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1、一、英文原文OFDM Channel Estimation in the Presence of Frequency Offset and Phase NoiseAbstract - 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 kno
2、wledge 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 wi
3、th the 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
4、 to filter the 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
5、.I. INTRODUCTIONOrthogonal 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
6、in wire and wireless 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 int
7、er-symbol interference (ISI) 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)
8、. The effect of phase noise has been analyzed in many papers 2-4. Many approacheshave 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
9、 methods cause loss of bandwidth efficiency, 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
10、 not be guaranteed for multipath fading channels. Later, asproposed 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 impo
11、rtant issue for OFDM systems. Blind channel estimation 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-
12、symbol-aided channel estimation methods can be applied in OFDM 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 an
13、d perform better than the ML/LS estimators. Nevertheless, its dependence 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 pe
14、rformance inevitably degrades. Without the assumption of perfect frequency synchronization, 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 funct
15、ion of the CIR length, the LS channel estimate results, which is based on the CP-based 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 fr
16、equency offset estimate, and filter the LS channel estimate reducing its sensitivity to 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
17、 estimation scheme. Section IV provides the numerical results to illustrate the effectiveness of the proposed scheme. The paper is concluded in Section V.II. OFDM SYSTEM MODELThe basic principle of OFDM is to divide each data symbol into N samples (subcarriers). The length N discrete Fourier transfo
18、rm (DFT) is applied to those samples and a cyclic prefix(CP) is added to eliminate ISI. Data is recovered at the receiver in reverse order. We define the length of CP asg 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.eL< g N .At OFD
19、M receiver, following the DFT and due to the presence of frequency offset and phase noise, the received kth sample of the mth symbol in frequency domain can be expressed byj2m N Ng nrm k F sm ngm n ewhere sm (n), gm (n) and n n denote the transmitted signal, the CIR and phase noise, respectively. m
20、(n) indicates the AWGN noise. & is the normalized frequency offset3. We assume 8 < 0.5 and the 3 dB Width of phase noise is much less than frequency offset. Equivalently, (1) can be given byN 1rm kxmkhmk Im 0Xmlhml Im l kZmkl 0l k(2)where x m (k ), h m (k) and z m (k ) are the corresponding f
21、requency domain expressions ofs m (n) , g m (n ) and 新(n ) respectively. I m (i)is a function of & and 师(n ) , given by:j2 m N Ng N eN 1j2 e(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 interf
22、erence (ICI) as well. For the mth symbol, Representing (2) by matrix yieldsr pxh z(4)The frequency offset and phase noise in P affects the accuracy of channel estimation. We cannot measure phase noise, but frequency offset can be estimated and compensated to reduce its effects on channel estimation.
23、 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 phase synchronization, reduces to identity matrix and therefore the performance of channel estimation can be guaranteed.FigP L Pmpo
24、号cdJi>r irtqucncy otlset and ch4inncl estimationIII. FREQUENCY OFFSET AND CHANNELESTIMATORSIn the presence of frequency offset and phase noise, both offset and channel responseshould be estimated to guarantee good receiver performance. Phase noise variance is assumed to be much less than unity.We
25、 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 scheme is shown in Fig.1.A. CP-based Frequency Offset EstimatorCP-
26、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 proposed in 10 improves that of 9 by considering CIR length and ta
27、king more symbols into consideration. However, when averaging frequency offset estimates obtained separately from each symbol, accumulated errors may be larger than expected.Moreover, its dependenceon the CIR length is quite a problem when channel statistics is not available. A different method is t
28、hus 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.( M 11,1*angler m k rm k NNg M m 0k Ng pwhere p is the CIR length which is unknown, M is the number o
29、f symbols used for averaging. The unknown parametep (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 EstimatorChannel estimation is quite crucial for OFDM systems. Also a
30、s 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 compensatefor the frequency offset, after which, we will
31、get from (4) thatn hr p pxh z(6) where P ptakes the same form ofP except thatI (i) is replaced byN 11j 2 (mN Ng n)(p)/N 2 m/N n(n)I p ieN n 0(6a)As shown in Fig. 1, channel estimate is easily obtained by using the LS method, which can be expressed byls 油cp p x rpThe LS method is quite sensitive to i
32、nterference 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 and ICI. Though CP
33、E might partially be compensatedby 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 response beyond
34、 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 yields 、h： WWH Bph；(8)p pTBp diag bpdiag bp 0 bp 1bp N 1 is an N XN diagonalmatrix defined by the window f
35、unction1i 0, pi p2 i pbm(i)0.42 0.5cos() 0.08cos() i p,2 PPP0 i 2p 1,N 1Note 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, Blackm
36、an function is used in (9), as the intermediate part of the designed window.C. The Most Likely CIR Length and Final SolutionThe most likely CIR length can be found by minimizing the cost function h hlsP2(10)To simplify the process, the window function is not used during the search, and we only have
37、to find the proper 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 (
38、5) becomes moreaccurate and the difference of channel estimates of (7) for adjacent p' 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 ofthe minimum of2 h Sp hs p 1|(11)In the presence
39、of AWGN noise, we have to assert that the value 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 whenp increases from 1 to the CIR length since the CIR e
40、ffects decreases. For increasing p, which is equivalent 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 grea
41、ter than the CIR length than when p is close to the CIR length. 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 followin
42、g criteria22-hlsp|hlsp 1 h1sp 22(12)I h lsp h lsp 1| h ls p 1 hls p|(13)To examine the effectiveness of the criteria, we resort to computer simulation. The final channel estimate is expressed by'lsH 1h： WBpW rP p(14)WhereP representsthe estimated value of p. Note that CIR length can found with o
43、nly a single search as in most cases it does not change even in a time variant channel and the result might be used for quite a few OFDM symbols.D. Interpolated Pilot SymbolsThe aforementioned channel estimator is for non-interpolation case. However, interpolation case is often used where pilot sign
44、als are multiplexed into the transmitted data stream, i.e., pilot signals are inserted into data stream every f D samples.Without loss of generality, we assume that f K = N D is integer, i.e., there areK pilot samples per symbol. In this case, the principle of the proposed scheme remains correct, ex
45、cept that the size of DFT matrix W and the window diagonal matrix p B become K XK diagonal matrices. The searching process forp remains the same, but the interpolation must be applied to the result of (8) to get the complete channel estimate.IV. NUMERICAL RESULTSmaxe/LT s ； maxrms 1 eThe proposed sc
46、heme was evaluated by simulation. Part of simulation parameters is based on IEEE 802.11a standard e.g., DFT length, CP length and sample periods T are 64, 16 and0.05 s , respectively. 3dB line width of phase noise equals 0.1% of subcarrier spacing. The actual frequency offset £ is set to 0.1382
47、. Number of symbols used to estimate frequency offset M equals 8. Exponential Rayleigh fading channel is used with the exponential power delay profile specified bywhere rms r s T and L are the mean delay spread, sample period andCIR length respectively. rms r is seto 0.05 sp, which equals s T . L is
48、 set to 6. There are 16 symbols per packet. The total energy of CIR has been normalized to one. Channel changes independently from symbol 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
49、the LMMSE channel estimator of 13 (which is termed conventional method) for both non-interpolation and interpolation cases. For fair comparison, the frequency offset estimator uses the firsM symbols of each packet with unknown CIR length. Simulation results are shown in Fig. 2-5. As can be seen from Fig. 2, the propos