Adaptive Sigle-Carrier Tramiio Uig QRM- ML Blok Sigal Detetio Tetuya YAMAMOTO ad Fumiyuki ADACI Dept. of Eletrial ad Commuiatio Egieerig, Graduate Sool of Egieerig, Tooku Uiverity 6-6-5 Aza-Aoba, Aramaki, Aoba-ku, Sedai, 98-8579 Japa yamamoto@mobile.eei.tooku.a.jp, adai@eei.tooku.a.jp Abtrat I ti paper, we propoe a adaptive igle-arrier (SC) tramiio uitable for ear maximum likeliood (ML) blok igal detetio uig QR deompoitio ad M-algoritm (QRM-MLBD). QRM-MLBD a igifiatly improve te bit error rate (BER) performae of SC blok tramiio. owever, i order to aieve a loe-to-ml performae, te ue of a fairly large umber M of urvivig pat i te M-algoritm i required ad ee it omputatioal omplexity i till ig. I ti paper, to redue te required umber of urvivig pat, we itrodue a uequal bit loadig i a SC tramiio blok. I QRM-MLBD, te igal-to-oie power ratio (SR) degrade for ymbol to be deteted at early tage i te M-algoritm. By otig ti fat, we propoe to load le umber of bit (i.e., lower modulatio level) o ymbol loe to te ed of blok. We ow tat te itrodutio of a uequal bit loadig to SC blok tramiio uig QRM-MLBD a aieve almot te ame BER performae a te ovetioal QRM-MLBD wile igifiatly reduig te required umber M of urvivig pat i te M-algoritm. Keyword-ompoet; Sigle-arrier, bit loadig, MLD, QR deompoitio, M-algoritm I. ITRODUCTIO Te broadbad wirele ael i araterized by a evere frequey-eletive ael ad terefore, te evere iter-ymbol iterferee (ISI) limit te tramiio performae []. Reetly, a ear maximum likeliood (ML) blok igal detetio uig QR deompoitio ad M- algoritm (alled te QRM-MLBD) wa propoed [, 3] for igle-arrier (SC) blok tramiio. It wa ow [3] tat QRM-MLBD a igifiatly improve te bit error rate (BER) performae of SC blok tramiio, ompared to te frequey-domai equalizatio (FDE) baed o te miimum mea quare error (MMSE) riterio [4, 5]. owever, i order to aieve te uffiietly improved BER performae, te ue of a fairly large umber M of urvivig pat i te M- algoritm i required ad ee te omputatioal omplexity i till very ig. I QRM-MLBD, te oateatio of te direte Fourier traform (DFT) ad te propagatio ael i regarded a a equivalet ael, to wi te QR deompoitio i applied. Te available igal power at ea tage i te M-algoritm deped o te elemet of a upper triagular matrix R, obtaied by QR deompoitio of te equivalet ael matrix. owever, te magitude of a elemet i loe-to-terigtmot olum of R may drop wit iger probability [6]. Terefore, te aievable igal-to-oie power ratio (SR) get lower at early tage i te M-algoritm ad ee te probability of removig te orret pat at early tage may ireae we maller M i ued. I ti paper, i order to redue te required umber M of urvivig pat for aievig te uffiietly improved BER performae, we propoe a adaptive SC blok tramiio for QRM-MLBD. Te probability of removig te orret pat deped o te SR of ea tage. I QRM-MLBD, if mall M i ued, te aievable SR ted to degrade for ymbol to be deteted at early tage i te M-algoritm. By otig ti fat, we propoe a uequal bit loadig i wi le umber of bit are loaded for ymbol loe to te ed of blok. Te remaider of ti paper i orgaized a follow. I Set. II, te ytem model of SC blok tramiio uig QRM- MLBD i preeted. I Set. III, te propoed adaptive SC tramiio eme i deribed. I Set. IV, we will ow ome imulatio reult. We will ow tat te propoed adaptive SC blok tramiio a redue te required umber M of urvivig pat (tereby a redue te omputatioal omplexity) wile aievig almot te ame performae a te ovetioal SC tramiio. Set. V offer ome oludig remark. II. SC BLOCK TRASMISSIO USIG QRM-MLBD A. Sigal Tramiio Model Te ytem model of SC blok tramiio uig QRM- MLBD i illutrated i Fig.. Trougout te paper, te ymbol-paed direte time repreetatio i ued. We oider blok data tramiio of ymbol. Te data ymbol blok i expreed uig te vetor form a d[d(), d(,...,d( )] T. Te lat g ymbol of ea blok are opied a a yli prefix (CP) ad ierted ito te guard iterval (GI) plaed at te begiig of ea blok ad a CPierted data blok of + g ymbol i tramitted. Te igal blok i tramitted over a frequey-eletive fadig ael. We aume a ymbol-paed frequeyeletive fadig ael ompoed of L propagatio pat wit differet time delay. Te GI-removed reeived igal blok y[y(),...,y(t),,y( )] T a be expreed uig te vetor form a y E / T d +, () were E ad T are repetively te ymbol eergy ad te ymbol duratio. [(),,(t),,( )] T i te oie vetor. Te tt elemet (t) of i te zero-mea additive wite Gauia oie (AWG) avig variae /T wit beig te oe-ided oie power petrum deity. i te ael impule repoe matrix give a
L L, () L L L were l i te omplex-valued pat gai wit E [ ]. L l Te lt pat time delay i aumed to be l ymbol. Te reeived igal blok after GI removal i traformed by - poit DFT ito te frequey-domai igal. Te frequeydomai reeived igal vetor Y[Y(),,Y(k),,Y( )] T i expreed a l Y Fy E / T Fd + F, (3) were F i te DFT matrix of ize give by ( ) j π jπ e e F. (4) ( ) ( ) ( ) jπ j π e e Sie te ael impule repoe matrix i a irulat matrix, te eigevalue deompoitio uig F a be applied [7]. We ave were FF diag (),, ( )], (5) ( k) L l l [ exp( jπkτ / l ), k~, ad (.) i te ermitia trapoe operatio. Uig Eq. (5), Eq. (3) a be rewritte a Y E / T Fd + E / T d +, (6) were F ad [(),..,(k),,( )] T are repetively te equivalet ael matrix ad te frequeydomai oie vetor. d +CP CP -poit DFT Cael etimatio Y Multipliatio of Q QR deompoitio Q Ŷ MLD uig M-algoritm Figure. Sytem model of SC blok tramiio uig QRM-MLBD. dˆ B. QRM-MLBD QRM-MLBD a be applied to te SC blok tramiio by treatig a oateatio of te frequey-domai ael ad DFT a te equivalet ael. Te QR deompoitio i applied to te equivalet ael matrix F to obtai QR, were Q i a uitary matrix ad R i a upper triagular matrix. Te traformed frequeydomai reeived igal obtaied a Yˆ Q Y ˆ E Rd + Q T ˆ(),..., ˆ( ˆ( T Y [ Y Y k),..., Y )] i R, R, R, d () R R. E d,, () + Q T R d, ( ) (7) From Eq. (7), te ML olutio a be obtaied by earig for te bet pat avig te miimum Eulidea ditae i te tree diagram ompoed of tage. I ea tage, te bet M pat are eleted a urvivig pat by omparig te pat metri baed o te quared Eulidea ditae for all urvivig pat ad are paed to te ext tage. Te pat metri e i, at te it tage (i,,..., ) i alulated a e i ˆ E R d j, i Y ( T j j, ( ) were d ( i te ymbol-adidate for d(. Te mot poible tramitted ymbol equee i determied by traig bak te pat wit te mallet pat metri at te lat tage (i )....8.6.4. 6 ρ3db ρdb 64 g 6 L6-pat expoetial power delay profile Stage idex, i 3 48 Figure. Reeived igal power available at ea tage of te M-algoritm. I QRM-MLBD, te reeived igal power aoiated wit te ymbol d( j) at te it tage, i,,, ad j,,,i, i te um of te quared magitude R i, j of te ( i, j)t elemet i R from i to. Terefore, te reeived igal power available at early tage likely beome low ie oly te elemet of matrix R loer to te lower rigt poitio are available. Furtermore, i te ae of SC blok tramiio, te magitude of elemet of matrix R loer to te lower rigt poitio may drop wit iger probability beaue te ael impule repoe matrix i irulat [6]. A a reult, te reeived igal power available at early tage igifiatly drop. Figure plot te ormalized reeived igal power available at ea tage of te M-algoritm for 64-ymbol blok tramiio i a L6- pat frequey-eletive ael wit expoetial power delay profile avig te deay fator ρdb. It a be ee from Fig. tat te reeived igal power available at early tage igifiatly drop ad te umber of te tage at wi te 64 (8)
reeived igal power igifiatly drop i larger a te ael frequey-eletivity get troger. Sie te available igal power i lower at early tage, te probability of removig te orret pat at early tage may ireae we maller M i ued. Terefore, a fairly large M mut be ued to aieve te uffiietly improved BER performae. owever, te ue of larger M ireae te omputatioal omplexity. III. ADAPTIVE SC BLOCK TRASMISSIO I ti paper, i order to redue te required umber M of urvivig pat for aievig te uffiietly improved BER performae, we itrodue a uequal bit loadig [8, 9] to te SC blok tramiio. It i deirable tat le umber of bit are loaded (i.e., lower modulatio level i ued) o te ymbol to be deteted at early tage i te M-algoritm. If lower modulatio level i ued, all poible pat likely a urvive at early tage eve we mall M i ued ad ee, te probability of removig te orret pat at early tage a be redued. Te umber of loaded bit i determied blok-byblok baed o te reeived igal power at ea tage of te M-algoritm. Te adaptive bit loadig to determie te modulatio level i baed o te Ceroff upper boud []. A. Adaptive Bit Loadig Seme A total umber of bit per SC tramiio blok i deoted by B. Te bit loadig i repreeted by {(),,(,,( )}, were ( repreet te umber of loaded bit for te t ymbol. Te adaptive bit loadig i doe a follow. Step ) Apply te QR deompoitio to te equivalet ael matrix at te reeiver. Step ) Selet te bit loadig {(),,(,,( )} tat miimize te BER averaged over blok. Step 3) Feedbak te bet ombiatio to te tramitter. Aumig Gray ode mappig, te oditioal BER P b, aoiated wit te t ymbol for te give et of modulatio level, SR, ad ael i give a [] γ( b, α( erf, (9) β( P were erf(.) deote te omplemetary error futio ad γ( repreet te aievable SR aoiated wit te t ymbol wi i itrodued i Set III-B. Table I ow te value of α( ad β( for QPSK, 6QAM ad 64QAM. I ti paper, te Ceroff upper boud of te BER i ued. Eq. (9) a be upper-bouded a [] γ( b, α( exp. () β( P Te upper bouded BER averaged over blok i give a γ ( P α b ( ( exp. () B β( Te bet bit loadig i foud uder te followig oditio. ( B () ( ( ). () Firt equatio mea tat adaptive bit loadig i doe uder te total umber of bit otrait. Te eod equatio mea tat le umber of bit (iludig o ymbol tramiio repreeted by () are loaded o ymbol to be deteted at early tage i te M-algoritm (ymbol ear te ed of blok) ie te igal power ted to drop at early tage i te M- algoritm a ow i Fig.. ote tat from te eod equatio i Eq. (), wat eed to feedbak to te tramitter i oly te modulatio boudary i a blok. For example, i te ae of {(6; ~4, (4; 5~6, (; 6, (; 6~63}, it i uffiiet to feedbak oly 5, 6, ad 6 to te tramitter. TABLE I. α( AD β( Data modulatio α( β( QPSK / 4 6QAM 3/8 64QAM 7/4 84 B. SR of Ea Symbol After QRM-MLBD Sie a exat aalyi of aievable SR aoiated wit te t ymbol after QRM-MLBD i quite diffiult if ot impoible, we ue a approximate SR expreio baed o te igal power of te M-algoritm give by I E γ( R i,, (3) i were te value of I i oe from [,,,]. If te reeived igal power of te t ymbol deped oly o te diagoal elemet R,, I ould be (i.e., γ ( ) (E / ) R, ). We M<X for X-QAM, te pat eletio otaiig te t ymbol greatly deped o te t diagoal elemet of R. Terefore, I may be oe. O te oter ad, we MX, te reeived igal power of te t ymbol deped ot oly o te diagoal elemet R, but alo o oter elemet, i.e., te ( ) to t elemet i te t olum. Terefore, I > i oe. It i expeted tat a M i larger, te value of I beome larger ad te optimum I doe exit, but it deped o te ael oditio, data modulatio, ad te umber M of urvivig pat. I ti paper, we foud, by prelimiary omputer imulatio, te bet I for ea M tat miimize te BER performae. IV. COMPUTER SIMULATIO We evaluate, by omputer imulatio, te aievable BER performae of adaptive SC blok tramiio uig QRM- MLBD ad ompare it wit te ovetioal SC blok tramiio uig QRM-MLBD. Te imulatio oditio i ummarized i Table II. Te blok ize i 64 ad CP legt i g 6. Te ael i aumed to be a frequey-eletive blok Rayleig fadig ael avig ymbol-paed L6- pat expoetial power delay profile avig te deay fator ρdb. Ideal ael etimatio i aumed. I te ae of te ovetioal SC blok tramiio, 6QAM i ued for all ymbol (te total umber of tramitted bit per SC blok i 4 ). I te ae of te adaptive SC blok tramiio, te total umber of bit per blok i et to B4 ad te umber of te loaded bit for ea ymbol i eleted from 4 adidate: (o ymbol tramiio, (QPSK), 4 (6QAM), ad 6 (64QAM) bit,
aordig to te ael oditio. Perfet feedbak of te modulatio boudary i aumed. (I 7). Ti i beaue te etimated SR of toe ymbol for M ad 4 i maller ta tat for M6. TABLE II. COMPUTER SIMULATIO CODITIO 6QAM for te ovetioal Data modulatio SC blok tramiio Tramitter Blok ize 64 CP legt g6 Fadig type Frequey-eletive blok Rayleig Cael Power delay profile L6 pat expoetial power delay profile Deay fator ρ, 3dB Reeiver Cael etimatio Ideal 3 4 M M4 M6 M64 B4 64, g 6 E b / 6dB A. Optimum I Figure 3 plot te average BER of adaptive SC blok tramiio uig QRM-MLBD a a futio of I for M, 4, 6, ad 64. Te average reeived bit eergy-to-oie power petrum deity ratio E b / i et to 6dB. Te deay fator ρ (orrepodig to te uiform power delay profile) i aumed. We M ad 4, I provide te miimum BER. Ti i beaue te pat eletio otaiig te t ymbol greatly deped o te t diagoal elemet of R. O te oter ad, we M6 ad 64, a I ireae, te BER dereae. Ti i beaue te reeived igal power of te t ymbol deped ot oly o te diagoal elemet but alo o oter elemet. owever, I i et to too large, te BER ireae beaue te optimum bit loadig aot be performed by overetimatig te SR. We te SR of te ymbol ear te ed of blok are overetimated, larger umber of bit i loaded for tee ymbol ad te, te probability of removig te orret pat at early tage ireae. We foud tat I 7 ad 9 miimize te BER we M6 ad 64, repetively. I te followig imulatio, we ue te bet I for ea M tat miimize te BER. B. Performae Te BER performae of adaptive SC blok tramiio uig QRM-MLBD i plotted i Fig. 4 a a futio of average reeived E b / for M, 4, 6, ad 64. For ompario, te BER performae of te ovetioal SC blok tramiio i alo plotted. It a be ee from Fig. 4 tat we mall M i ued, te aievable BER performae of te ovetioal blok tramiio degrade. O te oter ad, adaptive SC blok tramiio a aieve better BER performae eve if mall M i ued. We ρdb, adaptive SC blok tramiio wit M6(64) aieve almot te ame BER performae of te ovetioal SC blok tramiio uig M64(56). We ρ3db, adaptive SC blok tramiio wit M6 aieve almot te ame BER performae of te ovetioal SC blok tramiio uig M64. Figure 4 ow te probability of te umber of loaded bit for te t ymbol we E b / 6dB. Te deay fator ρ i aumed. It a be ee from Fig. 4 tat le umber of bit i likely to be loaded for ymbol to be deteted at early tage i te M-algoritm (ymbol ear te ed of blok). We le umber of bit i loaded for te ymbol to be deteted at early tage, all pat likely urvive at early tage eve we mall M i ued. Terefore, te BER performae doe ot degrade eve if mall M i ued. Te probability of le umber of bit i loaded for te ymbol to be deteted at early tage i te M- algoritm i iger for M ad 4 (I ) ta tat for M6 5 3 4 5 3 4 5 4 6 I Figure 3. Impat of I o average BER. QRM-MLBD M M4 M6 M64 M56 Adaptive SC blok tramiio Cov. SC blok tramiio 5 5 Average reeived E b / (db) QRM-MLBD M M4 M6 M64 (a) ρdb Adaptive SC blok tramiio Cov. SC blok tramiio 5 5 Average reeived E b / (db) (b) ρ3db Figure 4. performae. 8 B4 64, g 6 ρdb B4 64, g 6 ρ3db Fially, we diu te effet te umber of ymbol for wi bit are loaded adaptively. So far, te bet ombiatio of te umber of loaded bit wi miimize te BER averaged over blok i foud amog te all poible ombiatio wi atify te oditio expreed by Eq. (). owever, a te blok ize ireae, it require a proibitively large umber of earig. O te oter ad, it a be 5 5
udertood from Fig. 5 tat te ymbol for wi adaptive bit loadig i applied a be loalized oly ear te begiig ad ed of blok ad tat 6QAM i alway ued for te ymbol i te middle of te blok. Terefore, it i ot eeary to apply te adaptive bit loadig for all ymbol i a blok. Figure 6 ow BER depedey o te umber of ymbol for wi adaptive bit loadig i applied. Te bit are loaded adaptively for ymbol ear te begiig ad ed of blok. We ρdb, te adaptive bit loadig a be applied for at mot 6 ymbol ad ti aieve almot te ame BER a 64 ae (adaptive bit loadig i applied for all ymbol). We ρ3db, te umber of ymbol for wi adaptive bit loadig i applied a be furter redued. Ti i beaue te umber of te tage at wi te reeived igal power drop beome le a te ael frequey-eletivity get weaker a ow i Fig.. V. COCLUSIO I ti paper, i order to redue te required umber M of urvivig pat i QRM-MLBD for aievig te uffiietly improved BER performae, we propoed a adaptive SC blok tramiio uig QRM-MLBD. I te propoed eme, le umber of bit i loaded for te ymbol to be deteted at early tage i te M-algoritm, ad terefore, all pat likely urvive at early tage eve we mall M i ued. A a reult, te probability of removig te orret pat at early tage a be redued ad te BER performae doe ot degrade eve if mall M i ued. We owed tat adaptive SC blok tramiio uig QRM-MLBD a aieve almot te ame BER performae a te ovetioal SC blok tramiio uig QRM-MLBD wile reduig te umber M of urvivig pat to /4. REFERECES [] J. G. Proaki ad M. Salei, Digital ommuiatio, 5t ed., MGraw- ill, 8. [] K. agatomi, K. igui, ad. Kawai, Complexity redued MLD baed o QR deompoitio i OFDM MIMO multiplexig wit frequey domai preadig ad ode multiplexig, i Pro. IEEE Wirele Commuiatio ad etworkig Coferee (WCC 9), Apr. 9. [3] T. Yamamoto, K. Takeda, ad F. Adai, Sigle-arrier tramiio uig QRM-MLD wit atea diverity, i Pro. Te t Iteratioal Sympoium o Wirele Peroal Multimedia Commuiatio (WPMC 9), Sept. 9. [4] D. Faloer, S. L. Ariyaviitakul, A. Beyami-Seeyar, ad B. Edio, Frequey domai equalizatio for igle-arrier broadbad wirele ytem, IEEE Commu. Mag., Vol. 4, o. 4, pp. 58-66. Apr.. [5] F. Adai,. Tomeba, ad K. Takeda, Itrodutio of frequeydomai igal proeig to broadbad igle-arrier tramiio i a wirele ael, IEICE Tra. Commu., Vol.E9-B, o.9, pp. 789-88, Sept. 9. [6] K. Takeda,. Tomeba ad F. Adai, Joit Tomlio-araima Preodig ad Frequey-Domai Equalizatio for Broadbad Sigle- Carrier Tramiio, IEICE Tra. Commu., Vol.E9-B o., pp.58-66, Ja. 8. [7] G.. Golub ad C. F. va Loa, Matrix Computatio, 3rd ed. Baltimore, MD, Jo opki Uiv. Pre, 996. [8] K. Ba, M. Katayama, T. Yamazato, ad A. Ogawa, Joit optimizatio of tramitter/reeiver wit multiple tramit/reeive atea i badlimited ael, IEICE Tra. Commu., Vol. E83-B, o. 8, pp. 697-74, Aug.. [9] K. Ozaki, A. akajima, ad F. Adai, Frequey-domai eigebeam- SDM ad equalizatio for igle-arrier tramiio, IEICE Tra. Commu., Vol. E9-B, o. 5, pp. 5-53, May 8. Probability of te umber of loaded bit (%) Probability of te umber of loaded bit (%) 8 6 4 8 6 4 (6 (6 (4 4 6 8 55 57 59 Symbol idex, (a) M ad 4 (I ) (4 4 6 8 55 57 59 Symbol idex, ( ( 6 ((.3%) ( (b) M6 (I 7) Figure 5. Probability of te umber of loaded bit. 3 4 5 3 4 5 B4 64, g 6 E b / 6dB ρdb M4 M M6 4 6 8 4 B4 64, g 6 E b / 8dB ρ3db (a) ρdb M M4 M6 4 6 8 4 6 63 63 64 64 64 64 64 (b) ρ3db Figure 6. BER depedey o te umber of ymbol for wi adaptive bit loadig i applied. 6 6