A simple and efficient timing offset estimation for OFDM systems ...

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ers, Ng is the number of guard samples, j = fland the sampling period is Tu/N with l/Tu beingsubcarrier spacing. Consider a discrete-time channelcharacterized byK-1h(k) = hlb(k - 71)l=Owhere b(k) represents dirac-delta function, {hl} complexpath gains, (71) path time delays which are assumedin multiple of OFDM samples, and K the totalnumber of paths. Then the received samples, assumingperfect sampling clock, can be given byK-1r(k) = ezp(j2nkw/~) hls(k - 71 +E) + n(k)l=O(3)where w is the carrier frequency offset normalizedto subcarrier spacing, E is timing offset in units ofOFDM samples and n(k) is the sample of zero meancomplex Gaussian noise process with variance 0:.Suppose the sample indexes of perfectly synchronizedOFDM symbol be {-Ng, ..., -1, 0, 1, ..., N - 1)and maximum channel delay spread be T, . Then ifE E {-Ng +T,~~, -Ng + T + 1,. ~ .. ~ , 0}, the ~ orthogonalityamong the subcarriers will not be destroyedand the timing offset will only introduce a phase rotationin every subcarrier symbol Ym at the FFT outputasY, = e~p(j2~me/N)c,H~+n,, -Ng+rmaZ 5 E < 0(4)where m is subcarrier index, H, is channel frequencyresponse to the mth subchannel, i.e., {Hm} =DFT(h(lc)), and nm is complex Gaussian noise term.For coherent system, this phase rotation is compensatedin channel equalization which sees it as a channelintroduced phase shift. If the timing estimateis outside the above range, the orthogonality amongthe subcarriers will be destroyed by the resulting ISI.Then the FFT output subcarrier symbols are givenbYcapacity efficiency can be obtained. With this aspect,the performance of the symbol timing synchronizationalgorithms will be evaluated by the timingestimator variance.The symbol timing estimator finds the start of theOFDM symbol. The correct symbol timing point willbe taken as the start of the useful part of the OFDMsymbol (i.e., the start of FFT window for demodulation).Let the training symbol (excluding cyclic prefix)contain two identical halves in time domain eachhaving L = N/2 samples. At the receiver there willbe a phase difference between the samples in the firsthalf and their replica in the second half caused bythe carrier frequency offset. Training data is usuallya PN sequence. Then the Schmidl & Cox's timingestimator [14] takes as the start of the symbol themaximum point of the timing metric given bywhere d is a time index corresponding to the firstsample in a window of 2L samples and Pl(d) is correlationmetric given byL-1Pi(d)= xr*(d+m).r(d+m+L) (7)m=Owhere ()* represents conjugation and Rl(d) representsenergy of half OFDM symbol and is includedfor normalization of the correlation metric in accountfor the large fluctuation of the OFDM sample amplitudesand given byL-1Rl(d) = (r(d + m + L)12 (8)m=OThe timing metric reaches a plateau (see Fig. 2)which leads to some uncertainty as to the start ofthe frame. To alleviate this, [14] proposes an averagingmethod where the maximum point is first foundand then two points with 90% of the maximum value,one to the left and the other to the right of the maximumpoint, are found. The timing estimate is takenas the average of the two 90% points.111. Proposed Methodwhere the extra term IE(m) results from IS1 and interchannelinterference (ICI) caused by the loss oforthogonality among the subcarriers (see [13] for details).The useful term also suffers a slight gain lossbut for large N, it is negiligible. The main detrimentaleffect is caused by the extra ISI+ICI term.Thus, the guard interval should be long enough forthe timing estimate to lie within the range describedabove. On the other hand, longer guard intervaltranslates into more system capacity loss due to overhead.Hence high performance timing synchronizationis much desirable for higher system performanceand capacity efficiency. The smaller the variance oftiming estimator is, the shorter the guard interval canbe designed, and hence less overhead and increasedThe samples of the training symbol (excluding cyclicprefix) are designed to be of the forms=[AA -A -A] (9)where A represents samples of length L = N/4 generatedby N/4 point IFFT of N,/4 length modulateddata of a PN sequence. If desired, the abrupt amplitudechange due to sign conversion in the trainingsymbol can easily be avoided by modifying the PN sequencesuch that the sum of the corresponding modulateddata equals zero. The timing metric is givenby0-7803-5718-3/00/$10.00 02000 IEEE. 52VTC2000
correlationwindow ------A correlation window -.................... ..........CP + +CP = Cyclic Prefix(a) at exact symbol timing point(b) before exact symbol timing pointFig. 1. Correlation of the training symbol: Schmidl & Cox’s method (upper), Proposed method (lower)where1 L-1&(d) = r*(d+ 2Lk + m) . r(d+ 2Lk+ m + L)k=O 7n=Oand1 L-1R2(d) = Jr(d + 2Lk +VI + L)I2 (12)k=O m=OIn above equations, P2(d) and R2(d) can be calculatediteratively. The training symbol patterns of[14] and proposed method are shown in Fig.(l). Alsoshown is the elimination of timing metric plateau inherentin [14] by proper design of the training symbolin the proposed method. For training symbol of [14],the correlation metric has its peak for the whole intervalof cyclic prefix (under no noise and distortioncondition) while the proposed training symbol has itspeak of correlation metric at the correct FFT windowstarting point since off that point, correlation ofsome samples results in negative values as shown inFig. (1) hence eliminating the correlation metric peakplateau.The timing metrics of [14] (Eq. (6)), and the proposedmethod (Eq. (10)) are shown in Fig. 2 underno noise and distortion condition with the system parametersgiven in section IV. The correct timing point(index 0 in the figure) is taken as the start of theuseful part of training symbol (after cyclic prefix).As discussed previously, the proposed method doesnot have timing metric plateau and consequently itachieves better accuracy in timing offset estimation.In calculation of half symbol energy, N samples insteadof N/2 samples can be used to get more reliableresult. Then for both methods, Rl(d) and R*(d) canbe replaced by the following:, \ < .ow’ ’-400 -300 -200 -100 0 100 200 300 400Time (sample)Fig. 2. Timing metric under no noise and distortion conditionIV. SIMULATION RESULTS AND DISCUSSIONPerformance of the timing offset estimators have beeninvestigated by computer simulation for five cases: (I)Schmidl & Cox method (Eq. (6) with 90% maximumpoints averaging), (11) Schmidl & Cox method withmodified R(d) (same as (I), except that Rl(d) is replacedby R(d)), (111) the method from [16],(IV) ProposedMethod (Eq. (lo)), and (V) Proposed Methodusing R(d) in stead of R2(d). The system used is1000 subcarriers OFDM system with 1024 point FFT,10% guard interval (102 samples), carrier frequencyoffset of 12.4 subcarrier spacing, QPSK modulationand 10000 simulation runs. Two channel conditions0-7803-5718-3/00/$10.00 02000 IEEE. 53 VTC2000
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