Superimposed training designs for spatially correlated MIMO OFDM systems

Superimposed Training Designs for Spatially Correlated MIMO-OFDM Systems N N Tran and H D Tuan Ha H Nguyen School of Electrical Engineering and Telecommunications The University of New South Wales (UNSW) Sydney, NSW 2052, AUSTRALIA nam.nguyen@student.unsw.edu.au, h.d.tuan@unsw.edu.au Department of Electrical & Computer Engineering University of Saskatchewan Saskatoon, SK, S7N 5A9, CANADA ha.nguyen@usask.ca Abstract— Optimal training design and channel estimation for spatially correlated multiple-input multiple-output systems with orthogonal frequency-division multiplexing (MIMO-OFDM) is still an open research topic of great interest Only one asymptotic design for a special case of channel correlations was proposed in the literature To fill this gap, this paper applies tractable semidefinite programming (SDP) to obtain the optimal superimposed training signals for the general case of channel correlations To improve computational efficiency, an approximate design in closed-form is also proposed This approximate design is formed by minimizing an upper bound of the channel estimation mean-square error Since the superimposed training approach is taken, the derivation of an optimal non-redundancy precoder for data detection enhancement is also given Analytical and simulation results demonstrate the excellent performance of the proposed designs and their superior performance compared to the previously proposed design I I NTRODUCTION Obtaining channel state information (CSI) is a major challenge in the implementation of a multiple-input multiple-output with orthogonal frequency-division multiplexing (MIMO-OFDM) system Typically, training signals are used for identifying the unknown channel [1]–[5] and various designs of training signals have been proposed for OFDM and MIMO-OFDM systems in the last decade In most of the training designs for MIMO-OFDM, the transmit and receive antennas are assumed to be uncorrelated However, this assumption is impractical for certain environments [4]–[7] Moreover, the covariance matrices having Kronecker product structure are usually assumed for correlated MIMO channels [3]–[5] Estimation of the channel covariance matrix with such a structure is proposed in [8] Nevertheless, most of existing training designs are obtained for uncorrelated channels and they are not optimal when applied for correlated channels For spatially correlated flat fading MIMO channels, optimal designs have been proposed for time-multiplexing (TM) [3] and superimposed (SP) [5] training approaches For MIMO-OFDM operating over a spatially correlated frequencyselective fading channel, to our best knowledge, only reference [4] studies the training design and channel estimation However, [4] only provides the designs in some very special cases of correlation and only for TM training Furthermore, only asymptotic solutions at low and high signal-to-noise ratio (SNR) regions are given in [4] Although asymptotic design conditions are given, it is shown in [4] that sometime no training sequences can be found to satisfy the stated conditions This paper proposes training designs for spatially correlated MIMO-OFDM in the general case of channel correlation and arbitrary SNR Specifically, the approach of superimposed training, i.e., the training signal is superimposed on linearly precoded data, is considered It should be pointed out that SP training designs have been obtained for estimation of correlated flat fading MIMO channels in [5] and uncorrelated frequency-selective fading channels in [9], [10] Our previous SP training designs shall be extended to the more general and challenging case of spatially correlated MIMO-OFDM Since the system is correlated at both ends of the transmission, obtaining the exact closed-form solution to the training design problem appears intractable Therefore, we first propose and develop semi-definite programming (SDP) to design the optimal training signals Then an approximate solution in closed-form is derived to minimize an upper bound of the channel minimum mean-square error (MMSE), which is shown to be capable of efficiently identifying the MIMOOFDM wireless channel and effectively recover the OFDM data symbols In addition to employing a post-multiplying based precoder which introduces redundancy to guard against frequencyselective fading [9], [10], we further propose to use nonredundancy pre-multiplying based precoder to cope with the spatial correlation This type of non-redundancy precoder was shown to efficiently combat colored noise in [11] Moreover, a power allocation between training signal and data is given and shown to improve the detection performance Notation: Boldface upper and lower cases denote matrices and column vectors Superscripts T , H and ∗ mean transposition, Hermitian adjoint and complex conjugate operators, respectively IN is the N × N identity matrix E{·} is the expectation, tr{·} is the matrix trace operation and ⊗ means Kronecker matrix product The vectorization operator on a matrix to form a column vector by vertically stacking the matrix columns is denoted as vec{·} For any x, define x+ = max(x, 0), while diag{·} is a block diagonal matrix Furthermore, some properties of positive definite matrices used 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings in this paper are: N (P1) If < X ∈ CN ×N , then tr{X−1 } ≥ i=1 1/X(i, i) The equality holds iff X is a diagonal matrix (P2) If < X ∈ CN ×N , then tr{X−1 } ≥ N /tr{X} The equality holds iff X = κI, κ > II S PATIALLY C ORRELATED MIMO-OFDM S YSTEM Consider a frequency-selective Rayleigh fading channel, which stays constant during one block of duration M , but changes from block to block The channel has L delay paths The system has t transmit and r receive antennas and is assumed to have spatial correlation at both transmission sides The channel impulse response from the ith (i = 1, , t) transmit antenna to the jth (j = 1, , r) receive antenna in one OFDM block is defined as hji = [hji (0), , hji (L − 1)]T The channel gain matrix of the lth path can be defined as: H(l) = Σr (l)1/2 Hw (l)Σt (l)1/2 ∈ Cr×t (1) ⎤ ⎡ h11 (l) · · · h1t (l) ⎥ ⎢ =⎣ ⎦ , l = 0, , L − 1, (2) ··· hr1 (l) ··· hrt (l) where Σt (l) and Σr (l) are t × t and r × r known and invertible covariance matrices that capture the correlations of the transmit and receive antenna arrays, respectively The matrix Hw (l) is an r × t matrix whose entries are independent and identically-distributed (i.i.d.) circularly symmetric complex Gaussian random variables of variance σh2 , i.e., CN (0, σh2 ) For convenience we normalize σh2 = 1/L As a standard operation in OFDM, a cyclic prefix (CP) is used to eliminate inter-symbol-interference (ISI) between consecutive OFDM blocks Let N be the number of subcarriers Define the N × N discrete Fourier transform (DFT) N −1 √ Also, matrix as FN = 1/ N · exp(−j2πpk/N ) p,k=0 define FL as the matrix containing the 1st to Lth columns of FN So the input/output relationship corresponding to the transmission of one OFDM block of interval M is given by (after CP removal and DFT of the received signal): Y = DU + (Ir ⊗ FN )W under consideration, both non-redundancy pre-multiplying and redundant post-multiplying precoders are employed to handle antenna correlations and combat channel fading This is described further in the following Let S ∈ CN t×K denote the source symbol matrix, whose elements are modeled as zero-mean i.i.d random variables with variance σs2 This symbol matrix is first post-multiplied by the K × M linear precoder P, then pre-multiplied with the N t × N t square precoder F After that, this precoded data is arithmetically added to the N t × M SP training matrix C = CQH E to form the equivalent symbol matrix U = FSP + C of size N t × M Here the matrices F, P, C and QE are to be designed later, and they satisfy the following conditions: (C1) When the total transmit power is normalized to 1, the average training power is σc2 = tr{CCH }/(N M t) = − σs2 (C2) The matrix P is full rank and tr{PPH } = M (C3) tr{FFH } = N t This together with (C2) guarantees that the average power of the source is unchanged after precoding Since the linear precoder introduces redundancy, one has M > K and M − K is the number of redundant vectors introduced by linear precoding The larger M/K results in the better system performance but the less bandwidth efficiency Each data matrix U is then transmitted across t transmit antennas as described in (3) III C HANNEL E STIMATION AND S UPERIMPOSED T RAINING D ESIGN WITH SDP A Time-Domain Approach For channel estimation, the received signal Y in (3) is post multiplied by a matrix QE ∈ CM ×q chosen such that PQE = 0K×q and QH E QE = Iq By this operation, the training and data matrices are decoupled The decoupled signal for channel estimation phase is YQE = DCQE + (Ir ⊗ FN )WQE By changing variables Y = YQE ∈ CN r×q and N = (Ir ⊗ FN )WQE ∈ CN r×q , (5) can be rewritten as Y = DC + N (3) Here Y ∈ CN r×M , U ∈ CN t×M and W ∈ CN r×M are the received signal, transmitted signal and additive white Gaussian noise (AWGN), respectively The average power of the transmitted signals from all transmit antennas is normalized to ) Matrix D = unity The elements of W are i.i.d, CN (0, σw [Dji ], with j = 1, , r; i = 1, , t, is the N r ×N t channel matrix in the frequency domain, where Dji = diag{dji } with √ (4) dji = [dji (0), , dji (N − 1)]T = N FL hji The proposed SP training is used in conjunction with affine precoding Such an approach was proposed in [9], [10] for MIMO frequency-selective fading channel and in [2] for OFDM channel to improve detection performance Different from [5], [9], [10], for the MIMO-OFDM systems (5) (6) Since QH E QE = Iq , the matrix C is still constrained by the training power in (C1), tr{CCH } = tr{CCH } = N M tσc2 = PT , (7) and the noise statistic is unchanged after multiplying with QE Similar to [5], [9], [10], P and QE are designed as P = M K×M , QE = OH (K + : M ) ∈ CM ×q , K O(1 : K) ∈ C where O is an arbitrary M × M orthogonal matrix For a possible existence of P and QE , and for simplicity, we choose M ≥ K + q, and q = N t Rewrite Y, C, N in more detail as Y = [yjm ], C = T [cim ], N = [njm ] where yjm = [yjm (0), , yjm (N − 1)] , T cim = [cim (0), , cim (N − 1)] , and njm = [njm (0), , njm (N − 1)]T with i = 1, , t; j = 1, , r; m = 1, , q 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings Therefore, with n = 0, , N − 1, Equation (6) is rewritten as t dji (n)cim (n) + njm (n) (8) yjm (n) = i=1 The superimposed training design problem can now be formulated as follows Under the power constraint in (15), ¯ such find the training signal C, or equivalently the variable X, that the MMSE in (14) is minimized Stated mathematically: Then we can express (6) in the following new form: ¯ y ¯ +n ¯, = Cd (9) where, for n = 0, , N − 1, m = 1, , q, ¯ T (N − 1)]T ∈ CN rqì1 , y (n) = = [ ã y yT (0), , y T T ¯ q (n)]T ¯ m (n) = ∈ Crq×1 and y [¯ y1 (n), , y T [˜ y1m (n), , y˜rm (n)] ; ¯ = diag C(n) ¯ ∈ CN rq×N rt , • C ⊗ Ir n=0, ,N −1 ¯ q (n) T ∈ Cq×t and C ¯ m (n) = ¯ ¯ (n), , C C(n) = C T [˜ c1m (n), · · · , c˜tm (n)] Ctì1 ; T T T ã d = [d (0), , d (N − 1)] ∈ CN rt×1 , d(n) = [d11 (n), , dr1 (n), d12 (n) , dr2 (n), , d1t (n), , drt (n)]T ∈ Crt×1 ; ¯ T (N − 1)]T ∈ CN rq×1 , n ¯ (n) = ¯ = [¯ • n nT (0), , n T T ¯ q (n)]T ¯ m (n) = ∈ Crq×1 and n [¯ n1 (n), , n ˜ rm (n)]T ∈ Cr×1 [˜ n1m (n), , n Now write the frequency response dji (n) from the ith transmit antenna to the jth receive antenna √ at the nth sub-carrier L−1 in (4) in the following form dji (n) = N l=0 Fnl hji (l) ¯ Then d(n) can √ be rewritten in the new form as d(n)T = Fn h ¯ Here Fn = N [Fn0 Irt , , Fn(L−1) Irt ], h = [h (0), , hT (L − 1)]T , and h(l) = vec{H(l)} It follows from (1) that h(l) = vec{Σr (l)1/2 Hw (l)Σt (l)1/2 } = (Σt (l)1/2T ⊗ Σr (l)1/2 )hw (l), (10) where hw (l) = vec{Hw (l)} Therefore, d = [dT√(0), , ¯ with F ¯ = [F ¯T , , F ¯ T ]T = N FL ⊗ dT (N −1)]T = Fh, N −1 ¯ Irt Substituting d = Fh to (9) results in ¯ y = ¯ Fh ¯ +n ¯ C (11) ¯ H } be the covariance nn Let Rh = E{hhH } and Rn¯ = E{¯ matrices of the channel impulse response and additive noise, respectively Then it follows from (10) that Rh = σh2 diag{Σt (l)T h= ¯H ¯ H ¯ ¯ + 2F C CF σw (12) ¯H ¯ ¯ F XF σw −1 ¯ ≤ rPT (16) s.t tr{X} ¯ F ¯ have vastly different structures, Because Rh and X, a closed-form solution to Problem (16) appears intractable However, since the objective and constraint functions are both convex, we propose to use SDP to solve Problem (16) By introducing the new Hermitian matrix variable ≤ Y ∈ CLrt×Lrt , Problem (16) admits the following SDP: tr{Y} : ¯ X,Y Y Rh Rh + σw Rh ¯HX ¯ FR ¯ h Rh F ¯ ≤ rPT ≥ 0, tr{X} (17) There are many numerical methods to obtain the optimal ¯ from (17) Recomtraining, or equivalently, the variable X mended here is to use SeDuMi [13] for finding the solutions B Frequency-Domain Approach Given the one-to-one correspondence between the impulse and frequency responses, one expects that channel estimation and training design with SDP can be carried out in the frequency domain as well Recall that the channel’s frequency ¯ response is d = Fh, and the covariance of d is Rd = H H ¯ ¯ E{dd } = FRh F From (9), the LMMSE estimation [12] ¯ , where of d is d = Aopt y ¯ dC ¯ H +Rn )−1 ¯ H (CR y||2 } = Rd C Aopt = arg E{||d−A¯ A (18) The corresponding channel MMSE is1 [12]: ¯ dC ¯ d } ¯ H + Rn )−1 CR ¯ H (CR εd = tr{Rd − Rd C c (19) Under the power constraint (15), the problem is to find ¯ such that training signal C, or equivalently the variable C, the MMSE in (19) is minimum That is, ¯ dC ¯ H C} ¯ H +Rn¯ )−1 CR ¯ H (CR ¯ d } s.t tr{C ¯ ≤ rPT tr{Rd −Rd C −1 R−1 h + ¯H ¯ H ¯ F C y σw ¯H ¯ H ¯ ¯ F C CF σw (14) ¯ =C ¯ H C ¯ Then under (7), the power constraint Let ≤ X ¯ for new variable X is ¯ = rPT ¯ = tr{C ¯ H C} tr{X} ¯ : tr{Y} ¯ Y Rd (13) −1 (20) ¯ =C ¯ H C ¯ By introducing the Recall that Rn¯ = σw IN rq and X ¯ ∈ CN rt×N rt , Problem new Hermitian matrix variable ≤ Y (20) admits the following SDP: ¯ Y ¯ X, The corresponding MMSE, εh = E ||h − h||2 , is [12] εh = tr ¯ X≥0 R−1 h + ¯ C ⊗ Σr (l)}, l = 0, , L − On the other hand, since the noise elements are i.i.d, one ¯ H } = σw nn IN rt Therefore, with the linear has Rn¯ = E{¯ minimum mean-square error (LMMSE) estimator, the channel impulse response can be estimated from (11) as follows [12]: R−1 h tr (15) Rd + Rd ¯ d Rd XR σw ¯ ≤ rPT ≥ 0, tr{X} (21) The advantage of the formulation in (21) compared to (17) is that (21) leads to an approximate (relaxed) design problem that maximizes the upper bound of the objective function in (20) This approximate design problem can be solved much easier compared to SDP, especially when the number of variables is large It is pointed out that since R is not full rank, the inverse of R does not d d exist Since the matrix inversion formula is not applicable to pseudo-inversions [14], it is not possible to further simplify (19) to the compact form as in [12] 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings IV A N A PPROXIMATE T RAINING S IGNAL D ESIGN To design the training signal without using SDP, we shall minimize an upper bound of the objective in (20) The upper bound is realized by estimating the frequency responses from transmit antennas to each single receive antenna at each single sub-carrier Rewrite Equation (8) in the matrix form as follows: yj (n) = C(n)dj (n) + nj (n), (22) T T (n), , yjq (n)]T , nj (n) = [nTj1 (n), , where yj (n) = [yj1 T T T T njq (n)] , dj (n) = dj1 (n), , dTjt (n) , and ⎡ ⎤ c11 (n) · · · ct1 (n) ⎢ ⎥ q×t (23) C(n) = ⎣ ⎦∈C c1q (n) ··· ctq (n) Define the covariances of the frequency response and additive white Gaussian noise of the jth receive antenna at the nth sub-carrier as Rdj (n) = E{dj (n)dH j (n)} and Rnj (n) = H E{nj (n)nj (n)}= σw Iq With the LMMSE estimator, the frequency response at the jth receive antenna of the nth subcarrier dj (n) can be estimated from (22) as follows [12]: H −1 dj (n) = R−1 dj (n) + C (n)Rnj (n)C(n) −1 −1 yj (n), CH (n)Rnj and the corresponding MMSE is H −1 R−1 dj (n) + C (n)Rnj (n)C(n) εdj (n) = tr −1 Considering the estimation of all the frequency responses of the MIMO-OFDM system having r receive antennas and N sub-carriers as above, the total MMSE is N −1 r εd = tr R−1 dj (n) j=1 n=0 + CH (n)C(n) σn −1 (24) From (7) and (23), the power constraint for C(n) is N −1 H H n=0 tr{C(n)C (n)} = tr{CC } ≤ PT Therefore the design problem can be mathematically stated as r N −1 C(n) tr R−1 dj (n) + j=1 n=0 H C (n)C(n) σn2 (25) tr{C(n)CH (n)} ≤ PT n=0 By simple calculations, one can obtain the following covariance matrix Rdj (n) of the frequency response of the jth receive antenna at the nth sub-carrier: Rdj (n) = σh2 Fn (diag{Σt (l)T })FH n , l = 0, , L − 1, (26) √ N [Fn0 It Fn(L−1) It ] It can = where Fn be seen that Rdj (n) is independent of j, so for notation simplicity, Rd (n) shall be used instead of Rdj (n) As in [15], make the following singular-value −1 H decompositions (SVDs) R−1 d (n) = Ud (n)Λd (n)Ud (n), −1 t H Λd (n) = diag{[λd (n), , λd (n)]}, C (n)C(n) = Ux (n)Λx (n)UH x (n), Λx (n) = diag{[sn1 , , snt ]}, where Ud (n) and Ux (n) are unitary matrices Substitute (26) and the above SVDs into (25), Problem (25) becomes N −1 N −1 tr [Λ−1 d (n) + Z(n)≥0 n=0 N −1 Λx (n)≥0 N −1 tr{[Λ−1 d (n) + n=0 tr{Λx (n)} ≤ PT Λx (n)]−1 }s.t σw n=0 Applying the Karush-Kuhn-Tucker condition for optimality of convex programming [16] the closed-form solution is /µ − σ λi (n) σw w d sni,opt = N −1 + t where µ is found from n=0 i=1 /µ − σ λi (n) σw w d (28) + = PT , n = 0, , N − 1, i = 1, , t Remark: When the receive correlations at different delay paths are the same, except a scaling factor, i.e., Σr (l) = σl2 Σr , l = 0, , L − 1, a tighter upper bound of the objective function in (20) can be obtained by estimating the frequency responses of all transmit/receive antenna pairs, but at each single sub-channel individually However, the closed-form solution to this approximate problem is not available Instead, the iterative bisection procedure (IBP) can be used to find the exact optimal solution to such a relaxed problem (presented in detail in [17]) The simulation result for this special case is also given in Section VI V N ON -R EDUNDANCY P RECODER D ESIGN The design of a non-redundancy pre-multiplying precoder (NP) aims to minimize the average mean-square error (MSE) of LMMSE detection Since affine precoding technique is employed, the SP training signal needs to be removed from the received signal in (3) before data detection is carried out Making use of CPH = 0, and post-multiplying both sides of (3) by QD = PH (PPH )−1 gives −1 N −1 s.t H t×t where Z(n) = UH d (n)Ux (n)Λx (n)Ux (n)Ud (n) ∈ C Similar to [15], it can be shown that the optimal solution of (27) has to be diagonal This implies that Ux (n) = Ud (n) and Problem (27) simplifies to = YQD DFS + (Ir ⊗ FN )WQD (29) Let yi , si and qDi , i = 1, , K be the ith column of YQD , S and QD , respectively We have yi = DFsi + (Ir ⊗ FN )WqDi (30) Since the data symbols are assumed to be uncorrelated and the noise is white Gaussian, one has Rsi = E{si sH i } = σs IN t H and Rwi = E{[(Ir ⊗ FN )WqDi ][(Ir ⊗ FN )WqDi ] } = Kσw M IN r Let D denote the estimate of D, which is obtained from d = Aopt y, with Aopt given in (18) Based on D the LMMSE estimation of si is given by [12] H −1 −1 ˆsi = {R−1 (DF)H R−1 si + (DF) Rwi (DF)} wi yi Let si = si − ˆsi Then the corresponding MSE of the source symbols can be approximated as εsi = E{ si H H −1 −1 } ≈ tr{R−1 si + F D Rwi DF} (31) Thus, in order to improve the detection performance, it is tr{Z(n)} ≤ PT , (27) Z(n)]−1 : σw desirable to design the precoding matrix F to minimize (31) n=0 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings under the power constraint in (C3) Such a design, however, ˆ This yields a solution that depends on the channel estimate D is practically impossible because with affine precoding, the precoder F needs to be known before the channel can be estimated To circumvent this difficulty the precoder can be designed based on the channel statistics With this approach, one obtains the following approximately average MSE: ≈ H H −1 −1 tr{R−1 si + F E{D Rwi D}F} (32) Note that the above approximations are exact when the channel estimation is perfect Let Qi = E{DH R−1 wi D} The optimization problem can be stated as: H −1 s.t tr{FFH } ≤ N t tr{R−1 si + F Qi F} (33) F This convex optimization problem is a special case of a problem considered in [15, Equation 8], and its closed-form solution is given by [15, Theorem 1] It is summarized in the following Let R be the rank of Qi Make the following SVD of Qi = UH Q ΣQ UQ Here ΣQ = diag[ΣM Q 0] with ΣM Q > is a diagonal matrix having the eigenvalues of Qi on its main diagonal in decreasing order and UQ is the unitary matrix whose columns are the corresponding eigenvectors of Qi The optimal non-redundancy precoding matrix for MMSE detection of OFDM symbols is Fopt = UH Q diag ⎧ ⎨ ⎩ µ ¯−1/2 − γ(j) γ (j) ⎫ + 1/2 ⎬ ⎭ , j=1, ,R −4 M ×N −6 Before closing this section, it is relevant to point out that there is an important interaction between the channel estimation and detection phases via the total power constraint of σc2 + σs2 = A natural question is how to allocate the total power to training and data signals in order to optimize the system’s detection performance Unfortunately, such an optimal power allocation is yet to be found for the spatially correlated MIMO-OFDM systems Nevertheless, an ad-hoc power allocation scheme, which is basically the optimal solution obtained for the special case of uncorrelated systems (see [17]), is used in this paper for simulation in Section VI The allocation is σc,uncorr = + Lγσn2 − Lγσn2 (Lσn2 + N (Lrtσn2 − M γ) N M )(N rt + γ) VI S IMULATION R ESULTS Approximate, general case Approximate, special case Optimal, with SDP K=8 K=80 −2 where γ(j) = σs ΣM Q (j, j), UQ ∈ C with ∗]H , and µ ¯ is chosen such that UQ = [UH Q R −2 µ−1/2 γ(j) − 1)+ = N t j=1 ΣM Q (j, j)(¯ N Lrtσn2 Fig shows the MSE of the channel impulse response, ˆ }, for × MIMO-OFDM system having K = E{ h − h and K = 80 In order to apply the approximate design for the special case of receive correlation matrices, the parameters are chosen as ∆tl = 5.4(l + 2)o , ∆rl = 5.4o , dt = 0.5λ and dr = 0.3λ, which result in the same covariance matrix for all the delay paths at the receive antennas Such parameters are chosen quite arbitrarily for the illustration purpose but they yield high correlations at both sides of the transmission It can be seen from Fig that, at low SNR the optimal training design based on SDP produces the lowest MSE The approximate design using IBP for the special case performs worst than the SDP design, but outperforms the approximate design for the general case But more importantly, at high SNR, all the three proposed designs perform almost the same, especially in the case of K = 80 This observation suggests that the approximate design, given in closed-form in Section IV, is most attractive due to its good performance and simplicity (34) The one-ring model in [18, E.q (6)] is used to generate the elements of the covariance matrices Σr (l) and Σt (l) Specifically, Σt (l)(n, m) ≈ J0 ∆tl 2π λ dt |m − n| , Σr (l)(i, j) ≈ d |i − j| , where ∆ and ∆rl are the angle spreads J0 ∆rl 2π tl λ r (in radian) of the lth path at the transmitter and the receiver, respectively; dt and dr are the spacings of the transmit and receive antenna arrays, respectively; λ is the wavelength and J0 (·) is the zeroth-order Bessel function of the first kind Since the average transmitted power, including the training and data powers, is normalized to unity, the received SNR in dB is MSE (dB) ε si The system parameters are defined as SNR = −10log10 σw chosen as [1, Table I] with N = sub-carriers and L = taps A Estimation Performance −8 −10 −12 −14 −16 −18 −20 −8 −6 −4 −2 10 SNR (dB) Fig 1: Comparison of channel MSE of × MIMO-OFDM B Performance Comparison with [4] Comparisons of both channel estimation and BER performance between our proposed design and the design in [4] (which is for the cases of high or low SNR) are presented in Figs and 3, where the system is chosen to be the same as in [4], i.e., a × MIMO-OFDM system, N = 32 subchannels and L = taps Other parameters are ∆tl = 13.5o , ∆rl = 10(l + 2)o , dr = 0.5λ, dt = 0.3λ and K = For our proposed design, the source symbols, which are drawn from the quadrature phase shift keying (QPSK) constellation {± σs2 /2 ± j σs2 /2}, are precoded by the affine precoder and the precoded data matrix has size N t × M , with M = 5(K + q), q = N t For the TM system considered in [4], the N t-length training vector is sent first to estimate the fading channel Then the source symbol matrix of size N t × K (the same as in our SP training design) is linearly precoded 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings with the pre-multiplying non-redundancy precoder and postmultiplying precoder The designs of these precoding matrices are the same for SP and TM training Since in TM training, the first column of the transmitted signal matrix is used in the training phase, the size of the post-multiplying precoding matrix in the TM case is K × (M − 1) Furthermore, as in our proposed designs, the LMMSE estimation is considered and the average transmitted signal power is normalized to unity for the TM training design in [4] −5 Design in [4] Proposed approximate design −10 MSE (dB) −15 −20 R EFERENCES −25 −30 −35 −40 −45 10 15 20 SNR (dB) Fig 2: Comparison of channel MSE between the proposed design and the design in [4] 10 −1 10 BER also show that, with the proposed SP training and affine precoding designs, the system’s BER performance closely approaches that of the system with perfect channel knowledge at the receiver VII C ONCLUSIONS Using SDP, the optimal superimposed training signal for spatially correlated MIMO-OFDM has been designed Moreover, an approximate efficient design was proposed, which is given in a closed-form expression, for the general case of channel correlations The designs of pre-multiplying and postmultiplying precoding matrices to combat fading and exploit correlation of the wireless MIMO-OFDM channels were also considered Simulation results show that the proposed designs offer excellent estimation and detection performance and they significantly outperform the existing design −2 10 Design in [4] Proposed approximate design Estimated channel, without NP Estimated channel, with NP Perfect channel, without NP Perfect channel, with NP −3 10 −4 10 10 15 20 SNR (dB) Fig 3: Comparison of BER performance between the proposed design and the design in [4] Fig shows that the MSE of the TM training proposed in [4] is significantly higher than that obtained by our proposed SP training This translates into a much better BER performance of the proposed SP training design with affine precoder compared to the TM training system proposed in [4] For example, a power saving of about 10 dB is observed at the BER level of 10−2 It is also shown in Fig that the proposed non-redundancy precoder is very helpful in improving the detection performance of the spatially correlated MIMO-OFDM systems, for both TM and SP training (though the benefit with NP is more pronounced for SP training at high SNR) Such a performance improvement comes at no extra power nor transmission bandwidth Finally, the results in Fig [1] H Minn and N Al-Dhahir, “Optimal training signals for MIMO OFDM channel estimation,” IEEE Trans Wireless Commun., vol 5, pp 1158– 1168, May 2006 [2] S Ohno and G B Giannakis, “Optimal training and redundant precoding for block transmissions with application to wireless OFDM,” IEEE Trans Commun., vol 50, pp 2113–2123, Dec 2002 [3] J H Kotecha and A M Sayeed, “Transmit signal design for optimal estimation of correlated MIMO channels,” IEEE Trans Signal Process., vol 52, pp 546–557, Feb 2004 [4] H Zhang, Y G Li, A Reid, and J Terry, “Optimum training symbol design for MIMO OFDM in correlated fading channels,” IEEE Trans Wireless Commun., vol 5, pp 2343–2347, sept 2006 [5] V Nguyen, H D Tuan, H H Nguyen, 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correlated MIMO-OFDM systems,” submitted to IEEE Trans Wireless Commun, Mar 2008 [18] D Shiu, G J Foschini, M J Gans, and J M Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Trans Commun., vol 48, pp 502–513, Mar 2002 978-1-4244-2324-8/08/$25.00 © 2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings  ... detection performance of the spatially correlated MIMO- OFDM systems, for both TM and SP training (though the benefit with NP is more pronounced for SP training at high SNR) Such a performance improvement... optimal superimposed training signal for spatially correlated MIMO- OFDM has been designed Moreover, an approximate efficient design was proposed, which is given in a closed-form expression, for the... to training and data signals in order to optimize the system’s detection performance Unfortunately, such an optimal power allocation is yet to be found for the spatially correlated MIMO- OFDM systems