Robust MC-CDMA Channel Tracking for Fast Time ... - IEEE Xplore

4658

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 9, NOVEMBER 2010

LMMSE and MMSE-DFE have been proposed, where the estimation of the state vector (information symbols) can be recursively taken and becomes very computationally efficient. Analysis and simulation results show that DFE is more robust than linear equalization without much complexity increment. The proposed scheme can achieve considerable diversity gain with both time and frequency offsets and applies to frequency-selective fading channels. R EFERENCES [1] Y. Mei, Y. Hua, A. Swami, and B. Daneshrad, “Combating synchronization errors in cooperative relays,” in Proc. IEEE ICASSP, Mar. 2005, vol. 3, pp. 369–372. [2] S. Wei, D. Goeckel, and M. Valenti, “Asynchronous cooperative diversity,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1547–1557, Jun. 2006. [3] Y. Li and X.-G. Xia, “A family of distributed space-time trellis codes with asynchronous cooperative diversity,” IEEE Trans. Commun., vol. 55, no. 4, pp. 790–800, Apr. 2007. [4] Y. Shang and X.-G. Xia, “Shift-full-rank matrices and applications in space-time trellis codes for relay networks with asynchronous cooperative diversity,” IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3153–3167, Jul. 2006. [5] M. O. Damen and A. R. Hammons, “Delay-tolerant distributed-TAST codes for cooperative diversity,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp. 3755–3773, Oct. 2007. [6] X. Guo and X.-G. Xia, “Distributed linear convolutive space-time codes for asynchronous cooperative communication networks,” IEEE Trans. Wireless Commun., vol. 7, pt. II, no. 5, pp. 1857–1861, May 2008. [7] G. S. Rajan and B. S. Rajan, “OFDM based distributed space time coding for asynchronous relay networks,” in Proc. IEEE ICC, Beijing, China, May 2008, pp. 1118–1122. [8] H.-M. Wang, X.-G. Xia, and Q. Yin, “Distributed space-frequency codes for cooperative communication systems with multiple carrier frequency offsets,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 1045–1055, Feb. 2009. [9] Z. Li, D. Qu, and G. Zhu, “An equalization technique for distributed STBC-OFDM system with multiple carrier frequency offsets,” in Proc. IEEE WCNC, Apr. 2006, vol. 2, pp. 839–843. [10] F. Tian, X.-G. Xia, and P. C. Ching, “Signal detection in a spacefrequency coded cooperative communication system with multiple carrier frequency offsets by exploiting specific properties of the code structure,” IEEE Trans. Veh. Technol., vol. 58, no. 7, pp. 3396–3409, Sep. 2009. [11] X. Li, F. Ng, and T. Han, “Carrier frequency offset mitigation in asynchronous cooperative OFDM transmissions,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 675–685, Feb. 2008. [12] Z. Li and X.-G. Xia, “An Alamouti coded OFDM transmission for cooperative systems robust to both timing errors and frequency offsets,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1839–1844, May 2008. [13] D. Veronesi and D. L. Goeckel, “Multiple frequency offset compensation in cooperative wireless systems,” in Proc. IEEE Globecom, San Francisco, CA, Nov. 2006, pp. 1–5. [14] N. Benvenuto, S. Tomasin, and D. Veronesi, “Multiple frequency offsets estimation and compensation for cooperative networks,” in Proc. IEEE WCNC, Hong Kong, Mar. 2007, pp. 892–896. [15] H.-M. Wang, X.-G. Xia, and Q. Yin, “Computationally efficient equalization for asynchronous cooperative communications with multiple frequency offsets,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 648– 655, Feb. 2009. [16] R. E. Lawrence and H. Kaufman, “The Kalman filter for the equalization of a digital communications channel,” IEEE Trans. Commun. Technol., vol. COM-19, no. 6, pp. 1137–1141, Dec. 1971. [17] R. Chen, X. Wang, and J. S. Liu, “Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering,” IEEE Trans. Inf. Theory, vol. 46, no. 9, pp. 2079–2094, Sep. 2000. [18] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wesel, “Multiinput multi-output fading channel tracking and equalization using Kalman estimation,” IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1065–1076, May 2002. [19] B. Balakumar, S. Shahbazpanahi, and T. Kirubarajan, “Joint MIMO channel tracking and symbol decoding using Kalman filtering,” IEEE Trans. Signal Process., vol. 55, no. 12, pp. 5873–5879, Dec. 2007.

[20] L. Song and J. K. Tugnait, “Doubly-selective fading channel equalization: A comparison of the Kalman filter approach with the basis expansion model-based equalizers,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 60–65, Jan. 2009. [21] N. Al-Dhahir and J. M. Cioffi, “MMSE decision-feedback equalizers: Finite-length results,” IEEE Trans. Inf. Theory, vol. 41, no. 4, pp. 961– 975, Jul. 1995. [22] C. D. Meyer, Matrix Analysis and Applied Linear Algebra. New York: Cambridge Univ. Press, 2001. [23] S. Haykin, Adaptive Filter Theory, 4th ed. Englewood Cliffs, NJ: Prentice-Hall, 2001. [24] M. S. Grewal and A. P. Andrews, Kalman Filtering: Theory and Practice Using Matlab, 2nd ed. New York: Wiley, 2001.

Robust MC-CDMA Channel Tracking for Fast Time-Varying Multipath Fading Channel Chang-Yi Yang and Bor-Sen Chen, Fellow, IEEE

Abstract—An unscented Kalman filter (UKF)-based channel-tracking method is proposed for a fast time-varying multipath fading channel in a multicarrier code-division multiple-access (MC-CDMA) system. The mobile radio channel is modeled as an autoregressive (AR) random process. The parameters of the AR process and the channel gain are simultaneously estimated by the proposed method. One-step-ahead prediction can also be obtained during channel estimation. It is useful for the decision-directed channel-tracking design, particularly in the fast-fading channel. Meanwhile, the estimated parameters can enhance the minimum mean-square error (MMSE) equalizer for symbol detection. The simulation results show that the enhanced equalizer based on the proposed estimation algorithm performs much better than that based on the conventional channel estimators in symbol error rate. Index Terms—Autoregressive (AR) random process, decision-directed channel-tracking design, minimum mean-square error (MMSE) equalizer, multicarrier code-division multiple access (MC-CDMA), unscented Kalman filter (UKF).

I. I NTRODUCTION Multicarrier code-division multiple access (MC-CDMA) is a new communication technique that integrates the merits of both orthogonal frequency-division multiplexing (OFDM) and code-division multipleaccess (CDMA) schemes. It can be robust against the intersymbol interference (ISI), multiuser interference, and multipath fading [1]. To support high-bit-rate links in the future, the MC-CDMA system has been considered as one of the possible candidates for the next generation of wireless communications. The MC-CDMA system divides the available bandwidth into a large number of narrow subchannels and spreads each data symbol in the frequency domain by transmitting all the chips of a spread symbol at the same time but in different orthogonal subchannels [2], [3]. One of the properties of multicarrier transmission is that the Manuscript received March 25, 2010; revised July 19, 2010; accepted August 23, 2010. Date of publication September 27, 2010; date of current version November 12, 2010. The review of this paper was coordinated by Prof. J. Chun. C.-Y. Yang is with the Department of Computer Science and Information Engineering, National Penghu University of Science and Technology, Penghu 88046, Taiwan (e-mail: [email protected]). B.-S. Chen is with the Department of Electrical Engineering, National TsingHua University, Hsin-chu 300, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2010.2079962

0018-9545/$26.00 © 2010 IEEE


Fig. 1.

4659

Block diagram of the MC-CDMA transmitter.

channel gain of each subchannel is different from the other. The inner product of different spreading codes will no longer be zero since the MC-CDMA systems spread transmitted symbols in a nonflat-fading channel. This leads to the loss of orthogonality between different users. The multiple-access interference (MAI) is introduced, and the performance will be severely degraded in this case. To preserve the orthogonality between different users, the channel impairment should be precisely estimated and efficiently equalized. Different research approaches have been adopted for channel estimation and tracking. The pilot-symbol-aided channel-estimation methods in both the time and frequency domains have been proposed [4]. A channel-sounding approach is employed in which a train of pulses is periodically transmitted [5]. A multiple-channel model, which includes several possible channel models based on the different ranges of Doppler frequencies (or mobile velocities), is constructed to treat the timevarying fading channel [6]. In addition, a decision-directed channel estimation in the frequency domain using a Kalman-based filter has been proposed [1]. It requires no overhead symbols. However, the decision-directed scheme has a delay problem, which the current data detection must adopt the former channel-estimation result. This will lead to a deterioration of the detection performance in a fastfading channel because the previous channel is not suitable for the current data detection. Therefore, an accurate prediction is necessary for the decision-directed channel-tracking algorithm to improve the performance. A time-varying velocity of the mobile station communicated in the MC-CDMA system is considered in this paper. The spectral characteristics of the communication channel can be approximated as an autoregressive (AR) process with a white Gaussian process input [7]. The parameters of the AR process and the variance of the input process are determined by the velocity of the mobile station. Since the velocity is time varying and unavailable for a handset in the realistic application, a nonlinear state equation can be used to describe the MC-CDMA system. The extended Kalman filter (EKF) is suitable for nonlinear estimation. However, it is only the first-order approximation to the nonlinear estimation and needs the computation of Jacobians to linearize the process and measurement equations [8]. A new estimation method called unscented Kalman filter (UKF) has been widely and recently used in several nonlinear systems [9], [10]. It does not require the computation of Jacobians and can be accurate to the third-order

approximation for process and measurement noises that are Gaussian distributed. The motivation of this paper is to present a decision-directed channel-estimation algorithm based on the UKF. The proposed method cannot only simultaneously obtain the channel gain and dynamics of the channel but also get a one-step-ahead prediction during the prediction stage to overcome the delay problem of the decisiondirected scheme. A minimum mean-square-error (MMSE) equalizer is designed for signal detection based on the one-step-ahead prediction. The error covariance of channel estimation can be employed as a design factor to enhance the MMSE equalizer. Thus, the enhanced MMSE equalizer in consideration of the estimation error will be more robust to the fast-fading channel. The simulation results demonstrate the improvements of the proposed method over other methods. This paper is organized as follows. In Section II, the MC-CDMA system is described. The UKF-based decision-directed channeltracking scheme and the enhanced MMSE equalization are introduced for channel estimation in Section III. The computer simulations of the adaptive MC-CDMA detector are presented and compared in Section IV. The conclusions are summarized in Section V. In what follows, AT denotes the transpose of A, and AH denotes the Hermitian of A. II. M ULTICARRIER C ODE -D IVISION M ULTIPLE ACCESS C OMMUNICATION S YSTEM The models of the transmitter and receiver in an MC-CDMA system with Ns subchannels are described in this section. It is supposed that Nu users are simultaneously transmitting in the system. The nth multicarrier block symbol (duration T ) for user j is formed by taking μ symbols d1j (n), . . . , dμ j (n) in parallel, which are spread by the user’s spreading code cj with length N . Thus, the μ spread symbols are placed into Ns = μN available subchannels. The block diagram of the MC-CDMA transmitter is shown in Fig. 1. In the rest of this paper, for simplicity of notation, we concentrate only on the case where each user transmits one symbol (μ = 1) in each MC-CDMA block symbol. The transmitted symbol is simply represented as dj (n) for user j. Therefore, the total number of subchannels is Ns = N .

4660


Fig. 2. Signal-flow diagram of UKF-based decision-directed channel-tracking scheme.

The Walsh–Hadamard spread code cj for user j with length N is cj = [cj (0) cj (1)

...

cj (N − 1)]T

(1)

Yn (m) =DF T {yn (k)} = Sn (m)Hn (m)+Vn (m)

√ √ where cj (n) ∈ {(1/ N ), (−1/ N )}, and

cTi cj =

1, 0,

=

if i = j

(2)

otherwise.

cj (m)dj (n),

cj (m)dj (n)Hn (m)+Vn (m), m = 0, . . . ,N −1

j=1

where

m = 0, 1, . . . , N − 1.

(3)

j=1

The signal Sn (m) is then translated by the multicarrier modulation (i.e., IDFT), and the OFDM symbol sn (k) is produced as follows (see Fig. 1):

N −1

Hn (m) =

Nu

Sn (m) =

Nu

(6)

Therefore, the nth symbols with spread sequence of all Nu users transmitted on the mth subchannel can be written as

ever, channel variation during the successive symbol intervals is allowed. The received samples are demodulated by taking N -point DFT and can be represented as

hn (l)e−

j2πlm N

N −1

Vn (m) =

l=0

vn (k)e−

l=0

Hn (m) is the channel coefficients of the mth subcarrier at the nth MC-CDMA block symbol in the frequency domain, and Vn (m) is the frequency-domain noise. The covariance of Vn (m) can be expressed as

QVn (m) = E Vn (m)VnH (m) = N σv2 . j2πmk 1 Sn (m)e N , sn (k) = √ N m=0

j2πmk N

(7)

N −1

k = 0, 1, . . . , N − 1. (4)

The block diagram of the MC-CDMA receiver is illustrated in Fig. 2.

A guard interval is inserted between successive OFDM symbols to avoid the ISI effects by using a cyclic prefix technique [11]. In the downlink case, all users’ transmitting signals are synchronous and experience the same multipath Rayleigh-fading channel. After removing the guard interval, the received signal of the nth MC-CDMA symbol is given by

III. ROBUST U NSCENTED K ALMAN F ILTER C HANNEL T RACKING

L−1

yn (k) =

sn (k − l)hn (k, l) + vn (k),

k = 0, 1, . . . , N − 1

l=0

(5) where hn (k, l) and vn (k) denote the lth path sampling of the complex time-varying fading channel with length L and the additive noise at the kth instant of the nth MC-CDMA block symbol, respectively. vn (k) is assumed to be white Gaussian with zero mean and variance σv2 . For a high-data-rate transmission, it is reasonably assumed that the channel is time invariant during one MCCDMA block symbol interval T [1], i.e., hn (0, l) = hn (1, l) = · · · = hn (N − 1, l) for l = 0, 1, . . . , L − 1. The index k of hn (k, l) would be ignored in this case and can be simply rewritten as hn (l). How-

A. Subcarrier Channel Estimation Although the Jakes’ model is often utilized for the fading channel estimation, it is not easy to use this model to estimate channel variation. A higher order AR process can match the spectral characteristics of the Rayleigh-fading channel [7]. Here, we focus on the second-order AR model due to its simplicity and accuracy. The lth path of the channel impulse response at the nth MC-CDMA symbol can be modeled by an AR(2) process hn (l) = a1,n−1 hn−1 (l) + a2,n−1 hn−2 (l) + wn (l) l = 0, 1, . . . , L − 1 (8) where wn (l) is a complex zero-mean white Gaussian process. The parameters a1,n and a2,n are determined by the Doppler frequency shift due to the velocity change of the mobile station. They are defined as a1,n = 2rd cos(2πfD,n T ),

a2,n = −rd2

(9)


where fD,n is the maximum Doppler frequency shift in the nth symbol period, and rd is the pole radius that corresponds to the steepness of the peaks of the power spectrum [7]. The radius is chosen to be very close to 1 to model the spectral peaks at the maximum Doppler frequency of the fading channel. The channel-estimation method is performed in the frequency domain. Hence, the time-domain AR process of Rayleigh-fading channel must be transformed to a frequency-domain AR process. Taking the N -point DFT of hn (l), we have

N −1

Hn (m) =

hn (l)e

−j2πlm N

l=0

L−1

=

hn (l)e

4661

parameters related to dn in the UKF algorithm should be regenerated ˆn from d Sˆn (m) =

cj (m)dˆj (n).

The state-space model for the decision-directed channel tracking of the mth subchannel is rewritten as Xn = F (Xn−1 ) + Wn

(18)

ˆ H Xn Yn = C n

(19)

+ Vn

where Wn and Vn are assumed to have zero mean, and

= a1,n−1 Hn−1 (m) + a2,n−1 Hn−2 (m) + Wn (m) m = 0, 1, . . . , N − 1

ˆ n (m) = [ 0 C

Sˆn (m)

(10)

0 ]H

H

Q1 = E Wn Wn

Q2 = E |Vn |2 .

where

L−1

Wn (m) =

wn (l)e

−j2πlm N

.

l=0

Since the velocity of the mobile station is time varying and unavailable, the parameter a1,n is also time varying. For the best channel estimation, one way is to jointly estimate Hn (m) and a1,n . However, the joint estimation is nonlinear from (10) because of coupling. It is not easy for any linear estimation method to estimate the state. The UKF [12] is adopted in this paper to get a more accurate result than EKF for nonlinear parameter-estimation problem. From (6) and (10), the channel model can be described by the following state-space equations: Xn (m) = F (Xn−1 (m)) + Wn (m) Yn (m)

(17)

j=1

−j2πlm N

l=0

Nu

= CH n (m)Xn (m)

(11)

+ Vn (m)

(12)

where

F(Xn (m)) = [a1,n a1,n Hn (m)+a2,n Hn−1 (m) Hn (m)] Cn (m) = [0 Sn (m) 0]H Wn (m) = [wa,n Wn (m) 0]

The fundamental component of the filter is the unscented transformation. It provides a method for calculating the statistics of a random variable, which undergoes a nonlinear transformation [12]. This transformation uses a set of approximately weighted points to parameterize the mean and covariance information while permitting the direct propagation of that information through a set of nonlinear equations. Consider an M -dimensional random vector X through a nonlinear function F to generate an J-dimensional random variable z z = F (X).

ˆ χ0 =X

(13) T

(14) (15) (16)

wa,n is assigned to simulate the variation of velocity as a random walk, and F is a nonlinear function of the state Xn (m). The index m is omitted in the following computation since the procedure of UKF computation is the same for all subchannels.

(21)

ˆ and covariance matrix PX . It is assumed that X has the mean X The requirement is to compute zˆ = E{z} and PZ = E{(z − zˆ)(z − zˆ)T }. Based on the unscented transformation [12], a set of 2M + 1 sigma points {χi }2M i=0 are chosen so that the sample mean and sample ˆ and PX , respectively. covariance are X

ˆ+ χi =X

Xn (m) = [a1,n Hn (m) Hn−1 (m)]T

(20)

ˆ− χi =X (u)

R0 = (t)

R0 = (u)

Ri

(M +λ)PX

,

i = 1, 2, . . . , M

,

i = M + 1, M + 2, . . . , 2M

i

(M +λ)PX i

λ M +λ λ +(1−α2 +β) M +λ (t)

=Ri =

λ , 2(M +λ)

i = 1, 2, . . . , 2M

(22)

B. Decision-Directed Channel Tracking Based on UKF Because the transmitted signal is unknown for the receiver in the tracking mode, the detected data, instead of the transmitted signal, are fed to the UKF channel estimator for channel identification. This is the so-called decision-directed channel tracking. The occupancy of available bandwidth due to the transmission of the overhead symbol can be avoided in this method. The data-flow diagram of the proposed decision-directed channel tracking is shown as Fig. 2. The detected ˆ n is used to identify the channel in the tracking mode. (Since symbol d the MMSE data-detection scheme is introduced in Section III-C, ˆ n is available in this section.) Therefore, the it is assumed that d

where λ = α2 (M + κ) − M , ( (M + λ)PX )i is the ith column of (u) (t) the matrix square root of (M + λ)PX , and Ri and Ri are the weights for mean and variance, respectively. κ 0 is chosen to guarantee positive semidefiniteness of Pz . α is a small positive constant determining the spread of the sigma points around the mean value of X. β is a nonnegative weighting parameter, which incorporates on the distribution of X. The 2M + 1 sigma points are then propagated through the nonlinear transformations as z˜i = F (χi ),

i = 0, 1, . . . , 2M.

(23)

4662


TABLE I S UMMARY OF THE UKF A LGORITHM

TABLE II P OWER D ELAY P ROFILE IN THE T YPICAL U RBAN A REA

The design objective of the MMSE equalizer is to find an optimal ˆ n = diag{φˆn (0), φˆn (1), . . . , φˆn (N − 1)} based on the channel Φ prediction such that

on (m) = arg min E φ

2

Sn (m) − φ n (m)Yn (m) .

(26)

φn (m)

The optimal MMSE equalization for (26) with the channel prediction is [6] ˆH H n|n−1 (m)

on (m) =

φ

ˆ n|n−1 (m) 2 + βn|n−1 (m) + QV (m)/σ 2

H s n

ˆ n|n−1 (m) is where σs2 = E{|Sn (m)|2 }, the channel prediction H ˆ obtained from Xn|n−1 in Table I, and the covariance of channel pre˜ n|n−1 (m)|2 } = Pn|n−1,(2,2) (m) is diction error βn|n−1 (m) = E{|H the element at the second row and second column of Pn|n−1 (m) in Table I. Finally, the output of the MMSE equalizer will multiply ˆ n can with the Walsh–Hadamard code, and the MC-CDMA symbol d be detected [6]. It is worthy noting that the conventional decisiondirected scheme in [1] adopts the previous estimates to perform the current MMSE equalization. Its conventional MMSE equalizer cn (m) is φ

Finally, zˆ and Pz are obtained as follows: zˆ =

2M

(u)

Ri z˜i

i=0

Pz =

2M

(t)

Ri (˜ zi − zˆ)(˜ zi − zˆ)T .

(24)

i=0

For the nonlinear state-space model of (18) and (19) with additive noise, the UKF is a straightforward application of the unscented transformation. The overall procedure for the UKF is depicted in Table I. C. MMSE Equalizer An enhanced MMSE equalizer is developed based on the proposed UKF-based estimation method. The transmitted signal Sn (m) is replaced by the regenerated signal Sˆn (m) for channelparameter tracking since Sn (m) is unavailable for the receiver in the tracking mode. The regenerated signal Sˆn (m) is obtained from ˆ n in (17). That means the current channelthe detected data d parameter tracking can be done after the current data detected. However, the MMSE data detection needs to have the current channel parameters. This is a delay problem of the decision-directed channel-tracking scheme. However, the previous estimates will not be suitable for the current equalization when the channel variation is fast. Because the UKF algorithm provides the one-stepˆ n|n−1 during the prediction stage, a robust ahead prediction H MMSE equalization can be developed for the fast-fading channel case. The MMSE equalization for the MC-CDMA systems under the assumption of a perfectly known channel has been previously provided in [13]. A robust MMSE equalizer for each subchannel is developed here with consideration of channel estimation error. The prediction error for channel estimation is defined as ˆ n|n−1 (m) − Hn (m). ˜ n|n−1 (m) = H H

(27)

(25)

cn (m) =

φ

ˆH H n−1|n−1 (m)

.

ˆ n−1|n−1 (m) 2 + QV (m)/σs2

H n

(28)

It is based on the assumption that the channel variation is slow. However, the enhanced MMSE equalizer in (27) contains the one-step-ahead prediction and the covariance of prediction error. It can provide more precise detection than the conventional MMSE equalizer. IV. C OMPUTER S IMULATION A computer simulation has been conducted to confirm the performance of the proposed method. The urban case is considered. Table II shows the power delay profile in the urban area with the root mean square delay στ = 1 μs. The mobile velocity from 5 to 110 km/h is used to simulate different mobile environments. The central frequency fc is 2 GHz in the MC-CDMA system. The total bandwidth BW is 2.048 MHz, which is divided into 1024 subchannels. The subchannel spacing is then Δf = 2 kHz. An additional 8-μs guard-interval duration is used to provide protection from ISI due to the channel multipath delay spread. The length of the adopted Walsh–Hadamard code is N = 64 chips. Thus, the MC-CDMA system can support the maximum 64 active users. The AR(2) model parameters a1 and a2 are defined in (9). The parameter of the pole radius is chosen as rd = 0.998 for the simulation cases [7]. It is also assumed that the channel remains approximately constant during one MC symbol period. The data-modulation scheme is quadrature phase-shift keying in the following simulation. The performance of the systems is measured by the symbol error rate and the mean square error (MSE) of channel estimation, with


Fig. 3. MSE of estimation by different channel-tracking methods for the user with time-varying velocity.

4663

Fig. 4. MSE of prediction by different channel-tracking methods for the user with time-varying velocity.

simulation for 50 trials and 2000 MC-CDMA symbols per trial. The MSE of the channel estimation is defined as

N −1

MSE =

ˆ n (m) 2 . E Hn (m) − H

m=0

The user’s velocity is a random variable and varies with time in the realistic applications. Thus, an UKF has been proposed to jointly estimate the dynamics and path gains of the channel via nonlinear estimation. To simulate a velocity-varying mobile radio channel, the velocity of the mobile station is assumed to be a random variable uniformly distributed in the range of 5 km/h (fD T = 0.0047)– 110 km/h (fD T = 0.1034) with different signal-to-noise ratios in ˆ 0 and P0 are set to [1.85 0 0]T each trial. The initializations of X and diag([10−5 10−5 10−5 ]), respectively. The MSEs using different channel estimation methods are compared in Fig. 3. The 2-D pilotadded channel-tracking method is given in [14]. The time-domain pilots are spaced by a minimum coherence time Tcoh , which is equal to the reciprocal of a double maximum Doppler frequency Tcoh = (1/2fD ). Both the previously presented first-order Kalman filter [1] and the second-order Kalman filter [15] are failed because these two methods adopt fixed dynamics. The EKF is less accurate than the proposed UKF, although it can also be used for the nonlinear estimation. Therefore, the proposed UKF can get the best result for channel estimation among these methods. The main drawback of the 2-D pilot-added channel estimation is the occupancy of bandwidth, which is a valuable resource for the service providers. The decision-directed channel-tracking scheme is proposed to avoid this occupancy. However, the conventional decision-directed scheme can only adopt the previous estimate for current equalization and detection. The performance of detectors could be deteriorated by unsuitable estimates when the fading rate is high. Therefore, an accurate channel prediction is necessary to solve the delay problem of the decision-directed scheme. The MSEs of channel prediction error using different decision-directed channel-tracking algorithms are compared in Fig. 4. It can be seen that, without the overhead symbols, the EKF and the proposed UKF can still more accurately predict the channel than the 2-D pilot-added method when the velocity is time varying. Although the EKF can predict the next state using the nonlinear process function, it cannot achieve the accuracy of prediction

Fig. 5. SER of different channel-tracking methods for the user with timevarying velocity.

as UKF using the average of sigma points. The results of MMSE equalizers using different channel-tracking schemes are compared in Fig. 5, with the performance of a perfectly known channel as the lower bound. The performances of the two dynamic-fixed Kalman filters are bounded by the large estimation error. The proposed UKFbased channel-tracking algorithm can perform the best equalization and detection among these five methods because it can most accurately estimate and predict the channel. V. C ONCLUSION A robust channel-tracking method has been presented for fast timevarying multipath fading channel in an MC-CDMA system. The channel model is described by a nonlinear state-space equation. The state includes the gains and dynamics of the channel. The proposed UKF-based decision-directed channel-tracking scheme can estimate the state via a nonlinear estimation to achieve robustness for the time-varying velocity. In addition, it does not need a training signal periodically, so that the occupation of the available bandwidth is

4664


avoided. The delay problem of the decision-directed scheme can also be solved with the prediction ability of the proposed method. The MMSE equalizer can be enhanced using the covariance of estimation error, which is obtained from UKF. Simulation results indicate that the proposed method can more accurately track the channel than the other methods. Therefore, the proposed UKF-based decision-directed channel tracking for MC-CDMA systems is very useful for a user with time-varying velocity over a fast multipath fading channel. Since the MC-CDMA technique integrates the OFDM and CDMA schemes, the proposed method can be easily implemented to these two systems. The multi-input–multi-ouput OFDM (MIMO-OFDM) system has recently attracted much attention for research. Application of the proposed method to MIMO-OFDM system is a good topic for future study. R EFERENCES [1] D. N. Kalofonos, M. Stojanovic, and J. G. Proakis, “Performance of adaptive MC-CDMA detectors in rapidly fading Rayleigh channel,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 1375–1387, Mar. 2003. [2] N. Yee and J. P. Linnartz, “Wiener filtering of multi-carrier CDMA in Rayleigh fading channel,” in Proc. PIMRC, Sep. 1994, vol. 4, pp. 1344–1347. [3] A. B. Djebbar, K. Abed-Meraim, and A. Djebbari, “Blind and semi-blind equalization of downlink MC-CDMA system exploiting guard interval redundancy and excess codes,” IEEE Trans. Commun., vol. 57, no. 1, pp. 156–163, Jan. 2009. [4] A. Nagate and T. Fujii, “A study on channel estimation methods for timedomain spreading MC-CMDA systems,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5233–5237, Dec. 2008.

[5] L. Sanguinetti and M. Morelli, “Channel acquisition and tracking for MC-CDMA uplink transmissions,” IEEE Trans. Veh. Technol., vol. 55, no. 3, pp. 956–967, May 2006. [6] B. S. Chen and J. F. Liao, “Adaptive MC-CDMA multiple channel estimation and tracking over time-varying multiple fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 6, pp. 2328–2337, Jun. 2007. [7] P. H.-Y. Wu and A. Duel-Hallen, “Multiuser detectors with disjoint Kalman channel estimators for synchronous CDMA mobile radio channels,” IEEE Trans. Commun., vol. 48, no. 5, pp. 752–756, May 2000. [8] S. Haykin, Adaptive Filter Theory, 4th ed. Englewood Cliffs, NJ: Prentice-Hall, 2002. [9] D.-J. Lee and K. T. Alfriend, “Adaptive sigma point filtering for state and parameter estimation,” presented at the AIAA/AAS Astrodynamics Specialists Conf., Providence, RI, Aug. 2004, Paper AIAA 2004-5101. [10] Z. Ali, M. Deriche, and A. Andalusi, “A novel approach for multipath channel estimation in CDMA networks using the unscented Kalman filter,” in Proc. IFIP Conf. Wireless Opt. Commun. Netw., 2009, pp. 1–5. [11] V. Richard and P. Ramjee, OFDM Wireless Multimedia Communications. Norwood, MA: Artech House, 2000. [12] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE, vol. 92, no. 3, pp. 401–422, Mar. 2004. [13] A. Chouly, A. Brajal, and S. Jourdan, “Orthogonal multicarrier technique applied to direct sequence spread spectrum CDMA systems,” in Proc. IEEE GLOBECOM, Nov. 1993, pp. 1723–1728. [14] S. Kaiser and P. Hoeher, “Performance of multicarrier CDMA with channel estimation in two dimensions,” in Proc. IEEE Symp. PIMRC, 1997, vol. 1, pp. 115–119. [15] L. Lindbom, “Simplified Kalman estimation of fading mobile radio channels: High performance at LMS computational load,” in Proc. ICASSP, 1993, vol. 3, pp. 352–355.