Univrsity of Wollongong Rsarch Onl Faculty of Informatics - Paprs (Archiv Faculty of Engrg and Information Sccs 8 Mag carrir frquncy offst an advantag for orthogonal frquncy division multiplxg iaojg uang Univrsity of Wollongong huang@uow.du.au Publication Dtails. uang "Mag carrir frquncy offst an advantag for orthogonal frquncy division multiplxg" Communications and tworg Cha 8. Chacom 8. Third Intrnational Confrnc on 8 pp. 855-859. Rsarch Onl is th opn accss stitutional rpository for th Univrsity of Wollongong. For furthr formation contact th UOW Library: rsarch-pubs@uow.du.au
Mag carrir frquncy offst an advantag for orthogonal frquncy division multiplxg Abstract Contrary to th common blif that th carrir frquncy offst (CFO an orthogonal frquncy division multiplxg (OFDM systm would advrsly impact on systm prformanc this papr shows that th CFO actually has th ffct of lar prcodg among transmittd data symbols and hnc can b xploitd to improv th divrsity prformanc ovr frquncy-slctiv fadg channls. With both analysis and Mont Carlo simulation it is provd that an OFDM systm with CFO qual to half of th subcarrir spacg can potntially achiv th prformanc of divrsity ordr four by th maximum-lilihood dtction and dmonstrat a 5 db improvmnt usg th mimum man squard rror qualization. Discipls Physical Sccs and Mathmatics Publication Dtails. uang "Mag carrir frquncy offst an advantag for orthogonal frquncy division multiplxg" Communications and tworg Cha 8. Chacom 8. Third Intrnational Confrnc on 8 pp. 855-859. This confrnc papr is availabl at Rsarch Onl: http://ro.uow.du.au/fopaprs/39
Mag Carrir Frquncy Offst an Advantag for Orthogonal Frquncy Division Multiplxg iaojg uang School of Elctrical Computr and Tlcommunications Engrg Univrsity of Wollongong Wollongong Australia Abstract Contrary to th common blif that th carrir frquncy offst (CFO an orthogonal frquncy division multiplxg (OFDM systm would advrsly impact on systm prformanc this papr shows that th CFO actually has th ffct of lar prcodg among transmittd data symbols and hnc can b xploitd to improv th divrsity prformanc ovr frquncy-slctiv fadg channls. With both analysis and Mont Carlo simulation it is provd that an OFDM systm with CFO qual to half of th subcarrir spacg can potntially achiv th prformanc of divrsity ordr four by th maximumlilihood dtction and dmonstrat a 5 db improvmnt usg th mimum man squard rror qualization. Kywords carrir frquncy offst (CFO; orthogonal frquncy division multiplxg (OFDM; frquncy-slctiv fadg; multipath divrsity; lar prcodg. I. ITRODUCTIO Orthogonal frquncy division multiplxg (OFDM has bn widly usd today s digital communication systms du to its ffctiv tr-symbol trfrnc (ISI mitigation and simpl frquncy-doma channl qualization via fast Fourir transform (FFT. owvr as has bn commonly blivd it suffrs from som major disadvantags such as ( th larg pa-to-avrag powr ratio (PAPR ( th snsitivity to carrir frquncy offst (CFO [] and (3 th poor frquncy divrsity prformanc frquncy-slctiv fadg channls. Th first disadvantag is almost crta sc a larg PAPR th OFDM signal wavform not only drivs th dynamic rang rquirmnts for th digital-to-analog convrsion (D/A and analog-to-digital convrsion (A/D but mor importantly also rducs th transmittr and rcivr s powr amplifir fficcy. Th scond disadvantag is drawn basd on th obsrvation that th CFO causs tr-carrir trfrnc (ICI and thus frquncy synchronization/compnsation is ncssary. Th third disadvantag is a straightforward drivativ from th fact that th OFDM convrts frquncy-slctiv fadg to paralll flat fadg on orthogonal subcarrirs so that it only achivs divrsity ordr on and hnc prforms poorly frquncy-slctiv fadg channls. Channl codg has bn traditionally usd to improv th divrsity across frquncy and tim [3] and rcntly lar prcodg and bloc spradg for OFDM systms hav bn troducd to improv th frquncy divrsity prformanc [4-6]. Th ssnc of prcodg for OFDM is to troduc This rsarch is supportd by th Australian Rsarch Council Discovry Projct DP55845. corrlations among modulatd subcarrirs by applyg a unitary matrix to th data symbols to b transmittd to obta diffrnt lar combations of th origal data symbols. Aftr subcarrir mappg th prcodd data symbols ar sprad across th transmission frquncy band. Thus if a subcarrir xprcs a dp fad aftr transmittg ovr a frquncy-slctiv fadg channl th data symbol can b still rcovrd from othr subcarrirs so that th systm prformanc is improvd du to th crasd divrsity ordr [5]. Examg th ffct of CFO from this prcodg prcipl w s that th so-calld ICI causd by th CFO actually rflcts th corrlation among subcarrirs and hnc it should b prsrvd rathr than rmovd. With th right qualization and dtction tchniqus th CFO will no longr appar as a disadvantag but an advantag for OFDM. In this papr w illustrat how to dal with th CFO a diffrnt way not as an trfrnc mar but as an ffctiv mans to combat frquncy-slctiv fadg. Th convntional tim-doma CFO compnsation plus frquncy-doma qualization approach is rplacd by a nw frquncy-doma qualization plus trpolation approach. With both analysis and Mont Carlo simulation w also rval th prformanc lowr bounds by th maximum-lilihood ( dtction and th potntial prformanc improvmnt usg th mor practical mimum man squard rror ( qualization. Th rst of th papr is organizd as follows. In Sction II th rcivd OFDM signal modl with CFO is formulatd. In Sction III th convntional and nw approachs to dal with th CFO ar illustratd and compard. Sction IV is dvotd to th thortical analysis of th prformanc lowr bounds by th dtction and th prformanc usg th qualization. Mont Carlo simulation rsults ar givn Sction V. Fally conclusions ar drawn Sction VI. II. RECEIVED OFDM SIGAL MODEL WIT CFO An OFDM signal [ n] x is gnratd by prformg an - pot vrs discrt Fourir transform (IDFT on a bloc of data symbols aftr bary phas shift yg (BPSK quadratur phas shift yg (QPSK or any othr quadratur amplitud modulation (QAM constllation mappg of th put data bits i.. x = [ n] = π j n n. (
Bfor transmittg to a frquncy-slctiv multipath fadg channl with discrt channl impuls rspons h [ n] n = L whr L is th maximum multipath dlay sampls a cyclic prfix (CP of CP sampls CP L is srtd front of x [ n] to avoid adjacnt OFDM symbol trfrnc and turn th lar convolution of th transmittd signal with th channl to a circular on. Thus th total numbr of signal sampls an OFDM symbol bcoms + CP which corrsponds to an OFDM symbol duration ( + CP T aftr D/A whr T is th samplg priod. At th rcivr basband aftr A/D and CP rmoval th rcivd OFDM signal can b modld as r jωn [ n] = ( h[ n] x[ n] + v[ n] n ( whr dnots th circular convolution of lngth ω is a digital frquncy shift du to th CFO F btwn th transmittr and rcivr s local oscillators which can b xprssd as π ω = π FT = (3 whr = FT is dfd as th normalizd carrir frquncy offst with rspct to th subcarrir spacg T and v [ n] is th additiv zro-man whit Gaussian nois. From ( th -pot discrt Fourir transform (DFT of r n can b drivd as π π j ( j ( R = + V = + V (4 π π j ( j ( whr and rspctivly for simplicity ar sampld Fourir jω jω transforms ( and ( dnotd as and of h [ n] and x [ n] rspctivly and V is th -pot DFT of v [ n]. Whn = bcoms th -pot DFT of h [ n] and bcoms th -pot DFT of x [ n] i.. th transmittd data symbols. Accordg to th rlationship btwn th Fourir transform and th discrt Fourir transform of a fit lngth by squnc [7] can b trpolatd from π = Φ ( l (5 whr th trpolation function is dfd as Φ ( ω s ω s ω ( j ω =. (6 III. EW APPROACES DEALIG WIT CFO carrir frquncy offst compnsation is convntionally applid on To rcovr th transmittd data symbol jωn r [ n] th tim-doma first (i.. r[ n] is calculatd to shift by a frquncy offst and thn FFT is prformd to produc th product of with. Followg a simpl on-tap frquncy-doma qualization (dividg by for xampl is fally rcovrd. Th abov procss can b illustratd Fig. (a (b and (d for =7 = and =.5 (ignorg th nois for simplicity whr (a and (b show th frquncy-doma rprsntations of th rcivd OFDM signal bfor and aftr CFO compnsation rspctivly and (d shows th rcovrd OFDM data symbols aftr qualization. ormalizd Amplitud ormalizd Amplitud ormalizd Amplitud ormalizd Amplitud.5.5.5 3 4 5 6 (a.5.5 ( ( ((.5 3 4 5 6 (b.5.5.5 3 4 5 6 (c.5.5 ( (.5 3 4 5 6 (d Fig.. Frquncy-doma rprsntations of (a rcivd OFDM signal with CFO (b rcivd OFDM signal aftr CFO compnsation (c trpolatd data symbols aftr qualization without CFO compnsation and (d rcovrd data symbols (solid dots: DFT; solid l: Fourir transform; dottd l: trpolation functions. In a flat fadg channl th tim-doma CFO compnsation followd by th on-tap frquncy-doma qualization wors wll for th OFDM systm. owvr a frquncy-slctiv multipath fadg channl this approach can only achiv th prformanc of divrsity ordr on (i.. th prformanc for
flat fadg bcaus th data symbols modulatd on diffrnt subcarrirs ar dpndnt aftr CFO compnsation and hnc thr is no corrlation among subcarrirs to xplor for th rcovry of a dp fad on a subcarrir. In trms of troducg corrlation among transmittd data symbols th CFO now turns to b an advantag rathr than a disadvantag. This can b asily sn from (5 which clarly shows that whn is a lar combation of l. Thus stad of bg compnsatd th CFO should b pt. Thn nw qualization and dtction tchniqus can b dvlopd to xplor this advantag. An xampl of th nw approachs dalg with CFO can b illustratd usg Fig. (a (c and (d. Instad of bg shiftd by a frquncy offst via th tim-doma multiplication th DFT of th rcivd OFDM signal R = (ignorg th nois for simplicity shown Fig. (a is first (i.. zro-forcg qualization to produc dividd by th trpolatd data symbols shown Fig. (c. Thn is rcovrd s Fig. (d by trpolatg usg th vrs opration of (5. Compard with th convntional approach this nw approach rvrss th ordr of CFO compnsation and qualization and th CFO compnsation is rplacd by trpolation th frquncy-doma. Though th abov simpl zro-forcg qualization could possibly improv th divrsity prformanc th optimum tchniqu will b th dtction. In ordr to dscrib ths nw approachs bttr w now xprss th rcivd OFDM signal modl matrix form. By combg (4 and (5 togthr w hav R = U + V (7 whr R = ( R = ( = ( V column vctors = diag( ( matrix and U = ( u ( l is an V ar is an diagonal unitary matrix with th lmnt at th th row and th l th column as π u ( l = Φ ( l which satisfis th proprty U U = U U = I whr ( dnots matrix transposition and complx-conjugation opration and I is th idntity matrix of ordr. From (7 w s that whn thr is a carrir frquncy offst prsnt at th rcivr th OFDM systm is quivalnt to a prcodd systm with th prcodg matrix U as dfd abov. Th prcodd symbols U ar thn mappd onto subcarrirs qually spacd across th transmission band and xprc th channl fadg rprsntd by. Assumg prfct channl nowldg at th rcivr th stimat ˆ of th data vctor can b obtad by mimizg th quantity ( R U ˆ ( R U ˆ (8 through xhaust sarch from all possibl dat vctors. Du to th complxity of th optimum dtction a lar qualization is mor practical sc it can simply us a on-tap qualizr for ach subcarrir th frquncy-doma as has bn sn th prvious xampl. Th qualization and dtction procss can b gnrally dscribd as follows. Lt C dnot th on-tap qualizr coffict to b applid to R on th subcarrir and = diag C dnot an C ( diagonal matrix with diagonal lmnts C =. First applyg C to R producs th qualizd prcodd data vctor CR. Scond multiplyg U to rmov th prcodg yilds th dcision variabl vctor d = U CR. Fally an stimat ˆ of th transmittd data vctor is obtad aftr hard dcision. IV. BER AALYSIS I PRESECE OF CFO A. BER lowr bound of dtction Sc (8 can b xpandd as R R R R U ˆ + ˆ U U ˆ (9 and { } R R is dpndnt of ˆ th dtction can b carrid out by sarchg for ˆ to maximiz th quantity Ω ˆ = R R U ˆ ˆ U U ˆ. ( ( { } Lt th transmittd data symbol vctor b. Exprssg th stimat ˆ as ˆ = + whr is an rror vctor and substitutg (7 to ( yild Ω + Ω = R U V U U. ( ( ( { } W s that givn an rror vctor ( + Ω( Ω is a Gaussian distributd variabl with man δ and varianc σ V δ whr δ = U U is rfrrd to as th distanc btwn and V + and aftr prcodg and multipath channl σ is th varianc of th nois V. Suppos that w us th Gray-codd QPSK (i.. thr is only on bit diffrnc btwn two adjacnt constllation pots for th bit-tosymbol mappg bfor prcodg. Thus any on bit rror occurrg at th th data symbol ˆ will rsult an rror vctor ( T T = ± σ or = ( ± j σ th lmnt whr ( T dnots matrix transposition and ( th lmnt σ is th avrag powr of th data symbol. Th man and varianc of ( + Ω( Ω ar thrfor found to b δ and σ Vδ rspctivly whr δ = σ u ( l th dtction prcipl if ( + > Ω(. Accordg to Ω thn + will b dclard as th dtctd data vctor and hnc on bit rror occurs. Thrfor th probability for on bit rror is
valuatd as th probability with which Ω( + Ω( > i.. P Ω + ( ( Ω( > = = δ Q Q γ σ V u ( l (3 whr γ = σ σ is th put signal-to-nois ratio (SR and ( Q is th Q-function. V ot that th BER of th dtction is dtrmd by th mimum distanc δ m which rquirs an xhaust sarch of th rror vctor(s ladg to th mimum distanc. owvr sc th distanc givn by th rror vctor is always gratr than or qual to th mimum distanc th on bit rror probability (3 can srv as a lowr bound of th BER. Also not that th abov BER lowr bound rlis on a l ralization of th channl frquncy rspons l or quivalntly th channl impuls rspons h [ n]. Thus for a frquncy-slctiv channl th avrag BER lowr bound will b = P ( Eh Q γ l u l (4 = whr E {} dnots th nsmbl avragg ovr all h [ n]. h B. BER of Equalization For lar qualization whn th critrion is usd i.. dsigng C so that th man squard rror (MSE btwn d and ε = E d d (5 {( ( } is mimizd th diagonal lmnt C is found to b C = + γ (6 and th output SR th dcision variabl d for data symbol can b xprssd as [8] u ( l + γ γ ( =. (7 u ( l + γ W s that th output SR is also dtrmd by or quivalntly h [ n]. Assumg QPSK modulation for data symbols and mag a Gaussian distribution approximation for ISI th avrag BER of th qualizr for a givn ralization of = ( h [ n] can b valuatd as Q γ ( and consquntly th avrag BER for a frquncy-slctiv fadg channl is valuatd as P = Eh Q = ( ( γ. (8 V. MOTE CARLO SIMULATIO RESULTS Th prformanc P and P analyzd th abov sction ar valuatd usg th Mont Carlo simulation assumg that th channl has a full multipath divrsity of ordr L. That is all channl cofficts h [ n] n = L ar dpndnt and idntically distributd (i.i.d. complx Gaussian random variabls with zro-man and varianc. To valuat th nsmbl avrag ovr all L h [ n] w gnrat suffict ralizations of ths dpndnt Gaussian variabls calculat th BER for ach ralization and thn ta an avrag. To show th prformanc potntial of OFDM with CFO w first assum that th channl provids th maximum divrsity ordr i.. L = and valuat th BER lowr bounds of th dtction and th BERs of th qualization undr diffrnt CFOs. Th put SR γ is xprssd as E b for QPSK whr E b is th signal nrgy pr bit and is th nois powr spctral dnsity. Th rsults ar shown Fig. with =56. Diffrnt numbrs of such as 6 3 64 and 8 ar also tstd and th rsults ar all th sam. From Fig. w s that whn =±.5 th bst prformanc is achivd. Whn = th prformanc is th worst which is th sam as th on flat fadg (divrsity ordr on. For th dtction th bst prformanc achivs a divrsity ordr of four (s [9] and for th qualization th bst prformanc shows a mor than 4 db improvmnt at 5 db normalizd SR and 5 db at db normalizd SR. BER 3 4 = divrsity ordr four 5 5 5 ormalizd SR E b / (db Fig.. Prformanc potntial of OFDM with CFO (solid ls for =±.5; dashd ls for =±.375 or ±.65; dottd ls for =±.5 or ±.75.
Fig. 3 shows th prformanc for a mor practical systm sttg with L = / 4 and =8. As th divrsity ordr providd by th channl dcrass th prformanc is dgradd accordgly. For comparison purpos th prformanc curvs with L = ar also displayd. W s that th lowr bound has about 3dB loss but is still much bttr than th prformanc of divrsity ordr two [9]. Th qualization curs about db loss but still provid much bttr prformanc than that of th convntional OFDM without CFO or with CFO compnsation. Φ (π/ (.7.6.5.4.3.. =6 =3 =64 =8 BER 3 4 = L = L = /4 divrsity ordr two 5 5 5 ormalizd SR E / (db b Fig. 3. Prformanc of a practical OFDM with CFO =±.5 and =8. Th abov prformanc rsults suggst that whn th right qualization and dtction tchniqus ar usd th CFO an OFDM systm should b st to half of th subcarrir spacg ordr to achiv th bst divrsity prformanc. This obsrvation lads to th nw rcivr architctur to conduct frquncy synchronization an OFDM systm i.. aftr th CFO stimation th CFO should b adjustd to half of th subcarrir spacg rathr than b compnsatd. As a fal rmar on th rcivr complxity w pot out that th complxity would b th sam as that of a prcodd OFDM with th sam prcodg matrix siz. owvr by xploitg th charactristics of th trpolation function th complxity can b gratly rducd i.. th frquncy-doma trpolation can b simplifid by considrg only svral adjacnt subcarrirs. This can b asily sn from th trpolation function amplitud shown Fig. 4. Bcaus th significant valus of th trpolation function ar locatd around = a subcarrir can b simply trpolatd usg only svral adjacnt subcarrirs. Furthrmor if propr pr and post procssg is prformd bfor and aftr th frquncy-doma trpolation rspctivly a ral valud trpolation function can b usd and hnc furthr ovrall complxity rduction is possibl. Sc our purpos hr is to dmonstrat th prformanc potntial by xploitg th carrir frquncy offst th complxity rduction and how this rduction impacts on systm prformanc ar byond th scop of this papr. 3 3 Fig. 4. Amplitud of th trpolation function ( ω π ω = ( with =.5 for diffrnt. VI. COCLUSIOS Φ sampld at W hav shown that th CFO an OFDM systm troducs corrlation among modulatd subcarrirs. It achivs th sam ffct as lar prcodg but without xplicit prcodg opration at th transmittr. In trms of combatg frquncy-slctiv multipath fadg th CFO is actually bnficial rathr than dstructiv. Instad of bg compnsatd as impairmnt th CFO should b st to half of th subcarrir spacg ordr to ma full us of this divrsity advantag. REFERECES [] T. Pollt M. Vn Bladl and M. Monclay BER snsitivity of OFDM systms to carrir frquncy offst and Wr phas nois IEEE Transaction on Communications Vol. 43 Fbruary/March/ April 995 pp 9-93. [] C. Brrou A. Glaviux and P. Thitimajshima ar Shannon limit rror-corrctg codg and dcodg: Turbo-cods Procdgs of IEEE Intrnational Confrnc on Communications Gnva Switzrland May 993 Vol. pp. 64-7. [3] R. G. Gallagr Low-dnsity parity-chc cods IEEE Transactions on Information Thory Vol. 8 o. January 96 pp. -8. [4] Z. Wang and G. B. Giannais Larly prcodd or codd OFDM agast wirlss channl fads? Procdgs of Signal Procssg Advancs Wirlss Communications Worshop Taoyuan Taiwan 3 March pp. 67 7. [5] Z. Liu Y. and G. B. Giannais Lar constllation prcodg for OFDM with maximum multipath divrsity and codg gas IEEE Transactions on Communications Vol. 5 o. 3 March 3 pp. 46-47. [6] M. L. McCloud Analysis and dsign of short bloc OFDM spradg matrics for us on multipath fadg channls IEEE Transactions on Communications Vol. 53 o. 4 April 5 pp. 656-665. [7] A. V. Oppnhim and R. W.Schafr Digital Signal Procssg Prntic- all Inc. Englwood Cliffs w Jrsy 975. [8]. uang Multipath divrsity of prcodd OFDM with lar qualization prsntd at th 8 IEEE Intrnational Confrnc on Communications (ICC 8 Bijg Cha 9- May 8. [9] J. G. Proais Digital Communications Third Edition McGraw-ill Intrnational Editions 995.