Channel Estimation With Expectation Maximization and ... - IEEE Xplore

Received July 28, 2017, accepted August 13, 2017, date of publication August 29, 2017, date of current version February 1, 2018. Digital Object Identifier 10.1109/ACCESS.2017.2745708

Channel Estimation With Expectation Maximization and Historical Information Based Basis Expansion Model for Wireless Communication Systems on High Speed Railways XIYU WANG 1 School

1,2 ,

GONGPU WANG1,2 , RONGFEI FAN3 , AND BO AI2,4 , (Senior Member, IEEE)

of Computer and Information Technology, Beijing Jiaotong University, Beijing 100044, China 2 Beijing Engineering Research Center of High-Speed Railway Broadband Mobile Communications, Beijing, 100044, China 3 School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China 4 State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China

Corresponding author: Gongpu Wang ([email protected]) This work was supported in part by the ZTE Corporation, in part by the National Key Research and Development Program under Grant 2016YFB1200102-04, in part by the Natural Science Foundation of China under Grant U1334202, and in part by the National S&T Major Project under Grant 2016ZX03001021-003.

ABSTRACT This paper proposes a blind channel estimator based on expectation maximization algorithm and historical information-based basis expansion model for uplink wireless communication systems on high speed railways. The information of basis matrices is obtained from the uplink data of the past trains at the base station (BS). With the known basis matrices at the BS, our suggested estimator can estimate the basis coefficients and recover the channel parameters without requiring training symbols. The modified Cramer– Rao bound is derived for the estimated basis coefficients and the computational complexity of the proposed estimator is analyzed. Numerical results are then provided to corroborate our studies. It is shown that the proposed estimator outperforms existing data-aided estimators, including least square and linear minimum mean square error. INDEX TERMS Basis expansion model, channel estimation, expectation maximization, high speed railways.

I. INTRODUCTION

With the rapid development of high speed railways (HSRs) in China, there emerge numerous requirements for high data rate wireless access on HSRs [1]–[3]. HSR environment abounds in multipath channels [4]. Current wireless communication systems on HSR usually utilize orthogonal frequency-division multiplexing (OFDM) to switch the frequency-selective channels into a series of parallel, orthogonal frequency-flat channels [5]. A key challenge for OFDM systems on HSR is the timevarying channel estimation [6]. Existing estimators can be classified into two categories [7]. One approximates varying channels with the linear relations between OFDM symbols [8]–[10]. In [8], the time-varying channel impulse responses (CIR) are assumed to be linearly varied within OFDM symbols. In [9], the time variations of the wireless channels are approximated with a piece-wise linear model 72

with a constant slope over the symbol. Liu et al. [10] estimate the banded channel matrices by manipulating a variety of equalizations. Another type of estimators adopt basis expansion models (BEM) to convert the problem of estimating the CIR to the basis weights [11]–[17], which can decrease the number of parameters to be estimated. For example, in [11], the channel and inter-carrier interference (ICI) are considered simultaneously by developing the correlations in time and frequency domain, and the basis coefficients are estimated through the linear minimum mean square error (LMMSE) approach. A two-dimensional BEM is proposed in [12] to estimate the channel expansion for an OFDM frame and achieves the complexity of O(N 2 ). In [13], channel estimation is based on compressed sensing so as to utilize the channel sparsity and ICI is eliminated by exploiting the location information of trains.

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VOLUME 6, 2018

X. Wang et al.: Channel Estimation With EM and HiBEM for Wireless Communication Systems on HSRs

Typical BEMs include polynomial, wavelet, complex exponential bases, and Discrete Prolate-Sphere (DPS) sequence [14]. Nonetheless, when the characteristics of channels vary rapidly with the movement of the train on HSR, these BEMs fail to provide consistent performance due to the same basis matrices in approximating channels [18]. The work [15] studies the characteristic of HSR and proposes a historical information based BEM (HiBEM). It investigates the correlation matrix of channels within one transmitting block which would changes with the scenarios of the trains. Simulation results are provided to verify that HiBEM outperforms aforementioned BEMs [15]. In addition, expectation maximization (EM) algorithm, an iterative method to obtain the maximum likelihood (ML) estimate of the unknown parameters [19], has been incorporated with BEM to solve a series of problems including channel estimation [16], [17]. A joint carrier frequency offset (CFO) and channel estimator is designed in [16]. The authors explore the autoregressive (AR) model to approximate BEM coefficients and utilize the coefficients as latent variables to estimate CFO. In [17], the received signals are viewed as latent data and the transmitted signals are chosen as complete data to estimate basis coefficients. These works ( [16] and [17]) are data-aided channel estimators which require pilots. EM algorithm has also been utilized in the blind estimation [20] of the self-interference and communication channels in the fullduplex systems. However, the channels in [20] are supposed to be flat fading, which is not suitable for the HSR scenarios. In this paper, we address the problem of time-varying channel estimation on HSR. A proper channel estimator not only provides accurate channel state information but also requires less training symbols. Moreover, the estimator should have low complexity so as to accommodate the high mobility of HSR scenario. In order to meet the two criteria, we propose BEM-EM estimator, a EM-based blind channel estimator with the framework of BEM for the OFDM system. We adopt HiBEM to handle the special features of the HSR, of which the basis matrices can be calculated from uplink data of the past trains. Then the EM algorithm is exploited to estimate the basis weights. The transmitted signals are viewed as latent data so that it is not required to know the exact information of transmitted signal, which implies no need of training symbols. The main contributions of this paper can be summarized as follows: (1) We propose a BEM-EM estimator to obtain the timevarying channel parameters for wireless communication systems on HSR. The proposed BEM-EM does not need any training symbols and can outperform the existing data-aided estimators including least square (LS) and LMMSE. (2) We analyse the computational complexity of BEM-EM and show that BEM-EM estimator has lower complexity than the existing estimators. VOLUME 6, 2018

(3) We derive a modified Cramer-Rao bound (MCRB) of the estimate of the basis coefficients. The rest of paper is organized as follows. Section II introduces the HSR wireless communication system models. Section III describes the HiBEM as well as our proposed BEM-EM channel estimator and analyses the complexity of the estimator. Section IV derives the MCRB of the estimated basis coefficients. Simulation results are illustrated in section V. Finally, Section VI concludes the paper. Notations: Vectors and matrixes are boldface small and capital letters, respectively; (A)T , (A)H , (A)−1 , (A)∗ , (A)† denote the transpose, Hermitian, inverse, conjugate and pseudo-inverse of A, respectively; R{·}, J{·} represent the real and the imaginary part of the complex argument inside, respectively; the (i, j)th element of matrix A is denoted by Ai,j and the ith element of vector a is denoted by ai ; the Matlab notation A(n1 : n2 , m1 : m2 ) implies to extract a submatrix within A from row n1 to n2 and from column m1 to column m2 , and A(n1 : n2 , :) indicates that all the elements from row n1 to n2 are extracted. I is the identity matrix and d·e is the integer ceiling √ function; E[·] denotes the statistical expectation and  = −1 denotes the imaginary unit; Tr{·} is y the trace operation and ∇x represents the second-order partial derivative operator of the vector x and y.

FIGURE 1. Full-duplex relaying communication model on high speed railway.

II. SYSTEM MODEL

Consider a typical wireless communication system on HSR shown in Fig. 1 [21]. The base station (BS) is deployed along the railways and covers the communication within one cell. The relay station (RS) is mounted on the roof of the carriage to communicate with the BS and the user equipments in the train. Here we consider the wireless communication that the RS transmits data symbols to the BS and both of them are configured with a single antenna. The data symbols encounter reflections, refraction and diffraction caused by scatterers during the transmission. Nonetheless, there exists strong lineof-sight (LOS) component in most locations [4], [22]. Denote c(0), c(1), · · · , c(L − 1) as the frequency selective channels. The typical velocity of train can be up to 360 km/h, i.e., the speed v = 100 m/s. Suppose the carrier frequency fc is 3.5 GHz, which is widely accepted for wireless communications on HSR by most of countries [4]. The maximum 73


Doppler shift at the train antenna is v fd = fc · ≈ 1167Hz, (1) ν where ν is the light speed, i.e., ν = 3 × 108 m/s. Thus the coherence time can be computed as Tc = 1/(4 fd ) ≈ 0.214 ms. According to the long term evolution (LTE) standards [23], one OFDM symbol duration is approximately 0.0714 ms. Therefore, we assume that the multipath channels are time-invariant within one OFDM symbol but can vary from symbol to symbol.1

Taking the fast fourier transform (FFT) of both sides of (3) yields ˜ = Hx + w, y = F˜y = FCFH F˜x + Fw

(5)

where F is the N -point FFT matrix, H = FCFH denotes the channel matrix in the frequency domain, and w denotes the noise vector in frequency-domain. Since the channel is frequency selective, C is a cyclic matrix and H is a diagonal matrix. The ith diagonal element of H represents the channel frequency response (CFR) of the ith subcarrier. Defining h = diag{H} = [H1,1 , H2,2 , · · · , H(N −1),(N −1) ]T , we have 1 h = √ F(:, 1 : L)c, N

(6)

where c = [c(0), c(1), · · · , c(L − 1)]T is a L × 1 vector, and F(:, 1 : L) is the submatrix of F with the first L rows. It can be readily checked that (5) can be rewritten as y = Xh + w,

(7)

where X is an N × N matrix after diagonalization of x, i.e., X = diag{x}. III. BEM-EM ESTIMATOR

FIGURE 2. Discrete baseband equivalent system model.

Fig. 2 depicts the discrete baseband equivalent system model. The OFDM symbol in frequency domain has N subcarriers, denoted by x = [x(0), x(1), · · · , x(N − 1)]T . Then convert x to time domain through inverse fast Fourier transform (IFFT) x˜ = FH x,

(3)

˜ denotes the time-domain noise vector, and C is an where w N × N time-domain channel matrix which can be written as   c(0) 0 · · · c(L −1) · · · c(2) c(1)  c(1) c(0) ··· 0 · · · c(3) c(2)    .. .. .. ..  . . .. .. ..  .  . . . .     . . . .. .. 0 C = c(L −1) c(L −2) . . 0     . .. .. ..  .. .. ..  .. . . . . . .     0 0 · · · c(L −3) · · · c(0) 0  0 0 · · · c(L −2) · · · c(1) c(0) (4) where c(l) is the lth (0 ≤ l ≤ L − 1) path during one OFDM symbol period. 1 The channel is slow fading if the coherence time is much longer than the delay requirement of the application [5].

74

A. BEM CHANNEL MODEL

BEMs model the CFR of one OFDM symbol as h = Bλ =

(2)

where FH is the N point IFFT matrix. The transmitter adds a cyclic prefix (CP) of length L −1 to the x˜ . The received signal after discarding the CP in time domain is given as ˜ y˜ = C˜x + w,

In this section, we first introduce the principle of HiBEM and the estimation scheme which obtains the basis matrices from the past trains. Next we propose the BEM-EM estimator which employs EM algorithm to obtain the basis coefficients of BEM.

Q X

λq bq ,

(8)

q=1

where B = [b1 , b2 , · · · , bQ ] represents the basis matrix, bq (1 ≤ q ≤ Q) denotes the qth basis vector, λ = [λ1 , λ2 , · · · , λQ ]T is the BEM coefficients vector, and Q denotes the number of coefficients which should be no less than 2dfd NTs e in order to provide sufficient degrees of freedom. Substituting (8) into (7), the input-output equation can be written as y = XBλ + w.

(9)

The BEM scheme converts the estimate of CFR into the estimate of BEM coefficients λ. Clearly, the number of parameters to be estimated decreases from N to Q. B. HiBEM

Trains on HSR follow the fixed tracks. Accordingly, the reflectors and obstructions along the tracks stay the same at the same location. Thus channels in a fixed position of different trains are strongly related and the correlation may remain unchanged over time only if the environment and transceiver parameters are unvaried [24]. In our previous VOLUME 6, 2018


work [15], we proposed HiBEM which applied the Q eigenvectors of channel’s autocorrelation matrix as its basis matrix. The autocorrelation matrix is calculated from Rh = E{hhH }.

(10)

Take the singular value decomposition of Rh will yield Rh = U3V.

(11)

Hence we can construct the basis matrix from U, whose columns are the eigenvectors of Rh . The basis matrix of HiBEM, which is composed of the first Q eigenvectors of Rh , can be expressed as B = U(:, 1 : Q).

(12)

Therefore, the conditional probability function (PDF) of yi is given by |yi − xi Bi λ|2 1 exp − . f (yi |xi ; λ) = π σw2 σw2

density

(15)

The transmitted signal xi is the unobserved latent data. For the brevity of our discussion, we assume that the signals are binary phase shift keying (BPSK) and follow uniform distribution, i.e., p(+1) = p(−1) = 0.5. Next we will derive the BEM-EM estimate of λ. Note that our following analysis can be directly extended to multiple phase shift keying (MPSK). Suppose there are totally D possible values of a transmitted symbol in the case of MPSK, and denote xd (1 ≤ d ≤ D) as the transmitted symbol. Summing the joint PDF f (yi , xd ; λ) over all the possible value of xd , we can compute the marginal PDF of yi as X f (yi ; λ) = f (yi , xd ; λ) xd

=

X

f (yi |xd ; λ)p(xd )

xd

|yi − xd Bi λ|2 1 X . exp − = Dπ σw2 x σw2

(16)

d

The log-likelihood function of λ can be expressed as L(λ) = ln f (Y; λ) =

FIGURE 3. Scheme of historical information application on HSR.

k−1

(13)

i=1

to approximate the covariance matrix of the kth uplink channel, i.e., Rh ≈ R. C. CHANNEL ESTIMATION ALGORITHM

Given the system model in (9), the received signal of the ith subcarrier can be written as yi = xi Bi λ + wi ,

(14)

where xi is defined as the transmitted signal, Bi = B(i, :) is defined as the ith row of basis matrix,2 λ is the basis coefficient to be estimated and wi is complex Gaussian noise with zero mean and variance σw2 . 2 It is worth noticing that B i [Bi,1 , Bi,2 , · · · , Bi,Q ]. VOLUME 6, 2018

is a row vector, i.e., Bi

ln f (yi ; λ)

i=0

One key problem is the computation of Rh . Since channels on HSR has strong relativity within one OFDM symbol time, we take this characteristic into consideration and design the scheme of calculating the correlation matrix Rh . As shown in Fig. 3, supposing the current train is the kth train, we utilize the estimated channels of previous (k − 1) trains at the same position to compute the corresponding correlation matrices Rhi (1 ≤ i ≤ k − 1). Then the BS calculates 1 X R = E{Rhi } = Rhi , k −1

N −1 X

=

= −N ln(Dπ σw2 ) +

N −1 X i=0

ln

X xd

|yi − xd Bi λ|2 exp − . σw2

(17)

The ML estimate of λ can be found as λˆ = arg max L(λ). λ

(18)

Noting that the EM algorithm utilizes iteration to obtain ML estimation, we suppose the first estimate of λ through (18) is λ(1) , and the nth is λ(n) . The (n + 1)th iterative estimate can be computed by iterating the following two steps until desired convergence is realized. (1) Expectation step (E step): compute the expectation of log-likelihood function LB (λ|λ(n) ) , Ef (X|Y;λ(n) ) [ln f (Y, X; λ)].

(19)

(2) Maximization step (M step) : find the parameters of next iteration λ(n+1) which maximizes LB (λ|λ(n) ) λ(n+1) = arg max LB (λ|λ(n) ). λ

Our convergence criterion is LB (λ(n) ) − LB (λ(n+1) ) ≤ ,

(20)

(21)

where the threshold is a small constant. 75


TABLE 1. Computational complexity of BEM-EM algorithm.

The calculation of LB (λ|λ(n) ) in (E step) can be expressed as LB (λ|λ(n) ) N −1 X X = f (xd |yi ; λ(n) ) ln f (yi , xd ; λ)

=

i=0 xd N −1 X X i=0 xd

|yi − xd Bi λ|2 2 , f (xd |yi ; λ ) − ln(Dπ σw ) − σw2 (n)

(22) where f (xd |yi ; λ(n) ) denotes the posterior probability of xd when the received signal of each subcarrier and the latest estimate coefficient λ(n) are given. According to the Bayes formula [19], we can obtain f (yi |xd ; λ(n) )p(xd ) f (xd |yi ; λ ) = P (n) xd¯ f (yi |xd¯ ; λ ) o n (n) 2 d Bi λ | exp |yi −x−σ 2 w , Q(n) = i (xd ). P |yi −xd¯ Bi λ(n) |2 x ¯ exp −σ 2

Theorem 1: The basis coefficient of next iteration in (M step) is !−1 N −1 X X (n) (n+1) 2 H λ = Qi (xd )|xd | Bi Bi i=0 xd N −1 X X

(n)

Qi (xd )yi xd∗ BH i .

(24)

i=0 xd

Proof: See Appendix. Denote λopt as the estimated basis coefficient that satisfies the convergence criterion. The CFR of one OFDM symbol is given by hˆ = Bλopt

(25)

D. COMPLEXITY ANALYSIS

Here we briefly discuss the complexity of the proposed BEM-EM estimator in terms of the required complex multiplication operations. The BEM-EM estimator consists of two procedures: coefficient estimation and channel reconstruction. The complexity of the coefficient estimation part is for one iteration. It is computed by using (23) to calculate the posterior probability, (24) to obtain the coefficient of next iteration, and (22) for convergence criterion 76

CRB =

(23)

w

·

IV. MSE LOWER BOUND OF BASIS COEFFICIENT ESTIMATES

In this section we derive a lower bound of MSE for the estimation error of the basis coefficient λ. The standard form of Cramer-Rao bound (CRB) [26] is given by

(n)

d

judgement. And channel reconstruction can be computed by (25). The computational complexity of the proposed BEM-EM algorithm is summarized in Tabel 1. Each row of Table 1 displays the corresponding required operations of each step. Our BEM-EM estimator requires ND2 (3Q+6)+ND(Q2 + 5Q + 14) + Q3 complex multiplications per iteration. In a practical system, the value of system modulation order D and the number of basis coefficient Q are much lower than the number of subcarriers N [25]. Hence, the computational complexity for one iteration of BEM-EM estimator is lower than O(N 3 ).

1 1 , = λ I (f (y|λ)) Ey −∇λ ln f (y|λ)

(26)

where I (?) is the Fisher information matrix (FIM), Ey means the expectation is taken with respect to y, f (y|X; h) can be calculated via (15), and the second-order derivative operator with respect to the complex vector λ can be defined as ∂ ln f (y; λ) ∂ ln f (y; λ) H ∇λλ ln f (y|λ) = . (27) ∂λ∗ ∂λ∗ Unfortunately, in the non-training-symbol context, the calculation of FIM would be a tedious problem when taking the second-order derivative of f (y|λ) since the signals x are unknown. To tackle this problem, a modified CRB (MCRB) is proposed where the estimated parameters are assumed to be random with known priori information [27]. And the modified information matrix (MIM) is defined as h i M = EX [I (f (y|X; λ))] + Eλ −∇λλ ln p(λ) . (28) The first item of (28) is the modified FIM which can be computed as EX [I (f (y|X; λ))] =

N −1 X X

h i p(xd ) · Eyi −∇λλ ln f (yi |xd ; λ)

i=0 xd VOLUME 6, 2018


=

N −1 X X

p(xd )

i=0 xd

(

"

· −Eyi

∂ ln f (yi |xd ; λ) ∂λ∗

∂ ln f (yi |xd ; λ) ∂λ∗

N −1 1 X 2X T ∗ x = Bi Bi . Dσw2 x d d

H #)

(29)

i=0

For the second item of (28), we first obtain λ = (BH B)−1 BH h,

(30)

and calculate

FIGURE 4. Channel estimation MSEs versus SNR with different Q by BEM-LS and BEM-EM estimators.

h i Rλ = E λλH h i = E (BH B)−1 BH hhH B(BH B)−1 = (BH B)−1 BH Rh B(BH B)−1 .

(31)

Therefore, the PDF of λ can be expressed as n o 1 p(λ) = exp −λH R−1 (32) λ λ . π Rλ Taking the second-order derivative for (32) and averaging it with respect to λ, we can obtain the second part of (28) as h i Eλ −∇λλ ln p(λ) " # ∂ ln p (λ) H ∂ ln p (λ) = R−1 = Eλ − λ . (33) ∂λ∗ ∂λ∗ Consequently, using (29) and (33), the MCRB of the basis coefficients λ is given by n o MCRBλ = Tr M−1  !−1  −1  1 X NX  −1 2 T ∗ x B B + R = Tr . i i λ  Dσw2 x d  d

i=0

(34) V. SIMULATION RESULTS

In this section, we numerically investigate the performance of our proposed algorithm. For comparison, we also provide simulation results of other three traditional estimators: LS, LMMSE, and BEM-LS given as hˆ LS = X−1 y, hˆ LMMSE = Rh XH (XRh XH + σw2 I)−1 y, hˆ BEM−LS = (XB)† y,

(35) (36) (37)

respectively. To emulate the practical scenario on HSR, we investigate the multipath Rician channel models [5] which can be expressed as r r κ 1 ξl exp{ θ } + CN (0, ξl2 ), c(l) = δ(l − 1) κ +1 κ +1 (38) VOLUME 6, 2018

where c(l) denotes the CIR of the lth path, δ(·) is the Dirac’s delta function, κ is the Rician K-factor and ξl2 represents the power of the lth path in Watts, and CN (0, ξl2 ) indicates the complex Gaussian distribution. The phase shift caused by the high mobility of the train is defined as θ = 2π ε

n , N

(39)

where ε = fd /s is the relative Doppler shift, i.e., the ratio of Doppler shift fd to the subcarrier spacing s. The channel parameters in (38) is provided by measured parameters of tapped delay line model in the viaduct scenario. For each simulation run, we set 64 subcarriers within one OFDM symbol, i.e., N = 64. The carrier frequency is 3.5 GHz, the velocity of the train is set as 360 km/h, and the subcarrier spacing is set as 15 kHz. The transmitted symbols are generated by BPSK modulation, i.e. D = 2. The additive white Gaussian noise (AWGN) w is with zero mean and unit covariance. We utilize the coefficient estimated by BEM-LS to provide the initial information of BEM-EM. The number of pilots for the other three estimators are set as 16. Totally 104 Monte-Carlo runs are adopted for averaging. A. CHANNEL ESTIMATION WITH VARIED Q

In Fig. 4, we display the channel MSE of BEM-LS as well as BEM-EM estimators with the basis information perfectly known at the BS. Bringing in the configuration parameters, Q must be at least 2dfd NTs e = 1 in order to keep the sufficient degrees of freedom. We investigate the mean square error (MSE) for Q = 1, 2, · · · , 5. It can be observed from Fig. 4 that the proposed BEM-EM estimator has substantially lower MSEs than those of BEM-LS estimator for all cases of Q. It is also clear that the performance of the estimator decreases with the number of coefficients. This is partly because there exists strong LOS component in the viaduct scenario [24]. In addition, the CFRs within one OFDM symbol are highly correlated [4], thus the first eigenvector occupies almost all the energy. In the following example, we study the estimator with one basis coefficient, i.e., Q = 1. 77


FIGURE 5. Rician channel estimation MSEs versus SNR.

FIGURE 7. MSE comparison of basis coefficient between LS and EM approaches.

FIGURE 6. Channel estimation MSEs comparison for the proposed estimation scheme. FIGURE 8. BERs versus SNR of proposed blind estimator as well as traditional data-aided estimators.

B. CHANNEL ESTIMATION WITH PERFECT INFORMATION

We suppose the autocorrelation matrices of channel Rh are perfectly known at the BS. Fig. 5 compares the MSE performance of different estimators at the Rician channel condition. The black dotted line represents the standard CRB of estimated channels given by (26). As can be seen, LS estimator possesses the worst performance among all methods tested. BEM-LS and LMMSE have the similar estimation performance. And our proposed BEM-EM performs better than these data-aided estimators. In addition, there is an error floor in high SNR region. The reason is that the strong LOS components in the Rician channel model are intensely effected by Doppler frequency shift [28].

a comparison. It is found that the estimation MSE curves witness a downward trend with the increasing value of k and the performance when k = 4 approaches that of perfect channel knowledge. Consequently, utilizing the historical information of past trains is a feasible method of estimating channels on HSR. D. ESTIMATION OF BEM COEFFICIENT

C. CHANNEL ESTIMATION WITH HISTORICAL INFORMATION

Next we exploit the LS method and the EM method to estimate the basis coefficient, respectively. Fig. 7 illustrates the estimation MSEs of the coefficient versus SNR. For reference, the MCRB in (34) has been plotted. It can be seen that the EM algorithm outperforms LS method. The performance gap is pronounced at the low SNR region, while at high SNRs the performances of both estimators are nearly identical.

Noting that obtaining perfect knowledge of Rh is not practical, we study the estimation scheme described in Fig. 3. The covariance matrix Rh is calculated from the estimated channel of past (k − 1) trains. Then the BS uses the obtained Rh to estimate the channel of the kth train. Specifically, LS estimator is utilized to estimate channel when k = 1 for it requires no statistical information. MSEs of BEM-EM estimator are depicted in Fig. 6 for k = 1, 2, 3, 4 respectively. The case with perfect information of Rh is also evaluated as

Fig. 8 displays corresponding detection bit error rates (BERs) performance of the proposed blind estimator as well as the existing data-aided estimators. We also plot the BER curve with perfect channel state information as a reference. It is observed that the performances of LMMSE and BEM-LS are nearly identical and the proposed estimator is closer to the curve of perfect channel information.

78

E. DATA DETECTION

VOLUME 6, 2018


vector λ, we can obtain N −1 X h i X ∂g(λ) (n) = Qi −yi xd∗ BH + |xd |2 BH i Bi λ = 0. ∂λ x i=0

d

(A.2) By solving (A.2), it can be readily found that the stationary point is given by (24). REFERENCES

FIGURE 9. Average iteration times of BEM-EM estimator versus SNR for varied N.

F. ITERATION TIMES

Lastly, we examine the average iteration times of BEM-EM estimator for N = 16, 32, 64, and 128 respectively. From Fig. 9, we can find that it takes less iteration times with larger SNR and smaller number of subcarrier N . It can be seen from Fig. 9 that our BEM-EM estimator only needs two iteration times when SNR is large than 18 dB. VI. CONCLUSION

In this paper, we proposed a blind channel estimation algorithm based on EM algorithm and HiBEM for wireless communication systems on HSR. The suggested algorithm could estimate channel parameters without training sequence on condition than the channel covariance matrices are known at the BS. It was shown that the BEM-EM estimator had lower complexity than LS and LMMSE estimators. Numerical simulations were provided to illustrate that the proposed estimator could achieve better estimation performance of HSR channels than traditional data-aided estimators. APPENDIX PROOF FOR THE THEOREM 1

Let λ(n) be the coefficient vector obtained from the latest iteration. Then the M step is given by λ(n+1) = arg max LB (λ|λ(n) ) λ

= arg max λ

·

N −1 X X

(n)

Qi (xd )

i=0 xd

− ln(Dπ σw2 ) −

= arg min λ

N −1 X X

(n)

Qi (xd )|yi − xd Bi λ|2

i=0 xd

= arg min g(λ), λ

|yi − xd Bi λ|2 σw2

(A.1)

P −1 P (n) 2 where g(λ) = N xd Qi (xd )|yi − xd Bi λ| . Taking the i=0 first-order derivative of g(λ) with respect to the complex VOLUME 6, 2018

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XIYU WANG received the B.Eng. degree from North-East Electric Power University, China, in 2015. She is currently pursuing the master’s degree with the School of Computer and Information Technology, Beijing Jiaotong University. Her research interests include channel estimation, computer science, and 5G communication.

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GONGPU WANG received the B.Eng. degree in communication engineering from Anhui University, Hefei, China, in 2001, the M.Sc. degree from the Beijing University of Posts and Telecommunications (BUPT), China, in 2004, and the Ph.D. degree from the University of Alberta, Edmonton, AB, Canada, in 2011. From 2004 to 2007, he was an Assistant Professor with the School of Network Education, BUPT. After graduation, he joined the School of Computer and Information Technology, Beijing Jiaotong University, China, where he is currently an Associate Professor. His research interests include Internet of Things, wireless communication theory, and signal processing technologies.

RONGFEI FAN received the B.E. degree in electrical engineering from the Harbin Institute of Technology, Harbin, China, in 2007, and the Ph.D. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2012. Since 2013, he has been a Faculty Member with the Beijing Institute of Technology, Beijing, China, where he is currently an Assistant Professor with the School of Information and Electronics. His research interest includes cognitive radio, crosslayer design, radio resource management for wireless communications, and energy harvesting.

BO AI (M’00–SM’10) received the M.S. and Ph.D. degrees in wireless communications from Xidian University, Xi’an, China, in 2002 and 2004, respectively. He is currently a Professor and an Advisor of Ph.D. candidates with the Beijing Jiaotong University, Beijing, China, where he is also the Deputy Director of the State Key Laboratory of Rail Traffic Control and Safety. He has authored/ co-authored six books and 140 scientific research papers, and holds 26 invention patents in his research areas. His research interests include the research and applications of orthogonal frequency-division multiplexing techniques, high-power amplifier linearization techniques, radio propagation and channel modeling, global systems for mobile communications for railway systems, and long-term evolution for railway systems. He is a fellow of the Institution of Engineering and Technology. He is an Associate Editor of the IEEE TRANSACTIONS ON CONSUMER ELECTRONICS and an Editorial Committee Member of the Wireless Personal Communications Journal.

VOLUME 6, 2018