简单易懂的信道估计提供原文文章-Channel Estimation In OFDM systems.pdf

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简单易懂的信道估计提供原文文章-Channel Estimation In OFDM systems.pdf 简单易懂的信道估计(提供原文文章),基于BPSK 的OFDM。 有MSE,SER的仿真图做比较。
where QMmse can be shown to be X as=RgFx“XF2+Rg(Fx“xF) This MMSe channel estimator(9) has the form shown in Q Fig. 4. If g is not Gaussian, hmms is not necessarily a minimum mcan-square error estimator. It is however the best linear estimator in the mean-square error sense. In either case(g, gaussian or not) we will denote the channel estiMate as h The ls estimator for the cyclic impulse response g min mizes(y-XFg (y-XFgi and generate Fig. 5: Modified estimator structure hs= FQ FHX y, (11) L is a small fraction of N. Thus, the complexity of the MMSE estimator will decrease considerably where Although the complexity of the Ls estimator does not prompt for modifications, its perfora.lance in terms of (FX XF 12) f snr Note that h s also corresponds to the estimator structure by following the same general concept as above. The Ls in Fig 4. Since(11) reduces to estimator does not use the statistics of the channel. Intu- itivcly excluding low cncrgy taps of g will to some extent (13) Compensate for this shortcoming sinice the energy of g de creases rapidly outside the first L taps, whilst the noise the LS estimator is equivalent to what is also referred to energy is assumed to be constant over the entire range as the zero-forcing estimator Both estimators (9)and (13)have their drawbacks. The Taking only the first L taps of g into account, thus im MMSE eStimator suffers from a high complexity, whereas plicitly using channel statistics, the modified LS estimator the Ls estimate has a high mean-square error. In the next becomes section, we will address these draw backs and modify both hs= TQlTx y (10) nators Q S=T" XXT (17) B. Modified MMSE and is estimators The modified LS estinator also has the structure as shown The MMSE estimator requires the calculation of an N x N matrix QMMsE, which implies a high conplexity when N is large A straightforward way of decreasing the complexity is to reduce the size of QMmsp, As indicated in C. Estimator Complexity Fig.2, most of the energy in g is contained in, or near, the The complexity of the modified LS estimator(16) will first L=[Ial taps. Therefore we study a modification of be larger than that of the full LS estimator, since a simpli the MMSE estimator, where only the taps with significant fication as in(13)cannot be performed. Notice that while energy are considered. The elements in Rgg corresponding the full LS estimator(13)has much lower conplexity than to low energy taps in g are approximated by zero. the full MMSE estimator(9 ) the respective modified ver If we take into account the first L taps of g, and set sions(16 )and(14)are equally complex first L columns of the DFT-matrix F and Rpg denotes tion and noise variance. In practice these quantities e 3 p ly It should be noted that the mMse estimators have been educed to an L X L matrix. If the matrix T denotes the derived under the assumption of known channel correla the upper left L x L corner of rgg, the modified mMse and 2, are either taken fixed or estimated, possibly in an estimator becones adaptive way. This will increase the estimator complexity h39B=TQoT“x"y (14)and reduce the performance slightly In the special case where the channel (1)is T-spaced, where e, where Tm are integers, no leakage of energy to taps outside the interval [0, Ll will occur and the two modi- QMMSE=["XT)OrREE (TXXT) fied estimators(14)and(16) will not lose any information (15) about the channel. Thus, the modified Mmse estimato This modification is illustrated in Fig. 5. As mentioned ( 14)is equivalent to the MMSE estimator (9). The mod in Section I, an OFDM system is usually designed so that ified LS estimator(16) in this case will outperform the 817 LS estimator (13)for all SNRs, since the excluded taps 10° contain only noise. More generally, for non-T-spaced channels, any subset of the taps in g may be taken into account when modi- fying the MMSE and the LS estimator. Thc sizc of this subset determines the complexity for both types of modi fied estimators In scction iy we consider the case where this subset 10 consists of the taps 90.9L+K-1 and gN-K... gN-1,i.e the first i taps as well as K extra taps on each side 层103 s侣 E B-0 Ⅳ, SIMULA" TIONS + MMSE-10 4. Simulated channels In the simulations we consider a system operating with Average SNR dB bandwidth of 500 kHz. divided into 64 tones with a tota Fig. 6: Mean-square error for three modified MMSE estimators syMbol period of 138 us, of which 10 us is a cyclic prefix Sampling is performed with a 500 kHz rate. A symbol thus consists of 69 samples, five of which are contained il the cyclic prefix (i.e. L=5).50,000 channels are ran domized per average SnR, each consisting of five pulses of which four have uniformly distributed delays over the nterval 0-10 us, while one tap is al ways assumed to have a zero delay, corresponding to a perfect time synchroni- tants. The multipath intensit 102 profile is assumed to be p()Ne-r/rms, where Trms is 1/4 of the cyclic extension. We have used Monte-Carlo simu- lations to gencrate the Rgg for this channel model. This 2 MMSE covariance matrix together with the true noise variance o2 LS-O is used in the MMse estimations to follow The average -G-LS-10 SNR per symbol in Fig 3 is defined as flh Ink!h 10 since E[ckl) is normalised to unity. 30 he following estimators are used Average SNR dB Fig. 7: Mean-squarc error for thrce lS estimators Estimator Notation Taps used Sizc Q MMSE MMSE 0.63 64×64 dimensions of Q will give lower mean-square error for 0.63 NA Modified MMSE-00.4 Fig 7 shows the mean-square error versus average SNR MMSE MMSE 0.9,59.6315X15 or the mmse. ls. ls-o Ls-5 and Ls-10 estimators. Con- MMSE-100.14,54.6325×25 trary to the modification of the MMSE estimator, the Modified Ls-00.4 modification of the LS estimator reduces the mean-square LS-5 0...9, 59.[ x 15 error for a range of SNRS. However, Clre saue approxilnd- 0.14,546325×25 tion effect as in the modified MMSE estimators shows up t high SNRs. An interesting observation is, that for every B.Mem-qr℃eBO SNR there exists an optimal size of Qi, which gives th smallest mean-square error compared to the other modi- Fig. 6 shows t he mean-square error versus SNR for fied LS estimators the mmse. ls. Mmse-0, MMsE-5 and MMSE-10 esti mators. The difference between the modifed MMSe esti mators and the MmsE estimator is due to the fact that C, Symbol-error Rate parts of the channel statistics are not taken into account The symbol-error rate(SER) curves presented in this in the former. For lOw SNRs, this approximation effect section are based on the mean-square errors of the channel is small compared to the channel noise, while it becomes estimations presented in the previous section. For the dominant for large SNRs. The curves level out to a value calculation of SER, we have used the formulae presented determined by the energy in the excluded taps. Larger in [10. These formulae find the symbol error rate of a 10 8E LS MMSE k·MMSE ▲-MMSE5 10 P MMSE-10 10 20 30 10 Average SNR. [dB] Average SnR dBI Fig. 8: Symbol error rate for three modified MMSE estimators Fig 9: Symbol error rate for three modified Ls estimators 16-QAM system given a noisy estimate of the cha 10 We consider decision-directed estimation, without error propagation. As seen in Fig. 8, a gain in SNR up to 4 dB can be y obtained for certain S]Rs when using a modified MMSE tead of the ls esti ad- missible complexity. The same behaviour can be observed Fig 9. He in SNr is not as large as for the modified mMse esliInla- with the same size of the matrix Q. This is explicitly shown for MMSE-10 and LS-10 in Fig. 10. The difference y *K MMSD in snr between these two estimators is about 2 dB MMSE-10 Y, CONCLUSIO 10 15 20 Average SNR [ di ig. 10: Comparison of SF.R. between a. modified MMSE estimator The estimators in this study can be used to efficiently and the corresonding modified ls estimator. estimate the channel in an OFDM system given a certain knowledge about the channel statistics. The mmse esti- [2] M. Alard and R. Lassalle, "Principles of modulation and chan- mator assumes a priori knowledge of noise variance and nel coding for digital broadcasting for mobile recei channel covariance. Moreover, its complexi IS Iarge com- Review, no 224, pp 3-25, August 198 pared to the lS estimator. For high sNrs the ls estima- [3] B Marti et al., "European activities on digital television broad- tor is both simple and adequate. llowever. for low SNRs casting -from company to cooperative projects", EBU Techni cal Review, no. 256, pp. 20-29, 199 the presented Inlodilicalions of the MMSE and LS estirmd-[4 John Prcakis, Digital Comrnunications, McGraw-Hill,1989 tors will allow a compromise between estimator complex- 5] Sarah Kate Wilson, R. Ellen Khayata. and John M. Cioffi, ity and performance. For a 16-QAM signaling constella- 6-QAM alation frequent tion, up to 4 dB gain in SnR over the Ls estimator was tiplexing in a Rayleigh-fading environment", In Proc. VTC 1994, pp. 1660-1664, Stockholm, Sweden, June 1994 obtained, depending on estimator complexity. Even rela- [6] Peter Hoeher, " CM on frequency-selective land-mobile fading tively low-complex modified estimators, however, perforn channels", In Proc. of the 5th. Tirrenia International Work shop significantly better than the ls estimator for a range of o Digitcl Communications, Tirrenia, Italy, September 1991 SNRs In A C. Bingham, "Mult carrier modulation mission: an idea whose time has come IEEE Communication Magazine,28(5):5-14,May1990 REFERENCES 8 John M. Cioffi, Stanford University, Private communication [9 Louis L. Scharf, Statistical Signal Processing, Addison-Wesley, Leonard J Cimini, Jr, "Analysis and simulation of a digital mobile channel using orthogonal frequency-division multiplex [1O Sarah Kate Wilson, Digital Audio Broadcasting in a Fading and ing",IEEE Trans. Com., Vol. 33, no. 7, Pp 665-675, July Dispersive Channel, PhlD-thesis, Staniford University, Augus 1985 1994 819

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