## 5 Properties Analysis of the Functions \(\varphi_{m}^{s}\)  
For \(m>n\), we define a set of continuous variables \(\widetilde{p}_{n}^{s}=\widetilde{p}_{n+1}^{s}=\cdots=\widetilde{p}_{m}^{s}\) as \(\widetilde{p}\in[0,p^{s}]\). An auxiliary function is proposed to assist us in analyzing \(\varphi_{m}^{s}\):  
\[\varphi_{n}^{s}(\widetilde{p})\triangleq\sum\limits_{l=n}^{m}\varphi_{l}^{s} \widetilde{p}=\left\{\begin{array}{c}log_{2}((\widetilde{p}_{1}^{s}+ \widetilde{\sigma}_{\beta_{s}(1)}^{s})^{\alpha_{\beta_{s}(1)}})^{B_{S}},n=1\\ log_{2}((\frac{(\widetilde{p}_{n}^{s}+\widetilde{\sigma}_{\beta_{s}(n)}^{s})^{ \alpha_{\beta_{s}(n)}})^{B_{S}}}{(\widetilde{p}_{n}^{s}+\widetilde{\sigma}_{ \beta_{s}(n-1)}^{\alpha_{\beta_{s}(n-1)}})^{B_{S}}(n-1)}))^{B_{S}},n>1\end{array} \right.. \tag{24}\]  
Assuming that neither user \(n\) nor user (\(m-1\)) occupies resource block \(s\), these invalid users cannot allocate power from resource block \(s\): \(\widetilde{p}_{n}^{s}=\widetilde{p}_{n+1}^{s}=\cdots=\widetilde{p}_{m}^{s}\). Therefore, function \(\varphi_{n,m}^{s}\) can supersede the value of \(\sum_{l=n}^{m}\varphi_{l}^{s}\). Suppose that the upper limit for the number of valid users satisfies Condition \(w3^{*}\). In that case, each resource block can be allocated to at most \(A\) maritime users for multiplexing. Therefore, evaluating the calculative complexity of the target function in \(W_{RB}^{s}\) requires the execution of \(O(A)\) operations.  
Through the above simplification, the original optimization objective is transferred to the maximization of function \(\varphi_{n,m}^{s}\) with constraint conditions. We solve the extreme points and inflection points of the function by analyzing the first and second derivatives of \(\varphi_{n,m}^{s}\). Suppose that \(s\in S\), \(m\in\{1,2,\ldots,T\}\), and \(m>n\):  
* If \(n=1\) or the weight value satisfies \(\alpha_{\beta_{s}(m)}\geq\alpha_{\beta_{s}(n-1)}\), \(\varphi_{n,m}^{s}\) is monotonically increasing with convex in the interval \([0,\infty)\);
* If both the inequation of \(\alpha_{\beta_{s}(m)}<\alpha_{\beta_{s}(n-1)}\) and \(n>1\) are satisfied, \(\varphi_{n,m}^{s}\) shows a single peak. In this case, \(C_{1}\) refers to the extreme point. Function \(\varphi_{n,m}^{s}\) increases in interval \(\left(-\widetilde{\sigma}_{\tilde{p}_{s}(n-1)},C_{1}\right]\) but decreases in interval \([C_{1},+\infty)\). Let \(C_{2}\) denote the inflection point. \(\varphi^{s}_{n,m}\) is convex on \(\left(-\widetilde{\sigma}_{\tilde{p}_{s}(n-1)},C_{2}\right]\) and concave on \([C_{2},+\infty)\), where \(C_{1}\leq C_{2}\):  
\[C_{1}=\frac{\alpha_{\tilde{p}_{s}(n-1)}\widetilde{\sigma}_{\tilde{p}_{s}(m)}- \alpha_{\tilde{p}_{s}(m)}\widetilde{\sigma}_{\tilde{p}_{s}(n-1)}}{\alpha_{ \tilde{p}_{s}(m)}-\alpha_{\tilde{p}_{s}(n-1)}}, \tag{25}\]  
\[C_{2}=\frac{\sqrt{\alpha_{\tilde{p}_{s}(n-1)}\widetilde{\sigma}_{\tilde{p}_{s }(m)}-\sqrt{\alpha_{\tilde{p}_{s}(m)}}\widetilde{\sigma}_{\tilde{p}_{s}(n-1)} }}{\sqrt{\alpha_{\tilde{p}_{s}(m)}-\sqrt{\alpha_{\tilde{p}_{s}(n-1)}}}}. \tag{26}\]  
The pseudo-code provided in Algorithm 1 expresses the process of calculating \(argmax_{n,m}^{s}(\widetilde{p})\), aiming to find the corresponding solution \(\widetilde{p}\in[0,p^{s}]\) when function \(\varphi^{s}_{n,m}\) obtains the optimal value. The complexity of Algorithm 1 is \(O(1)\), since the calculation involves only a simple fixed number of fundamental operations. The works in this subsection facilitate the subsequent optimization for power and user allocation.  
```
1:Input \(n,m,\alpha,T,B_{S},G^{s}_{t}(t\in T),\sigma^{s}_{t}(t\in T)\), \(p^{s}\)
2:Function\(argmax_{\varphi}\)
3:\(x\leftarrow\tilde{p}_{s}(m)\)
4:\(y\leftarrow\tilde{p}_{s}(n-1)\)
5:If\(n=1\) or \(\alpha_{\tilde{p}_{s}(m)}\geq\alpha_{\tilde{p}_{s}(n-1)}\)then
6:Return\(p^{s}\)
7:Else
8:Return\(max\left\{0,min\left\{\frac{\alpha_{\tilde{p}_{s}}-\alpha_{\tilde{p}_{s}}}{ \alpha_{s}-\alpha_{s}},p^{s}\right\}\right\}\)
9:Endif
10:End Function
```  
**Algorithm 1** Compute \(argmax_{n,m}^{s}(\widetilde{p})\), \(\widetilde{p}\in[0,p^{s}]\)