# Multi-User Joint Detection Using Bi-Directional Deep Neural Network Framework in NOMA-OFDM System
## 3 System Model
### Signal and Channel Model  
An uplink multiuser NOMA-OFDM system is considered, where the system consists of a BS and \(UE_{i}\) where \(i=1,2,\ldots,N\). At the transmitter, a conventional OFDM-NOMA is used. The BS receives a superposition of symbols from multiple UE with additional channel noise. In the OFDM-NOMA system, a pilot signal is inserted to use CE and signal detection in favor of OFDM. After transmission and reception of data, a DNN model is trained and tested for channel estimation and signal detection. The overview of the proposed system is shown in Figure 2. The superposition coding symbol \(Y_{r}\) for \(N\) UE can be written as follows [15]:  
\[Y_{r}=\sum_{i=1}^{N}\sqrt{\omega_{i}}s_{i}, \tag{1}\]  
where \(\omega_{i}\) is the power allocation coefficient of \(UE_{i}\) and \(s_{i}\) is the baseband modulated symbols for \(UE_{i}\). Figure 1: Uplink multiuser communication system with BS. In the general OFDM system, before pilot signal insertion, at the beginning, superposition coding is converted into serial to parallel, and after that, it transformed through inverse discrete Fourier transform (IDFT) [22]. Therefore, the symbol of OFDM-NOMA can be expressed as follows:  
\[s_{i}(n)=IDFT\{Y_{r}(f)\}=\sum_{f=1}^{N_{g}}Y_{r}(f)e^{j(2\pi fn/N_{g})},n=1,2, \ldots,N_{g}, \tag{2}\]  
where \(N_{g}\) is the number of subcarriers in the frequency domain. In addition, the symbol in the \(fth\) subcarrier of OFDM is \(Y_{r}(f)\). To cancel the inter-carrier interference (ICI) among OFDM subcarriers, a cyclic prefix (CP) or guard interval is added to this signal. After adding CP, the OFDM symbol can be expressed as follows:  
\[ \begin{cases}s_{cp}(n)=s_{i}(N_{g}+n),n=-N_{cp},-N_{cp}+1,\ldots,-1\\ s_{i}(n),n=1,2,\ldots,N_{g},\end{cases}  \tag{3}\]  
where CP data length is \(N_{cp}\). After adding CP to OFDM, it is converted into serial form and is transmitted over the Rayleigh fading channel. However, the received signal at the terminal side is expressed as follows:  
\[y_{cpi}(n)=\sqrt{P}\omega_{i}*h_{i}(n)s_{cpi}(n)+G(n),i=1,2,\ldots,N, \tag{4}\]  
where the transmitted power of \(UE_{i}\) is \(P\), \(h_{i}(n)\) is the discrete Fourier transform of the impulse response of a multipath channel, and \(*\) denotes the convolution operation. The transmitted symbol with CP data for \(UE_{i}\) is represented by \(s_{cpi}(n)\). The additive white Gaussian noise (AWGN) \(G(n)\) at the receiver is represented as \(\mathcal{CN}(0,\sigma^{2})\). The \(G(n)\) can be expressed as follows [23]:  
\[G(n)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(n-y)^{2}}{2\sigma^{2}}}, \tag{5}\]  
where \(\sigma\) is the noise standard deviation and \(\mu\) is the mean value of the distribution. After the CP data is removed, the received signal can be written as follows:  
\[y_{i}(n)=y_{cpi}(n+N_{cp}),n=1,2,\ldots,N_{g}. \tag{6}\]  
Figure 2: The overview of the proposed DNN-based NOMA-OFDM estimation system. The received signal is transformed by discrete Fourier transform (DFT), and it can be expressed as follows:  
\[Y_{i}(f)=DFT\{y_{i}(n)\}=1/N_{g}\sum_{n=1}^{N_{g}}y(n)e^{j(2\pi f_{n}/N_{g})},f=1,2,\ldots,N_{g}. \tag{7}\]  
Therefore, in the OFDM system with \(N\) UEs per subcarrier, the received signal on subcarrier \(f\) can be written as follows:  
\[Y_{i}(f)=\sum_{f=1}^{N}\sqrt{P}\omega_{i}(f)*h_{i}(f)s_{i}(f)+G(f). \tag{8}\]  
The total power allocation coefficient is summed up to one and can be formulated as follows [24]:  
\[\sum_{i=1}^{N}\omega_{i}(f)=1. \tag{9}\]  
The scalar \(h_{i}(f)\) DFT of the impulse response of multipath channel \(H_{i}(t)\) for the \(U\!E_{i}\) can be expressed as follows [25]:  
\[H_{i}(t)=\sum_{r=1}^{R}v_{i,r}\eta(t-\tau_{i,r}), \tag{10}\]  
where the complex channel gain and corresponding time delay for the \(r\)th multipath parameters of the \(U\!E_{i}\) are represented by \(v_{i,r}\) and \(\tau_{i,r}\), respectively. In this proposed paper, the total number of resolved paths \(R\) is considered to be 20, and the channel is modeled by the Rayleigh fading.