## 4 Adaptive Modulation for the Single User  
When the available frequency windows are determined based on the transmission distance, the adaptive modulation problem can be formulated as follows:  
\[ \begin{split}&\mathbf{P}:\max_{\{\Delta f_{m,n},M_{m,n},P_{m,n}\}}R =\sum\limits_{m=1}^{W(d)}\sum\limits_{n=1}^{N_{\text{gr}}}\Delta f_{m,n}\log_{ 2}M_{m,n}\\ & s.t.\\ & C1:\sum\limits_{m=1}^{W(d)}\sum\limits_{n=1}^{N_{\text{gr}}} \Delta f_{m,n}\leq B(f,d)\\ & C2:\sum\limits_{n=1}^{N_{\text{gr}}}\Delta f_{m,n}\leq B_{m}(f,d),\forall m=1,2,\ldots,M(d)\\ & C3:\sum\limits_{m=1}^{W(d)}\sum\limits_{n=1}^{N_{\text{gr}}} \rho_{m,n}(\gamma_{m,n})\leq P_{T}\\ & C4:M(\gamma_{m,n})\leq 1+\frac{15 ln_{m,n}P_{m,n}(\gamma_{m,n})}{- \ln\big{(}5P_{m,n}\big{)}N_{0}\Delta f_{m,n}}\\ & C5:\Delta f_{m,n}\geq 0\\ & C6:P_{m,n}(\gamma_{m,n})\geq 0\\ & C7:M_{m,n}\in Z^{+}.\end{split}  \tag{12}\]  
The parameters \(W(d)\) and \(N_{m}\) denote the number of frequency sub-windows and the subcarriers in the \(m\)-th sub-window, respectively. Constraint C1 indicates that the sum of all subcarriers cannot exceed the available bandwidth \(B(f,d)\). Constraint C2 stipulates that the sum of subcarriers in each frequency sub-window should be less than the bandwidth of each sub-window. Constraint C3 asserts that the power of any subcarrier cannot exceed the total transmit power. Constraint C4 requires that the modulation order of each subcarrier meets the bit error requirements. Constraints C5 and C6 specify that the bandwidth and power must be positive. Constraint C7 stipulates that the modulation order must be an integer. The problem described is non-convex due to the coupling of variables \(\Delta f_{m,n}\) and \(M_{m,n}\) in the objective function and constraint C4, \(B(d)\) represents the available transmission bandwidth, while \(B_{m,n}(d)\) represents the bandwidth of the \(n\)-th subcarrier in the \(m\)-th sub-window at distance \(d\). Consequently, an effective algorithm needs to be proposed to transform this non-convex problem into a convex problem. Through the analysis of the problem structure, we propose the block-coordinated decent (BCD) method to solve it. The principle of the BCD is outlined as follows:  
\[ \begin{split}& f^{(0)}\rightarrow\left\{P^{(0)},M^{(0)}\right\} \rightarrow\ldots\to f^{(n-1)}\rightarrow\left\{P^{(n-1)},M^{(n-1)}\right\} \\ &\to f^{(n)}\rightarrow\left\{P^{(n)},M^{(n)}\right\} \rightarrow\ldots\to f^{*}\rightarrow\{P^{*},M^{*}\}.\end{split}  \tag{13}\]Particularly, a feasible initial solution \(\Delta f^{(0)}_{m,n}\) is firstly determined and then, the variables \(M_{m,n}\) and \(P_{m,n}\) are jointly optimized to obtain the initial \(M^{(0)}_{m,n}\) and \(P^{(0)}_{m,n}\). Based on the latest optimized \(M^{(n-1)}_{m,n}\) and \(P^{(n-1)}_{m,n}\), the variable \(\Delta f^{(n-1)}_{m,n}\) is updated to be \(\Delta f^{(n)}_{m,n}\). The optimization process is repeated until the rate difference between the two iterations is less than the predetermined threshold. According to the principle of the BCD algorithm, the data rate of each iteration will be greater than that of the previous one. Therefore, this algorithm is convergent. According to the BCD method, the prime non-convex optimization problem **P** is decomposed into two sub-problems **P1** and **P2** as following:  
\[ \begin{split}\textbf{P1}:&\max\limits_{\{M_{m,n},P_{ m,n}\}}R=\sum\limits_{m=1}^{W(d)}\sum\limits_{n=1}^{N_{m}}\Delta f_{m,n}\log_{2}M_{m,n} \\ s.t.&\text{C1}:&\sum\limits_{m=1}^{W(d )}\sum\limits_{n=1}^{N_{m}}P_{m,n}(\gamma_{m,n})\leq P_{T}\\ \text{C2}:& M(\gamma_{m,n})\leq 1+\frac{1.5l_{m,n}P_{ m,n}(\gamma_{m,n})}{-\ln(5l_{m,n})N_{0}\Delta f_{m,n}}\\ \text{C3}:& P_{m,n}(\gamma_{m,n})\geq 0\\ \text{C4}:& M_{m,n}\in Z^{+}\end{split}  \tag{14}\]  
and  
\[