# Joint Resource Allocation for Multiuser Opportunistic Beamforming Systems with OFDM-NOMA
## 2 System Model  
Let \(x\), \(\mathbf{x}\), \(\mathbf{X}\) be a variable, a vector and a matrix, respectively. \(\mathbf{x}^{\text{T}}\) and \(\mathbf{x}^{\text{H}}\) indicate transpose and conjugate transposes, respectively. \(\|\cdot\|\) represents the Frobenius norm of a matrix or a vector, and \(|\cdot|\) is the absolute value of a variable. \(\mathbb{C}\) denotes the complex space. \(\text{E}\{\cdot\}\) and \(\text{Var}\{\cdot\}\) represent expectation and variance, respectively. \(\lceil x\rceil\) denotes the minimum integer larger than \(x\). The meanings of the notations are listed in Appendix A. During the rest of this paper, we use \"NOMA\" to present \"power-domain NOMA\" for simplification. As it can be seen in Figure 1, a BS and \(U\) (\(U\geq 2\)) waiting users are involved in an OBON system. The BS has \(N_{T}\) (\(N_{T}\geq 2\)) transmit antennas, and each user has only one receive antenna. Assume that the locations of all the users are random. NOMA schemes are applied to multiplex users. Denote the number of subcarriers by \(N_{c}\) (\(N_{c}\geq 1\)). The maximum number of the multiplexed users on each subcarrier is \(K=2\), to reduce the complexity of NOMA scheme. Before transmitting users' signals, pilot signal is sent to evaluate all the subcarriers. To simplify the analyses, the processes of parallel-to-serial, serial-to-parallel, adding and removing cyclic prefix (CP) are ignored. Pilot vector is denoted by \(\mathbf{x}_{p}\), which is given by  
\[\mathbf{x}_{p}=\left[x_{p},x_{p},\cdots,x_{p}\right]^{\text{T}}\in\mathbb{C}^ {N_{c}\times 1}, \tag{1}\]  
where \(x_{p}\) represents the element of \(\mathbf{x}_{p}\), and \(\mathbf{x}_{p}\) satisfies \(\left\|\mathbf{x}_{p}\right\|^{2}=1\). Pilot vector \(\mathbf{x}_{p}\) is applied to estimate the received SNRs of users on each subcarrier. To preprocess the transmit signals, a random generated weight vector \(\mathbf{w}\) is introduced, which is given by  
\[\mathbf{w}=\left[w_{1},w_{2},\cdots,w_{n_{t}},\cdots,w_{N_{T}}\right]^{\text {T}}, \tag{2}\]  
where \(\mathbf{w}\in\mathbb{C}^{N_{T}\times 1}\) and \(\left\|\mathbf{w}\right\|^{2}=1\). \(n_{t}\) stands for the label of the transmit antenna and satisfies \(N_{T}\geq n_{t}\geq 1\). \(w_{n_{t}}\) is the \(n_{t}\)th element of \(\mathbf{w}\), which is achieved as  
\[w_{n_{t}}=\sqrt{\pi_{n_{t}}}e^{i\hat{y}_{n_{t}}}, \tag{3}\]  
Figure 1: The downlink OBON system, which contains both pilot and user’s signals transmissions. where \(\sqrt{\alpha_{n_{t}}}\), \(\phi_{n_{t}}\) and \(j\) represent the amplitude, phase and imaginary number symbol, respectively. Thus, the transmit signal of pilot signal is  
\[\mathbf{s}_{p}=\mathbf{w}\mathbf{\overline{x}}_{p}, \tag{4}\]  
where \(\mathbf{\overline{x}}_{p}\) represents the inverse discrete Fourier transform (IDFT) of pilot signal. Denote by \(h_{n_{t},u}\) the channel coefficient between the \(n_{t}\)th transmit antenna of the BS and the \(u\)th (\(U\geq u\geq 1\)) user, which is treated as a frequency selective fading channel. Assume that \(h_{n_{t},u}\) keeps constant during a bandwidth \(B_{sub}\) and coherent time, where \(B_{sub}\) is the same-sized bandwidth allocated to each subcarrier. The channel between the BS and the \(u\)th user can be expressed in a vector form as  
\[\mathbf{h}_{u}=\big{[}h_{1,u},h_{2,u},\cdots,h_{n_{t},u},\cdots,h_{N_{T},u} \big{]}^{\mathrm{T}}. \tag{5}\]  
The received signal of the \(u\)th user is  
\[\mathbf{y}_{p}=\mathbf{h}_{u}^{\mathrm{T}}\mathbf{s}_{p}+\mathbf{z}_{u}= \mathbf{h}_{u}^{\mathrm{T}}\mathbf{w}\mathbf{\overline{x}}_{p}+\mathbf{z}_{u }=\sum_{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{n_{t}}\mathbf{\overline{x}}_{p}+\mathbf{ z}_{u}, \tag{6}\]  
where \(\mathbf{z}_{u}\) represents additive white Gaussian noise (AWGN) vector, and equals to  
\[\mathbf{z}_{u}=z_{n_{t}}\times[1,1,\cdots,1]^{\mathrm{T}}\in\mathbb{C}^{N_{c} \times 1}. \tag{7}\]  
\(z_{n_{t}}\) is AWGN of the \(n_{t}\)th transmit antenna with variance \(\sigma^{2}\).