1 Introduction

Visible Light Communication (VLC) provides a wide range of spectrum, easy accessibility, high data rate, high power efficiency and other advantages that qualifies VLC to achieve the revolution in the next years. Several techniques in wireless communication have used VLC systems for its huge data rate and low bit error rate (BER).

Several researchers are interested in the field of NOMA-MIMO-VLC, where Dixit et al. demonstrated the analysis of BER for an angular diversity receiver utilizing MIMO-VLC (Dixit and Kumar 2021a, b). In Al-Nahhal et al. (2019), authors aimed to increase the spectral efficiency within MIMO-VLC using Orthogonal Frequency Division Multiplexing (OFDM). MIMO-OFDM-VLC was used in implementation of the wrap phase technique in Al-Nahhal et al. (2021). MIMO-OFDM-VLC system was used for optimization the spectral efficiency in Siddiqi et al. (2020). NOMA is a recent technique that used for providing extra spectral efficiency depending on Superposition Coding (SC) for multiplexing the transmitted signals in power domain. While at receiver, SIC with multiple users was applied to decode the received signal. The BER performance for NOMA-OFDM-VLC systems was analyzed in Lin (2017). Different modulation techniques were used for studying BER for NOMA-VLC as M-QAM, square QAM, M-PAM, and M-PSK in LiU et al. (2019), Almohimmah and Alresheedi (2020).

The SIC was also investigated to improve the performance of NOMA-VLC systems in Dixit and Kumar (2021a, b). To enhance the performance of NOMA-VLC, the MIMO technique was used to decrease the BER. In Chen et al. (2018), a NOMA-MIMO-VLC was presented using the power allocation method which improved the data rate. Also, in Mitra and Bhatia (2017), a recent precoder design was used to improve the performance of the NOMA-MIMO-VLC system. Recently, Artificial Intelligence (AI) has been used for improving the performance of several fields.

To study NOMA optical wireless communication systems, two major Machine Learning (ML) branches are now widely used. As shown in Bhatt et al. (2022), Deep Learning (DL) can be used to deal with the complexity of mathematical modeling and overcome nonlinear distortion in NOMA optical wireless communication systems for signals. By extracting features from large amounts of training data and optimizing system performance, DL, as a sub-branch of ML, can provide a data-driven solution to this problem (Tian et al. 2022). The DL network can replace the mathematical model in such a system. Although it is unclear whether DL-based wireless communication outperforms traditional wireless communication in terms of performance improvement and complexity, DL-based algorithms are thought to be faster and consume less power than the traditional ones.

Many fields have successfully applied DL technology, which can automatically learn more complex features from complex data structures, including signal classification, modulation recognition (O’Shea et al. 2016), sentiment analysis (Song et al. 2018), channel estimation (Soltani et al. 2019), and image processing (Ngo et al. 2021). Recently, a significant progress has been achieved in the quantization and feedback of MIMO channels, and certain channel state (CS) technology issues have been resolved; thanks to the DL approach (Qi and Su 2022; An et al. 2022; Kaur et al. 2022). The NOMA spectrum used in radio frequency (RF) for free space optical networks has been improved in Zaki et al. (2022), Upadhya et al. (2022). In terms of underwater networks, most NOMA research focuses on improving the receiving rates and performance, with channel studies occurring infrequently (Ai et al. 2022). DL was used to solve the NOMA scheme of an indoor VLC system (Ullah et al. 2022).

Our target in this article is to adopt AI to improve the NOMA-MIMO-VLC system performance.

The main aim of this paper is to design a NOMA-MIMO-VLC system, to achieve a low BER and a high data rate network. Moreover, the OFDM technique is used in the MIMO-VLC system that is proposed to maximize the spectral efficiency. Furthermore, a multi-user downlink NOMA-MIMO-VLC system is proposed with Equal Gain Combiner (EGC) at the receiver. The On–Off Keying (OOK) and L-Pulse Position Modulation (L-PPM) are adopted for performance analysis. The imperfect Successive Interference Cancellation (SIC) scenario is also considered to emphasize the practicability of the system, with both types: perfect and imperfect SIC. Our proposed DLMs are used to solve the complexity problem and over fitting, thereby improving our system performance. To evaluate the proposed model performance, the obtained data is divided into training and testing ones.

The regular steps to achieve our contribution of this research are as follows.

  1. 1.

    The NOMA-MIMO-VLC system in Dixit and Kumar (2022) is utilized to gain the advantages of transmitter and receiver diversity by applying the EGC techniques and the RC coding.

  2. 2.

    The SIC perfect and imperfect scenarios are considered.

  3. 3.

    Two modulation techniques are used; OOK and L-PPM.

  4. 4.

    The BER performance of the NOMA-MIMO-VLC system is used with system parameters as the diversity order, power allocation coefficient, α, and user count.

  5. 5.

    Two different DLMs are used to minimize the BER and maximize the spectral efficiency. Our proposed models are used to solve the complexity, over fitting problem, and to reduce the computational time, thereby improving our framework performance.

The organization of the paper is as follows. Section 2 describes the proposed system model. Section 3 lists the methodologies that are used. Section 4 displays and discusses the obtained results and analysis based on simulation and assessment parameters. Section 5 summarizes the findings and suggests some recommendations for a future research.

2 System model

Our proposed indoor VLC system is shown in Fig. 1. It consists of a room with dimensions (x, y, z) with two transmitters (LEDs) fixed on the ceiling at (x1, y1, h) and (x2, y2, h), where h is the height of the room. There is a K number of users and each user has M photodiodes (PDs) as receivers. Thus, each user is located at (xkm, ykm, hr) m, where k and m are the user number and PDs number, 1 ≤ k ≤ K, m ∈ {1, 2, … , M}, hr is the height of the receiver. By this consideration, we achieve the MIMO technique, multiple input (two transmitters) and multiple output (K receivers). A Line of Sight (LoS) signal is considered and the Non-Line of Sight (NLoS) one is neglected which contains a negligible power (Dixit and Kumar 2020).

Fig. 1
figure 1

NOMA-MIMO-VLC system model of user k

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The transmitted signal from all LEDs are identical and are transmitted in the same time. Thus, x1 = x2, where x1 is the transmitted signal from first LED and x2 is the transmitted signal from second LED.

The received signal at the user is the summation of received optical power from both LEDs. As discussed, each user has number M photodiodes, so, at mth PD, where m ∈ {1, 2, … , M), the total received power can be expressed as

$$P_{m}^{k} = \mathop \sum \limits_{n = 1}^{2} \frac{{P_{t} }}{2}h_{mn}^{k} = P_{avg} \cdot \frac{{P_{t} }}{2}$$
(1)

where Pt and Pavg = Pt/2 are, respectively, the total and average transmitted power per LED and \({h}_{m}^{k}\) is the channel gain at the mth PD, where

$${h}_{m}^{k}=\sum_{n=1}^{2}{h}_{mn}^{k}$$
(2)

2.1 NOMA technique

Each user takes its number as user 1, user 2, … , user K. These numbers are ascendingly sorted related to the summation of channel gains received by M PDs in the user. Related to the concept of NOMA, the optical power assigned to the user has a descended ratio according to its position order. This means applying the fixed power allocation where the kth user is assigned power with

$$P_{k} = \alpha P_{k} + 1$$
(3)

where α is the coefficient of power allocation, that is, \(0 \le \alpha \le 1\).

So, the average power is

$$\mathop \sum \limits_{k = 1}^{K} P_{k} = P_{avg}$$
(4)

2.2 Perfect SIC decoding

SIC means successive interference cancellation between the users in our case. For example, if we have two users U1 and U2, then, the perfect SIC aims to cancel the interference at U1 related to U2 perfectly as shown in Dixit and Kumar (2022), where authors discussed mathematically the BER analysis.

When U2 is correctly decoded at U1 and hence, interference related to U2 will be cancelled out.

Perfectly, the BER of this case is given by

$${\text{P}}( \in_{1}^{\left( 1 \right)} = \, 1||| \in_{2}^{\left( 1 \right)} = \, 0) = {\text{ P}}( \in_{1}^{\left( 1 \right)} = \, 1){\text{P}}( \in_{2}^{\left( 1 \right)} = \, 0) = {\text{ BER}}_{1}^{1} \left( {1 \, - {\text{ BER}}_{1}^{2} } \right)$$
(5)

where BER11 and BER21 are the BER of self-decoding and decoding U2 at U1, respectively.

We solved this by using threshold detection method from Dixit and Kumar (2022), as

$${\text{BER}}_{1}^{2} = \, 0.25 \times [2{\text{Q}}(\chi_{1} {\text{P}}_{2} ) \, + {\text{ Q }}(\chi_{1} \left( {{\text{P}}_{2} - \, 2{\text{P}}_{1} } \right)) \, + {\text{ Q }}(\chi_{1} \left( {2{\text{P}}_{1} + {\text{P}}_{2} } \right))],$$
(6)
$${\text{BER}}_{1}^{1} = {\text{ Q}}(\chi_{1} {\text{P}}_{1} )$$
(7)
$$\chi_{2} = {\text{Rh}}_{2} /\sqrt {{\text{M}}\sigma_{{\text{n}}} } .$$
(8)
$$\chi_{1} = {\text{Rh}}_{1 } /\sqrt {{\text{M}}\sigma_{{\text{n}}} } .$$
(9)

where M is the total number of transmitters, h1, h2 are the channel gains LOS link for Users 1 and 2, respectively, R is the responsibility of the receiver photodiode, and σn is the variance at the mth PD of the user.

2.3 Imperfect SIC decoding

On the opposite side, the imperfect SIC means that there are some interferences as a result of incorrect decoding at U1 related to U2. Thus, interference is generated due to the incorrect decoding of U2.

Also, authors in Dixit and Kumar (2022) discussed this case and analyzed the BER performance. The BER of this case is given by

$$\begin{aligned} {\text{P}}( \in_{1}^{\left( 2 \right)} = \, 1||| \in_{2}^{\left( 2 \right)} = \, 1) & = \, 0.25 \times [{\text{Q}}\left( {\chi_{1} \left( {{\text{P}}_{1} + \, 2{\text{P}}_{2} } \right)} \right){\text{Q}}\left( {\chi_{1} {\text{P}}_{2} } \right) \, + {\text{ Q}}(\chi_{1} ({\text{P}}_{1} - 2{\text{P}}_{2} )){\text{Q}}\left( {\chi_{1} {\text{P}}_{2} } \right) \\ & \quad + {\text{ Q}}(\chi_{1} ({\text{P}}_{1} - 2{\text{P}}_{2} )){\text{Q}}(\chi_{1} ({\text{P}}_{2} - 2{\text{P}}_{1} )) \, + {\text{Q}}\left( {\chi_{1} \left( {{\text{P}}_{1} + \, 2{\text{P}}_{2} } \right)} \right){\text{Q}}\left( {\chi_{1} \left( {2{\text{P}}_{1} + {\text{ P}}_{2} } \right)} \right)]. \\ \end{aligned}$$
(10)

3 Methodology

3.1 Dataset

Related to Dixit and Kumar (2022), we extracted the results from the figures as data sets to insert this data to DLM for training and adjusted the parameters to obtain the optimum performance according to a decreased BER with respect to signal to noise ratio (SNR). Table 1 demonstrates the extracted data sets with relation between BER and SNR.

Table 1 Data sets between BER and SNR (Dixit and Kumar 2022)
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3.2 Proposed DLMs

In our proposed framework, two different DLMs are performed to improve our framework performance through decreasing BER, complexity and increasing spectral efficiency. To solve various problems, pre-trained models are trained on a large benchmark dataset. In this paper, the DLMs are utilized to overcome the high complexity and over fitting, leading to improve the spectral efficiency. Different DLMs, ResNet50V2, and InceptionResNetV2 are introduced to enhance the performance.

3.2.1 Proposed DLMs, ResNet50V2, and InceptionResNetV2

The DLMs may have similar parameters. Many different layers’ types are used in DLMs; max pooling and convolutional layers. While the input layer maintains the raw input data, the convolutional layer performs the dot product operation across all filters and data patches to compute the output volume. The activation function layer subjects the output of the first layer or the convolutional layer element by element. Additionally, the pool layer is in charge of reducing the volume and improving processing performance; both of which are input to the DLMs with the main objective of avoiding any kind of over fitting.

ResNet50V2 (Rizos and Kalogeraki 2020) is the ResNet50 upgrade version. This architecture is based on skip connections, which allow us to feed activation from one layer to the next. Inception-ResNet-v2 (Wang et al. 2021) is the inception mutual architecture with residual connections. Average pooling 2D is used to calculate the average for each patch of the feature map during the training process for ResNet50V2 and InceptionResNetV2 models. ResNet50V2 is a modified version of ResNet50 that performs better on our dataset than ResNet50 and ResNet101. ResNet50V2 performs better on our dataset than ResNet50 and ResNet101. The propagation formulation of the links between blocks was changed in ResNet50V2. Moreover, ResNet50V2 employs a more straightforward, single-scale processing unit with our datasets pass-through connections. While Multiple sized convolutional filters are mixed with residual connections in the InceptionResNetV2 block. The introduction of residual connections not only solves the degradation issue caused by deep structures, but it also shortens the training time. Furthermore, InceptionResNetV2 separates processing by scale, combines the outputs, and then repeats the process. Inception generates 1536 characteristics, whereas ResNet50V2 generates 2048. Based on previous explanation, it is observed that our proposed framework achieves the best performance with least complexity.

The parameters of the DLMs may be comparable. Each layer of a DLMs has the ability to transform one volume into another using a differentiable function. In our suggested system, the input layer stores the raw input matrix data, and the convolutional layer computes the output volume by performing the dot product operation over all filters and input patches. The output of the first layer or the convolutional layer is subjected to the activation function layer element by element. Furthermore, the pool layer is in responsibility of lowering volume and boosting processing performance, both of which are sent into the DLMs with the goal of avoiding overfitting. Moreover, our DLMs are trained by our collected data by dividing our datasets into 70% for training and 30% for testing and validation. Also, in the training stage, the DLMs are trained based on BER and SNR datasets. While, the in the testing stage, the input is SNR to predict the corresponding BER.

Furthermore, overfitting is a key concern when training our proposed framework using sample data. When the number of epochs utilized to train the model exceeds what is required, the training model learns patterns that are very unique to the sample data. As a result, the model is unable to perform well on a new dataset. The models perform well on the training set (sample data), but fail to perform well on the test set. In other words, by overfitting the training data, the models lose generalization capacity. The model should be trained for an ideal number of epochs to reduce overfitting and increase the capacity. A portion of the training data is dedicated to model validation, which is used to assess the model performance after each session of training. Loss and accuracy on both the training and validation sets are tracked to determine the epoch number at which the model begins overfitting. Therefore, our ideal epochs number are proposed in Table 2.

Table 2 Parameters of DLMs
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The activations are then flattened to create a vectorized feature map, and two fully connected layers are connected: one with 128 nodes and the other with 2-class classification \((x,z)\). The second fully connected layer activations are then fed into a softmax layer, which calculates the probability for each coordinate \((x,z).\) The parameters of the DLMs are explained in Table 2.

4 Results and discussion

Related to the derivation for BER in Dixit and Kumar (2022), the MIMO-NOMA-VLC system for two users and three users are numerically evaluated. Table 3 demonstrates the parameters used in the NOMA-MIMO-VLC system.

Table 3 Parameters used for NOMA-MIMO-VLC system (Dixit and Kumar 2022)
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The users U1, U2, and U3 are placed indoor at the locations of (3, 3, 0.8) m, (3.5, 1.5, 0.8) m, and (1, 1, 0.8) m, respectively. The fixed power allocation coefficient, α = 0.2. The users channel gain combination are in the range 5 to 10 related to what used in Dixit and Kumar (2022).

In this work we use two DL techniques: ResNet50V2 and InceptionResNetV2, for improving the performance of the NOMA-MIMO-VLC system. Generally, ResNet50V2 outperforms InceptionResNetV2 for different systems by ~ 6% improvement in different cases. The BER performance for two users OOK modulation: SISO, 2 × 2 and 3 × 2 MIMO-NOMA-VLC systems are simulated by both ResNet50V2 and InceptionResNetV2. Figure 2 shows the BER performance BER based on ResNet50V2. By comparing the results with those given in Dixit and Kumar (2022), we dedicate the improvement percentage for utilizing ResNet50V2 is nearly 15% for U2 and 8.6% for U1. Repeating the procedure by applying the InceptionResNetV2, that is displayed in Fig. 3, improvement of 3.4% for U1 and 5.8% for U2 and are noticed.

Fig. 2
figure 2

BER performance for two users OOK modulation SISO and NOMA-MIMO-VLC system based on ResNet50V2

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Fig. 3
figure 3

BER performance for two users OOK modulation SISO and NOMA-MIMO-VLC system based on InceptionResNetV2

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The BER performance versus α of the proposed systems based on ResNet50V2 InceptionResNetV2, respectively, is shown in Figs. 4 and 5. Related to the results in Dixit and Kumar (2022), applying ResNet50V2 in the system enhances the performance ~ 7.3% for U1 and ~ 9% for U2, while using InceptionResNetV2 achieves 3.5% improvement for U1 and 7.25% for U2.

Fig. 4
figure 4

BER versus α of two users OOK modulated SISO and MIMO-NOMA-VLC systems based on ResNet50V2

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Fig. 5
figure 5

BER versus α of two users OOK modulation SISO and MIMO-NOMA-VLC systems based on InceptionResNetV2

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The performance of three users OOK modulation SISO-NOMA-VLC, (2 × 2) and (3 × 2) SISO-NOMAVLC systems is shown in Figs. 6 and 7, based on ResNet50V2 and InceptionResNetV2, respectively. The InceptionResNetV2 outperforms the results in Dixit and Kumar (2022). In Fig. 6, an improvement of 26%, 205%, and 12.7% is achieved for U3, U2, and U1, respectively. While the corresponding improvement when using InceptionResNetV2, Fig. 7, is 20.9%, 20.3% and 9% for U3, U2, U1, respectively.

Fig. 6
figure 6

BER performance of three users OOK modulation SISO and MIMO-NOMA-VLC systems based on ResNet50V2

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Fig. 7
figure 7

BER performance of three users OOK modulation SISO and MIMO-NOMA-VLC systems based on InceptionResNetV2

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Figures 8 and 9 show the BER performance versus α of three users OOK modulation SISO and NOMA-MIMO-VLC; (2 × 2) and (3 × 2), based on ResNet50V2 and InceptionResNetV2 respectively. Utilizing ResNet50V2 outperforms the results in Dixit and Kumar (2022) with 9%, 6%, and 2.8% improvement for U3, U2 and U1, respectively. This improvement decreases when using InceptionResNetV2, where the improvement is 6%, 3%, and 1.5% for U3, U2 and U1, respectively.

Fig. 8
figure 8

BER performance versus α of three users OOK modulation SISO and MIMO-NOMA-VLC systems based on ResNet50V2

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Fig. 9
figure 9

BER performance versus α of three users OOK modulation SISO and MIMO-NOMA-VLC systems based on InceptionResNetV2

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Figures 10 and 11 demonstrate a comparison for the BER performance of two users and three users OOK modulated NOMA-MIMO-VLC systems. Using ResNet50V2 and InceptionResNetV2 gain an improvement of 19% and 13%, respectively, if compared with the results (Dixit and Kumar 2022).

Fig. 10
figure 10

BER performance: a comparison of two users and three users OOK modulated NOMA-MIMO-VLC system based on ResNet50V2

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Fig. 11
figure 11

BER performance: a comparison of two users and three users OOK modulated NOMA-MIMO-VLC system based on InceptionResNetV2

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A BER comparison of two users OOK and L-PPM modulation NOMA-MIMO-VLC system is illustrated in Figs.12 and 13. The improvement percentage when using ResNet50V2, in Fig. 12, is 9.9% and is 5.3% when using InceptionResNetV2, Fig. 13.

Fig. 12
figure 12

BER comparison of two users OOK and L-PPM modulation NOMA-MIMO-VLC system based on ResNet50V2

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Fig. 13
figure 13

BER comparison of two users OOK and L-PPM modulation NOMA-MIMO- VLC system Based on InceptionResNetV2

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The procedure is repeated in Figs. 14 and 15, showing a comparison for the BER comparison performance of two users and three users 4-PPM modulation NOMA-MIMO-VLC systems. Based on ResNet50V2, Fig. 14, the improvement is 23% better than that found in Dixit and Kumar (2022) and is decreased to 15.8% when using InceptionResNetV2, Fig. 15.

Fig. 14
figure 14

BER performance: a comparison of two users and three users 4-PPM modulation NOMA-MIMO-VLC system VLC based on ResNet50V2

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Fig. 15
figure 15

BER performance: a comparison of two users and three users 4-PPM modulation NOMA-MIMO-VLC system VLC based on InceptionResNetV2

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Figures 16 and 17 display the BER performance versus α for two users and three users 4-PPM modulation NOMA-MIMO-VLC systems based on ResNet50V2 and InceptionResNetV2. The improvement achieved is 16% and 10%, respectively, when utilizing ResNet50V2 and InceptionResNetV2.

Fig. 16
figure 16

BER versus α for two users and three 4-PPM modulation NOMA-MIMO-VLC based on ResNet50V2

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Fig. 17
figure 17

BER versus α for two users and three 4-PPM modulation NOMA-MIMO-VLC based on InceptionResNetV2

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The BER performance BER for two users OOK and L-PPM modulated NOMA-MIMO-VLC system is shown in Figs. 18 and 19, with perfect and imperfect SIC, based on ResNet50V2 and InceptionResNetV2, respectively. For the OOK modulation, an improvement of ~ 31% and ~ 24% is obtained when applying ResNet50V2 and InceptionResNetV2, respectively. Also, the 8-PPM outperforms other modulation techniques, showing an improvement of 36% and 20.4%, when using ResNet50V2 and InceptionResNetV2, respectively.

Fig. 18
figure 18

BER performance for two users OOK and L-PPM modulation NOMA-MIMO-VLC system with perfect and imperfect SIC based on ResNet50V2

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Fig. 19
figure 19

BER performance for two users OOK and L-PPM modulation NOMA-MIMO-VLC system with perfect and imperfect SIC based on InceptionResNetV2

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5 Conclusion

In this article, we utilize DL techniques; ResNet50V2 and InceptionResNetV2, for improving the performance of the NOMA-MIMO-VLC systems. The obtained results reveal that, in general, ResNet50V2 outperforms InceptionResNetV2 by nearly 6%. The BER performance for two users and three users OOK modulation SISO, 2 × 2 and 3 × 2 NOMA-MIMO-VLC systems are simulated by both ResNet50V2 and InceptionResNetV2. The obtained results are compared with that previously published ones, showing an enhancement in the performance for two users and three users systems.

The BER performance versus α is also investigated for two users and three users OOK modulation SISO and (2 × 2), (3 × 2) NOMA-MIMO-VLC systems based on ResNet50V2 and InceptionResNetV2. For two and three users, the performance is better enhanced when using ResNet50V2 than using InceptionResNetV2. The same notice is obtained also for the two users and three users 4-PPM modulation NOMA-MIMO-VLC system.

The improvement of using DL techniques is 16% and 10% when using ResNet50V2 and InceptionResNetV2, respectively. The BER performance is studied for two users OOK modulated NOMA-MIMO-VLC system with perfect and imperfect SIC, resulting in better improvement based on ResNet50V2 than InceptionResNetV2. The same conclusion, for both DL techniques, is achieved also when using 8-PPM which outperforms other modulation techniques.