Coverage and Rate in Cellular Networks with Multi-User Spatial Multiplexing

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1 Covag and Rat in Cllula Ntwoks with Multi-Us Spatial Multiplxing Sjith T. Vtil, Kian Kuchi Dpatmnt of Elctical Engining Indian Institut of Tchnology, Hydabad Hydabad, India 55 {p, Anilsh K. Kishnaswamy, Radha Kishna Ganti Dpatmnt of Elctical Engining Indian Institut of Tchnology, Madas Chnnai, India 636 {8b7, Abstact In this pap w consid a multi-us spatial multiplxing (SM) cllula ntwok, wh stams a tansmittd to uss in th cll. Spcifically, w obtain th covag and at xpssions fo a systm mploying zo-focing (ZF) civ. Compad to singl stam tansmission (SST), it is intsting to s that SM dgads th at fo a notabl pcntag of uss. Fo th cas of two and fou civ antnnas, th incas in man at of SM is modst compad to singl stam tansmission (SST) whil SST povids a gain ov SM fo cll dg uss. I. INTRODUCTION Multipl-input multipl-output (MIMO) communications a now an intgal pat of cunt cllula standads. MIMO is a matu tchnology and th is an xtnsiv body of litatu fo vaious MIMO tchniqus [], []. In this pap, w consid opn-loop multi-us spatial multiplxing tchniqu fo cllula downlink. W analys th covag and avag godic at with a lina zo-focing civ in th psnc of intfnc fom oth clls. Spatial multiplxing has bn xtnsivly studid in th psnc of additiv Gaussian nois [3], [4], [5]. Fw sults xist that chaactiz th pfomanc of SM with xtnal intfnc [6]. Howv, intfnc is a pfomanc limiting facto in cunt cllula ntwoks, and hnc it is impotant to study th pfomanc of SM in th psnc of co-channl intfnc. In this pap, w modl th locations of th bas stations (BSs) by a spatial Poisson point pocss (PPP) and consid distanc dpndnt int-cll intfnc. Th PPP modl was usd fo bas station location modlling in [7] to analys th covag in cllula ntwoks with on antnna. PPP modl was also usd in [8], [9] and [] to analys spatial multiplxing with ZF in MIMO ad hoc ntwoks. In ad hoc ntwoks, an intf can b abitaily clos (much clos than th intndd tansmitt) to th civ in considation. This sults in intfnc that is havytaild. On th oth hand, in a cllula ntwok th us usually conncts to th closst BS and hnc th distanc to th nast intf is gat than th distanc to th sving BS. This lads to a mo tamd intfnc distibution compad to ad hoc ntwoks. Bcaus of this th insights obtaind in this pap diff fom [8], [9], []. In this pap, w xtnd th famwok in [7] to a cllula MIMO ntwok. W div gnal xpssions fo covag pobability, and godic at in a cllula downlink with SM and a ZF civ. W obsv that singl-stam tansmission povids a high at compad to SM with incasing tansmit antnna fo cll dg uss. II. SYSTEM MODEL W modl th locations of th bas stations (BSs) by a spatial Poisson point pocss [] of dnsity. Th mits and dmits of this modl fo BS locations hav bn xtnsivly discussd in [7]. W assum a nast BS connctivity modl, wh in a mobil tis to stablish a connction with its closst BS. This sults in a Voonoi tssllation of th plan cosponding to th BS locations, wh th svic aa of a BS is th Voonoi cll associatd with it. W assum that th BSs a quippd with antnna and th uss (UE) a quippd with N antnna. In this pap w focus on downlink and hnc th at th BSs a usd fo tansmission and th N antnna at th UE a usd fo cption. Fo convninc, w assum N n with n. W assum that all th BSs tansmit with qual pow which fo convninc w st to unity. Hnc ach tansmit antnna uss a pow of /. W assum th standad pathloss modl (x) x, >. Indpndnt Rayligh fading with unit man is assumd btwn any pai of antnna. W focus on th downlink pfomanc and hnc without loss of gnality, w consid and analys th pfomanc of a typical mobil us locatd at th oigin. Th N fading vcto btwn th q-th antnna of th BS x and th typical mobil at th oigin is dnotd by h x,q. W assum h x,q CN( N, I N ). W consid th cas wh ach BS uss its antnna to sv indpndnt data stams to uss in its cll. Lt ô dnot th BS that is closst to th mobil us at th oigin. W assum that th UE at th oigin is intstd in dcoding th k-th stam tansmittd by its associatd BS ô. Focusing on th k-th stam tansmittd by ô, th civd W mak th assumption that vy cll has at last uss. This is tu with high pobability whn th a lag numb of uss which is nomally th cas.

2 N signal vcto at th typical mobil us is y k a ô,k h ô,k + wh I( ) x \ô q,qk x hô,q aô,q + I( ) + w, () q h x,q a x,q, dnots th intfnc fom oth BSs. Th symbol tansmittd fom th th q-th antnna of th bas station x is dnotd by a x,q and E[ a x,q ]/. Th additiv whit Gaussian nois is givn by w CN( N, I N ). Th distanc btwn th typical mobil us at th oigin and its associatd (closst) BS is dnotd by ô. Obsv that is a andom vaiabl sinc th BS locations a andom. W now comput th post-pocssing SINR with a zofocing civ. Each UE has N n, n civ antnna. Hnc th civ antnna can b usd to cancl th slf-intfnc causd by th stams and n oth intfing BSs. Tchnically, som of th (n ) civ antnna can b usd fo divsity nhancmnt. Howv, in this pap w assum th (n ) civ antnna a ntily usd to cancl intfnc fom oth uss. This quis th civ to hav som capability to stimat th channl of th closst n intfs. Th civ filt v fo th typical us at th oigin is chosn othogonal to th channl vctos of th tansmitts that nd to b canclld out. W assum that th n intfs closst to th UE a canclld. Sinc th typical UE is intstd in th stam k, v is chosn as a unit nom vcto othogonal to th following vctos: hô,q : q,,.., k,k+,..,, h x,q : x {x,x,..., x n }, q,,..,, wh {x,x,..., x n } a th (n ) BSs closst to th typical UE in considation xcluding ô. Hnc at th civ, v y k a ô,k v hô,k + q,qk aô,q v hô,q + v I( ) + v w. Sinc v is dsignd to null th closst n intfs, v I( ) v I(ˆ) wh ˆ \{x,..., x n }. So w hav ỹ k a ô,k v hô,k + v I(ˆ) + v w. Lt S v hô,k and H x,q v h x,q. It can b shown that S and H x,q a i.i.d. xponntial andom vaiabls. Hnc th post pocssing zo-focing signal-to-intfnc-nois atio (SINR) is SINR + S x q H x,q Î(ˆ ). () III. COVERAGE In this sction, w analyz th covag using th ZF civ dscibd abov. A mobil us is said to b in covag if th civd SINR is gat than th thshold ndd to stablish th connction. Th pobability of covag is dnotd by P c (T,) and is givn by P c (T,) P[SINR >T]. (3) Fom th abov xpssion w s that covag pobability is th CCDF of th SINR. Th ZF civ is dsignd such that it can cancl intfnc fom (n ) BSs apat fom th sam cll intfnc. So w fist comput th distibution of distanc btwn th typical us and th (n ) th BS which will b usd in th analysis lat. A. Distanc to th sving BS and (n )-th BS. Rcall that dnots th distanc to th sving (nast) BS. W hav F ( )P[ > ] P[B(o, ) is mpty ], wh B(o, ) psnts a ball of adius aound th oigin. Hnc th nast nighbou PDF is f (),. (4) W now comput th distanc to th (n )-th closst BS conditiond on th distanc to th nast BS. Lt R dnot th distanc to th n -th BS. Hnc th vnt R R quals th vnt that th a at last n bas stations in th gion btwn two concntic cicls of adius and R cntd at oigin. Hnc F R (R )P [R R ] kn Hnc th conditional PDF is f R (R ) d dr F R (R ) (R ) [ (R )] k, R >. k! R (n )! (R ) (R ) n. B. Covag Pobability W fist povid th main sult which dals with th covag pobability fo a gnal, n> and. Thom. Th pobability of covag with ZF civ is givn by TNt L IR (T )f R (R )f ()drd, wh L IR (s) th conditional Laplac tansfom of th intfnc and is givn in (5). Poof: Th poof closly follows th main Thom in [7]. W only highlight th stps that diff significantly. W hav TNt L R (T )f R (R )f ()drd,

3 wh L IR is th Laplac tansfom of th intfnc conditiond on R. L IR (s) E sî(ˆ ) E xp s x H x,q. q Sinc H x,q a i.i.d xponntial, thi sum q H x,q is gamma distibutd. Using th Laplac tansfom of th gamma distibution, L IR (s) E E xp s x H x,q, q E ( + s x ), Nt appl appl (a) xp R ( + sx xdx, ) appl appl Nt xp R F appl, ; ; R s. wh (a) follows fom th pobability gnating functional (PGFL) of th PPP []. F (a, b, c, z) is th standad hypgomtic function. C. Spcial cas: Intfnc limitd. In this sction w focus on th covag pobability fo paticula valus of n, and in th absnc of nois. W bgin with th n cas. ) Cas n : Whn n, N and hnc only th slf intfnc can b canclld. In this cas R and th intgation with spct to R will not b ncssay. W hav L R (T ) ( F (, ; ; T) ). Substituting fo f (), th covag pobability ducs to F appl, ( F (, ; ; T)) d, ; ; T. Th covag pobability can b futh simplifid whn 4 and th covag sults a povidd in Tabl I. Whn nois is nglctd, w obsv that th covag pobability dos not dpnd on th dnsity of BSs. Th covag pobability is plottd 3. fo diffnt in Figu as a function of th SINR thshold T. (c) t F (a, b, c, z) b ( t) c b (b)(c b) ( tz) a dt. 3 Figu and Figu w gnatd in about 5 sconds ach in Mathmatica on a standad Dll dsktop. This is a vy shot tim compad to th tim takn if th cuvs w to b obtaind by Mont-Calo simulation of th nti systm. (5) Covag pobabilityp c(t,4) T T + +3 T tan T T (7T +9) 3 4 (T +) +5 T tan T T (T (57T +36)+87) 4 4(T +) T tan 8 T TABLE I: Covag pobability fo 4and fo diffnt. Sinc n, w hav N. Pc T,4. P c T,4 vsus T..5 x. x 3 x 3 4 x Thshold SINR : T Fig. : Covag pobability vsus T fo,, 3 3, 4 4 antnna configuations with and 4. ) Cas n : H N n, so th intfnc fom n BSs can b canclld. Stting and substituting fo f () and f R (R ) in Thom, th covag is db R ( F (, ; ; T R )) 4( )n (n )! (R ) n RdRd. Using th tansfomation R/ and t (which implis >), using th Jacobian fo chang of vaiabls w obtain 4( )n t ( F (, ; ; T )) (n )! t n ( ) n d dt. Exchanging th intgals and intgating with spct to t, th pobability of covag is (n ) n ( ) n F Nt, ; ; T n d. (6) W s that th covag pobability can b valuatd using a singl intgal. Lt R/ dnot th atio of th distanc of th n th closst BS of th typical UE to th distanc of its closst BS. It can b shown that th PDF of th andom vaiabl is g ( ) (n ) n ( ) n, >. Hnc fom (6), th covag pobability fo n> also quals E F appl, ; ; T n. (7)

4 Th atio of th distanc to th sving BS plays to th closst intf plays a cucial ol in dtmining th covag. Th avag valu of is givn by (n) E[ ] (n /) (n ). In Figu, th covag pobability givn by (6) is plottd Pc T,4 P c T,4 vsus T. x. x 4.5 x 4 x 8. 3 x 6 3 x Thshold SINR : T Fig. : Covag pobability vsus T fo, 4, 4, 8, 3 6, 3 9 antnna configuations with and 4. fo vaious antnna configuations. Whil a singl intf is canclld in, 4, 3 6 configuations, w obsv that has th bst pfomanc followd by 4 and 3 6. This is bcaus of th incasd agggat intfnc acoss th antnna as N incass. W also obsv that canclling mo intfs incass th covag. Also w can s that 8 has a simila pfomanc to, vn though th intfs a canclld in 8 compad to a singl intf in. Simila obsvation can b mad fo 4 and 3 9 configuations. db IV. AVERAGE ERGODIC RATE In this sction, w comput th godic data at achivabl ov a cll fo a givn us and th at CDF, assuming uss a svd by th BS in a cll. Fo computing th at, w consid th intfnc as nois. W also assum that th modulation and coding is chosn so that thy achiv Shannon bound log ( + SINR), by tating sidual intfnc as nois. A. Avag Achivabl Rat p us W bgin by th thom to find th godic capacity of typical mobil us and also consid som spcial cass of impotanc. Thom. Th avag godic at of a typical mobil us and its associatd BS in th downlink, in bits/sc/hz, is givn by C(,,,n) E [log ( + SINR)] P c ( t,)dt. (8) Poof: Th poof follows fom th CDF of th positiv andom vaiabl log ( + SINR). W now will discuss som spcial cass of dtmining avag achivabl at as in covag analysis. Whn n and, th avg at is C(,,, ) F appl, ; ; t dt, which can b simplifid whn 4. Fo n>, th avag at with is C(,,,n) (n ) n ( ) n F Nt, ; ;( t ) n d dt. B. Compaison with singl stam tansmission In ou systm modl, w considd a multi-us spatial multiplxing wh stams a tansmittd to th uss in th cll. In this sub-sction w want to compa this with a singl-stam tansmission (SST). In SST, th BS has only on antnna i..,. Hnc it can sv only on stam and hnc on us. So all th uss a svd by dividing th soucs ith in tim (TDMA) o fquncy (FDMA). Hnc in this cas, ach us has / tim o fquncy slic. Total at with SM: In SM ach us dcods a singl stam and hnc achivs an godic at C(,,,n), >. Hnc fo uss, th at CDF is givn by F SM (c) P( log ( + SINR(,n)) c), (9) wh SINR(,n) dnots th SINR with tansmit and n civ antnna. Th abov distibution can b asily computd fom th SINR CCDF in Thom. It is asy to s that th total avag downlink at is givn by C SM C(,,,n). Total at with SST: In SST, sinc th soucs hav to b dividd among th uss, ach us achivs an avag at Nt C(,,,n). Hnc fo uss th avag total downlink at achivd is Th at CDF is givn by C SST C(,,,n). F SST (c) P(log ( + SINR(,n)) wh SINR(,n) dnotd th SINR with tansmit and n civ antnna. Th abov distibution can b asily computd fom th SINR CCDF in Thom. Th avag godic ats achivabl fo vaious configuations is shown in th Tabl II. W mak th following obsvations. ) C(,,,n) incass as a function of. So adding mo tansmit antnnas at BSs impovs th man at, but th at pofil obtaind fom th at CDF givs us mo insight. ) Th man at fo 4 is.78 whil it is 4.3 fo 4 and 4.6 fo 44. Hnc th tuns a diminishing with c),

5 Two uss th cll N SM/SST Man 5% 5% 8% SST SM Fou uss in th cll N SM/SST Man 5% 5% 8% 4 SST SM SM Six uss in th cll N SM/SST Man 6 SST SM SM SM 5.74 TABLE II: Rat pofil fo vaious configuations. Th ats a in bits/sc/hz and a computd fo and 4. Fo a gnal n systm with k uss, th avag sum at fo compaison is Nt k C(,,,n)k C(,,,n). This follows fom th fact that th uss a dividd into k/ goup and soucs a dividd btwn thm. Each goup is svd using SM of stams. Fc C...5 Rat CDF Rat: C bits sc Hz Fig. 4: Rat CDF of 4, 4 and 4 4. x4 x4 4x, 4and cll intfnc. This might b bcaus of using ZF which is a suboptimal civ. It will b intsting to analys th pfomanc with MMSE civ. Fc C. Rat CDF ACKNOWLEDGMENT W would lik to acknowldg th IU-ATC pojct fo its suppot. This pojct is fundd by th Dpatmnt of Scinc and Tchnology (DST), India and Engining and Physical Scincs Rsach Council (EPSRC). W would also lik to thank th CPS pojct in IIT Hydabad fo suppoting this wok Rat: C bits sc Hz Fig. 3: Rat CDF of and with. x x, 4and incasing. Simila obsvation can b mad fo th k 6 fo k,, 3, 6. 3) Incasing and hnc incasing th numb of stams in SM dgads th ntwok pfomanc. It can b sn that th 5 pcntil at of th ntwok is btt fo and th 4 cass. Th sam is tu fo th cas of 5 pcntil point too. In both cass as is incasing, th at is highly ducd. But fo 8 pcntil and 4 a btt, but th ats a compaabl. This implis incasing and using th multipl tansmit antnna fo tansmitting mo stams will hut th cll dg uss. In Figus 3 and 4, th CDFs of th at a plottd fo vaious configuations. Th at pofil tlls that incasing not only povid lss incas in man at but also dgads th pfomanc of th ntwok. Fom th at CDF it is intsting to s that SM with ZF civs dgads th at fo mo than 6 pcntag of in th psnc of oth REFERENCES [] A. Paulaj, R. Naba, and D. Go, Intoduction to spac-tim wilss communications. Cambidg Univ P, 3. [] G. Foschini and M. Gans, On limits of wilss communications in a fading nvionmnt whn using multipl antnnas, Wilss psonal communications, vol. 6, no. 3, pp , 998. [3] Q. Spnc, A. Swindlhust, and M. Haadt, Zo-focing mthods fo downlink spatial multiplxing in multius MIMO channls, IEEE Tans. on Signal Pocssing, vol. 5, no., pp , fb. 4. [4] S. Jafa and A. Goldsmith, Isotopic fading vcto boadcast channls: Th scala upp bound and loss in dgs of fdom, IEEE Tans. on Info. Thoy, vol. 5, no. 3, pp , mach 5. [5] D. Gsbt, M. Kountouis, R. Hath, C.-B. Cha, and T. Salz, Shifting th MIMO paadigm, Signal Pocssing Magazin, IEEE, vol. 4, no. 5, pp , spt. 7. [6] R. Blum, MIMO capacity with intfnc, IEEE Jounal on Sl. Aas in Communications, vol., no. 5, pp , jun 3. [7] J. G. Andws, F. Bacclli, and R. K. Ganti, A tactabl appoach to covag and at in cllula ntwoks, IEEE Tans. on Communications, vol. 59, pp , Nov.. [8] N. Jindal, J. Andws, and S. Wb, Multi-antnna communication in ad hoc ntwoks: Achiving MIMO gains with SIMO tansmission, IEEE Tans. on Communications, vol. 59, no., pp , fbuay. [9] R. Vaz and R. Hath, Tansmission capacity of ad-hoc ntwoks with multipl antnnas using tansmit stam adaptation and intfnc cancllation, IEEE Tans. on Info. Thoy, vol. 58, no., pp , fb.. [] R. Loui, M. McKay, and I. Collings, Opn-loop spatial multiplxing and divsity communications in ad hoc ntwoks, IEEE Tans. on Info. Thoy, vol. 57, no., pp , jan.. [] D. Stoyan, W. S. Kndall, and J. Mck, Stochastic Gomty and its Applications, nd d., s. Wily sis in pobability and mathmatical statistics. Nw Yok: Wily, 995.