# Joint Resource Allocation for Multiuser Opportunistic Beamforming Systems with OFDM-NOMA
## 2 System Model
Passing a \(N_{c}\)-point discrete Fourier transform (DFT), the received signal is transformed to  
\[ \begin{split}\mathbf{\overline{y}}_{p}&=\mathbf{F} _{D}\left(\sum_{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{n_{t}}\mathbf{\overline{x}}_{p}+ \mathbf{z}_{u}\right)=\mathbf{F}_{D}\left(\sum_{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{ n_{t}}\mathbf{\overline{x}}_{p}+\mathbf{z}_{u}\right)\\ &=\sum_{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{n_{t}}\frac{1}{N_{c}} \mathbf{F}_{D}\mathbf{F}_{I}\mathbf{x}_{p}+\mathbf{F}_{D}\mathbf{z}_{u}=\sum _{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{n_{t}}\mathbf{x}_{p}+\mathbf{\overline{z}}_{ u},\end{split}  \tag{8}\]  
where \(\mathbf{\widetilde{z}}_{u}\) is also AWGN, due to the isotropy characteristic of AWGN. \(\mathbf{F}_{D}\) and \(\mathbf{F}_{I}\) denote DFT and IDFT matrixes, respectively. Define the equivalent channel by \(h_{u}^{\mathrm{equ}}\), which is given by  
\[h_{u}^{\mathrm{equ}}\overset{\Delta}{=}\sum_{n_{t}=1}^{N_{T}}h_{n_{t},u}w_{n _{t}}. \tag{9}\]  
Combine (8) and (9), we have  
\[\mathbf{\widetilde{y}}_{p}=h_{u}^{\mathrm{equ}}\mathbf{x}_{p}+\mathbf{ \widetilde{z}}_{u}, \tag{10}\]  
According to the known pilot signal, the \(u\)th user estimates its own equivalent channel \(h_{u}^{\mathrm{equ}}\) and then feeds \(h_{u}^{\mathrm{equ}}\) back to the BS. It is noted that compared to the conventional preprocessing schemes, the number of feedback bits and the complexity of feedback link in OBON scheme are greatly reduced, since each user is only required to return its equivalent channels \(h_{u}^{\mathrm{equ}}\) rather than the complete CSI of all the antennas. To obtain the channel coefficients of all the subcarriers, \(N_{c}\)-point DFT is operated for the equivalent channels \(h_{u}^{\mathrm{equ}}\), which is achieved as  
\[\widetilde{h}_{u}^{\mathrm{equ}}=[l_{0,u},l_{1,u},\cdots,l_{n_{c},u},\cdots,l_ {N_{c}-1,u}], \tag{11}\]  
where \(n_{c}\) is the label of subcarrier and satisfies \((N_{c}-1)\geq n_{c}\geq 0\). \(l_{n_{c},u}\) represents the channel coefficient of the \(n_{c}\)th subcarrier for the \(u\)th user. First, the BS randomly selects \(K=2\) users from \(U\) waiting users to be multiplexed on each subcarrier, known as \(k_{n_{c}}\)\((K\geq k_{n_{c}}\geq 1)\). The mapping between \(k_{n_{c}}\) and \(u\) is defined by \(k_{n_{c}}=\kappa(u)\). Then, a joint resource allocation is applied to determine the optimal user pair, subcarrier schedule and power allocation. Denote by \(P_{k_{n_{c}}}\) the power allocated to the \(k_{n_{c}}\)th user. Without loss of generality, let \(P_{1_{n_{c}}}\geq P_{2_{n_{c}}}\). Thus, the superposition transmit signal on the \(n_{c}\)th subcarrier is  
\[d_{n_{c}}=\sum\limits_{k_{n_{c}}=1}^{K}\sqrt{P_{k_{n_{c}}}}\times x_{k_{n_{c}}}, \tag{12}\]  
where \(x_{k_{n_{c}}}\) represents the desired signal of the \(k_{n_{c}}\)th user on the \(n_{c}\)th subcarrier and satisfies \(\left|x_{k_{n_{c}}}x_{k_{n_{c}}}^{*}\right|^{2}=1\). Thus, the received signal of the \(k_{n_{c}}\)th user on the \(n_{c}\)th subcarrier is given by  
\[ \begin{split} y_{k_{n_{c}}}&=l_{n_{c},k_{n_{c}}}d_{ n_{c}}+z_{k_{n_{c}}}=l_{n_{c},k_{n_{c}}}\sum\limits_{k_{n_{c}}=1}^{K}\sqrt{P_{k_{n_{ c}}}}x_{k_{n_{c}}}+z_{k_{n_{c}}}\\ &=l_{n_{c},k_{n_{c}}}\sqrt{P_{k_{n_{c}}}}x_{k_{n_{c}}}+l_{n_{c},k_ {n_{c}}}\sum\limits_{k_{n_{c}}^{\prime}=1,k_{n_{c}}^{\prime}\
eq k_{n_{c}}}^{K} \sqrt{P_{k_{n_{c}}^{\prime}}}x_{k_{n_{c}}^{\prime}}+z_{k_{n_{c}}},\end{split}  \tag{13}\]  
where the first, second and third terms are the \(k_{n_{c}}\) user's desired signal, interferences and AWGN, respectively. To detect the desired signal from the superposition signal \(y_{k_{n_{c}}}\), successive interference cancellation (SIC) is applied. The \(k_{n_{c}}\)th user first decides the signal with the largest power, then removes the influence of the decided signal, and repeat the previous processes until detecting its own desired signal.