# Deep Learning-Based Next-Generation Waveform for Multiuser VLC Systems
## 2 Materials and Methods
### OAM with Index Modulation (IM)  
This hybrid approach, i.e., combining IM with OAM, reduces power consumption and increases spectral efficiency by utilizing a certain number of OAM modes at a particular time instead of all available modes to store a modulated symbol, and additional information is transmitted by encoding the bits in the form of indices for the active OAM modes. Spectral efficiency depends on the available active modes out of the total active modes, \(M\), and is equal to the number of transmitting LEDs. In the current work, the number of transmitting and receiving resources is similar in a MIMO configuration. \(M_{a}\in[1,2,\cdots M]\) denotes the number of activated modes [4]. The OAM wave is called Mode Division Multiplexing (MDM), which has all the available modes activated, such as \(M=M_{a}\). OAM-IM's spectral efficiency for modulation order \(N\) is written as follows:  
\[\zeta=M_{a}\log_{2}N+\left[\log_{2}\!\left( \begin{array}{c}M\\ M_{a}\end{array} \right)\right], \tag{1}\]  
where \([. ]\) is the floor function, used to avoid errors due to the decimal in log operation. The number of bits used for modulation is \(M_{a}\log_{2}N\), and \(\left|\log_{2}\!\left( \begin{array}{c}M\\ M_{a}\end{array} \right)\right|\) shows the number of bits used for mode selection by index modulation as shown in Figure 3. The signal applied to the \(m^{\text{th}}\) LED is given as follows:  
\[x_{m}^{M_{a}}=\frac{1}{\sqrt{N}}p_{M_{a}}s_{M_{a}}\exp\left(-j2\pi\frac{mM_{a}}{ M}\right), \tag{2}\]  
where \(p_{M_{a}}\) is the power assigned to mode \(M\), and \(s_{M_{a}}\) is the modulated signal's \(M_{a}^{\text{th}}\) element. The vector \(\mathbf{s}\) comprises complex constellation symbols, \(d\in Se^{i\Theta i}\), as well as some zeros; the indices of nonzero items are kept in \(\mathcal{L}_{i}\), where \(\mathcal{L}_{i}=\{l_{1},l_{2},\cdots l_{M_{a}}\}\) is the \(i^{\text{th}}\) combination matrix of the activated modes depending on the mode selection bits. It is important to mention that \(p_{M_{a}}\) does not refer to the power factor used for NOMA; its sole purpose is to meet the power requirements for transmission. The signals from different active modes are linearly suppressed together [38]. The transmitted signal can be written as follows:  
\[x_{m}=\sum_{M_{a}=\mathcal{L}_{i}}x_{m}^{M_{a}}=\frac{1}{\sqrt{M}}p_{M_{a}}s_{ M_{a}}\exp\left(-j2\pi\frac{mM_{a}}{M}\right), \tag{3}\]  
\[p_{M_{a}}=\sqrt{\frac{\frac{P_{t}}{\delta_{M_{a}}}}{\frac{1}{\delta_{1}}+\frac {1}{\delta_{2}}+\cdots\frac{1}{\delta_{M-1}}}}, \tag{4}\]  
where \(P_{t}\) is the total power available, \(\delta_{i}\) is the \(i^{\text{th}}\) eigenvalue of VLC channel \(H\), and \(p_{M_{a}}\) is the power of each activated OAM mode. The MIMO VLC channel is defined in Section 2.3 Equations (2) and (3) are expressed as a vector as follows:  
\[\mathbf{x}=\text{{WPs}}, \tag{5}\]  
where \(\mathbf{W}\in\mathbb{C}^{M\times\text{M}}\) refers to the discrete Fourier transform matrix, \(\mathbf{P}\) represents the power matrix, and the vector \(\mathbf{s}\in\mathbb{C}^{M\times\mathbf{1}}\) contains complex constellation symbols. The indices of complex constellation symbols are kept in \(\mathcal{L}_{i}\). \(\mathbf{s}\) can have the following form:  
\[\mathbf{s}=[0\:d_{1}0\:d_{2}\cdots 0\:d_{Na}], \tag{6}\]  
\(\mathbf{x}\) is the final output of the modulation block. The signal \(\mathbf{x}\) is passed by the PWM-encoding block to convert the complex signal into 1s and 0s to drive the LEDs. Then, the signal is transmitted through the VLC channel. The signal \(\mathbf{y}\) is detected by the photodetector, and the signal is processed through PWM decoding to convert the 1s and 0s in complex symbols for further processing. Therefore, the signal \(\mathbf{y}\) is passed through the inverse  
Figure 3: System block diagram of OAM-IM. Only \(M_{a}\) subcarriers contain data in each subblock \(G\). Each subblock's output is transformed to serial mode and sent over the VLC channel.