Robust Fast Time-Varying Multipath Fading Channel ... - IEEE Xplore

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 4, MAY 2012

1599

Robust Fast Time-Varying Multipath Fading Channel Estimation and Equalization for MIMO-OFDM Systems via a Fuzzy Method Bor-Sen Chen, Fellow, IEEE, Chang-Yi Yang, and Wei-Ji Liao

Abstract—Channel estimation is an important issue for wireless communication systems. A channel estimation scheme using a Takagi-Sugeno (T-S) fuzzy-based Kalman filter under the timevarying velocity of the mobile station in a multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) system is proposed in this paper. The fuzzy technique is used to interpolate several linear models to approximate the nonlinear estimation system. A MIMO system with the orthogonal space-time block coding (OSTBC) scheme is considered, where the 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 be obtained during this estimation procedure. This is useful for the decision-directed channel-tracking design, particularly in the fast-fading channel. Furthermore, the robust minimum mean-square error (MMSE) equalization design can be achieved by considering the channel prediction error to improve the performance of symbol detection. To validate the performance of our proposed method, several simulation results are given and compared with those of other methods. When considering the time-varying velocity of the mobile station communication in the MIMO-OFDM system, the enhanced equalizer based on the T-S fuzzy-based Kalman filter performs better than those based on the conventional channel estimators in terms of symbol error rate. Index Terms—Autoregressive (AR) random process, decisiondirected channel-tracking design, multiple-input multiple-output (MIMO) communication system, orthogonal frequency-division multiplexing (OFDM) system, Takagi-Sugeno (T-S) fuzzy-based linear model.

I. I NTRODUCTION

W

ITH THE increasing demands for higher data rates and reliability in wireless communication systems, multiple-input multiple-output (MIMO) techniques have atManuscript received January 28, 2011; revised January 17, 2012; accepted January 18, 2012. Date of publication March 6, 2012; date of current version May 9, 2012. The work of C.-Y. Yang was supported by the National Science Council under Contract NSC100-2221-E-346-005. The review of this paper was coordinated by Dr. C. Cozzo. B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan (e-mail: [email protected]). C.-Y. Yang is with the Department of Computer Science and Information Engineering, National Penghu University of Science and Technology, Magong City 880, Taiwan (e-mail: [email protected]). W.-J. Liao was with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan. He is currently with Compal Communications Inc., Taipei 114, 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.2012.2188549

tracted tremendous attention. In a MIMO system, a high data rate is achieved by simultaneously transmitting data from several antennas. A typical scheme embedded in the MIMO system is orthogonal space-time block coding (OSTBC) [1]– [3]. It integrates the techniques of antenna array spatial diversity and channel coding to provide significant capacity gains in a wireless channel. Without loss of generality, previous studies have typically assumed that the channels are flat fading and time invariant over an OSTBC codeword period. Unfortunately, in a fast-fading channel, the MIMO system requires an efficient equalization technique to eliminate intersymbol interference (ISI), which is caused by high-data-rate applications or data transmissions over broadband frequency-selective channels [3]–[7]. To overcome this phenomenon, the orthogonal frequency-division multiplexing (OFDM) technique is used in MIMO systems [8]–[11]. The combination of MIMO and OFDM techniques, which is called a MIMO-OFDM system, has recently attracted much research [12]–[15]. Its advantages are antenna diversity, high data rate, elimination of the ISI effect, and low complexity. To approach the theoretical capacity and achieve maximum diversity gain, the channel state information (CSI) of the MIMO-OFDM system must be accurately estimated and tracked. For instance, the excellent performance of symbol detection at the receiver is based on the perfectly known state of the CSI [8]. However, perfect CSI is usually unknown to the receiver in wireless communication systems. Hence, the impact of channel estimation errors on the performance of MIMOOFDM detectors has recently attracted a significant amount of research interest [16]. Different research approaches have been adopted for channel estimation and tracking [13], [17]–[21]. An expectation–maximization (EM)-based channel estimation method has been proposed for channel estimation to detect the transmitted data in [22]. However, the EM algorithm needs numerous iterations to achieve optimal channel estimation results. Although an extended Kalman filter (EKF) with the pilotsymbol-aided method to track a MIMO-OFDM channel has been proposed in [23], channel estimation with a pilot-symbolaided method may occupy the available bandwidth (valuable for the service provider) and may degrade the data throughput [24], [25]. Other research works include the cyclic combtype training structure proposed in [26], in which all types of training symbols are cyclically transmitted at each antenna and estimated for the channel frequency response with weights obtained from the corresponding mean-square errors (MSEs) at

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the receiver. Iterative maximum-likelihood channel estimation for signal detection under a large variation of channel responses in the frequency domain was proposed in [27]. In addition, decision-directed channel estimation in the frequency domain using a Kalman-based filter was proposed in [28]. A time-varying velocity of the mobile station communicated in the MIMO-OFDM system is considered in this paper, in which the fading rate may be high or low for a user at any time instant. In this situation, the spectral characteristics of the mobile radio channel can be approximated as an autoregressive (AR) process [29], [30]. The parameters of the AR process are determined by the Doppler frequency (or the mobile velocity). Nevertheless, since the Doppler frequency is unavailable for a handset and the velocity is time varying in a realistic application, a nonlinear state equation can be used to describe the MIMO-OFDM system. The EKF is suitable for nonlinear channel estimation [31]. However, this is only the first-order approximation to the nonlinear channel estimation and needs the computation of Jacobians to linearize the process and measurement equations. In this regard, the fuzzy theorem has been recently applied in various fields, such as control systems [32], [33], signal processing [34], and communication systems [35]–[37]. The fuzzy systems are considered to be universal approximators for certain nonlinear systems in most of these applications. The Takagi-Sugeno (T-S) fuzzy model is frequently adopted due to its simplicity and efficiency [37]– [40]. Based on the T-S fuzzy model, a fuzzy Kalman filter is proposed [41] for nonlinear channel estimation and prediction without the computation of Jacobians. This can achieve accurate nonlinear channel estimation and prediction under the Gaussian-distributed process and measurement noises. The estimation method using overhead symbols is not efficient because it is necessary to transmit the maximum number of overhead symbols for the worst time-selective case, regardless of whether the fading rate is high or not. As opposed to the pilot-symbol-added [42]–[45] or channel-sounding approaches [10], [23], the decision-directed channel-tracking scheme does not require any overhead symbols [28], although an initial training period is required in practice. However, the decision-directed scheme has a delay problem in that the current data detection must adopt the estimate obtained from the previous detection [27]. This will lead to a deterioration of the detection performance in a fast-fading channel because the previous channel is unsuitable for the current data detection. Therefore, an accurate prediction is necessary for the decision-directed channel-tracking algorithm to improve the performance. The goal of this paper is to present a robust nonlinear decision-directed channel-estimation algorithm for the MIMOOFDM system. The T-S fuzzy filter, instead of the conventional filter, is employed to predict and estimate the channel response. It interpolates several linear channel estimation algorithms to approximate the nonlinear channel system. It can also efficiently mitigate the approximate error to achieve an accurate prediction and estimation of the channel response. A minimum MSE (MMSE) equalizer is designed for signal detection based on the channel prediction to overcome the delay problem.

Simulation results illustrate that the proposed method has better performance than other existing channel estimators. The contributions of this paper are stated as follows: To the best of our knowledge, this is the first study that simultaneously applies the fuzzy method to channel estimation and prediction in MIMO-OFDM systems. The estimated parameters include the channel impulse response and the mobile velocity. Channel prediction is very useful for designing the decision-directed equalizer in fast-fading channels. The covariance of the channel prediction error can also be employed as a robust design factor to enhance the MMSE equalizer. Thus, the enhanced MMSE equalizer, in consideration of the covariance of the channel prediction error, will be more robust for the fast-fading channel in a MIMO-OFDM system. This paper is organized as follows: In Section II, the MIMO-OFDM system is described. The T-S fuzzy-based channel-tracking scheme is introduced for channel estimation in Section III. The design of an enhanced MMSE equalizer based on the proposed method is presented in Section IV. Several computer simulations of the adaptive MIMO-OFDM detectors are presented and compared with the proposed method in Section V. Finally, the conclusions are summarized in Section VI. II. S YSTEM M ODELS OF MIMO-OFDM S YSTEMS A. Mathematical Notations In this paper, the notation “( )T ” denotes the transpose of a matrix or a vector, “( )H ” denotes the Hermitian transpose of a matrix or a vector, “E{ }” represents expectation, “Iq ” represents a q × q identity matrix, “vec{ }” refers to the vectorization operator (stacking all the columns of the matrix on top ˘ of each other), “ F ” denotes the Frobenius norm, and “A” denotes the operator for any complex matrix A as ˘ = vec{Re{A}} . A (1) vec{Im{A}} B. Model of a Transmitter A MIMO-OFDM system with Nt transmit antennas, Nr receive antennas, and Nc subcarriers is considered. As shown in Fig. 1, the information bits are first modulated by an M ary quadratic amplitude modulator (QAM). The modulated MQAM symbols are encoded by an OSTBC encoder and fed into an OFDM modulator. In the OSTBC scheme [1], the data stream to be transmitted is encoded in blocks, which are distributed among the spaced antennas and across time. According to the properties of the OSTBC, for any transmit signal Sn (m) = [Sn,1 (m), Sn,2 (m), . . . , Sn,K (m)]T , where K is the number of symbols per code, the OSTBC code matrix of the nth transmitted symbol at the mth subcarrier can be formed as C (Sn (m)) =

K

(Ak Re {Sn,k (m)} + jBk Im {Sn,k (m)})

k=1

(2)

CHEN et al.: TIME-VARYING MULTIPATH FADING CHANNEL ESTIMATION AND EQUALIZATION FOR MIMO-OFDM

Fig. 1.

1601

Block diagram of the MIMO-OFDM transmitter.

where {Ak , Bk } are the kth fixed Ts × Nt code matrices, and Ts represents the number of time slots for transmitting one block of symbols. These coded symbols are transmitted after the multicarrier modulation (i.e., inverse fast Fourier transform (FFT) and cyclic prefix). Lemma 1 ([46, Sec. 7.4, p. 102]): Assuming the unity of each transmitted signal power and considering the transmission matrix C(Sn (m)) in (2), then CH (Sn (m)) C (Sn (m)) = KINt

Hnj,i (m)

=

N c −1

− hj,i n (l)e

j2πlm Nc

.

(6)

l=0

The received signal described in (5) will be rewritten as rn (m) = Cn (m)hn (m) + vn (m)

(3)

holds for all complex Sn (m) if and only if {Ak , Bk } in (2) is an amicable orthogonal design, i.e.,

(7)

where rn (m) = vec {Rn (m)} Cn (m) = INr ⊗ C (Sn (m))

AH k Ak = INt

hn (m) = vec {Hn (m)}

BH k Bk = INt

vn (m) = vec {Vn (m)}

H AH k Aq = − Aq Ak H BH k Bq = − Bq Bk ,

The element Hnj,i (m) of the jth row and the ith column of matrix Hn (m) can be represented as [23]

and operator ⊗ denotes the Kronecker matrix product.

k = q

H AH k B q = B q Ak

for k = 1, . . . , K, and q = 1, . . . , K.

(4)

C. Model of a Receiver The time-domain channel impulse response from the jth transmit antenna to the ith receive antenna of the lth path sampling of the complex time-varying fading with length L at the kth instant is represented as hj,i n (k, l). For a highdata-rate transmission, it is assumed that the channel is time invariant during one OFDM block symbol [28], i.e., hj,i n (0, l) = j,i (1, l) = · · · = h (N − 1, l) for l = 0, 1, . . . , L − 1. The hj,i c n n index k of hj,i (k, l) would be ignored in this case and can be n (l). However, channel variation during simply rewritten as hj,i n the successive symbol intervals is allowed. At the receiver, after removing the cyclic prefix and passing FFT modulation, the received signal of the nth OFDM block symbol at the mth subchannel can be expressed as Rn (m) = C (Sn (m)) Hn (m) + Vn (m)

(5)

for m = 0, 1, . . . , Nc − 1, where Rn (m) ∈ CTs ×Nr is the received signal, Hn (m) ∈ CNt ×Nr is the channel frequency response of the nth OFDM block symbol on the mth subchannel, and Vn (m) ∈ CTs ×Nr is the additive white Gaussian noise, where the variance of each entry in the frequency domain is σv2 .

D. Problem Formulation The transmitted signal shall be regenerated by the equalizer from rn (m). Therefore, the equalizer requires the channel information to minimize the MSE between the transmitted signal and the regenerated signal. Since the channel information is unknown to the equalizer, the channel estimation of the MIMOOFDM system becomes an important issue. A robust fuzzybased decision-directed Kalman filter is proposed to estimate the channel information. The MMSE equalizer can be implemented according to the estimation result. The block diagram of the MIMO-OFDM receiver is shown in Fig. 2. III. F UZZY-BASED C HANNEL T RACKING A. Subcarrier Channel Estimation For the Rayleigh fading channel, an AR process can match its special characteristics [13], [20], [27], [47]. The second-order AR model [AR(2)] is adopted here due to its simplicity and accuracy. Therefore, the lth path of the time-domain channel impulse response from the jth transmit antenna to the ith receive antenna at the nth OFDM symbol can be modeled by j,i j,i j,i j,i j,i hj,i n (l) = a1,n−1 hn−1 (l) + a2,n−1 hn−2 (l) + wn (l)

(8)

for i = 1, . . . , Nr , j = 1, . . . , Nt , and l = 0, 1, . . . , L − 1, where wnj,i (l) is a zero-mean complex white Gaussian process. j,i Parameters aj,i 1,n and a2,n are determined by Doppler frequency

1602


Fig. 2. Block diagram of the MIMO-OFDM receiver with decision-directed channel tracking.

shift due to the velocity change of the mobile station. These parameters are defined as [48] j,i j,i 2 aj,i (9) 1,n = 2rd cos 2πfD,n T , a2,n = −rd

where

j,i is the maximum Doppler frequency shift in the nth where fD,n symbol period, and rd is the pole radius that corresponds to the steepness of the peaks of the power spectrum. Since the channel estimation method is performed in the frequency domain, (8) must be transformed into the frequency domain. From (6), we have

Hnj,i (m) =

N c −1

− hj,i n (l)e

j2πlm Nc

l=0

=

L−1

− hj,i n (l)e

j2πlm Nc

l=0

j,i j,i j,i j,i = aj,i 1,n−1 Hn−1 (m) + a2,n−1 Hn−2 (m) + Wn (m)

(10) for m = 0, 1, . . . , Nc − 1, where Wnj,i (m) =

L−1

wnj,i (l)e−

j2πlm Nc

.

l=0

We can find that the relationship of the AR(2) process in frequency domain is similar to that in time domain. A possible assumption that can be made is that the velocities of any antenna pair between the transmitter and the receiver are all equal in any time instant, i.e., aj,i 1,n = a1,n ∀ i, j. Since the velocity of the mobile station is time varying and unavailable, a1,n is also time varying. For the channel estimation, the best way is to jointly estimate a1,n and Hnj,i (m). However, the joint estimation in (10) is a nonlinear problem due to the coupling. It is not easy for any linear estimation method to simultaneously estimate them. The fuzzy-based state estimation is introduced in this paper to get more accurate results than the EKF for the nonlinear estimation problem. The channel gain and AR(2) coefficient estimation problem can be transformed into the following state-space equations: Xn (m) = G (Xn−1 (m)) + Wn (m)

(11)

rn (m) = Jn (m)Xn (m) + vn (m)

(12)

 a1,n Xn (m) =  vec {Hn (m)}  vec {Hn−1 (m)}   a1,n   G1,1 n     ..   .   t ,1   GN n     . .. G (Xn (m)) =      GNt ,Nr  n    H 1,1 (m)  n     ..   . HnNt ,Nr (m)   ωa,n  Wn1,1 (m)      ..   .   Nt ,1  Wn (m) =   Wn (m)    . ..      W Nt ,Nr (m)  n 0Nt Nr ×1 

j,i j,i Gj,i n = a1,n Hn (m) + a2,n Hn−1 (m)

Jn (m) = [ 0Nt Nr ×1

Cn (m)

0Nt Nr ]

(13)

where ωa,n is assigned to simulate the variation of velocity as a random walk, and Wn (m) and vn (m) are assumed to be zero mean and covariance Qw and Qv , respectively. The column vector Xn (m) ∈ C(2Nt Nr +1)×1 is a state variable that contains the parameter of the AR(2) process and the channel coefficients. If the velocity of the mobile station is known, the aforementioned state-space equation can be simplified as a linear model Xn (m) = Dn−1 Xn−1 (m) + Wn (m) rn (m) = Jn (m)Xn (m) + vn (m) where Dn−1



1 0  0 a1,n−1 INt Nr = .. . INt Nr

···

 0 a2,n−1 INt Nr  . 0Nt Nr

(14)


TABLE I SUMMARY OF THE FUZZY-BASED KALMAN FILTER

Because the mobile velocity is unknown to the system, the fuzzy-based Kalman filter is proposed for nonlinear state estimation. Index m is omitted in the next section since the procedure of the fuzzy-based Kalman filter computation is the same for all subchannels. B. Fuzzy-Based Kalman Filter for a Time-Varying Channel Estimation The T-S fuzzy system can be viewed as a somewhat piecewise linear function, where the change from one piece to the next is smooth rather than abrupt due to the interpolation of the weighting functions. Since the state-space model is linear if the velocity of the mobile station is known, the T-S fuzzy system can establish the rules based on the velocity of the mobile station. The nonlinear estimation problem is now interpolated by P linear fuzzy systems. The pth linear system for the pth rule is proposed in the following form:

If vˆn−1 is Fp

rn = Jn Xn + vn

Dp,n−1

1  = 0 .. .

0 a1,n−1 (p)INt Nr

(15)

...

 0 a2,n−1 INt Nr   0Nt Nr

INt Nr

vˆn−1 is the premise variable, a1,n (p) is the coefficient of the AR(2) process that corresponds to the velocity of the mobile station, and Fp is the fuzzy set. In the proposed fuzzy scheme, the estimated velocity of the previous time instant n − 1 by the AR(2) parameter can be used as the premise variable of the fuzzy model in (15) at time n. Owing to the fact that the velocity of the mobile station quickly varies, this information can be seen as prior information for the fuzzy model in (15) to estimate the current time channel gain. The state dynamics and the output of the fuzzy measurement system in (15) are implied as follows [32], [34], [37]: P vn−1 )(Dp,n−1 Xn−1 + Wn ) p=1 Fp (ˆ Xn = P vn−1 ) p=1 Fp (ˆ P

p=1

rn = Jn Xn + vn

(19)

µp (ˆ vn−1 )Dp,n−1 Xn−1 + Wn

(16)

p=1

rn = Jn Xn + vn

(17)

P p=1

µp (ˆ vn−1 )Dp,n−1 Xn−1 (20)

denotes the fuzzy approximation error between the nonlinear system and the fuzzy model. Suppose that the covariance of the fuzzy approximation error is bounded as

(21) E [∆G(Xn−1 )] [∆G(Xn−1 )]H ≤ QI2N tN r+1 . The upper bound Q can be regarded as the worst-case approximation error and shall be included into the covariance of the driving noise Wn in the sequel. Based on the fuzzy model (15) and the Kalman filter, the following fuzzy estimator is proposed to deal with the state n from the measurement rn in (12): estimation X Estimation Rule p: If vˆn−1|n−1 is Fp p,n|n = Dp,n−1 X n−1|n−1 + Kp,n en then X

n|n−1 en = rn − Jn X

(23)

and Kp,n is the corresponding fuzzy-based Kalman gain for the pth estimation rule, with p = 1, . . . , P . Here, en denotes the prediction error. The Kalman gain Kp,n is employed to minimize the covariance E{en eTn } of the prediction error. The overall procedure is depicted in Table I. If the state estimation has been performed by the fuzzy estimator in Table I, the corresponding fuzzy-based channel estimation is obtained by ˆ n|n = UX n|n h

(24)

where µp (ˆ vn ) = 1.

(22)

where

where Fp (ˆ vn ) is the grade of the membership function of vˆn in vn ) is denoted as Fp , and the fuzzy base µp (ˆ vn ) Fp (ˆ , vn ) = P µp (ˆ vn ) p=1 Fp (ˆ

P p=1

for p = 1, 2, . . . , P

=

Xn = G(Xn−1 ) + Wn P = µp (ˆ vn−1 )Dp,n−1 Xn−1 + ∆G(Xn−1 ) + Wn

∆G(Xn−1 ) = G(Xn−1 ) −

Then Xn = Dp,n−1 Xn−1 + Wn



The fuzzy model in (16) can be interpreted as an interpolation of P linearized systems according to P different mobile velocities vn ) to approximate the in (15) through the fuzzy bases µp (ˆ nonlinear system in (11). Therefore, (11) can be expressed as

where

Rule p :

where

1603

(18) U = [ 0Nt Nr ×1

INt Nr

0Nt Nr ] .

1604


Hence, the estimated channel state of the nth OFDM symbol of the mth subchannel can be obtained. Since the Kalman filter is a stable estimation method [31] and the fuzzy-based Kalman filter is a collection of several Kalman filters with different weights, the proposed estimation method is stable. IV. ROBUST F UZZY-BASED D ECISION -D IRECTED A LGORITHM IN THE T RACKING M ODE A. Decision-Directed Algorithm As the transmitted signal is unknown for the receiver in the tracking mode, the detected data (instead of the transmitted signal) are fed to the fuzzy-based Kalman filter for channel identification. This is known as decision-directed channel tracking. The occupancy of available bandwidth due to the overhead symbol can be avoided in this method. Based on the statespace model for the nonlinear estimation in (11) and (12), the decision-directed channel-tracking algorithm is proposed here. The block diagram of the proposed decision-directed channel tracking is shown in Fig. 2. In the tracking mode, the detected n is available (which will be stated in the next section) symbol S and can be used to identify the channel gain for the MMSE data-detection scheme. The state-space model for the decisiondirected channel tracking of the mth subchannel is rewritten as Xn (m) = G (Xn−1 (m)) + Wn (m) n (m)Xn (m) + vn (m) ˜rn (m) = J

(25)

Cn (m) = INr ⊗ C ( sn (m)) . n (m), instead of Jn (m), and Cn (m), instead of Cn (m), Thus, J are fed to the fuzzy-based Kalman filter. Decision-directed channel tracking is now complete. Then, the prediction channel gain is obtained as n|n−1 = X

p,n|n−1 µp (ˆ vn−1|n−1 )X

(26)

p=1

ˆ n|n−1 = UX n|n−1 . h

rn = O(Hn )sn + vn

(27)

Since the performance of the MMSE equalizer with the decision-directed scheme can be improved by reliable channel prediction, the predicted channel gain will be used for the MMSE equalizer. B. Robust MMSE Equalizer As the transmitted signal Sn (m) is unavailable for the receiver in the tracking model, it is directly replaced by the n (m) for channel parameter tracking. In this detected signal S way, the current channel-parameter tracking is done after the current data detection. However, the MMSE equalizer requires the current channel gain. There is a delay problem in the

(28)

˘ n , sn = S ˘ n , vn = V ˘ n , and the real matrix where rn = R O(Hn ) ∈ R2Nr Ts ×2K is given by O(Hn ) = A1˘Hn · · · Ak˘Hn j B1˘Hn · · · j Bk˘Hn . (29) It has been shown that, for any channel matrix Hn , matrix O(Hn ) satisfies the so-called decoupling property [46], i.e., the columns are orthogonal to each other and have an identical norm, which means OH (Hn )O(Hn ) = Hn 2F I2K .

n (m) is the augmented matrix, which is defined as where J n (m) = 0N N ×1 Cn (m) 0N N J t r t r

P

decision-directed tracking scheme. A conventional decision n−1 directed scheme [27], [28] adopts the previous estimator H for the current signal detection Sn . It is based on the assumption that the channel variation is slow. However, when the channel variation is fast, the previous estimates will be unsuitable for the current signal detection. To design the MMSE equalizer at n must be predicted time n, the channel frequency response H from the previous parameter estimation results to overcome the inherent delay problem of the decision-directed tracking scheme. Since the fuzzy-based Kalman filter can estimate the channel gain and AR(2) parameters, as illustrated in (26) and (27), a robust equalization scheme will be developed for the fast-fading channel situation. The received signal for each subcarrier in (5) can be rewritten as

(30)

Lemma 2: Let O(·) be described in (29) and the OSTBC be based on the amicable orthogonal design in (4). Denote Hn = n|n−1 , where H n|n−1 is the prediction error. Then, n|n−1 + H H the real channel gain after the OSTBC can be formulated as follows: n|n−1 ) + O(h n|n−1 ). O(Hn ) = O(h

(31)

Proof: The proof is given in Appendix A. According to Lemma 2, the received signal in (28) can be rewritten as rn = O(Hn )sn + vn n|n−1 )sn + O(H n|n−1 )sn + vn = O(H n|n−1 )sn + v ˜n = O(H

(32)

n|n−1 )sn + vn . The design objective of the ˜ n = O(H where v MMSE equalizer for each subcarrier is to find an optimal opt based on the channel prediction such that equalizer Φ n 2 n rn − sn opt = arg min E Φ Φ (33) . n n Φ Theorem 1: Considering a received signal in (32), the robust fuzzy optimal equalization for (33) with the channel prediction


is given as follows: opt = Φ n

P

+ p,n|n−1 ) µp (ˆ vp,n−1|n−1 )OH (H

p,n|n−1 ) + p,n|n−1 )O (H × O(H H

p=1

δp2 I2Nr T

−1

p,n|n−1 )+δ 2 I2N T p,n|n−1 )OH (H × O(H p r (34)

p,n|n−1 2 + σ 2 /K for p = 1, 2, . . . , P , and where δp2 = H v F p,n|n−1 2 can be obtained from the prediction error, H F Pp,n|n−1 in Table I, of the proposed fuzzy-based Kalman filter. Proof: The proof is given in Appendix B. The resulting MMSE estimate of sn is rn . ˆsn = Φ n opt

(35)

n = [IK jIK ]ˆsn The regenerated signal can be computed as S and be fed to the fuzzy-based Kalman filter for decisiondirected channel tracking. The fuzzy-based equalization in (34) is adopted here by the combination of each MMSE equalizer with different mobile velocities. It is worth noting that the conventional decision-directed scheme adopts the previous estimates to perform the current c is MMSE equalization. Its conventional MMSE equalizer Φ n −1 2 n−1 ) O(H n−1 )OH (H n−1 ) + σv I c = OH (H . Φ n−1 K (36) It is based on the assumption that the channel variation is slow. Since the design procedure considers the covariance of p,n|n−1 2 to be a kind of noise the channel prediction error H F in the design procedure to compensate for the prediction error in the channel estimation, the proposed equalization method in (34) is more robust than the conventional optimal equalizer. C. Performance Analysis The performance of the proposed symbol detection method for the MIMO-OFDM system is evaluated in this section. According to the results in Section IV-B, the detected signal ˆsn is given by opt rn = ˆsn = Φ n

P

p,n|n−1 ) µp (ˆ vn−1|n−1 )OH (H

p=1

p,n|n−1 )+δ 2 I2N T p,n|n−1 )OH (H × O(H p r n|n−1 )sn + × O(H

P

p,n|n−1 )+δ 2 I2N T p,n|n−1 )OH (H × O(H p r P p,n|n−1 ) = sn + µp (ˆ vn−1|n−1 )OH (H p=1

−1

−1

−1

˜ n . (37) v

The second term of (37) is the residual ISI, and the last term is the filtered additive Gaussian noise. Since the estimated term is assumed to be much larger than the estimation residual term, p,n|n−1 ) δ 2 I2N T , the residual ISI p,n|n−1 )OH (H i.e., O(H p r term can be ignored in comparison with the last term. The covariance of the residual ISI and the filtered noise can be approximated as δ 2 p,n|n−1 )Ξδ ΞH O(H p,n|n−1 ) µp (ˆ vp,n−1|n−1 )OH(H δ 2 p=1 (38) P

Cv˜ ≈

where −1 p,n|n−1 )OH (H p,n|n−1 ) + δ 2 I2N T Ξδ = O(H p r

(39)

and δ 2 = P vp,n−1|n−1 )δp2 . Therefore, the signal-top=1 µp (ˆ interference-noise ratio per subcarrier on the nth OFDM block is equal to the following equality after some derivations: ηf = where

Ωf = Tr δ 2

P

2KNr T Ωf

(40)

p,n|n−1 ) µp (ˆ vp,n−1|n−1 )OH (H

p=1

×

Ξδ ΞH δ O(Hp,n|n−1 )

.

By the definition in [46], the probability of a symbol error rate (SER) is given by Pe ≈ Q(ηf ) √ 2 ∞ where Q(x) = 1/ 2π x e−(t /2) dt for x > 0.

(41)

V. C OMPUTER S IMULATION

p,n|n−1 ) µp (ˆ vn−1|n−1 )OH (H

p=1

p,n|n−1 ) µp (ˆ vn−1|n−1 )OH (H

p=1

P

1605

In this section, extensive computer simulations have been run to confirm the performance of the proposed method. The parameters of the simulated MIMO-OFDM systems are described here before presenting the simulation results.

˜n v

−1 p,n|n−1 )+δ 2 I2N T p,n|n−1 )OH (H × O(H p r n|n−1 )−I sn O(H

A. Parameters of the MIMO-OFDM Systems Table II shows the power delay profile in the urban area with the root-mean-square delay στ = 1 µs. Mobile velocities from 5 to 200 km/h are used to simulate different mobile environments. The central frequency fc is 2 GHz in the MIMO-OFDM system. The available bandwidth BW is 2.048 MHz, which is divided into 512 subchannels. These correspond to a subcarrier

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TABLE II POWER DELAY PROFILE IN THE TYPICAL URBAN AREA

Fig. 4. NMSEs of channel estimation by different channel-tracking methods with time-varying velocity for a 2 × 2 MIMO-OFDM system in the Rayleigh fading channel.

Fig. 3. Membership functions of the fuzzy rules.

symbol rate of 4 kHz and an OFDM symbol duration of 312.5 µs. The number of cyclic prefixes (CPs) Ncp = 128 is used to provide protection from ISI due to the channel multipath delay spread. In the following simulations, two transmitter antennas and two receiver antennas are used (Nt = 2, Nr = 2), and the Alamouti code (T = 2) is adopted [2]. It is also assumed that the channel remains approximately constant during one OFDM symbol period. The data modulation scheme is 4-QAM in the succeeding simulations. B. Simulation Results By running the simulation for 50 trials and 1000 OFDM block symbols per trial in this example, the performance of the system is measured in terms of the SER and the normalized MSE (NMSE) of the channel estimation. The NMSE for the mth subcarrier is defined as 2 E Hn (m) − Hn (m)

. (42) NMSE(m) = E Hn (m)2 In all cases, we normalize the gains of delay paths, so that L−1

E |hn (l)|2 = 1.

l=0

The velocity of a mobile station is a random variable that varies with time in a real-world application. Thus, a fuzzybased Kalman filter has been proposed to jointly estimate the dynamics and path gains of the channel. In this simulation example, we consider three situations to be the fuzzy sets Fp (P = 3), as shown in Fig. 3, each representing one of three different mobile velocities: low, medium, or high speeds.

To simulate a velocity-varying mobile radio channel, we first assume that the velocity linearly varies in 1 s. Following this, the initial and final velocities in one second are random variables uniformly distributed in the range of 5 km/h (fD = 9.25 Hz) to 200 km/h (fD = 370 Hz) in each trial. The compared methods are the algorithms that can simultaneously provide the channel estimation and prediction. The NMSEs, using different channel estimation methods, are shown in Fig. 4. The results of the first-order Kalman filter [28] and the second-order Kalman filter [29] were failures because these two methods adopt fixed dynamics. The EKF is less accurate than the proposed fuzzy-based Kalman filter, although it can also be used for nonlinear channel estimation. Due to the fuzzy interpolation of the three local estimations at low-, medium-, and high-speed mobile velocities, the proposed method can obtain the best result for channel estimation among these methods. The main drawback of the pilot-symbol-aided channel estimation method is the occupancy of the bandwidth, which is a valuable resource for service providers [24], [25]. The conventional decisiondirected channel-tracking scheme can only adopt the previous estimate for current decoding and detection. The performance of these detectors could be deteriorated by unsuitable estimates suffering from the fast-fading rate [10]. Therefore, an accurate channel prediction is necessary to solve the inherent delay problem of a decision-directed scheme. The NMSEs of channel prediction using different decision-directed channel-tracking algorithms are compared in Fig. 5. It can be seen that, even without the overhead symbols, the proposed method can still predict better than the others when the velocity is time varying. To compare the SER, the semiblind approach [14] and the MAP-based method [43] are included for comparison. The results of MMSE equalizers using different channel-tracking schemes are compared in Fig. 6, with the performance of the perfectly known channel as the lower bound. The performances of these fixed dynamic methods are bounded by the large estimation error. We observed that the channel tracking of the proposed method can perform the best decoding and detection


Fig. 5. NMSEs of channel prediction by different channel-tracking methods with time-varying velocity for a 2 × 2 MIMO-OFDM system in the Rayleigh fading channel.

Fig. 6. SERs of different channel-tracking methods with time-varying velocity for a 2 × 2 MIMO-OFDM system in the Rayleigh fading channel.

of all these methods because it can most accurately predict and estimate the channel. The SER comparison between the proposed enhanced MMSE equalizer and the conventional MMSE equalizer is shown in Fig. 7. As mentioned in Section IV-B, the estimated CSI in the previous time is not suitable for the present symbol detection if the channel is fast fading. The proposed enhanced MMSE equalizer has better symbol detection performance than the conventional equalizer with high-signalto-noise-ratio (SNR) environment since the proposed channel estimator provides precise one-step-ahead prediction. VI. C ONCLUSION A robust fuzzy MIMO-OFDM channel estimation method with time-varying mobile velocity has been proposed in this paper. The channel model has been described by a nonlinear state-space dynamic equation. The states of the nonlinear channel parameter system include the time-varying channel

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Fig. 7. SER comparison of the proposed MMSE and conventional MMSE with time-varying velocity for a 2 × 2 MIMO-OFDM system in the Rayleigh fading channel.

gains and the dynamics of the channel. The proposed method can estimate the channel state by interpolating three linear parameter models at low, medium, and high mobile speeds to approximate the nonlinear channel parameter system. The decision-directed channel-tracking method via a fuzzy-based Kalman filter has been designed to avoid occupation of the available bandwidth because it does not need pilot assistance. The inherent delay problem of the decision-directed scheme can also be solved by the prediction ability of the proposed method. Furthermore, the robust MMSE equalizer has been proposed to improve symbol detection performance while factoring in the channel prediction error. Simulation results indicate that the proposed method can more accurately track the channel than existing methods. Therefore, the decision-directed channel tracking of the proposed method for MIMO-OFDM systems is efficient for a mobile station with time-varying velocity over a fast multipath fading channel.

A PPENDIX A P ROOF OF L EMMA 2 n|n−1 + H n|n−1 , so As mentioned in Section IV-B, Hn = H the matrix in (29) can be rewritten by the linearity of the matrix computation, i.e., n|n−1 ) n|n−1 + h O(Hn ) = O(H

  ˘ ˘ ˘ ˘ n|n−1 n|n−1 n|n−1 n|n−1 A1 (H +H ) · · · Ak (H +H )

 = ˘ ˘ ˘ ˘ n|n−1 n|n−1 n|n−1 n|n−1 +H ) · · · jBk (H +H ) jB1 (H ˘n|n−1 ˘n|n−1 ˘n|n−1 · · ·Ak H ˘n|n−1 ·A · ·k H A1 H A1 H + = ˘n|n−1 · · ·j Bk H ˘n|n−1 ˘n|n−1 · · ·j Bk H ˘n|n−1 j B1 H j B1 H n|n−1 ) + O(H n|n−1 ). = O(H The lemma is proven.

(A.1)

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A PPENDIX B P ROOF OF T HEOREM 1

and high speed of the mobile), the MMSE equalizer can be obtained as

When considering the modified receive signal equation (32), the goal of optimal equalization is to solve the following MMSE equalization problem: 2 n rn − sn n = arg min E (B.1) Φ . Φ n Φ Assume that sn and vn are independent. The MSE of each fuzzy rule in (B.1) can be written as

p,n rn − sn 2 E Φ 2 ˜ p,n − sn = E Φp,n O(Hp,n|n−1 )sn + Φp,n v

p,n|n−1 )sn sH OH (H p,n|n−1 )Φ H p,n O(H =E Φ p,n n H p,n O(H p,n|n−1 )sn sH + Φ p,n v H ˜ p,n v ˜H −Φ n p,n Φp,n −sn sn p,n|n−1 )Φ H + sn sH × OH (H p,n n

2 p,n|n−1 ) − I Φp,n O(H 2 p,n|n−1 2 + σv Φ p,n 2 + Kh F 2 2 ! 2 K σ v p,n|n−1 2 + I p,n O(H p,n|n−1 ) h = [I0] − Φ . F 2 K =

K 2

(B.2) Note that " # ˜H ˜ p,n v E v p,n H p,n|n−1 )sn + vn O(H p,n|n−1 )sn + vn = E O(H

# " p,n|n−1 )sn sH OH (H p,n|n−1 ) + E vn vH = E O(H n n 2 K σ2 (B.3) Hp,n|n−1 + v · I2N rT . F 2 2 p,n that By letting the gradient of (B.2) be zero, matrix Φ minimizes the MSE equation is equal to p,n|n−1 ) p,n = OH (H Φ −1 p,n|n−1 )OH (H p,n|n−1 ) + σ 2 I2N rT × O(H p

p = 1, . . . , P

(B.4)

p,n|n−1 2 + σ 2 /K. Note that H p,n|n−1 2 where σp2 = H v F F can be extracted from the prediction error Pp,n|n−1 of the proposed fuzzy-based Kalman filter in the pth fuzzy rule. In this paper, according to the proposed fuzzy-based Kalman filter via fuzzy bases of the premise situations (i.e., low, medium,

opt = Φ n

P

p,n µp (ˆ vn−1|n−1 )Φ

(B.5)

p=1

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Bor-Sen Chen (F’01) received the B.S. degree from Tatung Institute of Technology, Taipei, Taiwan, in 1970, the M.S. degree from National Central University, Chungli, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor with Tatung Institute of Technology from 1973 to 1987. He is currently the Tsing Hua Chair Professor of electrical engineering and computer science with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan. He is an Editor of the Asian Journal of Control and a Member of the Editorial Advisory Board of Fuzzy Sets and Systems and the International Journal of Control, Automotion and Systems; and was the Editor-in-chief of the International Journal of Fuzzy Systems from 2005 to 2008. He is currently the Editor-in-Chief of the International Journal of Systems and Synthetic Biology and a member of Editorial Board of BMC Systems Biology. His current research interests are control engineering, signal processing, and systems biology. Dr. Chen is a Research Fellow of the National Science Council of Taiwan and holds the excellent scholar Chair in engineering. He was an Associate Editor for the IEEE T RANSACTIONS ON F UZZY S YSTEMS from 2001 to 2006. He was the recipient of the Distinguished Research Award from the National Science Council of Taiwan four times, the Automatic Control Medal from the Automatic Control Society of Taiwan in 2001.

Chang-Yi Yang received the B.S. and M.S. degrees in control engineering from National Chiao Tung University, Hsinchu, Taiwan, in 1983 and 1988, respectively, and the Ph.D. degree in electriccal engineering from National Tsing Hua University, Hsinchu, in 1992. From 1985 to 1986, he was an Assistant Researcher with Mechanical Industry Research Laboratories, Industrial Technology Research Institute, Hsinchu. From 1992 to 1996, he worked on widearea networking with Southern Information Systems Inc., Hsinchu. He was with the Department of Electronic Engineering at Vanung University, Jhongli, Taiwan, from 1996 to 2004. Since 2004, he has been with the Department of Computer Science and Information Engineering, National Penghu University of Science and Technology, Magong City, Taiwan, where he currently is an Associate Professor. His research interests include communication theory and its applications to wireless networks.

Wei-Ji Liao received the B.S. degree from National Yunlin University of Science and Technology, Yunlin, Taiwan, in 2007 and the M.S. degree from National Tsing Hua University, Hsinchu, Taiwan, in 2010, both in electrical engineering. He is currently a Hardware Engineer with Compal Communications Inc., Taipei, Taiwan. His research interests include wireless communication, signal processing, and channel estimation for communication systems.