Exploiting transmission spatial diversity in frequency selective systems with feedback channel

EXPLOITING TRANSMISSION SPATIAL DIVERSITY IN FREQUENCY SELECTIVE SYSTEMS WITH FEEDBACK CHANNEL An onio Pascual Ise e, Ana I. Pi ez-Nei a, Miguel A. Lagunas He ndndez Depa men o Signal Theo y and Communica ions Poly echnic Uni e si y o Ca alonia (UPCj Telecommunica ions Technological Cen e o Ca alonia ClTC - Edi ici NEXUS I C/ Jo di Gi ona 1-3 (Campus No d UPC - mhdul DS), 08034 Ba celona (SPAIN) e-mai/:{ onip,anuskaj @gps. sc.upc.es. [email p o ec ed] , ABSTRACT In his pape we add ess he design o a mul iple ansmi an enna sys em in which he Channel S a e In o ma ion (CSI) a he ans- mi e is no pe ec . Two di e en app oaches a e analyzed one based on he Minimiza ion o he Mean Squa e E o (MMSEj and he o he based on he applica ion o he Maximum Likelihood Se- quence Es ima ion (MLSE). In bo h cases a Bayesian c i e ion is used in o de o ake in o accoun he e o be ween he CSI and he eal channel. Finally, some simula ion esul s and conclusions a e p o ided, showing which is he gain o hese app oaches when he e o be ween he CSI and he eal channel is ei he Gaussian o uni o m, whe e his las case co esponds o a quan iza ion o he channel ime esponse in o de o ansmi he CSI h ough a eedback channel om he ecei e o he ansmi e . 1. INTRODUCTION Spa ial di e si y is an e icien me hod so as o comba he impai - men s p esen in he wi eless channel. In cellula communica ions o Wi eless LAN's he ecei e an enna di e si y is no a ac i e o he downlink channel because he mobile s a ion should be equipped wi h mul iple ecei e an ennas. Fo his eason, he use o ansmi an enna di e si y o he downlink is mo e desi able. Exis ing ansmission schemes o exploi ing he po en ial o - e ed by ansmi an enna a ays a e gene ally conce ned wi h in- c easing he di e si y o de . The e a e se e al examples o such echniques, such as he delay di e si y s a egy, a special case o a mo e gene al solu ion p esen ed in [I]. O he possible app oaches ha inc ease he di e si y o de consis in he applica ion o spnce- ime coding, echnique p esen ed in wo ks such as [Z] and 131. Space- ime codes do no exploi channel knowledge a he ansmi c . In o ma ion abou he channel, i a ailable, should be used o imp o e he pe o mance by means o op imal e minal il e ing. I can be shown ha unde a ze o o cing c i e ion, he maximiza ion o he Signal o Noise Ra io (SNR) esul s in a de- coupled o spa ially scalable solu ion whe e each ansmi b anch can be designed independen ly. Fig. 1 shows a gene alized a chi- ec u e ha allows a no maliza ion o he il e s dynamic, while he beam o ming weigh s { uq):=, a e in cha ge o adjus ing he ansmi powe . Modula ion can be seen as a quan iza ion p ocess and i s e ec s can be s udied as quan iza ion noise. This quan i- za ion a he ou pu o each il e a oids ins abili ies, and so, IIR designs could also be used. By depa ing om his a chi ec u e, 0-7803-7663-3/03/$17.00 02003 IEEE IV - 85 FILTERING MODULATION BEAMFORMING 2, Q [.I WI .z Q b QPI b WQ 5(4 Fig. 1. T ansmission di e si y a chi ec u e. a clea ade-o be ween op imali y and c ucial aspec s like o- bus ness, p ac icali y, so -deg ada ion o he QoS, eliabili y, e c. can be easily aken in o accoun . Conce ning p ac ical conside - a ions, pa ial and quan ized Channel S a e In o ma ion (CSI) a he ansmi e can be in oduced in a na u al way by adap ing, o ins ance, each complex weigh wp o he s onges channel pa h a each b anch o , i a con olled uni a y dynamic is desi ed, jus by compensa ing he phase o he s onges pa h. To sum up, o- bus ness [4] implies insensi i i y o de ia ions om he heo e ical assump ions, being he impe ec CSI one o he possible sou ces o de ia ion. So a we ha e commen ed on ansmi e design based ei he on coding o linea p ocessing depending on he channel knowl- edge, and how impe ec ions o bad knowledge can be aken in o accoun by mo ling o obus a chi ec u es. Ano he al e na i e exis ing in he li e a u e o inco po a ing bad channel knowledge in ansmi space- ime p ocessing is by means o a Bayesian poin o iew, ha is, modeling he side channel in o ma ion using a pu ely s a is ical app oach. P e ious and ela ed wo k includes he pe o mance analysis o [SI o la ading channels o he design p oposed in [6] o OFDM sys ems. .In [7] he bene i s o ans- mi beam o ming and o hogonal space- ime block coding o la ading channels a e combined. [8] conside s he e o in he CSI om a MAXMIN poin o iew di e en om he Bayesian one. In his pape we analyze he gene al case o a equency selec- i e channel and p opose wo space- ime p ocessing solu ions ha ollow he Bayesian app oach in o de o inco po a e he e o in he CSI. Simula ions compa e he p oposed echniques wi h space- ime p ocessing designs ha assume pe ec CSI and schemes ha do no need CSI such as delay di e si y. Al hough he Bayesian app oach will ei he esul in spa ially non-scalable solu ions, o non- obus when he s a is ical assump ions a e no ue, i se es as a use ul benchma k o analyze and compa e he commen ed o- bus a chi ec u es. ICASSP 2003 Fig. 2. Gene al scheme o ansmi di e si y wi h eedback chan- nel and impe ec CSI a he ansmi e . 2. PRE-FILTERING WITH FEEDBACK CHANNEL In his sec ion we ocus ou a en ion on he case o a eal sys em in which a digi al eedback channel is implemen ed om he ecei e o he ansmi e . The single-an enna ecei e is esponsible o es ima ing he channel, quan ize his es ima e. code i in o a digi- al o ma and send i o he ansmi e ia he eedback channel. By means o his, he ansmi e has an es ima e o he channel, possibly impe ec CSI. In his wo k we exploi his CSI in o de o design linea il e s a he ansmi e om a Bayesian poin o iew and wi hou o cing he design o he spa ially scalable. The gene al scheme is p esen ed in Fig. 2. Ou goal is o design he il e s {z,},"=, o he Q ansmi an ennas aking in o accoun he impe ec ions in he CSI a he ansmi e . 2.1. Sys em and Signal Models Le us conside a equency selec i e channel wi h Q ansmi an ennas, whe e each o he channels in he Mul i-Inpu -Single- Ou pu (MISO) link has L aps. h, = [ h!p)hp). . . h.!) ep esen s he ime impulse esponse o he q h ansmi an enna. I is possible o collec all hese ime impulse esponses in a sin- gle ec o h by means o his no a ion: h = [ hTh .. . hz ]', whe e he numbe o componen s o h is K = QL. The channel is modeled as a complex andom Gaussian ec o , whe e i s co a i- ance ma ix Rh collec s he spa ial co ela ion and he powe delay p o ile o he channel. In case ha he e is a di ec line o sigh , hen hese andom ec o would ha e a ce ain mean m di e en om ze o. The e o e, he P obabili y Densi y Func ion (PDF) o he channel ollows he s a is ical law: h - G (m, Rh). Ou goal is o design he ansmi il e s while conside ing e y simple de ec o s a he ecei e . As i is seen in Fig. 2, wo possible ecei e s a e conside ed: a symbol-by-symbol de ec o and a Max- imum Likelihood Sequence De ec o (MLSE) based on he appli- ca ion o he Vi e bi algo i hm. Le zp = [ zj )z?) . . . zk) 1' be he ime M- aps impulse esponse o he q h ansmi il e . Once again, i is possible o ep esen in a single ec o z all hese il e s: z = [ zTzT. . . z: 1'. In he design o he ansmi il- e s, he ollowing ansmi powe cons ain mus be ul illed I' 112//2 = ZHZ = P, (1) - A he ansmi e side, only a channel es ima e h o pa ial CSI is a ailable. e ep esen s he e o be ween he channel es i- ma e h and he eal channel ealiza ion h: h = h + e. In he conside ed sys em, his channel e o is due o he own es ima ion p ocess a he ecei e and/o he quan iza ion o he channel es- ima e so ha i can be ansmi ed h ough he eedback channel om he ecei e o he ansmi e . In gene al, we model his e o s a is ically by means o i s PDF, which is assumed o be known: e(e). In case ha no quan iza ion is ca ied ou , hen he s a is ics o he e o would usually co espond o a Gaussian PDF, whe eas in case ha only quan iza ion is conside ed, he PDF would be uni- o m. By making use o his no a ion, i is possible o o mula e he PDF 0: G condi io_ned o he eal channel ealiza ion h as ollows: Glh(hlh) = c(h-h). - - 3. SYSTEM DESIGNS In his sec ion we p esen he wo conside ed design s a egies. The i s one co esponds o a symbol-by-symbol de ec o based on he Minimum Mean Squa e E o (MMSE) c i e ion, whe eas he o he one makes use o a MLSE de ec o by means o he ap- plica ion o he W e bi algo i hm. 3.1. Symbol-by-Symbol De ec o When applying a symbol-by-symbol de ec o a he ecei e , an ad- equa e design c i e ion is MMSE, as i akes in o accoun he noise powe and also he signal dis o ion o In e Symbol In e e ence (1%). In case ha he CSI was pe ec , he equi alen channel im- pulse esponse a he ecei e would be almos equalized. As his is no he case in a eal scena io, we add a il e hn a he ecei e esponsible o equalizing he esidual ISI. When designing his il e , he MMSE c i e ion is conside ed and i is assumed ha he eal channel impulse esponse h is known a he ecei e , as i is also assumed in 171. E(h, h) is he Mean Squa e E o (MSE) o a conc e e chan- nel h and o a conc e e collec ion o il e s z(G) and gain ac o a he ecei e aR(h): - whe e i is assumed ha he symbols s(n).a e no malized so ha E{ls(n)l'} = l,H = [ HlHz...Hq ] isama ixcan ain- ing all he Toepli z con olu ion ma ices {Hq}:=, co esponding o he Q channels {h,}$,, 1 is an all-ze os ec o excep a I in a posi ion ep esen ing he desi ed empo al esponse o he equal- ized channel akHz, and a$ ep esen s he powe o he Addi i e Whi e Gaussian Noise (AWGN) a he ecei e . The con olu ion ma ix H, is de ined as he (M+L- 1) x M dimensional Toepli z ma ix, whe e he i s ow is an all-ze os ec o excep he i s el- emen which is equal o hp), and he i s column is an all-ze os ec o excep he i s L elemen s which a e equal o he ec o h,. Ou goal is o design he il e s so as o minimize he MSE a e aged o e he eal channel s a is ics and he e o s a is ics. This can be exp essed as ollows: IV - 86 As h: il e s z and gain ac o an depend only on he channel es ima e h, he minimiza ion o E is equi alen o he minimiza ion o C(g) subjec o he ansmi powe Cons ain (I). The op imum solu ion co esponds o he ollowing equali ies: whe e a is a cons an such ha he ansmi powe cons ain (I) is ul illed, X(c) = E,,s {HHHIG} and M(G) = Eh,c {HI';}. In gene al i is di icul lo ob ain closed exp essions o X and M. In he Appendix we show how o ob ain a closed exp ession o he case in which he e o e is assumed o be Gaussian. 3.2. MLSE De ec o I is also possible o use o he kind o de ec o s wi h a highe com- pu a ional load bu wi h a be e pe o mance, such as he MLSE based on he Vi e bi algo i hm (see Fig. 2). This de ec o is he op imum one in case ha he channel is known wi h no e o a he ecei e , which co esponds wi h ou assump ions and as p e- sen ed in [7]. The pe o mance o his de ec o is di ec ly ela ed o he SNR, which is de ined as ollows: (6) -1 SNR(h,h) = -zH(c)HHHz(';) whe ei isassumed ha E{ls(n)l'} = 1andzandHa ede ined as in he p e ious subsec ion. Ou goal is o maximize he SNR a e aged o e he eal channel s a is ics and he e o s a is ics. This can be exp essed as ollows: ai SNR = g(g)SNR(K)& (7) s SNR(6) = Ehls {SNR(h,G)IK} = /SNR(h,K) ,,i;(hlc)dh The maximiza ion o he SNR is equi alen o he maximiza- ion o SNR(c) subjec o he powe cons ain (1). The solu ion o his op imiza ion p oblem is ound as an eigen ec o p oblem: z(G) = a" (8) x,,,u = xu, lull = 1 (9) - whe e he ma ix Xis de ined as in he p e ious subsec ion: X(h) = {HHHlc}. A closed exp ession o his ma ix is p esen ed in he Appendix when he e o is assumed o be Gaussian. I can be shown ha his design c i e ion is equi alen o he minimiza ion o he e o powe P, = E {Ie(n)I'}. The equi - alen ime impulse esponse o be used when applying he Vi e bi algo i hm is: hD = aRHz. The gain ac o a~ is a bi a y and does no a ec he pe o mance o he sys em. Usually, a.q is cal- cula ed so ha he mean powe a he inpu o he MLSE block is no malized o he uni y. I mus be said ha in his sec ion we assume ha he Vi e bi decode admi s any leng h o he equi a- len esponse ho, and he e o e, he complexi y and compu a ional load can be e y high. Fu he wo k will analyze o he kinds o de- ec o s based on MLSE bu wi h a lowe compu a ional complexi y by means o a sho ening o ho. . ;...;. A ., , ...... .. ' . . ..{ . < -. .~ Fig. 3. Simula ions esul s o he MMSE echnique. Fig. 4. Simula ions esul s o he echniques based on a MLSE ecei e . 4. SIMULATION RESULTS AND CONCLUSIONS In his sec ion we p esen some simula ions ha help o unde s and he bene i s o using he designs p esen ed in his wo k. We ha e simula ed no malized channels (E{~~hq~~z} = 1) wi h a delay sp ead o 3 symbol pe iods, an angula sp ead o 3(P and BPSK symbols. The leng h L o he channel o he MMSE echnique is 5, whe eas o he case o MLSE is 3. In Fig. 3 we p esen some esul s o he MMSE echnique, in which he ecei e is based on a symbol-by-symbol de ec o . The ansmi il e s zp ha e 7 aps, whe eas bq has IO aps. In hese simula ions, we ha e always made use o he ma hema ical exp essions p esen ed in he Appendix, i. e., we ha e assumed ha he e o is Gaussian. Two di e en si ua ions ha e been an- alyzed. The i s one co esponds o an e o which is ac ually Gaussian, whe eas in he second case he e o is due o he quan- iza ion o he channel impulse esponse. so, in his las case, he 1V - 87 s a is ical model o he e o does no co espond wi h he eali y. We ha e simula ed di e en powe s o he Gaussian e o and di - e en numbe o bi s o ca y ou he quan iza ion ( he numbe o bi s in he igu e ep esen s he numbe o bi s wi h which each ap o he channel is quan ized). As i can be seen, he solu ion ha does no ake in o accoun he e o in he CSI and assumes ha he channel es ima e is pe ec , is no able o dec ease he BER al- hough he SNR is inc eased, whe eas in he case o he solu ion based on he Bayesian poin o iew (Eq. (4)), he BER dec eases as he SNR inc eases. I can be also concluded ha , al hough o he case o quan iza ion he e o model does no co espond wi h he eal s a is ics, he Bayesian design is able o inc ease he pe - oimance in on o he non-s a is ical based solu ion, ha is, he solu ion ha assumes ha he CSI is pe ec . In Fig. 4 he equi alen esul s o he MLSE echnique a e p esen ed o 4 an ennas and a Gaussian channel es ima ion e o . The ansmi il e s ha e only I ap. We ha e also made compa - isons be ween his solu ion, which needs CSI. and he delay di- e si y echnique o I and 4 an ennas, also de ec ed by means o MLSE (Vi e bi). Delay di e si y is a linea p ecoding echnique ha does no need any CSI a he ansmi e . As i can be seen, in his case he gains ob ained by means o he Bayesian app oach a e less impo an han he ones ob ained wi h he MMSE ech- nique. The eason o i is ha he Vi e bi decode is no sensi- i e o non-equalized channels and ha he gains o mean SNR by means o he Bayesian app oach a e no ex emely impo an and do no ha e a di ec impac on he BER. I can be also concluded ha , al hough he noise powe in he CSI is e y high and he qual- i y o he channel es ima e is e y bad, he solu ion based on he MLSE echnique pe mi s inc easing impol an ly he pe o mance o he delay di e si y scheme. 5. APPENDIX In his Appendix we de9 he exp essions co esponding o he ma ices M(G) and X(h) when he e o e is assumed o be Gaus- sian wi h he ollowing PDF E - G (0, E). Unde his assump- ion i can be easily p o ed ha Glh - G (h, E) and hjc - G ( , C), whe e and C a e de ined as ollows: C = (Rhl+E-’)-’ (10) = C (- R,’m+E-’h -1 (11) Deduc ion o M: le us w i e he ec o = Eh,~ {hlc} de- ined in (11) as ollows: = [ y ?. - 6 ]’, whe e , = [ j‘l g).. . g’ 1’ and j‘) = Ehl~ {hj’)lG}. The ma ix M(G) is de ined as M = [ MiMz.. . MQ 1, whe e M, = Eh,i; {H&} is he condi ioned mean o he Toepli z con olu- ion ma ix H, associa ed o he channel co esponding o he q h ansmi an enna. The ma ix M, is he (M + L - 1) x M Toepli z con olu ion ma ix, whe e he i s ow is an all-ze os ec o ex- cep he i s elemen which is equal o @’, and he i s column is an all-ze os ec o excep he i s L elemen s which a e equal o he ec o ,. Deduc ion oJ X: le us w i e he ma ix de ined in (IO) C = Eh,,{(h- )(h- )HIK}as: By means o hese equali ies and ela ionships, he inal ex- p ession o he ma ix X(K) can be ob ained. 6. REFERENCES [I] G. W. Womell and M. T. 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