Robust Training Sequence Design for Spatially Correlated MIMO Channel Estimation

Robust Training Sequence Design for Spatially

Correlated MIMO Channel Estimation

Chin-Te Chiang and Carrson C. Fung, Member, IEEE

Abstract—A robust superimposed training sequence design is proposed for spatially correlated multiple-input–multiple-output (MIMO) channel estimation. The proposed scheme does not re-quire accurate knowledge of the spatial correlation matrix, and it is shown to outperform previously proposed robust correlated MIMO channel estimators, such as relaxed minimum mean square error (RMMSE) and least-square RMMSE. Since the training sequence is overlaid into the data stream, the spectral efficiency of the system is higher than those that use time-multiplexed pilots. A solution for the sequence can easily be obtained by using a projection on convex-set-based iterative algorithm that is guar-anteed to converge as long as the training sequence matrix is initialized to have full rank. Furthermore, it is shown that the proposed scheme is identical to the RMMSE-based schemes when the MIMO channel is spatially uncorrelated. The computational complexity of the proposed algorithm is also illustrated.

Index Terms—Affine precoder, majorization, multiple-input– multiple-output (MIMO), robust channel estimation, spatial cor-relation, superimposed training (SIT) sequence.

I. INTRODUCTION

T

HE DEVELOPMENT of single-user multiple-input– multiple-output (MIMO) systems [1]–[3] has spurred tremendous research effort in advancing techniques that maxi-mize diversity and spatial multiplexing gains [4], [5]. To realize such gains, channel state information (CSI) must accurately be obtained. Although techniques such as differential space–time coding [6], [7] and differential orthogonal space–time block coding (STBC) [8], [9] have been proposed to blindly de-modulate and decode the received signal, this degrades both performance (compared with coherent techniques) and spec-trum efficiency. Therefore, coherent detection is widely used in current MIMO systems where CSI is usually obtained using time-multiplexed pilot symbols.

Theoretically, MIMO performance gain depends only on the minimum between the number of transmit and receive antennas. However, propagation limitations such as channel rank loss and antenna correlation have to be properly handled before

Manuscript received July 13, 2010; revised April 29, 2011; accepted June 4, 2011. Date of publication July 14, 2011; date of current version September 19, 2011. This work was supported by the National Science Council under Grant 99-2219-E-009-011. This paper was presented in part at the 2010 IEEE International Conference on Communications. The review of this paper was coordinated by Dr. C. Cozzo.

C.-T. Chiang was with the Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan. He is now with the Intellectual Property Office, Ministry of Economic Affairs, Taipei 106, Taiwan (e-mail: jeremiah1214@gmail.com).

C. C. Fung is with the Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: c.fung@ieee.org).

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

Digital Object Identifier 10.1109/TVT.2011.2161640

such performance gains can be realized. Increased antenna correlation can be attributed to a reduction in antenna spacing or angular spread, which is caused by the lack of a rich scattering environment around the transceiver. Hence, some degree of spatial correlation will be experienced at the transmitter and/or receiver. Hence, it is vital to account for spatial correlation dur-ing the channel estimation process to maximize the decoddur-ing performance.

Many techniques have been proposed to tackle the prob-lem of correlated MIMO channel estimation. Reference [10] proposed using a state-space approach to estimate and track time-varying correlated MIMO channels, where the channel correlation matrix is estimated from the received data and treated as part of the state variable. In [11], a precoder-assisted linear minimum mean-square error (MMSE) estimator was proposed to estimate the channel. In [12], two channel estimators were derived under the MMSE and conditional mutual information criteria by exploiting the virtual channel representation. Unfortunately, there is no closed-form solution; thus, the solution has to be computed numerically. Another MMSE-based channel estimator was derived in [13] using structured correlation, which allows it to obtain better mean-square error (MSE) performance than the unstructured-based MMSE estimator. One major drawback shared among these estimators is that they require exact knowledge about the spatial correlation to outperform channel estimators that take no such correlation into account. Another disadvantage is that they were all derived under the premise that time-multiplexed pilot symbols are used, which can drastically reduce the transmission efficiency, particularly in cases where the channel is undergoing fast fading.

To bypass the second problem, a superimposed training (SIT) sequence-based channel estimation algorithm was proposed in [14], where the SIT sequence is arithmetically added into the transmitted signal, thus allowing the system to free up valuable time slots that were previously used by time-multiplexed pilot symbols. The training sequence can also be used to deal with the problem of synchronization [17]. Improved channel estima-tion algorithms based on the SIT sequence have since appeared in the literature [15]–[17]. The sequence itself can be extracted at the receiver by using first-order statistics [15], [16] or by using affine precoding [18]–[20]. However, the effectiveness of these algorithms still hinges on acquiring accurate estimates of the spatial correlation, making these methods somewhat infeasible in real situations. To combat against this problem, the relaxed MMSE (RMMSE) and least-square RMMSE (LS-RMMSE) algorithms that have recently been proposed by [21] can circumvent the dependency on the correlation matrix by

using diagonal loading. A different approach using basis ex-pansion was proposed in [22].

In this paper, a deterministic approach is taken in the design of SIT sequences that are robust toward spatial correlation un-certainty. Such a training sequence is then applied to the MMSE estimator to estimate the channel. The approach taken in this work allows for accurate and robust channel estimation without

a priori knowledge about the channel distribution model. For

ease of presentation, such an estimator will be called RoMMSE estimator (despite the fact that it is the sequence that is robust to spatial correlation error). The proposed design exploits the affine precoder scheme proposed in [20] and [24] to extract the training sequence for channel estimation. Since the training sequence is overlaid onto the information-bearing signal, it allows for greater spectral efficiency compared with conven-tional systems that use time-multiplexed pilot symbols. The simulation results will show that the proposed scheme performs extremely well against the estimator in [20], which does not take into account the spatial correlation estimate error. More-over, the RoMMSE estimator also outperforms the RMMSE and LS-RMMSE estimators when the MIMO channels are spatially correlated. Finally, the RoMMSE estimator will also be compared, analytically and by simulation, to the RMMSE and LS-RMMSE estimators for uncorrelated MIMO channels in which it is shown that the three estimators are identical. The rest of this paper is organized as follows: The system model and the affine precoding method are given in Section II, followed by a detailed description of the proposed RoMMSE algorithm in Section III. Simulation results are provided in Section IV, and this paper is concluded in Section V.

Notation: Upper (lower) boldface letters indicate matrices

(column vectors). SuperscriptH_{denotes Hermitian,}T _denotes

transposition, and ∗ denotes conjugation. E[·] stands for ex-pectation. N (A) denotes the null space of A. The operation vec(A) forms a column vector by vertically stacking the col-umn vectors of A. tr(A) denotes the trace of the matrix A. diag(x) denotes a diagonal matrix with x on its main diagonal, IN denotes an N× N identity matrix, and 0M×N denotes

an M× N all zero matrix. ⊗ denotes the Kronecker product.

AFdenotes the Frobenius norm of the matrix A.

II. SYSTEMMODEL ANDAFFINEPRECODING A. System Model

The system model used in [20] is adopted herein. For the sake of completeness, the model will also be described in the sequel. Consider a spatially correlated flat-fading MIMO channel with Nt transmit and Nr receive antennas, as shown

in Fig. 1. The information-bearing signal vector is denoted as u(k) = [u(kNs) u(kNs+ 1)· · · u(kNs+ Ns− 1)]T, where k

is the block index, and Nsdenotes the block size. Each block

of the signal is encoded using STBC, which can be used to increase the transmit diversity or multiplexing gain [23]. The STBC has Nt number of output vectors, with each vector

containing K≥ Ntsymbols as full rate STBC is assumed. This

can be represented in matrix form as X = [x1x2· · · xNt]

T _∈

CNt×K_{, where x}

i∈ CK, i = 1, 2, . . . , Ntdenotes the ith

out-Fig. 1. Block diagram of MIMO transceiver.

put vector. Each vector is then fed into the precoder P = [p1p2· · · pK]T ∈ CK×(K+L), which adds L≥ Ntredundant

symbols to each block of signal, resulting in the output signal vector di∈ CK+Lfor i = 1, 2, . . . , Nt. All Ntoutputs of the

precoder can be represented in matrix form as

D=Δ ⎡ ⎢ ⎢ ⎣ dT 1 dT 2 .. . dT Nt ⎤ ⎥ ⎥ ⎦ = XP = ⎡ ⎢ ⎢ ⎣ xT 1P xT 2P .. . xT NtP ⎤ ⎥ ⎥ ⎦ ∈ CNt×(K+L)_{. (1)} As seen in the sequel, the precoder is used to assist in the channel estimation [8]–[10] by eliminating the information-bearing signal at the receiver, thus leaving the SIT sequence intact for channel estimation. It was shown in [24] that the precoder can also be designed to improve the symbol detection rate or minimize the mean square error between the transmitted and recovered signals [8]. After precoding, the SIT sequence vector ci, i = 1, 2, . . . , Ntis added to di. Each vector is then

serialized before it is transmitted across the flat-fading MIMO channel, which is represented in matrix form as H∈ CNr×Nt_. Thus, the received signal can be written as

Y = H(C + D) + η = HC + HXP + η (2) where C=Δ ⎡ ⎢ ⎢ ⎣ cT₁ cT2 .. . cT Nt ⎤ ⎥ ⎥ ⎦ ∈ CNt×(K+L)_{, and η}∈ CNr×(K+L) are the SIT sequence matrix and the additive channel noise ma-trix, respectively. Notice in (2) that the received signal in space lies in the rows of Y. Thus, the rows of the information-bearing portion of the signal, i.e., xT_iP, i = 1, 2, . . . , Nt, belong to

the row space of P. Hence, the rows of HXP also belong to the same subspace. This is different from the conventional model used in [9], [25], and [26], where the information-bearing portion of the received signal is embedded inside the

range space of the unknown channel matrix, thus making it difficult for channel estimation using the SIT sequence. The affine precoding approach adopted herein eases the decoupling of the information-bearing signal and the training sequence because decoupling can now be done by postmultiplying Y by a decoupling matrix Q = [q1 q2. . . qK+L]T ∈ C(K+L)×Nt,

resulting in

YQ = HCQ + HXPQ + ηQ. (3)

Thus, by requiring the columns of Q to lie inN (P), i.e., PQ = 0K×Nt, then (3) becomes

YQ = HCQ + ηQ. (4)

In other words, the training sequence vector ci, i = 1,

2, . . . , Nt, should lie in the column space of Q. Therefore,

the condition that CPH= 0Nt×K guarantees the subspaces spanned by the vectors in P and C are complementary [20]. This suggests a simple way to design P and Q by extracting components off of an orthogonal matrix, i.e.,

P =  K + L K O(1 : K, :)∈ C K×(K+L) Q = (O ((K + 1) : (K + Nt), :))H ∈ C(K+L)×Nt.

Note that O(1 : K, :) and O((K + 1) : (K + Nt), :) keep only

rows 1 to K and rows (K + 1) to (K + Nt) of an orthogonal

matrix O∈ C(K+Nt)×(K+L)_{[24]. Hence, Q}H_{Q = I}

Ntso that noise amplification will not occur in the channel estimation process.

In addition to channel estimation, another decoupling matrix QD can be designed to maximize the symbol detection

per-formance. Such a decoupling matrix can be chosen to satisfy the condition QD= PH(PPH)−1, where P is designed such

that CPH = 0Nt×K. This ensures that the detection process is free of interference from the SIT sequence when QD is

postmultiplied to Y. Therefore PPH= QH_DQD

−1

=K + L

K IK

such that tr(PPH) = K + L. This is to ensure that the av-erage transmitted power of the information-bearing signal is unchanged after precoding.

According to the Kronecker model [23], the channel matrix can be decomposed as H = Σ 1 2 rHwΣ 1 2 t (5) where Σ1/2_r ∈ CNr×Nr _{and Σ}1/2

t ∈ CNt×Nt are the Cholesky

factors of the spatial correlation matrix of the receiver and transmitter, respectively. Hence, the overall spatial cor-relation is R = Σt⊗ Σr. The entries of Hw∈ CNr×Nt

are independent and identically distributed zero-mean com-plex Gaussian random variables with unit variance. Thus,

E[vec(Hw)vecH(Hw)] = INrNt.

B. MMSE Estimator and Training Sequence Design

To derive the proposed RoMMSE estimator, (4) is first vec-torized to obtain the received signal vector

y = Ch + n (6)

where y = vec(YQ)∈ CNrNt_{, C = (CQ)} T ⊗ I

Nr ∈ CNrNt×NrNt_{, h = vec(H)}∈ CNrNt_{, and n = vec(ηQ)}∈ CNrNt_{. E[nn}H_{] = σ}2

nINrNt. From the vectorized received signal y, the linear MMSE estimator of h is [27, p. 387]

h = RH yhR−1yyy = R CH  CR CH+ σ_nn2 INrNt ₋₁ y (7) where Ryy = E[yyΔ H], Ryh= E[yhΔ H], and R= E[hhΔ H]

are the autocorrelation matrix of the received signal y, the cross-correlation matrix of y and h, and the spatial correlation matrix of the channel, respectively. All the matrices are of size

NrNt× NrNt. Therefore, the optimal MMSE estimate of h

can be obtained by finding the optimal training sequence matrix 

C. Note that the mean-square error matrix between h and h is written as [27, p. 387] ξ = E  (h− h)(h − h)H  = R−1+ CH σ_nn2 INrNt −1 C ₋₁ . (8)

From (8), [20] proposed to design the optimal training sequence matrix C by minimizing the trace of ξ subject to the power constraint tr(CCH)≤ Nt(K + L)σ2cc

Δ

= PT, where σ2ccis the

average power of the training sequence. It was assumed in [20] that the average transmitted power, which includes the power of the information-bearing and training signals, is normalized as

σ2_xx+ σ2_cc= 1, where σ2_xx is the variance of the information-bearing signal. This assumption will also be applied to the pro-posed RoMMSE estimator. Since C = (CQ)T⊗ INr, using the properties that tr(AB) = tr(BA), (A⊗ B)(C ⊗ D) = (AC)⊗ (BD), (A ⊗ B)H_{= A}H_{⊗ B}H_{, and tr(A⊗ B) =}

tr(A)tr(B), the power constraint on C can be written as tr( C CH) = tr (CQ)T ⊗ INr   (CQ)T ⊗ INr H = tr (CQ)T ⊗ INr  [(CQ)∗⊗ INr] = tr (CQ)T(CQ)∗⊗ INr = tr (CQ)T(CQ)∗tr(INr) = Nrtr(QTCTC∗Q∗) = Nrtr(CTC∗) ≤ NrPT Δ = PT. (9)

The inequality is obtained because tr(CTC∗) = tr(CHC) = tr(CCH) =C2

F ≤ PT. It is important to note that the

per-formance of the RoMMSE estimator is dependent on the to-tal transmission power PT and not the number of redundant

Fig. 2. MSE versus SNR performance comparison between different numbers of redundant vectors with different PT for spatially correlated 2× 2 MIMO

system, Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

Fig. 3. MSE versus SNR performance comparison between different numbers of redundant vectors with fixed PT for spatially correlated 2 × 2 MIMO

system, Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

SIT sequence to be decoupled from the information-bearing signal at the receiver. Figs. 2 and 3 show the MSE performance of the proposed RoMMSE estimator (to be described in the next section) for 2× 2 correlated MIMO channels when L is increased with different and fixed PT, respectively. In the latter

case, σ2

ccis decreased while L is increased to keep PT constant.

From Fig. 2, it is clear that the MSE performance improves as L increases, whereas Fig. 3 shows that such performance improvement is due to the increase in PT, not just L, because

as PT is kept constant even while L is increased, there is no

change in MSE performance.

III. PROPOSEDROMMSE DESIGN A. Proposed Training Sequence Design

It is clear from (8) that exact knowledge of R is required at the receiver to obtain an accurate estimate of h using (7). However, in all likelihood, only an estimate of R can be obtained, for example, using the method proposed in [28]. To

desensitize the MSE from the estimation error of R, a novel SIT sequence design is proposed herein to incorporate such an estimation error. As the spatial correlation matrix is estimated at the receiver before it is fed back to the transmitter using a low-rate control channel, the quantization error will tend to color the spatial correlation mismatch between the estimated and actual spatial correlations. Hence, a deterministic approach is proposed herein to bound the error in a norm ball. Applying such a SIT sequence into the MMSE estimator in (7) allows the estimator to be more robust against estimation error in the spatial correlation than other MMSE-based estimators that do not take such error into account. Although the rate of change of the channel statistics is slower than that of the channel coefficients, imperfect channel statistics will still adversely affect the channel estimation performance and thus the bit error rate (BER), if not properly accounted for in the system design. Moreover, better robust channel estimation can be obtained if the spatial correlation and channel coefficient mismatches can separately be accounted for as the structure of the spatial correlation mismatch matrix will be different from that of the channel coefficient matrix.

Let

R = R + E (10)

where R denotes the estimate of R, and E is its corresponding spatial correlation mismatch matrix, respectively. In the present scheme, the error power is upper bounded such thatEF ≤ ε,

where ε is a predefined error power bound. In practice, a table of sequences for different ε can be computed a priori and placed in memory at the transmitter. An adaptive approach can then be used to choose a suitable sequence for channel estimation.

Using this bound with (9) and (10), the training sequence matrix C (or its equivalent C) can be designed by minimizing the maximum mean square error ξ, i.e.,

min C2 F≤PT max EF≤ε tr  ( R+E)−1+ CH σnn2 INrNt −1  C ₋₁ . (11) Note that (11) is not a convex problem with respect to E and C. However, the problem can be decomposed into two separate convex optimization problems: one with respect to E and the other to C. Furthermore, performing SVD on C, i.e.,



C = UCΣCV H



C and using the property tr(AB) = tr(BA),

the objective function of the maximization problem in (11) can be rewritten as tr( R + E)−1+ CH σ_nn2 INrNt −1  C ₋₁ = tr  ( R + E)−1+ σ_nn−2_VCΣH CΣCV H  C ₋₁ = tr  VH C( R + E) −1 VC+ σ −2 nnΣH_CΣ_C −1 . (12) Next, using the property tr(A + B) = tr(A) + tr(B) and the matrix inversion lemma (A + BCD)−1= A−1− A−1B (C−1+ DA−1B)−1DA−1, and letting A = σ−2nnΣH_CΣ_C,

C = VH C( R + E) −1 VC and B = D = INrNt, then (12) becomes tr  σ2_nnΣ−1 CΣ −H  C − σ 2 nnΣ−1_C Σ−H_C ×σ_nn2 Σ−1 CΣ −H  C + V H  C( R + E)VC −1 σ2_nnΣ−1 CΣ −H  C  = tr σ_nn2 Λ−1 C  − σ4 nntr  Λ−1 C  σ_nn2 Λ−1 C + V H  C( R + E)VC −1 Λ−1 C  = tr σnn2 Λ−1_C  − σ2 nntr  ΛC+ σ −2 nnΛ_CVH_C( R + E)V_CΛ_C ₋₁ = tr σ_nn2 Λ−1 C  − σ2 nntr   CHC + σ _nn−2CHC(  R + E) CHC ₋₁ (13) where ΛC Δ = ΣH

CΣC. Since the first term of (13) does not

depend on E, maximizing (11) is equivalent to minimizing the second term in (13). Therefore, the maximization problem in (11) becomes min EF≤ε tr   CHC + σ −2nnCHC(  R + E) CHC ₋₁ (14) and it can easily be solved using convex optimization toolbox such as cvx [29] since the ( R + E)−1term is eliminated.

Unfortunately, even if (14) is substituted into (11), it is difficult to find a closed-form solution for C. Therefore, the iterative algorithm in Fig. 4 is proposed. As seen from the figure, C is first initialized to be a full rank matrix satisfying the condition C(0)PH_{= 0. C(0) is then used in (17) [or,}

equivalently, (14)] to solve for a solution for E. This is then used in (18) to solve for C. This process will be repeated until

E(n) − E(n − 1)2_{/ε is less than some preset threshold α.} Note that C needs to be initialized to have full rank; otherwise, the inverse in (17) cannot be taken.

Assuming C has full row rank. Initializing C in the algorithm shown in Fig. 4 to be C(0), it is obvious that

C(0) = UC(0)[ΣC(0) 0Nt×(K+L−Nt)]V

H

C(0) (15)

where UC(0), VC(0), and ΣC(0)are the left and singular vector

matrix of C(0) and the invertible portion of the singular value matrix of C(0), respectively. Hence, to satisfy the condition that CPH= 0Nt×K, it is necessary that VC(0)= UQ, where UQ is the eigenvector matrix of QQH. That is

QQH= UQΛQUHQ = UQ  ΛQ 0Nt×(K+L−Nt) 0(K+L−Nt)×Nt 0(K+L−Nt)×(K+L−Nt)  UHQ

where Λ_Q∈ CNt×Nt _{is a diagonal matrix containing the} nonzero eigenvalues of QQH. Assume that the diagonal values

Fig. 4. Algorithm pseudocode for training sequence design.

of ΛQ are arranged in descending order. Hence, C2(n) =

(UC(n)ΛC(n)UHC(n))∗⊗ INr, n = 0, 1, . . . , n0, where n and n0 denote the iteration index and the iteration time whenE(n) − E(n − 1)2_{/ < α, respectively, and Λ}

C(n)=

ΣC(n)Σ∗C(n). Thus, VC(n)= VC(0)= UQ, n = 0, 1, . . . , n0, and the training sequence, when convergence has been reached, becomes C(n0) = UC(n0)  ΣC(n0) 0Nt×(K+L−Nt)  UH_Q (16) where UC(n0) is the singular vector matrix for C(n0), and

ΣC(n0)∈ C

Nt×Nt _{is a singular value matrix of C(n}

0) con-taining all nonzero singular values. This conforms with the structure previously derived in [20].

B. Convergence Analysis

Theorem 1: The iteration depicted in Fig. 4 will always

con-verge to the global optimal solution given that C is initialized as a matrix with full rank, where the constraint C(0)PH= 0Nt×K is satisfied.

Proof: Define the convex sets E = {( C, E)|EF ≤ ε}

and C = {( C, E)| C2

two tuples.1Express the objective function in (11) as f ( C, E). Given C, it is clear that max f ( C, E) with respect to E is a nonexpansive operator, i.e.,

0 = max E1F≤ε f ( C, E1)− max E2F≤ε f ( C, E2)   ≤ E1− E2. (19)

Similarly, given E, min f ( C, E) with respect to C is also a nonexpansive operator, i.e.,

0 =   _C1min2_F≤PT f ( C1, E)− min C22_F≤PT f ( C2, E)    ≤  C1− C2. (20)

Moreover, the solutions for (19) and (20) will always belong to E and C, respectively. Then, according to the theory of alternating projections [31], the algorithm depicted in Fig. 4 will always converge given appropriate initial conditions. Since the two sets are convex, there is a unique point of intersection, and thus, the solution obtained in Fig. 4 will always be the

global optimal solution. 

Note that it is possible for C(n) to lose rank when the SNR is low and when ε is sufficiently small (e.g., SNR = 0 dB and

ε = 0.1). To prevent this from occurring, C2_{(n) is diagonally} loaded, i.e., C2_{(n) = C}2_{(n) + ρI}

NtNr, where ρ is a small value compared with C2(n)F, e.g., ρ = 0.01 C2(n)F. C. Power Allocation

The use of the SIT sequence has an adverse effect on the recovery of the information-bearing signal at the receiver as the sequence reduces the power of the transmitted data signal. A suboptimal power allocation scheme for the training sequence was derived in [20], in which the effective SNR

SNReff = E   HX2 E   HX + NQD2  (21)

was maximized, which consequently maximizes the probability of detecting the correct signal at the receiver. A similar power allocation scheme can similarly be derived with the spatial correlation mismatch matrix E taken into account. The method is suboptimal due to the fact that the numerator in (21) can be written as E   HX2  = tr E[ HXXHHH]  = Kσ_x2tr E[HHH] + E[ H HH]  = Kσ_x2 tr( R + E) +   = Kσ_xx2 (NtNr+ ) (22)

where it has been assumed that tr(E[HHH]) = tr( R + E) =

NtNr, with  = tr(E[ H HH]) denoting the mean square error

1_{Both sets are convex because their respective constraints form a norm ball.}

of the channel. Note that the received SNR is defined as SNR =−10 log₁₀σ2

nnunder the assumption that the power of

the received signal is normalized to 1. Thus, trace( R + E) =

NtNr. In addition, it should be noted that there is an error in

the expression for E[ HX2] in [20] in which  was preceded by a minus sign, even if it should be preceded by a plus sign instead, as indicated in (22). Hence, SNReff becomes

SNReff = σ2 xx(NtNr+ ) σ2 xx + γ (23) where γ = Nrσ2nn(K/K + L). Using the property that if

tr(A) > tr(B), then tr(B−1) > tr(A−1), given that A and B are positive definite matrices. It then follows that  is upper bounded by (NrNt/Nr+ Nt)(σnn2 /σ2cc) = β(σ2nn/σ2cc).

Sub-stituting this upper bound into (22), the effective SNR is then lower bounded as SNReff σcc2 ≥ 1− σ2 cc  (NtNr+)σ2cc+ βσnn2  βσ2 nn− βσ2nnσ2cc+ γσcc2 . (24) The maximum of the effective SNR can then be achieved by maximizing the lower bound in (24), which can be accom-plished by differentiating the bound with respect to σ2

cc, setting

the result to zero, and solving for σ2

cc. This results in the

suboptimal power allocation for the SIT sequence

σ_cc,subopt2 = δβσ 2 nn−  δγβσ2 nn(δ− γ + βσnn2 ) δ (βσ2 nn) (25) where δ = NtNr. σcc,subopt2 in (25) is similar to the expression

derived in [20] except for the sign error, as previously indicated. The difference is due to the sign error in (22). However, the foregoing power allocation expression is derived directly with inclusion of the spatial correlation mismatch, thus generalizing the result previously reported in [20].

IV. SIMULATIONRESULTS

Monte Carlo simulations were used to demonstrate the ro-bustness of the proposed scheme. The channels used in all the simulations are assumed to be quasi-static block Rayleigh fading and spatially correlated, unless otherwise specified. The one-ring model [30] is used to generate entries of the Cholesky factors of the spatial transmit and receive correlation matrices

Σt(m, n)≈ J0  Δ2π λdt|m − n|  (26) Σr(i, j)≈ J0  2π λ dr|i − j|  (27) where dtand drare the spacing between transmit and receive

antennas, respectively. Δ denotes the angular spread, λ de-notes the carrier wavelength, and J0 is the 0th-order Bessel function of the first kind. The power allocation scheme in (25) for the training sequence is adopted. Quaternary phase-shift keying and Alamouti STBC are used for modulation of the information-bearing signals. In all the simulations, the threshold for the iteration algorithm is α = 10−6.

Fig. 5. MSE versus SNR performance comparison between RoMMSE and [20] for spatially correlated 2× 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

Fig. 6. MSE versus ε performance comparison between RoMMSE and [20] for spatially correlated 2× 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. SNR = 5 dB.

A 2 × 2 spatially correlated MIMO system with Δ = 5◦,

dt= 0.5λ, and dr= 0.2λ is considered in Fig. 5. The data

block size K is 60, and L = Nt= 2. When the correlation

matrix R is estimated perfectly, i.e., R = R, the sequence design in [20] outperforms the proposed RoMMSE algorithm with ε = 0.3. This is the case because the sequence design in [20] is MMSE optimal when perfect knowledge of R is available. However, when R is not estimated accurately, i.e., R = ˆR + E, the proposed RoMMSE estimator outperforms the estimator in [20] by as much as 8 dB.

Fig. 6 compares the MSE performance of the proposed scheme with that of [20] when the spatial correlation matrix error power is varied. The channel parameters in this figure are identical to those in Fig. 5 with SNR = 5 dB. For the case of “imperfect R”, i.e., E= 0NrNt×NrNt, the results for [20] are obtained by solving (8) with R = R. In the case of “perfect R”, i.e., E = 0NrNt×NrNt, the exact matrix channel correlation matrix is used to design the training sequence for both algorithms. It can be seen from the figure that the algorithm in [20] outperforms the proposed scheme when an

Fig. 7. MSE versus SNR performance comparison between RoMMSE and [20] for spatially correlated 4× 4 MIMO channel. Δ = 15◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

Fig. 8. MSE versus ε performance comparison between RoMMSE and [20] for spatially correlated 4× 4 MIMO channel. Δ = 15◦, dt= 0.5λ, and dr= 0.2λ. SNR = 5 dB.

accurate spatial correlation matrix is available for estimation. However, when R= R, then the proposed scheme outperforms [20]. Moreover, as the estimation error ε increases, the MSE of the RoMMSE estimator rises only gradually, whereas the MSE increases unbounded for [20].

Figs. 7 and 8 illustrate the same performance comparison as Figs. 5 and 6 but for 4 × 4 MIMO systems. The angular spread Δ is set to be 15◦, and the antenna spacing dtand drare

0.5λ and 0.2λ, respectively. K = 60, and L = Nt= 4. ε = 0.3

is used in Fig. 7, whereas SNR = 5 dB is used for Fig. 8. From Fig. 7, the performance of both algorithms for the 4× 4 system follows the same pattern as that of the 2× 2 system. Specifically, the proposed RoMMSE estimator outperforms the estimator in [20] by as much as 9 dB when MSE =−1 dB. In addition, unlike the algorithm in [20], the RoMMSE estimator performance does not flatten out as the SNR increases. This is because the inaccuracy in R has been taken into account during the channel estimation process. However, since there are more parameters to be estimated in the 4× 4 system compared with the 2× 2 system, there is a performance degradation not only

Fig. 9. BER versus SNR performance comparison between RoMMSE and [20] for spatially correlated 2× 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

Fig. 10. BER versus SNR performance comparison between RoMMSE and [20] for spatially correlated 2× 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.3.

in terms of the absolute MSE, but the rate of decrease of the MSE has also diminished.

In addition to MSE performance, Figs. 9 and 10 compare the BER performance of the RoMMSE algorithm and that of [20] when the estimate of spatial correlation is imperfect and when it is perfect, respectively. The 2× 2 MIMO systems are used. From Fig. 9, it can be seen that the RoMMSE algorithm outperforms the algorithm in [20] by 2 dB when the SNR is low. However, when the spatial correlation has been perfectly estimated, the RoMMSE algorithm and the algorithm in [20] render identical performance.

Figs. 11 and 12 compare the MSE performance for spatially correlated and uncorrelated MIMO channels of the RoMMSE estimator to the RMMSE and LS-RMMSE estimators in [21]. The spatial correlation in Fig. 11 is created by letting Δ = 5◦,

dt= 0.5λ, and dr= 0.2λ. The RMMSE uses diagonal loading

to derive an MMSE estimator that requires only knowledge of tr(R) instead of R to estimate the MIMO channel. LS-RMMSE further relaxes the requirement in LS-RMMSE by using

Fig. 11. MSE versus SNR performance comparison between RoMMSE, LS-RMMSE, and RMMSE [21] for spatially correlated 2× 2 MIMO system. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ.

Fig. 12. MSE versus SNR performance comparison between RoMMSE, LS-RMMSE, and RMMSE [21] for spatially uncorrelated 2× 2 MIMO system, i.e., R = INrNt. ε = 0.3.

TABLE I

NUMBER OFEIGENMODESUSEDDURINGCHANNELESTIMATION FOR

SPATIALLYCORRELATEDMIMO CHANNEL

the LS method to derive an MMSE estimator that no longer requires knowledge of tr(R). Instead, only knowledge about the Frobenius norm of the received signal matrix is required. As seen in Fig. 11, when spatial correlation exists, the pro-posed RoMMSE algorithm outperforms the RMMSE and LS-RMMSE algorithms in low SNR by 4 dB when ε = 0.05 but only by about 2 dB when  = 0.2. This shows that the error power bound cannot be too high; otherwise, the performance of the proposed scheme will degrade. This is so because as ε increases, E obtained from the iterative algorithm will decor-relate the spatial correlation more, thus adversely affecting the performance of the proposed scheme. This can be explained as follows. The RoMMSE estimator strives to minimize the

Fig. 13. MSE versus ε performance comparison between RoMMSE and RMMSE [21] for spatially correlated 2× 2 MIMO system. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. Actual correlation mismatch power is 0.3. (a) SNR = 0 dB. (b) SNR = 5 dB. (c) SNR = 10 dB. (d) SNR = 20 dB.

worst case MSE, as seen in (11). The worst-case MSE can be attained by increasing the number of parameters that needs to be estimated, the maximum being NrNt. In other words, the

present method attempts to increase the degrees of freedom in the correlated MIMO channel by reducing the spatial correla-tion. As ε increases,EFalso increases, which allows E more

freedom to zero out the off-diagonal elements of R, therefore lessening the spatial correlation. Complete decorrelation of R is attained as ε→ ∞. In addition, it has been observed that when the threshold α is met, E and C2 _{share the same} eigenvector matrix as R. This has been proven analytically in Appendix A. When there is no spatial correlation mismatch, it was shown in [12] that the transmitted signal corresponds to transmitting in specific eigenmodes of the spatial correlation, which determines which particular eigenmode of the channel will be estimated. Furthermore, the power on each eigenmode is determined by waterfilling solution based on certain optimiza-tion criteria, such as minimum MSE and maximum condioptimiza-tional mutual information. When the SNR is low, it was found that all the power will be allocated to the strongest eigenmode. However, when the SNR is high, the power is evenly distributed among all the eigenmodes. When spatial correlation mismatch

has been accounted for, it can be seen from the simulations that regardless of whether the system is operating under low or high SNR, the mismatch matrix E not only decorrelates the channel but equalizes all the diagonal values of R as well, such that tr( R + E) = NrNtgiven that ε is sufficiently large. Note

that this is also true even when tr( R + E) = NrNt is not a

constraint in (17). Hence, the robust training sequence evenly distributes power across all the eigenmodes. This is because the worst-case mismatch matrix Ewcan be obtained only when R,

E, and C2 _{are all diagonalized and because E and C}2 _share the same eigenvectors as R (see Appendix A). The constraint tr( R + E) = NrNtforces E to diagonalize R and equalizes

the diagonal values of R such that the constraint is satisfied. Hence, the mismatch matrix E will evenly distribute power across all the eigenmodes of R. If ε is not sufficient large, then E will not have enough degrees of freedom to diagonalize and equalize the diagonal values of R. This phenomenon has been summarized in Table I.

The MSE performance for different values of ε is compared between RoMMSE and RMMSE when the actual mismatch error power is equal to 0.3. The results in Fig. 13 suggest that

Fig. 14. MSE versus SNR performance comparison using time-multiplexed pilots between RoMMSE, LS-RMMSE, and RMMSE [21] for spatially corre-lated 2× 2 MIMO system. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ.

error power (if that is estimated a priori) as that will greatly and adversely affect the MSE. ε should not be chosen to be too high either, but that will be less detrimental as having a small

ε, particularly under low SNR condition. Hence, in the absence

of knowledge about the actual mismatch error power, a high value should be selected for ε. The results for higher SNR [see Fig. 13(b)–(d)] corroborate with those in Fig. 11, i.e., the MSE of the RMMSE and RoMMSE estimator converges as the SNR increases.

The performance of the RoMMSE estimator is also com-pared with that of RMMSE and LS-RMMSE when the MIMO channel is spatially uncorrelated. Fig. 12 indicates that in this situation all three estimators render similar MSE performance, which suggests that all three estimators are identical. This is indeed the case, and it is proven in Appendix B.

The data detection performance for the RMMSE and RoMMSE algorithms is also compared in the case of spatially correlated and uncorrelated channels. Since the RMMSE al-gorithm is proposed in a time-multiplexed pilot scheme, both MSE and BER are compared using time-multiplexed pilots. The channel estimation performance using time-multiplexed pilots is shown in Fig. 14, where the RoMMSE algorithm outperforms the RMMSE and LS-RMMSE algorithms, similar to the performance shown in Fig. 11. Notice that the estimation performance of the RoMMSE algorithm in Fig. 14 is worse than that shown in Fig. 11. This is because the power of the pilot is less than that of the SIT sequence. Next, the BER performance comparison is shown in Fig. 15. With ε = 0.05, RoMMSE outperforms the RMMSE by 2.5 dB in the low SNR region.

The BER performance of the RoMMSE algorithm versus different values of ε is shown in Fig. 16. Notice that a lower BER is obtained with increasing ε. As previously explained, this is because the mismatch matrix has more freedom to decorrelate the spatial correlation matrix as ε increases, which enhances the spatial diversity of the system, thus improving the BER performance.

Table II shows the number of iterations needed before the algorithm in Fig. 4 converges under different initial conditions

Fig. 15. BER versus SNR performance comparison using time-multiplexed pilots between RoMMSE and RMMSE [21] for spatially correlated 2 × 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ. ε = 0.05.

Fig. 16. BER versus SNR performance comparison using time-multiplexed pilots of RoMMSE with different  for spatially correlated 2 × 2 MIMO channel. Δ = 5◦, dt= 0.5λ, and dr= 0.2λ.

TABLE II

AVERAGENUMBER OFITERATIONSREQUIRED FORCONVERGENCE FOR THEPROPOSEDROMMSE ESTIMATOR. dt= 0.5λ, dr= 0.2λ, ε = 0.3,

ANDSNR = 5 dB, ANGULARSPREAD= 15◦(4× 4), 5◦(2× 2)

C(0). It shows that the proposed iterative algorithm always, on the average, converges faster if C(0) is initialized to be an orthogonal matrix than when it is initialized to be a random matrix. Although the table only shows the performance when the SNR is 5 dB, this convergence behavior has been observed for all the SNR values that have been tested. This speed up is due to the fact that the worst case MSE is achieved if the argument inside the trace operator in (17) forms a diagonal matrix. Hence, if C(0) is initialized to be an orthogonal matrix, then C(n), n > 0 will be closer to the optimal solution than when C(0) is initialized to be a random matrix, as it is already

TABLE III

AVERAGENUMBER OFITERATIONSREQUIRED FOR THE

IMPLEMENTATION OF THESDPT3 ALGORITHM FORROMMSEAND[20].

dt= 0.5λ, dr= 0.2λ, ε = 0.3, ANGULARSPREAD= 15◦(4× 4),AND

5◦(2× 2). ORTHOGONALMATRIXISUSED TOINITIATEE(0)

INROMMSE

equal to a diagonal matrix of the form αINrNt, for α being equal to an arbitrary constant.

Table III shows the average number of iterations required for the proposed RoMMSE and the algorithm in [20]. Note that the computational complexity in [12] is similar to that of [20]. The complexity figures depicted in Table III are the average number of iterations used in the primal–dual interior-point method in SDPT3 [33]. This is the default solver used in cvx, and it is also used to find solutions for (17) and (18). The entries in Table III are computed by noting the number of times (17) and (18) are invoked, multiplied by the number of iterations required to obtain optimal values in (17) and (18). The values obtained for the low and high SNR conditions are then averaged to obtain the entries in the table. The computational complexity for RoMMSE is understandably higher than those of [20] and [12] because the RoMMSE algorithm requires an extra step to solve for the worst-case E to compensate for the spatial correlation mismatch.

V. CONCLUSION

A robust SIT sequence design algorithm for spatially cor-related MIMO channel estimation has been proposed. The algorithm has been shown to be robust against error in the spatial correlation estimate. When the robust training sequence is inserted into the MMSE estimator, a robust MMSE, or RoMMSE, estimator is derived. Simulation results have shown that the proposed RoMMSE estimator not only outperforms the optimal MMSE estimator in [20] when the error in the spatial correlation exists, albeit at the cost of higher computational complexity, but it also outperforms other robust designs, such as RMMSE and LS-RMMSE [21]. Furthermore, it has been shown that the spatial correlation mismatch matrix decorrelates the spatial correlation matrix when the error power bound goes to infinity. Finally, the RoMMSE scheme is identical to the RMMSE-based schemes when the MIMO channel is spatially uncorrelated.

APPENDIXA DECORRELATION OFR

Insert (14) into (11), and note that the SVDs of CH_{C and  R}

are VC2Λ_C2V H  C2and URˆΛRˆU H ˆ

R, respectively. The sequence

design problem in (11) is equivalent to solving (17) and (18) iteratively. Using the SVD, (17) can be rewritten as

min EF≤ε tr  ΛC2+Λ_C2V H  C2 URΛRU H  R+E  VC2Λ_C2 ₋₁ . Define A_{= Λ}Δ C2+ Λ_C2V H  C2(URΛRU H  R+ E)VC2Λ_C2.

The objective function can then be written as f (λ(A)) = 

i(1/λi(A)), where λ(A) denotes a vector composed of

eigenvalues of A. Since φ(λi(A)) = 1/λi(A) is a convex

function, f (λ(A)) is Schur convex [12]. Moreover, A is a symmetric matrix. Therefore, f (λ(A)) majorizes f (d(A)), i.e., f (λ(A))≥ f(d(A)) [12], where d(A) denotes a vector that is composed of diagonal values of A. Since the equality will hold when A is a diagonal matrix, then the worst-case mismatch error will be Ew= V_CΛEVH

C, which ensures

that the lower bound of the MSE is reached. This implies that VH

C2URhas to be a diagonal matrix and that E shares the same

eigenvectors as R. The first condition can thus be achieved if 

R and C2_{also share the same eigenvector matrix.} APPENDIXB

COMPARISON OFROMMSEANDRELAXEDMINIMUM

MEANSQUAREERRORESTIMATORS

When the MIMO system is spatially uncorrelated, i.e., R = INrNt, the channel estimate from the RMMSE channel estima-tor in [21] becomes hRMMSE= CH   C CH+σ 2 nnNrNt tr( R) INrNt ₋₁ y = CH  C CH+ σ_nn2 INrNt ₋₁ y.

It is assumed that orthogonal sequences are employed for the RMMSE channel estimator [21], i.e., CH_{C = (P}

T/Nt)IK+L.

The estimate of the RoMMSE estimator is written as hRoMMSE = ( R+Ew)CH  C( R+Ew)CH+σ2nnINrNt ₋₁ y = (INrNt+Ew)C H _C(I NrNt+Ew)C H_+σ2 nnINrNt −1 y where Ewis the worst-case error of the estimated spatial

corre-lation. Let VC2and Λ_C2denote the eigenvector and eigenvalue

matrix of C2, respectively. From (13), (17) is equivalent to min EF≤ε tr  ΛC2+ Λ_C2V H  C2(INrNt+ E)V_C2Λ_C2 ₋₁ = min EF≤ε tr  ΛC2+ Λ 2  C2+ ΛC2V H  C2EVC2Λ_C2 −1 . (28) Define A_{= Λ}Δ _C2+ Λ 2  C2+ Λ_C2V H 

C2EVC2Λ_C2. The

objec-tive function can then be written as f (λ(A)) =_i(1/λi(A)),

where λ(A) denotes a vector composed of eigenvalues of A. Since φ(λi(A)) = 1/λi(A) is a convex function, f (λ(A)) is

Schur convex [12]. Moreover, A is a symmetric matrix. There-fore, f (λ(A)) majorizes f (d(A)), i.e., f (λ(A))≥ f(d(A)) [12], where d(A) denotes a vector that is composed of di-agonal values of A. Since the equality will hold when A is

a diagonal matrix, then the worst-case mismatch error will be Ew= V_CΛEVH

C, which ensures that the lower bound of

the MSE is reached. Since R = INrNt, from Section IV, Ew will have to be a diagonal matrix (or a linear combination of one) so that it will not minimize the degrees of freedom in the MIMO channel. This implies that either VC2 or ΛE

is an identity matrix. However, since C2 _{is not necessarily a} diagonal matrix, it is not necessary for VC2 to be an identity

matrix. This implies that ΛE must be either an identity or an

all zero matrix. Since the constraint tr( R + E) = NtNrmust

be satisfied, therefore, Ew must be an all zero matrix, i.e.,

Ew= 0NrNt×NrNt. Substituting Ew= 0NrNt×NrNtinto (11) and solving for C, C becomes an orthogonal matrix. That is, the optimal SIT sequence is an orthogonal sequence, which agrees with the conclusion in [21] that the optimal training sequence for spatially uncorrelated MIMO channel is an orthogonal sequence. Therefore, hRoMMSEbecomes

hRoMMSE= CH  C CH+ σ_nn2 INrNt ₋₁ y

which implies that the estimation performance of the proposed RoMMSE estimator and the RMMSE estimator is identical when the MIMO channel is spatially uncorrelated, thus agree-ing with the simulation results in Section IV.

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Chin-Te Chiang received the B.S. and M.S. degrees in electronics engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 2007 and 2010, respectively.

He is currently with the Intellectual Property Of-fice, Ministry of Economic Affairs, Taipei, Taiwan. His research interests include the signal processing aspects of wireless communication systems and bio-medical engineering.

Carrson C. Fung (M’91) received the B.S. degree in electrical engineering from Carnegie Mellon Uni-versity, Pittsburgh, PA, in 1994, the M.S. degree in electrical engineering from Columbia University, New York, NY, in 1996, and Ph.D. degree in elec-trical engineering from The Hong Kong University of Science and Technology, Kowloon, Hong Kong, in 2005.

He was the recipient of the prestigious Sir Edward Youde Ph.D. Fellowship in 2001–2002. From 1994 to 1999, he was a Member of Technical Staff with AT&T and Lucent Technologies Bell Laboratories, Holmdel, NJ, where he worked on video and audio coding. He was also a Researcher with the Hong Kong Applied Science and Technology Research Institute in 2005, where he worked on multiple-input–multiple-output orthogonal frequency-division mul-tiplexing systems, and a Senior DSP Engineer with Sennheiser Research Lab, Palo Alto, CA, in 2006, where he worked on microphone and microphone array technologies. Since 2006, he has been an Assistant Professor with the National Chiao Tung University, Hsinchu, Taiwan. His research interests include digital signal processing, communications, and biomedical engineering.