## 5 The Adjustment Probability of Modulation Strategy
\begin{split}& P_{2,prob}=\Pr\left(0<h_{m,n}<\frac{B+\sqrt{B^{2}+4 AC }}{2A}\right)\\ &=\int\limits_{-\infty}^{+\infty}F_{h_{m,n}}\left(\frac{B+\sqrt{B ^{2}+4 AC2}}{2A}\right)f_{\varepsilon}(\varepsilon)d\varepsilon\\ &=\int\limits_{-\infty}^{+\infty}F_{h_{m,n}}\left(\frac{B+\sqrt{B ^{2}+4 AC2}}{2A}\right)f_{\varepsilon}(\varepsilon)d\varepsilon\\ &=\int\limits_{-\infty}^{+\infty}F_{h_{m,n}}\left(\frac{B+\sqrt{B ^{2}+4 AC2}}{2A}\right)f_{\varepsilon}(\varepsilon)d\varepsilon\\ \end{split}  \tag{22}\]  
Case II: when \(C2>0\) and \(\Delta 2\leq 0\), \(P_{2,prob}=0\). Case III: when C2 \(<\) 0 and \(\Delta 2>\) 0, \(hb_{1}<\varepsilon<hb\) and \(lb<\varepsilon<lb_{1}\), where \(lb_{1}=-\sqrt{\frac{\gamma_{lb}\ln\left(5\gamma_{b,m,n}\right)N_{0}\left[\Delta f _{m,n}(\pm\Delta t)-\Delta f_{m,n}(\Delta t)\right]}{-15\gamma_{b,m}(\gamma_{b,m})}}\) and \(hb_{1}=\sqrt{\frac{\gamma_{lb}\ln\left(5\gamma_{b,m,n}\right)N_{0}\left\{\Delta f _{m,n}(\pm\Delta t)-\Delta f_{m,n}(\Delta t)\right\}}{-15\gamma_{b,m}(\gamma_{ b,n})}}\). Therefore, \(P_{2,prob}\) is calculated as  
\[ \begin{array}{l}P_{2,prob}=\Pr\left(\frac{B-\sqrt{B^{2}+4 AC}}{2\Delta}<h_{m, n}<\frac{B+\sqrt{B^{2}+4 AC}}{2\Delta}\right)\\ =\int\limits_{-\infty}^{+\infty}\left[F_{h_{m,n}}\left(\frac{B+\sqrt{\Delta}}{ 2\Delta}\right)-F_{h_{m,n}}\left(\frac{B-\sqrt{\Delta}}{2\Delta}\right)\right] f_{\varepsilon}(\varepsilon)d\varepsilon\\ =\int\limits_{lb}^{lb_{1}}\left[F_{h_{m,n}}\left(\frac{B+\sqrt{\Delta}}{2 \Delta}\right)-F_{h_{m,n}}\left(\frac{B-\sqrt{\Delta}}{2\Delta}\right)\right] f_{\varepsilon}(\varepsilon)d\varepsilon\\ +\int\limits_{lb_{1}}^{hb}\left[F_{h_{m,n}}\left(\frac{B+\sqrt{\Delta}}{2 \Delta}\right)-F_{h_{m,n}}\left(\frac{B-\sqrt{\Delta}}{2\Delta}\right)\right] f_{\varepsilon}(\varepsilon)d\varepsilon\\ \end{array}  \tag{23}\]  
Case IV: when C2 \(<\) 0 and \(\Delta 2\leq\) 0, \(P_{2,prob}=0\). To sum up, the total adjustment probability of the modulation scheme is given by \(P_{total}=P_{1,prob}+P_{2,prob}+\frac{1}{2}\left(1-P_{1,prob}-P_{2,prob}\right)\). In the third scenario, where the user remains unmoved with \(\Delta d>0\), the adjustment probability of the modulation scheme due to the channel time-varying, is calculated as  
\[ \begin{array}{l}P=\Pr\left(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq \gamma_{th}\right)\\ =\Pr\left(\left|\frac{1.5(h_{m,n}^{2}(t)-h_{m,n}^{2}(t+\Delta t))P_{m,n}(\gamma _{m,n})}{-\ln\left(5\gamma_{b,m,n}\right)N_{0}\Delta f_{m,n}}\right|\geq\gamma _{th}\right)\\ =\int\limits_{-\infty}^{0}F_{h_{m,n}}\left(-\frac{\varepsilon}{2}+\frac{ \gamma_{th}\ln\left(5\gamma_{b,m,n}\right)N_{0}\Delta f_{m,n}}{1.5\gamma_{b,m,n}(\gamma_{m,n})h_{1}^{2}2\varepsilon}\right)f_{\varepsilon}(\varepsilon)d \varepsilon\\ +\int\limits_{-\infty}^{+\infty}\left(1-F_{h_{m,n}}\left(-\frac{\varepsilon}{2 }-\frac{\gamma_{th}\ln\left(5\gamma_{b,m,n}\right)N_{0}\Delta f_{m,n}}{1.5 \gamma_{b,m}(\gamma_{m,n})h_{1}^{2}2\varepsilon}\right)\right)f_{ \varepsilon}(\varepsilon)d\varepsilon\end{array}  \tag{24}\]  
where \(h(t+\Delta t)=h(t)+\varepsilon\) with \(h(t)\) and \(\varepsilon\) independent of each other. The detailed derivation process is shown in Appendix A.