In wireless OFDM applications, since the radio channel is frequency selective and time-varying, a dynamic estimation of channel must be achieved before the demodulation of the transmitted OFDM signals. As an effective approach for solving the channel estimation problems, the pilot-assisted channel estimation technique has received considerable attention in recent years. In this paper, we investigate the channel estimation error in the existing pilot-assisted channel estimation approaches in detail, a new effective channel estimation approach with lower estimation error is proposed as well

Jiao, Bin

Wang, Jinlin

Xiao, Jitian

English

Research Online @ ECU

Edith Cowan University Research Online ECU Publications Pre. 2011 2005 Analysis of channel estimation error of OFDM systems in rayleigh fading Bin Jiao Institute of Acoustics, Chinese Academy of Sciences, P. R. China Jitian Xiao Edith Cowan University Jinlin Wang Institute of Acoustics, Chinese Academy of Sciences, P. R. China Follow this and additional works at: https://ro.ecu.edu.au/ecuworks  Part of the Computer Sciences Commons This is an Author's Accepted Manuscript of: Jiao , B., Xiao, J. , & Wang, J. (2005). Analysis of channel estimation error of OFDM systems in rayleigh fading. Proceedings of International Symposium on Signals, Circuits and Systems. (pp. 653-656). Iasi, Romania. IEEE Press. © 2005 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. This Conference Proceeding is posted at Research Online. https://ro.ecu.edu.au/ecuworks/2811 Analysis of Channel Estimation Error of OFDM Systems in Rayleigh Fading Bin Jiao', Jitian Xiaot, Jinlin Wang* *Networking & DSP Research Center, Institute of Acoustics Chinese Academy of Sciences, P. R. China. E-mail: jiaob,wangjl@dsp.ac.cn tSchool of Computer and Information Science Edith Cowan University, Australia. E-mail: j.xiaoBecu.edu.au Absht - In  wireless OFDM applications, since the radio channel is frequency selective and time-varying, a dynamic estimation of channel must be achieved before the demodulation of the transmitted OF1)M signah. As an effective approach for d v i n g  the channel esfimath problems, Ihe pilot-asisted channel estimation technique has received considerable attention in recent years. In this paper, we will investigate the channel estimation error in the existed pilobassisted channel estimation epprosches in detail, a new effective channel estimation approach with lower estimation error is proposed as well. I, INTRODUCTION In high speed' wireless OFDM systems, the channel es- timation can be achieved by the pilot-assisted scheme, this technique has been studied for several years [1][2]. In this scheme, the channel is usually assumed to have a finite-length impulse response, a cyclic prefix which is longer than the channel impulse response is put between consecutive symbol blocks for avoiding inter-channel interference (IC0 and pre- serving the orthogonality of the subcarriers. The pilot symbols are inserted at selected two-dimensional time-frequency pilot locations, the channel estimation is achieved by estimating the channel attenuations in frequency domain using least square (LS) method or linear minimum mean square error (LMMSE) method I?]. In this paper, we will present an analysis of the esti- mation error in the existed pilot-assisted channel estimation approaches for multipath Rayleigh fading channels. The paper is organized as follows. In Section 2, the OFDM systems and channel models are represented. The analysis of channel estimation is provided in Section 3, an improved channel estimation algorithm is also proposed in this section. The simutation results are presented in Section 4. 11. SYSTEM MODEL AND CHANNEL ESTIMATION A. OFDM System Model A baseband OFDM system model is shown in Fig. 1, where (S,} is the transmitted parallel complex symbol sequence with symbol rate fa = l /NT, .  The serial time-domain sequence {z,,} with data rate l /Tb is obtained by taking IDET on {X, , } .  The cyclic prefix (CP) of time length T, fig. 1. A baseband +I of OFDM system is used to eliminate inter-channel interference and preserve the orthogonality of the subcarriers, the D/A converter which contains ideal low-pass filter with bandwidth W, = 1/T, convegs the digital signal (5,) into the transmitted analog signal s(t ) .  At receiver, the input analog signal y ( t )  is discretized with sampling rate l/TB to obtain the digital data sequence {yn}. Removing the cyclic prefix from (y,} and then performing Dm on it, we obtain the received symbol sequence (Yn}. B. Channel Model The channel model considered throughout 'the paper is a multipath RayIeigh fading channel with time-varying finite impulse response [1][3], which can be written as M - 1  h(t ,T)  = hm(t)fi(7 - Tm(t)) (1) m-0 where A4 is the total number of the propagation paths of channel, h,(t) and .rm(t) are the attenuation coefficient arid excess delay, respectively, of the mth path component at time t. The excess delay r,(t) is required to satisfy the condition 0 5 ~ , ( t )  5 Tg, which ensures the entire impulse response lies inside the guard interval. The corresponding frequency response of channel is M-1 H ( t ,  f )  = hm(t)e-32=-(t)f (2) m=O Furthermore, the channel h(t, 7) is also assumed to be a wide- sense stationary uncorrelated scattering and slow fading chan- nel [4][5], in which' case the channel attenuation coefficient h,(t) and the excess delay ~,(t)  can remain constant during one symbol diration. Thus, the channel can be treated as a set of parallel independent stationary complex-valued Gaussian 0-7803-9029-6/05/$20.00 02005 IEEE 653 processes. The attenuation on the nth subcarrier with carrier frequency fn = fi during the pth symbol can be written as &k = 2, if and only if k = n K  or & = 0 elsewhere. Thus, z( t )  can be &so expressed as where T i s  the symbol duration with T = NT, + T,, and hp.m = h,(pT), T~,,,, = 7,(pT). For simplifying the discussion, we consider the case that only one symbol is transmitted in each sub-channel, thus we can drop the time (OFDM symbol) index p from the (3) to obtain 2nr n M - 1  H ,  = h,e-j* (4) m=O Consequently. the corresponding observed channel impulse response {h,) is the ~DFT of {H~}. 111. CHANNEL ESTIMATION A. Analysis of Chnnnel Estimation Error The pilot-assisted channel 'estimation problem is usually formulated to estimate the channel attenuatiok { H,} from the received data {Yn} and the transmitted data { X , }  [21[31. Since Y, = X,Hn + N,, where { N , }  is the additive Gaussian white noise of the channel, which is independent and urgorrelated dith the channel, the channel attenuation estimate {Hn} can then be written as - Yn Nll Hn = - = H, + - X n  Xn Thus, maximum likelihood (ML) method or LMMSE method can be adopted to estimate the channel attenuations. For holding the equation Y, = X n H ,  + N,, the channel excess delay T, is usually assumed to be equal to mT,. However, this assumption is seldom satisfied in practice [ 6 ] .  In that case, it will be demonstrated next that even for an ideal channel without noise. the proposed approaches based on (5) can not provide a good estimate of the channel attenuations. (5 )  Without loss of generality, consider a multipath fading channel whose excess delay 7, equals q, where K is an integer, then the channel impulse response can be represented by . Notice that the transmitted signal can be written as N - l  s(t )  = xn6( t  - nT,) n = O  (7) If we upsample the data sequence {zn} with the upsampling rate K ,  then we can obtain a new date sequence {%k} with K N - 1  At receiver, discretizing the received signal y ( t )  with the sampling rate E, then the obtained data sequence { y k }  can be written as y k  _ f k  @+, where-@ denotes the circular convolution. L e t  {Xk}. {&} and {Yk} be the DFTs of the data sequences { i k } ,  {h,} and {yk_),Lespeciively, with the length of KN, then we have Y k  = XkHk. For the upsampling transceiver system described above, the data sequences {g,}. which i s  obtained by discretizing the signal y(t) with sampling rate k ,  equals the sequence downsampled from { y k )  with downsampling rate K ,  i.e. ( iW yn = y k  if and only if n = k, and yn = x, @ h, where {hgK'} is the sequence downsampled from {hm} with downsampling rate K .  Let y(f)  = g ( Y n ) ,  PU) = (9) where ST(.) denotes discrete time Fourier uansform, then from [7], we have Discretizing the frequency at f = g, then we can obtain Similarly, it can also be proved that (12) h k = O  ' K Thus, for the system model presented in Section 2. it can be verified that in the case of ideal channel without noise, the channel attenuation estimate obtained in (5) is 113) On the other hand, substituting 7, = 9 into (41, we can see that the estimated channel attenuations should be From (12), we know that B, # HALK) for 0 5 n 5 N - 1 in general, it demonstrates that even for the case of ideal channel, the channel estimation approaches based on (5) can not give us an accurate estimate of the channel attenuation. 654 \ . .  . . . . . . ... . . . . , . - t L '  " " " " 1 '  ' " 0 1 2  3 4 5 6 7 8 9 1 0 1 1 i 2 1 3 i d 1 5  I x m m z  2 k 3 1  5 0  !-i Fig. 2. Corrparison between the channel alpnuation {Hn} and the estimate {A',,) obtained from the existed algorithm B. Improved Channel Estimate In the previous section, it has been shown that the channel estimation algorithms based on ( 5 )  m a y  have estimation errur even for ideal channels. In this section, we will investigate the methods to reduce this type of estimation error. Without loss of generalily, we can divide the channel models into two classes. For the first type of system modei, the channel excess delay 7, is assumed to equal For the second type of channel model, the channel excess delay r,,, is not a rational number. In this case, we can however always choose an integer h', such that IT, - 91 5 E << 0, where E is an arbitrary constant. Thus we can use as an approximation of im and ignore the difference between T,, and 9. In the previous section, it has been demonstrated that for the channel model with the channel excess delay T, : q, the channel attenuations ( H , )  are equal to the first N elements of {&}, so if we can get an accurate estimate of I&}, then he  ct"el attenuation (H,}  will be obtained. Suppose that the system has additive Gaussian white noise, where { Nk} .is the channel Gaussian noise. Let +. X =  I , i i = l  ! iz  i: 0 1 2  3 4 5 6 7 8 Q 1 0 1 1 1 2 1 3 1 4 1 5  -1 I x 25oKnz 2 f 1  ij-: 0 1 2 3 4 5 6 7 8 9 ' 1 0 t l 1 2 1 3 i 4 1 5  -2 x2MKHz Fig. 3. Comparison between the channel attenuation { Hn} and the estimate { H,,} obtained from the irrqaoved algorithm and where WFN = e-i %!F. Because ' G=Wh (18) Equation (15) can then be rewritten in matrix notation as 7 = 2Wh -I- 5 (19) Based on the above equation, the channel estimation problem in the sense of LS, can be formulated as the following optimization probIem min { (7 - %Wh)T(y - ZWh)} (20) Solving this LS problem using the algorithms in [8 ]  and substituting the solution h into (le), we can obtain an estimate of the channel attenuation {&I. Summarizing the discussion above, we propose the follow- ing algorithm for solving the channel estimation problem: Step 1: h Choose suitable upsampling rate K and then dis- cretize the received signal y(t) with sampling rate Step 2: Remove the cyclic prefix from {&} and then per- form DFT on it to obtain {Yh); Step 3: Solve the LS optimization problem120) to obtain h; Step 4: Substitute h into (18) to obtain the estimat of tu get b e  data sequence {ik}; channel attenuation {Hn}. f For the channel estimation approach based on the linear minimum mean square error aiterion, the channel attenuation 65 5 estimates {fin} can also be obtained from (191. Suppose the channel is Gaussian channel and uncorrelated with the channel noise, then channel attenuation estimate is where variance matrix of h, U: is the channel noise variance. denotes Hermitian transposition, Chh is the co- ( Iv. SIMULATION RESULTS In the simulation section, we consider a multipath time- variant Rayleigh fading channel. The number of propagation paths is M = 5, the channel impulse response and excess delay are h, = 0.5938, 0.2137, 0.3175, 0.7305, 0.4312 and T, = 0, 0.167, 0.51, 0.751 and 0.99~~. respectively. The OFDM system is a 16-QAM OFDM system whose 4MHz bandwidth i s  divided into 16 subchannels, each subchannel has a bandwidth of 250KHz, the corresponding time-interval of the transmitted data sequence is T, = 0.25ps, the guard interval Tg is Ips. The channel attenuation {H,} and its estimate { fi,) at 16 subcarrier frequency points are shown in Fig. 2, where the mth subcarrier frequencyf, is m x 250KHz. The estimate of channel attenuation { H , }  is obtained by solving the LS optimization problem based an (13). which implies that the data sequence (yn} is obtained by discretizing the analog signal y ( t )  with sampling rate $, the channel is assumed to be ideal channel without noise. from the simulation results shown in Fig, 2, it can be Seen that the difference between { H,} and { Hk}  is obvious. The channel attenuation estimate obtained by using the improved algorithm proposed in the paper 1s shown in Fig. 3. where the upsampling rate. K is chosen as K = 3. The simulation result shows that the estimate is very close to the actual channel attenuation, the maximum error between them is about 0.147, the corresponding maximum error in Fig. 2 is about 0.322. Comparing the simulation results shown in Fig.2 and Fig. 3, it can be seen clearly that the proposed approach given in the paper has better performance than the existed methods. V. CONCLUSION REFERENCES [l] L J .  V. fleek, 0. Edfors, and M. Sandell: "On channel esth"aion in OFDM system," IEEE Vehicul~r Technology Conjerence, pp. 8t5-819, 1995. [Z] R. Nilsson. 0. Edfors, M. Sandell, and P. 0. Borjesson. "An analysis of two-dimensional pilot-symbol assisted modulation for OFDM," ICPWC, [3] M. Chang and Y. T. Su. "Model-based channel estimation for OFDM signals in Rayleigh fading," IEEE Trans. Commun.. pp. 540-544, April 2002. [+ P. Hoeher, "A statistical dim&-time model for the WSSUS multipath channel," IEEE Trans. Veh. Ech., pp. 461-468, Nov. 1992. [ 5 ]  P. A. Bello. "Charactetization of randomly time-variant linear channels;' IEEE Trans. Commun. Syst., pp. 3 6 3 9 3 ,  Dec. 1963. [6] T. S. Rappaport. Wireless Communicntions P rincipfes md Procfice. Pearson Education Asia Limited. 2004. [7] P. P. Vaidyanathan. MuElirare System and Filrer Bur&, Englewood Cliffs, Prentice-Hall. 1993. 181 L. L. Scharf, Sfufisficai Signul Processing. Addison-Wesley, 1991. pp. 71-74. 1997. In this paper, the channel estimation error based on the pilot-assisted approaches has been investigated. A new OFDM channel estimation scheme based on the filter bank theory is introduced. With the proposed scheme, a new effective channel estimation approach has also been provided, It is demonstrated that the props& channel estimation method has lower estimation error than the existed approaches. 656 

Analysis of channel estimation error of OFDM systems in rayleigh fading

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Analysis of channel estimation error of OFDM systems in rayleigh fading

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