FREQUENCY OFFSET ESTIMATION FOR PCC-OFDM WITH SYMBOLS OVERLAPPED IN TIME DOMAIN

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1 FREQUECY OFFSET ESTIMATIO FOR PCC-OFDM WITH SYMBOLS OVERLAPPED I TIME DOMAI Jnwn Shntu an Jan Armstrong Dpartmnt of Ectronc Engnrng, La Trob Unvrsty Vctora 386, Austraa Ema: j.shntu@.atrob.u.au, j.armstrong@.atrob.u.au Abstract Ths papr prsnts a nw agorthm for frquncy offst stmaton for Poynoma Cancaton Co Orthogona Frquncy Dvson Mutpxng wth symbos ovrapp n th tm oman (Ovrap PCC-OFDM. Th agorthm xpots th Subcarrr Par Imbaanc (SPI caus by frquncy offst. Th stmaton s prform n th frquncy oman. o tranng symbos or pot tons ar rqur. Smuatons show that ths stmator s an approxmaty nar functon of frquncy offst. Thr ar thr ways to ruc th varanc of th stmaton: ncrasng th numbr of subcarrr pars, usng a two-mnsona Mnmum Man Squar Error (MMSE quar bfor th stmaton an usng PCC-OFDM pot symbos wth no ovrappng.. Ovrap PCC-OFDM Poynoma Cancaton Co Orthogona Frquncy Dvson Mutpxng (PCC-OFDM was sgn to ruc th Intrchann Intrfrnc (ICI caus by frquncy offst []. In PCC-OFDM, ach ata vau to b transmtt s mapp onto a group of subcarrrs. In ths papr, th cas whr ach ata vau to b transmtt s mapp onto pars of subcarrrs s consr. Dspt ts avantags, PCC-OFDM s not banwth ffcnt n ts smpst form []. On way to ovrcom ths rawback s to ovrap PCC-OFDM symbos n th tm oman [3]. Fgur shows th procur of symbo ovrappng. Wth ovrappng, ach ovrapp symbo conssts of thr parts, th currnt PCC-OFDM symbo, th scon haf of th prcng PCC-OFDM symbo an th frst haf of th foowng PCC-OFDM symbo. T hgh sp ata stram s f nto a sra to para convrtr an convrt nto n owr sp para substrams. Th th vctor to b transmtt s rprsnt by,. Thy ar mapp onto,, n, th vaus a, a, that mouat th subcarrrs n th th symbo pro. For convntona OFDM, whch has n an a k, k,, thr s a smp on-to-on mappng of ata vaus onto th subcarrrs. For PCC-OFDM, n an ar not qua. In ths papr, w hav n. In th chann, th sgna s ftr by th chann rspons h ( t an atv nos n ( t s njct. At th rcvr th th output vctor of th DFT s. Th mouat subcarrrs ar thn,, wght an a to gnrat th ata stmats v, v n,. For th subcarrr par M, an M,, an stmat s cacuat usng. To rcovr th transmtt ( v M, M, M, ata vaus from th ovrapp symbos, a twomnsona Mnmum Man Squar Error (MMSE frquncy oman quar can b us [3]. Hgh sp ata To MMSE Equar Dv nto ow bt rat strams v, v n,, n, Wght an a subcarrrs Mappng ata onto subcarrrs Transmttr Rcvr,, a, a, pont DFT y, y, pont IDFT b, b, Ftrng ADC Sra to Para Para to Sra ovrap DAC an ftrng xp( jπf t Chann h(t n(t BPF xp( jπ ( f f t c c T T 3T T Tm Fg.. Bock agram of a PCC-OFDM systm Fg. PCC-OFDM symbos ovrapp n th tm oman Fgur shows a smpf bock agram of an Ovrap PCC-OFDM communcaton systm. Th In [4], a bn frquncy offst stmator has bn prsnt for PCC-OFDM. As shown n [5], th frquncy offst stmator for PCC-OFDM can aso b us for PCC-OFDM wth symbos ovrapp n th tm oman. Rcnty, a nw MMSE frquncy

2 offst stmator has bn propos for PCC-OFDM [6]. In ths papr t w b shown that th sam form of th stmator can aso b us for Ovrap PCC- OFDM.. Subcarrr mbaanc n PCC-OFDM In Ovrap PCC-OFDM, ach mouat subcarrr contans ovrappng componnts from ajacnt PCC- OFDM symbos [4]. Th subcarrr par mbaanc s st fn as th amptu or powr ffrnc btwn two subcarrrs n a mouat subcarrr par. That s F( ft ( M, M, Each vau of mbaanc s a combnaton of th mbaanc of th PCC-OFDM subcarrr par an th mbaanc of th ovrappng componnts n th mouat subcarrr par. In th absnc of frquncy offst, two mouat subcarrrs n an Ovrap PCC-OFDM subcarrr par ar baanc n a partcuar way. Th subcarrr par mbaanc pns not ony on th frquncy offst an transmtt ata vaus n th currnt PCC-OFDM symbo, but aso on th ovrappng componnts from th prcng an foowng PCC-OFDM symbos. As for PCC-OFDM, th ata pnncy n Ovrap PCC-OFDM can b rmov by avragng ovr a numbr of subcarrr pars. Th avrag subcarrr par mbaanc pns on th frquncy offst, thrfor, w can us th avrag mbaanc for frquncy offst stmaton. Fg. 3. Subcarrr par mbaanc as a functon of frquncy offst Fgur 3 shows th avrag SPI as a functon of frquncy offst for 3. Th numbr of smuat subcarrrs was, an a subcarrr pars wr us. As for PCC-OFDM [6], Th fgur shows som ntrstng faturs: Crossng ro at ro frquncy offst. For <. 5, th mbaanc ncrass monotoncay as th frquncy offst ncrass. Atv Wht Gaussan os (AWG os not affct th ro crossng poston. Th frquncy stmaton rang can b xtn to on subcarrr spacng. 3. Exprsson for an Ovrap PCC-OFDM subcarrr Th k th samp n th th tm oman symbo n PCC-OFDM s gvn by [] b k, a, jπk ( whr a, s th th subcarrr n th th frquncy oman symbo. Th ovrappng componnts n th currnt symbo ar ntrouc from th scon haf of th ( th PCC-OFDM symbo an th frst haf of th ( th PCC-OFDM symbo. Th kth ovrappng componnt s gvn by [4] b k, jπk ( a, for k / jπk ( a, xp for / k (3 At th rcvr, th k th vau of th th nput ata bock to th FFT mouator s gvn by jπkε xp ( bk, bk, wk y k,, (4 whr ε s th frquncy offst, ε. w k, s th Gaussan nos. Th m th subcarrr n th th mouat symbo s thn gvn by Th m, k k jπkm yk, jπkm yk, k y k, jπkm (5 M th mouat subcarrr s gvn by M, ( CF ( L M CF ( L M L ( CS( L M CS( L M L ( c( L M c( L M L W M, (6

3 whr W M, s th FFT of w k,, c m ar compx coffcnts gvn by [] c m It s shown n [7], CF m an m CF CS sn( π ( m sn( π ( m xp( jπ ( ( m W M, CS ar gvn by m m k k (7 s Gaussan. Th coffcnts jπk ( m ε jπk ( m ε (8 (9 Smary, th ( M th mouat subcarrr can b obtan. ^ Fg. 4. Frquncy offst stmator as a functon of frquncy offst 4. MMSE frquncy offst stmator It s shown n Appnx A that quaton ( for Ovrap PCC-OFDM for 4QAM can b wrttn as F ( Ksn ( π M ( whr K s a constant gvn by [6] π K cos og ( k cos k π ( M s th rror trm. For sma frquncy offst, th rror trm can b approxmat as atv nos wth ro man an fnt varanc. Appyng MMSE tchnqus for quaton ( [8], w can obtan th frquncy offst stmator ˆ sn π KM M t F ( t ( whr M s numbr of subcarrr pars to b us for th frquncy offst stmaton. 5. Smuaton rsuts To vauat th prformanc of th MMSE stmator, smuatons wr prform. In th foowng smuatons, 4QAM s us for th mouaton schm wth M 5 an 8. Fgur 4 shows th stmator as a functon of th frquncy offst for E b B. It s vnt that th stmator has an approxmaty nar ratonshp wth frquncy offst. ft Fg. 5. Varanc of th stmator as a frquncy offst Fgur 5 shows th varanc of th frquncy offst stmator as a functon of th frquncy offst for an a chann. Th varanc os not chang sgnfcanty as frquncy offst ncrass. Ths mans that th ovrappng componnts n Ovrap PCC- OFDM ar th omnant factor for th varanc. Lowr varanc can b obtan by ncrasng M or usng a two-mnsona MMSE quar bfor th frquncy offst stmaton. Anothr way to ruc th varanc s to us PCC-OFDM pot symbos wth no ovrappng. Fgur 6 shows th varanc of frquncy offst stmator as a functon of E b. Th frquncy offst smuat was ro. Th varanc os not sgnfcanty chang as th chann nos ncrass. Th varanc s omnat by th ovrappng componnts.

4 Fg. 6. Varanc of th stmator as a functon of E b 6. Concuson An MMSE frquncy offst stmator for Ovrap PCC-OFDM has bn prsnt. A frquncy offst stmat s obtan by usng th avrag SPI at th output of th rcvr FFT. o tranng symbos or pot tons ar rqur. Th MMSE stmator has an approxmaty nar ratonshp wth frquncy offst. Thr ar thr masurmnts to ruc th varanc, on s to ncras th numbr of subcarrr pars, th scon s to us a two-mnsona MMSE quar bfor th stmaton an th thr s to us PCC-OFDM pot symbo wth no ovrappng componnts. Appnx A From (6, th powr of th by M, / / / ( LM M th subcarrr s gvn ( CF CF ( CS CS ( LM ( c c ( LM M, ( LM ( LM ( LM (A. Equaton (A. ncats that th powr of ach mouat subcarrr can b rprsnt n trms of th nvua powr componnts from th prcng, currnt an foowng PCC-OFDM symbos. ot for 4QAM wth ach subcarrr norma to unty, E b/, B. Th rror trm of (A. s thn gvn by M, / / ( c ( LM c ( LM c ( KM K L K ( c ( KM / / ( CF( LM CF( LM CF ( KM K L K / / ( CS ( LM CS( LM CS ( KM K L K K, ( CF ( KM ( CS ( KM K, K, / / ( c ( ( ( LM c ( LM CS KM CS( KM L, K, K / / ( CF( ( ( LM CF( LM c KM c( KM L, K, K / / ( c ( ( ( LM c ( LM CS KM CS( KM L, K, K / / ( CF( ( ( LM CF( LM c KM c( KM L, K, K / R{ W ( ( } M, CF KM CF( KM K, K / R{ W ( ( } M, CS KM CS( KM K, K / R{ W ( ( } M, c KM c ( KM K, W M, K (A. Bcaus th xpct vau of a cross trms s ro, th xpct vau of th rror trm s qua to th varanc of th nos. Smary, w can obtan th powr for th ( M th subcarrr. Usng L, for 4QAM, th subcarrr par mbaanc s thn gvn by M, M, M, / / ( CF ( ( ( LM CF( LM CS LM CS( LM / / ( CF ( ( ( LM CF( LM CS LM CS ( LM / / ( c ( ( ( L M c( L M c L M c( LM M, (A.3 It s shown n Appnx B that th combnaton of th frst four summatons n (A.3 quas ro. Thus w can obtan th subcarrr par mbaanc F ( ft S( (A.4 whr S( ft s gvn by [6] M, M, M, S( K sn( π (A.5 K s a factor fn n (. M, s th M th tota rror,. In Ovrap PCC-OFDM, th M, M, M, rror trm s mor compcat than n PCC-OFDM. Howvr, th omnant strbuton n th rror trm s st Gaussan. As for PCC-OFDM, M, can b

5 approxmat as AWG wth ro man. Th varanc s argr than that of PCC-OFDM bcaus of th ovrappng componnts. Appnx C shows th varanc of th coupng-crossng trms for M,. Thus, unr th conton of a sma frquncy offst, th ovra approxmat varanc s gvn by 4 (, 8 8σ σ (A.6 M, ~ n n whr "~" mans strbut as. ot that w hav us σ s for 4QAM n (A.6. Th SPI stmator s thrfor gvn by Appnx B ˆ sn π KM M F ( (A.7 Th mnt of th frst summaton of (A.3 s gvn by ( CF ( LM CF( LM πε πε sn cos π ( L ε π ( L ε sn sn sn( πε π sn π ( L ε π ( L ε sn sn (B. Th mnt of th scon summaton of (A.3 s gvn by ( CS ( LM CS( LM πε πε sn cos π ( L ε π ( L ε sn sn sn( πε π sn π ( L ε π ( L ε sn sn (B. Smary th thr an fourth trms can b rv. Substtutng th mnts rv to th combnaton of th frst four summatons of (A.3 gvs / / ( CF ( LM CF( LM CS( LM / / ( CF ( L ( M CF LM CS( LM ( CS ( LM ( CS ( LM πε πε cos cos ( ( π L ε π L ε sn sn (B.3 Appnx C Th coupng-cross trms CC, n (A. s gvn by CC M, M / / ( c( L M c( L M ( CS( K M CS( K M K, L K / / ( CF ( L M CF( L M ( c( K M c K M L (, K, L K / / ( c( L M c( L M ( CS( K M CS( K M K, L K / / ( CF ( L M CF( L M ( c( K M c( K M K, L K (C. Smary, th coupng-cross trms CC M, for th ( M th rror trm can aso b obtan. ot th frst trm an th thr trm ar mutuay conjugat, th scon trm an th fourth trm ar mutuay conjugat n (C.. Thus, CC M, CC M, can b wrttn as CC M, M, { M, ( CS ( K ( } M CS ( KM CS KM K, K { M, ( CF ( L ( M CF( LM CF LM } R R CC (C. whr R ( rprsnts th ra part of a compx numbr. For th frst summaton n (C., consrng th most sgnfcant compx coffcnts CS, w gt / ( CS ( ( K M CS ( K M CS K M M, K M, CS M, K, (C.3 Smary, w can aso smpfy th scon summaton n (C.. In aton, n th absnc of frquncy offst, th vau of th most sgnfcant compx coffcnt can b obtan from (7 an (8, that s CS CF.5. Thus (C. can b wrttn as

6 CC M, CC M, { } R{ } R M, M, M, M, (C.4 Th varanc of th ft s of (C.4 s gvn by E [ CC ] M, CC M, E[ 4( R( ( ( M, M, 4 R M, M, 4( R( ( R( ] 8σ 4 s M, M, M, M, (C.5 Whn a compx coffcnts ar consr, th varanc w b sghty argr than abov rsut. Pas not that ths rsut s obtan unr th assumpton of no frquncy offst. In th prsnc of a sma frquncy offst, th varanc cou b sghty argr. Rfrnc. J. Armstrong, "Anayss of w an Exstng Mthos of Rucng Intrcarrr Intrfrnc u to Carrr Frquncy Offst n OFDM", IEEE Transactons on Communcatons, March 999, VOL 47, o.3, pp J. Armstrong, P. M. Grant, an G. Povy, "Poynoma Cancaton Cong of OFDM to ruc Intrcarrr Intrfrnc u to Doppr spra", IEEE Gobcom, VOL. 5, pp , 8- ovmbr, J. Armstrong, J. Shntu an C. Tambura, Frquncy Doman Equaaton for OFDM Systms wth Mappng Data onto Subcarrr Pars an Ovrappng Symbo Pros. Procngs of th 5th Intrnatona Symposum on Communcaton Thory an Appcatons. Engan Juy 999, pp J. Shntu an J. Armstrong, "A w Frquncy Offst Estmator for OFDM", "Communcaton systms ntworks an gta sgna procssng", t by A. C. Boucouvaas, pp.3-6, Juy, UK. 5. J. Shntu an J. Armstrong, "Bn Frquncy Offst Estmaton for PCC-OFDM wth Symbos Ovrapp n th Tm Doman". Procngs of IEEE ISCAS, VOL. 4, pp , May 6-9, Syny. 6. J. Shntu, J. Armstrong an G. Tobn, MMSE Frquncy Offst Estmator for PCC-OFDM, accpt by IEEE MICC,. 7. A. Papous, "Probabty, Ranom Varabs an Stochastc Procsss," McGraw-H Intrnatona Book Company, A. Rona Gaant, "onnar Statstca Mos," John Wy an sons, 987.