MATCHED FILTER BOUND OPTIMIZATION FOR MULTIUSER DOWNLINK TRANSMIT BEAMFORMING

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1 MATCHED FILTER BOUND OPTIMIZATION FOR MULTIUSER DOWNLINK TRANSMIT BEAMFORMING Guspp Montalbano? and Drk T. M. Slock?? Insttut Eurécom 2229 Rout ds Crêts, B.P. 193, Sopha Antpols CEDEX, Franc E-Mal: fmontalba, Dpartmnto d Elttronca, Poltcnco d Torno, Torno, Italy ABSTRACT Ths papr s dvotd to th optmzaton of th matchd fltr bounds (MFB) of dffrnt co-channl usrs, usng adaptv antnna arrays at bas statons for downlnk transmt bamformng n cllular mobl communcaton systms. W manly consdr tm dvson multpl accss (TDMA) frquncy dvson duplx (FDD) basd systms. Not that n th cas of tm dvson duplx (TDD), undr crtan assumptons th downlnk channl can b assumd to b practcally th sam as th uplnk channl. On th contrary, whn usng FDD, th downlnk channl can not b drctly obsrvd and stmatd. That maks FDD basd systms most dffcult to dal wth, although th proposd crtra and mthods ar gnral and sutabl for both FDD and TDD. Problm formulatons ar provdd for both spatal dvson multpl accss (SDMA) and non-sdma spctrum rus tchnqus. Novl analytcal solutons and algorthms ar drvd, mplmntaton ssus ar dscussd and smulatons ar provdd n ordr to compar dffrnt approachs. 1. INTRODUCTION AND MOTIVATION Th us of adaptv antnna arrays at bas statons allows ncras th capacty of mobl rado ntworks by an mprovd spctrum ffcncy, n th uplnk as wll as n th downlnk. W nvstgat dffrnt approachs to optmz th wght vctors of adaptv antnna arrays at bas statons for multusr downlnk transmt bamformng. W addrss th problm manly n th contxt of tm dvson multpl accss (TDMA), frquncydvsonduplx (FDD) basd mobl communcaton systms. Th man dffculty n transmsson wth antnna arrays n FDD systms conssts n th lmtd knowldg at th bas staton of th downlnk channl, snc t can not b drctly obsrvd and thrfor stmatd. On th contrary, assumng th mobl vlocty low nough and th rcvr and transmttr appropratly calbratd, th uplnk and th downlnk channls can b consdrd to b practcally th sam n th cas of tm dvson duplx (TDD) basd systms. Whn consdrng th FDD downlnk, th bas staton nds fdback from th mobl about th downlnk channl to oprat n smlar condtons as for th TDD downlnk. Nvrthlss such a fdback nvolvs latncy prods, du to th mobl bas staton roundtrp tm and th procssng tm, rsultng n a dcras of th spctrum ffcncy. Morovr, n a typcal outdoor propagaton nvronmnt, du to th mobl vlocty, such latncy prods ar usually not compatbl wth th fdback rat rqurd to mak th systm rlabl. Ths problm was addrssd by Grlach [5] who proposd to fdback from th mobl only th nformaton rlatd to th downlnk channl covaranc matrx nstad of th channl mpuls rspons, n ordr to rduc th ncssary fdback rat. Howvr, no currnt or concvd cllular standard s dsgnd to support that fdback concpt. On th othr hand, f such fdback s not provdd, th downlnk channl charactrzaton can only b basd on th stmats of thos paramtrs rlatd to th uplnk channl, whch ar rlatvly frquncy ndpndnt and of whch th changng rat s slow wth rspct to th uplnk downlnk png-pong tm. In th prsnc of multpath such paramtrs ar usually th drctons of arrval (DOA), th powrs and th dlays assocatd to ach path (.g., s [2]). Unfortunatly, th phass of th paths ar strongly frquncy dpndnt so that thy cannot b stmatd from th uplnk channl. Actually n FDD mobl communcaton systms, n th absnc of fdback, t s possbl to stmat only th downlnk channl covaranc matrx avragd ovr th paths phass. That maks FDD transmsson mor dffcult to dal wth n practc, compard to TDD transmsson. Nvrthlss, snc th proposd optmzaton crtra assum only th knowldg of th covaranc matrx of th channl btwn ach usr and ach bas staton (vntually avragd n th tm or n th frquncy doman n FDD cas), thy apply for both FDD and TDD. Concrnng th spctrum rus tchnqu w consdrboth spatal dvson multpl accss (SDMA) and non-sdma. For SDMA d co-channl usrs ar allocatd n th sam cll n th prsnc of on bas staton, assumng ntrfrnc from othr clls s nglgbl. For non-sdma th sam usrs ar allocatd n d dffrnt clls,.., n th prsnc of d bas statons wth on usr pr cll. Thn th optmzaton goal s to maxmz th mnmum matchd fltr bound (MFB) at ach mobl rcvr among all th d usrs n ordr to mprov th mobl sgnal qualty and/or spctrum ffcncy. 2. GENERAL ASSUMPTIONS W consdr d co-channl usrs n both non-sdma and SDMA. In th frst cas w shall consdr d bas statons, ach on of thm ts wght vctor to on usr. In th scond cas only on bas staton transmts to all th usrs and optmzs all th corrspondng wght vctors. In addton for SDMA, w nglct th co-channl ntrfrnc. It would b possbl though to mx th SDMA and non-sdma cass. Th mobl s assumd to hav a sngl antnna lmnt whras ach bas staton has an array wth m lmnts. W us th trm downlnk transmt bamformng snc only spatal combnng s usd at a bas staton. Howvr, th mphass hr s not rally on formng bams to crtan drctons, but rathr on xplotng th dvrsty btwn multpl snsors. W assum multpath propagaton but th channl not ntroducng ntr-symbol ntrfrnc (ISI) (.., th dlay sprad ntroducd by th channl s lss than th duraton of a symbol prod). Whn ths assumpton holds th MFB rducs to th sgnalto-ntrfrnc plus nos-rato (SINR). Whn th channl ntroducs ISI, w wll just optmz th SINR nstad of th MFB.

2 3. MFB OPTIMIZATION PROBLEM FORMULATION 3.1. Non-SDMA cas Lt R j and w j dnot th covaranc matrx of th channl btwn th jth bas staton and th th usr (ncludng transmttr and rcvr fltrs), and th wght vctor of th jth bas staton rspctvly. Th SINR for th th usr s gvn by SINR = w H R w j=1;j6= wh j Rjwj + (1) whr = v 2 =a 2 and v 2, a 2 rprsnt th nos powr at th th mobl rcvr and th powr of th transmttd symbols rspctvly. W dnot SINR = for any, throughout th papr. Hnc th gnral optmzaton problm s max fw g mn f g (2) or mn fw g max f,1 g (3) Thn lt w = p p u, wth ku k 2 =1, th vctor of th nvrs MFB s,1 = [ 1,1 :::,1 d ]T and th vctor of th transmt powrs p =[p 1;:::; p d] T,whr T dnots transpos. For non-sdma w nd to lmt th maxmum transmt powr at ach bas staton,.., kpk 1 p max. Th crtron (3) can b rformulatd as mn p; fu g k,1 k 1 s.t. kpk 1 p max, ku k 2 =18 (4) Thn w dfn th normalzd powr dlvrd by th jth bas statontothth usr as For any w hav c j = u H j R ju j :,1 p c = X j6= p jc j + : (5) In ordr to account for all th usrs w ntroduc th matrx D c = dag(c 11; :::;c dd), th matrx C T dfnd as [C T ] j = cj for j 6= 0 for j = th vctor =[ 1 ::: d] T and th matrx P = dag(p). Thn w hav th followng quaton,1 = D,1 c P,1 [C T p + ] : (6) So th crtron (4) gnrally lads to a st of coupld problms whch cannot b solvd analytcally SDMA cas In th SDMA cas w hav only on bas staton n th prsnc of d co-channl usrs. Thn w shall rplac R j wth R for any. So that th SINR dfnton (1) bcoms SINR = w H R w j=1;j6= wh j Rwj + (7) In th prsnc of only on bas staton w nd to lmt th ovrall transmttd powr to b lss than or qual to p max. Thrfor th optmzaton crtron s mn p; fu g k,1 k 1 s.t. kpk 1 p max, ku k 2 =18 (8) Thn by rdfnng c j as c j = u H j R u j w obtan th xprsson of,1 n th sam form as (6). Du to th constrant on th powrs n th cas of SDMA t can b shown that th optmum (4) lads to th sam for all th usrs. Indd f th s, for =1; :::; dar not th sam, thn w can scal th powrs fp g to mprov mn (rfr to [1] for a dtald proof). On th contrary, th optmum gnrally dos not yld th sam for all th usrs n th cas of non-sdma bcaus w cannot arbtrarly scal th powrs p s (.., whn any p = p max no furthr ncras of p s possbl). 4. OPTIMIZATION IN THE ABSENCE OF NOISE In th absnc of nos th SINR bcoms qual to th sgnal-tontrfrnc rato (SIR) whch s nsnstv to th absolut powr lvl. Thn th constrant on th maxmum transmt powr s rrlvant wth rspct to th SIR optmzaton n both non-sdma and SDMA cass. In addton w obsrv that th optmzaton problms for non- SDMA and SDMA hav th sam form and both of thm lad to hav th sam SINR for all th usrs. So that n th squl of ths scton w provd a dtald analyss only for th cas of non-sdma, and th cas of SDMA wll rsult by rplacng th channl covaranc matrcs R j wth R. Th SIR for th th usr n non-sdma s dfnd as SIR = and th quaton (6) rducs to w H R w j=1;j6= wh j Rjwj (9),1 = D,1 c P,1 C T p () whr now = SIR for any. Consdrng th crtron (4) and th dfnton (9) t s straghtforward to s that th optmum s achvd whn all th ntr-usr-ntrfrnc (IUI) s zro so that =0for all s. Thn th optmum approach n th absnc of nos conssts n forcng to zro th IUI, whnvr possbl., Zro-Forcng (ZF) soluton In th absnc of nos at th rcvr th global optmum s achvd whn th followng ZF condtons ar satsfd max ku k 2 =1 uh R u s.t. X j6= p ju H j R ju j =0 (11) Not that th scond condton n (11) s quvalnt to a st of ZF condtons of th form u H R ju = 0,.., c j = 0,forj 6=. Practcally that lads to nullng th IUI whl maxmzng th sgnal-to nos rato (SNR) at th rcvr. In that cas C T = 0dd and th crtron (4) s satsfd whn all th bas statons transmt at th maxmum powr p max. Whn consdrng SDMA th crtron (8) s satsfd whn p = p max=d for all s. Unfortunatly to achv condtons (11) w nd a numbr of antnnas gratr than th sum of all th paths of all th usrs. In typcal mobl propagaton nvronmnts th numbr of all th paths of all th usrs s usually much largr than th numbr of antnnas at a bas staton so that ZF gnrally cannot b prformd wth purly spatal procssng.

3 4.2. Non-ZF solutons Whn th ZF condtons (11) cannot b achvd, othr approachs ar possbl but gnrally th optmzaton cannot b carrd out analytcally for both p and fu g at th sam tm. So w shall optmz p and fu g va a two-stp procdur vntually skng th global optmum usng an tratv algorthm. Bcaus of th nsnstvty to th absolut valu of th powrs w can smplfy th optmzaton problm by rqurng ach vctor u to hav unt norm n th mtrc R,.., u H R u =1for any, nstad of ku k 2 =1. Undr that assumpton, th powr p corrsponds to th powr at th th rcvr whras th actual powr at th rlatd transmttr s gvn by p ku k 2 2,forany. Also, D c = Id n that cas Powr assgnmnt optmzaton W optmz th vctor p assumng a gvn st of drcton vctors fu g. Not that snc th optmum nvolvs = for any, th quaton () rducs to,1 p = A T p (12) whr A T = D,1 c C T = C T s a non-ngatv matrx. Morovr p has to b a non-ngatv vctor and,1 has to b nonngatv as wll. On th bass of th followng thorms ([8],[1]) Thorm 1 For a non-ngatv matrx, th gnvalu of th largst norm s postv, and ts corrspondng gnvctor can b chosn to b non-ngatv. Thorm 2 For a non-ngatv matrx A, th non-ngatv gnvctorcorrspondng to th gnvalu of th largst norm s postv. Thorm 3 Gvn th matrx A thr xsts only on soluton to quaton (12). w can clam that th only postv gnvctor of A T s th on corrspondng to ts largst gnvalu whch s postv as wll. In othr words, that nsurs th xstnc of a postv,1 = max(a T ) and a unqu postv p = V max(a T ) Drcton vctors optmzaton Havng an stmat of p, w can optmz fu g. Indd th optmzaton crtron s gvn by whch s quvalnt to mn fu g max(at ) (13) mn fu g qt A T p s.t. u H R u =1 (14) whr q = V max(a). Th crtron (14) lads to a st of d dcoupld problms whos soluton s gvn by u = q, H R whr = V max(r ; j6= qjrj ) for any. Th nw st of drcton vctors fu g can b usd to r-optmz th powrs p accordng to (12) ka T k 1 mnmzaton basd soluton As ntalzaton w can us th followng crtron mn fu g kat k 1 s.t. u H R u =1 (15) Indd, max(a T ) wll b small whn A T s small. Ths approach has th advantag of optmzng th drcton vctors fu g ndpndntly from th powrs p. It lads to a st of d dcoupld mnmzaton problms whos soluton s gvn by u = q P d, whr, n ths cas, = V H max(r ; j=1 R Rj) for any. Not that th crtron (15) corrsponds to mnmzng th powr dlvrd to th undsrd usrs whl maxmzng th powr dlvrd to th dsrd usr, by ach bas staton n non-sdma, by ach ach wght vctor w n SDMA. Apart from a dffrnt normalzaton of ku k 2, ths crtron s quvalnt to th on proposd n [3, 4] max(a T ) mnmzaton basd algorthm Accordng to th prvous argumntaton, w propos th tratv procdur summarzd n Tabl 1 to fnd th global optmum. Tabl 1: max(a T ) mnmzaton basd algorthm () Intalz u usng (15) for any ; () Comput q = V max(a); P () Gvn q, comput = V max(r ; j6= (v) Comput u = q ; H R (v) Go back to () untl convrgnc; (v) Comput w = p p u,whrp = V max(a T ); k,1 k 1 mnmzaton basd algorthm qjrj ); Anothr gnrally sub-optmal crtron s th followng mn p; fu g k,1 k 1 (16) W rmark that th optmum ylds k,1 k 1 = dk,1 k 1 so that nar th optmum th crtron (16) s approxmatly quvalnt to th crtron (4). Wth (16) as optmzaton crtron w can asly drv an tratv two-stp optmzaton procdur, smlar to th on prvously dscrbd, summarzd n Tabl 2. Unfortunatly whn not nar th global optmum ths algorthm Tabl 2: k,1 k 1 mnmzaton basd algorthm () Intalz u usng (15) for any ; () Comput p = V max(a T ); P 1 () Gvn p, comput = V max(r ; j6= p j R j ); (v) Comput u = q ; H R (v) Go back to () untl convrgnc; (v) Comput w = p p u. can convrg to a local mnmum, rsultng n wors prformanc than th algorthm dscrbd n Tabl 1. An optmzaton crtron smlar to (16) was proposd n [5] assumng a dffrnt cost functon 1 to optmz th downlnk bamformng wghts at a bas staton n an SDMA contxt. Som varants ar possbl to mprov th prformanc and th numrcal robustnss of th algorthm (s [5] for furthr dtals), but convrgnc to th global optmum rmans not guarantd. 1 In [5] rprsnts th rato btwn th powr dlvrd to th th usr and th ntrfrnc gnratd to th othr usrs by th wght vctor w.

4 5. OPTIMIZATION IN THE PRESENCE OF NOISE W showd that whn th nos s prsnt, th dffrnt constrants on th transmt powrs mak th optmum stll ladng to th sam SINR for all th usrs n th cas of SDMA but not n th cas of non-sdma. Morovr, th nos maks th optmzaton of th drcton vctors fu g a st of coupld problms that dos not allow an analytcal approach to fnd a soluton. Although on can obsrv that th crtron (16) for a gvn p stll lads to th sam optmzaton problm for th drcton vctors fu g as n th noslss cas, nvrthlss w shall consdr that such crtron dos not guarant th convrgnc to th global optmum anyway. Thrfor, w suggst to comput th vctors fu g by usng th algorthm n Tabl 1 prvously drvd n th absnc of nos and thn optmz th powr assgnmnt accordng to th followng crtron Powr assgnmnt optmzaton for SDMA Assumng a gvn st fu g and all th s th sam, th xprsson (6) can b arrangd n ordr to nclud th constrant on th transmttd powr as follows whr ~p =[p T 1] T, A T B = 01d 0 B ~p =,1 G~p (17) G = Id g T 0d1,p max whr = D,1 c and g =[ku 1k 2 2 ::: ku dk 2 2] T s dfnd n ordr to hav g T p = p max. Thn as n to [1] snc G s nvrtbl w hav E ~p =,1 ~p; E = G,1 B = 4 AT g T A T g T pmax pmax (18) whch s a non-ngatv matrx. Hnc accordng to thorms 1 3, ~p can only b th gnvctor assocatd to th maxmum gnvalu of E. Furthr, not that w can always r-scal ~p n ordr to mak ts last lmnt qual to on Powr assgnmnt optmzaton for non-sdma In th cas of non-sdma w cannot nsur that th optmum ylds th sam SINR for all th usrs. So that th optmzaton problm cannot b tratd va any analytcal approach. As a possbl sub-optmal soluton, gvn th vctors u computd n th absnc of nos, w st p : kpk 1 p max n ordr to satsfy th crtron (4). 6. IMPLEMENTATION ISSUES W rmark that vn though th computaton of u rducs to a st of dcoupld problms as n th absnc of nos, th computaton of p gnrally rsults n a coupld problm, unlss ZF condtons (11) ar satsfd. Hnc n a non-sdma scnaro, dffrnt bas statons nd to communcat n ordr to choos th rspctv transmt powrs. On th contrary ths problm s not prsnt n SDMA. Whn th nos s prsnt, snc th bas staton cannot stmat th nos varanc 2 v at ach rcvr, unlss such an stmat s provdd by th mobl, th vctor cannot b stmatd. To rmdy ths drawback w shall proprly dfn th SNR at th rcvr. A possbl dfnton s gvn by SNR = p max(r ) for any. In practc w nd mnfsnr gsnr mn (19) whr SNR mn s a valu ncssary at th rcvr to work wth an outag probablty blow a spcfd maxmum. Assumng all th usrs usng th sam rcvr th worst cas for th th usr occurs whn p = p max whl = max = kk 1. Thrfor a suffcnt condton to satsfy th rqurmnt (19) s gvn by sttng SNR mn = pmax mnf max(r max )g () Gvn SNR mn and p max, max can b drvd. Thn sttng = max for all th s th condton (19) s satsfd. Fnally, not that for p max!1th optmum soluton s th on n th absnc of nos, for any max 6= SIMULATIONS In ths scton w consdr an SDMA scnaro n th prsnc of d =3co-channlusrs n th absnc of channl dlay sprad but n th prsnc of angular sprad du to multpath propagaton. Th numbr of all th paths of all th usrs s 14 (4 for th frst two usrs and 6 for th thrd, rspctvly). An antnna array wth m =8or m =6lmnts s assumd at th bas staton. Thn ZF condtons (11) cannot b appld. Fgurs 1 and 2 show th convrgnc curvs of th frst and th scond proposd algorthms, for m =8and m =6rspctvly, n th prsnc of th sam usr scnaro. As xpctd th scond algorthm prforms wors whn not nar th optmum. Th varant proposd by Grlach [5] n svral cass maks th scond algorthm prform mor closly to th frst on, but at th cost of an ncrasd computatonal complxty and a lowr convrgnc rat. SIR (db) No. of tratons Fgur 1: Convrgnc curvs for th two proposd algorthms, d =3and m =8 SIR (db) No. of tratons Fgur 2: Convrgnc curvs for th two proposd algorthms, d =3and m =6

5 SINR (db) var SNR mn (db) Fgur 3: Optmum SINR vs. SNR mn for th 1st, th 2nd and th varant n [5] of 2nd algorthm, for d =3and and m =8 SINR (db) var SNR (db) mn Fgur 4: Optmum SINR vs. SNR mn for th 1st, th 2nd and th varant n [5] of 2nd algorthm, for d =3and m =6 Fnally fgurs 3 and 4 show th prformancs of th two proposd algorthms and th Grlach s varant of th scond algorthm n th prsnc of nos, assumng th crtron (18) for th optmzaton of th powr assgnmnt. Onc agan w rmark that non of th prvous algorthms nsurs th convrgnc to th global optmum n th prsnc of nos. That xplans why th scond algorthm varant (crtron (16), can prform bttr than th frst on, as shown n fgur RELATION TO PREVIOUS APPROACHES Som smlarts among th approachs proposdhr and othr ons alrady dscrbd n th ltratur hav alrady bn pontd out. Svral approachs [2] [6] assumd th cost functon gvn by th SIR at th transmttr nstad of at th rcvr,.., SIR was dfnd as th th rato btwn th powr dlvrd to th th usr and th ntrfrnc gnratd to th othr co-channl usrs by th wght vctor w. That cost functon was both optmzd usr by usr [2, 3, 4, 6], wthout assumng any multusr powr assgnmnt optmzaton, ladng to th sam crtron as (15), and consdrng th multusr powr assgnmnt optmzaton [5]. Othr authors [7] consdrd th possblty of applyng ZF condtons to only th domnant path of ach usr. A fnal rmark concrns th powr assgnmnt optmzaton prsntd n [1]. Indd n that papr th authors consdrd only th spatal sgnaturs of th usrs nstad of th channl covaranc matrcs for th formulaton of th problm (18). Hnc, n th prsnc of channl dlay sprad th formulaton n [1] s not approprat and dos not lad to th optmzaton of th SINR of all th usrs. In ths papr w addrssd th problm of th optmzaton of th MFB wth rspct to th bamformng wghts at bas statons, for multusr downlnk transmsson n both non-sdma and SDMA spctrum rus tchnqus. A gnral problm formulaton yldd th dfnton of a propr cost functon to b mnmzd. Thn w consdrd th optmzaton problm n th absnc of nos. In that cas th ZF soluton rprsnts th optmum, but t can b achvd only undr crtan condtons usually not vrfd n practc. Thrfor a novl algorthm to fnd th global optmum n th absnc of nos has bn drvd by an analytcal approach whch dos not rqur th condtons ncssary to th ZF soluton. Th novl algorthm has shown to outprform othr sub-optmal algorthms basd on dffrnt optmzaton crtra, n trms of optmum MFB, convrgnc rat, numrcal robustnss and computatonal complxty. W rmarkd that th lattr sub-optmal ar vry smlar to othr ons alrady proposd n th ltratur [3, 4, 5], gnrally drvd startng from dffrnt cost functons. Th optmzaton problm n th prsnc of nos has also bn addrssd, consdrng th constrants on th transmt powr nhrnt to both SDMA and non-sdma spctrum rus tchnqus. At prsnt, w ar not awar of any analytcal approach to fnd a global optmum n that cas and w ar currntly nvstgatng non-analytcal optmzaton approachs. Only th assgnmnt of th transmttd powrs [1] can b asly optmzd n th SDMA cas for a gvn st of drcton vctors. So that for that am w propos to us th st of drcton vctors optmzd n th absnc of nos through th algorthm w dscrbd n Tabl 1. Fnally w analyzd va smulaton how optmum and sub-optmum algorthms drvd n th absnc of nos, prform n th prsnc of nos wth th approprat powr assgnmnt optmzaton.. REFERENCES [1] W. Yang and G. Xu, Optmal downlnk powr assgnmnt for smart antnna systms, Proc. ICASSP 98, Vol. 5, pp , May [2] G. G. Ralgh and V. K. Jons, Adaptv antnna transmsson for frquncy duplx dgtal wrlss communcaton, Proc. ICC 97, Vol. 2, pp , Jun [3] P. Zttrbrg Mobl cllular communcatons wth bas staton antnna arrays: Spctrum ffcncy, algorthms and propagaton modls, Ph.D. thss, Royal Inst. of Tchnology, Stockholm, Swdn, [4] J. Goldbrg and J. Fonollosa, Downlnk bamformng for cllular mobl communcatons, Proc. VTC 97, Vol. 2, pp , May [5] D. Grlach, Adaptv transmttng antnna arrays at th bas staton n mobl rado ntwork, Ph.D. Thss, ISL, Stanford Unvrsty, Stanford, USA, [6] G. G. Ralgh, S. N. Dggav, V. K. Jons, and A. Paulraj, A blnd transmt antnna algorthms for wrlss communcaton, Proc. ICC 95, Vol. 2, pp , Jun [7] G. Xu and H. Lu, An ffctv transmsson bamformng schm for frquncy-dvson-duplx dgtal wrlss communcaton systms, Proc. ICASSP 95, pp , [8] R. A. Horn and C. R. Johnson, Matrx analyss, Cambrdg Unvrsty Prss, CONCLUSIONS