Optimal superimposed training design for

() 3206 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 Optimal Superimposed Training Design for Spatially Correlated Fading MIMO Channels Vu Nguyen, Hoang D Tuan, Member, IEEE,[.]

3206 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 Optimal Superimposed Training Design for Spatially Correlated Fading MIMO Channels Vu Nguyen, Hoang D Tuan, Member, IEEE, Ha H Nguyen, Senior Member, IEEE and Nguyen N Tran, Student Member, IEEE Abstract—The problem of channel estimation for spatially correlated fading multiple-input multiple-output (MIMO) systems is considered Based on the channel’s second order statistic, the minimum mean-square error (MMSE) channel estimator that works with the superimposed training signal is first developed The problem of designing the optimal superimposed signal is then addressed and solved with an iterative optimization algorithm Results show that under the constraint of equal training power and bandwidth efficiency, our optimal design of the superimposed training signal leads to a significant reduction in channel estimation error when compared to the conventional design of time-multiplexing training, especially for slowly timevarying channels with a large coherence time The issue of power allocation between the information-bearing and training signals for detection enhancement is also investigated Simulation results demonstrate excellent bit-error-rate performance of orthogonal space-time block codes with our proposed channel estimation Index Terms—MIMO channel, spatial correlation, channel estimation, MMSE estimation, training signal, training design, time-multiplexing training, superimposed training I I NTRODUCTION T HE use of multiple antennas at both the transmitter and the receiver to create the so-called multiple-input multiple-output (MIMO) communication systems has been shown to greatly increase the data rate of the wireless transmission medium [21], [30] This is especially true when the channel fades among the transmitter-receiver pairs are independently Rayleigh distributed [5], [30], [38] In particular, it is shown in [30] that the capacity of a MIMO wireless channel increases linearly with the number of antennas The assumption of independent fades requires that the antennas be placed sufficiently far apart, both at the transmitter and the receiver In many practical applications, meeting such requirements might be very expensive and impractical (such as for the antennas in hand-held mobile units) It is therefore more practical and useful to consider spatial correlations Manuscript received March 2, 2007; revised August 1, 2007; accepted October 1, 2007 The associate editor coordinating the review of this paper and approving it for publication is D Dardari This work is supported by the Australian Research Council under grant ARC Discovery Project 0556174 A part of this work was presented at the IEEE Second International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, St Thomas, U.S Virgin Islands, USA, 12-14 December 2007 Vu Nguyen, Hoang D Tuan, and Nguyen N Tran are with the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, NSW 2052, Australia (e-mail: {q.nguyen, nam.nguyen}@student.unsw.edu.au, h.d.tuan@unsw.edu.au) Ha H Nguyen is with the Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, Canada S7N 5A9 (e-mail: ha.nguyen@usask.ca) Digital Object Identifier 10.1109/TWC.2008.070250 among different sub-channels of the MIMO channel matrix [5], [13], [28] Compared to an independent fading MIMO channel, the results in [3], [10]–[12], [28] show that the capacity of a spatially-correlated fading MIMO channel is substantially reduced Capacity reduction due to spatially-correlated fading can be partially alleviated by precoding the transmitted signal [20] This technique however requires the knowledge of the channel state information at the transmitter, which is not always available Furthermore, the MIMO channel capacity can be further reduced if inaccurate channel state information is obtained at the receiver [30] In other words, accurate channel estimation is very important to fully exploit the advantages of MIMO wireless communications The correlated fading channel is often estimated by a training sequence, which can be either time-multiplexing (TM) training (see e.g., [26], [32] for single-input multiple-output (MISO) channels and [7], [19] for MIMO channels), frequency-multiplexing [14], [18] or superimposed (SP) training (see e.g., [17], [36] for single-input single-output (SISO) channels and [33] for MISO channels) In superimposed traning, the training symbols are superimposed on the precoded data for transmission In fact, superimposed traning includes both time-multiplexing and frequency-multiplexing as special cases, which correspond to sending the non-zero training symbols when the data symbols are zero or sending the non-zero training symbols over subcarriers that are not occupied by the data symbols (i.e., pilot subcarriers) Because superimposed training is a general and powerful framework, it has recently received a growing interest in the research community [14], [15], [18], [33] In SP training, since the received signal is a superposition of the data-bearing signal, training signal and noise, a popular design approach is to decouple channel and symbol estimation [14], [18], [22], [37] This can be done by designing the precoding and training matrices so that the data-bearing and training signals belong to complementary signal subspaces Then, the data-bearing signal, which is considered as the unwanted noise in channel estimation, can be completely removed and channel estimation is carried out based on the training symbols An alternative approach is to perform joint channel/symbol estimation at the receiver and design the training signal accordingly Due to its more complicated processing and marginal advantage [37], joint channel/symbol estimation is less preferred than decoupled channel and symbol estimation This paper, therefore, is only concerned with c 2008 IEEE 1536-1276/08$25.00  NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS decoupled channel and symbol estimation approach For the estimation of independent Rayleigh fading channels, the works in [14], [18], [37] conventionally treat the received signal along time, i.e., as a concatenation of the columns of the received signal matrix The subspaces of the data-bearing and training signals thus depend on the unknown channel matrix Consequently, some special structures on both the precoding and training matrices are imposed This together with several complex arrangements make these matrices commutative with the channel matrix and hence, decouple the two subspaces of the data-bearing and training signals A novel approach has also been recently developed in [22], [25], [31], where the received signal is viewed along space, i.e., as a concatenation of the rows of the received signal matrix The most important consequence of this view is that the two subspaces of interest are independent of the channel matrix This means that, if designed properly, the orthogonality of the precoding and training matrices already guarantees subspace complementarity Therefore there is much more freedom in the optimal design of these matrices Indeed, the results in [22], [31] demonstrate that this design approach is superior than the designs in [14], [18], [37] in terms of estimation performance, symbol detectability and computational complexity This paper adopts the approach in [22], [31] to design the optimal superimposed training signal for the channel estimation of correlated block-fading MIMO channels For uncorrelated block-fading MIMO channels, precoding is well known to be useless [4], [6], [16], [18] For these MIMO channels, the optimal TM training and superimposed training can be easily seen to be the same with a scaled identity matrix as the optimal training matrix [24], [25] However, as pointed out in [9], the design of training signals for correlated fading MIMO channels is quite challenging in general This is due to the large number of channel parameters involved and the complex nature of the correlated channel coefficients In fact, while the design of TM training signal is quite straightforward for independent fading channels [8], it is still a difficult problem for the correlated fading channels [9] In particular, sub-optimal TM training signals for spatially-correlated fading channels were proposed in [9] for only two extreme cases of low and high signal-to-noise ratios (SNRs) It is still not clear what is the optimal TM training signal for a given SNR level Similarly, while the design of the optimal superimposed training signal is well understood for independent fading channels [14], [22], [31], [37], the design for spatially-correlated fading channels at any SNR level has not been addressed This paper solves this challenging design problem by developing an efficient iterative optimization algorithm to find the solution Results show that the proposed design of the superimposed training signal performs much better than the TM trainingbased estimation considered in [9] The remaining of this paper is organized as follows Section II introduces the system model and describes the design problem The optimal superimposed training design is presented in Section III with an iterative optimization algorithm The issue of optimal power allocation for the training signal is addressed in Section IV Section V provides numerical results to illustrate the advantages of the proposed design over the existing ones Section VI concludes the paper 3207 Notation: Boldface upper (lower) letters denote matrices (column vectors) The operation vec(·) means matrix vectorization which forms a column vector by vertically stacking the columns of a matrix For a matrix A, its transposition, Hermitian adjoint and trace are denoted by AT , AH and trace(A), respectively IN is the identity matrix of size N ×N , 0N ×M is the N × M zero matrix and (∗)N ×M stands for any matrix of size N × M UN and DN denote the sets of N × N unitary matrices and diagonal matrices, respectively For two Hermitian matrices X and Y, X ≤ Y means that Y − X is positive semi-definite Similarly, X < Y implies that Y−X is positive definite The symbol ⊗ is used for Kronecker matrix product, while E(A) is the expectation of random matrix A For any x, define (x)+ = max(x, 0) Furthermore, some properties of Kronecker product transformations and positive definite matrices used in this paper are as follows: (P1) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) (P2) (A ⊗ B)H = AH ⊗ BH (P3) (A ⊗ B)−1 = A−1 ⊗ B−1 (P4) trace(A ⊗ B) = trace(A)trace(B) (P5) vec(AXB) = (BT ⊗ A)vec(X) (P6) If UN ∈ UN and UM ∈ UM , then UM ⊗ UN ∈ UN M (P7) If < X < Y, then trace(X) < trace(Y) and trace(X−1 ) > trace(Y−1 ) (P8) If X ≥ 0, then X ⊗ IN ≥ 0, ∀N N  1/X(i, i) (P9) If < X ∈ CN ×N , then trace(X−1 ) ≥ i=1 II S YSTEM M ODEL Consider a narrowband frequency-flat block-fading MIMO channel with N transmit and M receive antennas (see Fig 1) The information-bearing symbols are grouped into blocks of size Ns , namely s(k) = [s(kNs ), s(kNs + 1), , s(kNs + Ns − 1)]T , where k denotes the block index Then each block s(k) is encoded and/or multiplexed in space and time, which is generally represented by a block labeled with space-time coding (STC) in Fig Thus the system under consideration can accommodate any specific space-time schemes such as the Alamouti’s orthogonal space-time block codes or the BLASTtype schemes [27] The output of the space-time encoder consists of N vectors, xi ∈ CK×1 , i = 1, , N , each having length K (K ≥ N ) symbols The information-bearing signal can therefore be represented by the following matrix: X = [x1 , x2 , , xN ]T ∈ CN ×K (1) Before directed to the transmit antennas, the signal matrix X is first precoded by post-multiplying with a precoding matrix P = [p1 , p2 , , pK ]T ∈ CK×(K+L) , where L ≥ N , to produce the following precoded signal matrix: ⎡ T⎤ ⎡ T ⎤ d1 x1 P ⎢ ⎥ ⎢ ⎥ (2) D := ⎣ ⎦ = XP = ⎣ ⎦ ∈ CN ×(K+L) dTN xTN P Here, L represents the number of redundant vectors resulted by precoding the transmitted signal In general, it is desirable to have L as small as possible in order to improve 3208 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 Superimposed training d1 x1 Spacetime coding (STC) Information symbols x2 Ant-1 c1 P/S Ant-2 c2 Precoding by postmultiplying with matrix P d2 P/S Ant-N cN xN dN P/S (a) Transmitter Ant-1 S/P y1 Ant-2 S/P y2 Space-Time Decoding Ant-M S/P Decoded symbols yM ˆ H Decoupling by post-multiplying with matrix Q Channel Estimator (b) Receiver Fig An equivalent discrete-time baseband MIMO system receive antenna arrays, respectively The matrix Hw is an M × N matrix whose entries are independent and identicallydistributed (i.i.d.) circularly symmetric complex Gaussian random variables of unit variance, i.e., CN (0, 1) In particular E[vec(Hw )vecH (Hw )] = IMN The known matrices Σr and Σt have the following forms: ⎡ ⎤ r12 · · · r1M ⎢ ∗ ⎥ ⎢ r12 r2M ⎥ ⎢ ⎥ Σr = ⎢ (4) ⎥ , ⎣ ⎦ ∗ ∗ r1M r2M ··· ⎡ ⎤ t12 · · · t1M ⎢ ∗ ⎥ ⎢ t12 t2M ⎥ ⎢ ⎥ Σt = ⎢ (5) ⎥ , ⎣ ⎦ t∗1M t∗2M · · · where tij (rnm , resp.) with i = j (n = m, resp.) reflects the correlated fading between the ith and the jth (nth and mth, resp.) elements of the transmit (receive, resp.) antenna array The elements of Σr and Σt can be specified, for example, by using the one-ring model in [28] The covariance matrix of the overall channel matrix H can be easily shown to be R = E[vec(H)vecH (H)] = Σt ⊗ Σr At the receiver, the received signal matrix is given as: Y = H(C + D) + N = HC + HXP + N, (6) the transmission efficiency Next, a training matrix C = M×(K+L) is the matrix of additive white [c1 , c2 , , cN ]T ∈ CN ×(K+L) is added (i.e., superimposed) where N ∈ C Gaussian noise (AWGN) samples Furthermore, the following to the precoded matrix D Finally, the nth row of D + C, assumptions are made for the input/output channel model in T T namely dn + cn is serially transmitted over the nth transmit (6): antenna, n = 1, , N It should be pointed out that the above superimposed training is performed after the space- (A1) The information-bearing symbols are independent, zeromean and with variance σx2 , i.e., E(XXH ) = Kσx2 IN time encoder Therefore our training design is flexible and Note that this assumption is valid if X is obtained can accommodate any space-time code by simply multiplexing the information symbol block Assume that the fading channel remains constant during s(k) in space and time When X is obtained by spaceevery block of (K + L) symbols, but changes independently time encoding the information block s(k), the correlation from block to block Typically, this assumption implies that the matrix E(XXH ) generally admits a different form Nevexact statistical behavior of the correlation in time is neither ertheless, the technique presented in this paper can be available nor exploited, but only the coherence time is used easily extended to cover any other form of E(XXH ) to determine the block length In practical systems, the speed (A2) The AWGN samples are also independent, zero-mean and of mobility and the transmission rate determine the coherence , i.e., E(NNH ) = (K + L)σn2 IM with variance σ n time (in units of information symbols) The coherence time in (A3) The average transmitted power, including the powers of turn dictates the block length [27] With this assumption and the information-bearing and training signals, is normalsince transmission and reception are conducted on a blockH ) 2 ized as σx + σc = 1, where σc2 = trace(CC by-block basis, the time index is omitted for convenience N (K+L) is the average power of the training signal Let H be the M × N MIMO channel matrix in an arbitrary transmission block To reflect spatially-correlated fading, the Moreover, the precoding matrix P is full rank and satisfies channel matrix is represented as follows [9]: trace(PPH ) = K + L (7) 1/2 1/2 (3) H = Σ r H w Σt , The above constraint is to ensure that the average transmitted where Σr and Σt are M × M and N × N known covariance power of the information-bearing signal is unchanged after matrices that capture the correlations of the transmit and precoding Mathematically, this is verified as trace E(DDH ) N σx2 trace(PPH ) For example, in a typical cellular system operating at a carrier frequency σd = = = σx2 N (K + L) N (K + L) of fc = 1.9 GHz, a mobile speed of v = 36 km/h translates to a coherence time of about Tc = c/(8fc v) ≈ 1.97 ms, where c = × 108 m/s is the speed of light If the data rate is 250 kbps and a 16-QAM constellation is used, then the block length would be about 123 QAM symbols In [14], [18], [37], the signal matrix X is precoded as PX, i of X is precoded by P or equivalently, each column x xi NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS Then the information-bearing signal HP xi at the receiver side belongs to a subspace governed by the unknown channel ci is the matrix H Similarly, the training signal H ci , where ith column of the training matrix C, belongs to a subspace that can only be determined by knowing H It follows that it is not easy to decouple these two unknown subspaces [14], [18], [37] for convenient and effective channel estimation Consequently, the optimal training matrix C cannot be readily derived On the contrary, it can be seen that our precoded signals, xTi P, i = 1, , N , belong to the subspace ΥP ⊂ CK+L spanned by the rows pTi , i = 1, , K, of the precoding matrix P Then the rows of the information-bearing part HXP in the received signal matrix Y also belong to ΥP , which is independent of the unknown channel matrix H Moreover, the rows of the training part HC belongs to the subspace ΥC ⊂ CN +L spanned by the rows of the training matrix C, which is also independent of H Therefore, in order to estimate the channel matrix H in an effective way, the received signal matrix Y is post-multiplied with the decoupling matrix Q = [q1 , q2 , , qK+L ]T ∈ R(K+L)×N , which is chosen such that (8) PQ = 0K×N The matrix Q for channel estimation is also full rank and satisfies QH Q = IN , which means that the noise is not enhanced by the decoupling operation Thus, the decoupled signal matrix for channel estimation is expressed as YQ = HCQ + HXPQ + NQ = HCQ + NQ, (9) which is free of the unwanted component HX, and hence H can be efficiently estimated Although (8) is the most important relationship between the precoding matrix P and the decoupling matrix Q for efficient channel estimation, the precoding matrix P can be further designed to improve the performance of symbol detection as follows [31] First, choose matrix for symbol the decoupling H detection as QD = P H PP −1 , where P is also chosen H such that CP = Then, by post-multiplying the received signal matrix in (6) with QD , the decoupled signal matrix for symbol detection can be expressed as YQD = HCQD + HXPQD + NQD = HX + NQD (10) i and qi be the ith columns of the estimated data matrix Let x  and QD , respectively Under the minimum mean-square X error (MMSE) criterion for symbol detection, the ith column  and the of X is recovered based on the channel estimate H matrix YQD as follows [8]: −1  1 H   H Yqi (11) i = H H H I+ x 2 σx σn ||qi || σn ||qi ||2 The total mean-square error (MSE) of symbol detection can be shown to be  −1  K  1  HH  H (12) εX (QD ) = I+ tr 2 σ σ ||q || i x n i=1 3209 H −1 and due to the the power conSince QH D QD = (PP ) straint in (7), the problem of precoding matrix design can be stated as  −1  K  1 H  H H tr I+ −1 }=K+L σx2 σn ||qi ||2 tr{[QH D QD ] i=1 (13) The closed-form solution to the above optimization problem has been shown in [31] to have the following structure: K +L IK (14) K Similar to [31], based on (8) and (14), the matrices P and Q are designed as follows:  K+L P = O(1 : K, :) ∈ CK×(K+L) (15) K H Q = O (K + 1) : (K + N ), : ∈ C(K+L)×N , where O(1 : K, :) and O (K + 1) : (K + N ), : keep only rows to K and rows (K + 1) to (K + N ) of an orthogonal matrix O ∈ C(K+N )×(K+L) As an example, O can be formed by keeping the first (K + N ) rows of an unitary UK+L ∈ UK+L , i.e., O = UK+L (1 : K + N, :) (16) −1 PPH = (QH = D QD ) Now, the key issue is how to design the superimposed training matrix C that results in the best estimation of the channel matrix H based on the input/output model in (9) When H is uncorrelated, this design is quite straightforward and a closed-form solution for the optimal C can be easily derived [31] In contrast, due to the spatial correlations among channels of different transmit-receive antenna pairs, the design under consideration is quite complicated Though a closedform solution is not yet available, the next section proposes an iterative optimization algorithm to effectively find the optimal solution III O PTIMAL D ESIGN OF THE S UPERIMPOSED T RAINING S IGNAL Let PT = N (K + L)σc2 According to (A3), the design of the training matrix C is subject to the following power constraint: (17) trace(CCH ) ≤ PT + n, where y = vec(YQ) ∈ CMN , Rewrite (9) as y = Ch T = (CQ)T ⊗ n = vec(NQ) = (Q ⊗ IM )vec(N) ∈ CMN , C MN ×MN MN IM ∈ C and h = vec(H) ∈ C Since QH Q = H H T IN , it follows that E[nn ] = (Q Q) ⊗ σn2 IM = σn2 IMN Based on the received signal vector y, the linear minimum mean-square error (MMSE) estimation of the channel vector h is: ˆ = RC H (CR C H + σn2 IMN )−1 y, h (18) where, recall that, R = Σt ⊗ Σr Furthermore, the covariance matrix of the estimation error vector is ˆ ˆ H] E : = E[(h − h)(h − h) −1 H (σn2 IM N )−1 C = R−1 + C −1 (19) = (Σt ⊗ Σr )−1 + CT H (QQH )T CT ⊗ IM σn 3210 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 The objective is to design C to further minimize (19) subject the the power constraint in (17) From (15), QQH admits the following singular value decomposition (SVD): QQH = UQ ΛUH (20) Q , UQ ∈ UK+L ,   IN 0N ×(K+L−N ) When O is where Λ = 0(K+L−N )×N 0K+L−N specified as in (16), UQ can be obtained by simply permuting the rows of UK+L Now, perform the following transformation: ¯ = UTQ CT C (21) ¯C ¯ H ) = trace(UT CT CT H CT H ) = One has trace(C Q H T trace((C C) ) = trace(CCH ) Thus, under the power constraint in (17), the optimal design of C ∈ CN ×(K+L) , that aims at minimizing E, can be formulated as the following ¯ ∈ C(K+L)×N : constrained optimization problem in C −1 ¯H ¯ ΛC ⊗ IM trace (Σt ⊗ Σr )−1 + C ¯ σn C ¯C ¯ H ) ≤ PT (22) subject to trace(C In what follows, the approach of matrix partition for matrix inequalities [23] is employed to derive the solu¯ tion to the above optimization problem Suppose that C ¯ is the optimal solution of Problem (22) Partition C =   CU ¯0 = , CU ∈ CN ×N , CL ∈ C(K+L−N )×N and define C  CL  CU ¯ = CH C U = ¯ H ΛC It can be verified that C U 0(K+L−N )×N H ¯ ¯ C0 ΛC0 On the other hand, whenever CL = 0, one has ¯C ¯ H ) = trace(CH CU ) + trace(CH CL ) > PT = trace(C U L H ¯ 0C ¯H trace(CU CU ) = trace(C ) It then follows that, whenever CL = there exists λ > such that PT = ¯ 0C ¯H trace(λ2 C ) Now, applying properties (P7) and (P8) yields −1 ¯H ¯ trace (Σt ⊗ Σr )−1 + C > opt ΛCopt ⊗ IM σn trace ¯ )H Λ(λC ¯ ) ⊗ IM −1 (23) (Σt ⊗ Σr )−1 + (λC σn ¯ is the optimal which contradicts with the assumption that C solution of Problem (22) The above result shows that the optimal solution of the optimization problem in (22) must have the following form:   CU ¯ opt = C (24) , CU ∈ CN ×N 0(K+L−N )×N and the optimization problem in (22) is equivalent to the following problem: −1 CU ⊗ IM trace (Σt ⊗ Σr )−1 + CH U σn CU ∈CN ×N subject to trace(CU CH U ) ≤ PT error does not depend on the actual value of L Thus, as far as channel estimation is concerned, choosing L = N is optimal to maximize the system’s bandwidth efficiency It should be noted, however, that choosing L > N affects the precoding operation and might lead to a better performance with respect to a different criterion (such as the bit-errorrate (BER) performance, or the effective SNR considered in Section IV) To find the solution to Problem (25), make the following SVDs: (25) Remark 1: The above optimization problem implies the following important consequence: As long as L ≥ N , performance of the channel estimation in terms of the mean-square Σt = Ut Λt UH t , Ut ∈ UN , Λt ∈ DN , Σr = Ur Λr UH r , Ur ∈ UM , Λr ∈ DM Then, the objective in (25) can be evaluated as follows: −1 H H −1 H trace (Ut Λ−1 t Ut ) ⊗ (Ur Λr Ur ) + (CU CH ) ⊗ IM σn −1 = trace (Λt ⊗ Λr )−1 + Z ⊗ IM , σn (26) where H N ×N Z = UH t CU CU Ut ∈ C (27) The power constraint in (25) can also be expressed in terms of the new variable Z as trace(CU CH H ) = trace(Z) ≤ PT With the expressions of the objective and constraint in Z, the equivalent optimization problem is: −1 trace (Λt ⊗ Λr )−1 + Z ⊗ IM σn Z∈CN ×N subject to trace(Z) ≤ PT (28) Using property (P9), we can easily see that −1 trace (Λt ⊗ Λr )−1 + Z ⊗ IM ≥ σn −1 trace (Λt ⊗ Λr )−1 + diag[Z(i, i)]i=1, ,N ⊗ IM σn and trace(diag[Z(i, i)]i=1, ,N ) = trace(Z) Here diag[Z(i, i)]i=1, ,N is the diagonal matrix with diagonal entries Z(i, i) This implies that the optimal solution of Problem (28) must be diagonal Consequently, the optimization problem in (28) can be reformulated as follows: −1 trace (Λt ⊗ Λr )−1 + ΛC ⊗ IM 0≤ΛC ∈DN σn subject to trace(ΛC ) ≤ PT (29) The following theorem summarizes the optimal design problem, based on (21), (24), (27) Theorem 1: The optimal solution ΛC,opt of Problem (29) provides the following optimal training signal:    ΛC,opt 0N ×(K+L−N ) UH (30) Copt = UHT t Q Remark 2: It is intuitively satisfying to observe that the precoding matrix P designed as in (15) and the optimal NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS training matrix Copt derived in (30) are orthogonal This can be shown as follows:    Copt PH = UHT ΛC,opt 0N ×(K+L−N ) UH PH t  Q  HT   0N ×K = Ut ΛC,opt 0N ×(K+L−N ) ∗ = 0N ×K (31) The above implies that the components in the received signal corresponding to the training signal (namely HCopt ) and the information-bearing signal (namely HXP) are guaranteed to be orthogonal, regardless of the unknown channel matrix H Unfortunately, a closed-form expression for the optimal solution of Problem (29) is not available Nevertheless, the following subsection provides an effective iterative algorithm to solve Problem (29) A Iterative Algorithm to Find the Optimal Solution of Problem (29) For convenience, define Note that the objective function f (s) in (33) is separable in sj , i.e., f (s) = = [s1 , s2 , , sN ]T = [ΛC (1, 1), , ΛC (N, N )]T s∈RN j=1 i=1 = i=1 δ(j−1)N +i + j=1 According to the Kuhn-Tucker condition for optimality of convex programming, the optimal solution of the optimization problem in (33) and the corresponding Lagrange multipliers must satisfy the following necessary and sufficient conditions: ∂L(s, µ, µ1 , , µj ) ∂fj (sj ) = + µ − µj = 0, ∂sj ∂sj (32) sj,opt = s+ j (µ) := max{sj (µ), 0}, j = 1, , N, where: ã sj (à) is the solution of the following nonlinear equation: M gj (sj ) := − ∂fj (sj )  = ∂sj σn [δ(j−1)N +i + i=1 δ(j−1)N +i + N  sj σn L(s, µ, µ1 , , µN ) ⎞ ⎛ N N   f (s) + µ ⎝ µj sj , µ ≥ 0, µj ≥ sj − PT ⎠ − j=1 sj σn , • subject to M  Furthermore, fj (sj ) is not only convex but also decreasing in sj The Lagrangian of (33) is Then Problem (29) can be equivalently re-expressed as N  M  fj (sj ), where fj (sj ) = j=1 = [(Λt ⊗ Λr )−1 (1, 1), , (Λt ⊗ Λr )−1 (M N, M N )]T , N  where µj sj = 0, j = 1, 2, , N Therefore, the optimal solution of (33) can be expressed as [δ1 , δ2 , , δMN ]T s 3211 sj ≤ PT , sj ≥ (33) j=1 When there is only one receive antenna, i.e., M = 1, the closed-form optimal solution of the above optimization problem is well-known to have the water-filling structure (see e.g [2]) The situation is quite different when M > and a closed-form solution is not expected Note that both the objective and constraint functions of the above problem are still convex in s In principle, the interior-point algorithms of convex programming (see e.g [2]) can be applied However, we shall exploit not only the convex structure of Problem (33), but also its monotonic structure and provide an efficient and fast computational algorithm to find its optimal solution Undoubtedly, convexity and monotonicity are the most useful properties in optimization [34], [35] Specifically, for a decreasing function h(t) the nonlinear scalar equation (34) h(t) = γ, t ∈ [t, t¯] can be solved online by the following iterative bisection procedure (IBP): • If h(t) < γ or h(t¯) > γ then there is no solution in [t, t¯] • For t = (t + t¯)/2 reset t = t if h(t) > γ and reset t¯ = t if h(t) < γ Repeat until h(t) = γ 2 sj ] σn = µ (35) Thus for each µ we can quickly locate s+ (µ) by the IBP j (µ) is decreasing in µ described before Obviously s+ j The scalar Lagrange multiplier µ > is such that g(µ) := N  sj,opt = N  + sj (µ) = PT (36) j=1 j=1 The function g(µ) is also decreasing in µ So again the IBP can be effectively used to locate the solution of (36) To summarize, an effective procedure for locating the optimal solution of the optimization problem in (33) is outlined below • Compute µ and µ such that the solution of (36) belongs to [à, ] ã Apply IBP to locate the solution of Equation (36) The subroutines include (i) the computations of sj (µ) and s¯j (µ) such that the solution of (35) belongs to [sj (µ), s¯j (µ)] and (ii) the application of IBP for locating the solutions sj (µ) of (35) To make the above procedure completely realizable, the expressions of sj (µ), s¯j (µ), µ and µ ¯ are given next Define δj,max = δj,min = i=1,2, ,M max δ(j−1)N +i , (37) δ(j−1)N +i , j = 1, 2, , N (38) i=1,2, ,M It is obvious that M M ≤ gj (sj ) ≤ 2 σn (δj,max + σ2 sj ) σn (δj,min + n 2 sj ) σn (39) 3212 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 Hence, for a fixed µ, the solution sj (µ) of (35) belongs to [sj (µ), s¯j (µ)] with  + M − δj,max , (40) sj (µ) := σn µσn2  + M s¯j (µ) := σn − δj,min (41) µσn2 Also, it must be true that µ ∈ [µ, µ ¯ ] for the solution of (36), ¯ are the solutions of where µ and µ N  + sj (µ) = PT (42) j=1 and N  + (¯ sj (µ)) = PT , (43) j=1 respectively In other words, {sj (µ), µ} and {¯ sj (¯ µ), µ ¯} are the water-filling structured optimal solutions and the corresponding Lagrange multipliers of the following optimization problems s∈RN and s∈RN N  j=1 N  j=1 M δj,max + M δj,min + sj σn sj σn : N  sj ≤ PT , sj ≥ (44) j=1 : N  sj ≤ PT , sj ≥ 0, (45) j=1 respectively The technique to find these solutions is quite standard and the details are omitted here IV P OWER A LLOCATION FOR D ETECTION E NHANCEMENT Until now, we have considered the problem of designing the optimal training signal C under its fixed power constraint as specified in (17) Under the total transmitted power constraint stated in (A3), the following tradeoff arises The performance of channel estimation can be improved by spending more power for the training signal This, however, comes at the expense of decreased transmitted power for the informationbearing signal, leading to performance degradation of signal detection This section considers this tradeoff problem and proposes a sub-optimal power allocation that maximizes the effective signal-to-noise ratio (SNR) The SNR is selected because any increase in the effective SNR translates to an increase in system capacity and/or quality of signal detection Maximizing the effective SNR is also very helpful for channel decoding if error control coding is implemented in the systems With the optimal training matrix Copt derived in the previous section by (30), Equation (6) is rewritten as Y = HCopt + HXP + N (46) Next, rewrite (10) to see the effect of channel estimation error on data detection as follows: YQD = = HX + NQD  + HX + NQD , HX (47) = H−H   is the estimated channel matrix and H where H is the channel estimation error matrix Since H in (47) is unknown and random and HX is uncorrelated to HX according to the orthogonality property [8], it is considered as noise Thus, the effective SNR of the input/output model in (47) is defined as  E HX (48) SNReff = , + NQD E HX Here,  = E HX = H H  H trace E HXX H H Kσx2 trace E HHH − E H = Kσx2 (M N − ǫ), H H , and where ǫ := trace E H + NQ E HX D H H H + trace E NQD QH H = trace E HXX DN = Kσx2 ǫ + trace E NPH (PPH )−1 (PPH )−H PNH = Kσx2 ǫ + M σn2 K2 K +L It follows that σ (M N − ǫ) , SNReff = x σx ǫ + γ with γ = M σn2 K K+L (49) Note that ǫ is exactly the optimal value of the objective function in Problem (29) As pointed out in Remark 1, ǫ does not depend on the actual value of L as long as L ≥ N On the other hand, it follows from (49) that the effective SNR increases with L, an intuitively satisfying result For simplicity and to maximize the system’s bandwidth efficiency, L = N is chosen in the remaining of this paper Furthermore, the following upper bound on ǫ can be easily derived: σn2 MN M N σn2 (50) = β , where β = ǫ≤ K + N σc2 σc2 K +N This is because ΛC = PNT IN = (K + N )σc2 IN is a feasible solution of (29) Then, by replacing σx2 with (1 − σc2 ), it is easy to see that SNReff in (49) is lower bounded as SNReff (σc2 ) ≥ (1 − σc2 )(M N σc2 − βσn2 ) βσn2 − βσn2 σc2 + γσc2 (51) Instead of maximizing SNReff , we maximize its lower bound as given by the right-hand-side of (51) This maximization leads to a sub-optimal power allocation as far as maximizing SNReff is concerned The sub-optimal solution of power allocation can be shown to be:  M N βσn2 − M N γβσn2 (−βσn2 + M N + γ) σc,sub-opt = M N (βσn2 − γ) (52) Since the total transmitted power is normalized to unity, the above expression essentially gives the fraction of the total power allocated to the training signal The above suboptimal power allocation is used to obtain the simulation results presented in the next section NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS 0 PSPT ESPT TMT K=10 K=60 PSPT ESPT TMT K=10 K=60 −2 Mean Square Error (dB) −5 Mean Square Error (dB) 3213 −10 −15 −4 −6 −8 −10 −20 −12 −25 10 SNR (dB) 15 20 −14 SNR (dB) Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems using different training signals: The proposed superimposed training (PSPT), the equal-powered superimposed training (ESPT) and the time-multiplexing training (TMT) Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems using different training signals: The proposed superimposed training (PSPT), the equal-powered superimposed training (ESPT) and the time-multiplexing training (TMT) V I LLUSTRATIVE R ESULTS for the uncorrelated MIMO channel The TMT design used for comparison in this section is the improved version of the design proposed in [9] (which applies to the case of high or low SNR only [9, Subsection IV-C]) The improved design is obtained by applying the iterative algorithm described in Section III to the optimization problem in [9, Equ (30)] As in [9], the linear MMSE estimator is used for identifying the wireless channel Furthermore, the length of the TMT signal is chosen to be the minimum length required for the channel estimation This minimum length is shown to be N symbols for a MIMO channel having N transmit antennas in [7] On the other hand, the use of precoding matrix P and decoupling matrix Q in this paper also introduces L = N redundant vectors per block Thus, the extra bandwidth consumption is the same for all the different training signals, namely the PSPT, ESPT and TMT signals Of course, the estimation performance of different training designs is compared based on the same training power and additive Gaussian noise environment The normalized mean-square errors (normalized by E[||h||2 ]), expressed in dB, of the channel estimation provided by the above three training signals are plotted versus SNR in Figs and for the × and × MIMO channels, respectively In these two figures, two different lengths of the information vector X, namely K = 10 and K = 60, are considered It should be noted that the implementation of TMT is independent of K as long as K ≥ N First, observe that at almost any SNR level (except for SNR > 10 dB in the × system), the mean-square error is significantly reduced with the use of the PSPT signal instead of the ESPT A more important observation is that, compared to PSPT, using the TMT signal results in a larger MSE at any SNR level for both cases of MIMO channels and for both block lengths considered As expected, the advantage of the superimposed training (including PSPT and ESPT) over the TMT becomes more evident for the system with a larger block length In fact, if the block length is not long enough, TMT can outperform ESPT as can be seen from Fig for the × channel This section provides simulation results to illustrate the performance of the proposed optimal training design In all simulations, the wireless channel model is assumed to be quasi-static block Rayleigh fading and spatially correlated as described in (3) The one-ring model in [28, E.q (6)] is used to generate the elements of the covariance matrices Σr and Σt Specifically,   2π (53) Σt (n, m) ≈ J0 ∆ dt |m − n| , λ   2π dr |i − j| (54) Σr (i, j) ≈ J0 λ where ∆ is the angle spread in the one-ring model; dt and dr are the spacings of the transmit and receive antenna arrays, respectively; λ is the carrier wave-length and J0 (·) is the zeroth order Bessel function of the first kind Note that the angle spread, ∆, and the antenna spacings, dt and dr , determine how correlated the fading is at the transmit and receive antenna arrays Unless stated otherwise, the values of ∆ = 50 , dt = 0.5λ and dr = 0.2λ are used in the simulation to create highly correlated fading Since the average transmitted power, including the training and data powers, is normalized to unity as in assumption (A3), the received SNR in dB is defined as SNR = −10log10 σn2 The power allocation for data and training signals in the proposed SP training follows (52) A Estimation Performance Two different designs of superimposed training and one conventional time-multiplexing training (TMT) design are investigated and compared The proposed superimposed training (PSPT) signal is obtained with the iterative algorithm described in Section III The equal-power superimposed training (ESPT) signal is chosen as a scaled identity matrix with power constraint PT , which is the optimal training scheme 3214 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 PSPT ESPT TMT Δ=15 Δ=30 Mean Square Error (dB) Mean Square Error (dB) −5 PSPT ESPT TMT Δ=15 Δ=30 −10 −15 −20 −2 −4 −6 −8 −10 10 SNR (dB) 15 −12 20 Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems for different angle spreads (∆ = 150 and ∆ = 300 ) and using different training signals (K = 60) and when K = 10 This particular case clearly shows the usefulness of our superimposed training design over the simple ESPT The difference in estimation performance between the PSPT and the TMT depends mainly on their length ratio, which ) ) is (K+N In general, the bigger the ratio (K+N is, the N N larger the performance difference between PSPT and TMT is, because the channel statistics is better incorporated for estimation by superimposed training This can be clearly seen from the results shown in Figs and for different values of K In practice, the value of K + N is determined by the coherence time of the channel In fact, when the coherence time is large, the estimation performance of TMT can be improved by extending the training length beyond the minimum required length of N symbols [7], [9] However, a direct consequence of extending the training length of TMT is a lower bandwidth efficiency Therefore, taking both estimation performance and bandwidth efficiency into account, PSPT is more attractive than TMT, especially for a slowly time-varying wireless channel whose coherence time can be very large Figures and illustrate the impact of having larger angle spreads, ∆ = 150 and ∆ = 300 , on the estimation performance of different training designs in both the × and × systems Here K = 60 is considered Several observations can be made from these two figures First, all training designs perform better when the angle spread increases This is expected since a larger ∆ makes the channel less correlated Second, based on the performance difference between PSPT and ESPT as well as the performance improvement when going from ∆ = 150 to ∆ = 300 , one concludes that the × MIMO channel can be considered spatially uncorrelated for ∆ ≥ 150 , while ∆ ≥ 300 makes the × MIMO channel uncorrelated Lastly, the PSPT scheme is seen to consistently outperform the other two training schemes in these two figures Next, the impact of antenna spacings on the estimation performance is illustrated in Figs and 7, where the mean- SNR (dB) Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems for different angle spreads (∆ = 150 and ∆ = 300 ) and using different training signals (K = 60) PSPT ESPT TMT d =0.5λ, d =0.2λ −5 Mean Square Error (dB) −25 t r d =0.2λ, d =0.1λ −10 t r −15 −20 −25 10 SNR (dB) 15 20 Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems for different antenna spacings and using different training signals (K = 60) squared errors of different training designs are plotted for two sets of antenna spacings, which are {dt = 0.5λ, dr = 0.2λ} and {dt = 0.2λ, dr = 0.1λ} These figures again confirm that all the training designs perform better in more correlated channels as the consequence of having smaller antenna spacings And at any SNR level, the estimation performance of PSPT is always the best for both MIMO channels B Impact of Power Allocation Fig plots the average training power (as a fraction of the total power) computed as in (52), that maximizes the lower bound of SNReff when K = 10 and K = 60 It can be seen that a higher training power is needed for channel estimation at a lower SNR level This is expected since the spatially correlated fading has a stronger effect on the quality of the channel estimation at the lower SNR level It is also evidenced from Fig that a larger portion of the total power is spent NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS 3215 −4 PSPT ESPT TMT d =0.5λ, d =0.2λ −6 d =0.2λ, d =0.1λ t t 20 r 15 r SNReff (dB) Mean Square Error (dB) −2 −8 −10 −12 −14 −16 10 Proposed allocation Lower bound 40% training power 50% training power 60% training power SNR (dB) Fig Comparison of the mean-squared errors in channel estimation of the × MIMO systems for different antenna spacings and using different training signals (K = 60) 10 SNR (dB) 15 20 Fig Plots of SNReff and its lower bound for the MIMO systems with different power allocations (K = 60) −1 10 −2 0.4 10 0.35 0.3 BER Average Training Power PSPT ESPT TMT Perfect 4x4 MIMO, K=10 4x4 MIMO, K=60 2x2 MIMO, K=10 2x2 MIMO, K=60 0.25 −3 10 −4 10 0.2 0.15 0.1 10 SNR (dB) 15 10 SNR (dB) 15 20 20 Fig Average training power that maximizes the lower bound of SNReff at different SNR levels Fig 10 BER performance of the × MIMO system using full-rate Alamouti OSTBC and QPSK: Comparison of PSPT, ESPT, TMT and perfect channel estimation (K = 60) C Bit-Error-Rate Performance for the training signal in the × MIMO system compared to that in the × MIMO system This is also expected since there are more channel parameters to be estimated in the × MIMO system than in the × MIMO system Furthermore, observe that the larger K is, the smaller the average training power becomes This is also reasonable since with a larger K the channel statistics is better incorporated for estimation by superimposed training The actual SNReff and its lower bound attained by the proposed power allocation are plotted as functions of the SNR in Fig for the case of × MIMO system having K = 60 Observe that the lower bound is very close to the actual SNReff , which suggests the tightness of the lower bound Moreover, shown in Fig are plots of SNReff achieved with several “ad-hoc” power allocation strategies It is obvious that failing to allocate the training power as proposed in (52) can significantly reduce SNReff The final aspect to be investigated is the bit-error-rate (BER) performance of the MIMO systems that employ the proposed superimposed training design To this end, orthogonal spacetime block codes (OSTBCs) together with the maximum likelihood (ML) decoding are incorporated in Fig For the × system, the full-rate Alamouti code [1] is selected, whereas a half-rate OSTBC [29, E.q (5)] is applied for the × system Both systems use QPSK modulation with Gray mapping and the length of the information signal vector is set to K = 60 The simulation results presented in Figs 10 and 11 were obtained with PSPT, ESPT, TMT and perfect channel estimation The BER performance with perfect channel estimation is shown to serve as the performance benchmark Consistent with the relative comparison of estimation performance made before for K = 60, the BER performance with PSPT is better than that with ESPT and TMT in both the × and × 3216 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 8, AUGUST 2008 PSPT ESPT TMT Perfect −3 BER 10 −4 10 −5 10 −6 10 SNR (dB) Fig 11 BER performance of the 4×4 MIMO system using half-rate OSTBC and QPSK: Comparison of PSPT, ESPT, TMT and perfect channel estimation (K = 60) systems Specifically, at the BER level of 10−4 , there are about 2.0 dB and 0.5 dB gains in SNR by employing PSPT over TMT for the × and × systems, respectively Note that, compared to the × system, the reduced SNR gain in the × is directly attributed to the reduction in the estimation performance This is because both systems have K = 60, but there is a larger number of channel parameters to be estimated in the × system compared to the × system, hence making the PSPT design less effective in the former system than the latter one Finally, for both systems, compared to the case of perfect channel estimation, our proposed superimposed training design experiences a performance loss of only about 0.5 dB at the BER level of 10−4 VI C ONCLUSION An MMSE channel estimator was developed for spatially correlated fading MIMO channels when superimposed training is used The main contribution is a novel design of the optimal superimposed signal with an iterative optimization algorithm Simulation results show that, when the coherence time of the channel is large and under the constraint of equal training power and bandwidth efficiency, the optimal superimposed training signal performs better than the conventional timemultiplexing training signal in terms of the channel estimation error A power allocation policy is also derived to maximize the lower bound on the effective signal-to-noise ratio at the channel output The excellent BER performance of MIMO systems that employ orthogonal space-time block codes and our proposed superimposed training design is also demonstrated R EFERENCES [1] S M Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J Select Areas Commun., vol 16, no 8, pp 1451–1458, Oct 1998 [2] S Boyd and L Vandenberghe, Convex Optimization Cambridge Univ Press, 2004 [3] C Chuah, D Tse, J M Kahn, and R Valenzuela, “Capacity scaling in MIMO wireless systems under correlated fading,” IEEE Trans Inform Theory, vol 48, no 3, pp 637–650, 2002 [4] G D Forney and M V Eyuboglu, “Combined equalization and coding using precoding,” IEEE Commun Mag., Dec 1991 [5] G J Foschini and M J Gans, “On limits of wireless communication in fading environment when using multiple antennas,” Wireless Personal Commun., vol 6, pp 311–335, 1998 [6] G B Giannakis, “Filterbanks for 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CORRELATED FADING MIMO CHANNELS [32] [33] [34] [35] [36] [37] [38] MIMO frequency-selective fading channels,” to appear in IEEE Trans Signal Processing J K Tugnait, L Tong, and Z Ding, “Single-user channel estimation and equalization,” IEEE Signal Processing Mag vol 17, pp 16–28, 2000 J K Tugnait and W Luo, “On channel estimation using superimposed training and first-order statistic,” IEEE Commun Lett., vol 7, pp 413– 415, Sept 2003 H Tuy, Convex Analysis and Global Optimization Kluwer Academic Press, 1999 H Tuy, “Monotonic optimization: problems and solution approaches,” SIAM J Optimization, pp 464–494, 2000 G T Zhou, M Viberg, and T McKelvey, “Superimposed periodic pilots for blind channel estimation,” in Proc 35th Annu Asilomar Conf Sig Systems Computers, Pacific Grove, CA, pp 653–657, 2001 A Vosoughi and A Scaglione, “Everything you always wanted to know about training: guidelines derived using the affine precoding framework and CRB,” IEEE Trans Signal Processing, vol 54, pp 940–954, 2006 J H Winters, J Salz, and R D Gitlin, “The impact of antenna diversity on the capacity of wireless communication systems,” IEEE Trans Commun., vol 42, pp 1740–1751, 1994 Vu Nguyen received the Bachelor degree in Electronic Engineering and Telecommunications from the University of Tasmania, Australia in 2003, and the Master degree in Telecommunications from the University of New South Wales, Australia in 2005 Mr Nguyen currently works for Optus Ltd., NSW, Australia Hoang Duong Tuan was born in Hanoi, Vietnam He received the diploma and the Ph.D degree, both in applied mathematics from Odessa State University, Ukraine, in 1987 and 1991, respectively From 1991 to 1994 he was a Researcher at Optimization and Systems Division, Vietnam National Center for Science and Technologies He spent academic years in Japan as an Assistant Professor at the Department of Electronic-Mechanical Engineering, Nagoya University from 1994 to 1999, and then as an Associate Professor at the Department of Electrical and Computer Engineering, Toyota Technological Institute, Nagoya from 1999 to 2003 Presently, he lives in Sydney, Australia, where he is an Associate Professor at the School of Electrical Engineering and Telecommunications, the University of New South Wales His research interests include several multi-disciplinary areas of control, signal processing, communications and bio-informatics 3217 Ha H Nguyen (M’01, SM’05) received the B Eng degree from Hanoi University of Technology, Hanoi, Vietnam, in 1995, the M Eng degree from Asian Institute of Technology, Bangkok, Thailand, in 1997, and the Ph.D degree from the University of Manitoba, Winnipeg, Canada, in 2001, all in electrical engineering Dr Nguyen joined the Department of Electrical Engineering, University of Saskatchewan, Canada in 2001 and become a Full Professor in 2007 He holds adjunct appointments at the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, and TRLabs, Saskatoon, SK, Canada and was a Senior Visiting Fellow in the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia during October 2007-June 2008 His research interests include digital communications, spread spectrum systems and error-control coding Dr Nguyen currently serves as an Associate Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS and the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY He is a Registered Member of the Association of Professional Engineers and Geoscientists of Saskatchewan (APEGS) Nguyen Nam Tran was born in Quang Nam, Vietnam He received the B.E degree in electrical engineering and telecommunications from Ho-ChiMinh City University of Technology in 2001, and the M.S.E degree in Physical Electronics from Ho-Chi-Minh City University of Natural Sciences in 2004 Since 2005 he has been pursuing the Ph.D degree with the School of Electrical Engineering and Telecommunications, University of New South Wales His research interests are MIMO and MIMO-OFDM wireless communications, including coding and signal processing techniques, and applications of convex optimization for training signal and precoder design under correlated channels and colored noise  ... TM training signal for a given SNR level Similarly, while the design of the optimal superimposed training signal is well understood for independent fading channels [14], [22], [31], [37], the design. .. describes the design problem The optimal superimposed training design is presented in Section III with an iterative optimization algorithm The issue of optimal power allocation for the training signal... that the precoding matrix P designed as in (15) and the optimal NGUYEN et al.: OPTIMAL SUPERIMPOSED TRAINING DESIGN FOR SPATIALLY CORRELATED FADING MIMO CHANNELS training matrix Copt derived