Expectation-Maximization (EM) Based Channel Estimation and Interpolation in OFDM Systems

Transcription

1 Expectation-Maximization (EM) Based Channel Estimation and Interpolation in OFDM Systems Eser Ustunel National Research Institute of Electronics & Cryptology, TUBITAK, Kocaeli, Turkey Department of Electrical and Electronics Engineering, Yeditepe University, Istanbul, Turkey Abstract Channel estimation and tracking methods are very critical parts of wireless Orthogonal Frequency Division Multiplexing (OFDM) systems in time and frequency selective communication channels. Interpolation has been popularly used in both domains when the channel is sampled with pilot subcarriers. The classical approach is to use the Least Square (LS) estimations to predict the channel for the pilot locations and then use Maximum Likelihood Estimation (MLE) and Minimum Mean Square Error Estimation (MMSEE) schemes to improve these initial estimations and interpolate for deriving a complete characterization of the channel. In this paper, in comb-type pilot arrangement, we propose to use the Expectation-Maximization (EM) algorithm to complement the LS scheme and propose two different approaches: (i) Using a sliding window approach based on EM estimators to interpolate, (ii) Using EM estimations on pilots instead of LS to enhance the MLE and MMSEE performance. Additionally, the necessary condition for EM to work properly is shown in the paper. Index Terms Channel estimation, tracking, and interpolation, EM Algorithm, LSE, MLE, MMSEE, OFDM. I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has been used in wireless communication systems due to its robustness to the multipath environment for high data rate communications. It has been used for various recent broadband communication standards such as IEEE a/g (WiFi), IEEE (WiMAX), and Long Term Evolution (LTE). In a real communication environment, reflected and scattered transmitted signals arrive at the receiver along multiple paths that cause frequency selective channel. Also, caused by the mobility of transmitters, scattered objects, or receivers, the channel can change fast in time. Therefore, a dynamic estimation and tracking of Channel Frequency Response (CFR) is necessary for coherent detection of OFDM symbols. In OFDM systems, general approach is using non-blind estimation techniques by pilots, which are known by the receiver, for fast fading channels. Generally, there are two types of pilots insertion methods, which are block-type and combtype pilot arrangements. In the block-type arrangement, all subcarriers are pilots that construct an OFDM training symbol. This type is considered for slow time variation of the channel. In the comb-type pilot arrangement, pilots are inserted among the data subcarriers with a proper consideration of time and frequency selectivity of the channel. In this type, channel is first estimated on the pilots and then interpolated between pilots to calculate the response over the data carriers. Different channel interpolation techniques are reviewed for comb type pilot arrangement in [1], where Low Pass (LP) filtering is offered to be the best interpolator. Maximum Likelihood Estimation (MLE) and Minimum Mean Square Error Estimation (MMSEE) are analyzed in detail for channels that have exponentially decaying power delay profiles in [2], which shows MLE achieves Cramer-Rao Lower Bound (CRLB) and MMSEE is a better estimator since it uses channel and noise statistics. Both techniques estimate the channel in time domain. Optimal rank reduction, which is less complex and robust to changes in channel correlation, to Linear MMSEE is offered in [3]. Adaptive Wiener filtering for actual channel characteristics is presented in [4]. Expectation-Maximization (EM) algorithm offers a lowcomplexity iterative approach based on likelihood maximization to estimate the channel. EM algorithm based LS polynomial fitting is offered in [5]. EM algorithm is showed to be robust to multipath channel delay profile variations with a high number of iterations for convergence in [6] for OFDM with transmitter diversity. Frequency and time domain EM based estimators are analyzed in [7]. A compact and explicit formula for the EM algorithm is driven in [8] and [9] for channel and noise variance estimations requiring a small number of iterations. Turbo coding output is used to feedback the EM algorithm iteratively in [9] to estimate one channel coefficient for a block of data with an assumption that channel is slowly changing within the block. Initial channel estimation, which is popularly done with Least Squares (LS) estimation, has much importance since it affects the subsequent estimation-interpolation stages e.g. MLE or MMSEE. Especially, when the pilot carrier is in deep fading, performance of MLE and MMSEE deteriorate. Therefore EM is a good candidate to mitigate the poor performance results of LS. In this paper, we develop an EM estimator on the pilot subcarriers using LS estimates for M-ary QAM and PSK modulation. EM estimation, which works on the neighborhood of a pilot subcarrier, finds one complex channel coefficient for that pilot. Size of the neighborhood window depends on the coherence bandwidth of the channel. Using the EM estimations on pilots, we propose two methods for channel estimation and interpolation: (i) An EM based interpolation by sliding the window, whose length is about the distance between pilots, around each pilot carrier can successfully interpolate the whole channel. At each slided window, the EM algorithm estimates one channel coefficient. (ii) Generally, LS is used as initial /12/$ IEEE

2 estimates for complex and efficient algorithms such as MLE and MMSEE in [1]. We propose a cascaded system, in which EM estimations instead of LS estimations on the pilots enter to MLE or MMSEE. This joint system improves channel estimation performance significantly. In addition, the necessary condition for EM algorithm is derived. By assuming a quasi-static channel with exponentially decaying power delay profile, simulation results show that EM based interpolation using sliding window performs better than both of the ML and MMSE estimators, which use LS estimates on the pilots. Moreover, cascaded joint channel estimation system, in which ML and MMSE estimators use EM estimates on the pilots, outperforms EM based interpolation. II. SYSTEM MODEL The baseband model of the system is shown in Fig.1. At the transmitter, N p pilot symbols and N N p data modulation symbols generate N subcarriers, X(k), in the frequency domain and k = 0, 1,.., N 1. Before entering Inverse Discrete Fourier Transform (IDFT), N p pilot symbols are evenly located between data symbols according to the combtype pilot arrangement, therefore pilot spacing is d = N/N p. Π(η) is an N p -by-1 pilot vector and its pilot subcarriers are located at k p = d.η, where η = 0, 1, 2,.., N p 1. OFDM sampling time is T s and total OFDM time is T ofdm = NT s, therefore subcarrier spacing is f = 1/T ofdm and pilot spacing is d f = f s, which is also a sampling rate of OFDM signal in frequency. Applying Nyquist sampling theorem to the signal, which is defined in the frequency domain, we get the popular Nyquist inequality 1 = NT s = N p T s LT s (1) f s d where L is maximum excess delay in terms of number of multitaps. LT s is the time bandwidth of the channel, h(n), which is causal and truncated at L i.e. h(n) = 0 for n < 0 and n L with n = 0, 1,.., N 1. Accordingly, number of pilots N p should be greater than or equal to L i.e. N p L, so that the variations of the channel in the frequency domain can be all captured. Frequency selective Channel Impulse Response (CIR) can be given as L 1 h(n) = e (n+1)/τ h n δ(n λ n ) (2) where λ n is the n th path delay normalized by the OFDM sampling time, T s. h n is independent zero-mean, unit-variance complex Gaussian random variable. Assuming static channel in time within an OFDM block, Inter-Carrier Interference (ICI) is avoided. The channel has an exponential decaying power delay profile with τ, which is used to normalize the channel impulse response energy to 1. N-point Discrete Fourier Transform (DFT) of the channel is H(k) = N 1 h(n)e j2πnk/n. (3) Fig. 1. Baseband OFDM system model After IDFT operation, OFDM signal is appended by cyclic prefix, whose length is N cp i.e. N cp L 1, and passes through the channel. The received signal is y(n) = x(n) h(n) + w(n) (4) where denotes linear convolution. Zero mean, complex Additive White Gaussian Noise (AWGN) is added to the faded signal in the receiver. After removing cyclic prefix and taking DFT of received signal, we have Y (k) = X(k)H(k) + W (k). (5) Basically, initial channel estimates are done on the known pilot symbols. Least Squares (LS) estimates of the channel are Ĥ LS (k p ) = Y (k p) X(k p ). (6) These noisy LS estimates are used as a basis step for the other more complex and powerful estimation algorithms. III. EXPECTATION-MAXIMIZATION (EM) ALGORITHM The EM estimation algorithm based on the Maximum Likelihood (ML) criterion is applied to estimate the complex channel coefficient. Basically, EM algorithm finds a channel coefficient that maximizes the expectation function Q(H Ĥq ) = ln[f(y X, H)]f(X Y, Ĥq ) dx (7) as a multiple integral in [9]. f(y X, H) is the vector likelihood function, which is the product of f(y k X k, H). K is the EM filter window, whose length is κ i.e. κ = d + 1, given as K = {k : k p d/2 k k p + d/2}. (8) The interpretation of Q(H Ĥq ) is that Ĥ q, which is the estimate of the channel at q th iteration, is the given information entering to the EM algorithm and H is the actual channel coefficient to be estimated by using the EM operation. For q = 0, Ĥ 0 = ĤLS is the initial estimate as in (6). The final iteration number is q fin = 2, which is enough to converge to the actual channel value. Therefore, Ĥ 2 is our final EM estimator. The window K is used for estimating the channel, H, on the pilot at k p. The reason why the window length, κ, is one more than the pilot spacing is to put the pilot in the middle of the filter window at k p i.e. equal number of left and right subcarriers of a pilot and including the pilot itself are used in /12/$ IEEE

3 the calculations of the EM estimator. We assume channel is slowly changing within d/2 subcarriers. By using the window (8), EM algorithm is applied for the pilot at k p to have better estimation than LS. A. Expectation Step At the k th subcarrier, the likelihood function of the channel parameter is f(y k X k, H) = 1 exp ( Y k HX k 2 ) πn 0 N 0 = 1 exp( Y k 2 + H 2 X k 2 2Re{Yk HX k } ) πn 0 N 0 where denotes conjugate of a complex number. Substituting (9) in (7), we get Q(H Ĥq ) = (9) [ κ ln(πn0) 1 Y k 2 (10) N 0 + H 2 X k 2 2Re{Y k HX k }] f(x k Y k, Ĥq ) dx k. Integration of the term H 2 X k 2 is H 2 X k 2 f(x k Y k, Ĥq ) dx k = H 2 X k 2 f(x k Y k, Ĥq ) dx k k = H 2 E[ X k 2 ] = H 2 X k 2 (11) since all integrations are 1 for k k. Similarly, integration of the term 2Re{Yk HX k } is 2Re{Yk HX k } f(x k Y k, Ĥq ) dx k = = 2Re{Y k HX k } (12) 2Im { Y k H } Im{X k } 2Re { Y k H } Re{X k } where E[X k ] = X k = X k kf(x k Y k, Ĥq ) dx k. The last raw of (12) will be used in complex derivatives in the following steps. Finally, the overall equation for (10) is Q(H Ĥq ) = κ ln(πn 0 ) 1 Y k 2 N 0 1 N 0 H 2 X k 2 2Re{Y k HX k } (13) where the first row of (13) does not depend on the channel parameter H. Therefore, derivative of the row with respect to H is zero. B. Maximization Step Maximization of the expectation, Q(H Ĥq ), can be done, if we take complex derivative of (13) with respect to H and set it to zero dq(h Ĥq ) dh = 1 N 0 H X k 2 Y k X k = 0. (14) Therefore, the EM channel estimator is at pilot, k p, Ĥ EM = 1 ( ) Y k Xk. (15) κ X k 2 This EM estimator can be used as a final estimator or can be inserted into the EM algorithm for an other iteration as in Fig.1. If we consider (15), Bayesian mean values of the data subcarriers around the pilot are used for noise averaging. Using the identity f(x k Y k ) = P r(x k )f(y k X k )/f(y k ), we get X k X k 2 = M i=1 X iexp( Y k Ĥq X i 2 /N 0 ) M i=1 X i 2 exp( Y k Ĥq X i 2 /N 0 ) (16) for the calculation of (15). X k is a modulation symbol and P r(x k ) = 1/M, where M is the number of different symbols in a modulation. For PSK modulation, we have X k 2 = X k 2 = 1. Considering (15) and (16), the important advantage of the EM algorithm is that it requires only probability density function of the noise. For zero mean AWGN case, only noise variance N 0 is required. C. Working Conditions of EM Algorithm There is a requirement for the initial channel estimates (6), which are going to be used in the EM algorithm. Consider the exponential term in the likelihood function i.e. exp( Y XĤLS 2 /N 0 ), where Y = XH+W. To maximize the likelihood, we need to maximize the exponential term exp ( [ X 2 ( H 2 + ĤLS 2 2Re{HĤ LS}) + W o ] /N0 ) (17) where W o is the other noise terms. Considering (17), it is maximized when Re{HĤ LS} = Re{H}Re{ĤLS} +Im{H}Im{ĤLS} > 0 (18) If the initial LS estimates are too noisy to cause Re{HĤ LS } < 0, Ĥ EM estimation converges to H instead of H. However, power of the EM estimation is that even if Re{ĤLS} has wrong sign comparing with Re{H}, larger Im{H}Im{ĤLS} term can compensate Re{H}Re{ĤLS} and after the iterations ĤEM converges to H, or vice versa. Small EM window size (small pilot spacing) decreases converging to H effect over the small number of data subcarriers. Also note that EM algorithm only affects a pilot individually since we have different LS estimates for the other pilots /12/$ IEEE

4 D. EM Based Interpolation We propose a new interpolation technique, which uses EM estimators in (15). We are going to use sliding EM window, which spans all frequency axis in the end by shifting window one subcarrier by one subcarrier. EM estimation is done for the pilot at location k = k p, then we shift the window left and then right (order is not important) K new = {k : k p d/2 k s k k p + d/2 k s } (19) where k s is the shift amount and maximum shift amount is d/2. minus sign in is for left shift and plus sign is for right shift. New EM estimation at the location k new = k p k s will be ĤEM (k new ) by using the new window. Ĥ EM (k p (k s 1)) is used for the initial estimate for the new shifted window since channel is correlated in the frequency domain. By this approach, we interpolate the neighborhood of each pilot that results in interpolation of all the frequency axis. IV. ML AND MMSE ESTIMATORS In this section Maximum Likelihood (ĤML) and Minimum Mean Square Error (ĤMMSE) estimators are given for the performance comparison with EM estimator. A. Maximum Likelihood (ML) Estimator The procedure for ML estimator is that: After zero padding between the LS estimates of pilots, we take IDFT and use an ideal Low Pass (LP) filter in time domain, whose pass band length is L for the positive time points since CIR is causal i.e. h(n) = 0 for n < 0. Timewidth of the LP filter is T W LP F = L since h(n) = 0 for n L. After LP filtering, we get Maximum Likelihood estimator, Ĥ ML, which is the same result with Fourier Transform of (13) in [2]. B. Minimum Mean Square Error (MMSE) Estimator Wiener Filter is used for MMSE estimations at the pilot subcarrriers and for interpolation at the data subcarrriers. Since CIR tap coefficients are zero mean complex Gaussian random variables, autocorrelation matrix of the Channel Frequency Response (CFR) is R(k, k ) = E[HH ] N,N (20) where denotes the Hermitian of a vector and H = [H(0)H(1)...H(N 1)] and k, k = 0, 1, 2,.., N 1. By using equations (2) and (3) with choosing λ n = n, (20) becomes L 1 R(k, k ) = 2(n + 1) exp[ τ + j2πn(k k) ]. (21) N Similarlay, autocorrelation matrix of the pilots is R Π (η, η ) = E[ΠΠ ] Np,N p (22) which is actually a 2 dimensional sampling of the matrix R with the intervals of pilots spacing. To interpolate the channel in the frequency domain, we need to find the cross correlation vectors of each sub-carrier with the pilots vector, Π, as P k = E[ΠH(k) ] Np,1. (23) Fig. 2. MSE vs SNR (db) for the channel only on N p = 64 pilot subcarriers Then, we can calculate the over all cross correlation matrix for all subcarriers as P = [P 0 P 1..P k..p N 1 ] Np,N. (24) For each subcarrier, Wiener filter coefficients are W Np,N = [R Π + N 0 I] 1 N p,n p P (25) where I is an N p -by-n p identity matrix. W (:, k) is the optimum weight column-vector for the k th subcarrier in the mean square error sense. Obtaining W is an offline process i.e. it is required to calculate once and can be used for other OFDM symbols. Wiener interpolation can be done as Ĥ MMSE = [ĤLS] t W (26) where t denotes transpose of a matrix. C. Improvement of ML and MMSE Estimators In general approach, (6) is used to implement ML and MMSE estimators. We prose that EM estimators (15) can be used for the basis of ML and MMSE estimators instead of LS estimates. By this way, the performance of the ML and MMSE estimators are improved. For ML estimator, EM algorithm based pilot estimates are zero padded between pilot locations as in the LS estimation based case. After IFFT and Low Pass filtering in time domain with the timewidth L, we have ĤML,em. Similarly, for EM based MMSE estimator Wiener coefficients are W em = [R Π + N 0 κ I] 1 N p,n p P (27) where noise variance is reduced by κ since we use EM estimator in (15). Accordingly, MMSE sense channel interpolater becomes Ĥ MMSE,em = [ĤEM ] t W em. (28) /12/$ IEEE

5 Fig. 3. MSE vs SNR (db) for the interpolated channel high SNR regime MLE performance converges to MMSEE since noise variance goes to zero in [2]. The proposed EM interpolation, by shifting EM window left and right, performance is better even than MMSEE ls in Fig.3. At MSE = 10 2, SNR gain is 5dB and 3dB comparing the MLE ls and MMSEE ls, respectively. The power of the EM estimation is subtle noise reduction, which calculates the Bayesian mean of the each subcarrier in the window in (15). It does not require maximum excess delay L, but it needs to know noise variance. We also give perfect EM scenario if we assume all subcarrier symbols are known instead of using their mean values. EM estimations on the pilot carriers can be used for MLE or MMSEE that is two estimation blocks are cascaded as in Fig.1. Since EM algorithm is an effective estimation, with using MLE or MMSEE jointly, performance increases much more. At MSE = 10 2, SNR gain is 9dB and 8dB comparing the MLE em -MLE ls and MMSEE em -MMSEE ls, respectively in Fig.3. V. SIMULATION RESULTS In our simulations, Binary Phase Shift Keying (BPSK) is used for modulation symbols with M = 2. Number of subcarriers is N = 1024 and number of uniformly distributed pilots is N p = 64, thus pilot spacing is d = N/N p = 16. Number of multitaps is L = 16 in the channel impulse response. Therefore required number of pilots is high enough for interpolation N p = 64 L = 16. Estimation window length is κ = d + 1 = 17 for EM algorithm. Initial estimates from LS estimates enter to the EM algorithm, which uses a κ length window K, for channel estimation on the pilot carriers. This EM estimation is inserted again into the EM algorithm and finds second EM estimation. Two EM iterations are enough since EM algorithm quickly converges to H. In Fig.2, performance of the EM estimator on pilots is given for the comparison with Least Squares (LS) estimation. At Mean Square Error MSE = 10 2, Signal-to-Noise Ratio (SNR) gain is 11dB for EM comparing the LS estimates on pilots. That shows EM algorithm significantly reduces the noise on pilots. We also give perfect EM scenario if we assume all data symbols around the pilot i.e. in the K neighborhood are known instead of using their mean values in (16). This case is when data subcarriers are perfectly obtained from a perfect error control coding. Note that for high SNR regimes, EM performance converges to the perfect EM performance. To estimate the channel frequency parameters on the data subcarriers, we use EM based sliding windows for channel interpolation. Similar processes are applied by shifting the window for a pilot to a new window K new for data symbols. For the new window, required initial estimate value comes from previous window s last EM iteration. In Fig.3, MMSEE ls is the MMSE estimator and MLE ls is the ML estimator that they are using LS for initial estimates. MMSE estimations are better than ML since it uses channel statistics and noise variance information (25). For both estimators, maximum excess delay L is required to be known. It is seen that for VI. CONCLUSION In this paper, EM based channel estimation and interpolation algorithm is derived for M-ary modulation symbols in OFDM systems. EM improves LS estimations significantly on the pilots, which are in comb-type arrangement. By sliding the EM window in the frequency axis, EM successfully interpolates the whole channel. Cascaded channel estimation-interpolation system is also proposed i.e. EM pilot estimates are used for ML or MMSE based interpolation. The cascaded system s gain can be reached to 9dB. EM dynamics are analyzed and the necessary working condition is given. Since only required parameter in EM is noise variance, noise variance estimation can be added to the system as a future work. REFERENCES [1] Coleri, S.; Ergen, M.; Puri, A.; Bahai, A., Channel estimation techniques based on pilot arrangement in OFDM systems, IEEE Trans. Broadcasting, vol.48, no.3, pp , Sep [2] Morelli, M.; Mengali, U., A comparison of pilot-aided channel estimation methods for OFDM systems, IEEE Trans. Signal Processing, vol.49, no.12, pp , Dec [3] Edfors, O.; Sandell, M.; van de Beek, J.J.; Wilson, S.K.; Borjesson, P.O., OFDM channel estimation by singular value decomposition, IEEE Trans. Commun., vol.46, no.7, pp , Jul [4] Necker, M.; Sanzi, F.; Speidel, J., An Adaptive Wiener-Filter for Improved Channel Estimation in Mobile OFDM-Systems, in Proc. IEEE Int. Symposium on Signal Processing and Information Theory, pp , Dec [5] Ma, X.; Kobayashi, H.; Schwartz, S.C., EM-based channel estimation for OFDM, in Proc. IEEE Pacific Rim Conf. Commun., Computers and Signal Processing, vol.2, pp , 2001 [6] Xie, Y.; Georghiades, C.N., An EM-based channel estimation algorithm for OFDM with transmitter diversity, in Proc. IEEE GLOBECOM, vol.2, pp , 2001 [7] Al-Naffouri, T.Y., An EM-Based Forward-Backward Kalman Filter for the Estimation of Time-Variant Channels in OFDM, IEEE Trans. Signal Processing, vol.55, no.7, pp , July 2007 [8] Cheng, S.; Valenti, M.C.; Torrieri, D., Turbo-NFSK: iterative estimation, noncoherent demodulation, and decoding for fast fading channels, in Proc. IEEE MILCOM vol.5, pp , Oct [9] Torrieri, D.; Ustunel, E.; Kwon H.; Min, S.; Kang, D.H., Iterative CDMA Receiver with EM Channel Estimation and Turbo Decoding, in Proc. IEEE MILCOM, pp.1-6, Oct /12/$ IEEE