Optimal SP training for spatially correlated MIMO channels under coloured noises

Optimal SP training for spatially correlated MIMO channels under coloured noises ✉ YQ = HXPQ + HCQ + NQ = HC + NQ (3) which is free of HX By multiplying the two sides of (1) with matrix Q, the received signal for data detection is decoupled as N.N Tran and H.X Nguyen Based on convex programming for optimisation, the optimal superimposed (SP) training signal design is proposed for spatially correlated multiple-input–multiple-output (MIMO) channels in the presence of correlated symbols and coloured Gaussian noises Simulation results show that the proposed training design can effectively estimate the channel and outperforms the existing designs Introduction: It is well known that multiple-input–multiple-output (MIMO) increases the wireless channel capacity [1] However, placing multiple antennas in the limited space of portable wireless communication devices is very challenging Spatial correlations [2] usually occur in practical MIMO systems and reduce the channel capacity To cope with this problem under uncorrelated source symbols and additive white Gaussian noise (AWGN), the optimal superimposed (SP) training signal [3, 4] has been designed in [3] Although AWGN has not existed in many practical cases (see, e.g [5–10]), and the training signal was examined for orthogonal frequency division multiplexing (OFDM) systems under coloured noises in [5, 6], there is no SP signal designed for spatially correlated MIMO channels under additive coloured Gaussian noises (ACGN) Moreover, in pracitice, under various signal processing techniques, correlated data symbols are usually transmitted over wireless channels [11] It is obvious that under the distortion effects of ACGN and correlated data, the design in [3] fails to neither efficiently estimate the wireless channel nor effectively recover the source symbols This leads to a need for an optimal SP design for correlated symbols and ACGN cases System model: The MIMO wireless communication system has N antennas at the transmitter and M antennas at the receiver The channel is frequency-flat block fading The source signal matrix is correlated and denoted as X = [x1 , , xN ]T [ CN ×K , with K ≥ N Before transmitting over the wireless channel, X is first multiplied by a precoder P [ CK×(K+L) to obtain XP Here, L ≥ N, and P = [p1, …, pK]T Then, a training matrix C [ CN ×(K+L) is superimposed to the precoded data So, the transmitted signal is now combined as XP + C Consider H [ CM ×N as the spatially correlated fading channel in an 1/2 arbitrary block H can be represented as [2, 3] H = S1/2 r Hw St , where Σ r is known with an M-dimension and Σ t is known with an N-dimension It has been shown in [2, 3] that Σ r and Σ t represent the transmit and receive correlations, respectively All entries of Hw are the unit variance of circularly symmetric complex Gaussian random variables with an independent and identical distribution By this well-known assumption, we have the expectation of vec(Hw )vecH (Hw ) being an identity matrix Moreover, Σ r and Σ t are constructed from the one-ring model as shown in [2] Let the MN-length channel vector h be the vectorisation of H, the overall channel covariance matrix is denoted as R = E{hhH } = St ⊗ Sr After transmitting the combined data over the above channel under ACGN, the received signal can be described as follows: Y = H(XP + C) = HXP + HC + N (1) where X is the correlated and N is the ACGN with zero mean and represented as N = Gn W [ CM ×(K+L) Here, W is the AWGN with zero mean and s2n -variance To keep the noise power unchanged after unexpected effects of colouring factors, the coefficient matrix Gn is normalised as tr{Gn GH n } = M The average transmitted power is also normalised as s2x + s2c = 1, where s2c = trace(CCH )/N (K + L) is the average training power and s2x is the average information-bearing power Let U be an orthogonal matrix with dimensions (K + N ) × (K + L) Let P and C be the to-be-designed square matrices with dimensions K × K and N × N, respectively Similar to [3, 4], we choose P = PU(1:K, :) [ CK×(K+L) with the matrix Q, the received signal for estimation is decoupled as Q = PH (PPH )−1 and Q = UH (K + 1):(K + N ), :) [ C(K+L)×N (2) C = CU((K + 1):(K + N ), :) = CQ H where U(1:K, :) has the first K rows and U((K + 1):(K + N ), :) has (K + 1) to (K + N ) rows of U, respectively It can be seen that Q HQ = IN, PQ = 0, CP H = and PQ = IK By multiplying the two sicks of (1) YQ = HXPQ + HCQ + NQ = HX + NQ (4) which is free of HC Although P could be further designed from (4) to enhance the detection performance, it is not the objective of this Letter because of the space limitation In the following Section, we design the SP training to optimally estimate the wireless channel H in (3) only Optimal SP training design: Since Q HQ = IN, from (2) we have H H trace(CCH ) = trace(CQH QC ) = trace(CC ) Let PT = N (K + L)s2c , the total design power of C is limited by the constraint H trace(CC ) = trace(CCH ) ≤ PT (5) For a correlated system with coloured noise, to efficiently estimate [12] channel H in (3), we have to vectorise the two sides of (3) Let y = vec(YQ) [ CMN n = vec(Gn WQ) = (QT ⊗ Gn )vec(W) [ CMN and C = C ⊗ I M [ CMN ×MN T We have H Rn = E[nnH ] = (QH Q)T ⊗ s2n Gn GH n = sn I N ⊗ Gn Gn By vectorising (3) as y = Ch + n and employing the linear minimum-mean-square error (MMSE) estimation [12], the channel estimate h is presented as follows: −1 H −1 hˆ = (R−1 + CH R−1 n C) C Rn y (6) with the covariance matrix of the estimation error vector being −1 = [(St ⊗ Sr )−1 + (1/s2n )C (R−1 + CH R−1 n C) TH T −1 −1 C ⊗ (Gn GH n) ] The training design problem is now how to optimise C for a minimal error: trace C[CN×N (St ⊗ Sr )−1 + s.t TH T −1 C C ⊗ (Gn GH n) s2n H trace(CC ) ≤ PT −1 (7) By applying the variable change X = C C [ CN ×N , problem (7) can be solved by the tractable semi-definite programming (SDP): TH T trace{Z} s.t trace(X) ≤ PT and Z,X ⎡ ⎤ Z R ⎣ ⎦≥0 −1 R R + R X ⊗ (Gn GH R n) sn (8) From the optimal value of X solved through SDP in (8), it is mathematically legal to obtain C = X 1/2T Then the training matrix C can be created from C easily as shown in (2) Simulation results: In this Section, channel estimation performances of the proposed SP training (SDP) are compared with those of the equalpower SP training (ESPT) and the iterative bi-section SP training (IBP) in [3] To have a fair comparison with [3], the channel is chosen the same as that in [3, Section V] Specifically, we use the one-ring model in [2, Equation (6)] with dr = 0.2λ and dt = 0.5λ The power between training and data is divided as [3, Equation (52)] The ESPT is the optimal solution for uncorrelated systems, i.e a scaled identity matrix with the fixed total power PT As it has been shown in [3] that the SP training outperforms the time-multiplexing (TM) training, a comparison of SP with TM training is unnecessary in this Letter However, to save transmission bandwidth, the minimum training symbols for the TM case, i.e L = N, was chosen for all simulations ELECTRONICS LETTERS 5th February 2015 Vol 51 No pp 247–249 −5 MSE, dB −6 −10 SDP IBP ESPT −12 Δ = 15º −8 SNR, dB 10 Conclusion: On the basis of tractable SDP, the optimal solution for SP training is derived for spatially correlated MIMO channels under ACGN and with correlated source data Simulation results have shown that the proposed SP design outperformed the previously known designs 10 SNR, dB 15 20 Fig MSE comparison of × MIMO having different designs: SDP, ESPT and IBP Acknowledgment: This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.02-2012.28 © The Institution of Engineering and Technology 2015 17 October 2014 doi: 10.1049/el.2014.3607 N.N Tran (Vietnam National University, HCMC, Vietnam) SDP IBP ESPT −5 MSE, dB −4 Fig MSE comparison of × MIMO having different angle spreads: Δ = 15° and Δ = 30° −20 ✉ E-mail: nntran@fetel.hcmus.edu.vn H.X Nguyen (Tan Tao University, Long An, Vietnam) References −10 −15 −20 −25 −2 –14 −15 −25 Δ = 30º SDP IBP ESPT −10 matter which angle spread is used, the proposed design outperforms the existing designs as shown in Fig MSE, dB The mean-square errors (MSE) are normalised to be E[||h||2 ], and then used as the main estimation comparison among the three training solutions in the case of K = 60 In Figs and 2, MSEs are illustrated for MIMO channels with × and × antennas, respectively For the × channel, Δ = 5°, the coloured factor Gn is randomly generated as Gn = [0.8189 0.5740; 0.5740 0.8189], while for the × channel, Δ = 3° and Gn is chosen as ⎡ ⎤ 0.6031 0.0666 0.6716 0.4252 ⎢ 0.3336 0.0628 0.8700 0.3574 ⎥ ⎥ Gn = ⎢ ⎣ 0.5301 −0.2113 0.7784 0.2616 ⎦ 0.4937 −0.1566 0.7080 0.4801 10 SNR, dB 15 20 Fig MSE comparison of × MIMO having different designs: SDP, ESPT and IBP It can be seen from Figs and that the proposed SDP design outperforms the IBP design in [3] at low and average SNR levels At higher SNR levels, the effect of correlation and coloured noise is minimal, and can be considered as an uncorrelated channel with white noise This gives a very similar performance for both SDP and IBP designs However, it is very important that for the × channel, SDP and IBP are significantly better than ESPT The impact of larger angle spreads, Δ = 15° and Δ = 30°, is illustrated in Fig with × antennas and K = 60 For Δ = 30°, the channel is almost uncorrelated, so the performance of the design in [3] is the same as that of the equal power training (which is the optimal design for uncorrelated systems) Although the channel can be considered as uncorrelated, the coloured noise still has an effect on the system performance It is easily seen that only the proposed SDP design can cope with ACGN, and thus yields a superior performance when compared with that of the other designs Moreover, it can be seen that the estimation performance of Δ = 15° is better than that of Δ = 30° as the channel is highly 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Estimation theory’ (Prentice-Hall, 1993) ELECTRONICS LETTERS 5th February 2015 Vol 51 No pp 247–249 Copyright of Electronics Letters is the property of Institution of Engineering & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use  ... IBP ESPT −12 Δ = 15º −8 SNR, dB 10 Conclusion: On the basis of tractable SDP, the optimal solution for SP training is derived for spatially correlated MIMO channels under ACGN and with correlated. .. 502–513 Nguyen, V., Tuan, H.D., Nguyen, H.H., and Tran, N.N.: Optimal superimposed training design for spatially correlated fading MIMO channels , IEEE Trans Wirel Commun., 2008, 7, pp 3206–3217... among the three training solutions in the case of K = 60 In Figs and 2, MSEs are illustrated for MIMO channels with × and × antennas, respectively For the × channel, Δ = 5°, the coloured factor