Optimal Linear Channel Prediction for LTE-A Uplink Under Channel Estimation Errors

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Fig. 3 indicates that the computational cost of iterative GBF is significantly lower than those of greedy bit filling [3] and the algorithm in [2], particularly in cases of larger N and smaller b. For instance, when N = 512 and b = 6 is the case, R1 varies from 88% to 98%

along with η, whereas the value of R2 is almost steadily 88%. For

given values of N and b, R1steadily increases along with η, whereas

R2slightly decreases. These observations reveal the appealing merits

of iterative GBF in comparison with earlier algorithms of O(N2₎

complexity.

Fig. 4 presents the improvement of iterative GBF compared with EBF in [1]. As can be seen, for each given pair of N and b, the corresponding curve of R3 has a sawtooth shape. It is due to the

fact that the number of comparisons saved by iterative GBF, i.e., d(N− 1), is proportional to d, which periodically varies from 0 to b− 1 along with η. Numerically, the improvement of iterative GBF can be significant when d is close to b− 1. For instance, R3

can be up to 11% and 9% for scenarios of N = 128 and N = 512, respectively, when b = 10 is the case. It can be also observed that the improvement of iterative GBF is comparatively more significant in scenarios of smaller N and larger b. These observations confirm that the computational cost of iterative GBF is lower than that of EBF.

Obviously, the same conclusions can be drawn for iterative GBR in comparison with greedy bit removal [5], the algorithm in [2], and EBR in [1].

VI. CONCLUDINGREMARKS

We have proposed r-GBF and r-GBR schemes for QAM-modulated OFDM systems, based on which a couple of iterative group-by-group bit-loading algorithms, i.e., iterative GBF and iterative GBR, are presented. In comparison with the original EBF and EBR in [1], the newly presented algorithms get rid of the dependence on conventional greedy algorithms thoroughly and converge to the optimal BAP with lower computational costs.

ACKNOWLEDGMENT

The authors would like to thank Prof. J.-Y. Chouinard and the ano-nymous reviewers for their constructive comments and suggestions.

REFERENCES

[1] D. Wang, Y. Cao, and L. Zheng, “Efficient two-stage discrete bit-loading algorithms for OFDM systems,” IEEE Trans. Veh. Technol., vol. 59, no. 7, pp. 3407–3416, Sep. 2010.

[2] N. Papandreou and T. Antonakopoulos, “A new computationally efficient discrete bit-loading algorithm for DMT applications,” IEEE Trans. Com-mun., vol. 53, no. 5, pp. 785–789, May 2005.

[3] E. Baccarelli and M. Biagi, “Optimal integer bit-loading for multicarrier ADSL systems subject to spectral-compatibility limits,” Signal Process., vol. 84, no. 4, pp. 729–741, Apr. 2004.

[4] A. Fasano, G. Di Blasio, E. Baccarelli, and M. Biagi, “Optimal discrete bit loading for DMT based constrained multicarrier systems,” in Proc. IEEE ISIT, Jul. 2002, p. 243.

[5] A. Fasano, “On the optimal discrete bit loading for multicarrier systems with constraints,” in Proc. IEEE VTC, Apr. 2003, pp. 915–919. [6] R. D. Dutton, “Weak-heap sort,” BIT, vol. 33, pp. 372–381, 1993.

Optimal Linear Channel Prediction for LTE-A Uplink Under Channel Estimation Errors

Jwo-Yuh Wu and Wen-Ming Lee

Abstract—This paper proposes an optimal design of the linear channel

prediction scheme for LTE-A uplink transmission, taking into account the effect of channel estimation errors. The optimization criterion is to minimize mean square prediction errors, averaged over the distributions of true channel coefficients and estimation errors. We derive a closed-form solution of the optimal channel predictor and provide exact analysis for the achievable mean square error (MSE). The performance gain, in terms of the amount of reduction in MSE, as compared with the conventional channel predictor, is also analytically characterized. Simulation results are used to validate our MSE analyses and to confirm the performance advantage of the proposed solution.

Index Terms—Channel estimation, channel prediction, Long-Term

Evolution/Long-Term Evolution Advanced (LTE/LTE-A), training over-head reduction.

I. INTRODUCTION

Low energy consumption has been a critical demand for the de-sign of next-generation wireless communications [1], [2]. This thus spurs the development of new energy-efficient green communication techniques from various aspects, e.g., efficient signal processing al-gorithms at the physical (PHY) layer, intelligent network resource management for improved medium access control, and cognitive ca-pability for better adaptation to the environments, to name just a few. From a PHY-layer perspective, typical approaches toward improved energy efficiency include, e.g., low-complexity algorithm implementa-tion and reduced overheads for various signaling protocols [2]. While the former has been a widely considered design target in the study and design of PHY-layer signal processing techniques, the latter has received relatively little attention.

Reliable acquisition of channel state information (CSI) at the re-ceiver is key to realizing various performance gains in modern wireless communications. The development of low-overhead CSI acquisition techniques is thus of great importance. In this paper, we consider the Long-Term Evolution Advanced (LTE-A) uplink, in which single-carrier frequency-domain multiple-access (SC-FDMA) modulation [3], [4] is employed. To reduce the training overhead for CSI acqui-sition, we propose to adopt the linear channel prediction technique at the base station (BS). The proposed approach is built on the fact that consecutive samples of real-world wireless channels are correlated in the time domain. Hence, as long as the BS has acquired a set of channel estimates during the past few training phases, it is then plausible to

Manuscript received October 30, 2012; revised February 5, 2013; accepted March 29, 2013. Date of publication April 12, 2013; date of current version October 12, 2013. This work was supported in part by the National Science Council of Taiwan under Grant NSC 97-2221-E-009-101-MY3, Grant NSC 99-2628-E-009-004, Grant NSC 2221-E-009-104-MY3, and Grant NSC 100-3113-P-009-001; by the Ministry of Education of Taiwan under the MoE ATU Program; and by the Telecommunication Laboratories, Chunghwa Telecom Co., Ltd., under Grant TL-101-G106. The review of this paper was coordinated by Dr. H. Lin.

The authors are with the Department of Electrical and Computer Engi-neering, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: jywu@cc.nctu.edu.tw; jerry90908@hotmail.com).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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exploit the channel temporal correlation to predict the latest CSI via certain channel prediction schemes. In this way, mobile users no longer need to frequently send training signals to the BS, and thus, the training overhead can be reduced. The main contributions of this paper can be summarized as follows.

1) Assume that during the past K training phases the channels are estimated at the BS based on the linear minimum mean-square-error (LMMSE) criterion [5]. Associated with each user, we propose to adopt a bank of K-tap linear finite-impulse response filters (one for each channel path) to linearly combine K channel estimates for future CSI prediction. The design criterion of the predictor coefficients is the minimization of the MSE between the predicted and the true channels, whereby the expectation is taken with respect to the statistics of both the true channels and estimation errors. We derive a closed-form solution of the proposed channel predictor. In addition, the achievable MSE of the proposed solution is also analytically derived.

2) In case channel estimation is perfect, the proposed channel predictor then reduces to the conventional solution widely seen in the literature, e.g., [6]. We also derive for this solution the formula of the achievable MSE under channel estimation errors. With the aid of this result, it is analytically shown that, compared with this conventional solution, the proposed channel predictor does yield a smaller MSE; in particular, the amount of reduction in MSE is analytically characterized. The computational com-plexity of the proposed robust channel prediction scheme and the conventional solution is also provided.

3) The proposed channel prediction technique can be utilized in conjunction with training-based channel estimation to develop a two-phase CSI acquisition protocol for training overhead re-duction. The training overhead reduction that is achieved by the proposed two-phase protocol is characterized. Simulation results are used to illustrate the resultant MSE and bit-error-rate (BER) performances.

We would like to remark that linear channel prediction has been considered for improving CSI quality in wireless communications, e.g., [7]–[16], to list just a few. Most of the existing works relied on the idealized assumption that channel estimation is perfect. To the best of the authors’ knowledge, our paper provides an original study of algorithm development and exact performance analyses for LMMSE channel prediction in LTE-A uplink under channel estimation errors. The rest of this paper is organized as follows. Section II is the preliminary. Section III shows the main results of this paper, including the development of the proposed channel prediction algo-rithm, characterization of the resulting MSE performance, complexity analysis, and a discussion on the issue of training overhead reduction. Section IV contains the simulation results. Finally, Section V is the conclusion.

II. PRELIMINARY A. System Model During the Training Phase

We consider the discrete-time baseband model for SC-FDMA trans-mission over an L-path frequency-selective fading channel [3], [4]. The communication environment is assumed to be quasi-static fading, in which the channel gains remain constant during one resource block and can independently vary from block to block. Let p[k]∈ CM _be the frequency-domain training symbol at the kth resource block; p[k] is mapped onto consecutive M out of a total number of N subcarriers to obtain ˜ p[k] = Ap[k] (2.1) where, for 0≤ μ ≤ N − M(N > M) A = ₀ μ×M IM 0(N−M−μ)×M ∈ CN×M _(2.2)

is the subcarrier mapping matrix, in which 0m×n and IM denote, respectively, the zero matrix of size m× n and the M × M identity matrix.1_{Before transmission, ˜}_{p[k] is first postmultiplied by the N}_×

N inverse fast Fourier transform matrix FH

N∈ CN×N to get

x[k] = FH

Np[k]˜ (2.3)

which is then appended with a cyclic prefix (CP) of length no less than L− 1 to combat ISI. Let

h[k]Δ= [ h0(k) h1(k) · · · hL−1(k) ] T _{∈ C}L

(2.4) be the channel impulse response vector during the kth resource block; the corresponding channel frequency responses are thus

g[k]= FΔ N h[k]T ₀ 1×(N−L) T ∈ CN_. (2.5) At the receiver side, with CP removal, N -point fast Fourier transform (FFT) processing, and subcarrier demapping, the frequency-domain training data read

y[k]= G[k]p[k] + w[k]Δ (2.6)

where

G[k]= diagΔ AT_g[k]_{∈ C}M×M _(2.7) is a diagonal matrix with the M selected channel frequency responses on the main diagonal, and w[k]∼ CN (0, σ2

wIM), the complex Gaussian distribution with zero mean and covariance σ2

wIN. To fa-cilitate channel estimation, let us rewrite (2.6) into

y[k] = P[k]·AT_g[k] _{+ w[k]} _(2.8) where

P[k] = diag{p[k]} ∈ CM×M _(2.9) is a diagonal pilot signal matrix. By invoking the definition of g[k] in (2.5), it is straightforward to obtain the following equivalent represen-tation of the training system (2.8):

y[k] = P[k] ˜FNh[k] + w[k] (2.10)

in which ˜FN∈ CM×L is obtained from FN by retaining the first L columns and M rows corresponding to the M selected subcarriers. The following assumptions are made throughout this paper.

Assumption 1: For each fixed k, channel gains hi(k)’s are inde-pendent and identically distributed such that hi(k)∼ CN (0, σh,i2 ), 0≤ i ≤ L − 1.

Assumption 2: For each fixed i, the temporal variation of the channel gains follows Jake’s model [17]; thus, the temporal correla-tion funccorrela-tion is given as E{hi(k)h∗i(m)} = σ2h,iJ0(2πfd(k− m)), where J0(·) is the zero-order Bessel function, and fdis the Doppler frequency in hertz.

1_{Application of the proposed approach to the case of distributed subcarrier} mapping is straightforward.

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B. Time-Domain LMMSE Channel Estimation

Based on (2.10), we consider the LMMSE estimation of the time-domain impulse response vector h[k]. By following the standard procedures, e.g., [5] and [18], the resultant channel estimate can be obtained as h[k] = h0(k) h1(k) · · · hL−1(k) T = σ−2w R−1_h + σ−2w F˜NF˜HN −1_˜ FHNP[k]−1y[k] (2.11) where Rh Δ = diagσ2 h,0 σ 2 h,1 · · · σ 2 h,L−1 . (2.12)

We assume that the receiver has acquired a sequence of K channel estimatesh[1], . . . ,h[K] according to (2.11) during K consecutive

training-based resource blocks, based on which channel prediction is then employed to predict the future CSI, which is denoted hereafter byh_P[K + n], for n≥ 1. This paper proposes a new linear channel prediction scheme, which takes into account the channel estimation errors, and studies the achievable MSE performance of the proposed solution. Based on the developed channel prediction technique, we further propose a channel acquisition scheme with reduced training overhead.

III. PROPOSEDCHANNELPREDICTIONSCHEME

This section introduces the proposed optimal channel predic-tion scheme. Secpredic-tion III-A derives the optimal channel predictor. Section III-B shows the achievable MSE performance. Complexity analysis in terms of flop count evaluation is given in Section III-C. Training overhead reduction achieved by the proposed channel predic-tion scheme is then discussed in Secpredic-tion III-D.

A. Algorithm Development

We assume that the receiver has acquired a sequence of K chan-nel estimatesh[1],· · · ,h[K] according to (2.11) during K consec-

utive training resource blocks. Based on the estimated channels, channel prediction is employed to predict the future CSI h_P[K + n] = [h0,P(K + n)h1,P(K + n)· · ·hL−1,P(K + n)]T, for n≥ 1.

For analytic simplicity, prediction is done on a tap-by-tap basis. Thus, associated with the ith path, 0≤ i ≤ L − 1, we consider a linear channel predictor of the form

hi,P(K + n) = K k=1 a(n)_i,kˆhi(K + 1− k), n≥ 1 (3.1) where a(n)_i,k’s are the predictor coefficients to be determined. The design of a(n)_i,k’s is done via minimizing the MSE, i.e.,

MSEi[n] Δ = E hi,P(K + n)− hi(K + n) 2 (3.2)

in which the expectation is taken with respect to the considered channel and noise statistics. By following the orthogonal principle [6], [18], the optimal solution can be obtained by solving the following equation, for 1≤ j ≤ K: E hi(K + n)− K k=1 a(n)_i,khi(K + 1− k) ˆ h∗i(K + 1− j) = 0 (3.3)

which can be rearranged as K k=1 a(n)_i,kE hi(K + 1− k)h∗i(K + 1− j) = E hi(K + n)h∗i(K + 1− j) , 1≤ j ≤ K. (3.4) Equation (3.4) defines a set of K linear equations with a(n)_i,k, 1≤ k ≤ K, as unknowns. Based on (3.4) and by invoking the assumptions about the channel and noise statistics, the optimal predictor coefficients are derived in the following theorem.

Theorem 3.1: Define R = ⎡ ⎢ ⎢ ⎢ ⎣ r[0] r[1] · · · r[K − 1] r[1] r[0] · · · r[K − 2] .. . ... . .. ... r[K− 1] r[K − 2] · · · r[0] ⎤ ⎥ ⎥ ⎥ ⎦ (3.5) r(n)_K = [ r[n] r[n + 1] · · · r[K + n − 1] ]T (3.6) where r[k− m]= JΔ 0(2πfd(k− m)). The optimal channel predictor ¯ a(n)i Δ = [¯a(n)i,1¯a (n) i,2 · · · ¯a (n) i,K] T_{∈ C}K_{is given by} ¯ a(n)_i = αiR−1i r (n) K (3.7) in which Ri Δ = βi· R + γi· IK (3.8) with αi Δ = σ2 h,i× e T iF˜ H NFÑ σ2 wR−1h + ˜F H NFÑ −1 ei (3.9) βi Δ = eT i σ2 wR−1h + ˜F H NFÑ −1_˜ FH NFÑRh × ˜FH NFÑ σ2wR−1h + ˜F H NFÑ −1 ei (3.10) γi Δ = σ2 we T i σ2 wR−1h + ˜F H NFÑ −1_˜ FH N × ˜FN σ2 wR−1h + ˜F H NFÑ −1 ei. (3.11)

Proof: See Appendix A.

It is noted that, with perfect channel estimation, viz., σ2

w= 0, it is easy to check that αi= βi= σh,i2 and γi= 0; as a result, the optimal predictor ¯a(n)_i in (3.7) is reduced to the well-known solution [6]

˜

a(n)_i =

˜

a(n)_i,1 ˜a(n)_i,2 · · · ˜a(n)_i,K T

Δ

= R−1r(n)_K . (3.12)

Moreover, while the proposed channel predictor (3.7) is different across the channel paths, it can be seen from (3.12) that ˜a(n)i is identical, irrespective of the paths.

B. MSE Analysis

To further validate the performance advantage of the proposed error-resistant predictor (3.7) as compared with (3.12), we will compare the achievable MSE of the two prediction schemes, which are defined respectively as MSE[n]=Δ L−1 i=0 E K k=1 ¯ a(n)_i,khi(K + 1− k) − hi(K + n) 2 (3.13)

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MSE[n]=Δ L−1 i=0 E K k=1 ˜ a(n)_i,khi(K + 1− k) − hi(K + n) 2 . (3.14) The following theorem provides analytic expressions for the two MSE formulas in (3.13) and (3.14).

Theorem 3.2: Let MSE[n] and MSE[n], which are defined in (3.13) and (3.14), respectively, be the MSEs achieved by channel predictors (3.7) and (3.12). Under the assumptions that L_i=1σ2

h,i= 1 and r[0] = 1, we have MSE[n] = 1− L−1 i=0 α2i r(n)T_K (βiR + γiIK)−1r (n) K (3.15) MSE[n] = MSE[n] + ˜a(n)H_i R−1_i × 1−αi βi 2 · Ri+ αiγi βi · IK 2 ˜ a(n)_i (3.16) where Ri, αi, βi, γi, and ˜a (n)

i are defined in (3.8)–(3.12), respectively.

Proof: See Appendix B.

From (3.16), it is easy to verify that MSE[n]≥ MSE[n]. Thus, the proposed solution, which takes into account the effect of channel estimation errors, does yield improved channel prediction accuracy; this will be further confirmed via numerical simulation in Section IV. C. Algorithmic Complexity

We go on to compare the computational complexity (measured by the number of flop counts of real addition and multiplication) of the proposed robust channel prediction scheme (3.7) and the conventional solution (3.12). First of all, the computations involved, as well as the required flop cost, for obtaining (3.7) are detailed as follows.

1) For each i = 1, . . . , L, compute αiin (3.9), βiin (3.10), and γi in (3.11), i = 1, . . . , L.

a) To obtain αi, computation of [(σw2R−1h + ˜F

H

NF˜N)−1ei] can be done via LU decomposition together with backward and forward substitutions [19]; this requires 4[(L(L−1)(L+1)/ 3)+L2_{] multiplications and (L(L}_{−1)(4L+1)/3)+4L}2₋

2L additions. Computation of the matrix–vector product σ2 h,i· [eT

iF˜HNF˜N]· [(σ2wR−1h + ˜F

H

NF˜N)−1ei] requires 4L + 4 multiplications and 4L additions.

b) To obtain βi, computation of RhF˜NHF˜N[(σw2R−1h + ˜F

H NF˜N)−1

ei] requires 4L2+ 4L multiplications and 4L2 additions. To obtain [eT i(σ2wR−1h + ˜F H NFÑ)−1F˜HNFÑ]× [RhF˜HNFÑ (σ2 wR−1h + ˜F H

NF˜N)−1ei], it requires 4L multiplications and 4L− 2 additions.

c) To obtain γi, it only requires 4L + 4 multiplications and 4L additions, since ˜FH NF˜N(σ2wR−1h+ ˜F H NF˜N)−1eiand (σ2wR−1h+ ˜ FH

NF˜N)−1eihave been available.

2) Once αi, βi, and γi are obtained, the predictor coefficients ¯

a(n)_i , i = 1, . . . , L, n≥ 1, can be computed by using the Levinson–Durbin algorithm [6], which can exploit the Toeplitz structure of the system matrix for complexity reduction. Based on [19, pp. 194–196], a total number of (2n− 1)K2

multiplica-tions and (2n− 1)K2_{additions are needed.}

3) Once ¯a(n)_i is available, the computation of the predicted channel taps [using (3.1)] calls for 4K multiplications and 4K− 2 additions.

To summarize the given results, the total flop counts required for obtaining the proposed scheme (3.7) are listed in Table I. By

TABLE I

FLOPCOUNTCOMPARISON FOR THEPROPOSEDROBUSTCHANNEL PREDICTIONSCHEME(3.7)AND THENONROBUSTSOLUTION(3.12) (n:THELENGTH OF THETRAININGPHASE, L:THENUMBER OF

CHANNELPATHS, K: PREDICTORORDER, M :THENUMBER OFOCCUPIEDSUBCARRIERS)

Fig. 1. Comparison of algorithmic complexity of the proposed robust channel predictor (3.7) with the conventional solution (3.12) with respect to different lengths of the channel prediction phase. The system parameters are set to be K = 3 and L = 6.

following similar analyses, the total flop counts needed for computing the conventional solution (3.12) are also given in the table. Fig. 1 plots the flop costs of the two channel prediction schemes (3.7) and (3.12) with respect to different lengths of the channel prediction phase (n); the system parameters are set to be L = 6 and K = 3. (The same setting is also used in the simulation section.) The figure shows that the algorithmic complexity of the proposed robust predictor (3.7) is higher; this is mainly due to the additional matrix inversion operations needed in computing coefficients αiand βi. Nevertheless, as will be seen in the following section, our robust scheme (3.7) can largely improve the MSE and BER performances as compared with the conventional solution (3.12).

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Fig. 2. (a) Schematic depiction of training-based channel estimation. (b) Schematic depiction of the proposed two-phase CSI acquisition protocol (K = 2 and n = 2).

D. Two-Phase CSI Acquisition Scheme for Training Overhead Reduction

The proposed robust channel prediction technique can be utilized for developing a reduced-overhead CSI acquisition scheme. Toward this end, we propose a two-phase CSI acquisition protocol, as shown in Fig. 2(b). The training phase consists of K consecutive training-based resource blocks, during each LMMSE channel estimation [5] is conducted. Once the K channel estimates in the training phase are available, a bank of K-tap channel predictors (one for each path) is em-ployed in the prediction phase to predict the CSI for the next n consec-utive resource blocks. Notably, since no training symbols are inserted in the resource blocks during the prediction phase, the training over-head is thus reduced.2_{Specifically, for a given n, which is the duration}

of the prediction phase, the proposed two-phase protocol can achieve η = [n/(K + n)]× 100% (3.17) reduction in the training overhead, as compared with the training-based protocol. In the simulation section, we will further compare the proposed two-phase protocol with the training-based scheme in terms of the MSE of the acquired CSI and the resultant BER.

IV. SIMULATIONRESULTS

Here, computer simulations are conducted to illustrate the perfor-mance of the proposed approach and to validate our theoretical MSE study. We consider an SC-FDMA system, in which the size of FFT is 1024, the number of channel paths is L = 6, and the length of CP is equal to L− 1 = 5; the sum of the channel delay power profile is normalized so that5_i=0σ2

h,i= 1. We compare the proposed robust channel prediction scheme (3.7) with two other methods: the con-ventional nonrobust solution (3.12) and the training-based LMMSE channel estimator [5]. For training-based channel estimation, pilot symbol placement in each time slot follows the LTE-A standard. The prediction-based scheme is implemented by using the two-phase protocol proposed in Section III-D. In our simulations, we set K = 3, and the velocity of the mobile user is 10 km/h.

Fig. 3 compares the MSE achieved by the three channel acquisition schemes; for the two-phase protocol, the duration of the prediction phase is set to be n = 1, 2, 3, which result in, respectively, 25%, 40%, and 50% reduction in the training overhead [cf. (3.17)]. We can first see in the figure that, for both prediction-based solutions, the theoretical MSE is very close to the simulated outcome. This thus validates our MSE analyses shown in Section III-B. Moreover, it can be seen that the proposed robust channel prediction scheme (3.7) outperforms the conventional nonrobust solution (3.12) and, surprisingly, achieves an

2_{It is noted that the amount of transmission energy dedicated to training} signals can be computed according to, e.g., [20], which takes into account the modulation types and various conditions of the propagation environment.

Fig. 3. MSE performances of the proposed robust predictor (3.7), the con-ventional nonrobust predictor (3.12), and the LMMSE training-based channel estimator [5].

Fig. 4. BER performances of the proposed robust predictor (3.7), the con-ventional nonrobust predictor (3.12), and the LMMSE training-based channel estimator [5] (QPSK modulation).

even smaller MSE as compared with the training-based approach. Figs. 4 and 5 then show the resultant BER performances for source symbols drawn from quaternary phase-shift keying (QPSK) and 16-quadrature amplitude modulation (QAM), respectively. As expected,

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Fig. 5. BER performances of the proposed robust predictor (3.7), the con-ventional nonrobust predictor (3.12), and the LMMSE training-based channel estimator [5] (16-QAM modulation).

the BER of the proposed solution (3.7) is substantially lower than that of the conventional predictor (3.12) and is very close to the error rate attained by the training-based scheme.

V. CONCLUSION

We propose a robust linear channel prediction scheme for LTE-A uplink transmission under channel estimation errors. The design crite-rion is to minimize the mean square prediction errors, averaged over the distributions of the true channel coefficients and estimation errors. We derive a closed-form solution of the optimal channel predictor and provide exact MSE analyses. The reduction in MSE as compared with the conventional channel predictor is also analytically characterized. Simulation results validate our MSE analyses. In addition, the pro-posed channel prediction scheme combined with the training-based channel estimation can realize a two-phase CSI acquisition protocol for training overhead reduction. Through computer simulations, it is seen that such a two-phase protocol favorably compares with the LMMSE-based training scheme in terms of MSE and BER, at reduced signaling overhead.

APPENDIXA PROOF OFTHEOREM3.1

We shall first evaluate the cross-correlation term E{hi(K+n)h∗i (K +1−j)} and auto-correlation term E{hi(K+1−k)h∗i(K+1−j)} involved in (3.4). Toward this end, let us first write

E hi(K + n)h∗i(K + 1− j) = eTiE h[K + n]hH[K + 1− j] ei. (A.1)

Based on (A.1), we have

eTiE h[K + n]ˆhH[K + 1− j]ei (a) = eTiE h[K + n]σw2 · R−1h + ˜F H NFÑ −1 FHN ×_F˜_N_{h[K + 1}_{− j] + X}−1 p [k]w[k] H ei (b) = eT iE h[K + n]hH_{[K + 1}_{− j]} × ˜FH NFÑ σ2w· R−1h + ˜F H NFÑ −1 ei (c) = r[n + j− 1] · eTiRhF˜ H NFÑ σ2w· R−1h + ˜F H NFÑ −1 ei (d) = r[n + j− 1] · σ2 h,i· e T iF˜ H NFÑ σ2 w· R−1h + ˜F H NFÑ −1 ei = r[n + j− 1] · αi (A.2)

where (a) follows by definition of ˆh[k] in (2.11), (b) holds since true

channel h[k] and noise w[k] are assumed to be uncorrelated, and (c) and (d) are obtained through straightforward manipulations. To determine the auto-correlation term, by following similar procedures, we have E hi(K + 1− k)h∗i(K + 1− j) = eT iE h[K + 1− k]hH_{[K + 1j]}_e i (e) = eTi σ2wR−1h + ˜F H NFÑ −1_˜ FHN × E_F˜ Nh[K + 1− k] + X−1p w[K + 1− k] ×hH[K + 1− j]˜FHN+ w H [K + 1− j]Xp × ˜FN σ2 wR−1h + ˜F H NFÑ −1 ei (f ) = eTi σw2R−1h + ˜F H NFÑ −1_˜ FHN ×r[j− k] · ˜FNRhF˜HN+ σ 2 w· δ[j − k] · IM × ˜FN σw2R−1h + ˜F H NFÑ −1 ei = r[j− k] · eTi σw2R−1h + ˜F H NFÑ −1_˜ FHNFÑRhF˜HNFÑ ×σ2 wR−1h + ˜F H NFÑ −1 ei + δ[j− k]eT i ×σ2 wR−1h + ˜F H NFÑ −1_˜ FH NFÑ σ2 wR−1h + ˜F H NFÑ −1 ei = r[j− k] · βi+ δ[j− k] · γi (A.3)

where (e) follows (2.11), and (d) holds due to the uncorrelated assump-tion between the channel and noise. Based on (A.2) and (A.3), the K linear equations in (3.4) can be put into a matrix–vector form as

(βiR + γiIK)· ¯a (n) i = αir

(n)

K . (A.4)

The assertion of the theorem directly follows from (A.4).

APPENDIXB PROOF OFTHEOREM3.2 To derive (3.15), let us expand MSE[n] in (3.13) into MSE[n] = L i=1 E|hi(K + n)|2 − 2 K k=1 ¯ a(n)_i,kRe E hi(K + n)h∗i(K + n− k)

(7)

+ K k=1 K j=1 ¯ a(n)_i,kE hi(K + n− k) ×h∗_i(K + n− j) a(n)_i,j . (B.1)

Since E{|hi(K + n)|2} = σ2h,iand using (A.3), (B.1) can be further rewritten as MSE[n] = L i=1 σ2 h,i− 2 K k=1 ¯ a(n)_i,kRe E hi(K + n)h∗i(K + n− k) + K k=1 K k=1 ¯ a(n)_i,k (r[j− k] · βi+ δ[j− k] · γi) ¯a (n) i,j ! (a) = 1− L i=1 2 K k=1 ¯ a(n)_i,kRe E hi(K + n)h∗i(K + n− k) + K k=1 K k=1 ¯ a(n)_i,k (r[j− k] · βi+ δ[j− k] · γi) ¯a (n) i,j ! (b) = 1− L−1 i=0 2αi¯a (n)T i r (n) K − ¯a (n)T i Ri¯a (n) i (c) = 1− L−1 i=0 α2i· r (n)T K (βi· RK+ γi· IK)−1r (n) K (B.2)

where (a) follows from the assumptionL_i=1σ2

h,i= 1, (b) is obtained through direct manipulations, and (c) holds by invoking ¯a(n)i =

αiR−1i r (n)

K together with some manipulations. Equation (3.15) is thus proved. To derive (3.16), let us define ui

Δ

= [hi[K] hi[K− 1] · · · hi[1]]Tand then rewrite MSE[n] in (3.14) as

MSE[n] = L i=1 E ˜a(n)T_i ui− hi(K + n) 2 . (B.3) Since, from (3.7) ¯ a(n)_i = αiR−1i r (n) K = αi(βi· R + γi· IK)−1· r (n) K =αi βi · R +γi βi · IK −1 · r(n) K (B.4)

using the matrix inversion lemma [18], we can obtain the following relation between ¯a(n)_i and ˜a(n)_i :

¯ a(n)_i =αi βi · R +γi βi · IK −1 · r(n) K =αi βi · R−1− R−1 R−1+βi γi IK −1 R−1 · r(n) K =αi βi · R−1− γi βi R +γi βi IK −1 R−1 · r(n) K =αi βi IK− γi βi R +γi βi IK −1 · R−1_{· r}(n) K =αi βi IK− γi βi RK+ γi βi IK −1 · ˜a(n) i . (B.5)

By defining Δ(n)_i = ˜Δa(n)_i − ¯a(n)_i and based on (B.5), we have Δ(n)_i = ã(n)_i − ¯a(n)_i = ã(n)_i −αi βi IK− γi βi R +γi βi IK −1 · ã(n) i = " 1− αi βi · IK+ αi· γi βi (βi· R + γi· IK)−1 # · ã(n) i = " 1− αi βi · IK+ αi· γi βi R−1_i # · ã(n) i . (B.6)

With the aid of (B.6), MSE[n] in (3.14) reads MSE[n] = L i=1 E ã(n)T_i ui− hi[K + n] 2 = L i=1 E ¯ a(n)i + Δ (n) i T ui− hi[K + n] 2 = L i=1 E a¯(n)T_i ui− hi[K + n] + Δ(n)T_i ui 2 (d) = L i=1 E a¯(n)Ti ui− hi[K + n] 2 + L i=1 E Δ(n)Ti ui 2 (B.7) where (d) holds because, with the orthogonality condition (3.3), it follows that E ¯ a(n)Ti ui− hi[K + n] Δ(n)Ti ui ∗ = E ¯ a(n)T_i ui− hi[K + n] · uH i · Δ(n) i = 0. (B.8) Finally, it remains to specify the term E{|Δ(n)T_i ui|2}, which can be expressed as E Δ(n)Ti ui 2 = Δ(n)H_i · EuiuHi · Δ(n) i = Δ(n)H_i · Ri· Δ (n) i = ã(n)H_i · " 1− αi βi · IK+ αi· γi βi R−1_i # × Ri " 1− αi βi · IK+ αi· γi βi R−1_i # · ã(n) i = ã(n)H_i · 1− αi βi 2 · Ri+ 2· 1− αi βi ·αi· γi βi · IK+ α2 i· γ 2 i β2 i · R−1 i · ã(n) i

(8)

= ã(n)H_i · R−1_i · 1−αi βi 2 · R2 i+ 2· 1−αi βi ·αi· γi βi · Ri+ α2 i· γi2 β2 i · IK · ã(n) i = ã(n)H_i · R−1_i · " 1− αi βi · Ri+ αi· γi βi · IK #2 · ã(n) i . (B.9)

Hence, (3.16) is thus proved.

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