Self-Cacellatio of Sample Frequecy Offset i OFDM Systems i the Presece of Carrier Frequecy Offset Zhe GAO, 2, Mary A Igram 2 School of Electroic ad Iformatio Egieerig, Tiaji Uiversity, Chia 372 2 School of Electrical ad Computer Egieerig, Georgia Tech, Atlata, GA 3332-25 zgao9@mail.gatech.edu, mai@ece.gatech.edu Abstract - A self-cacellatio scheme is proposed to cope with the sample frequecy offset (SFO) problem i OFDM systems i the presece of a carrier frequecy offset (CFO). Maig use of the symmetry betwee the phase shifts caused by SFO ad the subcarrier idex, we put the same costellatio symbol o symmetrical subcarriers, ad combie the pairs at the receiver coheretly. I this way, the SFO effects are approximately cacelled with the price of cuttig dow the badwidth efficiecy by half. However some array gai ad diversity gai are obtaied from the symmetrical combiig. Our scheme ca wor well together with the phase tracig for residual CFO, so that both SFO ad residual CFO ca be removed with low complexity. Simulatios show that, our scheme effectively removes the effect of SFO; the proposed system outperforms the ideal ormal OFDM systems (with o SFO) uder the same eergy efficiecy at high SR, so the proposed system will also outperform the ormal system that usig the same overhead for SFO estimatio. Fially, a mixed system is proposed to mitigate the drawbacs of our desig ad the ormal OFDM systems for the SFO compesatio. Our scheme may be helpful for the sychroizatio of multiple SFOs i the cooperative trasmissio. I. ITRODUCTIO The self-cacellatio scheme i this paper is to correct the phase shifts caused by samplig frequecy offset (SFO) i the orthogoal frequecy-divisio multiplexig (OFDM) system. The self-cacellatio schemes proposed before for the iter-carrier iterferece (ICI) caused by carrier frequecy offset (CFO), IQ imbalace or phase oise [-3] caot cacel the effect of SFO. We are the first to apply the self-cacellatio method for SFO, which is more difficult to have a accurate estimatio tha CFO. OFDM modulatio is preferable for high-speed trasmissio i frequecy selective chaels. However, because the subcarrier spacig is arrow, a OFDM system is sesitive to the frequecy sychroizatio errors, e.g. CFO ad SFO. For pacet-based OFDM trasmissio, CFO cacellatio has bee well studied i [4-7]. Compared with CFO, SFO is more difficult to estimate accurately. The effect of SFO o the performace of OFDM systems was first addressed by T. Pollet et al [8]. SFO itroduces two problems i the frequecy domai: iter-chael iterferece (ICI) ad phase rotatio of costellatios. As metioed i [8], [], [3] ad [6], the power of the ICI is so small that the ICI are usually tae as The authors gratefully acowledge the support of this wor from Chiese Scholarship Coucil (CSC), ad the atioal Sciece Foudatio uder CS -72296. Zhe GAO appreciates the importat advices from his advisor -- Professor GOG Ke of Tiaji Uiversity, P.R. Chia. additioal oise. So the removal of SFO is maily the correctio of phase rotatio. Geerally speaig, there are three methods to correct the SFO. The first is to cotrol the samplig frequecy of the ADC directly at the receiver [2-4]. However, accordig to [8], this method is ot preferable for low-cost aalog frot-eds. The secod method is iterpolatio/decimatio [9], [], [5-7]. The SFO is corrected by resamplig the base-bad sigal i the time domai. The problem of this method is that the complexity is too high for high-speed broadbad applicatios. The third method is to rotate the costellatios i the frequecy domai [2], [3]. The basis for this method is the delay-rotor property [3], which is that the SFO i the time domai causes phase shifts that are liearly proportioal to the subcarrier idex i the frequecy domai. The advatage of this phase rotatio method is its low complexity. However, the performace of such method relies o the accuracy of SFO estimatio. I previous wors, there are three methods for SFO estimatio. The first method is cyclic prefix (CP)-based estimatio []. The performace of this method relies o the legth of CP ad the delay spread of the multipath chael. The secod is the pilot-based method [9], [2] ad [7]. The problem with this method is that, because the pilots are just a small portio of the symbol, it always taes several te s of OFDM symbols for the tracig loop to coverge. The third is the decisio-directed (DD) method [4], [6]. The problem of this method is that whe SFO is large, the hard decisios are ot reliable, so the decisios ca be obtaied oly by decodig ad re-costructig the symbol, which requires more memory ad complexity. Because o estimatio method is perfect, the correctio method relyig o the estimatio will ot be perfect. So we cosider desigig a SFO self-cacellatio scheme, so that the effect of SFO ca be removed by itself without estimatio. Aother strog motivatio of our wor is the sychroizatio i cooperative trasmissio (CT). I a CT, multiple SFOs exist at the receiver, so all the methods metioed above caot solve the problem. Our SFO self-cacellatio scheme is composed of some liear operatios, so it will be very helpful for the cacellatio of SFOs at the receiver i a CT. I this paper, we propose a SFO self-cacellatio scheme for OFDM systems. I stead of focusig o the liearity betwee phase shifts caused by SFO ad subcarrier idex as usual, we mae use of the symmetry betwee them. We put the same costellatio o symmetrical subcarriers, ad combie each pair coheretly at the receiver, so that the phase shifts caused by SFO o symmetrical subcarriers approximately cacel each other. For the phase tracig for residual CFO, pilots are also iserted symmetrically i each OFDM symbol. The drawbac of this method is that the badwidth efficiecy is
cut dow by half because of the self-cacellatio ecodig. However, because of the combiatio, some diversity is obtaied, so the price is decreased. Simulatios show that our SFO self-cacellatio scheme ca wor well together with the usual residual CFO correctio, ad the performace of our system is better tha that of the ideal ormal OFDM system (with o SFO) uder the same eergy per bit coditio at high SR. This result tells that, the proposed system will outperform the ormal OFDM system i which half of the badwidth is used as pilots for SFO estimatio. This paper is orgaized as follows. Sectio II gives the OFDM sigal model. The SFO self-cacellatio algorithm is aalyzed i Sectio III. I Sectio IV, we give simulatio results that demostrate the performace of our algorithm. Sectio V gives same discussios about our scheme. Fially, coclusios are give i Sectio VI. A. Basic Model II. OFDM SIGAL MODEL The width of the iverse fast Fourier trasform (IFFT) (or umber of subcarriers) is, i which d subcarriers are used for data symbols ad q subcarriers are used for pilot symbols. The legth of CP is g, so the total legth of oe OFDM symbol is s = + g. We use f s to deote the samplig frequecy, the the sample duratio is T s = /f s. Assumig the costellatios of the m-th OFDM symbol are a m, ( = -/2, -/2+,, /2- is the subcarrier idex), the the trasmitted base-bad sigal is M 2 2π j ( m s g ) s[ ] = am, e rect( m s), () m= = 2 where, < rect( ) s =., else Lmax Assumig the multipath chael is h[ ] = hlδ ( l), where L max is the legth of the chael, the received base-bad sigal of the m-th OFDM symbol is Lmax r [ ] = h s[ m + + l] + w[ ], =,,..., (2) m l s g where w[] is the complex Gaussia oise. After FFT, the -th subcarrier of the m-th OFDM symbol is z = a H + w, (3) m, m, m, where H L max j2π l hle = is the frequecy domai respose. B. Itroductio of CFO ad SFO We assume f (Hz) is the CFO, ε = (T s '-T s )/T s is the SFO, ad T s ' is the sample duratio of the receiver. Accordig to [9], with the abbreviatio = m s + g +, the received samples ca be expressed as, (4) r e h a g ε l w j2 π '( + ε ) ft [ ] s m = l m, ( ',, ) + [ ] l m 2π j ( '( + ε ) ms g l ) where g( ', ε, l) = e rect( '( + ε ) m l). s After FFT, the -th subcarrier of the m-th OFDM symbol is jπφ j2 π (( ms + g )/ ) φ m, = ( )sic( πφ ) m, z e e a H π φ jπφi, j 2 (( ms + g )/ ) i ( e e )sic( πφi, ) am, ihi wm, i; i + + (5) with cross-subcarrier ad local subcarrier parameters φi, = ( + ε )( ft + i), (6) φ = ft + ε respectively, where T = T s. I (5), the first term is the expected sigal, ad the secod term is the ICIs from other subcarriers. As metioed i Sectio I, after the most part of ICIs is removed by coarse CFO sychroizatio, the ICIs left ca be tae as additioal oise w ICI. The the post-fft sigal becomes jπφ j2 π (( ms + g )/ ) φ z = ( e e )sic( πφ ) a H + w + w (7) m, m, ICI, m, j I (7), e πφ ad sic( πφ ) are the local phase icremet ad local amplitude gai, respectively, ad they are combied ito the estimated chael respose H '= j e πφ sic( ), πφ H. After chael equalizatio, (7) becomes j2 π (( ms + g ) / ) φ z = e a + w' + w', (8) m, m, ICI, m, where w' ICI, = wici, / H ' ad w' m, = wm, / H '. I (8), oly the accumulated phase j2 (( ms g ) / ) e π φ + eeds to be corrected. III. SELF-CACELLATIO OF SAMPLE FREQUECY OFFSET A. Idea Demostratio Our SFO self-cacellatio scheme is ispired by the relatioship betwee phase icremet ad the subcarrer idex. Fig. is a simulatio result that demostrates the phase shifts caused by residual CFO ad SFO. From the figure, we ca fid two truths. The first truth is that the phase shifts for the subcarriers i the middle are liearly proportioal to the subcarrier idex. This is the delay-rotor property metioed i Sectio I, ad has bee explored a lot for estimatio ad correctio of SFO. We also otice that the phase shifts for the edge subcarriers do ot obey the liearity. However, for the coveiece of desig of trasmit ad receiver filters, ad iter-chael iterferece suppressio, these subcarriers are usually set to be zeros [2]. The other truth is that the phase shifts caused by SFO are symmetrical relative to the costat phase shift caused by residual CFO (dotted horizotal lie i Fig. ), so if we put the same costellatio o symmetrical subcarriers, we may be able to combie the symbols at the receiver i a way such that the phase shifts o these two subcarriers caused by SFO ca approximately cacel each other. This mappig ca be called Symmetric symbol Repetitio (SSR), which is differet from other self-cacellatio techiques, Adjacet Symbol Repetitio (ASR), Symmetric Symbol Repetitio (SSR), Adjacet Cojugate Symbol Repetitio (ACSR), ad Symmetric Cojugate Symbol Repetitio (SCSR), proposed before i [-3].
Phase Shift caused by SFO ad residual CFO.4.2.8.6.4.2 Combie Symmetric Subcarriers B. Theoretical Aalysis Liear Part SFO = 5 ppm; 3 OFDM Symbols o-liear Part Phase shift caused by residual CFO -.2-32 -28-24 -2-6 -2-8 -4 4 8 2 6 2 24 28 3 Subcarrier Idex Fig.. Liearity ad Symmetry of the Phase shift. Assumig the same costellatio a m, is mapped o symmetrical subcarriers ad, the sigal (after chael equalizatio) o the pair ca be expressed, accordig to (8), as j 2 π (( m s + g ) / ) φ zm, = e am, ICI, m, j 2 π (( m s + g ) / ) φ zm, e = am, ICI, m,. (9) The the sum of z m, ad z m,- is z ' j2πg ft = 2cos(2 πgε ) e a + αw' + w'', () m, m, ICI, m, i which G = (m s + g )/. Because 2πGε <<, 2cos(2πGε) 2. I the same way, because the eergy of ICIs is maily from residual CFO, ad the ICIs caused by residual CFO are same for symmetrical subcarriers, the ICIs terms are also combied almost coheretly, which meas α 2. So the average SIR does ot chage after combiatio. w m, ad w m,- are idepedet, ad the fial oise term is w '' m, = w ' m, m,- = wm, / H ' + wm, / H '. () Assumig E{ a m, 2 } =, E{ H 2 } =, E{ w ICI, 2 } = σ 2 ICI, ad E{ w m, 2 } = σ 2, uder the assumptio that σ 2 ICI << σ 2, the average SIR before combiatio (from (7)) ad after combiatio (from ()) are, respectively, SIRBf = 2 2 2 σ ICI + σ σ. (2) 4 2 SIRAf 2 2 2 4σ ICI + 2σ σ The average SIR has bee improved by 3dB, which is just the array gai from combiatio. Also, because small values are more liely for 2 H tha for 2 ' 2 2 H (/ H ' + / ' ), some diversity gai is achieved. Fig. 2 shows that this diversity gai we get is about half of that of 2-brach MRC. I the figure, H, H 2 ad H are idepedet Rayleigh fadig radom variables. Simulatios i Sectio IV will also show the diversity gai. Pr(log(y) <= x).9 y = ( H.8 2 H 2 2 /( H 2 + H 2 2 )) / Media.7.6.5.4.3 y = ( H 2 + H 2 2 ) / Media y = 2 H 2 / Media Empirical CDF.2 Sigle Rayleigh path. SFO-SC MRC-2 - -5 5 x C. System Structure Fig. 2. Diversity gai from Symmetric Combiatio Fig. 3 (bottom of this page) gives the structure of the trasmitter ad receiver with our SFO self-cacellatio scheme. At the trasmitter, the Modulatio o Half Subcarriers ad Symmetrical Mappig blocs compose the Self-Cacellatio Ecodig module; at the receiver, the Chael Equalizatio ad Symmetrical Combiig blocs compose the Self- Cacellatio Decodig module. For the coarse CFO sychroizatio ad chael estimatio, repeated short traiig blocs ad repeated log traiig blocs compose the preamble. To remove the residual CFO, the phase shifts o pilots after the SFO self-cacellatio decodig are averaged to get oe phase shift, which is multiplied to all the data subcarriers after the self-cacellatio decodig. Fig. 4 shows how to do symmetrical mappig. To do phase tracig for residual CFO correctio, pilot symbols are also mapped symmetrically. For the coveiet of desig of trasmit filter ad receive filter, twelve subcarriers o the edge are set to be zeros. Fig. 4. Symmetrical Mappig. Fig. 3. Bloc diagram of the Trasmitter ad Receiver with the SFO Self-Cacellatio Scheme.
IV. SIMULATIOS MATLAB is used for the simulatio. Most of the parameters are listed i Table. ½ rate covolutioal codig ad bloc iterleavig apply to the bits sequece of oe pacet. The typical urba chael model COST27 [6] is used, ad the istataeous chael gai is ormalized to be uity. Each of the short traiig blocs ad each of the log traiig blocs are defied i the IEEE 82.a stadard [7]. I the simulatio, except ie short traiig blocs ad four log traiig blocs, each pacet icludes 5 OFDM symbols, so the total legth of oe pacet is 6 +8 4+8 5 = 448 samples, which is log eough for most applicatios. Cosiderig a reasoable value of SFO = 5 ppm [5], the accumulated time offset caused by SFO durig the trasmissio of oe pacet is 448 5-6 =.224, which meas o sample will be sipped or sampled twice due to SFO. I this case, o Sip/Dup operatio is eeded. A. Effectiveess of SFO Self-Cacellatio To show the effectiveess of our SFO self-cacellatio, we give the 6QAM costellatios i Fig. 5. I the simulatio, SR = 25dB, SFO = 5ppm, ad residual CFO exists after the coarse CFO correctio. Figures (a) ad (c) show the effect of residual CFO ad SFO separately. Figure (b) is the ideal case with o SFO ad residual CFO is corrected by phase tracig i ormal OFDM system. Figure (d) is the result after phase correctio (SFO self-cacellatio ad phase tracig for residual CFO) i our desig. From (d) we see that both residual CFO ad SFO are removed effectively. I additio, comparig with ideal case of ormal OFDM system (b), figure (d) loos eve better. This is because we get 3 db array gai ad some diversity gai from the symmetrical combiig. B. Pacet Error Rate TABLE I SOME PARAMETERS I THE SIMULATIO Meaig otatio Values i Simulatio Total bad width W (Hz) 8M umber of pilot subcarriers p 4 umber of data subcarriers d 48 umber of zero subcarriers z 2 Total umber of subcarriers = d + p+ z 64 Subcarrier Spacig df (Hz) 25 Legth * of CP L c 6 Legth of oe complete OFDM symbol L t= +L c 8 umber of short traiig blocs s Legth of oe short traiig bloc L s 6 umber of log traiig blocs ce 4 Legth of each log traiig bloc L t 8 *ote: the uit of legth i this table is sample. To show the performace of the whole system with our SFO self-cacellatio scheme, we compare the PER for various simulatio cofiguratios i Fig. 6. I the simulatios, CFO is.3 times the subcarrier spacig, ad SFO = 5 ppm. We give two referece systems that do ot do SFO self-cacellatio: 4QAM, uit trasmit power, all pilots i oe pacet are used for SFO estimatio ad correctio (4QAM-SC-Est-Uit, solid dot-lie); 4QAM, uit trasmit power with o SFO (4QAM-SC-Ideal-Uit, dash-lie), which is a ideal case. The Averaged Pacet Error Rate - - - SFO with residual CFO ucorrected - (a) CFO with SFO ucorrected -.5 - (b) PER Performace of SFO Self-Cacellatio Array Gai Diversity Gai -2 4QAM-SC-Est-Uit 4QAM-SC-Uit 4QAM-SC-Half 6QAM-SC-Uit 4QAM-SC-Ideal-Uit -3 5 2 25 3 Received SR (db) Fig. 6. PER Performace of the SFO Self-Cacellatio. systems with SFO self-cacellatio are: 4QAM, uit trasmit power (4QAM-SC-Uit, circle-lie); 4QAM,.5 trasmit power (4QAM-SC-Half, square-lie); 6QAM, uit trasmit power (6QAM-SC-Uit, triagle -lie). ote that 4QAM-SC-Half ad 6QAM-SC-Uit have the same eergy per bit with 4QAM-SC-Est-Uit ad 4QAM-SC-Ideal-Uit. From Fig.6, we ca see that, whe PER =.: () 4QAM-SC-Uit outperforms 4QAM-SC-Est-Uit by 8 db, ad outperforms 4QAM-SC-Ideal-Uit by 5.5 db (3dB array gai ad 2.5dB diversity gai); (2) 4QAM-SC-Half outperforms 4QAM-SC-Est-Uit for all SRs, ad outperforms 4QAM-SC-Ideal-Uit for high SR; (3) 6QAM-SC-Uit also outperforms 4QAM-SC-Est-Uit ad 4QAM-SC-Ideal.5.5 -.5 - SFO with residual CFO corrected After Phase Correctio - - (c) (d) Fig. 5. Effect of Phase Correctio with SFO Self-Cacellatio (6QAM): (a) Ideal OFDM before CFO correctio; (b) Ideal OFDM after CFO correctio; (c) SC before SFO correctio; (d) SC after SFO correctio. -
-Uit at high SR with the same data rate. Result () also tells that, i the ormal OFDM system, if half of the badwidth is used as pilots for SFO estimatio (the same badwidth efficiecy with our desig), the performace will be much worse tha the proposed system (> 5.5dB whe PER =.), because the SFO estimatio caot be perfect. The reaso we ca get some PER improvemet is that we get array gai (if the trasmit power is ot cut dow) ad some diversity gai from the self-cacellatio decodig. From the figure, we see that F, F ad F2 have larger slope tha R ad R2. Accordig to the power delay profile defied i [9], the delay spread σ τ 5us, so the 5% coherece badwidth is B c = /(5σ τ ) 4Hz, which is about 3.2 times the subcarrier spacig. This meas the four subcarriers i the middle are highly correlated, so the two combiatios of these two pairs will ot brig beefit of diversity, but oly array gai if the trasmit power is ot cut dow. V. DISCUSSIO As metioed above, our desig has two drawbacs. The first oe is its low badwidth efficiecy. Although the simulatios show that the performace of our desig is better tha or comparable with the ideal referece system uder the same eergy per bit at high SR, we still wat to improve the efficiecy. The secod drawbac is that our desig has ot cosidered the case whe the pacet is so log that oe sample may be sipped or sampled twice because of the accumulated timig offset. O the other had, for ormal OFDM systems, SFO caot be estimated accurately durig short period of time (this is oe of the motivatios of our wor). So, if we combie our desig with the ormal OFDM system smartly, the drawbacs of both systems metioed above ca be mitigated. The mixed system ca wor i the followig way. Because accurate SFO estimatio caot be obtaied durig the first te s of OFDM symbols, these symbols ca be trasmitted with our SFO self-cacellatio scheme. At the receiver, durig the decodig of these OFDM symbols, all the phase shifts o pilot subcarriers ca be collected together before symmetrical combiatio for SFO estimatio. After the trasmissio of these te s of OFDM symbols, a accurate SFO estimatio is already obtaied by the receiver, so the system ca switch to the ormal mode with a SFO tracig loop. VI. COCLUSIO A self-cacellatio scheme is desiged for SFO correctio i OFDM systems i the presece of CFO. Maig use of the symmetry betwee phase shifts caused by SFO ad subcarrier idex, each symmetrical pair of subcarriers carries the same symbol. I the receiver, each pair is combied coheretly so that the phase shifts caused by SFO o symmetrical subcarriers approximately cacel each other. I our desig, pilots are also iserted symmetrically i each OFDM symbol, so that residual CFO correctio by phase tracig ca wor as usual. Although the badwidth efficiecy of our desig is cut dow by half because of the self-cacellatio ecodig, array gai ad diversity gai result from the symmetrical combiig. Simulatios show that our scheme ca wor well with ormal CFO correctio, ad the performace of our system is eve better tha that of ideal ormal OFDM system with o SFO uder the same eergy efficiecy at high SR. Also, our system outperforms the ormal OFDM system usig the same overhead for SFO estimatio. Fially, a mixed system is proposed to compesate the drawbacs of our desig ad the ormal OFDM systems. Because our scheme oly ivolves liear operatios, it may be helpful for the correctio of multiple SFOs i the cooperative trasmissio. REFERECES [] Y. Zhao ad S. G. Haggma, Itercarrier iterferece self-cacellatio scheme for OFDM mobile commuicatio systems, IEEE Tras. Commu., vol. 49, pp. 85 9, Jul. 2. [2] Tag S., Gog K., Sog J., et al. Itercarrier iterferece cacellatio with frequecy diversity for OFDM systems. IEEE Tras. 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