à i.i.d Gaussienne, précisant cependant que nous référons seulement gt g aux M des ent rées dans chaque vect eur spécifique h t hˆ t ,t 0, et que nous ne suggérons pas que l’erreur des ent rées sont i.i.d. dans le t emps ou ent re les ut ilisat eurs.
7.10.4
R égion D oF des deux-ut ilisat eur M I SO B C
Nous procédons avec les principaux résult at s de DoF, dont il est prouvé dans la sect ion 2.4 (int érieure bound) et la sect ion 2.6 (ext erne lié). Nous rappelons ici que le lect eur des séquences (1)
(2)
{ ↵ t } nt= 1, { ↵ t } nt= 1 , {
(1) n t } t = 1, {
(2) n t } t= 1
de qualit é des exposant s, que ceux-ci ont ét é définis dans (7.8)-(7.11), ainsi que des moyennes correspondant es ↵¯ (1) , ↵¯ (2) , ¯ (1) , ¯ (2) de (7.12). Nous rappelons aussi au lect eur que nous considérons la communicat ion sur une grande durée n. Nous appelons désormais les ut ilisat eurs de sort e que ↵¯ (2) ↵¯ (1) . L’ext ension du t ravail dans [6] qui met l’accent sur CSIT avec une qualit é invariant e et symét rique, on procède d’abord par const ruire un nouveau DoF ext erne lié qui sout ient not re réglage. La preuve se t rouve dans la Sect ion 2.6. L em m a 7. La région DoF des deux-utilisateur MISO BC avec un procesˆ t ,t 0} nt= 1,t 0= 1 de qualité { (h t hˆ t ,t 0), (gt g ˆ t ,t 0)} nt= 1,t 0= 1, est sus CSI T { hˆ t ,t 0, g supérieure délimitée comme 1
(7.13)
2d1 + d2
2 + ↵¯ (1)
(7.14)
2d2 + d1
(2)
d1
1,
d2 2 + ↵¯
.
(7.15)
Le t héorème suivant fournit la DoF opt imale pour un large int ervalle de suffi samment bien diff éré CSIT . T heor em 8. La région optimal de la DoF des deux-utilisateur MISO BC ˆ t ,t 0} nt= 1,t 0= 1 de qualité { (h t avec un processus CSI T { hˆ t ,t 0, g hˆ t ,t 0), (gt
160
C hapt er 7
Fr ench Sum m ar y
ˆ t ,t 0)} nt= 1,t 0= 1 est donnée par g d1
1,
d2
1
(7.16)
2d1 + d2
2 + ↵¯ (1)
(7.17)
2d2 + d1
(2)
(7.18)
2 + ↵¯
pour tout processus CSI T suffi samment bien diff éré tels que min{ ¯ (1) , ¯ (2) }
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , }. 3 2
Figure 2.2 correspond au résult at principal dans le t héorème. Le résult at ci-dessus est complét é par la proposit ion suivant e. ˆ t ,t 0} nt= 1,t 0= 1 par min{ ¯ (1) , ¯ (2) } P r op osit ion 9. Pour un processus CSIT { hˆ t ,t 0, g ( 1)
( 2)
< min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ lygone décrit par
( 2)
} , la région DoF est intérieure délimitée par le po-
d1
1,
d2
1
(7.19)
2d1 + d2
2 + ↵¯ (1)
(7.20)
2d2 + d1
(2)
(7.21)
d1 + d2
2 + ↵¯
1 + min{ ¯ (1) , ¯ (2) } .
(7.22)
Figure 7.4 correspond au résult at de la Proposit ion 9. Avant de procéder àdes corollaires spécifiques qui off rent un meilleur aperçu, il vaut la peine de faire un comment aire sur le fait que t out e la complexit é du problème est capt uré au niveau de la qualit é des exposant s. Feedback sy m ét r ique et asy m ét r ique Nous part ons à la découvert e du cas part iculier de feedback symét rique où la qualit é de feedback accumulé est similaire ent re les ut ilisat eurs, c’est à dire, lorsque les liens de feedback de l’ut ilisat eur 1 et l’ut ilisat eur 2 part agent les mêmes moyennes des exposant s ↵¯ (1) = ↵¯ (2) = ↵¯ ,
¯ (1) = ¯ (2) = ¯ .
La plupart des ouvrages exist ant s, avec une except ion dans [33], relèvent ce paramèt re. Ce qui suit t ient direct ement du t héorème 8 et la Proposit ion 9. C or ollar y 8a (DoF en réact ion symét rique). La région de la DoF optimale pour le cas symétrique prend la forme d1
1,
d2
1,
2d1 + d2
2 + ↵¯ ,
2d2 + d1
2 + ↵¯
7.10
R ésum é du chapit r e 2
161
d2
d2
(1)
d2 2d1 2
d2 2d1
B
B
1
1
(1)
2
A
C
d1 2d2
C
( 2)
2
d1 2d2
2
( 2)
D
0
1
d1
(1)
2
0
2
2
(a) Case 1:
2
(1)
d1
(1)
2
1
(2)
(b) Case 2:
1
2
(1)
( 2)
1
Figur e 7.3 – Régions DoF opt imal pour les deux ut ilisat eurs MISO BC, ( 1) ( 2) ( 2) min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . Les point s d’angle pour le cas de min{ ¯ (1) , ¯ (2) } ( 2) prennent les valeurs suivant es : A = (1, 1+ ↵2¯ ), B = ( ↵¯ (2) , 1), ( 1) ( 2) ( 2) ( 1) C = ( 2+ 2¯↵ 3 ↵¯ , 2+ 2¯↵ 3 ↵¯ ), D = (1, ↵¯ (1) ).
d2
d2
B 1
B
1
E
E A
C
G
F D
d1
0
1
(a) Case 1:
2
(1)
(2)
1
d1
0
1
(b) Case 2:
2
(1)
(2)
1
Figur e 7.4 – Régions DoF pour les deux ut ilisat eurs MISO BC avec ( 1) ( 2) ( 2) min{ 1+ ↵¯ 3+ ↵¯ , 1+ ↵2¯ } . Les point s d’angle prennent les min{ ¯ (1) , ¯ (2) } valeurs suivant es : E = (2¯ ↵¯ (2) , 1 + ↵¯ (2) ¯ ), F = (1 + ↵¯ (1) ¯ , 2¯ ↵¯ (1) ), ( 1) ( 2) ( 2) G = (1, ¯ ), ¯ , min{ ¯ (1) , ¯ (2) , 1+ ↵¯ + ↵¯ , 1+ ↵¯ } . 3
2
162
C hapt er 7
Fr ench Sum m ar y
No CSI T Delayed CSI T [MAT]
d2 1
B=
Current + delayed CSI T
(1 2 ) / 3
Current + delayed CSI T
(1 2 ) / 3
,1
E
C= 2
3
,2
3
d1 d2 1
2/3
F D = 1,
0
d1
1
2/3
Figur e 7.5 – Région DoF deux ut ilisat eurs MISO BC avec feedback symét rique, ↵¯ (1) = ↵¯ (2) = ↵¯ , ¯ (1) = ¯ (2) = ¯ . La zone opt imale prend la forme d’un polygone avec des point s d’angle 1+ 2¯↵ { (0, 0), (0, 1), ( ↵¯ , 1), ( 2+3 ↵¯ , 2+3 ↵¯ ), (1, ↵¯ ), (1, 0)} pour ¯ 3 . Pour 1+ 2¯ ↵ ¯ < , la région dérivée prend la forme d’un polygone avec des point s 3 d’angle ¯ ), (1 + ↵¯ ¯ , 2 ¯ ↵¯ ), (1, ↵¯ ), (1, 0)} . { (0, 0), (0, 1), ( ↵¯ , 1), (2 ¯ ↵¯ , 1 + ↵¯ 1+ 2¯↵ ¯ < lorsque ¯ 3 , tandis que la région réalisable
1+ 2¯↵ 3
1
(7.23)
2d1 + d2
2 + ↵¯
(7.24)
2d2 + d1
2 + ↵¯ 1+ ¯.
(7.25)
d1
1,
cette région intérieure délimitée par
d2 + d1
d2
(7.26)
Figure 7.5 représent e la région DoF des deux-ut ilisat eur MISO BC avec feedback symét rique. Nous mesurons maint enant la mesure pour laquelle ayant un feedback symét rique off re un avant age sur le cas asymét rique où un ut ilisat eur a généralement plus de feedback que l’aut re. Diff érent s t ravaux ont ident ifié des cas où ayant des off res de feedback symét riques (gains de symét rie) par rapport au cas asymét rique (cf. [11], [33]). La comparaison générale qui suit se concent re sur le cas de parfait e CSIT diff érée, et oppose le cas symét rique ↵¯ (1) = ↵¯ (2) , au cas asymét rique ↵¯ (1) 6 = ↵¯ (2) , sous une cont raint e globale de feedback ↵¯ (1) + ↵¯ (2) = , pour t out 2 (0, 2]. La comparaison est en fonct ion de la somme-DoF opt imal d1 + d2,
7.10
R ésum é du chapit r e 2
163
où, nous rappelons que les ut ilisat eurs sont ident ifiés de sort e que ↵¯ (1) La preuve est direct e du t héorème 8 et le corollaire 8a.
↵¯ (2) .
C or ollar y 8b (Rét roact ion symét rique et asymét rique). Considérons tout , ↵¯ (1) + ↵¯ (2) fixe dans l’intervalle (0, 2]. Si 2¯↵ (1) ↵¯ (2) 1 0, ayant un feedback symétrique ( ↵¯ (1) = ↵¯ (2) ) n’off re pas un gain somme-DoF sur le cas de feedback asymétrique, tandis que si 2¯↵ (1) ↵¯ (2) 1 > 0, il ya un gain ( 1) ( 2) 1 somme-DoF symétrique et elle prend la forme 2¯↵ 6↵¯ . Exam ple 22. Par exemple, sous la contrainte que ↵¯ (1) + ↵¯ (2) = = 1, (1) (2) l’asymétrie de ↵¯ = 1, ↵¯ = 0 donne une somme optimale DoF de d1+ d2 = 3/ 2 (Theorem 8 avec une parfaite CSIT diff érée), tandis que le symétrique ↵¯ (1) = ↵¯ (2) = 0.5 donne d1 + d2 = 5/ 3, et un gain somme DoF de 5/ 3 3/ 2 = 1/ 6.
B esoin d’infor m at ions r et ar dé : C SI T I m par fait v s C SI T R et ar dé Nous mont rons ici que le CSIT imparfait ret ardé peut êt re aussi ut ile que le CSIT parfait ret ardé, et nous fournissons un aperçu de la qualit é de la réact ion globale (en t emps opport un et en diff éré) qui est nécessaire pour at teindre un cert ain rendement du DoF. C or ollar y 8c (CSIT imparfait vs parfait ret ardé). Un processus CSI T { hˆ t ,t 0, ˆ t ,t 0} nt= 1,t 0= 1 qui off re g min{ ¯ (1) , ¯ (2) }
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , } 3 2
(7.27)
Donne le même DoF qu’un processus CSI T qui off re un CSI T parfait retardé (1) (2) pour chaque réalisation de canal ( t = t = 1, 8t). Pour le cas symétrique, avoir 1 + 2¯↵ ¯ (7.28) 3 donne le même résultat. Il est int éressant de not er que les expressions dans le corollaire ci-dessus correspondent au mont ant du CSIT ret ardé utilisé par les schemas de t ransmission dans le passé, même si ceux-ci n’ont pas ét é conçus dans le but explicit e de réduire la quant it é de CSIT ret ardé. B esoin de C SI T pr édit Nous changeons maint enant l’accent du CSIT ret ardé pour l’aut re ext rême du CSIT prédit . Comme nous le rappelons, nous avons considéré un
164
C hapt er 7
Fr ench Sum m ar y
ˆ t ,t 0} t ,t 0, composé processus de canal { h t , gt } t et un processus CSIT { hˆ t ,t 0, g 0 ˆ 0 d’est imat ions h t ,t - disponibles en tout t emps t - du canal h t qui se mat érialise en t out t emps t. Nous avons également préconisé que nous pouvons supposer que E[||h t hˆ t ,t 0||2] E[||h t hˆ t ,t 00||2], pour t out t 0 > t 00, t out simplement parce que l’on peut revenir à des est imat ions ant érieures de st at ist iques de meilleure qualit é. Cet t e hypot hèse n’a cependant pas obst acle à l’ut ilit é possible des premières est imat ions (prévue), même si ces est imat ions sont généralement de moindre qualité que les est imat ions act uelles (de moins bonne qualit é que les est imat ions qui apparaissent pendant ou après le canal mat érialise). Ce qui suit donne un aperçu sur ce sujet . C or ollar y 8d (Besoin de CSIT prédit ). Si le CSI T retardé vérifie min{ ¯ (1) , ¯ (2) }
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , } 3 2
il n’y a pas de gain de DoF à utiliser du CSI T prédit. Plus précisément - pour une qualité de CSIT , et afin de parvenir au DoF optimal - la transmission à un certain moment t ⇤, n’a pas besoin d’examiner toute estimation hˆ t ,t 0 d’un canal à venir t > t ⇤, où cette estimation est devenu disponible - naturellement par prédiction - à tout moment t 0 t ⇤ < t.
7.10.5
CSI T evoluant per iodiquem ent
Nous nous concent rons ici sur des canaux variant s par block, et considérons une accumulat ion progressive de CSIT qui se t raduit par une qualit é de CSIT augment ant progressivement lorsque le t emps progresse à t ravers la période de la cohérence (Tc usages des canaux - CSIT courant ), ou à t out moment après la fin de la période de la cohérence (CSIT ret ardé). Cet t e améliorat ion progressive pourrait êt re recherchée dans les milieux FDD (Frequency Division Duplex) avec des liens de ret our à capacit é limit ée qui peuvent êt re ut ilisés plus d’une fois au cours de la période de cohérence afin d’affi ner progressivement le CSIT , ainsi que dans T DD (T ime Division Duplex) les paramèt res qui ut ilisent la réciprocit é basé sur l’est imat ion qui améliore progressivement au fil du temps. Dans ce cadre, le canal rest e le même pour une durée limit ée de Tc ut ilisat ions de canal, et l’indice de t emps est agencé de t elle sort e que h ` Tc + 1 = h ` Tc + 2 = · · · = h (` + 1)Tc g` Tc + 1 = g` Tc + 2 = · · · = g(` + 1)Tc pour un ent ier non négat if `. En out re la qualit é de CSIT est maint enant périodique, comme cela se reflèt e dans les exposant s de qualit é courant e CSIT où (i ) (i ) ↵ t = ↵ ` Tc + t , 8` = 0, 1, 2, · · · , i = 1, 2. (7.29)
7.10
R ésum é du chapit r e 2
165
Nous nous concent rons ici principalement sur le cas symét rique. Dans ce cadre la qualit é de rét roact ion est maint enant représent é par le Tc courant CSIT qualit é des exposant s { ↵ t } Tt =c 1 et par le ret ard CSIT exposant . Chaque ↵ t décrit la qualit é des est imat ions du CSIT act uel au t emps t Tc, alors capt e la qualit é des meilleures est imat ions de CSIT qui sont reçus après que le canal se soit écoulée, soit après la durée de cohérence du canal. Dans ce cadre nous avons que ↵1
0
···
↵ Tc
1
(7.30)
où une diff érence ent re les deux exposant s consécut ifs est due à l’informat ion qui a ét é reçu au cours de cet t e t ranche de t emps. Ce même cadre saisit aussi le moment de ce ret our périodique. Ayant par exemple ↵ 1 = 1 se réfère simplement pour le cas de parfait e (complèt e) CSIT , alors ayant ↵ Tc = 0 signifie simplement qu’aucune rét roact ion (ou t rès peu) est envoyée pendant la période de la cohérence du canal. Ayant même ↵ Tc = 0, 2 [0, 1] signifie simplement qu’aucune rét roact ion (ou t rès peu) est envoyée au cours de la première fract ion de la période de cohérence. Par exemple, avoir un processus de rét roact ion périodique qui envoie par exemple des informat ions deux fois par période de cohérence, pour t = 1Tc + 1, t = 2Tc + 1 et plus jamais sur ce même canal, résult e en avant comment aires
z }| 0 = ↵1 = · · · = ↵
{
1 Tc
A près la première réact ion
z ↵
1 Tc +
1
}| = ··· = ↵
↵ |
{
2 Tc 2 Tc +
1
= · · · = ↵ Tc = {z }
(7.31)
A près la deuxième réact ion
t andis que si le même syst ème de rét roact ion ajout e quelques informat ions après l’écoulement du canal, puis nous avons t out simplement que ↵ Tc . On peut not er que la réduct ion de ↵ Tc implique une quant it é réduit e d’informat ions, sur un canal spécifique, qui est envoyé au cours de la période de la cohérence de ce même canal, t andis que la réduct ion de implique une quant it é réduit e d’informat ions, au cours de ou après la période de la cohérence du canal. Le long de ces lignes, la réduct ion de ( ↵ Tc ) implique une quant it é réduit e d’informat ions, sur un coeffi cient d’évanouissement spécifique, qui est envoyé après la période de cohérence du canal. Les résult at s suivant suivent direct ement à part ir des résult at s ant érieurs dans ce t ravail, où nous avons maint enant simplement mis T
↵¯ =
1 Xc ↵t . Tc
(7.32)
t= 1
Ce qui suit t ient direct ement du corollaire 8a, dans le cas d’un processus de rét roact ion avec évolut ion périodiquement sur un canal quasi-st at ique.
166
C hapt er 7
Fr ench Sum m ar y
C or ollar y 8e (Comment aires sur l’évolut ion périodique). Pour un processus de rétroaction périodique avec { ↵ t } Tt =c 1 et parfait CSI T retardée (reçue à tout moment après la fin de la période de cohérence), la région de la DoF optimale sur un bloc-canal avec évanouissement est le polygone de sommets 2 + ↵¯ 2 + ↵¯ { (0, 0), (0, 1), ( ↵¯ , 1), ( , ), (1, ↵¯ ), (1, 0)} . (7.33) 3 3 Cette même région optimal peut en fait être atteint même avec retard CSIT 1+ 2¯↵ imparfait qualité, aussi longtemps que 3 . R em ar k 8 (Comment aires sur la dépendance qualit é versus quant it é). Bien que les résultats sont ici en termes de qualité de rétroaction plutôt qu’en termes de quantité de feedback, il ya des cas distincts où la relation entre les deux est bien défini. Tel est le cas lorsque les estimations de la CSI T sont calculés à l’aide de base - et pas nécessairement optimale - techniques de quantification scalaires [39]. Dans de tels cas, que nous mentionnons ici simplement pour off rir un aperçu, en consacrant ↵ log P de quantification par scalaire pour quantifier h en une estimation hˆ , permet à une erreur quadratique moyenne [39] . Ekh hˆ k2 = P ↵ . S’inspirant de cela, et de revenir à not re exemple précédent , considérons un processus de rét roact ion périodique qui envoie raffi nage informat ions deux fois par période de cohérence, en envoyant d’abord ↵ 0log P bit s d’informat ions par scalaire au t emps t = 1Tc + 1, puis l’envoi supplément aire ↵ 00log P bit s d’informat ions par scalaire au temps t = 2Tc + 1, et où il envoie finalement ( (↵ 0+ ↵ 00)) log P les bit s supplément aires de raffi nage comment aires par scalaire, à n’import e quel moment après la période de la cohérence d’un canal. Cela about irait à avoir Before feedback
z }| 0 = ↵1 = · · · = ↵
{ 1 Tc
0
A ft er first feedback
z ↵0= ↵
}| 1 Tc + 1 = · · · = ↵
{ 2 Tc
00
↵ + ↵ = ↵ 2 Tc + 1 = · · · = ↵ T c | {z }
| { z}
A ft er second feedback, before Tc
A ft er coherence period
. (7.34)
À t it re d’exemple, ayant évaluations périodiques qui envoie 49 log P bit s d’informat ions par scalaire au t emps t = 13 Tc + 1, puis envoie supplément aire 1 2 9 log P bit s d’informat ions par scalaire au t emps t = 3 Tc + 1, se t raduira par A ft er first feedback
Before feedback
z }| { 0 = ↵ 1 = · · · = ↵ 1 Tc 3
z }| { 4 = ↵ 1 Tc + 1 = · · · = ↵ 2 Tc 3 3 9 5 = ↵ 2 Tc + 1 = · · · = ↵ Tc 3 |9 {z } A ft er second feedback, before Tc
(7.35)
7.10
R ésum é du chapit r e 2
167
ce qui donne ↵¯ = (0 + 4/ 9 + 5/ 9)/ 3 = 1/ 3, qui à son t our donne une région d’DoF opt imale qui est définie par le polygone avec des point s d’angle { (0, 0), (0, 1), (1/ 3, 1), (7/ 9, 7/ 9), (1, 1/ 3), (1, 0)} .
(7.36)
On a sans dout e remarqué qu’il n’y avait pas besoin de bit s supplément aires de (ret ardée) comment aires après la fin de la période de cohérence. C’est parce que le mont ant et le calendrier act uel des bit s de rét roact ion déjà autorisées pour 1 + 2¯↵ 1 + 2/ 3 = ↵ Tc = 5/ 9 = = 3 3 que nous avons vu dans le corollaire 1f soit aussi bon que parfait réact ion ret ardée. Placer not re focus sur la qualit é des comment aires, nous procédons à un corollaire qui off re un aperçu sur la quest ion de ce qu’est la qualit é CSIT et le chronomét rage suffi sent pour at t eindre un cert ain rendement du DoF. Pour facilit er l’exposé, nous nous concent rons sur le point DoF plus diffi ciles à at t eindre d1 = d2 = d. C or ollar y 8f (Suffi samment d’informat ions pour cible DoF). Avoir ↵¯ 3d 2 avec 2d 1, ou d’avoir ↵¯ 3d 2 avec ↵ Tc 2d 1 (et pas de retour retardé en sus), suffi t à atteindre une cible symétrique DoF d1 = d2 = d. Un aut re aspect prat ique qui est abordé ici, a à voir avec les ret ards de rét roact ion. Ces ret ards peuvent ent raîner une dégradat ion des performances, ce qui pourrait êt re at t énué si le ret our, mais avec des ret ards, est de suffi samment bonne qualit é. Le corollaire suivant donne un aperçu sur ces aspect s, en décrivant les ret ards de rét roact ion qui permet t ent pour une cible donnée symét rique DoF d en présence de cont raint es sur les qualit és CSIT courant s et diff érés. Nous serons part iculièrement int éressés par le ret ard fract ionnaire admissible de rét roact ion (cf. [10]) , arg max {↵ 0 la fract ion
c
= 0}
= 0, ↵
Tc + 1
0T
(7.37)
1 pour lesquels ↵1 = · · · = ↵
Tc
> 0.
Nous sommes également int éressés de voir comment ce ret ard admissible réduit en présence d’une cont raint e ↵ t ↵ max 8 t sur la rét roact ion en t emps opport un, ou une cont raint e sur .
168
C hapt er 7
Fr ench Sum m ar y
C or ollar y 8g (Délai de ret our admissible). Sous une contrainte de qualité de la CSI T ↵ t ↵ max 8t, un DoF cible symétrique d peut être réalisé avec n’importe quel retard fractionnaire ( 1 1
3d 2 ↵ m ax
si d 2 [2/ 3, (2 + ↵ max )/ 3] si d 2 [0, 2/ 3]
alors que sous une contrainte ( 1 1 1 2 2d 1
1
max ,
il peut être réalisé avec toute
si d 2 [0, 1+ min{ 2m ax ,1/ 3} ] autre si d 2 [ 1+ min{ 2m ax ,1/ 3} , 1+
m ax
2
]
Enfin depuis ↵ max 1, ce qui précède montre que sous aucune max contrainte spécifique sur la qualité de la CSI T , d peut être réalisé avec ( 3(1 1
d)
if d 2 [2/ 3, 1] if d 2 [0, 2/ 3].
Nous allons maint enant voir comment le réglage de rét roact ion évoluer périodiquement , int ègre diff érent s paramèt res exist ant s d’int érêt .
7.10.6
L e r églage évolut ion p ér iodiquem ent comm e une génér alisat ion de par amèt r es exist ant s
Ce paramèt re de rét roact ion périodiquement l’évolut ion est puissant e car elle capt e les nombreuses opt ions t echniques que l’on peut avoir dans un cadre bloc-fading lorsque la nat ure des informat ions demeure largement inchangée à t ravers les périodes de cohérence. Il capt ure aussi et généralise les paramèt res exist ant s qui ont ét é d’un int érêt part iculier, comme le Maddah-Ali et le réglage Tse dans [4], le Yang et al. et Gou et le réglage Jafar dans [6,7], le réglage Lee et Heat h dans [10], et le réglage asymét rique dans [33]. Nous allons met t re en lumière cert aines de ces généralisat ions pour diff érent s paramèt res exist ant s d’int érêt . C SI T seulem ent r et ar dé - M addah-A li et Tse Le réglage Maddah-Ali et Tse dans [4] correspond à la mise en évolut ion avec ↵ t = 0, t = 1, 2, · · · , Tc et avec une parfait e CSIT ret ardée. Une généralisat ion direct e de [4], qui s’inscrit dans le cadre évolut if act uel, est de considérer CSIT ret ardé par diminut ion de la qualit é . De cet t e généralisat ion, nous savons maint enant que la même performance DoF de { (0, 0), (0, 1), ( 23 , 23 ), (1, 0)} , peut êt re at t eint même si la CSIT ret ardé envoyé, est de qualit é imparfait e, correspondant à t out e 1/ 3.
7.10
R ésum é du chapit r e 2
169
C SI T r et ar d avec C SI T act uel im par fait - Yang et al. et G ou et Jafar De même, le Yang et al. et Gou et la mise en Jafar [6, 7], correspond à la mise en évoluant avec ↵ t = ↵, t = 1, 2, · · · , Tc et avec une parfait e CSIT ret ardée. Comme ci-dessus, une généralisat ion direct e consist e à considérer la qualit é imparfait e ret ardé CSIT , et pour cela, nous savons que la DoF opt imal { (0, 0), (0, 1), ( 2+3 ↵ , 2+3 ↵ ), (1, 0)} peut êt re réalisé pour t out e combinaison des exposant s de qualit é CSIT qui donnent ↵¯ = ↵, et même avec la qualit é de la CSIT ret ard imparfait pour t out (1 + 2↵)/ 3. Pas si r et ar dé C SI T - L ee et H eat h La mise en [10] considère parfait e rét roact ion ret ardée et rét roact ion parfaite courant qui vient avec un ret ard fract ionnaire 2 [0, 1] de la période de la cohérence, c’est à dire, il considère que la rét roact ion parfait arrive t oujours Tc canal ut ilise après la réalisat ion du canal. Ce paramèt re - en met t ant l’accent ici sur le cas de deux ut ilisat eurs - puis des cart es au réglage de la CSIT évolut ion périodiquement avec une parfait e CSIT ret ard et avec ↵1 = · · · = ↵ {z |
Tc
= 0, ↵ } |
No feedback
Tc + 1
= · · · = ↵ Tc = 1 . {z }
(7.38)
Perfect CSI T
Cert aines généralisat ions prat iques ont ét é considérées dans le corollaire 1h qui décrit le ret ard maximal possible nécessaires pour réaliser une performance de DoF spécifique, sous des cont raint es sur la qualit é CSIT . C SI T r et ar dé par C SI T act uel p our un seul ut ilisat eur - M alek i, Jafar et Sham ai Le réglage évolut ion peut êt re nat urellement ét endu à la créat ion asymét rique où ↵¯ (1) 6 = ↵¯ (2) et où les exposant s de la CSIT ret ard (1) , (2) n’a pas besoin d’êt re égaux. Une t elle configurat ion asymét rique donnerait une généralisat ion pour le réglage asymét rique de Maleki et al. dans [33], où les ut ilisat eurs off ert s parfait CSIT ret ard, et où seul le premier ut ilisat eur off ert CSIT de courant parfait e, résult ant en une DoF opt imale correspondant à DoF point d’angle (1, 1/ 2) (somme DoF d1 + d2 = 3/ 2). Ce paramèt re correspond à la créat ion de la CSIT évolut ion périodiquement avec une parfait e CSIT ret ard et avec (1) (2) ↵ t = 1, ↵ t = 0, 8t. Le corollaire suivant propose une généralisat ion du résult at correspondant à [33]. La preuve est direct e depuis les suivant es adapt e simplement le résult at dans le t héorème principal, à la mise en évolut ion périodiquement . C’est nouveau pour la fixat ion d’assez bonne CSIT ret ardé par min{
(1)
,
(2)
}
min{
1 + ↵¯ (1) + ↵¯ (2) 1 + ↵¯ (2) , }. 3 2
170
C hapt er 7
Fr ench Sum m ar y
C or ollar y 8h (CSIT asymét rique et périodiques). La région de la DoF opti( 1) ( 2) ( 2) ( 1) mal est défini par des sommets B = ( ↵¯ (2) , 1), C = ( 2+ 2¯↵ 3 ↵¯ , 2+ 2¯↵ 3 ↵¯ ), ( 2) D = (1, ↵¯ (1) ) lorsque 2¯↵ (1) ↵¯ (2) < 1, sinon par des sommets A = (1, 1+ ↵2¯ ) Et B .
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