Comparative Study of SVD and QRS in Closed-Loop Beamforming Systems

1-4244- 1513-06 /07/$ 25.00 ©200 7 IEE E 1 of 4 COMPARATIVE STUDY OF SVD AND QRS IN CLOSED-L OOP BEAMFORMING SYSTE MS Ch au Y ue n , Sume i Sun Ji an- Kan g Zh ang In sti tute fo r In fo com m Re se arch (I 2 R), Sing a po r e McMaste r Un ive rs ity , Canad a {c y u en , sun sm }@i 2r.a-s ta r.e du .sg jk zhan g @m a il . e ce . mcm aste r.ca Abstract: We co mpa r e two clos ed -lo op b ea m f or min g algo rithms, one based on sing u lar val u e decom p osition (S V D) and the other based on equal diagon al QR decom position ( QRS). S V D has the adv antage of par alle lizing the MIMO ch annel, but ea c h of th e su b - c ha nn e ls ha s di f f er ent ga i n. QR S ha s the advantage o f having eq ual di agonal v a lue fo r the decom p osed ch a nnel, but the subc hannel s ar e no t fully pa ra ll eliz ed, h e nc e r e quir i n g s u cc ess i v e i nt er f er enc e can cellation or othe r techniq ues to p erform decoding . We co ns ider a cl os ed-l oop s yst em wh er e th e f eedbac k i nf or ma t ion is a un it a r y b ea mf or min g mat r ix . Du e t o the discrete and limi t ed mod u lation se t, SVD m a y ha v e i nf er i or p er f or ma nc e t o Q R S wh e n n o mo dulation set sele ct ion i s p erfo r me d . Howeve r, if the sele ction of mod u lation set i s pe rf o r med optim a lly, we sho w th at SV D can o utp e rform QRS. Keywords: c losed -loop beam for ming , singular value decom p osition , equal diagon al decom p osition , geom etry m e a n dec om p osition . I. I NTRODUCTIO N W e c o nsi d er a mult ip l e-t r a ns mit a n d mu lt ip l e- re ceiv e an ten n as w i rele ss co mm unic atio n s sys tem (deno t ed as MIMO). Such syst em is well-known f or ach ieving a highe r capa ci ty t han tradition a l single - tran s m it single -r e ceive an t enn a system s [1]. It has been ado pted in m a ny wirele ss c omm unication stand ar ds, w hich incl u de IEEE 802.11n fo r wi r eless loc al area ne twork (LA N) applic ation , 3GPP Long T er m E v olu t i on (L T E) for ce l lu la r c ommu n ic a t i o ns . De pending on the channel condi tion, the f irst gene r ation MIMO techniq u e aims at achieving a hi gh er da t a ra t e, s uc h a s s p a t ia l mu lt ip l ex i n g [2 ], or a high e r diversity , such as space- t ime codi ng [3]. These te chnique s do no t req u ire the knowledge of ch a nnel state in f o r m ation ( CSI) at th e tr ansm i tt e r. Due to t h e a dvanc ement i n t he c o mmun icat io n te chnique , limi t ed f eedback inform ation i s possi ble in the future wirele ss c omm unication s sy st em . It has been s hown in [ 4] t hat the feed ba ck of info r m ation on t h e CS I c a n gr ea t l y i mp r o v e t h e s ys t em p er f or ma nc e. In addition , closed-loop be a mfo r ming can a lso achie v e lowe r decoding comple x ity, as we will s how in the paper. A strai gh tfo rw ard be a m form ing algo ri th m is based on singular value dec omposition (SVD) t hat fully diagon alizes the channel. Howeve r, e ach of th e paralle l sub- channel s has di ffe r ent g a in. In o r de r to achiev e b ett er p erf or mance, d iffer ent modu la tion ca n be applied to diffe r en t stream of d ata, but th is req uir e s addi tion al feedback in f orm ation. Un fortun at ely , this solu t i o n is not op t i mal as th e modu la t i ons a r e “ dis cr et e” ( i. e. mo du la t io n s iz e is i n t h e f or m of p ow er of t w o). Moreove r, assigning the “a ppropriate” mo dul ation fo r e ach of the data stre a m in re al time is al so a ch alle ngin g task . In t his pape r, we conside r a no t her b e a m for ming algo rit hm bas ed on equal-diagonal Q R decom position (QRS) from [5] (also known a s geometry mean deco mposition , GMD, in [6]). Unlike S VD, QRS does no t fully diagonalize t he channel, but conve rts th e eq u ivalen t channel in t o an upper trian g ular matrix. C o mpa r e d wit h S VD, Q R S ha s t he a d va nt a g e of h aving eq ua l di agonal e lem ents for its eq u ivalen t ch a nne l. Thi s im plies th a t the sa me mod ulation ca n be applie d to all the data stream s. H ence it elimin a t e s the trouble of assignin g dif ferent mod u lation for di ffe r ent s t r ea m of da t a . It h as b een shown in [7] t hat by using a simple an d ef fi cie n t clo sed -loo p fe ed back , the pe rfo rm an ce of the MIMO- OFDM system f or wi rele ss L AN can b e g r eatl y improved . The e xtra cost is sim ply f ed back the un itary precode rs fr om S VD or QRS. Howeve r , no compa r ison b etween the S VD a nd QRS b e a m for ming al go rithm s can be fo und in t he li te ratu re . In t h i s pa pe r, we will provide a comparat ive study on SVD and QRS, w ith an d wi thout mo du lation set sele ct ion . The organization of this paper i s as follow s: w e fir s t int r o du c e t h e S VD a n d Q R S d ec o mp os it i on i n se ct i o n II , t h e de c o mp os it i o n w h en t h e nu mb er o f stre am s i s le ss th an the num be r o f tran smi t and re ce ive antenn a s for QRS scheme will al s o be di scussed. Ne xt, we w ill pe rfor m simul ation to com par e the two decom p osition sc heme s in Sec t ion I II. I n the first co mp a r is o n, we do n ot p er for m modu la t i o n s et sele ction , wh ile in the s e cond com parison , we perform 2 of 4 mod u lation s e t sele ction . We show t hat wi thout mod u lation s e t sele ction , beamfo r mi ng system bas ed on QRS can o utp erfo r m t he one based on SV D, a nd vice ve rsa if with mo dul ation se t s ele ction . Finally w e concl u de the pape r in Section IV. II. SVD AND QR S D ECOM POSITION Co nside r a poin t-to-point MI MO sy stem with M tran sm it and N re ceive ante nn as. Th e N –by–1 r eceive sign a l y can be modele d as foll ows: y= H x + n (1) w h er e H is th e N –by– M cha nn el co eff ici ent a nd x is the M –by–1 transmi t t ed s ignal with n being the A WGN noise. In this paper, we conside r Ra yleigh fading channel , so H c o ns is t s of z er o- mea n c o mp l e x Gaussian random variable s. To ach ie ve a b e tt e r de codin g pe rform ance and lowe r decoding com plexity, be a m for ming can be applied . We f irst de c ompose the channel m atrix H by SVD or QRS as follows: * * SVD: QRS : H = UDV H= Q R S (2) w h er e U , V , Q , S ar e all u n it a r y mat r ic es, D is a diagon al m atr ix with singular val u e s a s its eleme nts, while R is an uppe r tr ian g ular m atr ix w ith identic al diagon al ele me nts. Supe rscript * deno t e s con j u gate tran spose . T h e D and R are both of dime nsion N -by- M , and t h e r a nk, d , is li mit ed b y t he mi n i mu m nu mb er of M a nd N, i. e. () min , dM N ≤ (3) and for SVD, the diagonal m atrix D co ns is t s of t h e sing u lar value s of the channel: () 12 , , .. ., , 0, 0 d dia g δδ δ = D (4) w h er e δ 1 , δ 2 , … δ d are the singular v alues. W hile for QRS, the diagon al value, r , in R is t h e geomet r ic mea n of t h e s i ngu la r va lu e: () 1/ , 1 d k rk d δ =≤ ≤ ∏ (5) We con side r the clo sed-loop be a m forming system whe r e only a uni tar y m a trix, i.e . V in SVD and S in QR S , a nd t h e s e l ec t i o n of t h e mo du la t i on s et ca n b e fed back from t he re c eive r to the tr ansm i tt e r. A) SVD Beamform ing First c o ns ide r SVD beam forming , the transm i tt ed sig na l is : x=V u (6) w h er e V is t h e u nit a r y b ea mf or mi n g ma t r ix ob t a in e d fr om S VD d ec o mp os it i o n, a nd u is the intended d ata si gn al . The re cei ve r of the S V D be a m fo rm in g sy stem c an pe rfo r m th e fo llo wi ng : () =+ ** * * U y = U UDV Vu + U n Du n  (7) Wh er e t h e n o i s e n  is s t il l AGW N a s U is u nit a r y. Hence simple decoding can be ac h ieved , as D is a diagon al m a trix. Howeve r , each o f the sub-ch a nnel ha s a dif f er e nt ga i n i n t h is c a s e, a s t he d ia g o na l va lu e of D is r ela t e d t o t h e ei g en va lu e of t h e cha nn e l mat r ix H . As a re sult, to achie ve optim a l pe rform a nce , dif ferent d ata stre a m s sho uld use a diffe r en t modu la t i on, h ow e ver t h is w ou l d r equ ir e ext r a feed ba ck inform ation, i .e., in formatio n on D needs to be fed back to the transm itte r on to p o f V . Wh e n t h er e a r e f e w er nu mb er o f da t a st r ea ms t ha n th e max i mu m al l o wa b l e, i. e. th e mi n i mu m nu mb er of tr an sm it and rece iv e ante nn as, we can pe rf o rm the SVD de composi t ion based on the first few principle eigenve ctors. Fo r exam ple , in a fo ur tra nsm i t four re ceiv e an tenn a sy stem , i f we are only r eq ui red t o tran sm it n stream s, whe r e n < d , we can perfo rm SV D with V n , w h er e V n con s ists o f the first n pr in ic ip l e eigvenve cto rs of H . B) QRS Beamf orming N ext c o ns i d er Q R S b ea mf or mi n g, s i mi la r ly t h e t ra ns mitt ed s i g na l is : x=S u (8) w h er e S is t he u n it a r y b ea mf or mi n g mat r ix obt a in e d fr om QR S d ec omp os it i o n, a nd u i s th e in te n d e d d at a s i g na l . T h e r ec e i v er of t h e Q R S b ea mf o r mi n g s ys t e m c an pe rfo r m th e fo llo wi ng : () =+ ** * * Qy = Q Q R S S u + Q n Ru n  (9) Si n ce R is a n u p p er t r ia ngu la r mat r ix, su c cess i v e in t e rf e rence c anc ell ation (SIC) o r othe r a lgorithm ca n be perfo r med at the receive r t o decode data. The adv antage of QRS is th e diagonal elem ents o f R are iden tical; thi s im p lies th a t we ca n apply t he same mo dulation to all the data stre a m s. When we only need to tr a ns mit n stre a m s, whe r e 3 of 4 n < d , we ca n p er for m t h e Q R S de c o mp os it i o n b a s e d on the required eigen subchannels, H n , wh e r e H n is defined a s: nn = HH V (10) a nd V n consi sts of the fi rst n pr in iciple eigvenve ctors of H . III. S IMULATION R ESULTS We con side r a Ray leig h fl at fad ing u nco ded MIMO sy stem with f our transm it an d four re ceive ante nnas. In addi t i on, th r ougho ut the s imulation , it i s assu m ed th at th e chan ne l is e stim ated corre ctly , and the beamform ing m a trix is perfectly known at the t ra ns mit t er w it h out a ny d ela y. S I C w il l b e emp l oy e d fo r system with Q RS-based be a m forming. The tra ns mi ss io n is fix ed a t s pect r a l e ff ici enc y of 8 bps/Hz, w hich m a y be realized wit h two, th r ee or four da t a s tr ea ms. T he a va ila b l e mo du la t i o n s et s f or S VD a nd Q R S a r e a s fol l o ws : Modulation for SVD w ith 8bps/Hz: Two stream s: {QAM64- QPSK} Two stream s: {QA M16-QAM16} Thre e s tre am s: { QA M16 -Q PSK- QPSK } Th ree stre am s: { QAM 8- QA M8 -Q PSK} Modulation for QRS w ith 8bps /Hz: Two stream s: {QA M16-QAM16} Thre e s tre am s: { QA M16 -Q PSK- QPSK } Th ree stre am s: {QA M8 -QA M8 - QP SK } Fo ur st ream s: {QPSK - QPSK -Q PS K- QP SK} Due t o the uneven natur e on the diagon al values of t he e ff ec t ive channel on SV D, we have a two stream s u neven mod ulation QA M64-QPSK fo r SVD . In contrast, due t o the equal v alue on the diagon al val u e s of the effective ch a nnel on QRS, we h ave a fou r s t r ea ms equa l mo du la t i o n Q P S K for QR S. We com pare the BE R pe rf o rm ance in Fi g ure 1 when t he selec tion of mod ulation se t is no t allowed , i.e. same modul ation is used thro ughout the sim u latio n. For SV D, mod ulation se t with {16QA M- 16QA M} and {16QAM-QPSK-QPSK} g ive the best BE R p er for manc e. F or Q R S , modu la t i o n s et w it h {16QA M-16QAM} give the b e st BER p erfo r m a nce . S o fr o m Fi gu r e 1, we ca n c o nc lu de t ha t wit hout t h e modu la t i o n s et s e l ect i on, Q R S gr ea t ly ou t p er for ms S VD a t hig h S N R r egi o n, a b ou t 1 dB a t BE R of 1 0 -3 . 4 5 6 7 8 9 10 11 12 13 14 10 -4 10 -3 10 -2 10 -1 10 0 SN R BER SVD QA M64- QPSK SVD QA M16- QAM 16 SVD QA M16- QPSK -QPSK SVD QA M8- Q AM8 - Q PSK QRS QA M1 6-QA M 1 6 QRS QA M1 6-QP S K -QP S K QRS QA M8 - QA M 8 -QP S K Q R S Q PSK-Q PSK-Q PSK- QPSK Fig ur e 1 Sim u lated BER for 4tx-4 r x at 8bps /Hz wi tho ut mod ulation s et se lec tion We the n com pare BE R pe rfo r m an ce in Fig ure 2 whe n the s elec tion of mod u lation s et i s a llow ed . The sele ction is optim a l, as the selection is pe rf orm ed “of fline” , i.e. we p ick t he mod ulation se t th at gives the be st B ER. H ence th i s se rve s as a l owe r bo und for t he actual B ER perfo r m a nce wi t h “online ” mod u latio n set sele ction . F r o m F i g u r e 2 , w e o b s e r v e t h a t b y a l l o w i n g t h e sele ction of mod u lati on se t, SV D give s a bette r ove rall pe rfo rm an ce , abo ut 1d B at B ER o f 10 -3 . The gap be tween the sele ction of mod u lati on s e t versus the fi xed single mod ulation se t {Q AM16- QAM16} i s 4. 5 dB f or S VD a t BE R 10 -3 , while t he gap is 2.5d B fo r QRS at BER 10 -3 . T h is sug gests th a t the p e rfor m a nce o f SVD relie s heavily on the sele ction of the modu la t i on s et , w hic h r e qu ir es a dd it i o na l f ee db a c k in for m a tion . In addition , it stil l r em a ins un c le ar on th e o ptim a l w ay to selec t mod ulation se t in a pra cti cal MIMO O FDM syste m when i t involve s a large nu mb er of s u b -c a r r ier s . T he ga i n t ha t is d emo ns t r a t e d in Fig ur e 2 i s based on o ptim a l sele ct ion, the actual g ain would be r educed due t o non-o pt im a l s ele ction a nd d ela y i n f e edb a c k. 4 of 4 4 5 6 7 8 9 10 11 12 13 14 10 -5 10 -4 10 -3 10 -2 10 -1 SN R BER S V D QA M1 6-Q A M16 (repe at f rom p reviuos f igu re) QRS Q A M 16 -Q AM 16 ( repe at fro m pr eviuos figure ) S V D s ele c t in g t h e be s t mo dul at i o n s et QRS s el e c t i n g t he bes t mo dul at i o n s e t Fig ur e 2 Sim u lated BER for 4tx-4 r x at 8bps /Hz wi th mo dul ation se t s e lec t ion Fo r SVD , a te chnique called “powe r loading ” c a n hel p t o m i t igate the di sadvan tages o f disc r e t e modu la t i on, b ut it r equ ir es a ddit i o na l c o mpu t a t i o na l comp l exity a nd feedba c k over hea d. IV. C O NCLUSION W e ha v e s h ow n t ha t c l os ed- l oop M I M O s ys t e m with E qual-Diagon al QR decom pos ition (QRS ) m a y ou tp e rform the one based on SVD decom p osition . T his is du e t o fa ct t ha t t h er e is o n l y a l i mit e d c h oi c e o f modu la t i on s et s , a nd t h e a va ila b l e mo du la t io n s ets a r e disc rete , hence ge tt ing the optim a l m od u lation s e t m a tched to the ch annel gain provided by S V D m a y not be alway s p ossi ble. Howeve r , when the sele ct ion of mo du la t i o n s et is a l l o w ed, S VD ma y a c hi e v e a be tter pe rf orm ance . We f urther show th at the pe rfor m a nce of SV D r elie s he avily on the selec tion of the appropriate m odulati on s e t. In [8], pse u do-inve rs e i s pr oposed to achieve b ette r decoding p erforman ce . It would b e interesting t o look a t t he c ha n n e l dec o mp os it i o n of t h e equ i va l ent pse udo-i nve rse channel, for bo t h SV D and QRS. 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