1. Introduction

Figure 1. General indoor environment for collaborative tasks with LED arrays for transmission.

Figure 1. General indoor environment for collaborative tasks with LED arrays for transmission.

2. System Design

2.2. Simplified Gain Ratio Power Allocation (S-GRPA) for NOMA

Several agents’ data are combined using the power-domain NOMA, and data is then separated using SIC. NOMA is free of spectrum spreading or degraded SNR performance, in contrast to other multiple access methods such as OFDMA and CDMA [39]. Several agents using the same resource simultaneously in NOMA boosts system throughput, but each agent has a distinct power factor. In this study, we employ symbol-level NOMA. Figure 5 illustrates the NOMA framework for three agents.

Figure 4. 2D view of power distribution of VLC channel in an indoor environment with four transmitting locations.

Figure 4. 2D view of power distribution of VLC channel in an indoor environment with four transmitting locations.

Because of the relative difference in distance from the base station, the agent near the cell edge gets a lot more power than the agent at the center of the cell. The received signal can be expressed as follows [39]:

(5)

where $y_1$ and $y_2$ are, the suppressed information from all agents, received at each agent. The information from agent-1 is separated and demodulated as follows at the receiver end:

$${\widehat{s}}_1=\frac{y_1}{p_1}$$

(6)

After agent-1’s data has been retrieved, agent 2’s data is recovered by SIC reducing agent-1’s interference in the manner described below:

$$\begin{array}{c}\hfill \widehat{z}=y_2-p_1{\widehat{s}}_1\end{array}$$

(7)

$$\begin{array}{c}\hfill {\widehat{s}}_2=\frac{\widehat{z}}{p_2}\end{array}$$

(8)

This method can be expanded to a higher number of agents. As the power factor, p, for each agent depends on its location. The power distribution of the VLC system is shown in Figure 4, it can be seen that overall power distribution can be divided into zones, $n=1,2,\cdots ,N$, with radius, $r_N$. Here $r_n$ refers to the radius of each zone. Furthermore, zone N has the least illumination intensity, and zone 1 has the highest illumination intensity. The $n^{th}$ region can be defined as follows [37]:

$$\left\{\begin{array}{cc}[0,r_n]& n=1\\ r_{n-1},r_n]& 1<n<N\\ r_{n-1},r_e]& n=N\end{array}\right.$$

(9)

for

$$r_n=\sqrt{\frac{n{r}_e^2}{N}},n=1,2,\cdots ,N.$$

(10)

where $r_e$ is the maximum radius of the last zone. The relationship between the $k^{th}$ agent and the $k-1^{th}$ agent in terms of power allocation is described as follows [37]:

$$p_k={\left(\frac{H_1}{H_N}\right)}^k\times p_{k-1},$$

(11)

Here, the information on channel gain, H, is received by the indoor positioning algorithm and stored in the look-up table.

Figure 5. Illustration of NOMA method for VLC systems

Figure 5. Illustration of NOMA method for VLC systems

2.4. Support Vector Machine Algorithm

As with the RFR algorithm, the purpose of the Support Vector Machine (SVM) regression algorithm is to map the input $\mathbf{x}$ to the desired output y. Instead of minimizing the discrepancy between the estimated and true regression functions, $f_e$ and f, respectively, SVM penalizes the final result so that it can be used in linear regression. $y=w.x+b$. Let y be the true output, and inside an area defined as $y\pm ϵ$, SVM will make its predictions. The discrepancy between actual output, y, and expected output, z, is denoted by the $|z_i-y_i|<ϵ$. Two slack variables are added as a penalty if the projected output is outside the region $y\pm ϵ$. ${\zeta }^{+}$ indicates that the projected output lies above the region of $y+ϵ$, whereas ${\zeta }^{-}$ indicates that the predicted output lies below the region of $y-ϵ$. The following is the error function for the SVM regression algorithm: [43,44,45,46,47]:

$$minC\sum _{i=1}^L({\zeta }_i^{+}+{\zeta }_i^{-})+\frac{1}{2}{∥w∥}^2,$$

(13)

subject to:

$$\begin{array}{cc}\hfill & {\zeta }^{+}\ge 0,\hfill \\ \hfill & {\zeta }^{-}\ge 0,\hfill \\ \hfill & y_i\le z_i+ϵ+{\zeta }^{+}\hfill \\ \hfill & y_i\ge z_i+ϵ+{\zeta }^{-}\hfill \end{array}$$

(14)

C is the tunable variable that controls the penalty on slack variables and $ϵ$. To solve (13), following Lagrange multipliers are introduced [43,47]:

$$\begin{array}{cc}\hfill & {\alpha }_i^{+}\ge 0,{\alpha }_i^{-}\ge 0,\hfill \\ \hfill & {\mu }_i^{+}\ge 0,{\mu }_i^{-}\ge 0.\hfill \end{array}$$

(15)

This transforms (13) in following:

$$\begin{array}{cc}\hfill L_p=& C\sum _{i=1}^L({\zeta }_i^{+}+{\zeta }_i^{-})+\frac{1}{2}{∥w∥}^2-\sum _{i=1}^L({\mu }_i^{+}{\zeta }_i^{+}+{\mu }_i^{-}{\zeta }_i^{-})\hfill \\ \hfill & -\sum _{i=1}^L{\alpha }_i^{+}(ϵ+{\zeta }_i^{+}+z_i-y_i)-\sum _{i=1}^L{\alpha }_i^{-}(ϵ+{\zeta }_i^{-}-z_i+y_i)\hfill \end{array}$$

(16)

Now differentiating (16) to 0, with respect to each variable as below [43,47]:

$$\begin{array}{cc}\hfill & \frac{\partial L_p}{\partial w}=0\Rightarrow w=\sum _{i=1}^L({\alpha }_i^{+}-{\alpha }_i^{-}){\mathbf{x}}_{\mathbf{i}}\hfill \\ \hfill & \frac{\partial L_p}{\partial b}=0\Rightarrow \sum _{i=1}^L({\alpha }_i^{+}-{\alpha }_i^{-})=0\hfill \\ \hfill & \frac{\partial L_p}{\partial {\zeta }_i^{+}}=0\Rightarrow C={\alpha }_i^{+}+{\mu }_i^{+}\hfill \\ \hfill & \frac{\partial L_p}{\partial {\zeta }_i^{-}}=0\Rightarrow C={\alpha }_i^{-}+{\mu }_i^{-}\hfill \end{array}$$

(17)

The Dual form of the Primary $L_p$ can be defined as follows:

(18)

subject to:

$$\begin{array}{cc}\hfill & 0\le {\alpha }_i^{+}\le C,\hfill \\ \hfill & 0\le {\alpha }_i^{-}\le C,\hfill \\ \hfill & \sum _{i=1}^L({\alpha }_i^{+}-{\alpha }_i^{-})=0.\hfill \end{array}$$

(19)

The predicted output can be written as [47]:

$$z=\sum _{i=1}^L({\alpha }_i^{+}-{\alpha }_i^{-}){\mathbf{x}}_i.\mathbf{x}+b.$$

(20)

This concludes the regression algorithms to obtain the location of the agent based on power received at the receiver. However, in real-world settings, the direct receiver might not be able to receive the agent’s signal due to obstacles or the signal is highly disrupted due to noise. Therefore, for localization in collaborative indoor environments, the distance geometry problem plays an important role by obtaining the location of the particular agent using the mutual distances of all agents in the network. The next section discusses the distance geometry problem using Euclidian Distance Matrix (EDM).

2.5. The Distance Geometry Problem

The VLC systems for indoor environments with dynamic agents often face the loss of signal due to shadowing because of obstacles. To assist the localization in the event of signal loss in a multiagent collaborative environment, the Distance Geometry Problem (DGP) is used in conjunction with the machine learning algorithms discussed in the previous subsection. The objective of the DGP is to find the location of agents using their mutual distances. It is important to mention that agents do not only communicate with the base station during collaborative tasks but also communicate with each other, this provides extra information about the environment that agents transmit to the base station.

The mutual distances of agents are provided in the form of EDM. The solution to DGP for N agents in dimension, d, is a matrix, $S\in {\mathbb{R}}^{d\times N}=[s_1,s_2,\cdots ,s_N]$, here $s_i$ are the coordinates for $i^{th}$ agent. The mutual distance between agents can be referred as $D\in {\mathbb{R}}^{N\times N}=\left[d_{ij}\right]$, here $d_{ij}$ is the distance from $j^{th}$ agent to $i^{th}$ agent. To generate an initial estimate of the matrix $\widehat{S}$, multi-dimensional scaling is used. The estimated point cloud, $\widehat{S}$ can be mapped to the actual matrix, S, using rigid transformation in absolute coordinates. For this transformation, the Procrustes Analysis is performed which is spectral factorization. For this process, it is assumed that the location of some agents, $N_a<N$ is known, these agents are referred to as anchors. To find the location of missing agents in the network, DGP can be stated as static.

Table 1. Simulation Parameters.

Table 1. Simulation Parameters.

Parameters	Value
Dimension	6.3 x 2.5 x 2 $m^3$
Tx-Rx (LED arrays-PDs on agents) Configuration	4X4
No. of LED Arrays	4
No. of LEDs in single Array	60
Total Power	20 watts
Semi-angle at half power	70 degrees
PD field of view	60 degrees
Refractive Index	1.5
No. of Agents	5

The static DGP consists of three stages to obtain the matrix, S from the matrix, D. The first stage is to obtain a Grammian matrix, $G\in {\mathbb{R}}^{N\times N}=S^{T}S$, it has one-one relation with matrix, D. The Grammian matrix, G, can be obtained by following the optimization problem [48,49]:

$$\begin{array}{cccc}\hfill & \underset{G}{\mathrm{minimize}}\hfill & \hfill & \left|\right|\tilde{D}-W\circ \mathcal{K}\left(G\right){\left|\right|}^2\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\hfill & \hfill & \\ \hfill & \hfill & & G⪰0;G1=0;\mathrm{Rank}\left(G\right)\le d,\hfill \end{array}$$

where $\mathcal{K}(.)$ is a function, which maps the Grammian matrix, G, to the matrix, D. W is referred to as a binary mask matrix, the entries with zero values in this matrix show the missing measurement of the received power and ∘ is the Hadamard product. The next stage is to obtain the matrix, S, using the Grammian matrix, G. The estimate, $\widehat{S}$ can be obtained by the Singular Value Decomposition (SVD) method given that matrices, G, and S hold the mathematical relation, $G=S^{T}S$.

As mentioned earlier that matrix, $\widehat{S}$ can be mapped to the actual matrix, S, using rigid transformation along with the information of $N_a$ anchors. By denoting the columns of the matrix, S, related to anchors as $S_a$ and assuming $Y_a$ refers to the same columns in $\widehat{S}$. Furthermore, to make the $S_a$ and $Y_a$ centered at the origin, assume that $S_a$ and $Y_a$ are the translated version of S and Y. The transformation $\mathcal{R}$ can be defined as follows:

$$\begin{array}{c}\hfill \mathcal{R}=\underset{\mathcal{Q}:\mathcal{Q}{\mathcal{Q}}^T-I}{\arg \min }\left|\right|\mathcal{Q}{\overline{S}}_a-{\overline{Y}}_a{\left|\right|}_F^2\end{array}$$

The actual matrix, S can be calculated as follows:

$$S=\mathcal{R}(\widehat{S}-s_{a,c}{1}^T)+y_{a,c}{1}^T,$$

(21)

where $s_{a,c}$ and $y_{a,c}$ refer to the centroids of $S_a$ and $Y_a$ respectively.

This concludes the mathematical basis for the proposed framework for resource allocation using precise positioning of the agent in a collaborative indoor environment using VLC systems. The next section discusses the performance of indoor positioning using SVM and RFR algorithms, and bit-error-rate (BER) for NOMA-based VLC system.

3. Results

Figure 6. VLC indoor positioning using Random forest regression algorithm. actual data points (blue), predicted data points (red)

Figure 6. VLC indoor positioning using Random forest regression algorithm. actual data points (blue), predicted data points (red)

4. Conclusions

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