In SCMA, each user has a specific codebook, which contains constellation points of dimensions: χj={xj1,xj1,…,xjM}, among them, the steps of constellation mapping and sparse coding are combined, and each user sends the transmitted bit data signal through codebook coding mapping to obtain a 4-dimensional complex code words, in which the codebook is from literature [17], and then performs parallel serial conversion on the code words combined by 6 users, The obtained serial complex signal is modulated by ACO-OFDM to obtain a serial non negative real signal, and then the serial non negative real signal is converted into a parallel four-way non negative real signal, which is carried on the four primary color LEDs of R, G, B and Y. the signal is transmitted through four optical carriers. The number of subcarriers transmitted by OFDM is n, so the expression of the transmission signal carried by each subcarrier in the frequency domain is [16]: (1)Xn=∑j=1J∑diag(Pj)Xn,j

Here, n represents the nth subcarrier, (n = 1,…, N), and Pj represents the signal power of the jth user. The transmission signal after ACO-OFDM modulation is [6]: (2)Xt_ACO−OFDM=[0,X1,0,X2,…,0,XN,0,XN∗,0,XN−1∗,…,0,X1∗]T

The signal sent through the four basic color light carrier is: (3)xT=xR+xG+xB+xY

As shown in Figs. 2 and 3, the LED lamp carrying user information is fixed somewhere on the ceiling. Each receiving user uses a photo detector (PD) to receive the signal light. The separation between the plane of PD receiver and the LED lightis h. Here we assume that the radius of the receiver is r. After the user information is sent by the LED light, the optical signal received by the receiving end can be expressed as [13]: (4)y=hx+n

Here, h denotesthe channel gain of the whole downlink, x represents the signal sent by the LED, and n represents the additive Gaussian white noise (AWGN), which obeys the (0,σ2) normal distribution. The channel gain here h is the sum of direct channel (LoS) and indirect channel (NLoS). Nevertheless, in the case of indoor visible light communication scenarios, the signal of NLoS is very small compared with Los. Based on this, the expression of visible light channel gain here is as follows: (5)h=(m+1)2πnc2ArxTssin⁡(Ψ)2cos⁡(ϕ)mcos⁡(ψ)d2rect(ψΨ)

Here m represents the Lambert radiation order, m=−log⁡(2)log⁡(cos⁡(ϕ1/2)), ϕ1/2 represents the half power angle emitted from the LED, Arx denotes the PDreceiving area, nc represents the refractive index of receiving lens, Ψ is the PD detector field of view, d is the direct link separation between the receiverand the LED, ψ stands for the angle of incident of the light, rect(ψΨ) is the rectangular function among [ψ,Ψ], Ts is the gain coefficient of the filter, as illustrated in Fig. 1, cos⁡(ϕ)=Hd, d=r2+H2, so (6)h=(m+1)2πnc2ArxTssin⁡(Ψ)2Hmcos⁡(ψ)(r2+H2)(m+2)2rect(ψΨ)

The driver color mixed signal transmitted through the visible light channel is received by PD after passing through the filter. After parallel serial conversion, the signal received by the four parallel PD is demodulated by ACO-OFDM, and the expression of the received signal is [13] (7)y=ργ∑j=1Jdiag(hPj)xj+n

Here, γ is the responsiveness of PD detector and ρ is the photoelectric conversion factor; n represents additive Gaussian white noise, n∼𝒞𝒩(0,σn2I)

After channel equalization of the received signal, the received signal after equalization at this time is (8)y~=∑j=1Jdiag(Pj)xj+nργh

Here, because we only consider the case of direct channel, there is no influence of multipath effect, so the value of ργh is a fixed value, which is to say, nργh is still subject to white noise of Gaussian distribution, let n~=nργh, then n~∼𝒞𝒩(0,σn~2I)

Optical power received by the receiver [18,19] (9)Pr=hPt (10)Pt=PLED (11)SNRr=(ργPr()2)σtotal2 (12)SNRTX=(ρPt)2σtotal2

SNRr Is received signal-to-noise ratio and SNRTX is transmitted signal-to-noise ratio; Here, γ represents the responsiveness of PD detector and ρ represents the photoelectric conversion factor; (13)σ2total=σ2shot+σ2thermal+σ2am

σ2shot represents the variance of shot noise, (14)σ2shot=2qγPrB+2qγPbgI2B

Here, q represents the charge constant, B is the noise bandwidth, Pbg is the ambient noise power, and I2 is the noise bandwidth factor of square wave pulse forming at the sending end [20] (15)σ2thermal=8πkTAGηAI2B2+16π2kTAΓgmη2A2I3B3

Here TA denotes the absolute temperature of ambient environment, G is the gain of open-loop voltage, η is the unit area capacitance of the PD detector, Γ is the channel noise coefficient of FET (field effect transistor), gm is the trans conductance of FET field effect transistor, I3 is the noise bandwidth factor formed by the completely boosted cosine equalization pulse, and k is the Boltzmann constant [21,22]. σ2am represents the noise bandwidth of the amplification circuit at the receiver (16)σ2am=iam2Ba

iam represents the noise intensity of the amplifier and Ba is the response bandwidth of the amplifier;

Add signal to noise ratio of ambient noise (17)SNR′TX=Ptσtotal2+σbg2

Then SNR′TX=Pt/σtotal2σtotal2/σtotal2+σbg2/σtotal2=SNRTX1+σbg2SNRTX/Pt

Here, we use MPA (message passing algorithm) as the decoding method. According to the theoretical derivation in the literature, in the AWGN channel, the received signal expression and theoretical bit error rate formula of SCMA system based on the codebook of star QAM constellation points [16] is (18)yAWGN=∑j=1Jdiag(Pj)xj+n (19)pb_AWGN=12nπ∫02π−16π11M−12exp[−nEbN0sin2⁡8π11M−121−cos⁡8π11M−12cos⁡(8π11M−12+ϕ)]dϕ

Comparing the received signals y~ in the visible light channel, we can find that the types of y~ and yAWGN are consistent, so the theoretical average bit error rate in the visible light channel can be directly used in the theoretical bit error rate formula in the AWGN channel. In this way, we obtain the theoretical BER formula for 6 users and 4 resource blocks in the visible light channel as (20)pb_VLC=12nπ∫02π−16π11M−12exp[−nEbN0sin2⁡8π11M−121−cos⁡8π11M−12cos⁡(8π11M−12+ϕ)]dϕ

Here, we use the complex codebook of 4-dimensional columns, where, M=2n, M=2n{n=1,2,3,⋯} (21)EbN0=SNR−10log10⁡(3)dB

The Channel parameters and SNR parameters are shown in Tables 1 and 2, respectively.

From the Fig. 4, we can see that the simulation results of = 4,8,16 of codebook are roughly consistent with the theoretical analysis results without adding environmental noise, and we can see that m = 8 is that the simulation results are basically consistent with the theoretical analysis, which shows that the codebook we use is indeed feasible in visible light communication, and we have found the aco-ofd of SCMA using this codebook.

Figs. 5–7 are the comparison of the theoretical bit error rate and the simulation bit error rate with and without environmental noise at different m values, that is, under the constellation points of the codebook mapping that are not used. From the comparison results, with low transmission SNR, the theoretical BERand the simulation bit error rate with and without environmental noise are consistent. With the continuous increase of the transmission signal-to-noise ratio, The bit error rate with ambient noise is significantly lower than that without ambient noise. And we can see from the signal-to-noise ratio formula added with environmental noise that when the transmission signal-to-noise ratio increases to a certain extent, its simulation and theoretical bit error rate will tend to be stable, which also shows that we cannot onlyincrease the SNR to decrease the system’s error rate, but also must find other ways to improve the anti noise performance of the system.