Rayeigh Time-varying Channe Compex Gains Estimation and ICI Canceation in OFDM Systems Hussein Hijazi, Laurent Ros To cite this version: Hussein Hijazi, Laurent Ros. Rayeigh Time-varying Channe Compex Gains Estimation and ICI Canceation in OFDM Systems. European Transactions on Teecommunications, Wiey, 29, 2 8, pp.782-796. <ha-373794> HAL Id: ha-373794 https://ha.archives-ouvertes.fr/ha-373794 Submitted on 7 Apr 29 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.
EUROPEA TRASACTIOS O TELECOMMUICATIOS Rayeigh Time-varying Channe Compex Gains Estimation and ICI Canceation in OFDM Systems Hussein Hijazi and Laurent Ros E-mai: hussein.hijazi@gipsa-ab.inpg.fr, aurent.ros@gipsa-ab.inpg.fr Te: +33 4 76 82 71 78 and Fax: +33 4 76 82 63 84 GIPSA-ab, Departement Image Signa BP 46-3842 Saint Martin d Hères - FRACE SUMMARY In this paper, we consider an orthogona-frequency-division-mutipexing OFDM mobie communication system operating in downink mode in a time-varying mutipath Rayeigh channe scenario. We present a Mean Square Error theoretica anaysis for a mutipath channe compex gains estimation agorithm with inter-sub-carrier-interference ICI reduction using a comb-type piot. Assuming the presence of deayreated information, the time average of the mutipath compex gains, over the effective duration of each OFDM symbo, are estimated using LS criterion. After that, the time-variation of the mutipath compex gains within one OFDM symbo are obtained by interpoating the time-averaged symbo vaues using owpass interpoation. Hence, the channe matrix, which contains the channe frequency response and the coefficients of ICI, can be computed and the ICI can be reduced by using successive interference suppression SIS in data symbo detection. The agorithm s performance is further enhanced by an iterative procedure, performing channe estimation and ICI suppression at each iteration. Theoretica anaysis and simuation resuts show a significant performance improvement for high normaized Dopper spread especiay after the first iteration in comparison to conventiona methods. 1. ITRODUCTIO ORTHOGOAL frequency division mutipexing OFDM is widey known as the promising communication technique in the current broadband wireess mobie communication system due to the high spectra efficiency and robustness to the mutipath interference. Currenty, OFDM has been adapted to the digita audio and video broadcasting DAB/DVB system, highspeed wireess oca area networks WLA such as IEEE82.11x, fixed wireess access WiMax IEEE82.11e, 3GPP/LTE, HIPERLA II and mutimedia mobie access communications MMAC, ADSL, digita mutimedia broadcasting DMB system and muti-band OFDM type Part of this work was presented in 5-th IEEE GLOBECOM, Washington, USA, ovember 27 1] and in European Wireess Conference EW, Paris, FRACE, Apri 27 2] utra-wideband MB-OFDM UWB system, etc. However, OFDM system is very vunerabe when the channe changes within one OFDM symbo. In such case, the orthogonaity between subcarriers are easiy broken down resuting the inter-sub-carrier-interference ICI so that system performance may be consideraby degraded. A dynamic estimation of channe is necessary since the radio channe is frequency seective and time-varying for wideband mobie communication systems 5] 17] 19]. In practice, the channe may have significant changes even within one OFDM symbo. In this case, it is thus preferabe to estimate channe by inserting piot tones into each OFDM symbo which caed comb-type piot 6]. Assuming insertion of piot tones into each OFDM symbos, the conventiona channe estimation methods consist generay of estimating the channe at piot frequencies and next interpoating the channe
2 H. HIJAZI AD L. ROS frequency response. The estimation of the channe at the piot frequencies can be based on Least Square LS or Linear Minimum Mean-Square-Error LMMSE. LMMSE has been shown to have better performance than LS 6]. In 7], the compexity of LMMSE is reduced by deriving an optima ow-rank estimator with singuarvaue-decomposition. The interpoation techniques used in channe estimation are inear interpoation, second order interpoation, ow-pass interpoation, spine cubic interpoation, time domain interpoation and Wiener fitering as 2-D interpoation 19]. In 8], ow-pass interpoation has been shown to perform better than a the interpoation techniques. In 9] the channe estimator is based on a parametric channe mode, which consists of estimating directy the time deays and compex attenuations of the mutipath channe. This estimator yieds the best performance among a comb-type piot channe estimators, with the assumption that the channe is invariant within one OFDM symbo. For fast timevarying channe, many existing works resort to estimate the equivaent discrete-time channe taps which are modeed in a inear fashion 17] or more generay by a basis expansion mode BEM 15] 16]. The BEM methods 15] used to mode the equivaent discrete-time channe taps are Karhunen-Loeve BEM KL-BEM, proate spheroida BEM PS-BEM, compex-exponentia BEM CE-BEM and poynomia BEM P-BEM. In the present paper, we present an iterative agorithm for channe estimation with inter-sub-carrier-interference ICI reduction in OFDM downink mobie communication systems using comb-type piots. Our interesting is to estimate directy the physica channe instead of the equivaent discrete-time channe. That means estimating the physica propagation parameters such as mutipath deays and mutipath compex gains. By expoiting the nature of Radio-Frequency channes, the deays are assumed to be invariant over severa OFDM symbos and perfecty estimated, and ony the compex gains of the mutipath channe have to be estimated as we have aready done in CDMA context 3] 4]. otice that an initia very performant mutipath time deays estimation can be obtained by using the ESPRIT estimation of signa parameters by rotationa invariance techniques method 9] 11]. For a Jakes spectrum Rayeigh gain, we have showed that the centra vaue and the time averaged vaue over one OFDM symbo are extremey cosed even for high reaistic Dopper spread. So, for a bock of OFDM symbos, we propose to estimate the time average of the compex gains, over the effective duration of each OFDM symbo of different paths, using LS criterion. After that, the time variation of the different paths compex gains within one OFDM symbo are obtained by interpoating the time averaged symbo vaues using ow-pass interpoation. Hence, the channe matrix, which contains the channe frequency response and the coefficients of ICI, can be computed and the ICI can be reduced by using successive interference suppression SIS in data symbo detection. The present proposed agorithm, with ess number of piots and without suppression of interference, gives a good performance over the conventiona methods and performs better with starting interference suppression. This proposed agorithm can in fact be considered as a simpe extension of an agorithm for time-invariant channes but which brings, as we wi show, a significant gain in case of time-varying channes with reaistic Dopper spreads. Moreover, we give a theoretica and simuated Mean Square Error MSE mutipath channe compex gains estimation anaysis in terms of the normaized by the OFDM symbo-time Dopper spread. This further demonstrates the effectiveness of the proposed agorithm. This paper is organized as foows. Section II introduces the OFDM baseband mode and section III covers the mutipath compex gains estimation and the iterative agorithm. ext, Section IV presents some simuation resuts that demonstrate our technique. Finay, we concude the paper in Section V. otation: Superscripts T and H stand for transpose and Hermitian operators, respectivey., T r and E ] are the determinant, trace and expectation operations, respectivey. and are the magnitude and conjugate of a compex number, respectivey. and denote a vector and a matrix, respectivey. Am] denotes the mth entry of the vector A and Am,n] denotes the m,n]th entry of the matrix A. I is a identity matrix and diag{a} is a diagona matrix. J and J 1 denote the zeroth-order and the first-order Besse functions of the first kind, respectivey. denotes the convoution. δ k,m denotes the Kronecker symbo and k stands for the residue of k moduo. Letters between brackets d] or parentheses n denote that d and n are indexes or variabes. 2. SYSTEM MODEL Suppose that the symbo duration after seria-to-parae S/P conversion is T u. The entire signa bandwidth is covered by subcarriers, and the space between two neighboring subcarriers is 1/T u. Denoting the samping time by T s = T u /, and assuming that the ength of the cycic prefix is T g = g T s with g being an integer. The
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 3 duration of an OFDM symbo is T = + g T s. In an OFDM system, the transmitter usuay appies an -point IFFT to data bock normaized QAM-symbos {X n k]} i.e.,e X n k]x n k] ] = 1, where n and k represent respectivey the OFDM symbo index and the subcarrier index, and adds the cycic prefix CP, witch is a copy of the ast sampes of the IFFT output, to avoid inter-symbointerference ISI caused by mutipath fading channes. In order to imit the periodic spectrum of the discrete time signa at the output of the IFFT, we use an appropriate anaog transmission fiter G t f. As a resut, the output baseband signa of the transmitter can be represented as 9] 1]: xt = n= d= g x n d]g t t dt s nt 1 where g t t is the impuse response of the transmission anaog fiter and x n d], with d g, 1], are the + g sampes of the IFFT output and the cycic prefix of the nth OFDM symbo given by: x n d] = 1 2 1 m= 2 md j2π X n m]e 2 It is assumed that the signa is transmitted over a mutipath Rayeigh fading channe characterized by: ht,τ = L α tδτ τ T s 3 where L is the tota number of propagation paths, α is the th compex gains of variance σα 2 and τ is the th deay normaized by the samping time τ is not necessariy an integer. {α t} are wide-sense stationary WSS narrow-band compex Gaussian processes with the so-caed Jakes power spectrum of maximum Dopper frequency f d 1] and uncorreated with respect to each other. The average energy of the channe is normaized to one i.e., L σ2 α = 1. At the receiver side, after passing to discrete time through ow pass fitering and A/D conversion, the CP is removed assuming that its ength is no ess than the maximum deay. Afterwards, a -point FFT is appied to transform the sequence into frequency domain. The kth subcarrier output of FFT during the nth OFDM symbo is given by see Appendix A: Y n k] = 2 1 m= 2 X n m]g tm]g rm]h n k, m] + W n k] 4 where W n k] is white compex Gaussian noise with variance σ 2, G t m] and G r m] are the transmitter and receiver fiter frequency response vaues at the mth transmitted subcarrier frequency, and H n k,m] are the coefficients of the channe matrix from the mth transmitted subcarrier frequency to the kth received subcarrier frequency, and given by see Appendix A: H n k,m] = 1 L e j2π m τ q= ] α n m k j2π qt s e q 5 where k,m 2, 2 1] and {α n qt s } is the T s spaced samping of the th compex gain during the nth OFDM symbo. If we assume transmission subcarriers within the fat region of the frequency response of each of the transmitter and receiver fiters, then, by using the matrix notation and omitting the index time n, 4 can be rewritten as 1] 2]: Y = H X + W 6 where G t m] and G r m] are assumed to be equa to one at the fat region, where X, Y, W are 1 vectors given by: X = Y = W = X 2 ],X 2 + 1],...,X2 1] ] T Y 2 ],Y 2 + 1],...,Y 2 1] ] T W 2 ],W 2 + 1],...,W2 1] ] T and H is a channe matrix, which contains the time average of the channe frequency response Hk, k] on its diagona and the coefficients of the inter-carrier interference ICI Hk,m] for k m. otice that H woud be obviousy a diagona matrix if the compex gains were time-invariant within one symbo. 3. MULTIPATH COMPLEX GAIS ESTIMATIO AD THE ITERATIVE ALGORITHM In this section, we propose a method based on comb-type piots and mutipath time deays information to estimate the samped compex gains {α qt s ]} with samping period T s.
4 H. HIJAZI AD L. ROS Frequency 1 2 1 4 mse 2 1 6 L f Figure 1. Comb-Type Piot Arrangement with L f = 3 3.1. Piot Pattern and Received Piot Subcarriers Time The p piot subcarriers are fixed during transmission and eveny inserted into the subcarriers as shown in Fig. 1 where L f denotes the interva in terms of the number of subcarriers between two adjacent piots in the frequency domain. L f can be seected without the need of respecting the samping theorem in frequency domain as opposed to the methods shown in 9] 8]. However, as we wi see with equation 13, p must fufi the foowing requirement: p L. Let P denote the set that contains the index positions of the p piot subcarriers defined by: P = {p s p s = s L f 2, s =,..., p 1} 7 The received piot subcarriers can be written as the sum of three components 2]: Y p = X p H p + H pi X + W p 8 where the p p diagona matrix X p, and the p 1 vectors Y p and W p are given by: X p = diag{xp ],Xp 1 ],...,Xp p 1]} Y p = Y p ],Y p 1 ],...,Y p p 1] ] T W p = Wp ],Wp 1 ],...,Wp p 1] ] T H p is a p 1 vector and H pi is a p matrix with eements given by: H p p s ] = Hp s,p s ] = H pi p s,m] = L ps j2π α e τ { Hps,m] if m 2, 2 1] P if m P 1 8 mse 2 theoretica mse 2 simu 1 1.1.5.1.2.3.4.5 f d T Figure 2. MSE between α c and α for = 128 with α = 1 q= α qt s 9 α is the time average over the effective duration of the OFDM symbo of the th compex gain. The first component is the desired term without ICI and the second component is the ICI term. H p can be writen as the Fourier transform for the different compex gains time average {α }: H p = F p α 1 where F p and α are the p L Fourier transform matrix and the p 1 vector, respectivey, given by: F p = e j2π p τ 1 e j2π p τ L....... τ1 e j2π p p 1 e j2π p p 1 τl α = α 1,...,α L ] T 11 3.2. Estimation of Mutipath Compex Gains For a bock {α qt s, q =,..., 1} of T s -spaced samping of a gaussian compex gains with a frequency of the Jakes power spectrum f d, we have shown in Appendix B that: a The T s -spaced samping of the compex gain taken in the midde of the effective duration of the OFDM symbo α 2 T s is cosest to the compex gain time average over the effective duration α defined in 9 b For the whoe L gains, the mean square error MSE between exact averaged vaues α = α 1,...,α L ] T and exact centra vaues α c = α 1 2 T s,...,α L 2 T s ] T is
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 5 yk q ] F F T ˆ k Y d ˆ k, i H p I k Yp k, i 1 X d Channe Matrix Estimation, ˆ k i H k X p ˆ k, H i d SIS Detection ˆ k, X i d k Y p k, i H p I ˆ ˆ k, i 1 X d Suppression of k ICI p k X p k, i Y p LS Estimator, k i LS Low-pass Interpoation, k i qts ˆ ] Computing of Channe Matrix, Hˆ k i -1 z a CP b OFDM symbo B i The ith bock of K OFDM symbos B i - 1 B i + 1 LS LS LS Interpoation of K sampes estimated LS LS LS LS LS Interpoation of K sampes estimated Interpoation of K sampes estimated c Figure 3. The bock diagrams of the iterative agorithm: a the overa channe estimator and ICI suppression bock diagram; b the channe matrix estimation bock diagram; and c the diagram of compex gains estimator given by: mse 2 = E α α c H α α c ] L = σα 2 1 J 2πf d T s q 1 q 2 2 q= 2 q 1= q 2= J 2πf d T s q 2 + 1 12 otice that, for a normaized channe, mse 2 depends ony on f d T. Fig. 2 shows the evoution of mse 2 with f d T, obtained theoreticay from 12 and by Monte-Caro simuation. We can concude that, for reaistic normaized Dopper spread f d T <.5, the distance between α c and α is very negigibe. Hence, we can assume that an estimation of α is an estimation of α c. So, by estimating α for some OFDM symbos and interpoating them by a factor + g using ow-pass interpoation 8], we obtain an estimation of the samped compex gains {α qt s } at time T s during these OFDM symbos, for each path. The time average of the compex gains, over the effective duration of each OFDM symbo for the different paths, are estimated using the LS criterion. By negecting the ICI contribution, the LS-estimator of α, which minimizes Yp X p F p α H Yp X p F p α, is represented by: α LS = M Y p 1 with M = F H p X H p X p F H H p Fp X p 13 It shoud be noted that, the matrix F p H X p H X p F p in the expression of M 13 is not invertibe if p < L, since F p is a matrix of size p L. 3.3. Iterative Agorithm The iterative agorithm of channe estimation and ICI suppression is shown in Fig. 3. The whoe agorithm is divided into two modes: channe matrix estimation mode and detection mode, as shown in Fig. 3a. The first mode incudes estimating the samped compex gains {α qt s } at time T s via LS-estimator and ow-pass interpoation and computing the channe matrix as shown in Fig. 3b. The second mode incudes the detection of data symbos by using successive data interference suppression SIS
6 H. HIJAZI AD L. ROS scheme with one tap frequency equaizer see Appendix E. A feedback technique is used between these two modes, performing iterativey ICI suppression and channe matrix estimation. In this iterative agorithm, the OFDM symbos are grouped in bocks of K OFDM symbos each one. Each two consecutive bocks are intersected in two OFDM symbos as shown in Fig. 3c. For a bock of K OFDM symbos, the iterative agorithm proceeds as foowing: 1: Y p k,1 = Y p k 2: for i = 1 : iteration do 3: α k,i LS 4: {ˆα k,i qt s], = M Y p k,i k=2,...,k 1 q= g,..., } = interpαk,i LS, + g 5: compute using 5 the channe matrix Ĥk,i 6: remove the ICI of piots from the received data subcarriers Y d k in 31 7: detection the data symbos ˆXd k,i using SIS 8: Y p k,i+1 9: end for = Y p k Ĥk,i p I ˆXk,i where iteration is the number of iterations, interp denotes the interpoation Matab function and, i and k represent the iteration number and the number of OFDM symbo in a bock, respectivey. ote that, the steps 3 to 6 are executed without considering the first and the ast OFDM symbos i.e.,k = 2 to K 1 in order to avoid imiting effects of interpoation. 3.4. Mean Square Error MSE Anaysis The MSE of the LS-estimator of α is defined by: = E α LS α H α LS α ] 14 which gives see Appendix D: = Tr M R 1 + σ 2 I p M H 15 where the expression of the covariance matrix R 1 is detaied in Appendix D. otice that if ICI are competey eiminated then, R 1 is matrix of zeros. Thus, 15 becomes: without ICI = σ 2 Tr M M H 16 In genera, depends on the piot positions and the mutipath deays. It is cear that our LS-estimator is unbiased. So, the CRAMER-RAO BOUD CRB 14] is an important criterion to evauate how good the LS-estimator can be since it provides the MMSE bound among a unbiased estimators. We have shown in Appendix C that the Standard CRB SCRB for the estimator of α with ICI known is given by: 1 1 SCRBα = SR Tr F H p X H p X p F p 17 where SR = 1 σ is the normaized signa to noise ratio. 2 It is easy to show that with ICI > SCRBα 18 without ICI = SCRBα So, by iterativey estimating and removing the ICI wi be coser to SCRBα. The MSE of the assumption that α LS is an estimation of α c is given by see Appendix D: mse c = E α LS α c H α LS α c ] = + mse 2 + 2 + mse 21 19 where mse 2 is defined in 12 and, 2 and mse 21 are the cross-covariance terms, which are very negigibe, given by see Appendix D: 2 = E α LS α H α α c ] = Tr R 2 M H mse 21 = E α α c H α LS α ] = mse 12 where the matrix R 2 is computed in Appendix D. The MSE of the mutipath compex gain estimator at time T s is defined by: mse Ts = with α k q K 1 E k=2 q= g ] ˆα k q α k q H ˆα k q α k q = α k 1qT s,...,α k L qt s ] T 2 For a arge vaue K, assuming performant interpoator and respecting samping theorem in time domain f d T.5, we wi have see Appendix D: mse Ts mse c 21 We now study the MSE of the mutipath compex gain estimator and the interpoation method versus the
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 7 MSE 1 4 1 5 mse int exact centra vaues mse Ts exact averaged vaues mse + mse int 2 Tabe 1. Parameters of Channe Rayeigh Channe P ath umber Average P owerdb ormaized Deay 1-7.219 2-4.219.4 3-6.219 1 4-1.219 3.2 5-12.219 4.6 6-14.219 1 1 6 1 2 3 4 5 6 Interpoation Window Length K Figure 4. MSE of the compex gain estimator in terms of K for f d T =.1 MSE mean square error.25.2.15.1.5 with unknown ICI theoretica mse c with unknown ICI theoretica mse Ts with unknown ICI theoretica with unknown ICI simu mse c with unknown ICI simu mse Ts with unknown ICI simu.1.5.75.1.2.3 f T d Figure 5. Comparison between MSE for SR = 2dB OFDM bock ength K. Fig. 4 gives the mse Ts with exact averaged vaues and the mse int with exact centra vaues for f d T =.1. We notice that the interpoation error mse int decreases with K, whie the error of the estimator mse Ts is constant whatever the interpoation window ength K. This is due to mse 2 MSE between the centra vaue α c and the averaged vaue α which is dominant with respect to the interpoation error mse int. Moreover, we verify that mse Ts mse int + mse 2. This means that the cross-covariance terms are very negigibe. So, in genera, we may say that the interpoation window ength K is not necessary to be arge and it suffices to choose K such that K.T = T coh = 1 f d i.e., T coh is the coherence time in order to have strong correation between sampes α n qt s. For exampe f d T =.1, T coh = 1T, so we choose K = 1. 4. SIMULATIO RESULTS In this section, we verify the theory by simuation and we test the performance of the iterative agorithm. mean square error 1 1 1 2 SCRB with known ICI with unknown ICI theoretica with unknown ICI simu after one iteration simu after two iterations simu after three iterations simu 1 3.1.5.75.1.2.3 f T d Figure 6. The MSE of the LS-estimator for SR = 2dB The mean square error MSE and the bit error rate BER performances in terms of the average signa-tonoise ratio SR 9] 8] and maximum Dopper spread f d T normaized by 1/T for Rayeigh channe are examined. The normaized channe mode is Rayeigh as recommended by GSM Recommendations 5.5 12] 13], with parameters shown in the tabe beow 1 T s = 2MHz. A 4QAM-OFDM system with normaized symbos, = 128 subcarriers, g = 8 subcarriers, p = 16 piots i.e.,l f = 8 and K = 1 OFDM symbos in each bock is used. note that SRdB = E b db + 3dB. These parameters are seected in order to have some concordance with the standard WiMax IEEE82.16e same spacing between subcarriers about 1KHz for a carrier frequency f c = 2.5GHz and same rate between the symbo duration and the guard time. The BER performance of our iterative agorithm is evauated under a reativey rapid time-varying channe such as f d T =.5 and f d T =.1 corresponding to a vehice speed V m = 14km/h and V m = 28km/h, respectivey, for f c = 5GHz. Fig. 5 shows the MSE in terms of f d T for SR = 2dB. It is observed that, with a ICI, the MSE obtained by simuation agrees with the theoretica vaue of MSE. We notice that the difference between mse Ts and mse c
8 H. HIJAZI AD L. ROS mean square error 1 1 1 1 2 1 3 1 4 1 5 SCRB with known ICI with unknown ICI theoretica with unknown ICI simu after one iteration simu after two iterations simu after three iterations simu 1 6 5 1 15 2 25 3 35 4 45 SR db Figure 7. The MSE of the LS-estimator for f d T =.1 BER 1 1 1 1 2 1 3 perfect knowedge of channe and ICI SIS agorithm with perfect channe knowedge after one iteration 1 4 after two iterations after three iterations inverse diagona with a ICI LS piot with LPI LMMSE piot with LPI 1 5 5 1 15 2 25 3 35 4 SR db a 1 Rea Part of Compex Gains Imaginary Part of Compex Gains 1 1 2 4 6 8 1 12 2 2 2 4 6 8 1 12 2 2 2 4 6 8 1 12.5.5 2 4 6 8 1 12.5.5 2 4 6 8 1 12.5.5 1 2 4 6 8 1 12 1.5 2 4 6 8 1 12 1 1 2 4 6 8 1 12 1.5 2 4 6 8 1 12 1 1 2 4 6 8 1 12 BER 1 1 1 2 1 3 perfect knowedge of channe and ICI SIS agorithm with perfect channe knowedge after one iteration 1 4 after two iterations after three iterations inverse diagona with a ICI LS piot with LPI LMMSE piot with LPI 1 5 5 1 15 2 25 3 35 4 SR db b.5 2 4 6 8 1 12.2.4 2 4 6 8 1 12 exact time average compex gain LS estimated time average compex gain exact compex gain estimated compex gain Figure 8. The LS estimated compex gain of six paths over 8 OFDM symbos after one iteration with SR = 2dB and f d T =.1 increases in terms of f d T. This is due to the interpoation error which increases with f d T. In short, we can say that mse Ts mse c and especiay for f d T.1, which means our method is adequate over a bock of K OFDM symbos. We verify that mse 2 is negigibe with respect to see Fig. 2 and 5 and especiay for f d T.2, thus mse c. Fig. 6 gives the evoution of with the iterations in terms of f d T for SR = 2dB. We notice that, with a ICI, is far from SCRB and when we commence to reduce the ICI, by improving the estimation of ICI at each iteration, shows a significant improvement especiay after the first iteration and approaches the SCRB for f d T Figure 9. Comparison of BER: a f d T =.5; b f d T =.1.1. However, by increasing f d T, we show from mse 2 given in Fig. 2 that α moves away from α c. Hence, for f d T >.1, MSE of compex gains estimator is significant and the ICIs are not estimated and nor removed perfecty. Fig. 7 shows the evoution of mse Ts with the iterations in terms of SR for f d T =.1. ote that the x- axis represents the time axis normaized with respect to the sampe time T s. After one iteration, a great improvement is reaized and is very cose to the SCRB especiay in ow and moderate SR regions. This is because at ow SR, the noise is dominant with respect to the ICI eve, and at high SR ICI is not competey removed due to the data symbo detection error. For iustration, Fig. 8 gives the rea and the imaginary parts of the exact and estimated after one iteration mutipath compex gain. This is done for one channe reaization over 8 OFDM symbos with SR = 2dB and f d T =.1. otice how good is the estimation of mutipath compex gains for rapidy changing channes.
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 9 1 1 1 perfect knowedge of channe and ICI SIS agorithm with perfect channe knowedge after one iteration after two iterations after three iterations inverse diagona with a ICI LS piot with LPI LMMSE piot with LPI 1 1 1 1 2 perfect knowedge of channe and ICI SIS agorithm with perfect channe knowedge after one iteration after two iterations after three iterations BER BER 1 3 1 2 1 4 1 5 1 3 8 16 32 Figure 1. Comparison of BER for f d T =.1 and SR = 2dB p Fig. 9 gives the BER performance of our proposed iterative agorithm, compared to conventiona methods LS and LMMSE criteria with LPI in frequency domain 6] 8] and SIS agorithm with perfect channe knowedge for f d T =.5 and f d T =.1. As reference, we aso potted the performance obtained with perfect knowedge of channe and ICI. This resut shows that, with a ICI, our agorithm performs better than the conventiona methods. Moreover, when we start removing ICI our iterative agorithm offers an improvement in BER after each iteration because the estimation of ICI is improved during each iteration. After two iterations, a significant improvement occurs; the performance of our agorithm and the SIS agorithm with perfect channe knowedge are very cose. At a high SR, it is norma to not reach the performance obtained with perfect knowedge of channe and ICI because we have an error foor due to the data symbo detection error. Fig. 1 gives the BER in terms of p for f d T =.1 and SR = 2dB. It is obvious that when using more piots, performance wi be better. Moreover, the resuts show that, with ess piots and without interference suppression, our agorithm performs better than the conventiona methods and becomes better with starting interference suppression. Fig. 11 shows the BER performance of our proposed iterative agorithm, for c = 2 and f d T =.1 with IEEE82.11a standard channe coding 18]. The convoutiona encoder has a rate of 1/2, and its poynomias are P = 133 8 and P 1 = 171 8 and the intereaver is a bitwise bock intereaver with 16 rows and 14 coumns. It can ceary be seen that a significant improvement in BER occurs with channe coding, and that for high SR there is aways an error foor due to data symbo detection errors. 1 6 5 1 15 2 25 3 35 4 SR Figure 11. Comparison of BER, in the case of the IEEE82.11a convoutiona code, for f d T =.1 5. COCLUSIO In this paper, we have anayzed an iterative agorithm to estimate mutipath compex gains and mitigate the intersub-carrier-interference ICI for OFDM systems. The rapid time-variation compex gains are tracked by expoiting that the deays are assumed invariant over severa symbos and perfecty estimated. Theoretica anaysis and simuation resuts of our iterative agorithm show that by estimating and removing the ICI at each iteration, mutipath compex gains estimation and coherent demoduation can have a great improvement especiay after the first iteration for high reaistic Dopper spread. Moreover, our agorithm performs better than the conventiona methods and its BER performance is very cose to the performance of SIS agorithm with perfect channe knowedge. A. RECEIVED OFDM SYMBOL From equations 1 and 3 the received signa of the nth OFDM symbo at the output of the ow pass receiver fiter g r t is given by: y n t = d= g L x n d]α tβt dt s nt τ T s + wt where βt = g t g r t. After A/D conversion and removing the cycic prefix, the received sampes are given by:
1 H. HIJAZI AD L. ROS 1 2 L d= g y n q] = y n t t=qts+nt = x n d]α n qt sβ q d τ T s + w n qts where q, 1]. Using 2, the sampes of the FFT output are given by: msed] 1 3 1 4 1 2 1 m= 2 β Y n k] = X n m] q= kq j2π e q= md j2π q d τ T s e kq j2π y n q]e = L d= g + W n k] α n qt s 22 where k 2, 2 1] and W n k] = kq j2π w n qt s e. q= As a resut, for τ < g, we have: g L u= L α n mu j2π qt s]β u τ T s e α n qtsg t m]g r m]e j2π m τ 23 otice that strict equaity woud hod if u varies from to +. Inserting 23 into 22 yieds the resuts given in equations 4 and 5. B. CLOSEST SAMPLE TO TIME AVERAGED COMPLEX GAI Since α t is wide-sense stationary WSS narrow-band compex Gaussian processes with the so-caed Jakes msed] theoretica msed] simu axis of symmetric d = 63.5 1 5 63.5 127 d Figure 12. msed] with = 128 and f d T =.1 power spectrum 1] then: ] E α q 1 T s α q 2 T s = σ 2α J 2πf d T s q 1 q 2 24 Using 24, we can cacuate msed] as: msed] = L σα 2 1 2 q= 2 q 1= q 2= J 2πf d T s q 1 q 2 J 2πf d T s q d + 1 To find the cosest α d to α, we need to find d min that minimizes msed]. By using the derivative formua of the Besse function defined as: J t = J 1 t, we can cacuate the derivative of msed] as: mse d] = 4πf dt s = 4πf dt s L L σα 2 q= σ 2 α d u= d J 1 2πf d T s q d J 1 2πf d T s u Let α d = α 1 dt s,...,α L dt s ] T be a vector of sampes 25 of the compex gains taken at the time position d Since J 1 t is an odd function then, the soution of, 1] during the effective duration of the OFDM the equation mse d] = is obtained when the interva symbo. The MSE between α and α d is defined as: msed] = E α α d H α α d ] of the index u in 25 is centered at zero, thus d = = 2 1 2 not integer. It is easy to show that msed] is L ] symmetric with respect to d = 2 1 2 axis then, d min = E α α α α dt s α dt s α + α dt s α dt s 2 1 or 2. We denote the minimum of msed] by mse 2 = msed min ]. For iustration, Fig. 12 gives the curve of msed] theoretica and monte-caro simuation for = 128 and f d T =.1. It is we observed that d = 63.5 is an axis of symmetric and d min = 63 or 64.
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 11 C. CRB FOR THE ESTIMATOR OF α Assuming ICI p = H pi X in 8 are known then, the vector Y p for a given α is compex Gaussian with mean vector m = X p F p α + ICI p and covariance matrix S 1 = σ 2 I p. Thus, the probabiity density function p Y p α is defined as: p Y p α 1 = e 1 2Y p m H 1 S 1 Y p m 2πS 1 CRB 1 1 1 1 2 1 3 1 4 1 5 SCRB with known ICI BCRB with known ICI Since α is a compex Gaussian vector with zero mean and covariance matrix S 2 then, the probabiity density function of α is defined as: p α = 1 e 1 2πS 2 2 αh S 2 1 α where S 2 is a diagona matrix of eements S 2,], with 1,L], given by: S 2,] = E α α ] = σ2 α 2 q 1= q 2= The Standard CRB SCRB and the Bayesian CRB BCRB for the estimator of α are defined as 14]: SCRBα = Tr E 2 n p Y α 2 p α ] 1 BCRBα = Tr E 2 n p Y α 2 p,α ] 1 26 where p Y p,α = p Y p α p α is the joint probabiity density function of Y p and α and, the expectation is taken over Y p and α. otice that SCRB and BCRB are for the estimation of deterministic and random variabes, respectivey. The resuts of the second derivatives of n p Y p α and n p Y p,α with respect to α are given by: 2 α 2 n p Y p α = F p H X p H S 1 1 X p F p 27 2 α 2 n p Y p,α = F p H X p H S 1 1 X p F p S 2 1 1 6 5 1 15 2 25 3 35 4 45 5 SR Figure 13. SCBR and BCRC with = 128, p = 16 and f d T =.1 We notice that in our specific probem SCRB is independent of α. So, SCRB gives the ower bound if the priori distribution of α is not used in the estimation method, whereas BRCB takes this information into account. For iustration, Fig. 13 gives the SCRB and BCRB in terms of SR for the channe given in Tabe 1, = 128, p = J 2πf d T s q 1 q 2 16 and f d T =.1. It is observed that there is a sma difference between SCRB and BCRB at ow SR. So, we can compare the MSE of our LS-estimator of α to SCRB instead of BCRB. Moreover, with known ICI, the optima estimators of deterministic α and random Gaussian α are LS-estimator and maximum ikeihood ML estimator, respectivey. In our agorithm, the LS-estimator was used considering α deterministic because it requires ess information compared to ML-estimator. D. MEA SQUARE ERROR OF COMPLEX GAIS ESTIMATOR The MSE of the LS-estimator of α is given by: = E α LS α H α LS α ] = Tr M R 1 + σ 2 I p M H ] where R 1 = E H pi X X H H H pi and the expectation is taken over the data symbos, the noise and the compex gains, since the noise and the ICIs are uncorreated. The term ICI p = H pi X can be written as the sum of two Hence, substituting 27 in 26 yieds: components: 1 SCRBα = σ 2 Tr F H p X H p X p F p ICI p = H pp X p + H dd X d 1 1 BCRBα = Tr σ 2 F p H X H 1 where X d is the data symbos and, H pp and H dd are a p X p F p + S 2 p p and a p p matrices, respectivey, of
12 H. HIJAZI AD L. ROS p p 1 R pp k,m] = E = 1 R dd k,m] = E = u 1=p u 1 k p p 1 p p 1 u 2=p u 2 m p p 1 Xu 1 ]X u 2 ]Hk,u 1 ]H m,u 2 ] L 2 u 1=p u 2=p u 1 k u 2 m 2 1 u= 2 u p s L σα 2 Hk,u]H m,u] δ k,m p 2 σα 2 Xu 1 ]X u 2 ]e j2π u 1 u 2 τ 1 q1 q2 e j2π kq 1 mq 2 q 1= q 2= q 1=q 2= e j2π u 1 kq 1 u 2 mq 2 J 2πf d T s q 1 q 2 J 2πf d T s q 1 q 2 δ,q1 q 2 p 28 p p 1 R 2,k] = E = σ2 α 2 u=p u k p p 1 u=p u k X u]h k,u] α α 2 T s] X u]e j2π u τ q 1= q 2= e j2π u kq 1 J 2πf d T s q 1 q 2 J 2πf d T s q 1 2 29 eements given by: H pp k,m] = { Hk,m] if k,m P if k = m H dd k,m] = Hk,m] if k P, m 2, 2 1] P Hence, the matrix R 1 becomes: R 1 = R pp + R dd ] where R pp = E H pp X p X H H p H pp and R dd = ] E H dd X d X H H d H dd, since the data symbos and the coefficients Hk,m] are uncorreated. Since the data symbos are normaized i.e. E H X d X ] ] H d = I p then, R dd = E H dd H dd. Thus, the eements R pp k,m] and R dd k,m], with k, m P, are given by 28, shown at the top of the present page. The MSE of the assumption that α LS is an estimation of α c is given by: mse c = E α LS α c H α LS α c ] = + mse 2 + 2 + mse 21 where 2 and mse 21 are the cross-covariance terms given by: 2 = E α LS α H α α c ] = Tr R 2 M H mse 21 = E α α c H α LS α ] = mse 12 ] where R 2 = E α α c X H H H pp is a L p matrix of eements R 2,k], with 1,L] and k P, given by 29, shown at the top of the present page. otice that the eements of the matrix R 1 and R 2 depend on the known piot symbos and the mutipath deays.
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 13 The MSE of the mutipath compex gain estimator at time T s is defined by: mse Ts = K 1 E k=2 q= g ] e k H q e k q where e k q = ˆαk q αk q, with αk q = α k 1 qt s,...,α k L qt s ] T, is the error of the mutipath compex gain estimator at time T s. This error can be written as the sum of two errors: e k q = e k c + e int k where e k c is the error due to the assumption that α LS is an estimation of α c and e int k is the error due to the interpoation method which depends on the number K of OFDM symbos in each boc and the term f d T. For a arge vaue K, assuming performant interpoator and respecting samping theorem in time domain i.e. f d T.5, we wi have e int k,...,] T, and then: mse Ts mse c E. SUCCESSIVE ITERFERECE SUPPRESSIO METHOD The received data subcarriers are given by: Y d = H d X d + H d X p + W d 3 where X d the data transmitted, Y d the data received and W d the noise at data subcarrier positions are p 1 vectors, H d and H d are a p p and p p matrices, respectivey, of eements given by: H d k, m] = Hk, m] if k, m 2, 2 1 P H dk, m] = Hk, m] if k 2, 2 1 P, m P ote that in 3, the first component is the desired data term with ICI due to data symbos and the second component is the ICI term due to piot symbos. By successive interference suppression SIS scheme with optima ordering and one tap frequency equaizer the data wi be estimated. The optima ordering is cacuated from the arge to the sma magnitude of the diagona eements of the data channe matrix H d and given by: O = { O 1, O 2,..., O p } i < j if H d O i,o i ] > H d O j,o j ] The detection agorithm can now be described as foows: } 1: O = {O 1, O 2,..., O p 2: Y d1] = Y d = Y d H d X p 3: for i = 1 : p do 4: Xe d O i ] = Y di] O i]/h d O i,o i ] 5: ˆXd O i ] = QXe d O i ] 6: Y di+1] di] d O i ]H d Oi 7: end for where Y d is the received data subcarriers without contribution from piot subcarriers, Q. denotes the quantization operation appropriate to the consteation in use and H d Oi denotes the O i th coumn of the data channe matrix H d. ote that in our agorithm, we have used the minimum distance criterion as quantization method. REFERECES 1. H. Hijazi and L. Ros, Time-varying Channe Compex Gains Estimation and ICI Suppression in OFDM Systems in IEEE GLOBAL COMMUICATIOS Conf., Washington, USA, 27 to be appeared. 2. H. Hijazi, L. Ros and G. Jourdain, OFDM Channe Parameters Estimation used for ICI Reduction in time-varying Mutipath channes in EUROPEA WIRELESS Conf., Paris, FRACE, Apri 27. 3. E. Simon, L. Ros and K. Raoof, Synchronization over rapidy timevarying mutipath channe for CDMA downink RAKE receivers in Time-Division mode,in IEEE Trans. Vehicuar Techno., vo. 56. no. 4, Ju. 27 4. E. Simon and L. Ros, Adaptive mutipath channe estimation in CDMA based on prefitering and combination with a inear equaizer,14th IST Mobie and Wireess Communications Summit, Dresden, June 25. 5. A. R. S. Bahai and B. R. Satzberg, Muti-Carrier Dications: Theory and Appications of OFDM: Kuwer Academic/Penum, 1999. 6. M. Hsieh and C. Wei, Channe estimation for OFDM systems based on comb-type piot arrangement in frequency seective fading channes in IEEE Trans. Consumer Eectron., vo.44, no. 1, Feb. 1998. 7. O. Edfors, M. Sande, J. -J. van de Beek, S. K. Wison, and P. o. Brejesson, OFDM channe estimation by singuar vaue decomposition in IEEE Trans. Commun., vo. 46, no. 7, pp. 931-939, Ju. 1998.
14 H. HIJAZI AD L. ROS 8. S. Coeri, M. Ergen, A. Puri and A. Bahai, Channe estimation techniques based on piot arrangement in OFDM systems in IEEE Trans. Broad., vo. 48. no. 3, pp. 223-229 Sep. 22. 9. B. Yang, K. B. Letaief, R. S. Cheng and Z. Cao, Channe estimation for OFDM transmisson in mutipath fading channes based on parametric channe modeing in IEEE Trans. Commun., vo. 49, no. 3, pp. 467-479, March 21. 1. W. C. Jakes, Microwave Mobie Communications. Piscataway, J: IEEE Press, 1983. 11. R. Roy and T. Kaiath, ESPRIT-Estimation of signa parameters via rotationa invariance techniques in IEEE Trans. Acoust., Speech, Signa Processing, vo. 37, pp. 984-995, Juy 1989. 12. European Teecommunications Standards Institute, European Digita Ceuar Teecommunication System Phase 2; Radio Transmission and Reception, GSM 5.5, vers. 4.6., Sophia Antipois, France, Juy 1993. 13. Y. Zahao and A. Huang, A nove Channe estimation method for OFDM Mobie Communications Systems based on piot signas and transform domain processing in Proc. IEEE 47th Vehicuar Techno. Conf., Phonix, USA, May 1997, pp. 289-293. 14. H. L. Van Trees, Detection, estimation, and moduation theory: Part I, Wiey, ew York, 1968. 15. Z. Tang, R. C. Cannizzaro, G. Leus and P. Banei, Piot-assisted time-varying channe estimation for OFDM systems in IEEE Trans. Signa Process., vo. 55, pp. 2226-2238, May 27. 16. S. Tomasin, A. Gorokhov, H. Yang and J.-P. Linnartz, Iterative interference canceation and channe estimation for mobie OFDM in IEEE Trans. Wireess Commun., vo. 4, no. 1, pp. 238-245, Jan. 25. 17. M. K. Ozdemir and H. Arsan, Channe Estimation for Wireess OFDM Systems, IEEE Communications Surveys and Tutorias, vo. 9, pp. 18-48, Issue: 2, Second Quarter 27. 18. Y. Tang, L. Qian, and Y. Wang, Optimized software impementation of a fu-fate IEEE 82.11a compiant digita baseband transmitter on a digita signa processor in IEEE GLOBAL Teecommun. Conf., vo. 4, ov. 25. 19. P. Hoher, S. Kaiser and P. Robertson, Piot-symbo-aided channe estimation in time and frequency in Proceedings of IEEE Goba Teecommunications Conference, Communication Theory Mini Conference, pp. 996, Phoenix, USA, ov. 1997.
TIME-VARYIG CHAEL COMPLEX GAIS ESTIMATIO AD ICI CACELLATIO 15 AUTHORS BIOGRAPHIES Hussein Hijazi received the Ph.D. degree in signa processing and communications from the Institut ationa Poytechnique de Grenobe IPG, Grenobe, France, in 25 ovember 28, where he is currenty an Associate Professor. His dissertation focused on channe estimation in a high speed mobie receiver operating in an OFDM communication system. Prior to earning his MASTER Signa, Image, Speech, Teecom from IPG in 25, he was awarded the Dipoma in computer and communications engineering from the Lebanese University Facuty of Engineering, Beyrouth, Lebanon, in 24. His current research interests ie in the areas of signa processing and communications, incuding synchronisation, channe estimation and equaization agorithms for wireess digita communications. Laurent Ros received the degree in eectrica engineering from the Écoe Supérieure d Éectricité Supéec, Paris, France, in 1992 and the Ph.D. degree in signa processing and communications from the Institut ationa Poytechnique de Grenobe IPG, Grenobe, France, in 21. From 1993 to 1995, he was with France-Teecom R & D center, Lannion, France, where he worked in the area of very ow frequency transmissions for submarine appications, in coaboration with Direction des Constructions avaes, Touon, France. From 1995 to 1999, he was a Research and Deveopment Team Manager at Sodieec, Miau, France, where he worked in the design of digita modems and audio codecs for teecommunication appications. Since 1999, he has joined the Gipsa-ab/DIS ex Laboratory of Image and Signa, IPG, where he is currenty an Associate Professor. His genera research interests incude synchronisation, time-varying channe estimation and equaization probems for wireess digita communications.