Design of Protograph LDPC-Coded MIMO-VLC Systems with Generalized Spatial Modulation

Dai Lin ,Fang Yi,* ,Guan Yongliang ,Mohsen Guizani

1 School of Information Engineering,Guangdong University of Technology,Guangzhou 51006,China

2 The State Key Laboratory of Integrated Services Networks,Xidian University,Xi’an 710126,China

3 School of Electrical and Electronic Engineering,Nanyang Technological University,Singapore

4 Department of Machine Learning,Mohamed Bin Zayed University of Artificial Intelligence(MBZUAI),Abu Dhabi,United Arab Emirates

Abstract: This paper investigates the bit-interleaved coded generalized spatial modulation(BICGSM)with iterative decoding (BICGSM-ID) for multiple-input multiple-output(MIMO)visible light communications(VLC).In the BICGSM-ID scheme,the information bits conveyed by the signal-domain(SiD)symbols and the spatial-domain(SpD)light emitting diode(LED)-index patterns are coded by a protograph low-density parity-check(P-LDPC)code.Specifically,we propose a signal-domain symbol expanding and re-allocating(SSER) method for constructing a type of novel generalized spatial modulation (GSM) constellations,referred to as SSERGSM constellations,so as to boost the performance of the BICGSM-ID MIMO-VLC systems.Moreover,by applying a modified PEXIT(MPEXIT) algorithm,we further design a family of rate-compatible P-LDPC codes,referred to as enhanced accumulate-repeat-accumulate(EARA)codes,which possess both excellent decoding thresholds and linear-minimum-distance-growth property.Both analysis and simulation results illustrate that the proposed SSERGSM constellations and P-LDPC codes can remarkably improve the convergence and decoding performance of MIMO-VLC systems.Therefore,the proposed P-LDPC-coded SSERGSM-mapped BICGSMID configuration is envisioned as a promising transmission solution to satisfy the high-throughput requirement of MIMO-VLC applications.

Keywords: bit-interleaved coded modulation;generalized spatial modulation;multiple-input multipleoutput;protograph LDPC codes;visible light communication.

I.INTRODUCTION

As a promising complementary solution to alleviate the shortage of wireless spectrum resources,optical wireless communication (OWC) is an attractive technology which has drawn a great deal of attention from both the academic and industrial communities [1].Particularly,visible light communication(VLC)technology has been standardized in IEEE 802.15.7 [2],and hence could become a commercialized method for indoor communication scenarios due to the low infrastructure cost.VLC technology enjoys the characteristic that lighting and communication can coexist without causing interference for each other[3].Moreover,advantages in the terms of energy efficiency,wide bandwidth,license-free application and higher security make VLC a promising solution to be part of the 5G wireless networks [4].In indoor VLC systems,the light emitting diodes (LEDs) are the main medium used for data transmission by modulating the intensity of the emitted light because of their superior switching capabilities.However,the LEDs have quite low modulation bandwidth (i.e.,about 20 MHz) [5],which greatly limits the transmission rate of the VLC systems.Meanwhile,the conventional transmission framework typically adopts single-input-single-output(SISO)technique,which cannot provide spatial diversity gain.To deal with this problem,some technologies such as multiple-input multiple-output (MIMO)[6,7]have been applied to the VLC systems.MIMO is an effective technique to improve the transmission throughput of VLC systems by exploiting the existing LEDs in some indoor scenarios,such as office,hospital,aircraft cabins and so on[8,9].

The conventional MIMO technology activates all LED emitters at each transmission instant and transmits the same information simultaneously,which results in the inter-channel interference(ICI).For averting ICI,a multiple-LED modulation scheme,calledspatial modulation(SM),has been proposed in [10,11].However,SM would lead to low spectral efficiency (SE) because it does not efficiently utilize the transmitted resources.Therefore,some other schemes such as generalized spatial modulation (GSM) [12],quad-LED complex modulation(QCM)[13]and dual-LED complex modulation (DCM) [14] have been developed.As a spectral-efficient multiple-input multiple-output visible light communication(MIMOVLC) transmission scheme,the conventional GSM(ConGSM) simultaneously activates multiple LEDs and every activated LED is allowed to carry the different unipolar pulse amplitude modulated (UPAM)intensity level during each transmission instant [12].However,due to the channel correlation of the static MIMO-VLC channels under the line-of-sight (LOS)condition,the signal-domain(SiD)symbol design for the ConGSM is not the optimal.In [15],a new SiD symbol-design method foractivate-space collaborative constellation GSM(ASCCGSM) has been proposed by minimizing the power of the SiD symbols to improve the error performance of MIMO-VLC system.Beyond that,it is well known that the coded modulation(CM)scheme[16-20]plays an important role in improving the overall system performance.

In [17],a trellis coded spatial modulation (TCSM)scheme was proposed,in which only SiD bits are protected by a trellis code to improve the performance over correlated channels.After that,the authors in[21]have extended the TCSM scheme to an enhanced coded SM(ECSM)scheme which jointly encodes the bits conveyed in the spatial-domain (SpD) and SiD so as to achieve a higher coding gain in the indoor OWC systems.Furthermore,it has been pointed out in [22] that a trellis coded GSM (TGCSM) scheme is spectrally efficient and robust against the correlated MIMO channels.Additionally,a bit-interleaved coded modulation(BICM)combined with SM has been proposed as a promising solution,which provides performance improvements against the channel correlation effect [18].Nevertheless,the BICM is the suboptimal transmission scheme,whose performance can be significantly enhanced by introducing the iterative decoding (ID) at the receiver [23].In this paper,we consider the transmission scheme for BICM combined with GSM and ID,referred to asBICGSM-ID.Nevertheless,there exist some challenges in the design of the bit-interleaved coded generalized spatial modulation with iterative decoding(BICGSM-ID)scheme for the MIMO-VLC systems.

In communication systems,error-correction codes(ECCs) play a critical role in improving the system performance.In [21],the convolutional codes have been used for the ECSM-aided indoor OWC systems.Furtherfore,the polar codes have been applied into the GSM-aided MIMO systems [24,25].In addition,the non-binary low-density parity-check(LDPC)codes have been presented in coded spatial modulation(CSM)scheme for the SM-MIMO systems[26].As a class of capacity-approaching codes,protograph lowdensity parity-check (P-LDPC) codes [27] have been widely exploited in the wireless communication systems due to their noticeable performance and simple implementation.Especially,the accumulate-repeatby-4-jagged-accumulate (AR4JA) code [28],which possesses simple protograph structures to realize linear encoding and decoding,can achieve very desirable error performance in both low and high signalnoise-ratio (SNR) regions over additive white Gaussian noise (AWGN) channels.However,the conventional P-LDPC codes may not perform well in specific BICM-ID VLC systems.

To facilitate the design and analysis of the PLDPC-coded BICM systems,a variety of protographbased extrinsic information transfer (PEXIT) algorithms [29-32] have been developed to predict the system performance in the low SNR region.On the other hand,in[28,33],the asymptotic weight distribution(AWD)has been formulated to estimate the typical minimum distance ratio (TMDR),i.e.,the linearminimum-distance growth property,of a P-LDPC code so as to predict the system performance in the high SNR region.Note that the TMDR is only relevant to the structure of P-LDPC code,but irrelevant to the type of channel [33].Although the performance of the P-LDPC-coded BICM systems has been intensively studied,there is still a lack of design methodology for P-LDPC-coded BICGSM-ID scheme,especially in the MIMO-VLC systems.

Driven by the aforementioned motivations,we investigate the performance of the P-LDPC-coded BICGSM-ID MIMO-VLC systems.The main contributions are in two-fold and can be summarized as follows.1) A type of novel GSM constellations,referred to assymbol expanding and reallocating GSM(SSERGSM)constellations,is proposed through an SSER method.Meanwhile,we utilize the constellation-constrained average mutual information(AMI)to evaluate the achievable capacity of the proposed SSERGSM constellations in the MIMOVLC systems.2) Based on the modified protograph extrinsic information transfer (MPEXIT) algorithm,we put forward a novel family of rate-compatible P-LDPC codes,calledenhanced accumulate-repeataccumulate(EARA)codes.The proposed EARA codes can not only benefit from desirable decoding thresholds,but also possess the linear-minimumdistance-growth property in the BICGSM-ID MIMOVLC systems.Simulations are carried out to verify that the proposed P-LDPC-coded BICGSM-ID schemes with the SSERGSM constellations outperform the state-of-the-art counterparts in the MIMOVLC systems.

The remainder of this paper is organized as follows.Section II introduces the P-LDPC-coded BICGSM-ID MIMO-VLC system model and presents the information-theoretic methodology to calculate the AMIs of the GSM constellations.This section also describes the MPEXIT algorithm for the PLDPC-coded BICGSM-ID MIMO-VLC system.Section III proposes the SSERGSM constellation design,in which a new SSER method is exploited to design the SiD symbols.Section IV constructs a new family of ratecompatible P-LDPC codes.Section V provides various simulated results and discussions.Finally,Section VI concludes the contributions of this paper.

Notations:Boldface lowercase and uppercase letters denote vectors and matrices,respectively.Let (·)T,·」,‖·‖and ! denote the transpose operation,floor function,Frobenius norm and factorial operation,respectively.Besides,|·|represents the cardinality of a set.

II.SYSTEM MODEL AND PRELIMINARIES

2.1 Conventional GSM

In a conventional GSM (ConGSM) [12],we assumeNtandNa(i.e.,Na≥2) are the numbers of transmit LEDs and active LEDs at each transmission instant,respectively.It is apparent that the number of all possible LED activation patterns is Δ=Nt!/(Nt-Na)!Na!.To ensure that the number of LED activation patterns is a power of two,effective LED activation patterns are selected for theM-ary unipolar PAM (M-UPAM) intensity level transmission,whereρd=log2Δ」.TheM-UPAM intensity levels can be formulated as[12,34]

whereIais the average intensity level of each LED.

Consequently,there are two different types of constellations,i.e.,spatial-domain (SpD) constellation and signal-domain (SiD) constellation,involved in GSM.The SpD constellation set Ω is dependent on the effective LED activation patterns(i.e.,|Ω|=2ρd)and the SiD constellation set Λ is dependent on the orderMof UPAM andNa(i.e.,|Λ|=M·Na).As a result,at each transmission instant,the number of coded bits can be written as

In theseρcoded bits,the firstρdcoded bits(i.e.,SpD bits) are used to select the effective LED activation patterns,while the lastρs(ρs=Na· log2M) are regarded as the SiD bits.Moreover,every log2Mcomponent bits are independently mapped to anM-UPAMintensity level.More details of the ConGSM can be referred to[12].

2.2 System Model

The block diagram of a P-LDPC-coded BICGSM-ID MIMO-VLC system is shown in Figure 1,in which an indoor LOS MIMO-VLC link withNtLEDs andNrphotodetectors (PDs) at the transmitter and receiver,respectively,are considered.In such a scenario,we assume thatNa(i.e.,Na≥2) is the number of active LEDs at each transmission instant.In Figure 1,the input information-bit streamc=(c1,c2,...,ck)is first encoded to a length-ncodewordu(i.e.,codedbit stream) by exploiting a P-LDPC code.Further,the code-bit streamu=(u1,u2,...,un)is permuted by a random interleaver.Subsequently,everyρconsecutive coded bits are processed by a GSM mapper.More specifically,theρconsecutive coded bits are divided into two parts in a sequential order.The first part includingρdbits is used to select the effective LED activation patterns,while the second part containsρs=Nalog2Mbits,and every log2Mcomponent bits are independently mapped to an UPAM intensity level.

Figure 1.Block diagram of a P-LDPC-coded BICGSM-ID MIMO-VLC system.

After the GSM mapper,everyρ(i.e.,ρ=ρd+ρs) coded bits are carried by a GSM signal,and thus a length-(n/ρ) GSM symbol sequence=can be yieled.Finally,each GSM symbol is converted into a transmission vectorx=[x1,x2,···,xNt]T,which is propagated through a MIMO-VLC channel.Note that the MIMO-VLC channel has two types of links,i.e.,LOS and no LOS(NLOS).The LOS link is usually much stronger than the NLOS link in power[6].Therefore,in this work,we only consider the LOS link that is the same as in[7,15].The received signal can be given as

wherey=[y1,y2,···,]Tis a vector with size ofNr×1,andx=[x1,x2,···,]Tis the transmitted GSM signal vector.In particular,xconsists of theNt-Nazero elements representing the inactive LEDs,and theNapositive,real-valued elements choosing from the UPAM intensity symbol set.Moreover,wis the noise vector with size ofNr×1,each of which can be modeled as the independent and identically distributed(i.i.d)additive white Gaussian noise(AWGN)with zero mean and varianceσ2=N0/2,whereN0is the single sided noise power spectral density.His the LOS channel coefficient matrix with size ofNr×Nt,and the (i,j)-th element ofHcan be determined by[21]

wheredijrepresents the distance between thejth transmit LED and thei-th receive PD,i ∈(1,2,···,Nr) andj ∈(1,2,···,Nt).?ijandψijare the angle of irradiance and incidence from thej-th transmit LED to thei-th receive PD,respectively.η=-ln 2/ln(cos Φ1/2) represents the Lambert’s mode number,where Φ1/2is the transmit semi-angle of the LED.Also,ε,Aand Ψ1/2represent the PD responsivity,PD area and half-power field-of-view(FOV)angle of the PD,respectively.BecauseN0=2σ2,the average optical signal-to-noise ratio(OSNR)is defined as follows[35,36]

2.3 Achievable Average Mutual Information

Here,we discuss the AMI for the BICGSM transmission scheme in the MIMO-VLC systems.For the BICGSM scheme,the transmitted GSM symbol vectorxis determined jointly by the effective LED activation patternd ∈Ω and the SiD constellation symbols ∈Λ,in which each symbolscontainsNaUPAM intensity levels.

From the perspective of equivalent parallel channels,at each transmitted instant,the joint mapping for theρbits in the set Ω and Λ can be viewed asρparallel binary sub-channels which are independent and memoryless.Suppose that the receiver has ideal channel state information(CSI).The AMI can reflect the maximum information rate for error-free transmission,and thus the AMI can be used as a criterion for evaluating the channel transmission performance.We assume thatdandsare the independent uniformly distributed random variables in the sets Ω and Λ,respectively.The spatial-domain AMI (SiD-AMI) and signal-domain AMI(SpD-AMI),denoted byISpDandISiDrespectively,can be calculated as[39,40]

2.4 Modified PEXIT Algorithm

To the best of our knowledge,PEXIT algorithm can be used to track the evolution of the mutualinformation between the variable-node (VN) decoder and the check-node(CN)decoder[27].Through such a manner,PEXIT algorithm is able to evaluate the decoding threshold of a P-LDPC code so as to reveal the error performance at low SNR region under an iterative decoder.By modifying the traditional PEXIT algorithm[41],one can evaluate the convergence performance of the P-LDPC-coded BICGSM-ID scheme in MIMO-VLC systems.

To facilitate the treatment of the modified PEXIT(MPEXIT) algorithm for the P-LDPC-coded BICGSM-ID scheme,we first introduce the concept of P-LDPC codes.A protograph can be seen as a Tanner graph [42] containing a few variable nodes,check nodes and edges.In the Tanner graph,each edge connects a variable node and a check node and the parallel edges are allowed.Assume that a protograph possessesnvvariable nodes andnccheck nodes,which can be represented by annc×nvbase matrixB=(bi,j),bi,jis the number of edges connectingCiwithVj.Furthermore,a larger protograph graph corresponding to a P-LDPC code can be obtained from a given protograph(resp.base matrix)by a “copy-and-permute (lifting)” operation which is usually implemented by a modified progressive-edgegrowth(PEG)algorithm[43].

In the BICGSM-ID MIMO-VLC system,Suppose thatG1is the maximum number of inner iteration for the P-LDPC decoder whileG2is the maximum number of outer iteration between the GSM demapper and the P-LDPC decoder.The MPEXIT algorithm is shown as follows.

1.Initialization:Given an OSNR,initialize the incominga-prioriMI from thei-th variable node to thej-th check nodeIav(i,j)=0.

3.Channel MI calculation of variable nodes:The channel MI of thei-th variable nodeIch(i)is calculated by

Specially,Ich(i)=0 if thei-th variable node is punctured in a protograph.

4.Extrinsic MI update between variable nodes and check nodes:In the inner iteration process,theextrinsicMIs are iteratively updated between the variable nodes and the check nodes,where the detailed process is given in[41].

5.A-priori MI update for GSM demapper:Exploiting thea-prioriMI passed from check nodes to variable nodes,a-prioriMIfor the GSM dmapper can be measured by

where(i,j)is thea-prioriMI passed from thej-th check node to thei-th variable node,andJ(.)function is derived in[31].Thencan be further used for the next outer iteration.

Through this way,we can get the lowest OSNR value for which alla-posterioriMIs of the variable nodes converges to 1 by repeating Steps 2)～5)with different OSNR values.

Remark 1.The proposed MPEXIT algorithm is designed to investigate the convergence performance for the P-LDPC-coded BICGSM-ID MIMO-VLC systems,and thus considering the salient features of GSM and indoor MIMO-VLC channel,which leads to a different channel MI expression Ich(i)from that in the traditional PEXIT algorithm.In addition,the proposed MPEXIT algorithm assumes that a finite binary codeword including both0and1bits is transmitted when calculating the MI,while the traditional PEXIT algorithm assumes an infinite-length all-zero codeword to do so.Therefore,for a given P-LDPC code,the decoding threshold evaluated by the proposed PEXIT algorithm can predict the waterfall-region performance of the simulated BER result more accurately compared with the traditional counterpart.

III.DESIGN OF PROPOSED SSERGSM FOR BICGSM-ID MIMO-VLC SYSTEMS

3.1 Proposed SSERGSM Design Method

In the MIMO-VLC systems,SiD modulated symbols are the critical parameter for formulating the GSM constellation,which can be designed and optimized so as to boost the system performance.For the existing GSM constellations,i.e.,ConGSM [12] and ASCCGSM [15],the SpD bits allocated for all effective LED activation patterns share the sameM-UPAM symbol set.As a example,the mapping tables of ConGSM and ASCCGSM constellations for a 4×4 MIMO-VLC system withNa=2 andρ=4 (i.e.,M=2)are respectively shown in Table 1 and Table 2,where we choose(1,2),(1,3),(1,4)and(2,3)as the effective LED activation patterns,and the numbers 1,2,3 and 4 respectively represent the 1-th,2-nd,3-rd and 4-th LED indices.Given a MIMO-VLC channel (i.e.,a channel matrixH),the correlation property of each effective LED activation pattern can be measured by the corresponding normalized instantaneous channel correlation matrix(NICCM)[44].The larger the value of elements in NICCM is,the stronger the correlation of the corresponding LED activation pattern possesses.Due to the correlation property of the effective LED activation patterns,the ConGSM and ASCCGSM constellations with all effective LED activation patterns sharing the sameM-UPAM symbol set may make the extrinsic MIs output from the demapper unreliable,which degrades the system performance.Therefore,we put forward a signal-domain symbol expanding and re-allocating (SSER) method for constructing a type of novel GSM constellations to improve the performance of the BICGSM-ID MIMOVLC system.The detailed SSER design method is described as follows.

Table 1.Mapping table of the ConGSM constellation with Nt=Nr=4,Na=2 and ρ=4(i.e.,M=2).

Table 2.Mapping table of the ASCCGSM constellation with Nt=Nr=4,Na=2 and ρ=4(i.e.,M=2).

1.Expanding the number of the UPAM symbols:Given anM-ary UPAM symbol set S={I1,I2,···,IM},which must satisfyIt ∈(0,2Ia],?t ∈1,2,···,M,and the number of effective LED activation patternsβ=2ρd.First,we evenly divide the entire symbol space(0,2Ia]intoβsymbol subspaces,i.e.,(2Iaτ/β,2Ia(τ+1)/β],?τ ∈0,1,···,β-1.Subsequently,we chooseMUPAM symbols from each symbol subspace,and define the selection principle as

whereτ ∈0,1,···,β-1 andn′=1,2,···,M.Therefore,the new UPAM symbol setis generated.Taking the case ofρ=4 (i.e.,M=2) as a example,and the detailed design principle are shown in Figure 2.Referring to this figure,one can obtain 4 symbol subspaces,i.e.,(0,Ia/2],

Figure 2.Design principle of the proposed SSERGSM constellation with Nt= Nr=4, Na=2 and ρ=4 (i.e.,M=2).

(Ia/2,Ia],(Ia,3Ia/2] and (3Ia/2,2Ia].Then,we choose 2 UPAM symbols from each symbol subspace to form the new UPAM symbol set={Ia/6,2Ia/6,4Ia/6,5Ia/6,7Ia/6,8Ia/6,10Ia/6,11Ia/6}.Through this way,it is possible for all effective LED activation patterns to possess different UPAM symbol subsets.

2.Re-allocating the UPAM symbols for β effective LED activation patterns:For the expandedβMUPAM modulation symbols obtained from the first-step design,we first rank theβMsymbols in an ascending order of their corresponding intensity values.Then,we selectMdifferent UPAM modulation symbols at maximum equal-intensity interval for each effective LED activation patterns.For example,the effective LED activation patterns (1,2),(1,3),(1,4)and (2,3) possess the UPAM symbol subsets{Ia/6,7Ia/6},{2Ia/6,8Ia/6},{4Ia/6,10Ia/6}and{5Ia/6,11Ia/6},respectively.Finally,the SiD bits in each effective LED activation pattern are mapped to the corresponding UPAM modulation symbols.

Based on the above two steps,one can construct the SSERGSM constellation according to a given modulation orderMand the number of effective LED activation patternsβ,which can significantly enhance the demapper reliability so as to ensure excellent BICGSM-ID performance.As an example,the mapping table of SSERGSM constellation for a 4× 4 MIMO-VLC system withNa=2 andρ=4 (i.e.,M=2) is shown in Table 3.To verify the superiority of the proposed SSERGSM constellation,the convergence performance of BICGSM-ID will be investigated in the forthcoming subsections.

Table 3.Mapping table of the proposed SSERGSM constellation with Nt=Nr=4,Na=2 and ρ=4(i.e.,M=2).

3.2 Simulation Settings and BICGSM-AMI Analysis

3.2.1 Simulation Settings

Unless otherwise mentioned,we consider a 4×4 indoor LOS MIMO-VLC channel and assume that the geometric setup of the channel model is shown in Figure 3.In this figure,the LEDs and PDs are aligned in a quadratically 2×2 array which is located in the middle of a 5.0m×5.0m×3.0m room,thus the four LEDs and four PDs construct two squares of lengthsdtxanddrx,respectively.Moreover,the PDs are placed at the planez=0.75m(e.g.,the height of a table).Besides,the parameters utilized for transceiver are summarized in Table 4.Based on this model,we consider three different correlated MIMO-VLC channels by varying the value ofdtx(i.e.,dtx=0.3m,0.5m,and 0.7m),while the other system parameters remain unchanged.Note that the setting about the values ofdtxis the same as that in[21].The three different values ofdtxrepresent three different channel matrices,and the smaller the value ofdtxis,the stronger the correlation of the corresponding channel is.

Table 4.Parameter setting for the transceiver of an indoor MIMO-VLC system.

Figure 3.Geometric setup of the channel model for a 4×4 indoor LOS MIMO-VLC system.

3.2.2 BICGSM-AMI Analysis

According to Section 2.3,the achievable BICGSMAMI (i.e.,IBICGSM),SpD-AMI (i.e.,ISpD) and SiDAMI (i.e.,ISiD) for a given GSM constellation can be measured.Assuming that the numbers of coded bit at each transmitted instant are set asρ=4 (i.e.,M=2) andρ=6 (i.e.,M=4),the AMIs for the proposed SSERGSM,ConGSM[12],and ASCCGSM[15] over a MIMO-VLC channel withdtx=0.5m are depicted in Figure 4a and Figure 4b,respectively.As observed from Figure 4a,the proposed SSERGSM exhibits a significantly largerIBICGSMwith respect to the ConGSM and ASCCGSM.Meanwhile,ISpDandISiDof the proposed SSERGSM are larger than their corresponding counterparts.Therefore,it can be concluded that the BICGSM scheme with the proposed SSERGSM mapper can obtain better performance compared with ConGSM and ASCCGSM mappers.Similar phenomenon also occurs in the case ofρ=6(i.e.,M=4).

Figure 4.IBICGSM, ISpD and ISiD for the proposed SSERGSM, ConGSM, and ASCCGSM over a MIMO-VLC channel with dtx=0.5m,where the numbers of coded bits at each transmission instant are set as: (a) ρ=4 (i.e.,M=2)and(b)ρ=6(i.e.,M=4).

In order to show the influence of the spacing between two neighboring LEDs on the BICGSM capacity,we set three different values ofdtx,i.e.,dtx=0.3m,0.5m,0.7m in the cases ofρ=4(i.e.,M=2)andρ=6 (i.e.,M=4).Then,the correspondingBICGSM capacities for the proposed SSERGSM are illustrated in Figure 5.One can observe that the proposed SSERGSM has the largest BICGSM capacity whendtx=0.7m,while it can achieve the lowest BICGSM capacity whendtx=0.3m whether the case ofρ=4 orρ=6.This indicates thatdtxis a critical parameter to determine the performance of the proposed SSERGSM constellations.Hence,the proposed SSERGSM can achieve better performance for a largerdtxover the correlated MIMO-VLC channels.

Figure 5.IBICGSM for the proposed SSERGSM over MIMO-VLC channels with dtx=0.3m, dtx=0.5m and dtx=0.7m, where the cases of ρ=4 (i.e., M=2) and ρ=6(i.e.,M=4)are considered.

3.3 Convergence-Performance Analysis

3.3.1 Decoding Thresholds In order to further demonstrate the merit of the proposed SSERGSM in the BICGSM-ID MIMO-VLC systems,we conduct the convergence-performance analysis by means of the MPEXIT algorithm.Suppose that all information bits used in the above scenarios are encoded by the rate-1/2 AR4JA code[28].Unless otherwise stated,the information block length is 3600 bits,and the maximum numbers of inner and outer iterations are assumed asG1=20 andG2=4,respectively.Furthermore,we consider three different values ofdtx,i.e.,dtx=0.3m,0.5m,0.7m,in the BICGSM-ID MIMO-VLC systems,where the cases ofρ=4 andρ=6 are involved.

In Table 5,we analyze the decoding thresholds of rate-1/2 AR4JA code[28]in the BICGSM-ID MIMOVLC systems with three different GSM constellations,i.e.,the proposed SSERGSM,ConGSM[12]and ASCCGSM[15].As can be seen from Table 5,in the case ofρ=4,the P-LDPC-coded BICGSM-ID MIMOVLC system with the proposed SSERGSM constellation achieves the smaller decoding threshold compared with other two GSM constellations,i.e.,ConGSM[12]and ASCCGSM [15] constellations,whendtxare respectively set to be 0.3m,0.5m and 0.7m.Moreover,the decoding thresholds of the AR4JA-coded BICGSM-ID schemes with the three different GSM constellations are reduced asdtxvaries from 0.3m to 0.7m,which indicates that system performance is optimal whendtx=0.7m.Similar conclusions can be drawn from the case ofρ=6.We have also analyzed the decoding thresholds for the proposed SSERGSM constellation in the AR4JA-coded BICMGSM systems without ID,and have found that the ID framework converges faster than the non-ID framework.Thus,the SSERGSM constellation can obtain better performance than other two GSM constellations in the P-LDPC-coded BICGSM-ID MIMO-VLC systems.

Table 5.Decoding thresholds(dB)of the AR4JA-coded BICGSM-ID schemes with three different GSM constellations over MIMO-VLC channels with three different values of dtx,where the cases of ρ=4 and ρ=6 are considered. The maximum numbers of inner and outer iterations are assumed as G1=20 and G2=4,respectively.

3.3.2 Extrinsic MI of GSM Demapper

To elaborate a little further,we analyze the extrinsic MIs output from the demappers corresponding to three different GSM constellations in the cases ofρ=4 andρ=6 for the AR4JA-coded BICGSMID MIMO-VLC systems withdtx=0.5m,and show the results in Figure 6.As observed,in the case ofρ=4 (see Figure 6a),when OSNR is varying from 4.05 dB to 4.75 dB,the proposed SSERGSM constellation obtains greater extrinsic MI compared with other two GSM constellations,implying that using the SSERGSM constellation can provide more reliable information for the demapper during the iterative process with respect to other two GSM constellations.It is well known that more reliable demapping information can accelerate the convergence of the PLDPC-coded BICM-ID scheme.Similar conclusion can be found in Figure 6b with the case ofρ=6.It can be easily observed that the extrinsic-MI analyses agree well with their corresponding decodingthreshold analyses.Therefore,given an OSNR range,the extrinsic MI of the GSM demapper can serve as a useful performance metric of a GSM constellation over the MIMO-VLC channels.

Figure 6.Extrinsic MIs for the proposed SSERGSM,Con-GSM and ASCCGSM in a P-LDPC-coded BICGSM-ID MIMO-VLC system with dtx=0.5m, where the numbers of coded bits at each transmission instant are set as: (a)ρ=4(i.e.,M=2)and(b)ρ=6(i.e.,M=4).

IV.DESIGN OF P-LDPC CODES FOR BICGSM-ID MIMO-VLC SYSTEMS

In order to improve the error correction performance of the BICGSM-ID MIMO-VLC systems,a type of rate-compatible P-LDPC codes is designed in this section.

4.1 Design of P-LDPC Codes

It is well known that the conventional AR4JA code[28]has excellent error performance over the AWGN channels because it enjoys two desirable properties,i.e.,low decoding threshold and linear minimum distance growth(the minimum Hamming distance growing linearly with the codeword length)[45].The base matrix corresponding to a rate-(e+1)/(e+2)AR4JA code is given by

whereeis the number of extension patterns(i.e.,e=0,1,2,···),the matrix size is 3×(2e+5),and the degree-6 variable node(i.e.,the second column ofB)is punctured.

Typically,the performance of P-LDPC codes depends on not only the structure of codes,but also the type of channel.The conventional P-LDPC codes,such as AR4JA code,that performs well over AWGN channels may not suitable for the usage in the BICGSM-ID MIMO-VLC systems.Besides,by using the PEXIT analysis,we found that the P-LDPC codes with the lowest-degree variable node punctured can achieve better convergence performance with respect to the counterparts with other puncturing rules in the BICGSM-ID MIMO-VLC systems.Therefore,we aim to design a type of rate-compatible P-LDPC code with puncturing the lowest degree variable node.

Considering a given codeword length,the lifting factor(i.e.,“copy-and-permuting”times)for P-LDPC code will becomes smaller if the protograph size increases,which makes it more difficult to avoid error floor.Furthermore,the larger the protograph is,the more complex the corresponding optimization algorithm will be.Therefore,we first consider a rate-1/2 mother base matrix with the same size as that of the conventional AR4JA code,which includes a punctured variable node.To facilitate the code design,we impose some constraints on the rate-1/2 P-LDPC code in order to guarantee that its mother base matrix possesses the linear-minimum-distance-growth property and a relatively low decoding threshold,as follows.

1) Initialize a protograph with a degree-1 variable node being the punctured variable node,which corresponds to the second column in the mother base matrix.

2) Impose empirical constraints for the linearminimum-distance-growth property on the base matrix,i.e.,only one degree-2 variable node is allowed in the mother base matrix.

3)Set the maximum number of parallel edge to be 3 in the mother base matrix so as to maintain the low encoding and decoding complexity.This is because the encoding and decoding complexity of a P-LDPC code can be reflected by the number of edges in its corresponding protograph.Besides,for a fixed codeword length,if the number of parallel edges increases,the likelihood of short cycles will increase.

Based on the above rules,the mother base-matrix structure of the proposed rate-1/2 P-LDPC code can be formulated as

To assign the first column in the mother base matrixB1/2as the largest degree variable node,we add an additional constraint,i.e.,b1,1+b2,1+b3,1＞b1,3+b2,3+b3,3andb1,1+b2,1+b3,1＞b1,4+b2,4+b3,4.After a exhaustive search using the proposed MPEXIT algorithm,one can obtain the optimal mother base matrix with lowest decoding threshold and linear-minimumdistance-growth property,as

However,for higher-rate P-LDPC code design,the search space becomes larger,hence the corresponding exhaustive-search method is more complex.Therefore,we resort to a pattern-extension method,which is similar to the construction of the AR4JA codes,to design the higher-rate P-LDPC code.In particular,the pattern-extension method can be realized by repeatedly adding two variable nodes into the resultant mother protograph (i.e.,base matrix).It has been demonstrated in [28,33] that adding degree-3 variable node into a rate-1/2 protograph can maintain the lowest-complexity feature for its corresponding higher-rate counterparts without deteriorating the linear-minimum-distance-growth property.

For the above reason,a type of enhanced ratecompatible P-LDPC codes with the code rateR=(e+1)/(e+2) (e=0,1,2,···),referred toEARA codes,is obtained.The protograph structure of the proposed EARA code is illustrated in Figure 7,where the dark circles denote transmitted variable nodes,the white circle denotes the punctured variable node,and the plus circle are check nodes.As a result,the corresponding base matrix is given as

Figure 7.Protograph structure of the proposed EARA PLDPC code.

4.2 Asymptotic Performance Metrics

4.2.1 Decoding-Threshold Analysis To verify the convergence performance of the proposed EARA P-LDPC code,we take the rate-2/3 AR4JA code [28],accumulate-repeat-by-4-accumulate (AR4A) code [46],improved-AR4JA(IAR4JA) code,improved-AR4A (IAR4A) code and regular-(3,9) code [28] as benchmarks in the proposed SSERGSM-aided BICGSM-ID MIMO-VLC systems with three different values ofdtx,i.e.,dtx=0.3m,0.5m,0.7m,and show their corresponding decoding thresholds in Table 6.It is noted that the IAR4JA code and IAR4A code respectively represent two modified P-LDPC codes possessing the same base matrix as the AR4JA code and the AR4A code,but only with puncturing the lowest degree variable nodes.As shown,whendtx=0.3m,the proposed EARA PLDPC code possesses the lowest decoding threshold among the six types of P-LDPC codes in the cases ofρ=4 andρ=6.Similar observations can be found whendtx=0.5m anddtx=0.7m.This implies that the proposed EARA P-LDPC code can achieve better error performance in low-OSNR region compared with its counterparts in the proposed SSERGSM-aided BICGSM-ID MIMO-VLC systems.

Table 6.Decoding thresholds(dB)of six rate-2/3 P-LDPC codes,i.e.,the proposed EARA code,AR4JA code,AR4A code,IAR4JA code,IAR4A code and regular-(3,9)code,in the proposed SSERGSM-aided BICGSM-ID MIMO-VLC systems with three different values of dtx,where ρ=4 and ρ=6 are considered. The maximum numbers of inner and outer iterations are assumed as G1=20 and G2=4,respectively.

4.2.2 AWD Analysis

By exploiting AWD function[28,33],we measure the TMDRs for rate-2/3 P-LDPC codes,i.e.,the proposed EARA code,AR4JA code,AR4A code,IAR4JA code,IAR4A code and regular-(3,9)code and give the corresponding results in Table 7.Referring to this table,except that the AR4A code and IAR4A code,the other four types of P-LDPC codes possess effective TMDR values(i.e.,positive TMDR values).

Table 7.TMDRs of rate-2/3 P-LDPC codes,i.e.,the proposed EARA code,AR4JA code,AR4A code,IAR4JA code,IAR4A code and the regular-(3,9)code,in the BICGSM-ID MIMO-VLC systems.

Based on the above decoding-threshold and AWD analyses,it is affirmed that the proposed EARA ensemble can not only perform best in the low-OSNR region among the six types of P-LDPC codes,but also exhibit desirable error performance in the high-OSNR region in the proposed SSERGSM-aided BICGSM-ID MIMO-VLC systems.

Remark 2.We have also evaluated the six types of P-LDPC codes with other code rates(e.g.,R=1/2,3/4)and have observed that the EARA codes also obtain the lowest decoding threshold and have effective TMDR values.

4.3 Complexity Analysis

In this subsection,we analyze the computational complexity of the P-LDPC codes for successful decoding under the serially concatenated ID framework.In this ID framework,GSM demapper and P-LDPC decoder impose the main computational complexity.Therefore,Therefore,based on the Max-Log-Map demapping algorithm [37] and the LLR-based belief propa-gation (LLR-BP) decoding algorithm [47],we evaluate the numbers of real addition (RA) and real multiplication(RM)operations of a P-LDPC code for successful decoding under the ID framework.The results are shown in Table 8,wherenandmare the number of VNs and CNs,respectively;pis the number of punctured VNs;anddenote the average degrees of VNs and CNs,respectively;anddenote the average numbers of inner and outer iterations.As a further investigation,Table 9 presents the average numbers of RA and RM operations of different P-LDPC codes for successful decoding in the SSERGSM-aided BIGSMID MIMO-VLC systems withdtx=0.3m andρ=4.As can be seen from this table,the proposed EARA P-LDPC code possesses a lower computational complexity compared with its counterparts under the ID framework.

Table 8.Average number of operations of a P-LDPC code for successful decoding under the ID framework.

Table 9.Average Numbers of RA and RM operations of different P-LDPC codes for successful decoding under the ID framework.

V.SIMULATION RESULTS

To validate our proposed SSERGSM constellation and constructed EARA P-LDPC code,we present various simulation results on the BICGSM-ID MIMO-VLC systems in this section.To be specific,a 4×4 indoor LOS MIMO-VLC channel model is considered.Furthermore,we assume that the information block length of P-LDPC codes is 3600 bits.In addition,unless otherwise stated,the maximum numbers of inner iterations and outer iterations areG1=20 andG2=4,respectively.

5.1 BER Performance of Proposed SSERGSM Constellation

Figure 8 depicts the bit-error-rate (BER) curves of rate-1/2 the AR4JA-coded BICGSM-ID schemes with three different types of GSM constellations,i.e.,the proposed SSERGSM constellations,ConGSM [12],and ASCCGSM[15]over a MIMO-VLC channel withdtx=0.5m,where the maximum numbers of outer iterations are set asG2=0,2,4.For the case ofρ=4(i.e.,M=2),as observed from Figure 8a,the proposed SSERGSM constellation can achieve better error performance compared with the two counterparts in the BICGSM-ID MIMO-VLC systems.To be specific,whenG2=4,the proposed SSERGSM constellation exhibits remarkable gains of about 1.9 dB and 2.4 dB over the ASCCGSM and the ConGSM constellations,respectively,at a BER of 7×10-6.Furthermore,whenG2=0 (i.e.,BICGSM system),it is evident that the proposed SSERGSM constellation can also obtain larger gains compared with its counterparts,indicating that the proposed SSERGSM constellation is suitable for not only ID scenario but also non-ID scenario.Additionally,it can be easily observed that,at a BER 5× 10-6,the proposed SSERGSM constellation withG2=4 obtains about 0.3-dB gain over that withG2=2,while the latter obtains about 0.56-dB gain over that withG2=0.Similar observations can be also found in the case ofρ=6 (i.e.,M=4)(see Figure 8b).Based on the above discus-sions,the proposed SSERGSM constellation will become better as the maximum number of out iterations(i.e.,the value ofG2)increases,while the relative performance between the proposed SSERGSM constellation and the benchmarks remains the same.

Figure 8.BER curves of the AR4JA-coded BICGSM-ID schemes with the proposed SSERGSM, ASCCGSM, and ConGSM constellations over a MIMO-VLC channel with dtx=0.5m,where(a)ρ=4(i.e.,M=2)and(b)ρ=6(i.e., M=4) are considered. The maximum numbers of outer iterations are set as G2=0,2,4.

5.2 BER/FER Performance of Proposed EARA P-LDPC Code

As a further investigation,we perform the BER performance for three different types of GSM constellations,i.e.,the proposed SSERGSM constellations,ConGSM[12],ASCCGSM[15]in the AR4JA-coded BICGSMID MIMO-VLC systems with three different values ofdtx(i.e.,dtx=0.3m,0.5m,and 0.7m) in Figure 9.Referring to Figure 9a corresponding to the case ofρ=4 (i.e.,M=2),at a BER of 7× 10-6,the proposed SSERGSM constellation only requires about a 4.8-dB OSNR whendtx=0.7m,while it needs about 5.68-dB and 6.88-dB OSNR whendtx=0.5m and 0.3m,respectively.We also observe that,aiming to achieve a BER of 7× 10-6,the proposed SSERGSM constellation benefits from performance gains of about 3 dB and 3.64 dB compared with the ASCCGSM and ConGSM constellations whendtx=0.3m.Whendtx=0.5m,the proposed SSERGSM constellation also significantly outperforms the ASCCGSM and ConGSM constellations about 1.9-dB and 2.4-dB gains,respectively.Whendtx=0.7m,this performance gains of the proposed SSERGSM constellation with respect to the ASCCGSM and Con-GSM constellations are only about 0.45 dB and 0.6 dB,respectively.Likewise,one can observe from Figure 9b that the relative performance among the three different types of GSM constellations in the case ofρ=6(i.e.,M=4)is identical to that in the case ofρ=4(i.e.,M=2).

Figure 9.BER curves of the AR4JA-coded BICGSM-ID schemes with the proposed SSERGSM, ASCCGSM, and ConGSM constellations over MIMO-VLC channels with(a)ρ=4 (i.e., M=2) and (b) ρ=6 (i.e., M=4).The spacings between two neighboring LEDs of are set as dtx=0.3m,0.5m,and 0.7m.

These phenomenons suggest that the P-LDPCcoded BICGSM-ID MIMO-VLC system performance is continuously improved asdtxincreases from 0.3m to 0.7m,while the relative performance-gain between the proposed SSERGSM constellation and the benchmarks becomes smaller simultaneously.The reason is the fact that the parameterdtxplays an important role in affecting the correlation of the MIMO-VLC channel.Specifically,a smallerdtxproduces a higher correlated MIMO-VLC channel.It manifests that the proposed SSERGSM constellation has greater advantage in the highly-correlated MIMO-VLC channels.

Here,we carry out some BER/frame-error-rate(FER) performance simulations on the rate-2/3 proposed EARA code,the AR4JA code[28],AR4A code[46],IAR4JA code,IAR4A code and the regular-(3,9) code [28] in the SSERGSM-aided BICGSMID MIMO-VLC systems withdtx=0.3m,where the cases ofρ=4(i.e.,M=2)andρ=6(i.e.,M=6)are considered.The results are illustrated in Figure 10.According to Figure 10a,at a BER of 7×10-6,the proposed EARA P-LDPC code outperforms the IAR4A code and the IAR4JA code by about 0.67 dB and 0.78 dB,respectively.Moreover,the proposed EARA PLDPC code achieves more than 0.4-dB gain over the regular-(3,9) code,while the regular-(3,9) code further respectively performs better than the AR4A code and the AR4JA code by about 1.15 dB and 1.69 dB in the case ofρ=4 (i.e.,M=2).Referring to Figure 10b for the case ofρ=6 (i.e.,M=4),at OSNR=9.0 dB,the proposed EARA P-LDPC code can achieve a BER of 6× 10-6,while the AR4JA code,AR4A code,IAR4JA code,IAR4A code and the regular-(3,9) code accomplish BERs of 7.4×10-2,5.2×10-2,5.8×10-3,4.1×10-3and 3.7×10-4,respectively.Importantly,there is no error-floor phenomenon for the proposed EARA P-LDPC code when the BER reduces to 10-6.Based on the above observations,the proposed EARA P-LDPC code exhibits the best performance among the six types of P-LDPC codes.

Figure 10.BER/FER curves of six types of P-LDPC codes in the SSERGSM-aided BICGSM-ID MIMO-VLC systems with dtx=0.3m, where (a) ρ=4 (i.e., M=2) and (b)ρ=6(i.e.,M=4)are considered.

Remark 3.We have also carried out simulations with other system setups(e.g.,different values of Nt,Nr,Na,G1,G2and ρ),and have found that the relative performance between the proposed SSERGSM constellation and the benchmarks remains the same,which verifies the robustness of our proposed design.

VI.CONCLUSION

In this paper,we presented a new BICGSM-ID transmission scheme over the MIMO-VLC channels.The design and optimization of GSM constellations and PLDPC codes for use in the BICGSM-ID have been carefully investigated.To be specific,we modified the PEXIT algorithm according to the characteristics of GSM and MIMO-VLC channel.Next,we put forward an SSER design method to formulate a type of SSERGSM constellations tailored for the P-LDPC-coded BICGSM-ID MIMO-VLC systems,which exhibits better performance than other GSM constellations in this scenario.By exploiting the MPEXIT algorithm,we also constructed a family of EARA P-LDPC codes,which possess lower decoding thresholds than other P-LDPC codes and desirable linear-minimum-distance-growth property in the SSERGSM-aided BICGSM-ID MIMO-VLC systems.Simulation results validated that the joint deployment of SSERGSM constellations and EARA PLDPC codes allow the BICGSM-ID MIMO-VLC systems to achieve very desirable error performance.In addition,we analyzed the effect of the correlation of MIMO-VLC channel on the system performance.Owing to these advantages,the proposed P-LDPC-coded BICGSM-ID is expected to promise a good solution for high-throughput MIMO-VLC applications.

ACKNOWLEDGEMENT

This work was supported in part by the NSF of China under Grant 62322106,62071131,the Guangdong Basic and Applied Basic Research Foundation under Grant 2022B1515020086,the International Collaborative Research Program of Guangdong Science and Technology Department under Grant 2022A0505050070,in part by the Open Research Fund of the State Key Laboratory of Integrated Services Networks under Grant ISN22-23,and the National Research Foundation,Singapore University of Technology Design under its Future Communications Research&Development Programme“Advanced Error Control Coding for 6G URLLC and mMTC”Grant No.FCP-NTU-RG-2022-020.