Efficient power allocation for downlink MIMO-NOMA-based visible light communication systems

Abstract

Visible light communication (VLC) networks are emerging as a viable option to meet the ever-growing wireless data demand, primarily for indoor environments in recent years. However, developing a high data rate VLC system using off-the-shelf light emitting diodes (LEDs) is difficult due to the narrow modulation bandwidth of the white light emitting diodes. The existing studies have integrated Multiple-Input Multiple-Output (MIMO) and Non-Orthogonal Multiple Access (NOMA) schemes into the downlink VLC system so as improve its performance. Despite its apparent advantages, the NOMA scheme suffers from multi-user interference problem. For an efficient NOMA scheme, which uses successive interference cancellation (SIC) technique to decode the superimposed received signal of multipleusers, adaptive and fair power allocation methods are required. The existing power allocation schemes are inefficient when the number of users increases. In this work, a low complexity and efficient power allocation strategy is proposed for MIMO-NOMA based VLC systems. The proposed parametric power allocation strategy is based on the order of channel gain among end-users and overcomes the limitations of the existing schemes. The performance of the proposed power allocation scheme in an indoor 4$\times$4 MIMO-NOMA-based multi-user VLC was analyzed through simulation. A zero-forcing detection mechanism is used in the analysis. Simulation results using MATLAB depict the effectiveness of the proposed power allocation strategy. It achieves higher performance than the existing power allocation schemes, gain ratio power allocation (GRPA) and normalized gain difference power allocation (NGDPA) when the number of users are greater than two. Specifically, the proposed power allocation scheme improves the achievable sum rate of the NGDPA and GRPA schemes by 18.36% and 65.98%, respectively, in 4$\times$ 4 MIMO-based visible light communication networks with eight uniformly distributed users.

Keywords:

• Non-Orthogonal multiple accesses
• multiple input multiple output
• visible light communication
• MIMO
• NOMA

Reviewing Editor:

• Qingsong Ai; Wuhan University of Technology; China

Subjects:

• Communications & Information Processing
• Electrical Engineering Communications
• Radio
• Satellites
• Television & Audio
• Telecommunication

1. Introduction

As the demand for wireless data services has exponentially increased in recent years, it has also posed several challenging issues to the existing wireless technologies to meet the demands. In the last few decades, we have seen an abrupt increase in the number of mobile users requiring access to high data rate wireless services all over the world. This ultimate growth motivates mobile operators and standardization bodies to continuously find and develop new transmission technologies, network infrastructure solutions, protocols, and standards to enhance and achieve mobile subscribers’ demands. Furthermore, future wireless technologies, including cognitive networks, wireless sensor networks, delay tolerant networks, and vehicular communications networks, will require reliable and high data rate connections (Arnon et al., 2012).

VLC is a promising technology complementary to Radio Frequency (RF) spectrum (3 kHz–300 GHz), which operates in the visible light spectrum band (400 THz–780 THz), for future indoor wireless communications. The available bandwidth of visible light is 10,000 times larger than that of the microwave RF band (Parikh et al., 2013). Therefore, to alleviate spectrum shortage in the radio frequency band, we need to take advantage of the ubiquitous capacity of the visible light spectrum as a viable solution. It has the advantages of strong immunity to electromagnetic interference, license-free spectrum, high security, and low-cost front-ends. However, developing a high-capacity VLC system is difficult due to the limited modulation bandwidth of light-emitting diodes (LEDs). Various techniques, such as Multiple Input Multiple Output (MIMO) (Zeng et al., 2009), Orthogonal Frequency Division Multiple Access (OFDMA) (Sung et al., 2015), and Non- Orthogonal Multiple Access (NOMA) (Ding et al., 2014), were proposed in the literature to overcome the limitations of VLC systems.

In (Zeng et al., 2009), the MIMO method is proposed to achieve a high data rate. However, achieving a high data rate is challenging due to the low modulation bandwidth of the sources (several MHz). Still, the brightening levels indicated for occupation ensure that a very high signal-to-noise ratio (SNR) is available. Furthermore, the availability of many high SNR channels with low bandwidth makes MIMO techniques an appealing option for achieving high data rates. In Chen, Zhong, and Wu (2017), the authors proposed MIMO techniques to increase system capacity and improve indoor visible light communications coverage.

Orthogonal frequency division multiple access (OFDMA) has also been applied to VLC systems (Sung et al., 2015). It has been shown that the OFDMA based VLC system can effectively use the bandwidth of the available LED sources. However, the achievable data rate employing OFDMA is reduced due to the spectrum partitioning.

NOMA was found to achieve higher performance in high signal-to-noise ratio (SNR) situations because of the short distance between the transmitter and the receiver (Ding et al., 2014) in VLC networks. From this fact, it is helpful to apply NOMA in downlink VLC systems. In (Chen, Zhong, & Wu, 2017; Guan et al., 2016; Marshoud et al., 2016), the performance of NOMA-based VLC has been examined. In (Marshoud et al., 2016), NOMA is recommended as a potential candidate for high-speed VLC systems with gain ratio power allocation (GRPA) strategy. Moreover, more advanced power allocation techniques for NOMA-based VLC systems were proposed in (Chen, Zhong, & Wu, 2017), with moderately high computational complexity. In (Guan et al., 2016), a phase pre-distortion approach was proposed to improve the error rate performance of uplink NOMA-based visible light communication systems.

In (Lin et al., 2017), a VLC system based on MIMO-NOMA was experimentally verified. The authors experimented with a single carrier mode of transmission, using the frequency domain successive interference cancellation (SIC), but without considering the power allocation problem. While applying MIMO, the power allocation method of the single LED NOMA-VLC systems can’t be straightforwardly adopted in MIMO-NOMA-based VLC systems.

The authors of Chen, Zhong, Yang, et al. (2017) proposed a power allocation algorithm called normalized gain difference power allocation (NGDPA) approach aimed at reducing the complexity and increasing the efficiency of $2\times 2$ MIMO-NOMA-VLC systems with multiple users. The authors applied the NGDPA technique for a small number of users. The authors considered only two and three users, which is not practically applicable.

It is essential to have efficient power allocation techniques to improve the end user performance in VLC systems. In NOMA schemes, which use SIC to decode the superimposed users’ signal, optimal performance is obtained when there is a large channel gain difference between the multiple users (Yang et al., 2017). Unfortunately, since the VLC cells in indoor communication has a small coverage area, the users in the same cell may have similar or close channel gains (Marshoud et al., 2018). This increases the interference between the signals of the users, resulting in a reduction in the performance of the system. Moreover, the performance of existing power allocation schemes, NGDPA (Chen, Zhong, Yang, et al., 2017) and GRPA (Marshoud et al., 2016), decreases as the number of users increases in indoor VLC systems. For example, when the channel gains of all the users are equal, the GRPA scheme allocates equal power to all users which makes the interference between users significantly high and makes it difficult to decode the user signals. In this case, the NGDPA scheme allocates all the available power to only one user, the user whose signal is first decoded, and the signals of the remaining users are lost due to noise.

In this study, unlike the previous works which considered $2\times 2$ MIMO-NOMA-VLC system (Chen, Zhong, Yang, et al., 2017), a $4\times 4$ MIMO-NOMA-VLC system is considered for downlink communication. Along this, to enhance the performance of end users, a new parametric power allocation strategy is proposed. The proposed power allocation scheme uses channel gains as information to systematically allocate power in the NOMA method. It allocates different power to different users even when all the users have the same channel gains. This power allocation scheme outperforms the existing power allocation schemes when the number of users are greater than two.

The remaining sections of the paper are organized as follows. In Section 2, the proposed system model is presented. In Section 3, different power allocation schemes are discussed. In Section 3.1, the analysis of the signal-to-noise ratio and achievable sumrate of the proposed VLC system is given. The simulation results and discussions are provided in Section 4. Finally, the concluding remarks are provided in Section 5.

2. The 4 × 4 MIMO-NOMA-based VLC system model

Multiple-input multiple-output (MIMO) is a technique used in wireless communication systems to improve the performance of the system by using multiple antennas at both the transmitter and the receiver. In visible light communication (VLC) systems, MIMO can also be used to improve the performance of the system by increasing the data rate, improving the coverage area, and increasing the robustness of the system to interference and fading. Overall, the use of MIMO in VLC systems can significantly improve the performance and reliability of the system, making it a valuable technique for a wide range of applications (Chen et al., 2019).

In order to realize a multi-user VLC system, a multiple access (MA) scheme is required. Many MA schemes have been proposed for VLC such as time division multiple access (TDMA) (Abdelhady et al., 2019), space division multiple access (SDMA) (Chen & Haas, 2015), code division multiple access (CDMA) (Qiu et al., 2018), and orthogonal frequency division multiple access (OFDMA) (Bawazir et al., 2018). One of the recently proposed MA schemes is the non-orthogonal multiple access (NOMA) which is characterized by its high spectral efficiency and its ability to support more users (Liu et al., 2019). The performance of NOMA scheme is highly dependent on the power allocation scheme.

In this work, the downlink of an indoor MIMO-NOMA-based VLC network with multiple LEDs simultaneously serving multiple users is considered. Assume that the LED transmitter is installed on the ceiling, which has i down confronting LEDs and communicates with K users; every user has j up-facing photodetectors (PDs). The configuration in Chen, Zhong, Yang, et al. (2017), which is a $2\times 2$ MIMO-VLC system, is extended to a $4\times 4$ MIMO-VLC network. Without loss of generality, assume four LEDs $(i=1,2,3,4)$ on the ceiling serving K number of users, and each user is equipped with four PDs $(j=1,2,3,4).$ The parameters of all LEDs and PDs are assumed to be identical. Therefore, the system considered represents a typical MIMO-VLC system model. Multiple PDs are equipped with each NOMA user who can utilize the entire available modulation bandwidth of LEDs. The block diagram of such a system is shown in Figure 1. The transmitter consists of DC-biased Optical OFDM, Multiplexer module and LEDs. The receiver part of the system is made up of PDs, demultiplexer, successive interference cancellation (SIC) and DC offset optical OFDM demodulator. The components of the proposed VLC system model are described in the following sections.

Figure 1. The proposed VLC system block diagram.

2.1. DC-biased optical OFDM method

In the Intensity-Modulated/Direct Detection (IM/DD) optical system, the light intensity can neither be complex nor negative. Therefore, the input feeding the Inverse Fast Fourier Transform (IFFT) during the OFDM generation flow is constrained to have a Hermitian symmetry to deliver a real-valued signal (Deng et al., 2019). In addition, a DC bias is usually applied to ensure a positive output (Li et al., 2012).

In this work, a 4 QAM-based DC biased optical OFDM modulation is considered. According to the power domain multiplexing NOMA principle, the $i^{\mathit{\text{th}}}$ transmitter LED simultaneously transmits the message data $s_i$ to all K users by using all the bandwidth through a superposition coding technique at the transmitter side. Thus, after DC biasing optical OFDM modulation, the transmitted signal of $i^{\mathit{\text{th}}}$ LED is the combined signals of K users, which can be expressed as (Chen, Zhong, Yang, et al., 2017): (1) $$\begin{array}{c}{x}_{i,k}=\sum _{k=1}^K\sqrt{{\rho }_{i,k}}{s}_{i,k}(t)+I_{\mathit{\text{DC}}}\hfill \end{array}$$(1)

Where $s_{i,k}(t)$ is the modulated message signal intended for the $k^{\mathit{\text{th}}}(k=1,2,\dots ,K)$ user from the $i^{\mathit{\text{th}}}$ LED, ${\rho }_{i,k}$ is the electrical power allocated for the $k^{\mathit{\text{th}}}$ user in the $i^{\mathit{\text{th}}}$ LED, and $I_{\mathit{\text{DC}}}$ is the DC bias provided to each LED to ensure that the instantaneous transmitted signal remains to be positive. This superimposed multi-user signal is transmitted by LEDs through the optical channel and received by PDs.

2.2. LOS propagation channel model for VLC

The summed-up Lambertian radiation pattern model was applied to find the LED’s line-of-sight (LOS) irradiance. The geometric model of LOS transmission is shown in Figure 2. Compared to non-line-of-sight (NLOS), the line-of-sight provides a much higher intensity of light in the VLC system (Zeng et al., 2009). Therefore, the line-of-sight channel gain of the optical link between $i^{\mathit{\text{th}}}$ LED and the $j^{\mathit{\text{th}}}$ PD of the $k^{\mathit{\text{th}}}$ user is given by Kahn and Barry (1997): (2) $$\begin{array}{c}{h}_{\mathit{\text{ji}},k}=\{\begin{array}{cc}\frac{(m+1)A_{\mathit{\text{PD}}}}{2\pi d^2}{T}_s(\psi )g(\psi ){\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}}^m(\varphi )\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}(\psi ),\hfill & 0\le \psi \le {\psi }_c\hfill \\ 0,\hfill & \psi >{\psi }_c\hfill \end{array}\hfill \end{array}$$(2) (3) $$\begin{array}{c}m=-\frac{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}2}{\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}(\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}({\Phi }_{1/2}))}\hfill \end{array}$$(3)

Figure 2. Geometric model of LOS transmission.

Where d is the LOS distance between the $i^{\mathit{\text{th}}}$ LED and the $j^{\mathit{\text{th}}}$ PD of the $k^{\mathit{\text{th}}}$ user, m is the lambertian emission order and ${\Phi }_{1/2}$ is the semi-angle at half power of the LED, $A_{\mathit{\text{PD}}}$ is an active area of PD, $\varphi$ is the transmitter viewing angle, $\psi$ is the angle of incidence with respect to the receiver axis, ${\psi }_c$ is the concentrator field of view (FOV), n is the internal refractive index of the concentrator, $T_s(\psi )$ is the gain of the optical filter adopted at the receiver and $g(\psi )$ represents the gain of the non-imaging concentrator, which is given by Kahn and Barry (1997): (4) $$\begin{array}{c}g(\psi )=\{\begin{array}{cc}\frac{n^2}{{\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}}^2({\psi }_c)},\hfill & 0\le \psi \le {\psi }_c\hfill \\ 0,\hfill & \psi >{\psi }_c\hfill \end{array}\hfill \end{array}$$(4)

After going through a free-space optical wireless channel, the optical intensity signal is changed at the PD receivers of the $k^{\mathit{\text{th}}}$ user into a current signal by means of photo-electric conversion. Further, the constant DC offset $I_{\mathit{\text{DC}}}$ is removed in the electrical domain. In general, noise is assumed to be introduced in the electrical domain. As a result, the electrical signal received at the $j^{\mathit{\text{th}}}$ PD of the $k^{\mathit{\text{th}}}$ user is expressed as: (5) $$\begin{array}{c}{y}_{j,k}=\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}\sum _{i=1}^4{h}_{\mathit{\text{ji}},k}{x}_i(t)+n_{j,k}\hfill \end{array}$$(5)

Where $\mathfrak{R}$ is responsivity of PD, $P_{\mathit{\text{opt}}}$ is optical output power, $\mu$ is modulation index, $n_{j,k}$ is the Additive Gaussian Noise with zero mean and variance ${\sigma }_{j,k}^2,$ and $x_i(t)$ is the transmitted signal. The received electrical signal of $k^{\mathit{\text{th}}}$ user can be expressed in matrix form as: (6) $$\begin{array}{c}{Y}_k=\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}{H}_{k}X+N_k\hfill \end{array}$$(6)

Where $X={[x_1 x_2 x_3 x_4]}^T$ is transmitted electrical signal vector, $H_k$ is channel gain matrix and $N_k$ is Additive White Noise vector. The channel gain matrix can be given as: (7) $$\begin{array}{c}{H}_k=\left[\begin{array}{cccc}{h}_{11}& h_{12}& h_{13}& h_{14}\\ h_{21}& h_{22}& h_{23}& h_{24}\\ h_{31}& h_{32}& h_{33}& h_{34}\\ h_{41}& h_{42}& h_{43}& h_{44}\end{array}\right]\hfill \end{array}$$(7)

2.3. Zero forcing (ZF) MIMO receiver

Zero Forcing (ZF) detection is a viable and straightforward technique for recovering multiple data streams at the receiver. But it needs knowledge of channel state information. In this work, ZF-MIMO de-multiplexing is used to retrieve the transmitted data with primary channel inversion due to its low complexity (Burton et al., 2014). The normalized estimated electrical signal vector at the output of the ZF-MIMO receiver of the $k^{\mathit{\text{th}}}$ user is given as (Chen, Zhong, and Wu, 2017): (8) $$\begin{array}{c}{\tilde{X}}_k=\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}{H}_k}{Y}_k=X_k+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{\widehat{H}}_k{N}_k\hfill \end{array}$$(8)

Where ${\widehat{H}}_k\stackrel{\Delta }{=}{(H_k^T{H}_k)}^{-1}{H}_k^T$ is the pseudo inverse of $H_k$ and ${(.)}^T$ denotes conjugate transpose operation. In VLC channels, the diversity gain is severely limited because of the high correlation between signal paths; hence conventional MIMO-VLC systems are symmetric (i.e. It is $4\times 4$ in this case). Therefore, the pseudo inverse of $H_k$ is simplified to ${\widehat{H}}_k\stackrel{\Delta }{=}{H}_k^{-1}$ and the above equation (8)(8) $$\begin{array}{c}{\tilde{X}}_k=\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}{H}_k}{Y}_k=X_k+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{\widehat{H}}_k{N}_k\hfill \end{array}$$(8) is simplified to: (9) $$\begin{array}{c}{\tilde{X}}_k=X_k+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{W}_k\hfill \end{array}$$(9)

Where $W_k\stackrel{\Delta }{=}{H}_k^{-1}{N}_k.$ As a result, the estimated signal received by $k^{\mathit{\text{th}}}$ user from $i^{\mathit{\text{th}}}$ LED is: (10) $$\begin{array}{c}{\tilde{x}}_{i,k}(t)=\sum _{k=1}^K\sqrt{{\rho }_{i,k}}{s}_{i,k}(t)+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{w}_{i,k}+I_{\mathit{\text{DC}}}\hfill \end{array}$$(10)

Where $w_{i,k}$ is the $i^{\mathit{\text{th}}}$ element of the vector $W_k.$ From the above Equation (10)(10) $$\begin{array}{c}{\tilde{x}}_{i,k}(t)=\sum _{k=1}^K\sqrt{{\rho }_{i,k}}{s}_{i,k}(t)+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{w}_{i,k}+I_{\mathit{\text{DC}}}\hfill \end{array}$$(10) , it is obvious that the estimated signal contains a message signal from all users. In order to extract the desired message signal, successive interference cancellation (SIC) is applied at the receiver. To apply SIC, the decoding order of the users must be determined for every LED. Sorting the users is done by finding the optical channel’s sum of every user for every LED. Without loss of generality, it is assumed that K users for the $i^{\mathit{\text{th}}}$ LED are sorted in descending order of their sum optical channel gains as follows: (11) $$\begin{array}{c}{h}_{1i,1}+h_{2i,1}+h_{3i,1}+h_{4i,1}>h_{1i,2}+h_{2i,2}+h_{3i,2}+\hspace{0.17em}{h}_{4i,2}>\dots >h_{1i,K}+h_{2i,K}+h_{3i,K}+h_{4i,K}\hfill \end{array}$$(11)

In SIC, the strong users decode the signals with high power and subtract them from the received signal until the desired signal is decoded. With reference to the $i^{\mathit{\text{th}}}$ LED, the decoding order is then set to be (Chen, Zhong, Yang, et al., 2017). (12) $$\begin{array}{c}{O}_{i,1}<O_{i,2}<\dots <O_{i,K}\hfill \end{array}$$(12)

Finally, at $k^{\mathit{\text{th}}}$ user, the estimated message signal from $i^{\mathit{\text{th}}}$ LED can be expressed as (Yin et al., 2016): (13) $$\begin{array}{c}{\tilde{s}}_{i,k}(t)=\sqrt{{\rho }_{i,k}}{s}_{i,k}(t)+\sum _{k\hspace{0.17em}=\hspace{0.17em}k\hspace{0.17em}+\hspace{0.17em}1}^K\sqrt{{\rho }_{i,k}}{s}_{i,k}(t)+\frac{1}{\mu \mathfrak{R}{P}_{\mathit{\text{opt}}}}{w}_{i,k}\hfill \end{array}$$(13)

3. The proposed power allocation scheme

In order to achieve high throughput and minimize the unfairness problem among end-users, optimal power allocation is critical in NOMA systems. Therefore, different power allocation strategies have been proposed for VLC networks. Some of them are fixed power allocation (FPA), gain ratio power allocation (GRPA) (Marshoud et al., 2016), and normalized gain difference power allocation (NGDPA) (Chen, Zhong, Yang, et al., 2017). The main target behind these different power allocation strategies are due to the fact that a minimum power level is enough to decode signals for good channel condition users, we need to assign a small fraction of power for those users, and allocating a large fraction of power for bad channel condition users helps to filter out and decode their information from the superimposed signal using SIC technique. In the GRPA, the power allocation technique is determined by the optical channel gains of users compared to the gain of the first sorted user. In the NGDPA, the optical channel gain difference is used to allocate power. These algorithms have not been evaluated for multi-user for a VLC system comprising 4 × 4 MIMO and NOMA for a large number of users.

In this work, a parametric new power-splitting strategy is proposed to enhance the power allocation efficiency among end-users. Based on the channel gain order presented in the Equation (12)(12) $$\begin{array}{c}{O}_{i,1}<O_{i,2}<\dots <O_{i,K}\hfill \end{array}$$(12) , the relationship between electrical powers allocated to user k and $(k+1)$ following $i^{\mathit{\text{th}}}$ LED is given by the following relationship. (14) $$\begin{array}{c}{\rho }_{i,k}={\left(\beta +\frac{h_{1i,k\hspace{0.17em}+\hspace{0.17em}1}\hspace{0.17em}+\hspace{0.17em}{h}_{2i,k\hspace{0.17em}+\hspace{0.17em}1}\hspace{0.17em}+\hspace{0.17em}{h}_{3i,k\hspace{0.17em}+\hspace{0.17em}1}\hspace{0.17em}+\hspace{0.17em}{h}_{4i,k\hspace{0.17em}+\hspace{0.17em}1}}{h_{1i,1}\hspace{0.17em}+\hspace{0.17em}{h}_{2i,1}\hspace{0.17em}+\hspace{0.17em}{h}_{3i,1}\hspace{0.17em}+\hspace{0.17em}{h}_{4i,1}}\right)}^{-k}{\rho }_{i,k+1}\hfill \end{array}$$(14)

Where $\beta$ is a parameteric constant whose value is given by $\beta \ge 1.$ The addition of $\beta$ makes the proposed power allocation a parametric gain ratio power allocation (PGRPA) scheme. This helps us to control the power allocation fairness among users in various channel conditions. Let us compare the proposed PA algorithm with the existing benchmarks in certain intuitive scenarios. When the channel gains of all the users are equal, GRPA assigns equal power to all the users and NGDPA assigns all the available power to the users with the worst channel condition. In this case, GRPA multi-user interference to all the users becomes high to decode the signals using SIC. Similarly, in NGDPA, the user with the worst channel condition is decoded seamlessly, whereas the signals of the other users are lost due to noise. In the proposed PA method, even when the channels gains of all users are equal, different power is allocated based on their order of decoding. This makes the proposed algorithm robust to varying channel conditions in VLC networks.

To maintain overall electrical power for the $i^{\mathit{\text{th}}}$ transmitter LED to be constant, the power allocation should satisfy the following constraint (Chen, Zhong, Yang, et al., 2017). (15) $$\begin{array}{c}\sum _{k=1}^K{\rho }_{i,k}=\mathit{\text{Pe}}=1\hfill \end{array}$$(15)

Where Pe is total electrical power. When applying SIC decoding at the receiver, the users with lower decoding order should handle higher values of interference and also encounter poorer channel conditions. Therefore, to ensure fairness among users, the users with lower decoding order are allocated higher power. The relationship between power allocated to successive users for $(k>1)$ is given by: (16) $$\begin{array}{c}{\rho }_{i,k}={\alpha }_{i,k}{\rho }_{i,k-1}\hfill \end{array}$$(16)

Where $0\le {\alpha }_{i,k}\le 1$ is the power allocation coefficient of the $k^{\mathit{\text{th}}}$ user at the $i^{\mathit{\text{th}}}$ LED. Thus, depending up on the above constraint in Equation (15)(15) $$\begin{array}{c}\sum _{k=1}^K{\rho }_{i,k}=\mathit{\text{Pe}}=1\hfill \end{array}$$(15) , the power allocated to the first user ($k=1$) is given by: (17) $$\begin{array}{c}{\rho }_{i,1}=\frac{\mathit{\text{Pe}}}{1+\sum _{j=2}^K{\prod }_{m=1}^{j-1}{\alpha }_{i,m}}\hfill \end{array}$$(17)

From Equations (16)(16) $$\begin{array}{c}{\rho }_{i,k}={\alpha }_{i,k}{\rho }_{i,k-1}\hfill \end{array}$$(16) and (17)(17) $$\begin{array}{c}{\rho }_{i,1}=\frac{\mathit{\text{Pe}}}{1+\sum _{j=2}^K{\prod }_{m=1}^{j-1}{\alpha }_{i,m}}\hfill \end{array}$$(17) equations, power allocated to the $k^{\mathit{\text{th}}}$ user ($k>1$) can be generalized as: (18) $$\begin{array}{c}{\rho }_{i,k}=\frac{\mathit{\text{Pe}}{\prod }_{m=1}^{k-1}{\alpha }_{i,m}}{1+{\sum }_{j=2}^K{\prod }_{m=1}^{j-1}{\alpha }_{i,m}}\hfill \end{array}$$(18)

This ensures that the proposed power allocation scheme more realistically allocates power for each transmitter based on the relative channel gain ratio which improves the performance of the power domain NOMA system.

3.1. Performance metrics

In this section, the metrics used to assess the performance of the proposed system are presented. These metrics are signal-to-noise ratio (SNR), achievable sum rate and achievable data rate as illustrated in the following.

In the VLC systems, the noise vector is composed of the shot noise and the thermal noise. It is expected that the total noise is dominated by the White Gaussian component (Cui et al., 2010). Thus, the noise generated has a Gaussian distribution of zero mean and variance of ${\sigma }_{\mathit{\text{nk}}}^2$ (Kahn & Barry, 1997). (19) $$\begin{array}{c}{\sigma }_{\mathit{\text{nk}}}^2={\sigma }_{\mathit{\text{sh}}}^2+{\sigma }_{\mathit{\text{th}}}^2\hfill \end{array}$$(19)

Where ${\sigma }_{\mathit{\text{nk}}}^2$ is total noise variances, ${\sigma }_{\mathit{\text{sh}}}^2$ is shot noise variances and ${\sigma }_{\mathit{\text{th}}}^2$ is thermal noise variances. The shot noise variances at $k^{\mathit{\text{th}}}$ the user is determined by Komine and Nakagawa (2004): (20) $$\begin{array}{c}{\sigma }_{\mathit{\text{sh}}}^2=2\times q\times \left(P_{\mathit{\text{rx}}}+I_{\mathit{\text{bg}}}\times I_2\right)\times B\hfill \end{array}$$(20)

Where q is the electronic charge, $P_{\mathit{\text{rx}}}$ is the received optical power by $k^{\mathit{\text{th}}}$ the user, B is the equivalent noise bandwidth, $I_{\mathit{\text{bg}}}$ is the photocurrent due to background radiation and $I_2$ is the noise bandwidth factor. The thermal noise variances at $k^{\mathit{\text{th}}}$ user is determined by Komine and Nakagawa (2004): (21) $$\begin{array}{c}{\sigma }_{\mathit{\text{th}}}^2=8\times \pi \times K\times T\times \eta \times A_{\mathit{\text{PD}}}\times B^2\hfill \\ \times \left(\frac{I_2}{G}+2\times \pi \times \gamma \times \eta \times A_{\mathit{\text{PD}}}\times I_3\times \frac{B}{g_m}\right)\hfill \end{array}$$(21)

Where K is Boltzmann constant, T is Temperature in Kelvin, G is open-loop voltage gain, $\eta$ is fixed capacitance, $\gamma$ is Field Effect Transistor (FET) channel noise factor, $I_3$ = 0.0868 (Komine & Nakagawa, 2004) and $g_m$ is FET Transconductance. Signal to Noise Ratio (SNR) is determined as follows: (22) $$\begin{array}{c}{\mathit{\text{SNR}}}_{i,k}=\frac{P_{\mathit{\text{rx}}}}{{\sigma }_{\mathit{\text{nk}}}^2}\hfill \end{array}$$(22)

The sum rate (SR) of $4\times 4$ MIMO-VLC system of K users is given by: (23) $$\begin{array}{c}S{R}_K=\left(\sum _{k=1}^K\sum _{i=1}^4\frac{\frac{\mathit{\text{Bw}}}{2}{\hspace{0.17em}\mathrm{\text{log}}}_2(1+{\mathit{\text{SNR}}}_{i,k})}{{10}^6}\right)\hfill \end{array}$$(23)

Where Bw is the transmission bandwidth, ${\mathit{\text{SNR}}}_{i,k}$ is the signal-to-noise ratio for each link between the $i^{\mathit{\text{th}}}$ LED and the $k^{\mathit{\text{th}}}$ user.

The achievable data rate of $k^{\mathit{\text{th}}}$ user ($R_k$) is calculated by considering the power received from all LEDs and it is determined as follows: (24) $$\begin{array}{c}{R}_k=\left(\sum _{i=1}^4\frac{\frac{\mathit{\text{Bw}}}{2}{\hspace{0.17em}\mathrm{\text{log}}}_2(1+{\mathit{\text{SNR}}}_{i,k})}{{10}^6}\right)\hfill \end{array}$$(24)

The sum rate gain of the proposed NOMA (PNOMA) over NGDPA (${\mathit{\text{SRG}}}_{\frac{\mathit{\text{PNOMA}}}{\mathit{\text{NGDPA}}}}$) is given by: (25) $$\begin{array}{c}\mathit{\text{SRG}}{}_{\frac{\mathit{\text{PNOMA}}}{\mathit{\text{NGDPA}}}}=\left(\frac{S{R}_{\mathit{\text{PNOMA}}}-S{R}_{\mathit{\text{NGDPA}}}}{S{R}_{\mathit{\text{NGDPA}}}}\right)\times 100\hfill \end{array}$$(25)

Where $S{R}_{\mathit{\text{NGDPA}}}$ is a sum-rate gain of NGDPA based power allocation NOMA scheme. Similarly, the sum rate gain of the proposed NOMA (PNOMA) scheme over the GRPA scheme (${\mathit{\text{SRG}}}_{\frac{\mathit{\text{PNOMA}}}{\mathit{\text{GRPA}}}}$) is given by: (26) $$\begin{array}{c}\mathit{\text{SRG}}{}_{\frac{\mathit{\text{PNOMA}}}{\mathit{\text{GRPA}}}}=\left(\frac{S{R}_{\mathit{\text{PNOMA}}}-S{R}_{\mathit{\text{GRPA}}}}{S{R}_{\mathit{\text{GRPA}}}}\right)\times 100\hfill \end{array}$$(26)

Where $S{R}_{\mathit{\text{PNOMA}}}$ is the sum-rate gain of the proposed NOMA scheme, and $S{R}_{\mathit{\text{GRPA}}}$ is the sum-rate of the GRPA scheme.

4. Simulation results and discussions

4.1. Simulation setup

The performance, in terms of the total sum rate, sum rate gain, and achievable rate, of different power allocation techniques for NOMA-based MIMO-VLC systems have been analyzed. In the simulations, the system is set up with four LED on the ceiling used as a transmitter, and each user is equipped with four PD inside a typical room as shown in Figure 3. The normalized offset has been defined for system coverage. The distance from user one $(k=1),$ which has the largest channel gain sum, to user K, which has the smallest channel gain sum, is fixed as $r(m).$ The maximum distance from the user $(k=1)$ to the edge of the system is $R(m).$ All users are uniformly distributed over the coverage area. The normalized offset of user K with respect to the user $(k=1)$ is defined as $\frac{r}{R},$ and the normalized offset of user k with respect to the user K is defined as $\frac{(k-1)r}{(K-1)R}.$ The parameters used for the performance analysis of the proposed system using simulation in matlab are given in Table 1.

Figure 3. Illustration of a $4\times 4$ MIMO-NOMA based VLC system with K users.

Table 1. Simulation parameters.

Parameter	Symbol	Value
Number of LEDs	i	4
Number of PDs	j	4
System Coverage	R	4 m
Number of User	K	$2-10$ (Variable)
Receiver height	h	0.85 m
Transmitted power	$P_{\mathit{\text{opt}}}$	10 W
Modulation index	$\mu$	0.5
Transmitter semi angle	${\Phi }_{1/2}$	${60}^{°}$
Separation between LED	$d_{\mathit{\text{LED}}}$	1 m
Electronic charge	q	$1.6\times e^{-19}C$
Noise bandwidth factor	$I_2$	0.562
Boltzmann constant	K	$1.38\times e^{-23}$
Absolute temprature	T	300K
Fixed capacitance	$\eta$	$112e^{-12}/1e^{-4}(F/m)$
Open loop voltage gain	G	10
FET transconductance	$g_m$	$30e^{-3}S$
FET channel noise factor	$\gamma$	1.5
Receiver field of view	${\psi }_c$	$72°$
Total bandwidth	Bw	10 MHz
Photodiode responsivity	$\mathfrak{R}$	0.53 A/W
Refractive index	n	1.5
Separation between PD	$d_{\mathit{\text{PD}}}$	4 cm
Gain of optical filter	$T_s(\psi )$	0.9
Physical area of a PD	$A_{\mathit{\text{PD}}}$	1 $c{m}^2$
The power allocation parameter	$\beta$	1

4.2. Results and discussions

4.2.1. Power allocation coefficient

The power allocated to five users by different power allocation schemes and the proposed scheme are shown in Figure 4. The figure shows the power allocated to each user using the proposed power allocation scheme with different parameter values, GRPA and NGDPA for comparison. From the figure, we can deduce that when the $\beta$ value is small (close to 1), the power allocation scheme resembles GRPA. Whereas, when the $\beta$ is greater than 1.5, the PGRPA scheme resembles that of NGDPA scheme.

Figure 4. Power allocated for each user of a VLC system with five users.

Table 2 shows power allocation coefficients for LED 1, LED 2, LED 3, and LED 4 by using GRPA, NGDPA, and the proposed NOMA techniques for five ($k=5$) users in the considered system coverage. The power allocation coefficient is calculated according to the constraint explained in Equation (15)(15) $$\begin{array}{c}\sum _{k=1}^K{\rho }_{i,k}=\mathit{\text{Pe}}=1\hfill \end{array}$$(15) . Fair power allocation is essential among users. Therefore, from the table, the proposed NOMA fairly allocates power among users. The impact of this fair power allocation coefficients on achievable sum rate, achievable average sum rate, and sum-rate gain is illustrated in the following sections.

Table 2. Comparison of power allocation coefficients for five users.

	Power Allocation Coefficient
	LED 1 and LED 3			LED 2 and LED 4
User No	GRPA	NGDPA	Proposed	GRPA	NGDPA	Proposed
1	2.4415$\times e^{-7}$	0.0050	0.0288	0.0023	2.7887$\times e^{-6}$	0.0054
2	4.3278$\times e^{-7}$	0.0201	0.0505	0.0028	2.7907$\times e^{-5}$	0.0102
3	3.4052$\times e^{-6}$	0.0815	0.1140	0.9726	0.9150	0.7737
4	3.2713$\times e^{-4}$	0.2587	0.2619	0.0039	0.0028	0.0369
5	0.9996	0.6345	0.5447	0.0185	0.0822	0.1738

4.2.2. Achievable sum rate vs. normalized offset

Figure 5 shows the achievable sum rate of proposed NOMA, NGDPA, and GRPA versus normalized offset for user five $(k=5)$ and user six $(k=6).$ In both cases, the proposed NOMA has achieved a higher sum rate than the others. The proposed NOMA achieves a sum rate of 264.89 Mbit/s and 261.89 Mbit/s for $(k=5)$ and $(k=6),$ respectively at the edge of the system. NGDPA achieves a sum rate of 254.45 Mbit/s and 235.28 Mbit/s for $(k=5)$ and $(k=6),$ respectively at the edge of the system. GRPA achieves a sum rate of 156.59 Mbit/s and 152.27 Mbits/s for $(k=5)$ and $(k=6),$ respectively at the edge of the system.

Figure 5. Achievable Sum rate of Proposed NOMA, NGDPA and GRPA vs. r/R ($k=5,6$).

Figure 6 depicts the achievable sum rate of the proposed NOMA, NGDPA, and GRPA versus normalized offset for seven $(k=7)$ and eight $(k=8).$ In both cases, the proposed NOMA achieves a higher sum rate. The proposed NOMA achieves a sum rate of 252.55 Mbit/s and 228.85 Mbit/s for $(k=7)$ and $(k=8)$ respectively at the edge of the system. NGDPA achieves a sum rate of 223.49 Mbit/s and 193.36 Mbit/s for $(k=7)$ and $(k=8)$ respectively at the edge of the system. GRPA achieves a sum rate of 148.30 Mbit/s and 137.88 Mbits/s for $(k=7)$ and $(k=8)$ respectively at the border of the system as shown in Table 3.

Figure 6. Achievable Sum rate of the proposed NOMA, NGDPA and GRPA vs. r/R ($k=7,8,9,10$).

Table 3. Achievable sumrate comparison of the proposed NOMA scheme with GRPA and NGDPA for 7 and 8 users.

	$S{R}_{\mathit{\text{GRPA}}}$		$S{R}_{\mathit{\text{NGDPA}}}$		$S{R}_{\mathit{\text{PNOMA}}}$
r/R	k = 7	k = 8	k = 7	k = 8	k = 7	k = 8
0.1	268.94	268.94	265.67	265.65	267.17	266.64
0.2	268.93	268.90	261.12	260.72	265.11	263.54
0.3	268.70	267.90	256.69	255.48	263.23	259.97
0.4	265.18	257.68	251.31	249.83	261.10	256.33
0.5	249.74	243.48	246.44	243.28	261.70	253.07
0.6	233.17	229.36	244.18	238.47	263.63	251.90
0.7	215.52	213.52	240.32	232.66	263.38	250.45
0.8	198.26	197.79	232.71	223.17	259.27	246.34
0.9	178.72	171.84	229.57	207.13	257.53	236.53
1	148.30	137.88	223.49	193.36	252.55	228.85

Figure 6 also shows the achievable sum rate of proposed NOMA, NGDPA, and GRPA versus normalized offset for nine $(k=9)$ and ten $(k=10)$ users. In both cases, proposed NOMA achieves a higher sum rate. The proposed NOMA achieves a sum rate of 208.03 Mbit/s and 182.19 Mbit/s for $(k=9)$ and $(k=10)$ respectively at the edge of the system. NGDPA achieves a sum rate of 181.80 Mbit/s and 165.44 Mbit/s for $(k=9)$ and $(k=10)$ respectively at the edge of the system. GRPA achieves a sum rate of 143.34 Mbit/s and 138.77 Mbits/s for $(k=9)$ and $(k=10)$ respectively at the edge of the system as detailed in Table 4.

Table 4. Achievable sum rate comparison of proposed NOMA, NGDPA and GRPA (k = 9 and k = 10).

	$S{R}_{\mathit{\text{GRPA}}}$		$S{R}_{\mathit{\text{NGDPA}}}$		$S{R}_{\mathit{\text{PNOMA}}}$
r/R	k = 9	k = 10	k = 9	k = 10	k = 9	k = 10
0.1	268.94	269.33	265.65	265.65	266.33	266.15
0.2	268.84	268.78	260.63	260.62	262.68	262.11
0.3	265.18	261.39	254.57	254.15	258.24	257.08
0.4	252.60	250.64	248.57	247.42	253.27	251.32
0.5	241.07	239.67	241.55	240.22	248.04	245.00
0.6	227.17	226.08	235.30	233.26	244.25	239.68
0.7	212.88	212.83	225.83	222.98	237.47	231.33
0.8	197.66	195.04	216.80	209.02	231.76	219.41
0.9	171.13	173.56	199.85	194.82	219.53	207.89
1	143.34	138.77	181.80	165.44	208.03	182.19

Tables 3 and 4 show a comparison of the achievable sum rates of different user numbers for the proposed NOMA, GRPA, and NGDPA. From both tables, we can deduce that the proposed NOMA is more effective than other power allocation techniques. Mainly, as normalized offset increases, the GRPA techniques perform poorly. The sum-rate achieved by NGDPA is also lower than the proposed NOMA system.

The abbreviations used in this work are summarized in (Table 5) for clarity.

Table 5. Notations and its definition.

Notation	Definition
DC	Direct Current
DD	Direct Detection
FET	Field Effective Transistor
FPA	Fixed Power Allocation
FOV	Field-of-View
GRPA	Gain Ratio Power Allocation
IFFT	Inverse Fast Fourier Transform
IM	Intensity Modulation
LED	Light Emitting Diode
LOS	Line-of-Sight
MA	Multiple Access
MIMO	Multiple-Input-Multiple-Output
NGDPA	Normalized Gain Difference Power Allocation
NLOS	Non-Line-of-Sight
NOMA	Non-Orthogonal Multiple Access
OFDMA	Orthogonal Frequency Division Multiple Access
PD	Photo Detector
QAM	Quadrature Amplitude Modulation
RF	Radio Frequency
SIC	Successive Interference Cancellation
SNR	Signal-to-Noise Ratio
VLC	Visible Light Communication
ZF	Zero Forcing

4.2.3. Achievable average sum rate vs. number of users

The average sum rate of different power allocation techniques was analyzed over given system coverage R. The numerical simulation results show that as the number of users increases in the given area, the total sum rate decreases. When we compare power allocation strategies versus a number of users, the proposed NOMA improves the average sum rate of edge users as the number of users increases. Specifically, the proposed NOMA gives a significant improvement of sum rate over GRPA strategies. Especially, when the number of users is greater than four ($k>4$), the proposed NOMA has a greater average sum rate than both GRPA and NGDPA, as shown in Figure 7. Therefore, the proposed NOMA has improved the performance of edge users than the other conventional power allocation techniques (GRPA and NGDPA).

Figure 7. Achievable Average Sum Rate vs. Number of Users.

4.2.4. Sum rate gain vs. normalized offset

The sum rate gain of proposed NOMA over NGDPA and GRPA techniques are analyzed using simulation. Figure 8 shows the sum-rate gain vs. normalized offset of proposed NOMA over NGDPA for user numbers five, six, and seven ($k=5,6,$ and 7). It is obvious that sum-rate gain increases significantly for large value of the normalized offset. This shows that the performance of edge users is improved. The achievable sum rate gain is 4.097%, 11.31%, and 13% for $k=5,6,$ and 7, respectively.

Figure 8. Sum rate gain of Proposed NOMA over NGDPA vs. Normalized offset ($k=5,6,$ $\&7$), ($k=8,9,$ $\&10$).

Figure 8 also illustrates sum-rate gain versus normalized offset of proposed NOMA over NGDPA for user numbers eight, nine, and ten (k = 8, 9, and 10). The achievable sum rate gain is 18.36%, 14.43%, and 10.12% for k = 8, 9, and 10, respectively.

Figure 9 shows the sum-rate gain vs. normalized offset of proposed NOMA over GRPA for user numbers five, six, and seven (k = 5, 6, and 7). The achievable sum rate gain is 69.16%, 71.99%, and 70.3% for k = 5, 6, and 7, respectively.

Figure 9. Sum rate gain of Proposed NOMA over GRPA vs. Normalized offset ($k=5,6,$ $\&7$), ($k=8,9,$ $\&10$).

Figure 9 also shows the sum-rate gain vs. normalized offset of proposed NOMA over GRPA for eight, nine and ten (k = 8, 9, and 10) number of users. The achievable sum rate gain is 65.98%, 45.13%, and 31.29% for k = 8, 9, and 10, respectively. Generally, we can deduce that there is a high percentage of sum rate gain for a larger number of users. Hence, the proposed power allocation method within the proposed NOMA scheme improves the performance of VLC systems significantly.

5. Conclusions

In this work, the performance of a $4\times 4$ downlink MIMO-NOMA-based VLC system for a multi-user indoor network was studied using MatLab simulations. A new power allocation strategy of NOMA in VLC systems was considered and compared with exisitng schemes. The new power splitting strategy was also found to improve the performance of edge users.

The numerical results of simulations show that the proposed NOMA system enhances the sum rate of the overall system. Comparison of different power allocation techniques, including GRPA, NGDPA, and proposed NOMA, shows that the newly proposed NOMA has achieved the highest sum rate for a higher number of users among the three methods. The proposed NOMA attained an achievable sum rate of 18.36% and 65.98% compared with the NGDPA approaches and GRPA approaches, respectively, in the $4\times 4$ MIMO-based VLC networks with eight uniformly distributed users. Generally, as the number of users increases in the considered system environment, the total sum rate declines. However, the proposed NOMA has superior performance than GRPA and NGDPA. The results obtained in this show that MIMO-NOMA is a promising technique for multi-user VLC systems.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors did not receive support from any organization for the submitted work.

Notes on contributors

Gemechu Tesfaye Tola

Gemechu Tesfaye Tola received BSc. degree in Electrical and Computer Engineering from Jimma University, Jimma, Ethiopia in 2017, the MSc. degree in Communication Engineering from Jimma University, Jimma, Ethiopia, in 2021. Since then he is working in the department of Electrical and Computer Engineering, Salale University, Ethiopia. His research interests include Visible Light Communication, Wireless Communication and Antenna Designing.

Kinde Anlay Fante

Kinde Anlay Fante received the B.Sc. degree in electrical engineering from Bahir Dar University, Bahir Dar, Ethiopia, in 2008, the M.Tech. degree in electronics and computer engineering from Addis Ababa University, Ethiopia, in 2010, and the Ph.D. degree in electrical engineering from Indian Institute of Technology Delhi, India, in 2016. Since 2016, he has been working in the Faculty of Electrical and Computer Engineering, Jimma Institute of Technology, Jimma University, Ethiopia. His research interests include VLSI design for image and video processing, digital signal processing, antenna design, wireless communication, and machine learning.

Sherwin Nogueras Catolos

Sherwin Nogueras Catolos received the B.Sc. Degree in Electronics and Communications Engineering at Cagayan State University, Carig Campus, Tuguegarao, Philippines in 2001, the Masters of Engineering degree Major in ECE at University of Saint Louis (USL) Tuguegarao in 2008, and he is currently pursuing Ph.D. degree in Engineering Management at University of Saint Louis Tuguegarao (2022). He started working with the School of Engineering, Architecture and Information Technology Education (SEAIT E) as a faculty at USLTuguegarao Philippines since 2001. He also worked as an Electronics engineer at ESCO Micro, P T E, LTd based in Singapore in 2009. He also joined the group of expatriates to teach at Jimma Institute of Technology, Jimma University, Ethiopia from 2012-2021. His research interests include antenna design, digital signal processing, and in wire and wireless communication.