SINR Analysis for V-BLAST with Ordered MMSE-SIC Detection

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1 SINR Aalysis for V-BLAST wih Ordered MMSE-SIC Deecio Roald Böhke ad Karl-Dirk Kammeyer Deparme of Commuicaios Egieerig Uiversiy of Breme Oo-Hah-Allee NW 839 Breme, Germay {boehke, ABSTRACT A ew way o deermie he exac layer-wise SINR disribuio for V-BLAST wih successive ierferece cacellaio a he receiver is preseed. I coras o previous publicaios, we do o resric o zero-forcig, bu also cosider miimum mea square error ierferece suppressio. I is show aalyically ha a opimized deecio order has a eve larger impac i his case. Numerical examples provide deeper isighs io he uderlyig effecs. Caegories ad Subjec Descripors: G.3 [Probabiliy ad Saisics]: Disribuio Fucios Geeral Terms: Theory. Keywords: V-BLAST, SINR Disribuio, Zero-Forcig, MMSE, Ordered Successive Ierferece Cacellaio.. INTRODUCTION I is well kow ha very high specral efficiecies ca be achieved by usig muliple aeas a he rasmier ad he receiver []. Oe cadidae for fuure mobile commuicaio sysems is he V-BLAST archiecure [], where idepede daa sreams are rasmied from differe aeas. A deailed performace aalysis for simple liear as well as opimal maximum-likelihood receivers ca be foud i [3]. However, higs become more complicaed for he ordered successive ierferece cacellaio (SIC) proposed i []. I [4], i was show ha wihou orderig he diversiy order of he k-h layer is give by N R N T + k, where N T ad N R deoe he umber of rasmi ad receive aeas. The imporace of a opimized deecio order for he iformaio ouage probabiliy was highlighed i [], where a uiform power ad rae allocaio amog a subse of rasmi aeas was cojecured o be opimal. However, he required disribuio of he layer-wise sigal o oise raio (SNR) wih opimal orderig was oly approximaed by Moe-Carlo simulaios. The exac expressio for he case of wo rasmi aeas was deermied Permissio o make digial or hard copies of all or par of his work for persoal or classroom use is graed wihou fee provided ha copies are o made or disribued for profi or commercial advaage ad ha copies bear his oice ad he full ciaio o he firs page. To copy oherwise, o republish, o pos o servers or o redisribue o liss, requires prior specific permissio ad/or a fee. IWCMC 6, July 3 6, 6, Vacouver, Briish Columbia, Caada. Copyrigh 6 ACM /6/7...$.. i [6] usig he disribuio of he agle bewee wo complex Gaussia vecors, which was also used o derive loose bouds for N T > i [7]. A aleraive approach based o he ivered complex Wishar disribuio was recely preseed i [8]. The above meioed publicaios assume perfec suppressio of he remaiig ierferece by a zero-forcig (ZF) filer. I his paper, we describe aoher ad somewha more direc mehod o calculae he SNR disribuio of ordered ZF-SIC, ad exed i aferwards o aalyze also he sigal o ierferece ad oise raio (SINR) if a miimum mea square error (MMSE) filer is applied isead. Expec for he simple case of usored ZF-SIC we will resric o oly wo rasmi aeas; alhough our basic approach may possibly be geeralized, he resulig expressios become quie ivolved. Several illusraios help o gai iuiio io he fudameal effecs of opimizig he deecio order ad MMSE filerig.. PRELIMINARIES AND NOTATION Throughou his paper, vecors (marices) are represeed by bold lower (upper) case leers. I is a ideiy marix of size, E { } he expecaio operaor, ad ( ) T ad ( ) H deoe raspose ad Hermiia raspose, respecively. We will frequely use he regularized icomplee gamma fucios [9] Γ(, x) Γ(, x) = = e d = e x x Γ() x Γ() m= m () m! γ(, x) γ(, x) = = x e d = Γ() Γ() Γ(, x) () for ieger argume, ad he iegrals a e Γ() b c as well as (for b ad b c) a b γ(,[ c]a) e u, c [ c] du d = e a e b, c =! e b γ(, c) d = eab γ(, ac) b c γ(,[c b]a) [c b] b will be required. Fially, he abbreviaios cdf ad pdf sad for cumulaive disribuio fucio ad probabiliy desiy fucio. (3) (4)

2 3. SYSTEM MODEL Cosider he equivale basebad model of a sigle-user muliple aea sysem wih N T rasmi ad N R N T receive aeas. The chaels are ucorrelaed ad fla Rayleigh fadig. Hece, he N R N T chael marix H cosiss of idepede circularly symmeric complex Gaussia eries wih zero mea ad ui variace. The receive vecor is give by y = Hx +, () where he vecor x = [x,..., x NT ] T wih covariace marix E xx H =I NT coais idepede rasmi symbols, ad represes circularly symmeric ad whie complex Gaussia oise wih variace σ. Perfec chael sae iformaio is assumed a he receiver. 4. ZF-SIC WITHOUT ORDERING For each layer (i.e., he daa sream of oe specific rasmi aea), he followig wo seps are performed: The ierferece caused by already deeced layers is subraced from he receive sigal. The ierferece of he remaiig layers is suppressed by a liear filer. The required filer marices follow from he QL decomposiio of he chael marix H = QL, where he N R N T marix Q has orhogoal colums wih ui orm ad L is lower riagular wih real-valued ad oegaive diagoal elemes []. Muliplyig y wih Q H resuls i z = Q H y = Lx + ñ m. (6) The oise a he filer oupu ñ = Q H is sill whie wih variace σ. For k =,..., N T, he esimae ˆx k of x k ca be obaied by quaizig x k = k k l km ˆx (7) l kk z m= k l km =x k + (x m ˆx m) + l kk m= ñk (8) l kk o he discree symbol alphabe. Assumig correc decisios i all previous deecio seps (i.e., ˆx m = x m), he SNR of he k-h layer is give by SNR k = l kk σ, (9) which is ideical o he SINR, because he ZF filer compleely removes all ierferece. 4. SNR Disribuio The QL decomposiio may be ierpreed as a special chage of he coordiae sysem i which he marix H is represeed, wih he colums of Q beig he orhogoal base vecors. From he roaioal ivariace of he mulivariae Gaussia disribuio of h k i ca be deduced ha he elemes of L are idepede ad l mk is complex Gaussia for m > k. O he oher had, he squared colum orm N T h k = h mk = lkk + l mk () N R m= m=k+ σ ϑ () p l (u) Figure : O he calculaio of P SNR (ϑ l = ) from he joi pdf of l ad l for ZF-SIC. PSNR(ϑ l = ) σ ϑ Figure : Codiioal cdf P SNR (ϑ l = ) for ZF- SIC. is a sum of N R expoeial radom variables; cosequely, l kk mus be χ -disribued wih (N R N T + k) degrees of freedom []. This leads o he pdf s kk () = N R N T +k e Γ(N R N T + k), () p lmk () = e, m > k () ha are zero for <. From (9) ad () we ca ow immediaely obai he cdf of he SNR o he k-h layer P SNRk (ϑ) = Pr l kk < σ ϑ = γ(n R N T +k ϑ). (3) As already meioed i he iroducio, we will focus o he case of wo rasmi aeas i he followig. I order o deermie he disribuio of SINR for a opimized deecio order as well as MMSE ierferece suppressio, i eeds o be codiioed o he secod layer, as a sar. For he usored ZF-SIC cosidered here, his is of course ideical o (3) due o he saisical idepedece of l ad l. Thus, he codiioal cdf show i Fig. for N R = receive aeas does o deped o a all. Fig. addiioally demosraes how (3) ca be calculaed by iegraig over he joi pdf of l ad l. These figures serve as a referece for laer compariso o illusrae he impac of sorig ad MMSE filerig. u

3 . ZF-SIC WITH OPTIMIZED ORDER The order of deecio is crucial for he performace of SIC. I ca be opimized by exchagig elemes of he rasmi vecor x ad he correspodig colums of H. For some permuaio marix Π, we defie ˇx = Π T x, Ȟ = HΠ, Ȟ = ˇQĽ. (4) σ ϑ () p l (u) I ca easily be verified ha Ȟˇx = Hx, because Π is orhogoal, so he receive vecor i () is o affeced. I was show i [] ha he miimum SNR amog all rasmied sreams is maximized by a greedy approach which always chooses he bes layer o be deeced ex. Assumig ha he firs k colums of Ȟ have already bee deermied, ȟ k mus be seleced from he remaiig colums of H such ha ľ kk is as large as possible. As he SNR s afer liear ZF filerig wih he pseudo-iverse Ȟ + = Ľ ˇQ H are iversely proporioal o he row orms of Ǧ = Ľ, he codiios ǧ kk m =k ǧ m m > k () have o be saisfied. For wo rasmi aeas, () becomes Ǧ = ľ ľ ľ ľ ľ ľ ľ + ľ. (6) Noe ha his crierio is also employed by he efficie orderig algorihm proposed i [], where he diagoal elemes ľkk are miimized i he order hey are calculaed durig he orhogoalizaio process (k = N T,..., ) isead of maximizig hem i he opposie order. However, for N T > his approach is o loger guaraeed o be opimal, hough he loss is usually raher small.. SNR Disribuio From (6) ad (), i ca be cocluded ha for he opimal permuaio he squared secod diagoal eleme is ľ = mi h, h (7) Hece, we ca apply order saisics [] o obai he correspodig cdf P ľ () = Pr h k = Γ(N R, ) = γ(n R, ) [ γ(n R, )], (8) ad, similar o (3), he disribuio of SNR is give by P SNR (ϑ) = P ľ (σ ϑ). Thus, he ouage probabiliy of he secod layer is approximaely doubled by sorig if σ ϑ is small, as already oed i [6]. Uforuaely, he diagoal elemes of Ľ are o idepede aymore. However, as boh deecio orders are equiprobable, he cdf of ľ codiioed o ľ = is ideical o ha of l uder he assumpio ha he aural orderig Π = I is opimal for l =. Wih (6) ad (9), his leads o he codiioal cdf P SNR ϑ ľ = =P l σ ϑ l + l. (9) Exploiig he saisical idepedece of l ad l, we Figure 3: O he calculaio of P SNR (ϑ ľ = ) from he joi pdf of l ad l for ordered ZF-SIC. PSNR(ϑ ľ = ) σϑ Figure 4: Codiioal cdf P SNR (ϑ ľ = ) for ordered ZF-SIC. sar by calculaig he joi probabiliy Pr l < σϑ, l + l = σ ϑ ( ) + () p l (u)du d () = σ ϑ NR e Γ(N R ) Pr l < σϑ, l + l ( ) + e u dud, () where he shorhad oaio (x) + = max{x,} was iroduced. The limis of he iegrals are illusraed i Fig. 3. Compared o Fig., a sigifica par of he joi pdf of l ad l is disregarded i () due o he orderig crierio. For σ ϑ >, we jus eed o subrac he coribuio of he whie riagle from (3), while (4) ca be applied oherwise o ge = (σϑ) NR e /Γ(N R) ϑ γ(n R ϑ) γ(n R, ) ϑ >. u () From (), we ca already deduce he iuiive resul ha he impac of orderig o he SNR disribuio of he firs layer is mos proouced if is large; for =, he cosrai (6) is o effecive a all. O he oher had, akig

4 he limi σϑ yields Pr l + l = γ(n R, ) = Γ(N R, ), (3) which is he complemeary cdf of h k, so he codiioal cdf i (9) is give by P SNR (ϑ ľ = ) = Pr l < σϑ, l + l Pr {l + l } = (σϑ) NR e /Γ(N R, ) ϑ Γ(N R ϑ)/ Γ(N R, ) ϑ >. (4) Fig. 4 agai shows a example for N R =. Comparig his o Fig., we fid ha opimal orderig reduces he ouage probabiliy of he firs layer for ay give >, ad especially for large values, as expeced. Hece, he ucodiioal cdf of SNR, which ca be compued by averagig (4) over ľ, will also be decreased. The required pdf of ľ is obaied by akig he derivaive of (8) p ľ () = N R e Γ(N R, ). () Γ(N R) Wih his, we fially arrive a P SNR (ϑ) = = (σ ϑ/) N R Γ(NR,σ ϑ) Γ(N R) P SNR (ϑ ľ = ) p ľ () d (6) + Γ(NR ϑ) Γ(N R ϑ) γ(n R ϑ). (7) The firs erm belogig o he case σ ϑ i (4) resuls from () afer he subsiuio =, ad he oher oes immediaely follow from (8) ad (). Noe ha (7) ca also be rewrie i erms of expoeial fucios ad polyomials usig he relaios i () ad (). 6. MMSE-SIC WITHOUT ORDERING Followig he approach of [], we firs defie he QL decomposiio of he exeded chael marix H = H σ I NT = Q Q L. (8) The, usig H = Q L i ca easily be verified ha he liear MMSE filer for he sysem model () is give by (H H H + σ I NT ) H H = (H H H) H H = L Q H. (9) Hece, MMSE-SIC is ideical o he ZF-SIC described i Secio 4 wih Q ad L beig replaced by Q ad L, respecively, ad he filer oupu sigal becomes z = Q H y = Lx σ Q H x + Q H ñ ; (3) he secod erm i (3) represes a bias ad remaiig ierferece. Usig he relaios Q H Q + Q H Q = I NT ad Q = σ L, which follow from he properies of he QL decomposiio i (8), we fid ha he SINR of layer k assumig o errors i previous deecio seps is give by SINR k = l kk σ /l kk σ σ 4 /l kk = l kk. (3) σ This of course equals he SNR for he secod layer, as he ierferece has already bee subraced compleely. Thus, we ca use he resuls derived for ZF-SIC ad cocerae σ ϑ () p l (u) (+σ )ϑ u Figure : O he calculaio of P SINR (ϑ l = ) from he joi pdf of l ad l for MMSE-SIC. PSINR(ϑ l = ) σ ϑ Figure 6: Codiioal cdf P SINR (ϑ l = ) for MMSE-SIC. o he firs layer. Observig ha h k = h k + σ ad h H h = l l, we obai l = h h H h / h (3) =l + σ + σ l l +, (33) σ which ca be plugged io (3) o ge SINR = l + l σ l +. (34) σ Noe ha (34) correspods o he SNR of ZF-SIC i (9) for l, while i has he same disribuio as SNR for l =. Thus, ulike he liear ZF filer, he MMSE filer beefis from small SNR s o he secod layer, because less ierferece eeds o be suppressed. 6. SINR Disribuio From (34) i follows ha he SINR disribuio of he firs layer codiioed o l = ca be calculaed by iegraig he joi pdf of l ad l over he regio skeched i

5 Fig.. Wih () ad (3) we ge ϑ P SINR (ϑ l = ) = Pr l + l < σ + σ σ ϑ () p l (u) = σ ϑ NR e Γ(N R ) [+σ ]ϑ +σ σ e u dud (3) )ϑ γ(n = γ(n R ϑ) e (+σ R, ϑ). (36) ( /σ) N R I lie wih he above observaios, he firs erm is he cdf of SNR for ZF-SIC, while he oher oe vaishes for, ad becomes e σ ϑ (σ ϑ) N R /Γ(N R) for, so ha he cdf coverges o γ(n R ϑ). A example for N R = ad σ = is show i Fig. 6. Here, he codiioal ouage probabiliy decreases for small values of. This is jus opposie o he case of ordered ZF-SIC i Fig. 4, where a large improves he performace of he firs layer. Noig ha (4) eds o c /([c b] b) for b < c < ad a, he ucodiioal cdf P SINR (ϑ) = P SINR (ϑ l = ) () d = γ(n R ϑ) (σ ϑ) NR e σ ϑ Γ(N R) [ϑ + ] (37) ca also easily be deermied. I coras o ZF ierferece suppressio i does o oly deped o he produc σ ϑ, bu also o ϑ iself. 7. MMSE-SIC WITH OPTIMIZED ORDER We ow fially ur o he case of ordered MMSE-SIC. Similar o Secio, he permuaio mus be chose such ha he diagoal eleme ľ is as small as possible i order o maximize SINR. However, his ow addiioally meas ha less ierferece eeds o be suppressed by he liear MMSE filer i he firs sep. Hece, i may already be cojecured ha he impac of sorig is much more proouced for MMSE-SIC. The derivaio of he SINR disribuio is similar o hose for ordered ZF-SIC, so we will subsequely focus o he mai resuls ad omi iermediae seps. = 7. SINR Disribuio Combiig Fig. 3 wih Fig., we fid ha he joi pdf of l ad l mus be iegraed over he regio depiced i Fig. 7 i order o obai he joi probabiliy, [ + σ]ϑ [ (+σ )ϑ]n R e Γ(N R ) ( /σ )N R e (+σ )ϑ γ(n R, [+σ ]ϑ) ( /σ )N R ϑ < [ + σ]ϑ γ(n R ϑ) γ(n R, ) e (+σ )ϑ γ(n R, ϑ) ( /σ )N R ϑ >. Pr l + l < ϑ, l σ + σ + l (38) I he secod case, he wo boudaries ( + σ )(ϑ /σ ) ad iersec, while he hird case direcly follows from () ad (36). For σ ϑ we ge (3) agai. Thus, dividig (38) by Γ(N R, ) yields he cdf of SINR wih opimal (+σ )ϑ Figure 7: O he calculaio of P SINR (ϑ ľ = ) from he joi pdf of l ad l for ordered MMSE-SIC. PSINR(ϑ ľ = ) σ ϑ = Figure 8: Codiioal cdf P SINR (ϑ ľ = ) for ordered MMSE-SIC. sorig codiioed o ľ = P SINR (ϑ ľ = ) (39), [ + σ]ϑ [ (+σ )ϑ]n R e Γ(N R,) ( /σ )N R e (+σ )ϑ γ(n R, [+σ ]ϑ) Γ(N R,) ( /σ )N R ϑ < [ + σ]ϑ Γ(N R ϑ)/ Γ(N R, ) e (+σ )ϑ γ(n R, ϑ) Γ(N R,) ( /σ )N R ϑ >. I order o avoid umerical problems, he icomplee gamma fucios wih egaive secod argume i (39) should be replaced by he polyomial represeaio i (), ad he expoeial relaio e a e b = e a+b should be applied. From Fig. 8 i ca be observed ha he ouage probabiliy is sigificaly reduced for all values of. For small his is maily a cosequece of MMSE filerig, while for large sorig plays a major role. The ucodiioal cdf is agai calculaed by akig he expecaio of (39) over ľ. Wih adequae subsiuios we ca use (4) o obai afer some u

6 simplificaios P SINR (ϑ) = Γ(NRϑ) Γ(N R ϑ) γ(n Rϑ) + ( σ ) N R Γ(N R) ( σ ϑ ) NR e σ ϑ (4) Γ(N R) ( ϑ)n R e σ ϑ γ(n R ϑ + ϑ) + ( ϑ)n R e σ ϑ ϑ N R (ϑ + ) Υ N R ϑ ϑ wih Υ(, x) = γ(, x), x Γ(, x), x >. (4) The wo differe cases mus be disiguished, because he firs codiio i (39) is ever fulfilled for ϑ, which affecs he limis of he iegrals. 8. NUMERICAL RESULTS Fig. 9 depics he layer-wise SINR disribuios for a muliple aea sysem wih wo rasmi ad receive aeas. The hi lies represe he secod layer wih o, opimal, or ivered sorig, i.e. ľ = max{ h, h }. As already oed i Secio, opimal sorig approximaely doubles he ouage probabiliy i his deecio sep. I coras, he curve of he firs layer is shifed o he righ by 3 db for ZF-SIC. The hree differe curves for MMSE- SIC belog o icreasig oise variaces (from lef o righ). For srog oise, he ierferece ca be egleced, ad he SINR disribuio is close o ha of he secod layer wih he iverse orderig applied. O he oher had, for small values of σ, MMSE-SIC performs similar o ZF-SIC over a wide rage, bu eveually decreases for sufficiely small ϑ. Wih opimal sorig, he ouage probabiliy rapidly coverges o he ierferece-free case for ϑ <, while a cosa gap remais wihou opimized deecio order. PSINR k (ϑ) Layer, w/o Layer, op. Layer, iv. ZF SIC, w/o ZF SIC, op. MMSE SIC, w/o MMSE SIC, op. 3 σϑ i db Figure 9: SINR disribuios for N T = N R = ad σ {.,., } (required oly for MMSE-SIC). 9. CONCLUSION We have preseed a ew uified approach o he SINR aalysis of V-BLAST wih (ordered) ZF- or MMSE-SIC deecio. Based o geomerical cosideraios, he SINR disribuio of he firs layer was calculaed for differe receiver archiecures by firs codiioig o, ad he averagig over he effecive chael gai of he secod layer. The effecs of a opimized deecio order ad MMSE ierferece suppressio were ivesigaed separaely ad visualized by meas of he codiioal cdf of SINR. Furhermore, i was show aalyically ha he opimal orderig is eve more impora i combiaio wih MMSE filerig.. REFERENCES [] E. Telaar, Capaciy of Muli-aea Gaussia Chaels, Europea Trasacios o Telecommuicaios, vol., o. 6, pp. 8 9, November-December. [] P. W. Woliasky, G. J. Foschii, G. D. Golde, ad R. A. Valezuela, V-BLAST: A Archiecure for Realizig Very High Daa Raes Over he Rich-Scaerig Wireless Chael, i Proc. ISSSE, Pisa, Ialy, Sepember 998. [3] M. Kießlig, Saisical Aalysis ad Trasmi Prefilerig for MIMO Wireless Sysems i Correlaed Fadig Eviromes, Ph.D. disseraio, Isiu für Nachricheüberragug, Uiversiä Sugar, 4. [4] G. J. Foschii, G. D. Golde, A. Valezela, ad P. W. Woliasky, Simplified Processig for High Specral Efficiecy Wireless Commuicaios Emplyig Muli-Eleme Arrays, IEEE Joural o Seleced Areas i Commuuicaios, vol. 7, o., pp. 84 8, November 999. [] A. Kaa, B. Varadaraja, ad J. Barry, Joi Opimizaio of Rae Allocaio ad BLAST Orderig o Miimize Ouage Probabiliy, i Proc. IEEE Wireless Commuicaios ad Neworkig Coferece, New Orleas, USA, March. [6] S. Loyka, V-BLAST Ouage Probabiliy: Aalyical Aalysis, i Proc. IEEE Vehicular Techology Coferece (VTC), Vacouver, Caada, Sepember. [7] S. Loyka ad F. Gago, Aalyical Framework for Ouage ad BER Aalysis of he V-BLAST Algorihm, i Proc. Ieraioal Zurich Semiar o Commuicaios, Zurich, Swizerlad, February 4. [8] R. Xu ad F. Lau, Aalyical Approach of V-BLAST Performace wih Two Trasmi Aeas, i Proc. IEEE Wireless Commuicaios ad Neworkig Coferece, New Orleas, USA, March. [9] I. Gradshey ad I. Ryzhik, Table of Iegrals, Series ad Producs, 6h ed. Sa Diego, CA: Academic Press,. [] D. Wübbe, R. Böhke, J. Rias, V. Küh, ad K. D. Kammeyer, Efficie Algorihm for Decodig Layered Space-Time Codes, IEE Elecroics Leers, vol. 37, o., pp , Ocober. [] A. Papoulis, Probabiliy, Radom Variables, ad Sochasic Processes, 3rd ed. New York: McGraw-Hill, 99. [] R. Böhke, D. Wübbe, V. Küh, ad K. D. Kammeyer, Reduced Complexiy MMSE Deecio for BLAST Archiecures, i Proc. IEEE Global Commuicaios Coferece (Globecom 3), Sa Fracisco, Califoria, USA, December 3.