Bounds on the Second-Order Coding Rate of the MIMO Rayleigh Block-Fading Channel

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1 Bouds o th Scod-Od Codig Rat of th MIMO Rayligh Block-Fadig Chal Jakob Hoydis Bll Laboatois, Alcatl-Luct Stuttgat, Gmay jakob.hoydis@alcatl-luct.com Romai Couillt Dpt. of Tlcommuicatios SUPELEC, Fac omai.couillt@suplc.f Pablo Piataida Dpt. of Tlcommuicatios SUPELEC, Fac pablo.piataida@suplc.f Abstact W study th scod-od codig at of th multipl-iput multipl-output MIMO Rayligh block-fadig chal via statistical bouds fom ifomatio spctum mthods ad adom matix thoy. Basd o a asymptotic aalysis of th mutual ifomatio dsity which cosids th simultaous gowth of th block lgth ad th umb of tasmit ad civ atas K ad N, w div closd-fom upp ad low bouds o th optimal avag o pobability wh th cod at is withi O/ K of th asymptotic capacity. A Gaussia appoximatio is th usd to stablish a upp boud o th o pobability fo abitay cod ats which is show by simulatios to b accuat fo small N, K, ad. I. INTRODUCTION Tha chal capacity dscibs th maximal at at which data xchag with vaishig o pobability is possibl, povidd that th lgth of ach codwod is allowd to gow without limit. By focusig o this asymptotic limit, th thoy of ifomatio igos th ol of dlay as a impotat paamt. Although th chal capacity ad th latd Shao thoy costitut a scitific succss stoy [], i may al-wold applicatios, lativly shot block lgths a quid du to ith dlay ad/o complxity costaits. Thus, it is impotat to aalyz th quid back-off fom capacity to guaat a ctai o pobability fo a giv block lgth. Ufotuatly, i th fiit block-lgth gim, th a o xact tactabl fomulas to facilitat th aalysis. This is i paticula th cas fo pactical quasi-static fadig multipl-iput multipl-output MIMO chals. Amog th fist to ivstigat bouds o th o pobability fo a giv codig at w Fisti ad Shao [], [] who stablishd th covgc of th optimal at to th capacity with gowig block lgth. Ths bouds a basd o th fudamtal latio btw th ifomatio dsity [3], th codig at, ad th o pobability. Th optimal xpotial at of dcas of th o pobability was divd by Gallag [4]. Howv, his sult dos ot povid th bst upp boud o th avag o pobability i chal codig wh th tasmissio at is lag tha th capacity [5]. Focusig o th chaactizatio of chal capacity via this statistical appoach, Stass [6] divd a gal xpssio fo th disct mmoylss chal with ucostaid iputs, wh th otio of omal appoximatio of th scod-od codig at was itoducd fo th fist tim. Ufotuatly, Stass s appoach caot b galizd to chals with iput costaits.g., th Gaussia ad th fadig MIMO chals. Futh wok o th asymptotic block-lgth gim via ifomatio-spctum mthods stablishd th most gal capacity fomula [7] which quid a ovl statistical boud [8], [9] i th covs poof. Th authos of [0], [5] ivstigatd th fiit blocklgth gim ad th scod-od codig at of sval chal modls i psc of cost o iput costaits. Alog th sam lis, th scala additiv whit Gaussia ois AWGN block-fadig chal was addssd i th coht ad o-coht sttigs i [] ad [], spctivly. I this pap, w ivstigat th fiit block-lgth gim of th MIMO Rayligh block-fadig chal. Th cas of study is mad difficult bcaus th chal dos ot satisfy th godic quimts to apply th usual tools fo th aalysis of th scod-od codig at. Followig ou fist cotibutio i [3], w thfo study th asymptotic bhavio of th o pobability wh th codig at is a ptubatio of od O/ K of th asymptotic capacity whil th block-lgth, ad th umb of tasmit ad civ atas K ad N, spctivly, gow ifiitly lag at th sam at. I this asymptotic gim, w stablish a w low ad upp boud o th optimal avag o pobability ad apply thm to obtai a upp boud appoximatio fo fiit. II. SYSTEM MODEL AND PROBLEM STATEMENT Cosid th followig MIMO fadig chal: y t = K H x t + σw t, t =,..., } wh y t C N is th chal output at tim t, H is a alizatio of th adom chal matix H C N K whos tis a idpdt ad idtically distibutd i.i.d. CN 0, ad th idx is usd to mid that H is costat fo th duatio of chal uss, x t is th alizatio of th adom chal iput x t C K at tim t, ad σw t is th alizatio of th adom ois vcto σw t C N at tim t whos tis a i.i.d. CN 0, σ. Th tasmitt has oly statistical kowldg about H whil th civ kows H pfctly. I paticula, w will assum H, x t, ad w t to b idpdt fo ach t. W dfi X = x x C K, W = w w C N, ad Y = y y C N, ad thi adom coutpats X = x x C K,

2 W = w w C N, ad Y = y y C N. Fo Γ > 0, w futh dfi } SΓ = X C K K t X X H Γ i.., th st of iputs X with gy costait Γ. Th mutual ifomatio dsity of P Y X, i.., th pobability masu of Y coditiod o X, is dfid by [3] I = K log P Y XdY X P Y dy wh, fo a giv X, P Y X X /P Y is th Rado Nykodym divativ of P Y X, X with spct to P Y if P Y X, X P Y ad is st to othwis. Of paticula impotac is th cas of idpdt Gaussia iputs, i.., x t CN 0, ΓI K, fo which th mutual ifomatio dsity taks th fom wh C = K log dt R = K t 3 I = C + R 4 I N + Γ σ K H H H Γ K H H H + σ I N Y Y H 5 W W H]. 6 Not that such iputs do ot spct th gy costait. Dfiitio : A P, M, Γ-cod C fo th chal modl with pow costait cosists of: A cod mappig: ϕ : M C K. Th tasmittd symbols a X m = ϕm SΓ fo vy mssag m uifomly distibutd ov th st M =,..., M } of mssags. A dcod mappig: φ H : C N M } which poducs th dcod s dcisio ˆm = φ H Ym, Ym = K H X m + σw, o th tasmittd mssag m, o th o vt. Fo a cod C with block lgth, codbook siz M, cod ϕ, ad dcod φ H } H CN K, th avag o pobability is dfid as P = P C = M M m= P [ ˆm m m ] wh th pobability is tak ov th adom vaiabls W ad H. Lt suppc dot th codbook ϕ,..., ϕm }. Th optimal avag o pobability fo th at R with gy costait Γ is dfid as P R = if C :suppc S Γ P 7 C log M } K R. 8 Th xact chaactizatio of P R fo fixd, K, N is gally itactabl. I th cas of statioay disct mmoylss sigl-iput sigl-output SISO additiv whit Gaussia ois AWGN chals, th hav b ct ffots [5] s also [0] ad [6] to stablish o pobability appoximatios wh th codig at is withi O/ of th godic capacity as gows lag. Howv, immdiat xtsios of ths sults to block-fadig chals, as cosidd i this pap, a ot possibl bcaus th godic capacity fo ths chals is ot dfid. To cicumvt this issu, [] assums codig ov a lag umb of idpdt alizatios of icasigly lag block-fadig chals, which maks th ovall chal godic th aticl is actually stictd to SISO chals but would asily adapt to th MIMO cas. I th pst aticl, w tak th appoach of iducig godicity by gowig th chal matix dimsios. Idd, lttig K, N, th i.i.d. stuctu of H maks th chal godic i th limit. Pcisly, w assum h that K, N, whil K = β ad N K = c fo som costats β, c > 0. This will b dotd by β,c. I this limitig gim, th p-ata capacity of th chal covgs fo almost vy chal alizatio to a asymptotic limit C. W ca th chaactiz th o pobability i th scod-od codig at, i.., wh th codig at is withi O/ K of th limitig capacity C. With ths assumptios, simila to [5], [0], P R is placd by th followig tactabl limitig o pobability: Dfiitio : Th optimal avag o pobability fo th scod-od codig at with iput gy costait Γ is wh P β, c, Γ = if C } = suppc S lim sup P C β,c } lim if K β,c K log M C 9 C = lim if E[C ]. 0 β,c Without loss of gality ad fo simplicity, w tak Γ = ad dot S = S ad P β, c = P β, c,. III. MAIN RESULTS A. Bouds o th optimal avag o pobability Thom Bouds o P β, c: Fo x > 0 ad c > 0, dfi δ 0 x = c x c + + x + 4cx x with divativ δ 0x = δ 0x + δ 0 x c + x + xδ 0 x ad dot, fo σ > 0, θ = β log c = β log c δ 0 σ + δ 0 σ δ 0 σ + δ 0 σ + c + σ 4 δ 0σ ] + c σ δ 0 σ ].

3 Th, fo th chal modl with uit iput gy costait, th optimal avag o pobability P β, c satisfis: If 0, P β, c θ If > 0, P β, c. Poof: Th full poof is povidd i [4]. Although th thom povids closd-fom bouds o P β, c, it must ot hid th fact that its poof is quit ivolvd. Ou appoach follows closly Hayashi s mthod [5]. Th majo difficulty ad tchical cotibutio li i th thoough aalysis of th asymptotic statistics of I ud difft assumptios o th distibutio P X of X. W mak xtsiv us of tools fom adom matix thoy, spcially th chaactistic fuctio appoach du to Pastu, s,.g., [5], alog with th itgatio by pats fomula fo Gaussia vctos ad th Poicaé Nash iquality. I cotast to th usual sttig of lag adom matix thoy, bcaus of, X X H is oly boudd i tac ath tha i spctal om. This complicats th aalysis at may occasios. I a utshll, fo th low boud, w pov that I X wh IX is I coditiod o a paticula X has a vaiac which scals as O+ K t A with A = I X X H, which ca th gow ifiitly lag o ot, dpdig o X S. W th show that if ε K t X X H, fo som ε small, I, wh poply ctd ad scald, satisfis a ctal limit thom CLT. Th miimizatio of th cospodig asymptotic o pobability th bigs th limitig ma C ad vaiac θ fo < 0 o fo 0. Fo th upp boud, w us a squc of Gaussia iputs X with vaiac lss tha but abitaily clos to. I this cas, w pov that th adom vaiabl I satisfis a CLT with asymptotic ma C ad vaiac θ+. Thom idicats that, fo sufficitly lag chal dimsios ad block lgth, th optimal o pobability wh codig clos to th godic capacity is cotaid withi two xplicit bouds which dpd oly o c, β, ad σ. I th AWGN scaio of [5], [0], th cospodig bouds w foud to dpd oly o σ. Not that, fo cod ats abov th godic capacity limit i.., fo > 0, th low boud is vy pssimistic ad ca b fa fom th upp boud. I cotast, fo < 0, both bouds a gally vy clos to o-aoth. This ca b s fom Fig, which dpicts th bouds o th optimal avag o pobability fo vayig scod-od codig ats ad difft SNR valus dfid as SNR = σ, icludig also th xtm high- ad low- SNR cass. W choos c = ad β = 6. Fo gativ scod-od codig ats, th gap btw th upp ad low bouds is ath small ad dcass with ith gowig o dcasig SNR. Rmak : O ca show that fo vy c, β, σ > 0, > θ, < 0 >, > 0. Bouds o P β, c / /θ SNR = SNR = 0 db SNR = 0 db SNR = 0 db SNR = Scod-od codig at Fig.. Bouds o th optimal avag o pobability as a fuctio of th scod-od codig at fo difft SNRs ad th paamts c = ad β = 6. Apat fo = 0, th low ad upp bouds o th optimal avag o pobability a thfo v qual. This is i shap cotast to [5], [0] wh, fo SISO AWGN chals, th bouds a povd to b qual. Th aso fo this discpacy lis i th psc of th adom chal H which atually iducs a dpdc of th scod od statistics of I o th fouth od momt E[K t X X H ] of P X. Th wak low boud / fo > 0 is i paticula a cosquc of th impossibility to boud th fouth od momt of P X fom abov ud th sol costait. I [5], [0], oly scala scod od momts of P X play a ol i th scod od statistics of I. Ths a asily cotolld by. Rmak : I [5, Thom ], it was show that E [C ] = C + O wh th limitig mutual ifomatio C is giv as C = log + δ 0 + c log + suppc S σ + δ 0 δ 0 + δ 0 3 with δ 0 = δ 0 σ as dfid i Thom. Thus, th optimal avag o pobability may b altativly witt as P β, c = if lim sup P C } C = β,c sic lim if K β,c K log M E[C ] } 4 lim if K E[C ] C = 0. 5 β,c

4 I th fiit N, K, -gim, w may thfo s th optimal avag o pobability as a appoximatio of th optimal achivabl o ud th at costait K log M E[C ] +. 6 K Not that th latio 5 is fudamtally dpdt o th Gaussiaity of H. Idd, is a much stog sult tha th wll-kow covgc of th p-ata mutual ifomatio to its asymptotic limit s,.g., [6] which holds fo chals composd of abitay i.i.d. tis with fiit scod-od momt. It was pcisly show i [7, Thom 4.4] that, whv th tis [ of H hav a ozo fouth-od cummulat κ = E H 4], a bias tm B popotioal to κ aiss such that 5 must b modifid to K E[C ] C B as β,c. I this cas th quivalc of 4 ad 9 dos ot hold. Fo Gaussia chals sic κ = 0 ad th B = 0, howv, th asymptotic mutual ifomatio is achd at a sufficitly fast at of O [5]. Rmak 3: W may also cosid th scod-od outag pobability β, c fo th at, which w dfi as β, c = if lim sup P C :suppc S } C = β,c } lim if K β,c K log M C. Not that β, c = P β β, c. This dfiitio allows us to study th bhavio of β, c fo gowig β. I th fiit dimsioal sttig, this cospods to icasig th block lgth whil maitaiig N ad K ad thus th capacity KC fixd. This caot b pfomd o P β, c sic, by gowig, KC gows as wll. Fom th abov dfiitio ad Thom, w hav mi θ out, } β, c θ+ out 7 wh θ out = β θ ad θ+ out = β. Itstigly, fo 0, as β, w cov th limitig outag pobability of MIMO Gaussia fadig chals,.g., [7], with θ out = lim β, c = β θ out 8 log δ 0 σ ] c + δ 0 σ. 9 Although both sults coicid, th is a fudamtal diffc i th way thy a obtaid. I [7], th block lgth is assumd to b ifiitly lag fom th stat ad th th limit is tak i N ad K. I cotast, w hav obtaid 8 by chagig th od of both limits. Figu dpicts th bouds o β, c i 7 as a fuctio of β fo difft valus of c, assumig SNR = 0 db Bouds o Pout β, c c = c = c = β /θ out /θ out + Fig.. Bouds o th scod-od outag pobability as a fuctio of β fo difft valus of c, =, ad SNR = 0 db. Th limitig outag pobability is P out =, c. ad = fixd. Fo ach valu of c w also povid th limitig outag pobability as giv i 8. Th upp ad low bouds a s to appoach th outag pobability at a at Oβ as β gows, which ca b asily povd. B. Fiit dimsioal appoximatio W ow povid a upp boud appoximatio o th optimal avag o pobability fo abitay codig ats R i th fiit dimsioal gim. W assum tasmissios with a avag gy costait ath tha a pak gy P costait ad dfi, M, -cods as th quivalt to P, M, -cods with iput distibutio P X satisfyig a uit avag gy costait, i.., P X S, wh [ ] } S = P X E K t X X H. 0 W th dfi th optimal avag o pobability P R fo at R ud uit avag gy costait as } P R = if P C :P X S C K log M R P wh C is th avag o pobability fo a giv P, M, -cod. Bfo w cotiu, w d to itoduc a auxilliay lmma which is a simpl galizatio of Fisti s lmma [] to abitay iput distibutios: Lmma Vaiatio of Fisti s lmma: Lt b a itg ad lt P X A b a abitay pobability masu wh A PC K. Dot by Y th output fom th chal P Y X cospodig to th iput X. Th, th xists a block lgth codbook of siz M that, togth with th maximum a postioi MAP dcod, foms

5 P R Upp bouds o K =, N = 4, = 3 C, 0.5 K = 4, N = 8, = 64 K = 8, N = 6, = Tagt at R Fisti Thom Thom Fig. 3. Upp-bouds o P R as a fuctio of th codig-at R fo SNR = 0 db ad difft valus of K, N, ad. a cod C whos avag o pobability P C satisfis P if γ>0 C P [ log P Y X dy X P Y dy log γ Poof: Th full poof is povidd i [4]. ] + M }. γ Sic Lmma holds i paticula fo A = S it ca b usd to pov th followig sult: Thom Appoximatio of Fisti s uppboud: Lt R } = b a al squc. Th, th xists a al squc l } = such that P R K R C + δ with l 0 as β,c, wh δ + Kδ + l = C R + θ+ [ C R + K θ+ log πkθ+ C R + θ+ ad wh C is giv by 3 ad is dfid i Thom. Poof: Th full poof is povidd i [4]. Not that Thom fully xploits Lmma i th ss that, fo all fiit, th optimal choic fo γ whos ol is playd by δ h i is cosidd. This quatity is kow to b zo i th asymptotic limit, so that it dos ot appa i Thom. Nothlss, sic w caot obtai th covgc at of l to zo with spct to that of δ, th pottial gais of Thom caot b aalytically assssd. This is i cotast to [0] wh a By Ess iquality is usd to show that, fo R = C + K /, δ = O log, whil l = O. Figu 3 povids th compaativ pfomac of Thom ad Thom as a appoximatio of Fisti s upp boud Lmma fo P R. Pcisly, th cuvs of Figu 3 a associatd to th followig appoximatios of th upp boud o P R: [ ] } if δ>0 P I R + δ + Kδ Fisti K R C + δ + Kδ Thom K R C Thom wh I is th mutual ifomatio dsity fo Gaussia iputs X dfid i 4 with Γ =. W cosid th difft sts of paamts K, N,. As xpctd, th lag all ths paamts, th small th gap btw th bouds of Thom ad Thom. Fo small valus of ths paamts, th appoximatio by Thom povids a much btt appoximatio of Lmma du to a o-gligibl valu of δ. REFERENCES [] C. E. Shao, Pobability of o fo optimal cods i a Gaussia chal, Bll Syst. Tch. J., vol. 38, o. 3, pp , 959. [] A. Fisti, A w basic thom of ifomatio thoy, IRE Tasactios o Ifomatio Thoy, pp. 0, 954. [3] T. S. Ha, Ifomatio-Spctum Mthods i Ifomatio Thoy. Spig-Vlag, 003. [4] R. Gallag, A simpl divatio of th codig thom ad som applicatios, IEEE Tas. If. Thoy, vol., o., pp. 3 8, Ja [5] M. Hayashi, Ifomatio spctum appoach to scod-od codig at i chal codig, IEEE Tas. If. Thoy, vol. 55, o., pp , Nov [6] V. Stass, Asymptotisch Abschätzug i Shao s Ifomatiosthoi, i Poc. 3d Pagu Cof. If. Thoy, Czchoslovak Acadmy of Scics, Pagu, Czch Rpulic, 96, pp [7] S. Vdú ad T. S. Ha, A gal fomula fo chal capacity, IEEE Tas. If. Thoy, vol. 40, o. 4, pp , Jul [8] J. Wolfowitz, Nots o a gal stog covs, Ifomatio ad Cotol, vol., o., pp. 4, 968. [9] T. S. Ha ad S. Vdú, Appoximatio thoy of output statistics, IEEE Tas. If. Thoy, vol. 39, o. 3, pp , May 993. [0] Y. Polyaskiy, H. V. Poo, ad S. Vdú, Chal codig at i th fiit blocklgth gim, IEEE Tas. If. Thoy, vol. 56, o. 5, pp , May 00. [] Y. Polyaskiy ad S. Vdú, Scala coht fadig chal: Dispsio aalysis, i IEEE It. Symp. If. Thoy ISIT, Aug. 0, pp [] W. Yag, G. Duisi, T. Koch, ad Y. Polyaskiy, Divsity vsus chal kowldg at fiit block-lgth. [Oli]. Availabl: [3] J. Hoydis, R. Couillt, P. Piataida, ad M. Dbbah, A adom matix appoach to th fiit blocklgth gim of MIMO fadig chals, i IEEE It. Symp. If. Thoy ISIT, Cambidg, MA, US, Jul. 0. [4] J. Hoydis, R. Couillt, ad P. Piataida, Th scod-od codig at of th MIMO Rayligh block-fadig chal, to b submittd. Availabl upo qust. [5] W. Hachm, O. Khouzhiy, P. Loubato, J. Najim, ad L. Pastu, A w appoach fo mutual ifomatio aalysis of lag dimsioal multi-ata chals, IEEE Tas. If. Thoy, vol. 54, o. 9, pp , 008. [6] S. Vdu ad S. Shamai, Spctal fficicy of CDMA with adom spadig, IEEE Tas. If. Thoy, vol. 45, o., pp , Ma [7] W. Hachm, P. Loubato, ad J. Najim, A CLT fo ifomatio thotic statistics of Gam adom matics with a giv vaiac pofil, Th Aals of Pobability, vol. 8, o. 6, pp , Dc. 008.