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MIMO-OFDM Wireless Channel Prediction by Exploiting Spatial-Temporal Correlation Lihong Liu, Hui Feng, Student Member, IEEE, Tao Yang, Member, IEEE, and Bo Hu, Member, IEEE

Abstract—Channel prediction is an appealing technique to mitigate the performance degradation due to the inevitable feedback delay of the channel state information (CSI) in modern wireless systems. We first propose a general MIMO-OFDM channel prediction framework, which exploits both the spatial and temporal correlations among antennas. Then we derive two predictors which select data for auto-regressive (AR) predictors in different ways based on the proposed framework. The first predictor chooses the data set via minimizing the mean square error (MSE) of prediction model. The second predictor chooses the data in a heuristic way, which aims to reduce the computational complexity. Our algorithms can be applied to improve the precoding performance in multi-user MIMO-OFDM systems. Simulation results show that the proposed methods can overcome the feedback delay effectively, even when the channel changes rapidly. Index Terms—MIMO-OFDM, channel prediction, spatialtemporal correlation, AR model.

I. I NTRODUCTION

M

ULTIPLE input multiple output orthogonal frequency division multiplexing (MIMO-OFDM) is considered to be a promising technique for reliable high data-rate wireless transmission systems, which can provide high spectral efficiency and high data rate transmission over frequency selective channels [1]. Recently, adaptive multi-user resource allocation [2] and precoding [3] techniques are introduced to modern MIMO-OFDM systems to further improve the spectral efficiency and the system performance. However, the benefit of these techniques significantly rely on the accurate (to some level) channel state information (CSI) at the transmitter [4]. In frequency division duplex (FDD) systems, CSI can only be estimated at the receiver and then be fed back to the transmitter. While in mobile environments with the timevarying channel, the CSI fed back to the transmitter would be outdated due to the feedback delay, which results in significant performance degradation. An effective mean to overcome the feedback delay is the channel prediction technique discussed in this paper, which predicts future channel coefficients based on the history data. Manuscript received March 14, 2013; revised July 29 and October 21, 2013; accepted October 22, 2013. The associate editor coordinating the review of this paper and approving it for publication was G. Yue. This work was supported by the NSF of China (Grant No. 60972024), and the NSTMP of China (Grant Nos. 2011ZX03003-001-02, 2012ZX03001007003, 2013ZX03003006-003). This work has been presented in part at the International Conference on Wireless Communications and Signal Processing (WCSP), Huangshan, October 2012. The authors are with the Department of Electronics Engineering, Fudan University, Shanghai, China (e-mail: {06300720084, hfeng, taoyang, bohu}@fudan.edu.cn). Digital Object Identifier 10.1109/TWC.2013.112613.130455

Channel prediction for single-input single-output (SISO) flat-fading channel has been investigated extensively in the past. Most methods in literatures fall into two categories: the parametric autoregressive model and the parametric radio channel model. Conventional approaches based on the wellknown autoregressive (AR) model treat the channel as a widesense stationary (WSS) stochastic process [5]–[11]. The AR coefficients can be computed using the minimum mean square error (MMSE) criterion, which needs the knowledge of the channel correlation function [5]. However, the correlation function may be unknown and even time-variant in reality [7]. Adaptive filtering techniques, such as least mean squares (LMS) [8], recursive least squares (RLS) [7] and Kalman filter [9], are developed to track the changes of channel coefficients. These methods can be broadly classified as the parametric autoregressive model [12]. Another attractive model called parametric radio channel begins with an approximation of the physical propagation process. In such way, the channel is modeled as a sum of complex sinusoids, which can be estimated by frequency estimation techniques, e.g., multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRIT) [10], [11]. Recently, both of the parametric autoregressive model and parametric radio channel model have been studied for multicarrier SISO systems. The authors predict the channel coefficient in time domain by MMSE [13], which has better performance in frequency domain [14]. A robust frequency domain channel estimator for OFDM systems is presented in [15], which makes use of temporal and frequency correlation and is insensitive to the channel statistics. These approaches are all based on the parametric autoregressive model. The idea of parametric radio channel model is also fully extended to time and frequency selective channels. A 2-D ESPRIT channel prediction algorithm is proposed to extract the 2-D sinusoidal parameter in [16]. The authors in [17] develop a low-complexity algorithm by 2-step 1-D ESPRIT predictions. Meanwhile, several studies have been reported on the channel prediction in single-carrier correlated MIMO systems [18]. When no correlation exists among the antennas in MIMO systems, it doesn’t need the signals from other antennas to improve the channel prediction. In such case, the CSI in MIMO systems can be predicted in an analogous SISO’s way. However, the spatial correlation does exist in reality. For example, insufficient spacing between antennas in deployment may bring in inevitable spatial correlation [19]. Spatial correlation has been used to refine the channel estimation performance in MIMO-OFDM systems [20]. However, the method in [20] does not intend to predict the channel coefficient. In

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

TABLE I P REVIOUS W ORKS ON C HANNEL P REDICTION Single-carrier system

Multi-carrier system

SISO

[5]–[11]

[13], [14], [16], [17]

MIMO

[18]

[22]–[27] (assembling of flat SISO approach) [28]

applications in multi-user precoding methods are presented in Section V. Performance analysis and simulation results are provided in Section VI. Finally, we conclude the paper with some remarks in section VII. II. MIMO-OFDM S YSTEM A ND C HANNEL M ODEL A. OFDM System Model

fact, the spatial correlation can be utilized to improve the performance of channel prediction as indicated in [21]. A reduced complexity 2-step prediction approach which exploits the spatial correlation is proposed for single-carrier correlated MIMO systems in [18]. More recently, the channel prediction has been extended to the MIMO-OFDM cases. A simplified adaptive channel prediction scheme is proposed in [22], which assumes that the MIMO channels are independent with each other and only considers the temporal correlation. Similarly, the prediction methods in [22]–[25] for MIMO-OFDM systems treat every subcarrier of MIMO-OFDM as a narrowband SISO channel, which only consider the temporal correlation like the algorithms mentioned above for flat-fading channels. These approaches, in fact, can be thought of as a simple assembling of the flat SISO approach, which do not fully exploit the correlations in space and time domains. The MIMO-OFDM channel estimation method in [26], [27] firstly derives the time domain channel taps, and then uses the time correlation to refine the estimation results, which also ignores the spatial correlation. In contrast, a joint spatio-temporal (JST) filtering based MMSE channel estimator is derived in [28], which exploits both the temporal and spatial correlations. However, the method in [28] encounters the high computational complexity of the matrix inversion operation. (Table I summarizes previous works referred above). We expect to develop a MIMO-OFDM prediction framework to exploit both the temporal and spatial correlations to improve the prediction performance without increasing computational burden a lot. In this paper, we first propose a novel MIMO-OFDM channel prediction framework developed from [13], which takes both spatial and temporal correlations of multiple antennas into account. Then, we propose two order-reduced autoregressive (AR) prediction algorithms to reduce the computational complexity under the proposed framework. The first predictor chooses the data via minimizing the MSE of prediction model, while the second predictor chooses the data in a heuristic way. Both of the two proposed algorithms can exploit spatial and temporal correlations effectively while maintaining a low computational complexity. The proposed algorithms are an extension work from [29]. We give more detailed analysis and simulation results in this paper. Furthermore, we present an appealing application of the proposed prediction algorithms, which can improve the precoding performance in MU-MIMO systems. The rest of this paper is organized as follows. Section II describes the MIMO-OFDM system and channel model used in this paper. In section III, we develop a general framework for spatial-temporal MIMO-OFDM channel prediction. In section IV, two concrete prediction algorithms are presented. Their

Consider a MIMO-OFDM system with M transmit antennas, N receive antennas, and K subcarriers. At the transmitter, the transmitted symbol Xm (i, k) is transformed into the time domain signal at the m-th transmit antenna, i-th symbol time and the k-th subcarrier using IFFT [29], [30]. Then, a cyclic prefix (CP) is inserted to avoid inter-symbol interference. At the receiver, the CP is removed before the FFT process. We assume that the CP is greater than the maximum delay spread of channel, and the time and frequency synchronization is perfect, such that the received symbol at the n-th receive antenna can be represented as Yn (i, k) =

M

Hn,m (i, k)Xm (i, k) + Zn (i, k),

(1)

m=1

where Hn,m (i, k) is the frequency response of the channel impulse response (CIR) at the k-th subcarrier and the i-th symbol time for the (m, n)-th antenna pair. Zn (i, k) is the background noise plus interference term of the n-th receive antenna, which can be approximated as a zero mean additive 2 white Gaussian noise (AWGN) with variance σZ . B. Channel Model The impulse response of the wireless channel can be represented as Ln,m −1

hn,m (t, τ ) =

hn,m (t, l)δ(τ − τn,m (l)),

(2)

l=0

where Ln,m is the number of multiple radio path of the (m, n)th antenna pair, δ(·) is the Kronecker delta function, τn,m (l) and hn,m (t, l) are the delay and complex-value CIR at time t of the l-th path from the (m, n)-antenna pair respectively. Let Hn,m (t, f ) be the frequency response of the time domain CIR hn,m (t, τ ). The frequency domain CIR with symbol time Ts and subcarrier spacing fs can be represented in a discrete manner as Hn,m (i, k) = Hn,m (iTS , kfs ). (3) Similarly, the time domain CIR is hn,m (i, l) = hn,m (iTS , l).

(4)

Hereafter, a random Rayleigh fading channel model satisfying the wide-sense stationary uncorrelated scattering (WSSUS) assumption is used as [13], E{hn,m (i + Δi, l)hn,m (i, l )} = rt (Δi)δ(l − l ),

(5)

where rt (Δi) is the channel’s time-delay correlation function. Due to the WSSUS property, hn,m (i+Δi, l ) and hn,m (i, l) are uncorrelated for l = l [31]. Here, we extend this idea to the MIMO case.

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Suppose that there are L scatterers in total and hn,m (i, l) is caused by the l-th scatterer, i.e., hn,m (i+Δi, l) and hn ,m (i, l) are both caused by the l-th scatterer. As the multipath components caused by different scatterers are uncorrelated in MIMO systems [31], hn,m (i + Δi, l) and hn ,m (i, l) are uncorrelated for l = l . The channel correlation for MIMO system can be represented as [13] E{hn,m (i + Δi, l)hn ,m (i, l )} = rt (Δi)rs (n, m, n , m )δ(l − l ),

(6)

where n, n ∈ {1, 2, . . . , N }, m, m ∈ {1, 2, . . . , M } and rs (·) denotes the spatial correlation function. Since the transmitter is far away from the receiver in general, the spatial correlation function can be decomposed into two correlation part at the transmitter and the receiver as rs (n, m, n m ) = rsr (n, n )rst (m, m ).

(7)

In other words, the spatial correlation matrix of the MIMO channel is given by [32] RMIMO = RMS ⊗ RBS ,

(8)

where ⊗ represents the Kronecker product, RMS and RBS are spatial correlation matrices at the mobile station (MS) and the base station (BS) respectively. C. Pilot Pattern and Least-Squares Channel Estimation The pilot pattern in the 3rd generation partnership project long term evolution (3GPP LTE) [33] standard is adopted in this paper, where pilots and data symbols are sent exclusively in the time-frequency domain [30] as ⎧ ⎪ ⎨dm (i, k), data, Xm (i, k) = pm (i, k), pilot at m-th transmit antenna, ⎪ ⎩ 0, pilot at other transmit antenna. Suppose there are D pilot subcarriers for each transmit antenna, namely, km,1 , . . . , km,j , . . . , km,D , where km,j is the j-th pilot subcarrier for the m-th transmit antenna. We perform a least square (LS) channel estimation at the pilot locations using the received symbol Yn (i, k) and the known pilot symbol Xm (i, k), given as ˆ n,m (i, km,j ) = Yn (i, km,j )/Xm (i, km,j ) H

= Hn,m (i, km,j ) + Zn,m (i, km,j )

Fig. 1. System model of the spatial-temporal MIMO-OFDM channel prediction.

(9)

where Hn,m (i, km,j ) is the ideal channel coefficient and Zn,m (i, km,j ) is the estimation error denoted as a zero mean AWGN with variance σz2 . III. S PATIAL -T EMPORAL C HANNEL P REDICTION F RAMEWORK As indicated in [12], the time-domain predictor in OFDM systems has better MSE performance than the frequency domain predictor while maintaining a low complexity. Thus, we adopt the time-domain prediction approach (cf. Fig. 1) in this paper, which is an extension for MIMO-OFDM channel from the SISO model in [13].

Firstly, the time-domain channel coefficient of every channel pair (m, n) can be estimated by conventional methods, e.g. IFFT with interpolation, reduced LS and LMMSE [34], [35]. For instance, a K-points IFFT can be used to do the frequency-time transformation job in Fig. 1, K−1

ˆ n,m (i, l) = 1 ˆ n,m (i, k)ej2πlk/K h H K k=0 hn,m (i, l) + zn,m (i, l) l = 0, . . . , L − 1 = zn,m (i, l) l = L, . . . , K − 1. (10) Here, we suppose the channel delay τn,m (l) of every tap is an integer multiple of sampling interval. Then, all of the energy from the path will be mapped to the zero to L − 1 taps. In that case, it can be easily proved that zn,m (i, l) is also a zero mean AWGN with variance

β 2 = σz2 /K.

(11)

Subsequently, a MIMO predictor is performed to predict the time domain channel impulse response hn,m (i + p, l) for each delay l = 0, . . . , L − 1, where L − 1 represents the channel’s maximum delay, and p denotes the prediction length. Due to the WSSUS property, as mentioned earlier, hn,m (i + Δi, l) and hn ,m (i, l ) are uncorrelated for l = l . Therefore, the MIMO predictor for the l-th tap only needs to consider the corresponding tap of every channel pair (m, n) (as other taps are uncorrelated with the predicted tap, they are useless for prediction). Thus the multi-carrier MIMO prediction problem is transformed to the single-carrier MIMO prediction problem in this step. pre ˆ n,m Finally, the frequency domain channel coefficient H (i + p, k) is obtained from the predicted time-domain impulse ˆ pre (i + p, l) via K-points FFT. response sample h n,m IV. P REDICTION A LGORITHMS The MIMO predictor showed in the red box of Fig.1 is discussed in detail in this section. We begin with the AR model, which can capture most of the fading dynamics [13]. In fact, a thorough comparison of different channel prediction algorithms is performed in [12]. Their conclusion is that



the AR approach outperforms other prediction modeling for measured channels. Define p as the prediction length. For every tap of the channel, Q current and previous estimated coefficients of the channel are considered. Denote ˆ n,m (i, l) h ˆ n,m (i, l), ˆ ˆ n,m (i − Q + 1, l)]T . = [h hn,m (i − 1, l), . . . , h (12) ˆ pre (i + p, l), the data set h ˆ is utilized, where To predict h n,m ˆ 1,2 (i, l)T , . . . , h ˆ 1,M (i, l)T , ˆ = [h ˆ 1,1 (i, l)T , h h ˆ 2,1 (i, l)T , . . . , h ˆ N,1 (i, l)T , . . . , h ˆ N,M (i, l)T ]T . h

(13)

A. Extreme predictors We first introduce two extreme prediction methods, which will be helpful to introduce our algorithms. 1) SISO predictor: A traditional prediction algorithm is called the SISO predictor [26], which ignores the spatial correlation and only considers the temporal correlation. To ˆ pre (i + p, l), the data set h ˆn,m (i, l) is used with a predict h n,m Q-order MMSE filter ws as ˆ pre (i + p, l) = wH h ˆ h n,m s n,m (i, l),

(14)

where ˆ n,m (i, l)2 }. (15) ws = arg min E{hn,m (i + p, l) − wsH h ws

According to the orthogonal principle, the MMSE filter should satisfy E{(hn,m (i + p, l) − ˆ n,m (i, l))∗ h ˆ n,m (i, l)} = 0. After some simplification, wsH h (15) can be written as ws = (Rs + β 2 I)−1 rs ,

(16)

where Rs is the Hermitian-symmetric and Toeplitz Q × Q temporal autocorrelation matrix with entries [Rs ]i,j = rt ((i − j)Ts ), T

rs = [rt (pTs ), . . . , rt ((P + Q − 1)Ts )] .

(17)

The MSE is given by εs = rt (0) − rTs (Rs + β 2 I)−1 rs .

(18)

2) All-correlation predictor: Then, we introduce the prediction algorithm at the other extreme: all-correlation predictor, which exploits all the possible spatial-temporal correlations [18]. This method is also a general case of JST filtering method [28]. The JST filtering in [28] assumes that different transmit antennas are uncorrelated and only considers the spatial correlation of receive antennas, while the all-correlation predictor considers both the temporal and spatial correlations. ˆ pre (i + p, l), where a The data set h is used to predict h n,m M × N × Q-order MMSE filter w2D is applied as ˆhpre (i + p, l) = wH h. ˆ n,m 2D

(19)

Similarly, we can get w2D in the SISO predictor’s way. Using the orthogonal principle, the all-correlation predictor

H ˆ ∗ˆ should satisfy E{(hn,m (i + p, l) − w2D h) h} = 0. After some simplifications, we have (N ∗(m−1)+n)

w2D = (Rs ⊗RMIMO +β 2 I)−1 (rs ⊗RMIMO (N ∗(m−1)+n) where RMIMO of RMIMO . And the

), (20)

denotes the (N ∗(m−1)+n)-th column MSE is (N ∗(m−1)+n) T

ε2D = rt (0) − (rs ⊗ RMIMO 2

−1

+ β I)

) (Rs ⊗ RMIMO (N ∗(m−1)+n)

(rs ⊗ RMIMO

).

(21)

Since the all-correlation predictor exploits all the possible spatial and temporal correlations, we can expect that it outperforms the SISO predictor when the spatial correlation is present. However, the M × N × Q-order AR model requires a huge computation compared with the Q-order AR model. So we need a tradeoff between the prediction precision and the computational complexity, which is the motivation of the proposed algorithms in the following. B. Forward-stepwise subset (FSS) predictor A reduced order AR predictor is proposed in this subsection, which aims to reduce the computational burden of the allcorrelation predictor. In this algorithm, the observations are not considered to be equally important for prediction. Some observations just offer little information for the AR model. Particularly, if a datum is independent with the predicted datum, then the datum has no help for prediction. Therefore, we come up with the idea that the most helpful data can be chosen to create the AR prediction model, considering both the spatial and temporal correlations. The key problem remained is how to measure the helpfulness of each datum. Suppose we have already chosen Q ˆ where Q is a tradeoff between the prediction data from h, ˜ The precision and complexity, to form a Q × 1 vector h. prediction AR model is ˆ pre (i + p, l) = wH h. ˜ h n,m B

(22)

According to the MMSE criterion, we get H˜ 2 h }. wB = arg min E{hn,m(i + p, l) − wB wB

(23)

Similarly, using the orthogonal principle, (23) can be written as ˜h ˜ H ]−1 E[hn,m (i + p, l)∗ h], ˜ wB = E[h (24) where the MSE is εB = rt (0)

˜ ∗ ]T E[h ˜h ˜ H ]−1 E[hn,m (i + p, l)h ˜ ∗ ]. − E[hn,m (i + p, l)h (25)

Intuitively, the prediction model with the lowest MSE is the ˜ choose Q best. Thus we derive the criterion for choosing h: data point so as to get the minimal MSE. According to this criterion, using a greedy algorithm, the prediction algorithm is displayed in Algorithm 1. Since the proposed prediction algorithm exploits the most useful observations, it outperforms the SISO prediction algorithm even using the same order of AR model. However, the process of data choosing is a huge computational burden. An


5

Algorithm 1 The FSS predictor 1. 2.

˜ old = h ˜1 Find the datum which is most correlated with hn,m (i + p, l) and denote by ˜ h1 , initialize j = 1, h Minimize MSE to find ˜ j = arg min (rt (0) − E[hn,m (i + p, l)h ˜ ∗ ]T E[h ˜ new h ˜ ∗ ]), ˜ H ]−1 E[hn,m (i + p, l)h h new new new ˆ h ˜ h˜j ∈h\ old

3. 4.

T ˜ new = [h ˜T , ˜ where h old hj ] . ˜ old = h ˜ new .If j < Q , turn to step 2. j = j + 1,h H˜ ˜ which is a Q × 1 vector. Then the desired Q order AR prediction model is ˜ The final selected data set is h hpre n,m (i + p, l) = wB hnew , H −1 ∗ ˜ ˜ ˜ where wB = E[hnew h ] E[hn,m (i + p, l) hnew ]. new

inversion of k × k matrix is needed to measure the MSE in the k-th step. In total, (2M N Q − Q )Q /2 computations are needed to find the optimal model.

C. Reduced-complexity FSS predictor In this subsection, a prediction algorithm is introduced which aims to further reduce the computational complexity of FSS method. The key idea is to select the data incrementally for prediction from the correlation’s view. If the new observation has a high correlation with the selected data in previous steps, then the new observation cannot provide more new information and may help little. Therefore, the considering ˜ k in the k-th step can be chosen by the analysis of value h ˜ 1, . . . , ˜ the selected data [h hk−1 ]. Based on this idea, the k-th element is chosen as follows. First-step: according to MMSE criterion and AR model ˆ pre (i + p, l) = wH h, ˜ = ˜ we get wR , where h h n,m R T ˜ 1, . . . , ˜ hk−1 ] . [h H˜ Second-step: define residual = hn,m (i + p, l) − wR h, ˜ the hk is the selected data which is most correlated with the residual. The complete prediction algorithm which aims to create the Q -order desired AR model is showed in Algorithm 2. The proposed reduced-complexity FSS predictor reduces the times of computing MSE to Q , and only needs one time inversion of a Q × Q matrix in each selection.

V. T HE A PPLICATION OF P ROPOSED P REDICTION M ETHODS To further investigate the application of the proposed prediction methods, the multi-user MIMO-OFDM (MU-MIMOOFDM) system with precoding is considered in this section. In MU-MIMO-OFDM systems, precoding is proven to be an effective way to increase the system performance and eliminate the interferences among users. Unfortunately, CSI can only be feedback from the receiver in FDD systems, thus resulting in unavoidable outdated problem. In this section, we will apply the proposed reduced-complexity FSS predictor to overcome the outdated problem (cf. Fig. 2). Suppose there are P users communicating with the BS simultaneously. User p(p = 1, . . . , P ) has Np receive antennas. The predicted channel matrix of BS to the p-th user at the k-th subcarrier and the i-th symbol time can be represented

Fig. 2.

A MU-MIMO-OFDM system with predicted precoding.

by the Np × M channel matrix ⎛

⎞ 1 ˜ p,1 ˜ 1 (i, k) H (i, k) . . . H p,M ⎜ ⎟ .. .. ⎟ .. ˜ p (i, k) = ⎜ H ⎜ ⎟ . . . ⎝ ⎠ ˜ Np (i, k) ˜ Np (i, k) . . . H H p,1 p,M

(26)

˜ j (i, k) denotes the predicted frequency response of where H p,m the channel impulse response (CIR) at the k-th subcarrier, the i-th symbol time and the (j, m)-th antenna pair of user p. Define the whole predicted channel as ˜ k) = [H ˜ T (i, k), . . . , H ˜ T (i, k)]T , H(i, 1 P

(27)

and P(i, k) is the precoding matrix at the k-th subcarrier and the i-th symbol. In order to validate the performance improvement of the proposed prediction method, three classic MU-MIMO precoding schemes [36] are used as follows based on the predicted channel coefficients.

A. Zero Forcing In the ZF scheme, P(i, k) is chosen such that each user receives no interference from other users, which can be written in the form ˜ k)H (H(i, ˜ k)H(i, ˜ k)H )−1 . P(i, k) = H(i,

(28)

B. MMSE The MMSE precoding scheme can be considered as an improvement of ZF, which is ˜ k)H (H(i, ˜ k)H(i, ˜ k)H + βI)−1 , P(i, k) = H(i,

(29)

where β is a regularization factor whose value is given in [36].



Algorithm 2 Reduced-Complexity FSS Predictor ˜ old = h ˜1. Find the datum which is most correlated with hn,m (i + p, l) and denote by ˜ h1 , initialize j = 1, h pre H H −1 ∗ ˆ n,m (i + p, l) = w h ˜ old h ˜ old , where wR = E[h ˜ ] E[hn,m (i + p, l) h ˜ old ]. Create AR model h

1. 2.

R

old

˜ j according to h ˜ = arg min E[(hn,m (i + p, l) − wH h ˜∗ ˜ ˜T ˜ T ˜ Find h R new )hj ], where hnew = [hold , hj ] . ˆ h ˜ ˜ j ∈h\ h old

˜ old = h ˜ new . If j < Q , turn to step 2. j = j + 1, h H˜ ˜ is a Q × 1 vector. Then the desired Q order AR prediction model is h ˆ pre The final selected data set is h n,m (i + p, l) = wB hnew ˜ H ]−1 E[hn,m (i + p, l)∗ h ˜ new ]. ˜ new h where wB = E[h

3. 4.

new

TABLE II S IMULATION PARAMETERS

C. Block Diagonalization The block diagonalization (BD) scheme can be considered as a generalization of channel inversion for situations with multiple antennas per user. Define ¯ p (i, k) H ˜ Tp−1 (i, k), H ˜ Tp+1 (i, k), . . . , H ˜ TP (i, k)]T , ˜ T1 (i, k), . . . , H = [H (30) whose singular value decomposition (SVD) is

Item

Value

Item

Value

Bandwidth

10 Mhz

Subcarrier spacing

15 kHz

Carrier Frequency

2.6 Ghz

CP length

4.6 μs

LTE frame length

10 ms

Symbol period Ts

1 ms

Subframe length

1 ms

Prediction length 1 ms ⎛ ⎞ ⎞ 1 α 1 β ⎝ ⎝ ⎠ ⎠ RBS = , RM S = α∗ 1 β∗ 1 ⎞ ⎛ 1 α1/9 α4/9 α ⎟ ⎜ ⎜α1/9∗ 1 α1/9 α4/9 ⎟ ⎟ ⎜ RBS = ⎜ ⎟ 1/9 ⎟ ⎜α4/9∗ α1/9∗ 1 α ⎠ ⎝ α∗ α4/9∗ α1/9∗ 1 ⎛ ⎞ 1 β ⎝ ⎠ RM S = β∗ 1

Correlation matrix (2 × 2)

¯ p (i, k) = U ¯ p (i, k)D ¯ p (i, k)(V ¯ (1) (i, k)V ¯ (0) (i, k))H , (31) H p p ¯ p (i, k) denotes the right singular matrices correwhere V sponding to the zero singular values. ˜ k)V ¯ p(0) (i, k) is The SVD of H(i, (0)

˜ k)V ¯ (0) (i, k) H(i, p ⎛ ⎞ Dp (i, k) 0 ⎠ (Vp(1) (i, k)Vp(0) (i, k))H , = Up (i, k) ⎝ 0 0 (1)

P

P

TABLE III C ORRLEATION S CENARIOS

(32)

where Vp (i, k) denotes the right singular matrices corresponding to the non-zero singular values. Suppose that D(i, k) is a diagonal matrix whose elements are Dp (i, k) . Then the precoding matrix is ¯ (0) (i, k)V(1) (i, k), P(i, k) = [V 1 1 ¯ (0) (i, k)V(1) (i, k)]C(i, k)1/2 . ...,V

Correlation matrix (4 × 2)

(33)

where C(i, k) is the optimal power loading matrix determined by the water filling on the diagonal element of D(i, k) as in [36]. VI. P ERFORMANCE A NALYSIS A ND S IMULATIONS We consider the typical downlink scenarios specified in 3GPP LTE standard in our simulations [33]. Fundamental characteristics of MIMO-OFDM FDD system are listed in Table II, which are based on the 3GPP LTE specification [33]. We simulate the delay spread and delay profiles of channel using EVA [33]. For each tap in the power delay profile, the fading channel is generated by the modified Jakes’ method [37] with Rp = 16 rays per path, which is sufficient to match the desired second-order statistics [17]. The Doppler frequencies are generated randomly per realization as fr,p = ±fmax cos((2πr + θ − π)/(4Rp )), r = 1, . . . , Rp , where fmax = ν/(3 × 108 /Fc ) is the maximum Doppler frequency, θ follows uniform distribution U [−π, π), and ν is the mobile

⎛

Low correlation

Medium correlation

High correlation

α

β

α

β

α

β

0

0

0.3

0.9

0.9

0.9

speed. The Doppler frequency is 70 Hz unless indicated otherwise. Assume that the noise variance β 2 and the correlation matrix E{hn,m (i + Δi, l)hn ,m (i, l)} are known in all simulations. The normalized mean square error (NMSE) is used to measure the prediction accuracy, which is defined as ˆ pre (i + p, l)2 } E{hn,m (i + p, l) − h n,m }, E{hn,m (i + p, l)2 } (34) For comparison, we choose Q = Q = 30. The MIMO channel model is referred to [32]. The correlation matrix is showed in Table II. The correlation coefficients α and β for different correlation types are given in Table III [38]. The NMSE is averaged over 10000 frames. We next analyze the performance of our proposed prediction algorithms, and compare it with SISO, all-correlation and 2step algorithms [18] in this section. N M SE = 10 log{

A. Prediction Performance of Different Spatial Correlation Cases We investigate the effect of spatial correlation on predictors in this subsection. A 2 × 2 MIMO-OFDM system


−10

−20

SISO All−correlation FSS Reduced−complexity FSS 2−step

−20

NMSE(dB)

−15 NMSE(dB)

−15

SISO All−correlation FSS Reduced−complexity FSS 2−step

7

−25

−30

−25 −35

−30 0

5

10 SNR(dB)

15

20

Fig. 3. Prediction performance of a high correlation scenario (α = 0.9, β = 0.9), M = 2, N = 2, doppler frequcency is 70 Hz.

−10

SISO All−corelation FSS Reduced−complexity FSS 2−step

NMSE(dB)

−15

−20

−25

−30 0

5

10 SNR(dB)

15

20

Fig. 4. Prediction performance of a medium correlation scenario (α = 0.9, β = 0.9), M = 2, N = 2, doppler frequcency is 70 Hz.

is considered in Fig. 3-6. In order to show the effect of the spatial correlation, we consider the high, medium and low correlation cases one by one. As the velocity increases, the time correlation decreases and the spatial correlation is more significant for the prediction method. The performance difference of SISO method and the proposed methods in Fig. 3 where Doppler frequency is 70 Hz is bigger compared with Fig. 5 whose Doppler frequency is 5 Hz. To show the benefit of spatial correlation brings, we choose a middle velocity scenario whose Doppler is frequency 70 Hz. In all these simulations, it can be seen that the FSS predictor and the reduced-complexity FSS predictor have almost the same performance, the performance curves of the two overlap each other. In Fig. 3, Fig. 4, and Fig. 5, the all-correlation approach has the lowest NMSE among all the algorithms for all parameter configurations. Our proposed prediction methods and 2-step method outperform the SISO approach, which proves that the spatial correlation is really helpful. It is also observed that the proposed methods perform better than 2-step algorithm, particularly when the SNR is low. The reason is that

−40 0

5

10 SNR(dB)

15

20


the data in our methods is selected flexibly from both spatial and temporal domains. In contrast, the 2-step algorithm fixes the prediction order, which first exploits temporal correlation by a SISO MMSE time-domain predictor, and then follows by an MMSE spatial predictor to exploit the spatial correlation. Furthermore, the AR order is adaptively tuned in our methods, which is also helpful to the prediction performance. In Fig. 3, the channel correlations are high (α = 0.9, β = 0.9). The performance of the two proposed methods and 2step method are much better than the SISO method, especially when the SNR is low. It can be seen that the performance of the proposed algorithms are nearly the same with the allcorrelation method, and are better than 2-step method. It is concluded that exploiting the spatial correlation can effectively improve the prediction performance. Medium correlation scenario (α = 0.9, β = 0.9) is considered in Fig. 4. It can be seen that two proposed algorithms and the 2-step method outperform the SISO method slightly, and the performances of the proposed algorithms lie between the 2-step and all-correlation methods. It is observed that exploiting the spatial correlation has a little help to improve the prediction performance, when the spatial correlation is low. Furthermore, when the spatial correlation is zero and only the temporal correlation exists, the all-correlation, 2-step and proposed methods degrade to the SISO method. B. Prediction Performance of Different Doppler Frequency Cases We next study the effect of mobile velocity on prediction algorithms. Two typical scenarios are simulated as in [39] , which is showed in Fig. 3 and Fig. 5. The middle velocity scenario with the Doppler frequency 70 Hz is considered in Fig. 3, while the slow velocity scenario with the Doppler frequency 5 Hz is considered in Fig. 5 [39]. It is illustrated that the MSE of all predictors increase for high velocity, which is consistent with intuition since fast fading channels are difficult to predict. The performance difference of the proposed approaches with the SISO method is smaller in Fig.5 compared with which in Fig.3. It is an expected behavior

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

−10

FSS 2−step SISO All−correlation Reduced−complexity FSS

MMSE(dB)

−15

−20

−25

0.4

BD unpredicted BD predicted MMSE unpredicted MMSE predicted ZF unpredicted ZF predicted

0.35 0.3 Uncoded BER

8

0.25 0.2 0.15 0.1

−30 0.05 −35 0

5

10 SNR(dB)

15

20

BD unpredicted BD predicted MMSE unpredicted MMSE predicted ZF unpredicted ZF predicted

Uncoded BER

0.15

0.1

0.05

0

2

4

6

8 SNR(dB)

5

10

15

SNR(dB)


0.2

0 0

10

12

14

Fig. 7. Performance of precoding with predicted channel coefficient of a high correlation scenario (α = 0.9, β = 0.9), P = 2, M = 4, Np = 2(p = 1, 2), doppler frequcency is 5 Hz.

since time correlation is weaker for high velocity, the effect of spatial correlation will be more significant. C. Prediction Performance of Different Antenna Case This subsection aims to investigate the effect of antenna number. As the performance of the FSS predictor and the reduced-complexity FSS predictor are nearly the same in the 2 × 2 MIMO-OFDM system, the 4 × 2 case is further simulated to investigate the performance differences of the two proposed algorithms. In the simulation, the Doppler frequency is 70 Hz, and high correlations are considered. It is showed that the FSS predictor offers better prediction performance compared with the reduced-complexity FSS predictor in Fig. 6. Nevertheless, the reduced-complexity FSS predictor still outperforms the 2-step and SISO method. It is concluded that the proposed reduced-complexity FSS predictor can offer comparable performance while maintain a low computational complexity, and it can be recommended as a good choice especially when the number of antenna is small.

Fig. 8. Performance of precoding with predicted channel coefficient of a high correlation scenario (α = 0.9, β = 0.9), P = 2, M = 4, Np = 2(p = 1, 2), doppler frequcency is 70 Hz.

D. Uncoded BER Performance of Precoding With the Reduced-complexity FSS Predictor We finally investigate the uncoded bit error rate (BER) performance of the overall MU-MIMO-OFDM system with precoding using predicted channel coefficients of the proposed reduced-complexity FSS predictor, the results are showed in Fig.7 and Fig.8. The outdated error caused by the feedback delay is acceptable, when the channel changes slowly. However, when the channel changes rapidly, the outdated error results in significant performance degradation, So, we expect to improve the precoding utilizing the predicted channel information. Three classic precoding methods (ZF, BD, and MMSE) with the reduced-complexity FSS predictor are tested to evaluate the performance improvement In the simulations, two users are considered to form a MU-MIMO group, and each user has two receive antennas. The base station is equipped with eight transmitted antennas. Slow velocity scenario with Doppler frequency 5 Hz is used in Fig. 7. It is showed that the performance of all the considered precoding methods (ZF, BD, and MMSE) which exploit the predicted channel coefficient are only slightly better than those corresponding unpredicted precoding methods’. However, in Fig. 8 with the Doppler frequency 70 Hz, it is illustrated that the precoding schemes which using the predicted channel coefficient outperform than those corresponding unpredicted precoding methods. E. Complexity Analysis The computational complexities of the predictors discussed are listed in Table IV. In the initialization stage, both of the two proposed algorithms are more complex than other compared prediction methods when Q > M N . However, as the number of antenna increases such as Q < M N , the reduced-complexity FSS predictor is only slightly more complex than the SISO method and much less complex than the all-correlation method. In all cases, the reduced-complexity FSS predictor is less complex than the FSS predictor. Since the initialization is performed only once in advance, we focus on the complexity of the prediction stage which is performed


TABLE IV C OMPUTATIONAL C OMPLEXITIES OF VARIOUS M ETHODS Algorithm

Initialization

Prediction

All-corrlation

O(M 2 N 2 Q2 )

O(M N Q)

O(Q2 )

SISO

O(Q2

2-step

+ M 2N 2)

O(M N Q5 )

FSS Reduced-complexity FSS

O(Q4

+

M N Q2 )

O(Q) O(Q + M N ) O(Q) O(Q)

in each symbol interval. Since Q = Q in all simulations, the prediction complexity of both two proposed algorithms are the same as the SISO’s, which are much less than 2-step and all-correlations. Moreover, we can choose Q adaptively according to the prediction accuracy. VII. C ONCLUSION There are two main contributions in our work. First, we derive a novel channel prediction framework for MIMOOFDM systems which takes both spatial and temporal correlations into account. Second, we propose two MIMO prediction algorithms which select the useful data for AR modeling. The FSS predictor employs the optimal data selection strategy, which requires huge computations. In contrast, the reducedcomplexity FSS predictor chooses the data in a heuristic way, whose computational complexity is quite low. The performance of the two proposed algorithms is nearly the same in 2×2 antenna case. As the antenna increase up to 4×2, the FSS predictor offers slightly better performance compared with the reduced-complexity FSS predictor. Simulation results show that the prediction performance can be effectively improved by exploiting the spatial correlation, especially when the spatial correlation is relatively high. ACKNOWLEDGMENT We thank the anonymous reviewers for their comments to improve the quality of this paper. We also thank Mr. Yiling Yuan for his professional work on LATEX typesetting for this paper. R EFERENCES [1] H. Sampath, S. Talwar, J. Tellado, V. Erceg, and A. Paulraj, “A fourth generation MIMO-OFDM broadband wireless system design, performance, and field trial results,” IEEE Commun. Mag., pp. 143– 149, Sept. 2005. [2] X. Wang and G. B. Giannakis, “Resource allocation for wireless multiuser OFDM networks,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4359–4372, July 2011. [3] M. Joham, P. M. Castro, W. Utschick, and L. Castedo, “Robust precoding with limited feedback design based on predcoding MSE for MU-MISO systems,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 3101–3111, June 2012. [4] C. Shen and M. P. Fitz, “MIMO-OFDM beamforming for improved channel estimation,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 958–959, Aug. 2008. [5] A. Du-Hallen, S. Hu, and H. Hallen, “Long-range prediction of fading signals: enabling adaptive transmission for mobile radio channels,” IEEE Signal Process. Mag., vol. 17, no. 3, pp. 62–75, May 2000. [6] S. Prakash and I. McLoughlin, “Predictive transmit antenna selection with maximal ratio combining,” in Proc. 2009 IEEE GLOBECOM, pp. 1–6.

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[33] 3GPP, “3rd generation partnership project (3GPP); evolved universal terrestrial radio access (E-UTRA); physical channels and modulation (release 10),” Tech. Rep. TS 36.211 V9.1.0, Dec. 2010. [34] H. Zhang, J. T. Y. Li, and A. Reid, “Channel estimation for MIMOOFDM in correlated fading channels,” in Proc. 2005 ICC, pp. 2626– 2630. [35] J. J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson, “On channel estimation in OFDM systems,” in Proc. 2010 IEEE VTC, pp. 528–531. [36] P. Liu, M. Q. Wu, C. X. Xu, and F. Zheng, “Multi-user MIMO linear precoding schemes in OFDM systems,” in Proc. 1995 ICCSIT, pp. 815– 819. [37] Y. R. Zheng and C. Xiao, “Simulation models with correct statistical properties for rayleigh fading channels,” IEEE Trans. Commun., vol. 51, no. 6, pp. 920–928, June 2003. [38] 3GPP, “3rd generation partnership project (3GPP); evolved universal terrestrial radio access (E-UTRA); user equipment (UE) radio transmission and reception (release 10),” Tech. Rep. TS 36.101 V10.4.0 (2011-09), Dec. 2010. [39] ——, “3rd generation partnership project (3GPP); evolved universal terrestrial radio access (E-UTRA); user equipment (UE) radio transmission and reception (release 10),” Tech. Rep. TS 36.211 V9.1.0, Dec. 2010. Lihong Liu received the B.Sc. and M.Sc. degrees in electronic engineering from Fudan University, Shanghai, China in 2010 and 2013, respectively. She is currently an engineer with INTEL Corporation, Visual and Parallel Computing Group, Shanghai, China. Her research focuses on the channel model and processing of wireless networks.

Hui Feng (S’09) received the B.Sc. and M.Sc. degrees in electronic engineering from Fudan University, Shanghai, China in 2003 and 2006, respectively. He is currently a Lecturer with the Department of Electronic Engineering, Fudan University, where he has also been working toward the Pd.D degree since September 2008. His research focuses on distributed optimization and signal processing.

Tao Yang (M’07) received the B.S. degree from Shaanxi Institute of Technology in 1994 and the M.S. degree from Shandong University in 2000, both in automation. He received the Ph.D. degree in control theory and application from Shanghai Jiao Tong University in 2004. In Jan. 2007, He joined the Department of Electronics Engineering at Fudan University, where he is currently an associate professor. Prior to that, he was a Postdoctoral Researcher with Fudan University. His research interests are in general area of application of intelligent signal processing, network information sensing and fusion. Bo Hu (M’00) received the B.Sc. and Ph.D. degrees in electronic engineering from Fudan University, Shanghai, China in 1990 and 1996, respectively. He is currently a Professor with the Department of Electronic Engineering, Fudan University. His research interests focuses on digital image processing, digital communication, and digital system design.