Research ArticleMin-Max-MSE Transceiver Design for MU-MIMO VLC System

Zhirong Zeng 1 Huiqin Du 12 and Zujian Wu 3

1College of Information Science and Technology Jinan University Guangzhou China2National Mobile Communications Research Laboratory Southeast University Nanjing China3Jinan University-University of Birmingham Joint Institute at Jinan University (J-BJI) Guangzhou China

Correspondence should be addressed to Huiqin Du thuiqindujnueducn

Received 3 February 2018 Revised 12 April 2018 Accepted 15 April 2018 Published 15 May 2018

Academic Editor Imran S Ansari

Copyright copy 2018 Zhirong Zeng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper considers a multiuser (MU) multiple-input multiple-output (MIMO) visible light communication (VLC) system withbroadcast channels Due to the simultaneous transmission of the different source data to the receivers coveredwith the light rays theinteruser interference (IUI)may degrade the system performanceWe strive to suppress the IUI byminimizing themaximummeansquare errors (MSE) under the assumption of the perfect knowledge of channel state information (CSI) However since the CSImaynot be perfectly known in practice a robust design is required against the channel uncertainties Additionally the nonnegativityand the limited linear range of the optical signals have been taken into account in the VLC transceiver designs Simulation resultsvalidate that the proposed min-max-MSE approach can provide fair transmission compared with the minimization of sum-MSEapproach Furthermore it is demonstrated that the robust scheme is capable of providing robustness and gaining a considerable biterror rate (BER) performance

1 Introduction

With the rapid development of the light-emitting diode(LED) a novel communication technology known as VLCgives rise to researchersrsquo interests Conventional bulbs arereplaced by LEDs which can be used for both room illumi-nation and high speed data communication [1 2] Besidesvisible light cannot penetrate walls because of its highfrequency as a result indoor communications will not sufferfrom the intercell interference and security can be provided[3]

In an indoor environment multiple lighting sourcesare often utilized for the whole room illumination and ahigher transmission rate can be achieved through spatialmultiplexing by constructing the MIMO VLC system [4ndash6]Moreovermultiple users accessing the system simultaneouslybecomes a promising technique [7] Due to users receivingall light rays from multiple lighting sources the IUI mayexist which degrades the system performance The IUIcancellation methods have been extensively investigated inMU radio frequency (RF) systems Nevertheless they cannotbe applied directly to VLC systems because of the followingreasons

(a) The VLC system adopts the intensity modulationand direct detection (IMDD) scheme where the transmittedsignals should be unipolar and nonnegative real values ratherthan bipolar and complex signals in RF systems

(b) Owing to the VLC specific feature of providing stablebrightness andmeanwhile conveying information the opticalsignal intensity ismore limitedwith norm 1 constraint insteadof the electronic power constraint in RF communication

As a result the above properties must be taken intoaccount in the research of VLC systems Many efforts areproposed to suppress the IUI for MU-MIMO VLC appli-cations such as sum-rate maximization [8ndash11] sum-MSEminimization [12] and max-MSE minimization [13] How-ever the sum-rate maximization approach or the sum-MSEminimization method may introduce unfair transmissionbecause the total sum rate and the total MSE are optimizedTo consider the fairness among the multiple users theminimization of the maximum MSE algorithm is adoptedover the multiple-input single-output (MISO) VLC system[13] under the assumption of the perfect CSI However theCSI may not always be perfectly known and the channeluncertainties could lead to severe performance degradationwhen the transceiver design is only based on the estimated

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 3080643 10 pageshttpsdoiorg10115520183080643

2 Wireless Communications and Mobile Computing

Modulation

Modulation

Modulation

LEDArray 1

LEDArray 2

LEDArray N

PD

PD

PD

PD

Demodulation

Demodulation

1st userrsquosdata

data

data

2nd userrsquos

1st userrsquosdata

s1

s2

sK

Precoding

Wk

x1

x2

xN

p1

p2

pN

H1

MIMO Channel

MIMO Channel

HK

Kth userrsquos data

n1

n1

nM

nM

EqualizerG1

EqualizerGK

r1

rK

W1 W2

Kth userrsquos

Figure 1 Schematic model of the proposed MU-MIMO VLC system

channel information [14] Therefore it is critical to considerthe channel imperfection in the VLC transceiver design

In this work we consider the indoor MU-MIMO VLCsystem and strive to eliminate the IUI by minimizing themaximum MSE Due to the nonconvexity of the underlyingproblem with respect to multiple variables the alternatingminimization approach over variables is proposed to achievethe optimal solutions The main contributions of this papercan be summarized as follows

(1) Considering that the output signal of LED must benonnegative and limited within the linear dynamic range weimpose the optical power constraint in the proposed designand the direct current (DC) bias is applied to provide stablebrightness

(2) Our proposed transceiver design optimizes the totalMSE while providing throughput fairness for all users Thesimulation and comparison results confirm the effectivenessof the proposed algorithm for guaranteeing the fairness withrespect to the MSE performance

(3) Besides the perfect CSI case the CSI with uncer-tainties is also considered in our work and the robustdesign is developed The numerical results demonstrate thatthe robust scheme outperforms the nonrobust scheme andachieves considerable BER performance under the channelimperfection

(4) Finally we also investigate the impact of differentgeometricmetrics including usersrsquo relative locations heightsand angles The results imply that channel correlation is themost decisive factor for the system performance

The rest of the paper is organized as follows Section 2presents the indoor MU-MIMO VLC system model Sec-tion 3 investigates the optimal transceiver with the perfectchannel information and the robust scheme with imperfectCSI is developed in Section 4 The simulation results arediscussed in Section 5 to verify the performance of theproposed iterative algorithm Finally the conclusions aredrawn in Section 6

Notation 1 In this paper uppercase and lowercase lettersdenote matrices and vectors respectively The operatorsvec(sdot) (sdot)119879 and (sdot)minus1 stand for the vectorization transposeand inverse of a matrix or a vector respectively | sdot | representsthe absolute operator sdot 1 denotes the 1198711 norm of vectors

and sdot F is the Frobenius norm EXsdot denotes the statisticalexpectation overX tr(sdot) is the trace of matrix I is the identitymatrix with corresponding dimensions (sdot)12 represents thesquare root of a square matrix The operator otimes denotes theKronecker product

2 System Model

The indoor VLC system consisting of 119873 transmitting LEDarrays and 119870 user terminals equipped with multiple LEDsandmultiple photodetectors (PD) respectively is consideredFigure 1 illustrates the schematic model of the proposedVLC system where the On-Off Keying (OOK)modulation isapplied LED arrays cooperate to transmit all data to multipleusers through the light channels and users can detect all thelight rays leading to the IUI that may cause severe systemperformance degradation Consequently it is necessary tosuppress the interference

21 Transmitter Model The white LED which has the Lam-bertian radiation pattern is chosen as the lighting sourcefor the indoor VLC application The radiant intensity can beexpressed as

119877119900 (120601) = (119898 + 1)2120587 cos119898 (120601) (1)

where 119898 = ln 2 ln (cosΦ12) is the order of Lambertianemission determined by the semiangle for half illuminanceof the LED Φ12 and 120601 is the angle of emission relative to theoptical axis of the LED array

The LED output signals can be written as

x = 119870sum119896=1

W119896s119896 + p (2)

where the original signal for 119896th user s119896 isin R119897times1 is independentand identically distributed (iid) and bounded in [minus1 1]withthe covariance matrix Rs119896 = Es119896s119879119896 = 1205902119904 I 1205902119904 is the varianceof the input signal and 119897 represents the length of data streamW119896 isin R119873times119897 denotes the transmitter precoder and the DC biasvector p = [1199011 119901119899 119901119873]119879 isin R119873times1 is added to satisfythe nonnegativity and linear-range requirement of LED inputsignals

Wireless Communications and Mobile Computing 3

Let 119901min and 119901max be the lower and upper bound of LEDlinear region the optical power constraints for transmittersignals can be written as

119901119899 minus119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 ⩾ 119901min119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 + 119901119899 ⩽ 119901max

119899 = 1 119873

(3)

where w119899119896 isin R1times119897 denotes the 119899th row vector for 119896th precod-ing matrixW119896

As a result the optical power constraint imposed ontransmitting precoder can be rewritten as

119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 =119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (4)

where 119901119905 = min119901119899 minus 119901min 119901max minus 119901119899 and e119899 denotesa zero vector with 119899th element being one that is e119899 =[01times(119899minus1) 1 01times(119873minus119899)]11987922 VLC Channel Model Several well-known channel mod-els are used in the optical wireless transmission such as thelognormal model [15 16] and the indoor practical opticalchannel model [6 17 18] The lognormal model is used tomodel free space optical (FSO) channel In this work theindoor practical optical channel model is adopted

Assuming that there are no walls in the research model(eg an inner part of a larger indoor environment) and nobarriers between transmitters and receivers the reflectionsfrom walls can be ignored and only the light-of-sight (LOS)links are considered Therefore the VLC channel matrixmodel can be described as

H119896 =[[[[[[[[

ℎ11989611 ℎ11989612 ℎ1198961119873ℎ11989621 ℎ11989622 ℎ1198962119873 d

ℎ1198961198721 ℎ1198961198722 ℎ119896119872119873

]]]]]]]]

(5)

where each element ℎ119896119898119899 of channel matrix H119896 isin R119872times119873

denotes the DC gain between119898th PD of the 119896th user and 119899thLED array It can be presented as

ℎ119896119898119899

= sum 119860119877119900 (120601119898119899)

1198892119898119899 119892 (120595119898119899) cos (120595119898119899) 0 ⩽ 120595119898119899 ⩽ Ψ0 120595119898119899 gt Ψ

(6)

where 119860 is the detector area of the receiver and 119889119898119899 is thedistance between the central point of the 119899th LED array and119898th PD 120595119898119899 denotes the incidence angle relative to PD and

Ψ represents the optical field of view (FOV) of PD The gainof optical concentrator 119892(120595119898119899) is defined as

119892 (120595119898119899) =

1205812sin2 (Ψ) 0 ⩽ 120595119898119899 ⩽ Ψ0 120595119898119899 gt Ψ

(7)

with the refractive index of concentrator 120581For convenience of analysis the arrayed LED transmitter

is simplified as a point lighting source having the same powerwith the actual setting where the LED spacing is fixed as1 cm for simulationsMoreover it is validated that even in thelimit case with 24 cm LED spacing the maximum deviationof optical path loss is still acceptable [19] Therefore thesimplification of arrayed LED transmitters for VLC channelmodeling in this work is reasonable

23 Receiver Model At the 119896th user terminal the receivedsignal vector can be written as

y119896 = H119896x + n119896 = H119896W119896s119896 +H119896sum119895 =119896

W119895s119895 +H119896p + n119896 (8)

where n119896 isin R119872times1 is the additive white Gaussian noise withzero mean and covariance matrix Rn119896 = En119896n119879119896 = 1205902119899I 1205902119899is the receiver noise variance The second term denotes theIUI which involves the interference signal s119895 (119895 = 119896) Thecorresponding received data of the 119896th user can be presentedas

r119896 = G119896y119896 = G119896H119896119870sum119895=1

W119895s119895 + G119896H119896p + G119896n119896 (9)

where G119896 isin R119897times119872 is defined as the linear equalizer

3 Optimal Transceiver Design with PerfectChannel State Information

In order to recover the original source information andsubtract the influence of the IUI efficiently the precoder andequalizer are jointly optimized based on MMSE criteria Thecorresponding MSE of the 119896th user can be written as

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896 1003817100381710038171003817r119896 minus G119896H119896p minus s119896

10038171003817100381710038172F= tr ((G119896H119896W119896 minus I)Rs119896 (G119896H119896W119896 minus I)119879)+ sum119895 =119896

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879)

+ tr (G119896Rn119896G119879119896 )

4 Wireless Communications and Mobile Computing

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

(10)

In this work we strive to minimize the maximum MSEover all users under the optical power constraint (4) that is

minW119895G119896

max119896=1119870

MSE119896 (11a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (11b)

However the objective function is nonconvex because ofthe multiple optimum variables W119895 and G119896 As a resultthe alternating minimization approach is introduced anddescribed briefly in the following steps

(a) By fixing the precoders W119895 the optimal equalizerGlowast119896 can be obtained by taking the differential of the objectivefunction (11a) with respect to G119896 that is

Glowast119896 = Rs119896 (H119896W119896)119879sum119870119895=1H119896W119895Rs119895 (H119896W119895)119879 + Rn119896

(12)

(b) From the relationship between the Kronecker productand the vector operator vec(BAC) = (C119879 otimes B)vec(A) thefirst term of the MSE function (10) can be reformulated asa quadratic form that is

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879)= tr (W119895Rs119895W

119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 vec (Rs119895W119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

(13)

(c) Assuming that the maximumMSE value of all K usersis 120576 and combining (10) and (13) the optimization problem(11a) and (11b) can be rewritten as follows

minW119895120576

120576 (14a)

st 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + tr (Rs119896)

+ tr (G119896Rn119896G119879119896 ) ⩽ 120576 119896 = 1 119870

(14b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (14c)

which is convex with respect to W119895 and can be solveddirectly by CVX Details of the reformulation process can befound in Appendix A

The iterative algorithm is stopped when the differencebetween the optimal solutions of two successive iterationsis less than a predefined threshold or the iteration numberreaches the predefined default maximum number

Each step of the proposed approach is summarized inAlgorithm 1 explicitly

4 Robust Transceiver Design with ImperfectChannel State Information

In practice the CSI may not be perfectly known and theestimation errors may exist which could lead to the severedegradation of systemperformance As a result it is necessaryto develop a robust design against imperfect CSI over theindoor MU-MIMO VLC systems

Considering the channel imperfection the channel can beexpressed as

H = H + ΔH (15)

where H isin R119872times119873 and ΔH isin R119872times119873 represent the esti-mated channel information and the channel uncertaintyrespectively A Gaussian-Kronecker model is used to presentthe channel uncertainty ΔH that is

ΔH simN (0 1205902119890Σ otimesΨ) (16)

with the variance 1205902119890 of channel imperfection the rowcovariance matrix Σ isin R119872times119872 and the column covariancematrix Ψ isin R119873times119873 Hence the channel matrix H satisfiesthe same model given by H sim N(H 1205902119890Σ otimes Ψ) which hasa property of matrix variate normal distribution [20] that is

EH (HAH119879) = 1205902119890 tr (A119879Ψ)Σ + HAH119879 (17)

with any constant matrix A isin R119873times119873

Wireless Communications and Mobile Computing 5

(1) Initialize precoders W0119895 within constraint (4) and set the DC bias p Let 119894 = 0(2) Repeat(3) Update G119894119896 with given W119894119895 according to (12)(4) With G119894119896 solve problem (14a) (14b) and (14c) by using CVX to obtain the optimal precoders W119894+1119895 (5) 119894 = 119894 + 1(6) Until W119894119896 minusW119894minus1119896 2F ⩽ 120598 or 119894 = 119894max where 120598 is a predefined threshold and 119894max is the predefined default max iteration number

Algorithm 1 Iterative algorithm for joint design of precoders and equalizers

Under the assumption of channel imperfection the MSEcan be written as follows

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896H119896 10038171003817100381710038171003817r119896 minus G119896H119896p minus s119896

100381710038171003817100381710038172

F

= EH119896119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)

+ tr (G119896ΔH119896pp119879ΔH119879119896G119879119896 ) + tr (G119896Rn119896G119879119896 )

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

(18)

The IUI can be suppressed by minimizing the maximumMSE under the power constraint that is

minW119895G119896

max119896=1119870

MSE119896 (19a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (19b)

which is nonconvex due to the simultaneous optimization ofW119895 andG119896 Following the iterative algorithm in the perfectCSI scenario the optimal solutions can be achieved by thefollowing steps

(a) With the fixed W119895 the equalizerGlowast119896 for 119896th user cansimilarly rewritten as

Glowast119896 = Rs119896 (H119896W119896)119879 (Q + 1205902119890 tr (pp119879Ψ)Σ + Rn119896)minus1 (20)

where

Q = 119870sum119895=1

(1205902119890 tr (W119895Rs119895W119879119895Ψ)Σ + H119896W119895Rs119895W

119879119895 H119879119896) (21)

(b) Under a similar transformation the first and secondterms of (18) can be rewritten as

tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

= tr (W119895Rs119895W119879119895Ψ) tr (G119896ΣG119879119896 )

= vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 ) (22)

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 ) (23)

respectively(c) Introducing the maximum MSE value 120576 of K users

and substituting (22) and (23) into (18) the robust max-MSEminimization problem can be presented in (24a) (24b) and(24c) and solved via the CVX tool A detailed reformulationprocess can be found in Appendix B

minW119895

120576 (24a)

st 119870sum119895=1

1205902119890 vec (W119895119879)119879 (Ψ119879 otimes Rs119895) vec (W119895119879) tr (G119896ΣG119879119896 ) +119870sum119895=1

vec (W119895119879)119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879)

+ tr (Rs119896) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G119879119896 ) ⩽ 120576

(24b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (24c)

6 Wireless Communications and Mobile Computing

Table 1 Simulation parameters

Description ValueNumber of LEDs per array 3600LED pitch 001mData rate 100MbitsModulation OOKLED array pitch 2mLED optical linear region [0 100W]Transmitter semiangle 70 degReceiver field of view (half angle) 70 degNumber of PDs per user terminal 2Refractive index of concentrator 120581 15Physical area of PD 119860 10 cm2

PD pitch 001m

Table 2 The location examples of two users (m)

Case 1 [25 25 08] [49 49 08]Case 2 [15 15 08] [35 35 08]Case 3 [25 25 08] [35 35 08]

The iterative process continues until it converges or themaximum predefined number is achieved

5 Simulation Results

This section presents the simulation results of the proposeddesign under a 5 times 5 times 3m3 inner part of a large indoor spaceenvironment There are four LED arrays on the ceiling withthe positions [15 15 3] [35 15 3] [15 35 3] [35 25 3]and two stationary users in the room The entries of thecovariance matrices in (16) can be chosen as [16]

[Ψ]119894119895 = 120572|119894minus119895| 119894 119895 = 1 119873[Σ]119894119895 = 120573|119894minus119895| 119894 119895 = 1 119872

(25)

where 0 lt 120572 120573 lt 1 are correlation coefficients Unlessotherwise specified the rest of the parameters used andthe user location examples are specified in Tables 1 and 2respectively

51 Convergence Performance The convergence of the pro-posed iterative algorithm is presented in Figure 2 with thelocation of case 2 The optical power constraint is set as119901119899 = 10W and other parameters are set as 120572 = 05 120573 =06 1205902119904 = 1 and 1205902119899 = 1 The proposed iterative algorithmconverges to the optimum point with limited iterations It isobvious that the sum-MSE performance with perfect CSI isbetter than the one with imperfect CSI under the same powerconstraint

52 Robustness Performance Figure 3 investigates the robust-ness of the proposed algorithm with the same parametersDue to the fact that the channel coefficients between arrayedLED transmitters and receivers are in the order of 10minus1the estimation error variances can be chosen from 10minus4 to

Iteration index

Sum

MSE

101

100

10minus1

10minus2

5 10 15 20 25

Robust max-MSE minimization with 2e = 001

Max-MSE minimization with 2e = 0

Figure 2 Convergence performance of the proposed iterative algo-rithm

Sum

MSE

Robust max-MSE minimization with imperfect CSINon-robust max-MSE minimization with imperfect

101

102

100

10minus1

10minus3 10minus210minus2

10minus1

Channel estimation error 2e

CSI

Figure 3 The robustness of the proposed algorithm with 120572 = 05120573 = 06 and 119901119899 = 10W

1 for this simulation When the channel estimation error1205902119890 is small the robust scheme and the nonrobust schemehave almost the same performance With the increase of thechannel estimation error the sumMSE of the robust schemeand the nonrobust scheme increase However the robustscheme outperforms the onewithout considering the channelimperfections It is concluded that the MSE performance issensitive to the large CSI error and the robust scheme canprovide robustness against the channel uncertainties

53 MSE Fairness Performance The sum-MSE performancesover case 2 are illustrated in Figure 4 where the shortforms ldquomin-max-MSErdquo ldquomin-sum-MSErdquo and ldquoBD schemerdquorepresent the proposed max-MSE minimization approachthe sum-MSE minimization algorithm [12] and the max-MSE minimization scheme with the assistance of block

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

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Wireless Communications and Mobile Computing 3

Let 119901min and 119901max be the lower and upper bound of LEDlinear region the optical power constraints for transmittersignals can be written as

119901119899 minus119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 ⩾ 119901min119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 + 119901119899 ⩽ 119901max

119899 = 1 119873

(3)

where w119899119896 isin R1times119897 denotes the 119899th row vector for 119896th precod-ing matrixW119896

As a result the optical power constraint imposed ontransmitting precoder can be rewritten as

119870sum119896=1

1003816100381610038161003816w1198991198961003816100381610038161003816 =119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (4)

where 119901119905 = min119901119899 minus 119901min 119901max minus 119901119899 and e119899 denotesa zero vector with 119899th element being one that is e119899 =[01times(119899minus1) 1 01times(119873minus119899)]11987922 VLC Channel Model Several well-known channel mod-els are used in the optical wireless transmission such as thelognormal model [15 16] and the indoor practical opticalchannel model [6 17 18] The lognormal model is used tomodel free space optical (FSO) channel In this work theindoor practical optical channel model is adopted

Assuming that there are no walls in the research model(eg an inner part of a larger indoor environment) and nobarriers between transmitters and receivers the reflectionsfrom walls can be ignored and only the light-of-sight (LOS)links are considered Therefore the VLC channel matrixmodel can be described as

H119896 =[[[[[[[[

ℎ11989611 ℎ11989612 ℎ1198961119873ℎ11989621 ℎ11989622 ℎ1198962119873 d

ℎ1198961198721 ℎ1198961198722 ℎ119896119872119873

]]]]]]]]

(5)

where each element ℎ119896119898119899 of channel matrix H119896 isin R119872times119873

denotes the DC gain between119898th PD of the 119896th user and 119899thLED array It can be presented as

ℎ119896119898119899

= sum 119860119877119900 (120601119898119899)

1198892119898119899 119892 (120595119898119899) cos (120595119898119899) 0 ⩽ 120595119898119899 ⩽ Ψ0 120595119898119899 gt Ψ

(6)

where 119860 is the detector area of the receiver and 119889119898119899 is thedistance between the central point of the 119899th LED array and119898th PD 120595119898119899 denotes the incidence angle relative to PD and

Ψ represents the optical field of view (FOV) of PD The gainof optical concentrator 119892(120595119898119899) is defined as

119892 (120595119898119899) =

1205812sin2 (Ψ) 0 ⩽ 120595119898119899 ⩽ Ψ0 120595119898119899 gt Ψ

(7)

with the refractive index of concentrator 120581For convenience of analysis the arrayed LED transmitter

is simplified as a point lighting source having the same powerwith the actual setting where the LED spacing is fixed as1 cm for simulationsMoreover it is validated that even in thelimit case with 24 cm LED spacing the maximum deviationof optical path loss is still acceptable [19] Therefore thesimplification of arrayed LED transmitters for VLC channelmodeling in this work is reasonable

23 Receiver Model At the 119896th user terminal the receivedsignal vector can be written as

y119896 = H119896x + n119896 = H119896W119896s119896 +H119896sum119895 =119896

W119895s119895 +H119896p + n119896 (8)

where n119896 isin R119872times1 is the additive white Gaussian noise withzero mean and covariance matrix Rn119896 = En119896n119879119896 = 1205902119899I 1205902119899is the receiver noise variance The second term denotes theIUI which involves the interference signal s119895 (119895 = 119896) Thecorresponding received data of the 119896th user can be presentedas

r119896 = G119896y119896 = G119896H119896119870sum119895=1

W119895s119895 + G119896H119896p + G119896n119896 (9)

where G119896 isin R119897times119872 is defined as the linear equalizer

3 Optimal Transceiver Design with PerfectChannel State Information

In order to recover the original source information andsubtract the influence of the IUI efficiently the precoder andequalizer are jointly optimized based on MMSE criteria Thecorresponding MSE of the 119896th user can be written as

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896 1003817100381710038171003817r119896 minus G119896H119896p minus s119896

10038171003817100381710038172F= tr ((G119896H119896W119896 minus I)Rs119896 (G119896H119896W119896 minus I)119879)+ sum119895 =119896

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879)

+ tr (G119896Rn119896G119879119896 )

4 Wireless Communications and Mobile Computing

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

(10)

In this work we strive to minimize the maximum MSEover all users under the optical power constraint (4) that is

minW119895G119896

max119896=1119870

MSE119896 (11a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (11b)

However the objective function is nonconvex because ofthe multiple optimum variables W119895 and G119896 As a resultthe alternating minimization approach is introduced anddescribed briefly in the following steps

(a) By fixing the precoders W119895 the optimal equalizerGlowast119896 can be obtained by taking the differential of the objectivefunction (11a) with respect to G119896 that is

Glowast119896 = Rs119896 (H119896W119896)119879sum119870119895=1H119896W119895Rs119895 (H119896W119895)119879 + Rn119896

(12)

(b) From the relationship between the Kronecker productand the vector operator vec(BAC) = (C119879 otimes B)vec(A) thefirst term of the MSE function (10) can be reformulated asa quadratic form that is

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879)= tr (W119895Rs119895W

119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 vec (Rs119895W119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

(13)

(c) Assuming that the maximumMSE value of all K usersis 120576 and combining (10) and (13) the optimization problem(11a) and (11b) can be rewritten as follows

minW119895120576

120576 (14a)

st 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + tr (Rs119896)

+ tr (G119896Rn119896G119879119896 ) ⩽ 120576 119896 = 1 119870

(14b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (14c)

which is convex with respect to W119895 and can be solveddirectly by CVX Details of the reformulation process can befound in Appendix A

The iterative algorithm is stopped when the differencebetween the optimal solutions of two successive iterationsis less than a predefined threshold or the iteration numberreaches the predefined default maximum number

Each step of the proposed approach is summarized inAlgorithm 1 explicitly

4 Robust Transceiver Design with ImperfectChannel State Information

In practice the CSI may not be perfectly known and theestimation errors may exist which could lead to the severedegradation of systemperformance As a result it is necessaryto develop a robust design against imperfect CSI over theindoor MU-MIMO VLC systems

Considering the channel imperfection the channel can beexpressed as

H = H + ΔH (15)

where H isin R119872times119873 and ΔH isin R119872times119873 represent the esti-mated channel information and the channel uncertaintyrespectively A Gaussian-Kronecker model is used to presentthe channel uncertainty ΔH that is

ΔH simN (0 1205902119890Σ otimesΨ) (16)

with the variance 1205902119890 of channel imperfection the rowcovariance matrix Σ isin R119872times119872 and the column covariancematrix Ψ isin R119873times119873 Hence the channel matrix H satisfiesthe same model given by H sim N(H 1205902119890Σ otimes Ψ) which hasa property of matrix variate normal distribution [20] that is

EH (HAH119879) = 1205902119890 tr (A119879Ψ)Σ + HAH119879 (17)

with any constant matrix A isin R119873times119873

Wireless Communications and Mobile Computing 5

(1) Initialize precoders W0119895 within constraint (4) and set the DC bias p Let 119894 = 0(2) Repeat(3) Update G119894119896 with given W119894119895 according to (12)(4) With G119894119896 solve problem (14a) (14b) and (14c) by using CVX to obtain the optimal precoders W119894+1119895 (5) 119894 = 119894 + 1(6) Until W119894119896 minusW119894minus1119896 2F ⩽ 120598 or 119894 = 119894max where 120598 is a predefined threshold and 119894max is the predefined default max iteration number

Algorithm 1 Iterative algorithm for joint design of precoders and equalizers

Under the assumption of channel imperfection the MSEcan be written as follows

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896H119896 10038171003817100381710038171003817r119896 minus G119896H119896p minus s119896

100381710038171003817100381710038172

F

= EH119896119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)

+ tr (G119896ΔH119896pp119879ΔH119879119896G119879119896 ) + tr (G119896Rn119896G119879119896 )

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

(18)

The IUI can be suppressed by minimizing the maximumMSE under the power constraint that is

minW119895G119896

max119896=1119870

MSE119896 (19a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (19b)

which is nonconvex due to the simultaneous optimization ofW119895 andG119896 Following the iterative algorithm in the perfectCSI scenario the optimal solutions can be achieved by thefollowing steps

(a) With the fixed W119895 the equalizerGlowast119896 for 119896th user cansimilarly rewritten as

Glowast119896 = Rs119896 (H119896W119896)119879 (Q + 1205902119890 tr (pp119879Ψ)Σ + Rn119896)minus1 (20)

where

Q = 119870sum119895=1

(1205902119890 tr (W119895Rs119895W119879119895Ψ)Σ + H119896W119895Rs119895W

119879119895 H119879119896) (21)

(b) Under a similar transformation the first and secondterms of (18) can be rewritten as

tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

= tr (W119895Rs119895W119879119895Ψ) tr (G119896ΣG119879119896 )

= vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 ) (22)

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 ) (23)

respectively(c) Introducing the maximum MSE value 120576 of K users

and substituting (22) and (23) into (18) the robust max-MSEminimization problem can be presented in (24a) (24b) and(24c) and solved via the CVX tool A detailed reformulationprocess can be found in Appendix B

minW119895

120576 (24a)

st 119870sum119895=1

1205902119890 vec (W119895119879)119879 (Ψ119879 otimes Rs119895) vec (W119895119879) tr (G119896ΣG119879119896 ) +119870sum119895=1

vec (W119895119879)119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879)

+ tr (Rs119896) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G119879119896 ) ⩽ 120576

(24b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (24c)

6 Wireless Communications and Mobile Computing

Table 1 Simulation parameters

Description ValueNumber of LEDs per array 3600LED pitch 001mData rate 100MbitsModulation OOKLED array pitch 2mLED optical linear region [0 100W]Transmitter semiangle 70 degReceiver field of view (half angle) 70 degNumber of PDs per user terminal 2Refractive index of concentrator 120581 15Physical area of PD 119860 10 cm2

PD pitch 001m

Table 2 The location examples of two users (m)

Case 1 [25 25 08] [49 49 08]Case 2 [15 15 08] [35 35 08]Case 3 [25 25 08] [35 35 08]

The iterative process continues until it converges or themaximum predefined number is achieved

5 Simulation Results

This section presents the simulation results of the proposeddesign under a 5 times 5 times 3m3 inner part of a large indoor spaceenvironment There are four LED arrays on the ceiling withthe positions [15 15 3] [35 15 3] [15 35 3] [35 25 3]and two stationary users in the room The entries of thecovariance matrices in (16) can be chosen as [16]

[Ψ]119894119895 = 120572|119894minus119895| 119894 119895 = 1 119873[Σ]119894119895 = 120573|119894minus119895| 119894 119895 = 1 119872

(25)

where 0 lt 120572 120573 lt 1 are correlation coefficients Unlessotherwise specified the rest of the parameters used andthe user location examples are specified in Tables 1 and 2respectively

51 Convergence Performance The convergence of the pro-posed iterative algorithm is presented in Figure 2 with thelocation of case 2 The optical power constraint is set as119901119899 = 10W and other parameters are set as 120572 = 05 120573 =06 1205902119904 = 1 and 1205902119899 = 1 The proposed iterative algorithmconverges to the optimum point with limited iterations It isobvious that the sum-MSE performance with perfect CSI isbetter than the one with imperfect CSI under the same powerconstraint

52 Robustness Performance Figure 3 investigates the robust-ness of the proposed algorithm with the same parametersDue to the fact that the channel coefficients between arrayedLED transmitters and receivers are in the order of 10minus1the estimation error variances can be chosen from 10minus4 to

Iteration index

Sum

MSE

101

100

10minus1

10minus2

5 10 15 20 25

Robust max-MSE minimization with 2e = 001

Max-MSE minimization with 2e = 0

Figure 2 Convergence performance of the proposed iterative algo-rithm

Sum

MSE

Robust max-MSE minimization with imperfect CSINon-robust max-MSE minimization with imperfect

101

102

100

10minus1

10minus3 10minus210minus2

10minus1

Channel estimation error 2e

CSI

Figure 3 The robustness of the proposed algorithm with 120572 = 05120573 = 06 and 119901119899 = 10W

1 for this simulation When the channel estimation error1205902119890 is small the robust scheme and the nonrobust schemehave almost the same performance With the increase of thechannel estimation error the sumMSE of the robust schemeand the nonrobust scheme increase However the robustscheme outperforms the onewithout considering the channelimperfections It is concluded that the MSE performance issensitive to the large CSI error and the robust scheme canprovide robustness against the channel uncertainties

53 MSE Fairness Performance The sum-MSE performancesover case 2 are illustrated in Figure 4 where the shortforms ldquomin-max-MSErdquo ldquomin-sum-MSErdquo and ldquoBD schemerdquorepresent the proposed max-MSE minimization approachthe sum-MSE minimization algorithm [12] and the max-MSE minimization scheme with the assistance of block

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

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4 Wireless Communications and Mobile Computing

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

(10)

In this work we strive to minimize the maximum MSEover all users under the optical power constraint (4) that is

minW119895G119896

max119896=1119870

MSE119896 (11a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (11b)

However the objective function is nonconvex because ofthe multiple optimum variables W119895 and G119896 As a resultthe alternating minimization approach is introduced anddescribed briefly in the following steps

(a) By fixing the precoders W119895 the optimal equalizerGlowast119896 can be obtained by taking the differential of the objectivefunction (11a) with respect to G119896 that is

Glowast119896 = Rs119896 (H119896W119896)119879sum119870119895=1H119896W119895Rs119895 (H119896W119895)119879 + Rn119896

(12)

(b) From the relationship between the Kronecker productand the vector operator vec(BAC) = (C119879 otimes B)vec(A) thefirst term of the MSE function (10) can be reformulated asa quadratic form that is

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879)= tr (W119895Rs119895W

119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 vec (Rs119895W119879119895H119879119896G119879119896G119896H119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

(13)

(c) Assuming that the maximumMSE value of all K usersis 120576 and combining (10) and (13) the optimization problem(11a) and (11b) can be rewritten as follows

minW119895120576

120576 (14a)

st 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + tr (Rs119896)

+ tr (G119896Rn119896G119879119896 ) ⩽ 120576 119896 = 1 119870

(14b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (14c)

which is convex with respect to W119895 and can be solveddirectly by CVX Details of the reformulation process can befound in Appendix A

The iterative algorithm is stopped when the differencebetween the optimal solutions of two successive iterationsis less than a predefined threshold or the iteration numberreaches the predefined default maximum number

Each step of the proposed approach is summarized inAlgorithm 1 explicitly

4 Robust Transceiver Design with ImperfectChannel State Information

In practice the CSI may not be perfectly known and theestimation errors may exist which could lead to the severedegradation of systemperformance As a result it is necessaryto develop a robust design against imperfect CSI over theindoor MU-MIMO VLC systems

Considering the channel imperfection the channel can beexpressed as

H = H + ΔH (15)

where H isin R119872times119873 and ΔH isin R119872times119873 represent the esti-mated channel information and the channel uncertaintyrespectively A Gaussian-Kronecker model is used to presentthe channel uncertainty ΔH that is

ΔH simN (0 1205902119890Σ otimesΨ) (16)

with the variance 1205902119890 of channel imperfection the rowcovariance matrix Σ isin R119872times119872 and the column covariancematrix Ψ isin R119873times119873 Hence the channel matrix H satisfiesthe same model given by H sim N(H 1205902119890Σ otimes Ψ) which hasa property of matrix variate normal distribution [20] that is

EH (HAH119879) = 1205902119890 tr (A119879Ψ)Σ + HAH119879 (17)

with any constant matrix A isin R119873times119873

Wireless Communications and Mobile Computing 5

(1) Initialize precoders W0119895 within constraint (4) and set the DC bias p Let 119894 = 0(2) Repeat(3) Update G119894119896 with given W119894119895 according to (12)(4) With G119894119896 solve problem (14a) (14b) and (14c) by using CVX to obtain the optimal precoders W119894+1119895 (5) 119894 = 119894 + 1(6) Until W119894119896 minusW119894minus1119896 2F ⩽ 120598 or 119894 = 119894max where 120598 is a predefined threshold and 119894max is the predefined default max iteration number

Algorithm 1 Iterative algorithm for joint design of precoders and equalizers

Under the assumption of channel imperfection the MSEcan be written as follows

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896H119896 10038171003817100381710038171003817r119896 minus G119896H119896p minus s119896

100381710038171003817100381710038172

F

= EH119896119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)

+ tr (G119896ΔH119896pp119879ΔH119879119896G119879119896 ) + tr (G119896Rn119896G119879119896 )

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

(18)

The IUI can be suppressed by minimizing the maximumMSE under the power constraint that is

minW119895G119896

max119896=1119870

MSE119896 (19a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (19b)

which is nonconvex due to the simultaneous optimization ofW119895 andG119896 Following the iterative algorithm in the perfectCSI scenario the optimal solutions can be achieved by thefollowing steps

(a) With the fixed W119895 the equalizerGlowast119896 for 119896th user cansimilarly rewritten as

Glowast119896 = Rs119896 (H119896W119896)119879 (Q + 1205902119890 tr (pp119879Ψ)Σ + Rn119896)minus1 (20)

where

Q = 119870sum119895=1

(1205902119890 tr (W119895Rs119895W119879119895Ψ)Σ + H119896W119895Rs119895W

119879119895 H119879119896) (21)

(b) Under a similar transformation the first and secondterms of (18) can be rewritten as

tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

= tr (W119895Rs119895W119879119895Ψ) tr (G119896ΣG119879119896 )

= vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 ) (22)

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 ) (23)

respectively(c) Introducing the maximum MSE value 120576 of K users

and substituting (22) and (23) into (18) the robust max-MSEminimization problem can be presented in (24a) (24b) and(24c) and solved via the CVX tool A detailed reformulationprocess can be found in Appendix B

minW119895

120576 (24a)

st 119870sum119895=1

1205902119890 vec (W119895119879)119879 (Ψ119879 otimes Rs119895) vec (W119895119879) tr (G119896ΣG119879119896 ) +119870sum119895=1

vec (W119895119879)119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879)

+ tr (Rs119896) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G119879119896 ) ⩽ 120576

(24b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (24c)

6 Wireless Communications and Mobile Computing

Table 1 Simulation parameters

Description ValueNumber of LEDs per array 3600LED pitch 001mData rate 100MbitsModulation OOKLED array pitch 2mLED optical linear region [0 100W]Transmitter semiangle 70 degReceiver field of view (half angle) 70 degNumber of PDs per user terminal 2Refractive index of concentrator 120581 15Physical area of PD 119860 10 cm2

PD pitch 001m

Table 2 The location examples of two users (m)

Case 1 [25 25 08] [49 49 08]Case 2 [15 15 08] [35 35 08]Case 3 [25 25 08] [35 35 08]

The iterative process continues until it converges or themaximum predefined number is achieved

5 Simulation Results

This section presents the simulation results of the proposeddesign under a 5 times 5 times 3m3 inner part of a large indoor spaceenvironment There are four LED arrays on the ceiling withthe positions [15 15 3] [35 15 3] [15 35 3] [35 25 3]and two stationary users in the room The entries of thecovariance matrices in (16) can be chosen as [16]

[Ψ]119894119895 = 120572|119894minus119895| 119894 119895 = 1 119873[Σ]119894119895 = 120573|119894minus119895| 119894 119895 = 1 119872

(25)

where 0 lt 120572 120573 lt 1 are correlation coefficients Unlessotherwise specified the rest of the parameters used andthe user location examples are specified in Tables 1 and 2respectively

51 Convergence Performance The convergence of the pro-posed iterative algorithm is presented in Figure 2 with thelocation of case 2 The optical power constraint is set as119901119899 = 10W and other parameters are set as 120572 = 05 120573 =06 1205902119904 = 1 and 1205902119899 = 1 The proposed iterative algorithmconverges to the optimum point with limited iterations It isobvious that the sum-MSE performance with perfect CSI isbetter than the one with imperfect CSI under the same powerconstraint

52 Robustness Performance Figure 3 investigates the robust-ness of the proposed algorithm with the same parametersDue to the fact that the channel coefficients between arrayedLED transmitters and receivers are in the order of 10minus1the estimation error variances can be chosen from 10minus4 to

Iteration index

Sum

MSE

101

100

10minus1

10minus2

5 10 15 20 25

Robust max-MSE minimization with 2e = 001

Max-MSE minimization with 2e = 0

Figure 2 Convergence performance of the proposed iterative algo-rithm

Sum

MSE

Robust max-MSE minimization with imperfect CSINon-robust max-MSE minimization with imperfect

101

102

100

10minus1

10minus3 10minus210minus2

10minus1

Channel estimation error 2e

CSI

Figure 3 The robustness of the proposed algorithm with 120572 = 05120573 = 06 and 119901119899 = 10W

1 for this simulation When the channel estimation error1205902119890 is small the robust scheme and the nonrobust schemehave almost the same performance With the increase of thechannel estimation error the sumMSE of the robust schemeand the nonrobust scheme increase However the robustscheme outperforms the onewithout considering the channelimperfections It is concluded that the MSE performance issensitive to the large CSI error and the robust scheme canprovide robustness against the channel uncertainties

53 MSE Fairness Performance The sum-MSE performancesover case 2 are illustrated in Figure 4 where the shortforms ldquomin-max-MSErdquo ldquomin-sum-MSErdquo and ldquoBD schemerdquorepresent the proposed max-MSE minimization approachthe sum-MSE minimization algorithm [12] and the max-MSE minimization scheme with the assistance of block

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

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Wireless Communications and Mobile Computing 5

(1) Initialize precoders W0119895 within constraint (4) and set the DC bias p Let 119894 = 0(2) Repeat(3) Update G119894119896 with given W119894119895 according to (12)(4) With G119894119896 solve problem (14a) (14b) and (14c) by using CVX to obtain the optimal precoders W119894+1119895 (5) 119894 = 119894 + 1(6) Until W119894119896 minusW119894minus1119896 2F ⩽ 120598 or 119894 = 119894max where 120598 is a predefined threshold and 119894max is the predefined default max iteration number

Algorithm 1 Iterative algorithm for joint design of precoders and equalizers

Under the assumption of channel imperfection the MSEcan be written as follows

MSE119896 (r119896 s119896 W119895119895=1119896119870 G119896)= Es119896n119896H119896 10038171003817100381710038171003817r119896 minus G119896H119896p minus s119896

100381710038171003817100381710038172

F

= EH119896119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)

+ tr (G119896ΔH119896pp119879ΔH119879119896G119879119896 ) + tr (G119896Rn119896G119879119896 )

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

(18)

The IUI can be suppressed by minimizing the maximumMSE under the power constraint that is

minW119895G119896

max119896=1119870

MSE119896 (19a)

st 119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (19b)

which is nonconvex due to the simultaneous optimization ofW119895 andG119896 Following the iterative algorithm in the perfectCSI scenario the optimal solutions can be achieved by thefollowing steps

(a) With the fixed W119895 the equalizerGlowast119896 for 119896th user cansimilarly rewritten as

Glowast119896 = Rs119896 (H119896W119896)119879 (Q + 1205902119890 tr (pp119879Ψ)Σ + Rn119896)minus1 (20)

where

Q = 119870sum119895=1

(1205902119890 tr (W119895Rs119895W119879119895Ψ)Σ + H119896W119895Rs119895W

119879119895 H119879119896) (21)

(b) Under a similar transformation the first and secondterms of (18) can be rewritten as

tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

= tr (W119895Rs119895W119879119895Ψ) tr (G119896ΣG119879119896 )

= vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 ) (22)

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896)

= vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 ) (23)

respectively(c) Introducing the maximum MSE value 120576 of K users

and substituting (22) and (23) into (18) the robust max-MSEminimization problem can be presented in (24a) (24b) and(24c) and solved via the CVX tool A detailed reformulationprocess can be found in Appendix B

minW119895

120576 (24a)

st 119870sum119895=1

1205902119890 vec (W119895119879)119879 (Ψ119879 otimes Rs119895) vec (W119895119879) tr (G119896ΣG119879119896 ) +119870sum119895=1

vec (W119895119879)119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119895119879)

+ tr (Rs119896) minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879) + 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G119879119896 ) ⩽ 120576

(24b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (24c)

6 Wireless Communications and Mobile Computing

Table 1 Simulation parameters

Description ValueNumber of LEDs per array 3600LED pitch 001mData rate 100MbitsModulation OOKLED array pitch 2mLED optical linear region [0 100W]Transmitter semiangle 70 degReceiver field of view (half angle) 70 degNumber of PDs per user terminal 2Refractive index of concentrator 120581 15Physical area of PD 119860 10 cm2

PD pitch 001m

Table 2 The location examples of two users (m)

Case 1 [25 25 08] [49 49 08]Case 2 [15 15 08] [35 35 08]Case 3 [25 25 08] [35 35 08]

The iterative process continues until it converges or themaximum predefined number is achieved

5 Simulation Results

This section presents the simulation results of the proposeddesign under a 5 times 5 times 3m3 inner part of a large indoor spaceenvironment There are four LED arrays on the ceiling withthe positions [15 15 3] [35 15 3] [15 35 3] [35 25 3]and two stationary users in the room The entries of thecovariance matrices in (16) can be chosen as [16]

[Ψ]119894119895 = 120572|119894minus119895| 119894 119895 = 1 119873[Σ]119894119895 = 120573|119894minus119895| 119894 119895 = 1 119872

(25)

where 0 lt 120572 120573 lt 1 are correlation coefficients Unlessotherwise specified the rest of the parameters used andthe user location examples are specified in Tables 1 and 2respectively

51 Convergence Performance The convergence of the pro-posed iterative algorithm is presented in Figure 2 with thelocation of case 2 The optical power constraint is set as119901119899 = 10W and other parameters are set as 120572 = 05 120573 =06 1205902119904 = 1 and 1205902119899 = 1 The proposed iterative algorithmconverges to the optimum point with limited iterations It isobvious that the sum-MSE performance with perfect CSI isbetter than the one with imperfect CSI under the same powerconstraint

52 Robustness Performance Figure 3 investigates the robust-ness of the proposed algorithm with the same parametersDue to the fact that the channel coefficients between arrayedLED transmitters and receivers are in the order of 10minus1the estimation error variances can be chosen from 10minus4 to

Iteration index

Sum

MSE

101

100

10minus1

10minus2

5 10 15 20 25

Robust max-MSE minimization with 2e = 001

Max-MSE minimization with 2e = 0

Figure 2 Convergence performance of the proposed iterative algo-rithm

Sum

MSE

Robust max-MSE minimization with imperfect CSINon-robust max-MSE minimization with imperfect

101

102

100

10minus1

10minus3 10minus210minus2

10minus1

Channel estimation error 2e

CSI

Figure 3 The robustness of the proposed algorithm with 120572 = 05120573 = 06 and 119901119899 = 10W

1 for this simulation When the channel estimation error1205902119890 is small the robust scheme and the nonrobust schemehave almost the same performance With the increase of thechannel estimation error the sumMSE of the robust schemeand the nonrobust scheme increase However the robustscheme outperforms the onewithout considering the channelimperfections It is concluded that the MSE performance issensitive to the large CSI error and the robust scheme canprovide robustness against the channel uncertainties

53 MSE Fairness Performance The sum-MSE performancesover case 2 are illustrated in Figure 4 where the shortforms ldquomin-max-MSErdquo ldquomin-sum-MSErdquo and ldquoBD schemerdquorepresent the proposed max-MSE minimization approachthe sum-MSE minimization algorithm [12] and the max-MSE minimization scheme with the assistance of block

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

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6 Wireless Communications and Mobile Computing

Table 1 Simulation parameters

Description ValueNumber of LEDs per array 3600LED pitch 001mData rate 100MbitsModulation OOKLED array pitch 2mLED optical linear region [0 100W]Transmitter semiangle 70 degReceiver field of view (half angle) 70 degNumber of PDs per user terminal 2Refractive index of concentrator 120581 15Physical area of PD 119860 10 cm2

PD pitch 001m

Table 2 The location examples of two users (m)

Case 1 [25 25 08] [49 49 08]Case 2 [15 15 08] [35 35 08]Case 3 [25 25 08] [35 35 08]

The iterative process continues until it converges or themaximum predefined number is achieved

5 Simulation Results

This section presents the simulation results of the proposeddesign under a 5 times 5 times 3m3 inner part of a large indoor spaceenvironment There are four LED arrays on the ceiling withthe positions [15 15 3] [35 15 3] [15 35 3] [35 25 3]and two stationary users in the room The entries of thecovariance matrices in (16) can be chosen as [16]

[Ψ]119894119895 = 120572|119894minus119895| 119894 119895 = 1 119873[Σ]119894119895 = 120573|119894minus119895| 119894 119895 = 1 119872

(25)

where 0 lt 120572 120573 lt 1 are correlation coefficients Unlessotherwise specified the rest of the parameters used andthe user location examples are specified in Tables 1 and 2respectively

51 Convergence Performance The convergence of the pro-posed iterative algorithm is presented in Figure 2 with thelocation of case 2 The optical power constraint is set as119901119899 = 10W and other parameters are set as 120572 = 05 120573 =06 1205902119904 = 1 and 1205902119899 = 1 The proposed iterative algorithmconverges to the optimum point with limited iterations It isobvious that the sum-MSE performance with perfect CSI isbetter than the one with imperfect CSI under the same powerconstraint

52 Robustness Performance Figure 3 investigates the robust-ness of the proposed algorithm with the same parametersDue to the fact that the channel coefficients between arrayedLED transmitters and receivers are in the order of 10minus1the estimation error variances can be chosen from 10minus4 to

Iteration index

Sum

MSE

101

100

10minus1

10minus2

5 10 15 20 25

Robust max-MSE minimization with 2e = 001

Max-MSE minimization with 2e = 0

Figure 2 Convergence performance of the proposed iterative algo-rithm

Sum

MSE

Robust max-MSE minimization with imperfect CSINon-robust max-MSE minimization with imperfect

101

102

100

10minus1

10minus3 10minus210minus2

10minus1

Channel estimation error 2e

CSI

Figure 3 The robustness of the proposed algorithm with 120572 = 05120573 = 06 and 119901119899 = 10W

1 for this simulation When the channel estimation error1205902119890 is small the robust scheme and the nonrobust schemehave almost the same performance With the increase of thechannel estimation error the sumMSE of the robust schemeand the nonrobust scheme increase However the robustscheme outperforms the onewithout considering the channelimperfections It is concluded that the MSE performance issensitive to the large CSI error and the robust scheme canprovide robustness against the channel uncertainties

53 MSE Fairness Performance The sum-MSE performancesover case 2 are illustrated in Figure 4 where the shortforms ldquomin-max-MSErdquo ldquomin-sum-MSErdquo and ldquoBD schemerdquorepresent the proposed max-MSE minimization approachthe sum-MSE minimization algorithm [12] and the max-MSE minimization scheme with the assistance of block

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wireless Communications and Mobile Computing 7

BD scheme [10]min-max-MSEmin-sum-MSE [12]

4 6 82 12 14 16 18 20 2210

DC bias pn (W)

10minus2

10minus1

100

Sum

MSE

Figure 4 SumMSE versus optical power

min-max-MSE BD scheme [10] min-sum-MSE [12]0

01

02

03

04

05

MSE

user 1 in case 1user 2 in case 1user 1 in case 2user 2 in case 2user 1 in case 3user 2 in case 3sum MSE in case 1sum MSE in case 2sum MSE in case 3

MSEcase 1case 2case 3

Fairness index

09

095

1

Fairn

ess i

ndex

Figure 5 MSE distribution and fairness performance of differentalgorithms

diagonalization (BD) algorithm [10] respectively The sim-ulation parameters are set as 120572 = 05 120573 = 06 and1205902119890 = 0001 It is shown that the sum-MSE minimizationalgorithm can achieve the lowest sum MSE since it isdesigned to minimize the total MSE without any fairnessconstraints while the proposed max-MSE minimizationapproach sacrifices a little MSE to balance the fairness Onthe other hand the BD scheme has the highest sum MSEsince it suppresses the interference by imposing the IUIon the channelrsquos null space under the limited degrees offreedom

Furthermore Figure 5 demonstrates the fairness supe-riority via the MSE distribution performances with thethree mentioned algorithms Three location configurations

Robust min-max-MSE with 4-PPMNon-robust min-max-MSE with 4-PPMRobust min-max-MSE with OOKNon-robust min-max-MSE with OOK

4 6 8 10 12 14 16 18 200 2

DC bias pn (W)

10minus5

10minus4

10minus3

10minus2

10minus1

100

Aver

age B

ER

Figure 6 BER performance of the different modulation schemeswith 1205902119890 = 0005

specified in Table 2 are considered with 119901119899 = 10W Jainrsquosfairness index is adopted as the criterion to compare thefairness which is defined as 119891(x) = (sum119870119894=1 119909119894)2(119870sum119870119894=1 1199092119894 )with 119909119894 ⩾ 0 [21] The value of 119891(x) indicates the fairness ofthe system More specifically the maximum fairness index is1 when the resources are allocated equally among all usersIf only one user dominates the resource while others receivenone the fairness index will reduce to 1119870 Therefore thefairness index is bounded between the intervals of [1119870 1]In Figure 5 the proposedmin-max-MSE algorithm performsslightly worse than the min-sum-MSE algorithm in termsof sum MSE but it offers fairer transmission for differentlocation settings The fairness index of the proposed min-max-MSE algorithm is highest which indicates that theproposed algorithm provides the best fairness performanceamong the three approaches

54 BER Performance In our simulations OOKmodulationis applied and the theoretical BER can be calculated by [22]

BER = 119876 (radicSINR) (26)

where SINR = (sumMSE119896)minus1 minus 1 [23] and 119876(sdot) representsthe 119876-function Figures 6 and 7 investigate the BER per-formance with the same parameters in Section 53 TheBER performances of OOK modulation and 4-level pulse-position modulation (4-PPM) are compared in Figure 6The BER decreases with the increase of the optical powerand the robust scheme performs better than the nonro-bust scheme The curves of nonrobust scheme becomeflat and the gaps between robust scheme and nonrobustscheme become larger This is because without consideringthe channel imperfection the DC bias estimation errorterm 1205902119890 tr(G119896tr(pp119879Ψ)ΣG119879119896 ) increases when optical power pbecomes largerThePPMmodulation technique improves thepower efficiency of the OOK modulation to achieve a given

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

8 Wireless Communications and Mobile ComputingBE

R

10minus4

10minus3

10minus2

10minus1

100

pn = 15７ 2e = 0005

pn = 5７ 2e = 0005

pn = 15７ 2e = 0001

pn = 5７ 2e = 0001

100 101 10210minus2 10minus1

Noise variance 2n

Figure 7 BER performance of the robust designwith different noisevariances

BER On the other hand the PPM modulation requires boththe slot and the symbol-level synchronization which mayincrease the receiver complexity while the OOKmodulationis simple to be implemented leading to its wide use in opticalwireless systems

Figure 7 presents the BER performance with variousnoise variance values to further investigate the performanceof the proposed robust design with the OOK modulationAs we expected the BER decreases with the reduction ofnoise variance With fixed noise variance the BER withhigher optical power is better than that with the lowerpower It is clear that under the low noise variancethe BER is only dominated by the channel uncertaintiesand its performance is degraded with the large channelimperfection

55 The Impact of Different Geometric Metrics on SystemPerformance Finally we investigate the impact of differentgeometricmetrics including usersrsquo relative positions heightsand angles in Figures 8 and 9 It is assumed that oneuser is fixed at the position of [25 25 08] and anotheruser moves around on the same receive plane As expectedthe MSE performance becomes worse when two users aregetting closer shown in Figure 8 due to the negative channelcorrelation

Besides Figure 9 illustrates the MSE performance withvarying height of the receiver plane and receiverrsquos FOV Theusersrsquo horizontal coordinates are set as [25 25] and [45 45]In this situation it is nearly impossible to receive opticalsignals with small angles or on the plane of high level wherethe communication link is interrupted When the FOV islarge enough to receive all the transmitted optical links itis hard to distinguish the signals for different users in thereceivers which reduces the ability for signal recovery anddegrades the MSE performance

0

01

5

02

03

4

MSE

04

5

05

3

06

Room Y-axis (m)

42 3

Room X-axis (m)21 10 0

Figure 8 MSE distribution of different positions

Users height (m)

0

10

20

30

40

50

60

70

80

90

Rece

iver

fiel

d of

vie

w (d

eg)

004

005

006

007

008

009

01

011

0 05 1 15 2 25 3

Figure 9 Impact of userrsquos height and angles on MSE performance

6 Conclusion

In this work the optimal transceivers are designed for theindoor MU-MIMO VLC system where the max-MSE min-imization approach is developed to suppress the IUI Due tothe channel uncertainties in practice the mismatch betweenthe estimated channel information and the perfect channelinformation may severely degrade the system performanceTherefore the robust design is investigated against the chan-nel uncertainties The simulation results demonstrate thatthe robust design can provide robustness against the channelimperfections and gains a considerable BER performanceMoreover the proposed algorithm is verified to achievealmost the same performance of the sum-MSE minimizationalgorithm but guaranteeing the fairness among the multipleusers The influences of different geometric metrics are alsodiscussed in simulation results and it is demonstrated thatthe channel correlation the distance between transmittersand receivers and the receiver FOV are the decisive factorsfor performance improvement

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Wireless Communications and Mobile Computing 9

Appendix

A Formulation of Problem (14a)(14b) and (14c)

By introducing an auxiliary variable 120576 that serves an upperbound on the MSE value of 119896th user the optimizationproblem (11a) and (11b) can be transformed as

minW119895120576

120576 (A1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (A1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (A1c)

Using the transformation of (13) the MSE can be rewrit-ten as

MSE119896

= 119870sum119895=1

tr (G119896H119896W119895Rs119895 (G119896H119896W119895)119879) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ tr (G119896Rn119896G

119879119896 ) + tr (Rs119896)

(A2)

Applying (A2) on (A1a) (A1b) and (A1c) we canobtain optimization problem (14a) (14b) and (14c)

B Formulation of Problem (24a)(24b) and (24c)

Similar to the formulation of problem (14a) (14b) and (14c)the robust max-MSE minimization problem (19a) and (19b)can be rewritten as

minW119895120576

120576 (B1a)

st MSE119896 ⩽ 120576 119896 = 1 119870 (B1b)

119870sum119896=1

10038171003817100381710038171003817W119879119896 e119899100381710038171003817100381710038171 ⩽ 119901119905 119899 = 1 119873 (B1c)

where the corresponding MSE can be reformed as

MSE119896

= 119870sum119895=1

1205902119890 tr (G119896tr (W119895Rs119895W119879119895Ψ)ΣG119879119896)

+ 119870sum119895=1

tr (G119896H119896W119895Rs119895W119879119895 H119879119896G119879119896) + tr (Rs119896)

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

= 119870sum119895=1

vec (W119879119895 )119879 (Ψ119879 otimes Rs119895) vec (W119879119895 ) tr (G119896ΣG119879119896 )

+ 119870sum119895=1

vec (W119879119895 )119879 (H119879119896G119879119896G119896H119896 otimes Rs119895) vec (W119879119895 )

minus tr (G119896H119896W119896Rs119896) minus tr (Rs119896 (G119896H119896W119896)119879)+ 1205902119890 tr (G119896tr (pp119879Ψ)ΣG119879119896 ) + tr (G119896Rn119896G

119879119896 )

+ tr (Rs119896)

(B2)

leading to (24a) (24b) and (24c)

Data Availability

All relevant data are within the paper

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

The work of Zhirong Zeng and Huiqin Du is supported bythe National Natural Science Foundation of China (Grant no61401178) and the Open Research Fund of National MobileCommunications Research Laboratory Southeast University(Grant no 2017D12) The work of Zujian Wu is supportedby the National Natural Science Foundation of China (Grantno 61602209) and the Fundamental Research Funds for theCentral Universities (Grant no 21615316)

References

[1] P H Pathak X Feng P Hu and P Mohapatra ldquoVisible lightcommunication networking and sensing a survey potentialand challengesrdquo IEEE Communications Surveys amp Tutorials vol17 no 4 pp 2047ndash2077 2015

[2] S Zhao ldquoA serial concatenation-based coding scheme fordimmable visible light communication systemsrdquo IEEE Commu-nications Letters vol 20 no 10 pp 1951ndash1954 2016

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Wireless Communications and Mobile Computing

[3] R Shaaban and S Faruque ldquoA survey of indoor visible lightcommunication power distribution and color shift keyingtransmissionrdquo in Proceedings of the 2017 IEEE InternationalConference on Electro Information Technology EIT 2017 pp149ndash153 Lincoln Neb USA 2017

[4] M K Jha A Addanki Y V S Lakshmi and N Kumar ldquoChan-nel coding performance of optical MIMO indoor visible lightcommunicationrdquo in Proceedings of the International Conferenceon Advances in Computing Communications and InformaticsICACCI 2015 pp 97ndash102 Kochi India 2015

[5] K D Dambul D C OrsquoBrien and G Faulkner ldquoIndooroptical wireless MIMO system with an imaging receiverrdquo IEEEPhotonics Technology Letters vol 23 no 2 pp 97ndash99 2011

[6] L Zeng D C OrsquoBrien H Le Minh et al ldquoHigh datarate Multiple Input Multiple Output (MIMO) optical wirelesscommunications using white LED lightingrdquo IEEE Journal onSelected Areas in Communications vol 27 no 9 pp 1654ndash16622009

[7] HMarshoud P C Sofotasios SMuhaidat andG K Karagian-nidis ldquoMulti-user techniques in visible light communicationsA surveyrdquo in Proceedings of the International Conference onAdvanced Communication Systems and Information Security(ACOSIS) Morocco 2016

[8] H Shen Y Deng W Xu and C Zhao ldquoRate-maximized zero-forcing beamforming for VLC multiuser MISO downlinksrdquoIEEE Photonics Journal vol 8 no 1 2016

[9] H Shen Y Deng W Xu and C Zhao ldquoRate maximization fordownlinkmultiuser visible light communicationsrdquo IEEE Accessvol 4 pp 6567ndash6573 2016

[10] T V Pham H Le Minh Z Ghassemlooy T Hayashi and AT Pham ldquoSum-rate maximization of multi-user MIMO visiblelight communicationsrdquo in Proceedings of the IEEE InternationalConference onCommunicationWorkshop ICCW2015 pp 1344ndash1349 London UK June 2015

[11] Z Zeng and H Du ldquoRobust precoding scheme for multi-user MIMO visible light communication systemrdquo in Proceed-ings of the 2017 25th European Signal Processing Conference(EUSIPCO) pp 2546ndash2550 Kos Greece August 2017

[12] H Ma L Lampe and S Hranilovic ldquoRobust MMSE linear pre-coding for visible light communication broadcasting systemsrdquoin Proceedings of the IEEE Globecom Workshops GC Wkshps2013 pp 1081ndash1086 Atlanta Ga USA 2013

[13] B Li J Wang R Zhang H Shen C Zhao and L Hanzo ldquoMul-tiuserMISO transceiver design for indoor downlink visible lightcommunication under per-LED optical power constraintsrdquoIEEE Photonics Journal vol 7 no 4 2015

[14] H Marshoud D Dawoud V M Kapinas G K KaragiannidisS Muhaidat and B Sharif ldquoMU-MIMO precoding for VLCwith imperfect CSIrdquo in Proceedings of the 4th InternationalWorkshop on Optical Wireless Communications IWOW 2015pp 93ndash97 Istanbul Turkey 2015

[15] K-H Park Y-C Ko and M-S Alouini ldquoOn the power andoffset allocation for rate adaptation of spatial multiplexingin optical wireless MIMO channelsrdquo IEEE Transactions onCommunications vol 61 no 4 pp 1535ndash1543 2013

[16] K Ying H Qian R J Baxley and S Yao ldquoJoint Optimization ofPrecoder and Equalizer in MIMO VLC Systemsrdquo IEEE Journalon Selected Areas in Communications vol 33 no 9 pp 1949ndash1958 2015

[17] A Vavoulas H G Sandalidis T A Tsiftsis and N VaiopoulosldquoCoverage Aspects of Indoor VLC Networksrdquo Journal of Light-wave Technology vol 33 no 23 Article ID 7299610 pp 4915ndash4921 2015

[18] A Burton H Le Minh Z Ghassemlooy E Bentley andC Botella ldquoExperimental demonstration of 50-Mbs visiblelight communications using 4 times 4 MIMOrdquo IEEE PhotonicsTechnology Letters vol 26 no 9 pp 945ndash948 2014

[19] J Ding K Wang and Z Xu ldquoImpact of different LED-spacingin arrayed LED transmitter on VLC channel modelingrdquo inProceedings of the 2014 6th International Conference on WirelessCommunications and Signal Processing WCSP 2014 pp 1ndash6Hefei China 2014

[20] A Gupta and D Nagar Matrix Variate Distribution Chapmanamp HallCRC London UK 2000

[21] R Jain D Chiu and W Hawe ldquoA quantitative measure offairness and discrimination for resource allocation in sharedcomputer systemsrdquo Digital Equipment Corporation ResearchTechnical Report 1984

[22] Z Ghassemlooy W Popoola and S Rajbhandari OpticalWireless Communication System and Channel Modelling withMatlab CRC publisher USA 2012

[23] D Perez Palomar M Bengtsson and B Ottersten ldquoMinimumBER linear transceivers for MIMO channels via primal decom-positionrdquo IEEE Transactions on Signal Processing vol 53 no 8pp 2866ndash2882 2005

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

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