Optimal SP training for spatially correlated MIMO channels under coloured noises

Nguyen Based on convex programming for optimisation, the optimal superim-posed SP training signal design is prosuperim-posed for spatially correlated multiple-input–multiple-output MIMO

Optimal SP training for spatially correlated

MIMO channels under coloured noises

N.N Tran✉and H.X Nguyen

Based on convex programming for optimisation, the optimal

superim-posed (SP) training signal design is prosuperim-posed 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

communi-cation 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

cor-related 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 precoderP [ CK ×(K+L)to obtainXP Here, L ≥ N, and P = [p1,…,

pK]T Then, a training matrixC [ CN ×(K+L)is superimposed to the

pre-coded data So, the transmitted signal is now combined asXP + C

ConsiderH [ CM ×N as the spatially correlated fading channel in an

arbitrary block.H can be represented as [2,3]H = S1 /2

r HwS1 /2

t , where

Σris known with an M-dimension andΣtis known with an N-dimension

It has been shown in [2,3] thatΣrandΣtrepresent the transmit and

receive correlations, respectively All entries ofHware 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,ΣrandΣtare constructed from the one-ring model as shown

in [2] Let the MN-length channel vectorh 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)

whereX is the correlated and N is the ACGN with zero mean and

rep-resented asN = GnW [ CM ×(K+L) Here,W is the AWGN with zero

mean ands2-variance To keep the noise power unchanged after

unex-pected effects of colouring factors, the coefficient matrix Gnis

normal-ised as tr{GnGH

n}= M The average transmitted power is also

normalised ass2+s2= 1, wheres2= trace(CCH)/N(K + L) is the

average training power and s2 is the average information-bearing

power LetU be an orthogonal matrix with dimensions (K + N) × (K

+ L) LetP 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) and Q = PH(PPH)−1

Q = UH(K+ 1):(K + N), :) [ C(K +L)×N

C = CU((K + 1):(K + N), :) = CQH

(2)

whereU(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 QHQ = IN,

PQ = 0, CPH= 0 andPQ = IK By multiplying the two sicks of (1)

with the matrixQ, the received signal for estimation is decoupled as

YQ = HXPQ + HCQ + NQ = HC + NQ (3) which is free ofHX By multiplying the two sides of (1) with matrix Q, the received signal for data detection is decoupled as

YQ = HXPQ + HCQ + NQ = HX + NQ (4) which is free ofHC 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 channelH in (3) only Optimal SP training design: Since QHQ = IN, from (2) we have trace(CCH)= trace(CQHQCH

)= trace(CCH

) Let PT= N(K + L)s2, the total design power ofC is limited by the constraint

trace(CCH)= trace(CCH)≤ PT (5) For a correlated system with coloured noise, to efficiently estimate [12] channelH in (3), we have to vectorise the two sides of (3) Let

y = vec(YQ) [ CMN

n = vec(GnWQ) = (QT⊗ Gn)vec(W) [ CMN and

C = CT

⊗ IM [ CMN ×MN

We have

Rn= E[nnH]= (QHQ)T⊗s2GnGH

n =s2IN⊗ GnGH

n

By vectorising (3) as y =
Ch + n and employing the linear minimum-mean-square error (MMSE) estimation [12], the channel estimateh is presented as follows:

ˆh = (R−1+
CHR−1

n
C)−1
CHR−1

with the covariance matrix of the estimation error vector being (R−1+
CHR−1

n
C)−1= [(St⊗ Sr)−1+ (1/s2

)CTH

CT

⊗ (GnGH

n)−1]−1 The training design problem is now how to optimiseC for a minimal error:

min

C[C N×Ntrace (St⊗ Sr)−1+s12CTH

CT

⊗ (GnGH

n)−1

s.t trace(CCH)≤ PT (7)

By applying the variable changeX = CTH

CT

[ CN×N, problem (7) can

be solved by the tractable semi-definite programming (SDP):

min

Z,X trace{Z} s.t trace(X) ≤ PT and

R R + R s12X ⊗ (GnGH

n)−1

R

⎡

⎣

⎤

⎦ ≥ 0 (8)

From the optimal value ofX solved through SDP in (8), it is mathemat-ically legal to obtain C = X1/2T Then the training matrix C can be created fromC 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 equal-power 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 thefixed total power PT As it has been shown in [3] that the SP training outperforms the time-multiplexing (TM) train-ing, 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 3 pp 247–249

The mean-square errors (MSE) are normalised to be E[||h||], and

then used as the main estimation comparison among the three training

solutions in the case of K = 60 In Figs.1and2, MSEs are illustrated

for MIMO channels with 2 × 2 and 4 × 4 antennas, respectively For

the 2 × 2 channel,Δ = 5°, the coloured factor Gnis randomly generated

as Gn= [0.8189 0.5740; 0.5740 0.8189], while for the 4 × 4

channel,Δ = 3° and Gnis chosen as

Gn=

0.6031 0.0666 0.6716 0.4252

0.3336 0.0628 0.8700 0.3574 0.5301 −0.2113 0.7784 0.2616

0.4937 −0.1566 0.7080 0.4801

⎡

⎢

⎣

⎤

⎥

⎦

−25

−20

−15

−10

−5

SNR, dB

SDP IBP ESPT

Fig 1 MSE comparison of 2 × 2 MIMO having different designs: SDP, ESPT

and IBP

−25

−20

−15

−10

−5

0

SNR, dB

SDP IBP ESPT

Fig 2 MSE comparison of 4 × 4 MIMO having different designs: SDP, ESPT

and IBP

It can be seen from Figs.1and2that the proposed SDP design

out-performs 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 4 × 4 channel, SDP and IBP

are significantly better than ESPT

The impact of larger angle spreads,Δ = 15° and Δ = 30°, is illustrated

in Fig.3with 4 × 4 antennas and K = 60 ForΔ = 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 ofΔ = 15° is better than that of Δ = 30° as

the channel is highly correlated in the former case Nevertheless, no

matter which angle spread is used, the proposed design outperforms the existing designs as shown in Fig.3

–14

−12

−10

−8

−6

−4

−2 0

SNR, dB

IBP ESPT

Δ = 15º

Δ = 30º

Fig 3 MSE comparison of 4 × 4 MIMO having different angle spreads:

Δ = 15° and Δ = 30°

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 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)

✉ E-mail: nntran@fetel.hcmus.edu.vn H.X Nguyen (Tan Tao University, Long An, Vietnam) References

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ELECTRONICS LETTERS 5th February 2015 Vol 51 No 3 pp 247–249

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