Lattice-Reduction-Aided Equalization and Generalized Partial- Response Signaling for Point-to-Point Transmission over Flat- Fading MIMO Channels

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1 Lttice-Reductio-Aided Equliztio d Geerlized rtil- Respose Siglig for oit-to-oit Trsmissio over Flt- Fdig MIMO Chels Robert F.. Fischer Lehrstuhl für Iformtiosübertrgug, Friedrich Aleder Uiversität Erlge Nürberg, Cuerstrße 7/LIT, Erlge, Germ, Emil: Abstrct oit-to-poit commuictio over flt-fdig MIMO chels is coside. Kow equliztio d preequliztio schemes re reviewed d comp. I prticulr, lttice-uctio-ided techiques re coside, d strtig from them ew pproch is derived. This proposed techique c be iterpreted s geerliztio of prtil-respose siglig, kow from itersmbol-iterferece chels. The mi dvtge of this lowcompleit scheme is tht ol ver little chel-stte iformtio is requi t the trsmitter d performce close to mimum-likelihood detectio is possible. Moreover, chel codig c directl be pplied. 1 Itroductio Commuictio over MIMO chels, e.g., b usig te rrs, is ver iterestig becuse of the ver high spectrl efficiecies d hece dt rtes which c be chieved. The multi-te iterferece preset i such scerio hs to be combtted with some mes of equliztio. Over the lst ers, equliztio techiques kow from itersmbol-iterferece (ISI) chels hve bee trsfe to the MIMO settig, cf., e.g., [3]. Besides this, ew techiques bsed o lttice bsis uctio hve bee recetl developed [1], [15]. I this pper, equliztio schemes for poit-to-poit trsmissio over flt-fdig MIMO chels, i prticulr lttice-uctio-ided techiques, re reviewed d comp. Equliztio/pre-equliztio structures beig the respective couterprt to ech other re idetified. Besides this, ew pproch which c be iterpreted s geerliztio of prtil-respose siglig (well kow o ISI chels) is derived. These schemes require ol ver limited chel kowledge t the trsmitter side but perform rther close to much more comple mimum-likelihood detectio. I Sectio the chel model is give d Sectio 3 briefl reviews covetiol (pre-)equliztio schemes. Sectio 4 discusses lttice-uctio-ided detectio d presets geerlized prtil-respose siglig for MIMO chels. Numericl results re give d discussed i Sectio 5; Sectio 6 drws some coclusios. Chel Model Cosider multiple-te poit-to-poit trsmissio over flt-fdig chels. Over ech of the N T trsmit tes dt smbols c,µ, µ =1,,...,N T, tke from M -r squre QAM sigl costelltio A c (vrice σ def =E{ c,µ }, µ), re trsmitted. Vritio of the costelltio (rte/power lodig, dptive modultio) is ot coside here. The dt smbols re combied ito the vector 1 c =[ c,1,..., c,nt ] T. The chel is chrcterized i the equivlet comple bsebd [1] b the fdig coefficiets h c,kl, collected i the N R N T chel mtri c =[h c,kl ], betwee ech pir of trsmit/receive te. Sice, due to the flt-fdig ssumptio, ech smbol itervl c be processed o its ow, we do ot itroduce time ide. The receive smbols c,k t the N R receive tes (combied ito the vector c = [ c,1,..., c,nr ] T ) re give b the usul iput/output reltio c = c c + c. c =[ c,1,..., c,nr ] T is the vector cotiig the comple-vlued oise smbols d is epected to be white with vrice σ per compoet, i.e., E{ c c } = σi. Throughout the pper we ssume N T = N R ; the geerliztio is esil possible. The comple-vlued MIMO chel model is equivletl writte s K-dimesiol (K =N def T ) relvlued MIMO chel model ccordig to [ ] [ ][ ] [ ] R c R c I = c R c R c + I c I c R c I c I c (R d I deote rel d imgir prt, respectivel), or i short usig obvious defiitios = +. (1) Due to the ssumptio of squre QAM costelltios A c, ech compoet µ of is idepedetl drw 1 Nottio: A T : trspose of mtri A; A : ermiti (i.e., cojugte) trspose. I: Idetit mtri. E{ }: epecttio.

2 from oe-dimesiol M-r ASK costelltio A = {±1/, ±3/,...,±(M 1)/}. All subsequet discussios re purel bsed o the rel-vlued MIMO chel model (1). Lier equliztio techiques (icludig sigulr-vlue decompositio) d mimum-likelihood pproches perform ectl the sme whe derivig them for the comple or rel chel descriptio, respectivel. owever, due to lrger umber of degrees of freedom, successive decodig or ecodig schemes show better performce whe strtig from the rel-vlued model, cf. [5]. 3 Equliztio Strtegies We ow briefl review trsmissio schemes for poitto-poit trsmissio over MIMO chels. Cotrr to multipoit-to-poit (multiple ccess; uplik) or poitto-multipoit scerios (brodcst chel; dowlik) here processig c be either doe t receiver or trsmitter d splittig betwee both sides is possible. For tht we ssume throughout tht perfect chel stte iformtio (CSI) (the chel mtri ) is vilble t the receiver side. If (prtil) CSI is lso requi t the trsmitter side (TX CSI), this fct is eplicitel stted. lese ote tht ll equliztio techiques re optimized ccordig to the zero-forcig (ZF) criterio. I ech cse optimiztio ccordig to the miimum me-squ error (MMSE) criterio is possible, too. This strteg i fct offers dditiol gi but o ew isight ito the fudmetl properties of the equliztio schemes d their compriso is provided. 3.1 Lier Equliztio d Lier re- Equliztio I lier equliztio t the receiver side the decisio vector is geerted ccordig to r = 1. It is well-kow tht lier equliztio suffers from oise ehcemet d hece hs poor power efficiec. If the chel mtri is kow to the trsmitter (e.g., b commuictig it to the trsmitter vi bckwrd chel) the dt strems c be seprted vi lier pre-equliztio [4], [8]. ere, the vector of chel smbols µ, which re fed ito the chel isted of the dt smbols µ itself, is geerted s = ς 1. The rel costt ς is chose such tht verge trsmit power equls tht of direct trsmissio of the dt vector (short term power costrit). ς is compested t the receiver side b proper sclig (utomtic gi cotrol, AGC). Eforcig the short term power costrit, lier pre-equliztio shows (ecept mior differeces) the sme performce s lier receiver side equliztio. This strteg is lso kow uder the me chel iversio (e.g., [8]) d the mbiguous deomitio zero-forcig, which, however, should ol be used s ddedum to emphsize the optimiztio criterio. Figure 1 (top) shows the trsmissio schemes emploig lier equliztio d lier pre-equliztio. 3. Decisio-Feedbck Equliztio d recodig erformce c be improved if decisio-feedbck equliztio (DFE) is pplied. ere, decisios l vilble re used for iterferece ccelltio. DFE is lso kow s successive ccelltio i multiuser detectio [] d is the equliztio priciple emploed i the (V)BLAST (Bell Lbortories Le Spce- Time) scheme [19], [7]. The process of successive ccelltio (subtrctio) of iterferece of l detected dt smbols, lier filterig for suppressio of iterferece cotributed b ot et detected dt smbols, d detectio (threshold decisio) c be depicted s i the middle of Figure 1. The receive vector is processed b feedforwrd mtri F, which hs the tsk to trsform the chel ito lower trigulr shpe with uit mi digol. Due to this structure, the smbols c be detected i sequece. The iterferece of l detected smbols is subtrcted b the strictl lower trigulr mtri, where B deotes the feedbck mtri. I geerl, for optimum performce the dt smbols should be detected i optimized order [7], described b permuttio mtri. I the lst step, vi, the origil order is reestblished. Give, the mtrices re clculted such tht [4], [16] = F 1 B, () i.e., sorted QR-tpe fctoriztio 3 hs to be performed. Defiig criterio of optimlit (usull mimum sigl-to-oise rtio i ech detectio step), the fctoriztio is uique d c be performed s, e.g., give i [19], [7]. DFE suffers from error propgtio i the feedbck loop d requires immedite decisios, which complictes the pplictio of chel codig. Both problems c be voided if DFE is replced b trsmitter side precodig. Bsicll, Tomliso-rshim-tpe precodig [4] is derived from DFE b flippig the etire structure, see Figure 1. ere, the dt smbols re first permuted b d the processed b the olier feedbck loop with feedbck mtri B. The olier modulo opertio, which hs to mtch up with the ctul sigl costelltio A, restricts the output smbols b ddig iteger multiples of M to the support regio [ M/, M/) 3 Give the permuttio, the requested fctoriztio c be performed usig stdrd QR decompositio, i.e., M = QR, where Q is uitr (QQ = I) d R is upper (right) trigulr. Usig the ti-digol idetit mtri J (idetit mtri I with reversed colum order; J = I), we perform: J = QR. ece: = QJ JRJ = def Q L, d usig D = dig(l 11,l,...), with L =[L ij ], we rrive t = Q D D 1 L = def F 1 B, where F 1 hs orthogol colums d B is lower trigulr with uit mi digol.

3 r 1 ς 1 F ςf Algorithm Algorithm d ς 1 Fig. 1. Trsmissio sstems for poit-to-poit trsmissio over flt-fdig MIMO chels. Left colums: receiver side equliztio; right colum: trsmitter side techiques. Top to bottom: lier equliztio/pre-equliztio; decisio-feedbck equliztio / precodig; mimumlikelihood detectio / vector precodig. of A. Due to the modulo uctio, the iitil sigl set A is eteded periodicll; ll sigl poits which differ b multiples of M represet the sme dt. The chel iput smbols re obtied b feedforwrd filterig with ςf. At the receiver side, ol sclig (AGC) b ς 1 remis, followed b threshold decisio which tkes the modulo cogruece ito ccout. Detils o precodig c be foud, e.g., i [4]. Give, the mtrices re ow obtied ccordig to 4 [4], [16] = BF 1, (3) where the mtrices hve the sme properties s bove d ς is djusted such tht the short-term power costrit is met. Ecept smll icresed umber of erest eighbor sigl poits d (usull) egligible precodig loss (smbols t the output of the modulo device hve slightl lrger vrice comp to the iitil dt smbols), precodig performs s good s DFE would without error propgtio. 3.3 Mimum-Likelihood Detectio d Vector recodig All bove metioed schemes perform smbol-bsmbol decisios d hve moderte compleit. Best performce, t the price of highest cost, is obtied b mimum-likelihood detectio (MLD). ere, o Gussi chels lgorithm serches through the dt vectors for which the (oiseless) chel output hs lest (squ) Euclide distce to the observed receive vector, mthemticll [1] ( deotes Euclide orm) =rgmi A K. (4) 4 This fctoriztio c be obtied b QR decompositio of T = def QR. With D = dig(l 11,l,...), L =[L ij ],we hve: ( T ) = = R Q = def LQ = LD 1 DQ = def BF 1. MLD c be efficietl implemeted usig so-clled sphere decoder [1]. The trsmissio scheme usig MLD is visulized i Figure 1, too. vig CSI t the trsmitter, the lgorithmic serch c be trsfe to the trsmitter, resultig i combied precodig/shpig techique sometimes clled vector precodig (V) [11], cf. lso [13] d [4, Sectio 5.3]. ere, give the dt vector, lgorithm (lttice decoder) serches through vectors d M K, for which the (o-ormlized) vector = 1 ( + d) of trsmit smbols hs lest orm (power), see Figure 1. ς is the gi chose to gurtee fied (short-term) power. Cotrr to lier schemes d DFE/precodig (diversit order 1), MLD d V chieve the full diversit order (K/) of the chel. owever, this improvemet i performce is ped tpicll with (much) higher compleit. 4 Lttice-Reductio-Aided Detectio d recodig Recetl, low-compleit equliztio schemes, bridgig the gp betwee DFE/precodig d MLD/V, hve bee proposed [1], [15], [18], [0]. The ide is to combie priciples kow from lttice theor i prticulr lttice (bsis) uctio (the choice of more suited represettio of lttice) which hs to be doe ol oce i pre-processig step with covetiol equliztio schemes such s lier equliztio or DFE discussed bove. Astoishigl, these cocepts chieve the full diversit orders [14], i.e., the error rte curves whe usig such detectors ru prllel to those for MLD/V. 4.1 Bsic Opertio Usull (ecept SK), the sigl poits i ech qudrture compoet re drw from ( trslte of) the i-

4 1 Z 1 F Z 1 1 F Z 1 ς 1 ςf Fig.. Trsmissio sstems usig lttice-bsis-uctio-ided equliztio. Top to bottom: receiver side LRA techique (lier equliztio/dfe), o TX CSI requi; prtil respose LRA techique (lier/dfe), prtil TX CSI requi; trsmitter side LRA techique (lier/precodig), full TX CSI requi. (The requi shift of the sigls to the iteger grid is ot show.) teger lttice. ece, t the receiver side (disregrdig the oise) the lttice K is preset. Applig lttice bsis uctio, e.g., b usig the LLL lgorithm [10], [6], the chel mtri m be fcto s = Z, (5) where Z is mtri with iteger etries tht hs uit determit, i.e., Z 1 lso cotis ol iteger etries. is more suited chel descriptio s it specifies the sme lttice of chel output sigl poits, K K, but its colums re closer to orthogol. Isted of performig equliztio of the etire chel, ol the fctor is equlized which cuses less oise ehcemet. The, sice Z K = K, idividul threshold decisio i ech compoet c be performed. To recover dt, vi Z 1 estimtes µ of the iitil dt smbols re geerted. Thereb, error multiplictio will occur. Lier equliztio of c be replced b DFE s give bove. For tht, is decomposed ccordig to (). The trsmissio scheme usig (upper brch) lttice-bsis-uctioided (LRA) lier equliztio d LRA DFE (lower brch) is depicted o top of Figure. 4. rtil-respose Siglig I poit-to-poit trsmissio, ssumig pproprite CSI, we hve the freedom to choose t which side equliztio is performed. A iterestig pproch is to move the iteger mtri Z 1 to the trsmitter. Thereb, error multiplictio i dt recover is voided. owever, direct pplictio of Z 1 would icrese verge trsmit power. The solutio is to use modulo uctio similr to Tomliso-rshim precodig. The proposed trsmissio scheme either usig lier prtil equliztio or DFE is depicted i the middle row of Figure. Sice K (igorig the shift b 1/) d Z is uimodulr mtri, Z 1 K. The modulo opertio (fter gi itroducig the trsltio) results smbols µ drw from the sme sigl set A s the dt smbols µ, formll =mod M ( Z 1 ( + M 1 ) 1) M 1 1, (6) where 1 is the ll-oe vector d mod M ( ) is the covetiol modulo uctio of ech compoet to the itervl [0, M). At the slicer (lier equliztio) the vector M 1 (Z I)1 + M K + 1 is preset. After elimitig the offset (secod term) threshold device tkig the periodic etesio (cused b +M K ) ito ccout c recover the dt. Similr cosidertios hold for DFE. This scheme, up to ow ot preset i literture, c be iterpreted s geerliztio of prtil-respose (R) siglig [9], populr o ISI chels. At the trsmitter, ol the kowledge of Z is requi (prtil TX CSI), which c be commuicted ver efficietl due to its iteger coefficiets. Additioll, the chel smbols µ re tke from the sme costelltio s the dt smbols µ ; hece o icrese i verge trsmit power is cused. Fill, o error multiplictio occurs t the receiver d, sice ol periodic etesio is preset, chel codig c be pplied immeditel whe lier equliztio is performed or the code words re rrged i time directio (cf. -BLAST).

5 BER lier pre-equl. lier equliztio SVD DFE precodig MLD vector precodig log 10 (Ēb/N 0)[dB] BER LRA lier equl. LRA DFE LRA lier pre-eq. LRA precodig referece log 10 (Ēb/N 0)[dB] Fig. 3. Bit error rte over Ēb/N 0 (i db). Top to bottom: lier preequliztio, lier equliztio, SVD, DFE (dotted: geie-ided), precodig, vector precodig, MLD. K =8; 4-r ASK trsmissio per rel compoet. Fig. 4. Bit error rte over Ēb/N 0 (i db). Top to bottom: LRA lier equliztio, LRA DFE (dotted: geie-ided), LRA lier preequliztio, LRA precodig. Gr: referece lier equl./mld. K =8; 4-r ASK trsmissio per rel compoet. 4.3 LRA re-equliztio d recodig The ide of lttice-bsis-uctio-ided equliztio c lso be pplied to pre-equliztio techiques [15], [18] ere, (5) hs to be replced b = Z, (7) i.e., lttice uctio is performed o T. Usig precodig, is dditioll fcto ccordig to (3). The respective LRA schemes which re bsicll obtied b flippig the structures for receiver side equliztio re depicted i the bottom row of Figure. ere, complete chel kowledge is requi t the trsmitter side (full TX CSI). 5 Numericl Results d Discussio The performce of the vrious schemes is ow ssessed b umericl simultios. We ssume N T = N R =4(K =8), d the coefficiets of the chel mtri c re chose i.i.d. comple Gussi with uit vrice. Ech of the 8 prllel rel dt strems uses ucoded (M =4)-r ASK siglig. The results re verged over lrge umber of chel reliztios. I ech cse (chel reliztio/trsmissio scheme) the trsmissio burst (10000 smbols) is scled for fied verge power (sme s for direct trsmissio of the dt smbols). The bit error rte (BER) results re displed over the rtio of the verge trsmitted eerg per bit Ē b d the (oe-sided) oise power spectrl desit N 0. I the preset cse we hve Ē b /N 0 = σ /(log (M )σ )=σ /(4σ ). Clssicl Schemes: I Figure 3 the performce of clssicl trsmissio schemes is comp. As o rte or power lodig is ctive, lier receiver side equliztio, lier pre-equliztio d sigulr vlue decompositio (SVD) show lmost the sme performce. I prticulr, the diversit order (egtive slope of the error rte curves i double-logrithmic scle) is ol 1. Usig DFE (with optiml orderig) some gi i SNR c be chieved but the diversit order does ot chge. Moreover, much gi is lost due to error propgtio i the feedbck loop s the compriso with the geie-ided (perfect feedbck) DFE revels. recodig performs close to geie-ided DFE; the error rte is somewht icresed due the lrger umber of erest eighbor sigl poits. Iterestigl, MLD d V lso perform ver similr. Agi, the slightl higher error rte c be eplied b the periodic etesio of the sigl set d hece lrger umber of erest eighbors. Both schemes chieve full diversit order; 4 i the preset cse. LRA Schemes: Net, LRA equliztio schemes re ssessed i Figure 4. The curves of lier equliztio (worst cse) d MLD (best cse) re repeted for referece. The performce dvtge of LRA schemes (lier d DFE) is clerl visible. I prticulr, the full diversit order is chieved, cf. [14], however, sigifict gp to the MLD curve b up to 4 db remis. LRA DFE offers ol gi of 1 db over LRA lier equliztio; sice is close to orthogol, error propgtio i the DFE feedbck loop is of mior iterest comp to error multiplictio t Z 1. Trsferrig (prts of) the equliztio i prticulr the iteger mtri Z 1 to the trsmitter is clerl dvtgeous. Now, error multiplictio whe recoverig dt vi Z 1 is voided. LRA Tomliso-rshimtpe precodig performs ver close to MLD d V (cf. lso [18]) followed b LRA lier pre-equliztio. owever, i these cses full TX CSI is requi. Implemetig ol Z 1 t the trsmitter (prtil respose siglig) offers ver good performce but requires ol little TX CSI, see Figure 5. R LRA lier equliztio performs slightl worse th LRA lier pre-equliztio but eve better th LRA DFE. The slight loss comp to pure trsmitter side techiques is due to the fct tht whe implemetig

6 BER Equliztio schemes for poit-to-poit commuictio over flt-fdig MIMO chels hve bee reviewed d comp. Strtig from lttice-uctio-ided techiques, ew pproch is derived, which c be iterpreted s geerlized prtil-respose siglig. This low-compleit scheme requires ol ver limited TX CSI d performce is close to MLD. The combitio with chel codig is strightforwrd. Sice the trsmitter opertes ol o iteger coefficiets w, the R LRA schemes re robust gist certi degree of chel vritios. I decisio directed mer, the receive filters c esil b dpted to the ctul situtio. Ol if the chel chges sigifictl, ew iteger mtri Z hs to be commuicted to the trsmitter. ece, such schemes re well suited for mobile pplictios with slowl to modertel vrig chel coditios R LRA lier eq. R LRA DFE referece log 10 (Ēb/N 0)[dB] Fig. 5. Bit error rte over Ēb/N 0 (i db). Top to bottom: prtil respose LRA lier equliztio, prtil respose LRA DFE (dotted: geie-ided). Gr: referece (LRA curves from Figure 4). K =8; 4-r ASK trsmissio per rel compoet. (or F ) t the trsmitter isted of the receiver, some form of power lodig over the prllel chel is ctive (lso vlid for the covetiol trsmitter side techiques). All prllel chels ehibit the sme error rte, wheres i pure receiver side equliztio the prllel dt strems usull hve differet error rtes d the worst cse error rte domites. owever, sice iteger ture of the sigls i R LRA schemes is requi, o trsmitter side sclig for power djustmet is possible. Nevertheless, R LRA DFE is ttrctive trsmissio scheme: it chieves ver good performce with ol little TX CSI. Fill it should be oted tht, give the chel reliztio, it would be possible to decide whether equliztio is preferbl doe t the trsmitter or receiver side. E.g., i DFE or precodig, the feedforwrd mtri F c be implemeted either t the trsmitter or receiver, cf. [17]. Eve though o verge both pproches perform ectl the sme, gi is possible if for give the istteous better versio is chose. This however, requires full TX CSI. Due to the iteger structure of the sigls, such procedure is ot possible i LRA schemes. ere, feedforwrd mtri d feedbck structure lws hve to be implemeted t the sme side; o splittig is llowed. 1 6 Coclusios Refereces [1] E. Agrell, T. Eriksso, A. Vrd, K. Zeger. Closest oit Serch i Lttices. IEEE Tr. If. Theor, pp , Aug. 00. [] A. Duel-lle. A Fmil of Multiuser Decisio-Feedbck Detectors for Aschroous Code-Divisio Multiple-Access Chels. IEEE Tr. Comm., pp , Feb./Mr./Apr [3] R.F.. Fischer, C. Widpssiger, A. Lmpe, J.B. uber. Spce-Time Trsmissio usig Tomliso-rshim recodig. ITG Cof. Source d Chel Codig, Berli, Germ, J. 00. [4] R.F.. Fischer. recodig d Sigl Shpig for Digitl Trsmissio, Joh Wile & Sos, New York, 00. [5] R.F.. Fischer, C. Widpssiger. Rel- vs. comple-vlued equlistio i V-BLAST sstems. Electroics Letters, pp , Mr [6] J. v.z. Gthe, J. Gerhrd. Moder Computer Algebr. Cmbridge Uiversit ress, Cmbridge, UK, d editio, 00. [7] G.D. Golde, G.J. Foschii, R.A. Vlezuel,.W. Wolisk. Detectio lgorithm d iitil lbortor results usig V- BLAST spce-time commuictio rchitecture. Electroics Letters, pp , J [8] T. ustei, C. v. elmolt, E. Jorswieck, V. Jugickel, V. ohl. erformce of MIMO Sstems with Chel Iversio. VTC Sprig 00, Birmighm, Albm, M 00. [9] Kbl, S. supth. rtil-respose Siglig. IEEE Tr. Comm., pp , Sep [10] A.K. Lestr,.W. Lestr, L. Lovász. Fctorig polomils with rtiol coefficiets, Mth. A., pp , 198. [11] C.B. eel, B.M. ochwld, B.M., A.L. Swidlehurst. A Vector-erturbtio Techique for Ner-Cpcit Multite Multiuser Commuictio rts I d II. IEEE Tr. Comm., pp , J. 005, d pp , Mr [1] J.G. rokis. Digitl Commuictios. McGrw-ill, New York, 4. editio, 001. [13] D. Schmidt, M. Johm, W. Utschick. Miimum Me Squre Error Vector recodig. IMRC 05, Berli, Germ, Sep [14] M. Therzdeh, A. Mobsher, A. Khdi. LLL Lttice-Bsis Reductio Achieves Mimum Diversit i MIMO Sstems. IEEE ISIT 05, pp , Adelide, Austrli, Sep [15] C. Widpssiger, R.F.. Fischer. Low-Compleit Ner-Mimum-Likelihood Detectio d recodig for MIMO Sstems usig Lttice Reductio. IEEE If. Theor Workshop 003, pp , ris, Frce, Mr./Apr [16] C. Widpssiger. Detectio d recodig for Multiple Iput Multiple Output Chels. Disserttio, Erlge, Jue 004. [17] C. Widpssiger, R.F.. Fischer, T. Vecel, J.B. uber. recodig i Multi-Ate d Multi-User Commuictios. IEEE Tr. Wireless Comm., pp , Jul 004. [18] C. Widpssiger, R.F.. Fischer, J.B. uber. Lttice-Reductio-Aided Brodcst recodig. IEEE Tr. Comm., pp , Dec [19]. Wolisk, G. Foschii, G. Golde, R. Vlezuel. V- BLAST: A Architecture for Relizig Ver igh Dt Rtes Over the Rich-Sctterig Wireless Chel. ISSSE 98, ise, Itl, Sep [0] D. Wübbe, R. Böhke, V. Küh, K.D. Kmmeer. Ner-Mimum-Likelihood Detectio of MIMO Sstems usig MMSE- Bsed Lttice Reductio. IEEE ICC 004, pp , ris, Frce, Jue 004. [1]. Yo, G.W. Worell. Lttice-Reductio-Aided Detectors for MIMO Commuictio Sstems. IEEE Globecom 00, Tipei, Tiw, Nov. 00.

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