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1、附 录 一、英文原文 OFDM Channel Estimation in the Presence of Frequency Offset and Phase Noise Abstract In this paper,we consider OFDM channel estimation in the presence of frequency offset and phase noise.In the literatures,most channel estimation methods assume perfect frequency synchronization and the kn

2、owledge of channel statistics.Phase noise and residual frequency offset cause inter-carrier interference(ICI),which consequently impairs the accuracy of channel estimation.The lack of knowledge of channel statistics can make channel estimation much harder.To resolve these problems,we propose with th

3、e aid of cyclic prefix(CP)based frequency offset estimation statistics-independent channel estimation.We iteratively search for the most likely channel impulse response(CIR)length,and use it not only for the optimum compensation of frequency offset,but also for finding the optimum window to filter t

4、he least square(LS)channel estimate which further suppress the effects of ICI and noise.The proposed scheme is compared with conventional methods for both non-interpolation and interpolation cases2.Numerical results are presented to illustrate the effectiveness of the proposed scheme.I.INTRODUCTION

5、Orthogonal frequency division multiplexing(OFDM)is a bandwidth efficient transmission technique which provides high bandwidth efficiency and is quite effective in handling time dispersion of multipath fading channels.It has been chosen as the transmission method of many standards in wire and wireles

6、s communications,such as Digital Subscribe Line(DSL),European Digital Audio and Video Broadcasting(DAB/DVB),IEEE 802.11a and European HIPERLAN/2 for wireless local area network(WLAN)etc.Based on multi-carrier modulation 1,OFDM has symbol period long enough to eliminate inter-symbol interference(ISI)

7、caused by time dispersive channels.Nevertheless,the multicarrier modulation is also sensitive to frequency offset and phase noise.Frequency offset and phase noise cause loss of orthogonality among subcarriers and consequently introduce inter-carrier interference(ICI).The effect of phase noise has be

8、en analyzed in many papers 2-4.Many approaches have also been proposed to analyze,estimate and compensate frequency offset 25-10.Though it is impossible to estimate random phase noise,frequency offset estimation can be achieved by using pilot signals 56.As these methods cause loss of bandwidth effic

9、iency,non-pilot-aided frequency offset estimation has be used 7-10.The cyclic prefix(CP)based method,initially proposed in 9,is quite attractive among non-pilot-aided approaches due to its simplicity.Nevertheless,the accuracy of the CP-based method could not be guaranteed for multipath fading channe

10、ls.Later,as proposed in 10,the method of 9 was improved by considering the channel impulse response(CIR)length.The proposed method in 10,however,is not feasible in cases when the CIR length is unknown.Furthermore,channel estimation is a very important issue for OFDM systems.Blind channel estimation

11、is a desirable approach as it does not require pilot signals.It does require,however,a large amount of data and thus higher computational complexity.With perfect frequency synchronization(without residual frequency offset),different pilot-symbol-aided channel estimation methods can be applied in OFD

12、M 11-14.The maximum likelihood/least square(ML/LS)estimators of 11 and 12 can readily be implemented without knowing channel statistics.The minimum mean square error(MMSE)estimators in 12-14,however,are more robust against noise and perform better than the ML/LS estimators.Nevertheless,its dependenc

13、e on channel statistics and the operating signal to noise ratio(SNR)makes it disadvantageous.Despite its robustness against mismatch 1314,when there is no a priori knowledge of channel statistics and the operating SNR,the performance inevitably degrades.Without the assumption of perfect frequency sy

14、nchronization,the performance may further degrade due to frequency offset and phase noise.In this paper,we consider statistics-independent channel estimation in the presence of frequency offset and phase noise.As a function of the CIR length,the LS channel estimate results,which is based on the CP-b

15、ased frequency offset estimation and compensation,are used to search for the CIR length iteratively.The minimization of channel estimation errors leads to the most likely CIR length,which is then used to optimize frequency offset estimate,and filter the LS channel estimate reducing its sensitivity t

16、o noise and ICI.Thus better performance is achieved.The paper is organized as follows.The OFDM system model is introduced in Section II.Section III presents and analyzes the proposed frequency offset and channel estimation scheme.Section IV provides the numerical results to illustrate the effectiven

17、ess of the proposed scheme.The paper is concluded in Section V.II.OFDM SYSTEM MODEL The basic principle of OFDM is to divide each data symbol into N samples(subcarriers).The length N discrete Fourier transform(DFT)is applied to those samples and a cyclic prefix(CP)is added to eliminate ISI.Data is r

18、ecovered at the receiver in reverse order.We define the length of CP as g N and the length of CIR as L,and further assume that CIR is finite and its length is less than that of CP,i.e.,L g N.At OFDM receiver,following the DFT and due to the presence of frequency offset and phase noise,the received k

19、th sample of the mth symbol in frequency domain can be expressed by 2Fgm nm N NnjNmmmm nrksngn e (1)where sm(n),gm(n)and nn denote the transmitted signal,the CIR and phase noise,respectively.m(n)indicates the AWGN noise.is the normalized frequency offset3.We assume 0.5 and the 3 dB line width of pha

20、se noise is much less than frequency offset.Equivalently,(1)can be given by 1mm0I0INmmmmmmll krkxk hkxl hllkzk (2)where x m(k),h m(k)and z m(k)are the corresponding frequency domain expressions of s m(n),g m(n)and m(n)respectively.I m(i)is a function of and m(n),given by:2120gn njm NNNNjn iNmneIieN

21、(3)where i=0,.,N 1.From(2)together with(3),frequency offset and phase noise cause the common phase error(CPE)and introduce inter-carrier interference(ICI)as well.For the mth symbol,Representing(2)by matrix yields rpxhz (4)The frequency offset and phase noise in P affects the accuracy of channel esti

22、mation.We cannot measure phase noise,but frequency offset can be estimated and compensated to reduce its effects on channel estimation.The effects of phase noise and residual frequency offset(due to estimation errors)can possibly be suppressed by filtering channel estimate.For perfect frequency and

23、phase synchronization,P reduces to identity matrix and therefore the performance of channel estimation can be guaranteed.III.FREQUENCY OFFSET AND CHANNEL ESTIMATORS In the presence of frequency offset and phase noise,both offset and channel response should be estimated to guarantee good receiver per

24、formance.Phase noise variance is assumed to be much less than unity.We a new scheme with which,by iteratively searching for the most likely CIR length and using it for both frequency offset and channel estimation,performance is greatly improved in comparison with conventional approaches.The proposed

25、 scheme is shown in Fig.1.A.CP-based Frequency Offset Estimator CP-based frequency offset estimator in 9 is quite simple and bandwidth efficient,but it does not consider the effects of multipath fading and estimation results may not be accurate as it is based on the CP of one symbol only.The method

26、proposed in 10 improves that of 9 by considering CIR length and taking more symbols into consideration.However,when averaging frequency offset estimates obtained separately from each symbol,accumulated errors may be larger than expected.Moreover,its dependence on the CIR length is quite a problem wh

27、en channel statistics is not available.A different method is thus proposed in this paper to solve these problems.Like 10,several symbols are used to estimate frequency offset,but it does not accumulate errors by using the following expression for estimation.11*0112gMmmgmkNppanglerk r k NNM (5)where

28、p is the CIR length which is unknown,M is the number of symbols used for averaging.The unknown parameter p(as we will show later)can be set initially to one and be found by iteration.Therefore,we can still get the accurate estimate of(5)even without channel statistics.B.Channel Estimator Channel est

29、imation is quite crucial for OFDM systems.Also as stated earlier,LS method is advantageous over MMSE method due to its simplicity and independence of channel statistics.Hence assuming in this paper unknown channel statistics,we will focus on LS method.The estimate of(5)can be used to compensate for

30、the frequency offset,after which,we will get from(4)that prp xhz (6)where pPtakes the same form of P except that I(i)is replaced by n12(mn)(p)/N 2m/N(n)01gNjNNpnIieN (6a)As shown in Fig.1,channel estimate is easily obtained by using the LS method,which can be expressed by 1lspppx r (7)The LS method

31、is quite sensitive to interference and noise.Therefore without perfect frequency and phase synchronization,the effects of frequency offset and phase noise become worse.Furthermore,there would still be residual frequency offset even after compensation,which,together with phase noise,introduces CPE an

32、d ICI.Though CPE might partially be compensated by channel estimation itself,ICI will definitely affect the accuracy of such estimation.Therefore,some method must be introduced to reduce the sensitivity of channel estimation to interference and noise.As CIR has a finite length in time domain,the res

33、ponse beyond this CIR length is thus due to ICI and noise.Hence,a window function may be used to filter out these effects of ICI and noise on channel estimates.In time-domain,using a window function on(7)yields HslslppphWW B h (8)011Tppdiagdiagbbpbp NpBbis an N N diagonal matrix defined by the windo

34、w function10,2(i)0.420.5cos()0.08cos(),2021,1mipipipbippppipN (9)Note that rectangular window is not used here as it introduces more high frequency components than is tolerated which causes a distortion of channel frequency response.Instead,due to its excellent descending properties,Blackman functio

35、n is used in(9),as the intermediate part of the designed window.C.The Most Likely CIR Length and Final Solution The most likely CIR length can be found by minimizing the cost function 2lsphh (10)To simplify the process,the window function is not used during the search,and we only have to find the pr

36、oper p that produces the frequency offset estimate minimizing(10).Unfortunately,there are two unknown parameters,h and in(10),which makes such direct minimization difficult.However,we notice that,in the absence of AWGN noise,as p increases,frequency offset estimate of(5)becomes more accurate and the

37、 difference of channel estimates of(7)for adjacent ps values becomes smaller,and minimum when p is greater than or equal to CIR length.Therefore,the minimization of(10)can be obtained by the first occurrence of the minimum of 21lslspp hh (11)In the presence of AWGN noise,we have to assert that the v

38、alue p that minimizes(10)is the same as that of(11)before we can use(11).Statistically the minimum of(11)would occur when p is close to the CIR length when noise is not so high.(11)decreases when p increases from 1 to the CIR length since the CIR effects decreases.For increasing p,which is equivalen

39、t to using fewer samples(see(5),the frequency offset estimation becomes less accurate and so does the channel estimate.Thus when p becomes greater than CIR length but less than CP,the difference of(11)is statistically higher when p is greater than the CIR length than when p is close to the CIR lengt

40、h.Hence,the minimum of(11)occurs with high probability at the point where p is equal to the CIR length.Hence,the most likely CIR length can be found by varying p between 1 and g N,and choosing the value which satisfies the following criteria 22112lslslslspppphhhh (12)2211lslslslspppphhhh (13)To exam

41、ine the effectiveness of the criteria,we resort to computer simulation.The final channel estimate is expressed by 1HlsppPhWB W r (14)WherePrepresents the estimated value of p.Note that CIR length can found with only a single search as in most cases it does not change even in a time variant channel a

42、nd the result might be used for quite a few OFDM symbols.D.Interpolated Pilot Symbols The aforementioned channel estimator is for non-interpolation case.However,interpolation case is often used where pilot signals are multiplexed into the transmitted data stream,i.e.,pilot signals are inserted into

43、data stream every f D samples.Without loss of generality,we assume that/f K=N D is integer,i.e.,there are K pilot samples per symbol.In this case,the principle of the proposed scheme remains correct,except that the size of DFT matrix W and the window diagonal matrix p B become K K diagonal matrices.

44、The searching process for p remains the same,but the interpolation must be applied to the result of(8)to get the complete channel estimate.IV.NUMERICAL RESULTS The proposed scheme was evaluated by simulation.Part of simulation parameters is based on IEEE 802.11a standard e.g.,DFT length,CP length an

45、d sample period s T are 64,16 and 0.05 s,respectively.3dB line width of phase noise equals 0.1%of subcarrier spacing.The actual frequency offset is set to 0.1382.Number of symbols used to estimate frequency offset M equals 8.Exponential Rayleigh fading channel is used with the exponential power dela

46、y profile specified bymaxmax1sLTrmsee where rms ,s T and L are the mean delay spread,sample period and CIR length respectively.rms is set to 0.05 s,which equals s T.L is set to 6.There are 16 symbols per packet.The total energy of CIR has been normalized to one.Channel changes independently from sym

47、bol to symbol,but remains static within a symbol.16QAM is used to examine our scheme.The proposed scheme is compared with the frequency offset estimator of 10 plus the LMMSE channel estimator of 13(which is termed conventional method)for both non-interpolation and interpolation cases.For fair compar

48、ison,the frequency offset estimator uses the first M symbols of each packet with unknown CIR length.Simulation results are shown in Fig.2-5.As can be seen from Fig.2,the proposed scheme performs quite well in estimating frequency offset.For SNR 5dB,the mean square error of the estimation is of the o

49、rder of 103 or less.The accurate frequency offset estimate also reflects the fact that the proposed scheme is quite successful in searching for the CIR length.From Fig.3,the most likely CIR length is 5 and an estimated length between 5 and 7 accounts for over 80percent of total possibilities,which i

50、ndicates the effectiveness of CIR length searching method.Note that,since the AWGN noise affects the searching process and we use the exponential power delay profile with the maximum delay spread of 6 s T,the earch result in between 5-7 is quite reasonable.With the estimated CIR length obtained,the