## Appendix A Proof of Convergence of Algorithm 2  
From the optimization problem \(P_{1}\), we have \(R\left(f_{m,n}^{(0)},P_{m,n}^{*(1)},M_{m,n}^{*(1)}\right)\geq R\left(f_{m,n}^{ (0)},P_{m,n},M_{m,n}\right)\). From optimization problem \(P_{2}\), we have  
Figure 7: The modulation order comparison of different modulation schemes.  
\[R\Big{(}f_{m,n}^{*(1)},P_{m,n}^{*(1)},M_{m,n}^{*(1)}\Big{)}\geq R\Big{(}f_{m,n}^{ (0)},P_{m,n}^{*(1)},M_{m,n}^{*(1)}\Big{)}\). After updating \(f_{m,n}\) by replacing \(f_{m,n}^{*(1)}\) with \(f_{m,n}^{(0)}\) from optimization problem \(P_{1}\), we have \(R\Big{(}f_{m,n}^{*(1)},P_{m,n}^{*(2)},M_{m,n}^{*(2)}\Big{)}\geq R\Big{(}f_{m,n }^{*(1)},P_{m,n}^{*(1)},M_{m,n}^{*(1)}\Big{)}\). By replacing \(\{P_{m,n}^{*(1)},M_{m,n}^{*(1)}\}\) with \(\{P_{m,n}^{*(2)},M_{m,n}^{*(2)}\}\), from optimization problem \(P_{2}\), we have \(R\Big{(}f_{m,n}^{*(2)},P_{m,n}^{*(2)},M_{m,n}^{*(2)}\Big{)}\geq R\Big{(}f_{m,n }^{*(1)},P_{m,n}^{*(2)},M_{m,n}^{*(2)}\Big{)}\geq R\Big{(}f_{m,n}^{*(1)},P_{m, n}^{*(1)},M_{m,n}^{*(1)}\Big{)}\). The process repeats until the convergence condition is satisfied. According to the recursive algorithm, we can obtain that \(R\Big{(}f_{m,n}^{*(n)},P_{m,n}^{*(n)},M_{m,n}^{*(n)}\Big{)}\geq R\Big{(}f_{m,n }^{*(n-1)},P_{m,n}^{*(n)},M_{m,n}^{*(n)}\Big{)}\geq R\Big{(}f_{m,n}^{*(n-1)},P_ {m,n}^{*(n-1)},M_{m,n}^{*(n-1)}\Big{)}\) at the \((n+1)\)-th iteration of the iterative method. Since the solution region is compact due to the bounded \(M_{m,n}\), \(P_{m,n}\) and \(f_{m,n}\), the iterative method proposed in Algorithm 2 can finally converge to the optimal point \(\{M_{m,n}^{*},P_{m,n}^{*},f_{m,n}^{*}\}\), which completes the proof.  
Proof of Adjustment Probability.: According to the adjustment rules in Table 2, the adjustment probability of modulation scheme is expressed as  
\[\begin{split}& P=\text{Pr}(|\gamma(t)-\gamma(t+\Delta t)|\geq \gamma_{th})\\ &=\text{Pr}\bigg{(}\bigg{|}\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n}( \gamma_{m,n})}{-\ln(5P_{m,n})\ln 3\Delta f_{m,n}(d+\Delta t)}\bigg{|}\geq\gamma_{th} \bigg{)}\\ &\qquad\qquad-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+\Delta d)P_{m,n }(\gamma_{m,n})}{-\ln(5P_{m,n})\ln 3\Delta f_{m,n}(d+\Delta t)}\geq\gamma_{th} \bigg{)}\\ &\qquad\qquad-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+\Delta d)P_{m,n }(\gamma_{m,n})}{-\ln(5P_{m,n})\ln 3\Delta f_{m,n}(d+\Delta t)}\leq-\gamma_{th} \bigg{)}\\ &\qquad\qquad+\text{Pr}\bigg{(}\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n }(\gamma_{m,n})}{-\ln(5P_{m,n})\ln 3\Delta f_{m,n}(d+\Delta t)}\leq-\gamma_{th} \bigg{)}\\ &\qquad\qquad=P_{1,prob}+P_{2,prob}\end{split} \tag{16}\]  
By introducing the parameters as (18), \(P_{1,prob}\) and \(P_{2,prob}\) are expressed as  
\[P_{1,prob}=\text{Pr}((1.5P_{m,n}(\gamma_{m,n})(\Delta f_{m,n}(d+ \Delta t)-\Delta f_{m,n}(d))) \tag{17a}\] \[\times h_{m,n}^{2}(t,d)-3\Delta f_{m,n}(d)P_{m,n}(\gamma_{m,n}) eth_{m,n}(t,d)\] \[\geq 1.5\Delta f_{m,n}(d)\epsilon^{2}P_{m,n}(\gamma_{m,n})\] \[-\gamma_{th}\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d)\Delta f_{m,n}(d +\Delta d))\] \[=\text{Pr}\Big{(}Ah_{m,n}^{2}(t,d)-Bh_{m,n}(t,d)-C\geq 0\Big{)}\] \[P_{2,prob}=\text{Pr}((1.5P_{m,n}(\gamma_{m,n})(\Delta f_{m,n}(d+ \Delta d)-\Delta f_{m,n}(d)))\] \[\times h_{m,n}^{2}(t,d)-3\Delta f_{m,n}(d)\epsilon P_{m,n}(\gamma _{m,n})h_{m,n}(t,d)\] \[\leq 1.5\Delta f_{m,n}(d)\epsilon^{2}P_{m,n}(\gamma_{m,n})\] \[+\gamma_{th}\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d)\Delta f_{m,n}(d +\Delta d))\] \[=\text{Pr}\Big{(}Ah_{m,n}^{2}(t,d)-Bh_{m,n}(t,d)-C2\leq 0\Big{)}\]  
According to the solution existence conditions of the inequalities \(Ah_{m,n}^{2}-Bh_{m,n}-C\geq 0\) and \(Ah_{m,n}^{2}-Bh_{m,n}-C2\leq 0\), the calculation of \(P_{1}\) and \(P_{2}\) is classified into several cases under three different scenarios: the user moving farther from the base station with \(\Delta d\geq 0\), the user moving closer to the base station with \(\Delta d\leq 0\), and the user remaining stationary with \(\Delta d=0\), as shown in Equations (16)-(23).