## 4 Analysis of DoF and User Scaling
### User Scaling Law  
For uplink transmission, it is obvious that the sum DoF of \(M\) is achievable by using the ZF receiver at the BS. Thus, we focus on analyzing how to achieve the sum DoF of \(M\) for downlink transmission. When the sets of the selected downlink and uplink MSs, denoted by \(\mathcal{S}^{(d)}=\{\pi_{1},\cdots,\pi_{M}\}\) and \(\mathcal{S}^{(u)}=\{\phi_{1},\cdots,\phi_{M}\}\), respectively, are determined, the received signal at MS \(\pi_{i}\) for downlink transmission is rewritten as  
\[y_{\pi_{i}}^{(d)} = \beta_{\pi_{i}}^{(d)}{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}{ \mathbf{s}^{(d)}}+\sum_{j\in\mathcal{S}^{(u)}}\beta_{\pi_{ij}}^{(M)}h_{\pi_{i} j}s_{j}^{(u)}+n_{\pi_{i}}^{(d)} \tag{8}\] \[= \beta_{\pi_{i}}^{(d)}{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}{ \mathbf{v}_{i}^{(d)}}x_{i}^{(d)}+\sum_{k=1,k\
eq i}^{M}\beta_{\pi_{i}}^{(d)}{ \mathbf{h}_{\pi_{k}}^{(d)}}^{\dagger}{\mathbf{v}_{k}^{(d)}}x_{k}^{(d)}+\sum_{ j\in\mathcal{S}^{(u)}}\beta_{\pi_{ij}}^{(M)}h_{\pi_{ij}}s_{j}^{(u)}+n_{i}^{(d)}.\]  
Thus, from Equation (8), the received signal-to-interference-plus-noise ratio (SINR) at MS \(\pi_{i}\) is given by  
\[ \begin{array}{rcl}\text{SINR}_{\pi_{i}}^{(d)}&=&\frac{\text{SNR}\left(\beta_ {\pi_{i}}^{(d)}\right)^{2}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}{ \mathbf{v}_{i}}\right|^{2}}{\text{SNR}\sum_{k=1,k\
eq i}^{M}\left(\beta_{\pi_{ i}}^{(d)}\right)^{2}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}{\mathbf{v}_{k} }\right|^{2}+\text{SNR}\sum_{j\in\mathcal{S}^{(u)}}\left(\beta_{\pi_{ij}}^{(M) }\right)^{2}\left|{n_{\pi_{ij}}}\right|^{2}+1}\\ &=&\frac{\text{SNR}\left(\beta_{\pi_{i}}^{(d)}\right)^{2}\left|{\mathbf{h}_{ \pi_{i}}^{(d)}}^{\dagger}{\mathbf{v}_{i}^{2}}\right|^{2}}{\mathcal{I}_{\pi_{i }}^{(d)}+\mathcal{I}_{\pi_{i}}^{(u)}+1},\end{array}  \tag{9}\]  
where \(\mathcal{I}_{\pi_{i}}^{(d)}=\text{SNR}\sum_{k=1,k\
eq i}\left(\beta_{\pi_{i}}^ {(d)}\right)^{2}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^{\dagger}{\mathbf{v}_{k}} \right|^{2}\) and \(\mathcal{I}_{\pi_{i}}^{(u)}=\text{SNR}\sum_{j\in\mathcal{S}^{(u)}}\left(\beta _{\pi_{ij}}^{(M)}\right)^{2}\left|{h_{\pi_{ij}}}\right|^{2}\) denote the interference caused by other generated beams (i.e., the downlink interference) and the interference from the selected uplink MSs to MS \(\pi_{i}\) (i.e., the MS-to-MS interference), respectively. Then, using the received SINR in Equation (9), the achievable sum-rate for downlink is given by  
\[R^{(d)}=\sum_{i=1}^{M}\log_{2}\left(1+\text{SINR}_{\pi_{i}}^{(d)}\right). \tag{10}\]  
Now, the following theorem establishes the DoF achievability of the proposed hybrid opportunistic scheduling method presented in Section 3. Theorem 1: _For the multi-antenna full-duplex system in Section 2, the optimal DoF of \(2M\) is achievable with high probability if_  
\[N=\omega\left(SNR^{M}\right). \tag{11}\]  
Proof: For uplink transmission, it is obvious that the sum DoF of \(M\) is achievable by using the ZF receiver at the BS. Thus, we focus on the achievable DoF for downlink. Let us define \(P_{d}\) and \(P_{u}\) by the probabilities that the downlink interference and the MS-to-MS interference at all the selected downlink MSs are less than or equal to \(\tilde{\epsilon}_{1}>0\) and \(\tilde{\epsilon}_{2}>0\), respectively, where \(\tilde{\epsilon}_{1}\) and \(\tilde{\epsilon}_{2}\) are small constants independent of SNR. Then, \(P_{d}\) and \(P_{u}\) can be expressed as  
\[P_{d}=\lim_{SNR\rightarrow\infty}\text{Pr}\left\{\text{SNR}\sum_{k=1,k\
eq i} ^{M}\left(\beta_{\pi_{i}}^{(d)}\right)^{2}\left|{\mathbf{h}_{\pi_{i}}^{(d)}}^ {\dagger}{\mathbf{v}_{k}}\right|^{2}\leq\tilde{\epsilon}_{1},\;\forall i\in\{1,\cdots,M\}\right\} \tag{12}\]  
and  
\[P_{u}=\lim_{SNR\rightarrow\infty}\text{Pr}\left\{\text{SNR}\sum_{j\in\mathcal{ S}^{(u)}}\left(\beta_{\pi_{ij}}^{(M)}\right)^{2}\left|{n_{\pi_{ij}}}\right|^{2} \leq\tilde{\epsilon}_{2},\;\forall i\in\{1,\cdots,M\}\right\}, \tag{13}\]respectively.