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基于OFDM系统信道估计以及帧同步算法研究

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Signal Processing 81 (2001) 16951704/locate/sigproAnalysis of a new frequency synchronization scheme inOFDM systemsS. Zazo, Jos. e Manuel P. aez-BorralloETS Ingenieros de Telecomunicaci? on, Universidad Polit? ecnica de Madrid, Ciudad Universitaria s=n, Madrid 28040, SpainReceived 22 March 2000; received in revised form 25 November 2000AbstractThis paper deals with the analysis of a new method which allows the unambiguous frequency o7set acquisition inN-OFDM systems exploiting the knowledge of only one symbol. This new approach provides an important improvementin those systems where fast and accurate time-frequency synchronization is a critical point. In our proposal, the limitof the acquisition range for the carrier frequency o7set can be extended toN=2 the subcarrier spacing rather than1=2 the subcarrier spacing as previous methods. A common formulation of related methods and the calculus of theprobability of detection of our present work are also provided. Finally, several computer simulations in particularscenarios with both deterministic and random frequency o7sets support our proposal.?2001 Elsevier Science B.V.All rights reserved.Keywords: OFDM; Frequency synchronization1. IntroductionResearch and development are taking place allover the world to de=ne the next generation ofwireless broadband multimedia communicationssystems that may create the global informationvillage. This system is expected to provide itsusers with customers premises services that haveinformation rates exceeding 2Mbps. The mostsuitable modulation choice seems to be orthogonalfrequency division multiplexing (OFDM) as a spe-cial case of multicarrier (N) transmission where asingle data stream is transmitted over a number oflower rate subcarriers 8.Correspondingauthor.Tel:+34-91-549-5700;fax:34-91-3367350.E-mail address: santiagogaps.ssr.upm.es (S. Zazo).One of the main reasons to use OFDM is to in-crease the robustness against frequency selectivefading because in fact there is no intersymbol in-terference (ISI) at all if a long enough cyclic pre-=x (CP) is included. This e7ectively simulates achannel performing cyclic convolution which im-plies orthogonality over dispersive channels 1. Letus observe that in a single carrier system, a singlefade can cause the entire link to fail, meanwhile in amulticarrier system, only a small percentage of thesubcarriers will be a7ected. Error correction codingcan then be used to correct for the few erroneoussubcarriers.From the point of view of synchronization re-quirements, two main applications can be distin-guished: on one hand, broadcasting (digital audioor video broadcasting DAB, DVB) with a continu-ous data stream where synchronization is based on0165-1684/01/$-see front matter?2001 Elsevier Science B.V. All rights reserved.PII: S0165-1684(01)00080-91696S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 16951704the cyclic extension with a long acquisition time;on the other hand, burst operation as in TDMAradio access (recall that all European ACTS (ad-vanced communications technologies and services)are TDMA) requires a rapid time-frequency syn-chronization using the shorter number of specialtraining symbols.We are focused on the latter application where atthe present, several sophisticated methods providethe frame and carrier recovery at least in two sym-bols, independently of the particular channel. Ourproposal deals with this issue and investigates thereasons of the inherent ambiguity of these meth-ods. The main idea is based in the transmissionof a time-duplicated symbol (simply transmit-ting by only the odd (or even) carriers), wherethe sample-wise comparison between both halvesshould provide a robust estimation of the o7set,regardless of the channel impairments 7,4. How-ever, there is an intrinsic ambiguity related withrelative distance between both halves. A simplesolution that provides multiple replications in thissymbol in order to close up the relative phase com-parisons must be discarded because the multiplereplication is related with the transmission of idlecarriers increasing signi=cantly the estimator vari-ance 4. Other methods include a second symbolin order to perform the correct frequency acquisi-tion 7. By our own, we propose that a very simpleapproach using only one symbol can be developedby time oversampling which duplicates contigu-ous samples therefore increasing signi=cantly theacquisition range. In our proposal, the limit of theacquisition range for the carrier frequency o7setcan be extended toN=2 the subcarrier spacingrather than1=2 the subcarrier spacing as relatedmethods 7,4.Our approach formulates the performance ofmethods dealing with the duplication of the =rstsymbol in two identical halves from the point ofview of matrix notation and the concept of decima-tion in frequency DFT algorithms 5. This notationimproves the knowledge over the performance ofthese methods and also allows a new synchroniza-tion scheme by means of the concept of Decima-tion in Time DFT Algorithms. This new schemeextends the acquisition range over the transmissionbandwidth by just oversampling the received se-Fig. 1. Synchronization scheme in OFDM systems.quence by a factor of two. This method was pointedout by the authors 9,6 and in the present paperwe will develop it from both an intuitive point ofview and from theoretical analysis. Finally, severalcomputer simulations supporting our approach areincluded.2. Formulation of the OFDM synchronizationsystemLet us consider the following block diagram inFig. 1 which remarks the main topics concernedwith the generation, transmission, and synchro-nization of a standard N-OFDM system in theabsence of noise (N means the number of sub-carriers).In Fig. 1 X means the transmitted symbol by allsubcarriers, r is the received time domain symboland Y represents the received symbol by each sub-carrier after synchronization (observe that we haveassumed bold letters for the time domain repre-sentation and capital bold letters for the frequencydomain representation). Let us now describe theseoperators:(a) Input vector X can be decomposed in twoparts: =rst part is the S (N1) vector which rep-resents the OFDM symbol to be transmitted; fromthe point of view of frequency synchronization,this symbol has a particular pattern which willbe discussed later in the sequel; second part is azero-padding vector 0 of length N(R1) whichallows a simple formulation of the time oversam-pling technique (where R means the oversamplingfactor).X=?S0?:(1)S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517041697(b) Matrix H is a diagonal matrix which repre-sents the attenuation complex factor at each subcar-rier, as the channel transfer function H(!) at thecorresponding subcarrier frequency. This diagonalmatrix originates from the use of a long pre=x, aswas explained in the previous section.diag(H)=?H(0) H?2?RN?H?2?(RN1)RN?(2)(c) Matrix T (T1) represents the DFT (IDFT)operator in matrix form. Usually, in the analysis,we will include a subscript remarking its dimensionfor the sake of clearness.(d) Matrix E is also a diagonal matrix which rep-resentsthefrequencyo7sete7ectasalinearincreas-ing phase o7set in the time domain; this matrix isparametrized by ? as the relative frequency o7set ofthe channel (the ratio of the actual frequency o7setto the intercarrier spacing).diag(E)=?1ej(2?=RN) ej(2?(RN1)=RN)?(3)Assuming perfect time synchronization is nowr=ET1HX:(4)After frequency estimation, in the frequency do-main we obtainY=TE1ET1HX:(5)Obviously, if matrixE1E?=I (identity matrix),intercarrier interference (ICI) appears.For convenience, let us de=ne the followingnotation:givenanarbitraryvectorz=z(1);z(2);:;z(N)H, (superscriptHmeans Hermitianand N is even) we deriveza=z(1);z(2); :; z(N=2)H;zb=z(N=2 + 1);z(N=2 + 2);:;z(N)H;zo=z(1);z(3);:;z(N1)H;ze=z(2);z(4);:;z(N)H;(6)Fig. 2. Duplication of the training symbol.where basically subscript a means the =rst half ofthe given vector and subscript b, means the secondhalf.Also,subscriptomeanstheoddcomponentsofthe original vector and subscript e means the evencomponents. This de=nition can be also be derivedfor diagonal matrices by direct application to themain diagonal vector.The main results related with practical frequencysynchronization methods 2,4 can be formulatedfrom the point of view of the decomposition of thesynchronization symbol into two identical halves,and by comparison of the relative phase shiftingbetween identical symbols. In Fig. 2, we show thereceived sequence at the synchronization symbolwith both halves ra;rb.By using the concept of decimation in frequencyDFT algorithm, each half can be expressed asfollows in terms of both even and odd carriersymbols.ra=12Ea(T1RN=2HeXe+ W1T1RN=2HoXo);rb=12Eb(T1RN=2HeXeW1T1RN=2HoXo);(7)where W is a diagonal frequency shifting matrixwhose main diagonal is as follows:diag(W)=?1ej(2?=RN) ej(2?(RN=21)=RN)?:(8)Using either even or odd carriers, angle oper-ator retrieves the o7set information (the limitof the acquisition range is1=2 the intercarrierspacing):(a)Even carriers: ?=angle?rHarb?=?:(b)Odd carriers: ?=angle?rHarb?=?1:(9)In order to solve the ambiguity, 7 includes asecond symbol di7erentially modulated by the eval-uation of a functional spanning the range of possi-ble frequency o7sets.1698S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 16951704In a similar way, 4 presents an analysis per-forming the cross correlation in the frequency do-main. This approach is identical to the later due tothe orthogonal property of the DFT operator. Bothhalves in the frequency domain areRa=Tra;Rb=Trb:(10)After the cross correlation, it is obtained the samestatistic as in the time domain approach 7.RHaRb=rHaTHTrb=rHarb:(11)The basic strategy for initial frequency o7setacquisition is to shorten the dfts and use largercarrier spacing such that the phase shift does notexceed?. Therefore, several symbols with di7er-ent lengths are required with the consequent esti-mation variance increase with the carrier spacing(Fig. 3).3. Increasing the acquisition range byoversamplingThe main idea of our proposal lies in fact thatmethods described in Section 2 are based in thecomparison between two identical halves which al-lows the estimation of the frequency o7set. Let usregard that oversampling does really introduce asequence replication in this case between adjacentsamples: the main fact to be remarked is that phasecomparison between odd and even samples in thetime domain training symbols will also provide theo7set information. More indeed, it is intuitive thatthe comparison between samples much closer in thetime domain will increase signi=cantly the acquisi-tion range.It is well known that oversampling in the time orfrequency domain is related with zero-padding inthe other domain (in this case, we only analyze anoversampling by a factor of two).In a generic case, the frequency o7set is greaterthan one intercarrier spacing, that is:? = ?k+ ?f= k + ?f;(12)where ?fmeans the fractional part of the frequencyo7set and ?k=k is the integer part (k-intercarrierspacing where kZ). Therefore, the o7set matrixE can be expressed as the product of these twodiagonal matrices:E=EkEf;(13)diag(Ek)=?1ej(2?k=RN) ej(2?k(RN1)=RN)?;diag(Ek)=?1ej(2?f=RN) ej(2?f(RN1)=RN)?:(14)Let us suppose that method described in Eq. (9)performs optimally, and therefore, we are able toestimate with very high accuracy the fractional fre-quency o7set:EfEf:(15)Inthefrequencydomain,thee7ectoftheambigu-ous determination of the integer intercarrier spacingis expressed byY = TE1fET1HXTEkT1HX = IkHX;(16)where Ikrepresents a matrix as the k-circulant iden-tity matrix. The result of Eq. (16) lead us to a cyclicrotation of vector HX, that is no ICI appears butthere is an unknown shifting between the originaland the actual location of the intercarrier symbolswhich is the responsible of the frequency ambigu-ity.The time oversampling problem can be formu-lated in an appropriate form by the concept of deci-mation in time DFT algorithms in terms of =rst andsecond halves in the frequency domainXa=TNxe+ WTNxo;Xb=TNxe+ WTNxo:(17)Therefore, the received sequence can be ex-pressed in the following way:re=12EeT1N(HaXa+ HbXb);ro=12EoT1NW1(HaXaHbXb):(18)S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517041699Fig. 3. Replication of the synchronization symbol.Imposing Xb=0 in Eq. (18) because the secondhalf of subcarrier symbols are =ctitious, and realiz-ing thatEo=ej(2?=2N)Ee(19)the correlation between odd and even samplesyieldsrHero=ej(2?=2N)XHaHHaW1HaXa:(20)It can be observed that Eq. (20) (direct crosscorrelation) does not provide the desired frequencyacquisition due to the presence of matrix W, but wecan propose an alternative metric:rHeAro;(21)where matrix A is de=ned as follows:A=4EefTHBTEHef(22)andEefis performed by the estimation of the frac-tional part of the frequency o7set by method de-scribed in Eq. (9). The structure of matrix B willbe developed in the sequel. After some algebra, Eq.(21) yieldsrHeAro=XHaHHaTEHEefTHBTEHefET1W1HaXaej(2?=2N)(23)In terms of the k-circulant identity matrix, Eq.(23) becomesrHeAro=XHaHHaIHkBIkW1HaXaej(2?=2N)(24)If we consider matrix B as a parametric matrixBswhere s is an arbitrary shifting parameter for thes-circulant identity matrix:Bs=IsWIHs:(25)Let us observe that the central factors in Eq. (24)yields in the following form (or a cyclic rotation ofit),IHkBsIkW1=?IN(ks)xN(ks)00I(ks)x(ks)?ej(2?(sk)=2N);(26)providing a diagonal matrix with positive (N(ks) and negative (ks) elements. Finally, the pro-posed method yieldsrHeAsro=XHaHHa?IN(ks)xN(ks)00I(ks)x(ks)?ej(2?(sk)=RN)HaXaej(2?(k+?f)=RN)=?N(ks)?l=1|Xa(l)|2|Ha(l)|2N?l=N(ks)+1|Xa(l)|2|Ha(l)|2ej(2?(s+?f)=RN):(27)Eq.(27)statesanimportantconclusion:theangleofthismetricisblindtothetrueshiftingvaluek,butthis information is included into the modulus: letus observe that the condition of maximum modulusis obtained when s=k; therefore, the evaluation ofthe modulus of metric described by Eq. (27) forarbitrary shifting values of s1; 2; :; Nwillprovide the integer intercarrier spacing (k) at itsmaximum conditionk =maxs?rHeAsro?:(28)1700S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517044. Analysis of the probability of detectionIn the present section, we provide the calculus ofa bound of the probability of detection of the properintegershiftinginamultipathscenariowithadditivewhite Gaussian noise. Recalling the expressions ofthe odd and even parts of the oversampled sym-bol given by Eq. (18) including the additive noisee7ectre=12EeT1HaXa+ ne;ro=12EoT1W1HaXa+ no;(29)where EnenHe=EnonHo=?2nI.The detection process is based on the compara-tive of N random variablesZss1; 2; :; Ngiven byZs=?rHeAsro?(30)for simplicity reasons we will denote in the sequelre=CeXa+ ne;ro=CoXa+ no:(31)The probability of detection can be stated by thefollowing N-dimensional problem:Pd=PN?s=1s?=kZkZs;(32)assuming an arbitrary k as the right integer shift.In order to reach a simpler solution, let us =ndan appropriate bound for this probability. It iswell known that Eq. (32) can be expressed asfollows:Pd=pN?s=1s?=kZkZs=1PN?s=1s?=kZkZs;(33)which is still a N-dimensional problem but also itcan be found an appropriate bound for the errorprobability Peas the complementary probability ofthe probability of detection Pd3:Pe=1Pd=PN?s=1s?=kZkZs6N?s=1s?=kPZkZs:(34)Let us now discuss about the calculus of the indi-vidual probabilities. A generic random variable Zscan be decomposed in the following terms:Zs=|?XHaCHe+ nHe?As(CoXa+ no)|=|XHaCHeAsCoXa+ nHeAsCoXa+XHaCHaAsno+ nHeAsno|=|Zs1+ Zs2+ Zs3+ Zs4|;(35)where Zs1is a deterministic value as a mean value,Zs2, Zs3are Gaussian random variables as the linearcombination of Gaussian variables, and Zs4followsa Chi-squared distribution.Assuming a normal situation for proper detectionwhere the noise power should be of low variance,Zs4can be neglected in front of the mean value Zs1.In this situation, Zscan be easily identi=ed as aRice distribution. Also, let as point out another fea-sible approximation that simpli=es signi=cantly theproblem. Zscan be decomposed into two orthogo-nal components, one in the direction of Zs1(let uslabel it ? as you can identify in Eq. (28) and theother in the quadrature directionZs=?(inP?Zs1+Zs2+Zs3)2+(inQ?Zs2+Zs3)2;(36)where notationinP?aninQ?mean the in phaseand quadrature component of an arbitrary angle ?,respectively. This approach has been suggested in7 but also it has been adopted in other applicationsas is shown in 2.Assuming again the hypothesis of high SNR, itcan be approximated byZs inP?Zs1+ Zs2+ Zs3;(37)S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517041701Fig. 4. (a) Frequency estimation and frequency error for the deterministic sweeping.SNR=10dB, (b) Frequency estimation andfrequency error for the deterministic sweeping.SNR=20dB.1702S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 16951704where the statistics of the involved random vari-ables are identical. The key point of this analysis isthe fact that now Zsis a Gaussian random variable.Other point of view of this approach is the fact thata Rice distribution becomes a Gaussian distributionfor high mean.Finally, the probability can be expressed asPZkZs=PZkZs0=PZks0;(38)where the new random variable Zksis also Gaussianwith mean and variance given by?ks=EZks=N?l=N(ks)+1|Xa(l)|2|Ha(l)|2?2ks=E(Zks?ks)2=?XHaCHo(AkAs)H(AkAs)CoXa?2n+(XHaCHe(AkAs)(AkAs)HCeXa)?2n: (39)Finally, we obtainPe6N?s=1s?=kPZkZs=N?s=1s?=kFZks(0)=N?s=1s?=k?1Q?ks?ks?=N?s=1s?=kQ?ks?ks?(40)expressed in terms of the Q-function as is custom-ary in engineering. A =nal comment can be consid-ered: most of the terms in Eq. (40) do not contributesigni=cantly to the =nal error probability. In partic-ular, those terms with high ?ks, which mean an errorof several subcarriers, can be neglected in front ofthe adjacent cases with one or two subcarrier spac-ings. Also, in these terms the approximation of highSNR is more feasible.5. Computer simulationsIn order to support our theoretical derivationswe have developed several simulations of the fre-quency o7set correction algorithm as is describedin Eqs. (9) and (28) for a 32-OFDM system: ourproposal is decomposed in two steps; =rst step isFig. 5. (a) Error Histogram forSNR=10dB, (b) Error His-togram forSNR=20dB.devoted to the fractional frequency o7set by appli-cation of metric described in Eq. (9). Afterwards,second step uses this previous result in order to ap-ply the metric discussed in Section 3 (Eq. (28) toobtain the integer intercarrier spacing. The globalresult is the addition of results obtained by bothsteps.Our =rst scenario considers a AWGN channelwith several signal to noise ratios (SNR): low SNR(10dB) and high SNR (20dB). The =rst set ofresults is performed by the deterministic sweep-ing of the nominal frequency o7set in the rangeS. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517041703Table 1Mean and standard deviation of the error distributionSNR(dB)MeanSigma105:581042:7102201:581041:110215?15; Figs. 4a and b shows the average fre-quency estimation and the error estimation for bothcases.On the other hand, we have performed 1000 in-dependent frequency o7sets randomly distributed(uniformly) in the range15?15; afterwardswe present the error histogram for the SNRs men-tioned before in front of the intercarrier spacing,assuming that the =rst step behaves properly (seeFig. 5a and b) Let us remark that the error is mostlywithin the recommended 5% even in the 10dB case.Additionally, an estimation of the mean (in terms ofthe intercarrier spacing) and the standard deviation(?) are also provided Table 1.Fig. 6. Probability of fail detection for randomly chosen frequency o7set in the range15?15.Finally,wehaveconsideredthepreviousscenario in order to evaluate the probabilityofdetectionofourmethod.Fig.6consid-ers that the application of both schemes, frac-tional and integer part, do perform properlyif the estimation error is lower the intercar-rier spacing. From our present point of view,we are not focused on the estimation errorwithin the intercarrier spacing. This topic hasbeen evaluated in 7 and also by the authorsin 5,9. Let us =nally remark that the theo-retical bound matches well with the simulationresult.Our method will also work in mobile radiochannels if it guaranteed that the channel do notchange signi=cantly within one OFDM symbol. Inthis situation, we have shown theoretically that itsperformance is independent of the channel state.However, the analysis in a real time-varying en-vironment will be very dependent of the timesynchronizer behavior and will be treated in aforthcoming paper.1704S. Zazo, J.M. P? aez-Borrallo/Signal Processing 81 (2001) 169517046. ConclusionsIn this paper, we have presented a new schemefor frequency synchronization of OFDM systemsbased on the time oversampled received signal. Anew metric exploiting this formulation provides anunambiguous estimation in the rangeN=2 the sub-carrier spacing rather than1=2 the subcarrier spac-ingaspreviousmethods.Aderivationofthemethodand the analysis of the probability of detection areincluded.References1 J.A.C.Bingham,Multicarriermodulationfordatatransmission: an idea whose time has come, IEEE Comm.Mag. 28 (5) (May 1990) 514.2 G.W. Lank, I.S. Reed, G.E. Pollon, A semicoherentdetection and Doppler estimation statistic, IEEE Trans.Aerospace Electron. Systems, AES-9(2) (March 1973)151165.3 E.A. Lee, D.G. Messerschimitt, Digital Communication,Kluwer Academic Publishers, Dordrecht.4 P. Moose, A technique fo
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