## 5 The Adjustment Probability of Modulation Strategy
By introducing the auxiliary parameters,  
\[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]2 \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]3 (18b) \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]4 (18c) \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]5 \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]6 (18d) \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]7 \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]8 (18e) \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]9  
The calculation of \(P_{1,prob}\) and \(P_{2,prob}\) can be classified into several cases under three different scenarios, namely, 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 \(\Delta d=0\). In the first scenario, where the user moves farther from the base station with \(\Delta d>0\), \(f(d)-f(d+\Delta d)>0\), \(A<0\), \(C>0\) and \(\Delta 2>0\), the calculation of \(P_{1,prob}\) and \(P_{2,prob}\) are classified into two cases, respectively. The calculation of \(P_{1,prob}\) is shown as following  
Case I: when \(\Delta\leq 0\), the probability \(P_{1,prob}=0\). Case II: when \(\Delta>0\), the probability \(P_{1,prob}=\Pr(h_{m,n}>0)=1-F_{h_{m,n}}(0)\), where \(F(\cdot)\) represents the cumulative probability distribution function (CDF). \(h_{m,n}\) denotes the channel power gain of \(n\)-th sub-frequency in the \(m\)-th sub-window. The probability \(P_{2,prob}\) is derived as follows:  
Case I: when \(C2>\), \(P_{2,prob}=1\). Case II: when \(C2<0,lb<\varepsilon<hb\), where the \(lb=-\sqrt{\frac{-\gamma_{lb}\ln\left(5P_{1,prob,n}\right)N_{0}\Delta f_{m,n}( \ell+\Delta d)}{1.5P_{m,n}(\gamma_{m,n})}}\) and \(hb=\sqrt{\frac{-\gamma_{lb}\ln\left(5P_{1,prob,n}\right)N_{0}\Delta f_{m,n}( \ell+\Delta d)}{1.5P_{m,n}(\gamma_{m,n})}}\), and \(h_{m,n}>\frac{B-\sqrt{B^{2}+4 AC2}}{2A}\). \(P_{2,prob}\) is thus calculated as  
\[ \begin{split}& P_{2,prob}=\Pr\left(h_{m,n}>\frac{B-\sqrt{B^{2}+4 AC2}}{2A }\right)\\ &=\int\limits_{lb}^{hb}\left(1-F_{h_{m,n}}\left(\frac{B-\sqrt{B^ {2}+4 AC2}}{2A}\right)\right)f_{\varepsilon}(\varepsilon)d\varepsilon.\end{split}  \tag{19}\]  
To sum up, the total adjustment probability of modulation scheme is written as  
\[P_{total}=P_{1,prob}+P_{2,prob}+\frac{1}{2}\left(1-P_{1,prob}-P_{2,prob}\right). \tag{20}\]  
According to Table 2, the adjustment occurs when \(\Delta\text{SNR}\geq\gamma_{th}\) or \(\Delta\text{SNR}<\gamma_{th}\) and \(\Delta B>B_{min}(d)\). The probability of the first case is \(P_{1,prob}+P_{2,prob}\) and the probability of the second case is \(\frac{1}{2}\left(1-P_{1,prob}-P_{2,prob}\right)\), where we assume the probabilities of the bandwidth difference being no larger and larger than the threshold are equal due to the random mobility of the user. In the second scenario, where the user moves closer to the base station with \(\Delta d<0\), \(f(d)-f(d+\Delta d)<0\), \(A>0\), \(C>0\), and \(\Delta>0\), the calculation of \(P_{2}\) is classified into four cases. The \(P_{1,prob}\) is derived as  
\[ \begin{split}& P_{1,prob}=\Pr\left(h_{m,n}\geq\frac{B+\sqrt{B^{2}+4 AC }}{2A}\right)\\ &=\int\limits_{-\infty}^{+\infty}\left(1-F_{h_{m,n}}\left(\frac{B +\sqrt{B^{2}+4 AC}}{2A}\right)\right)f_{\varepsilon}(\varepsilon)d\varepsilon \end{split}  \tag{21}\]  
There are four cases for the calculation of \(P_{2,prob}\):  
Case I: when \(C2>0\) and \(\Delta 2>0\), \(-\infty<\varepsilon<-\sqrt{\frac{-\gamma_{lb}\ln\left(5P_{1,prob,n}\right)N_{0} \Delta f_{m,n}(d+\Delta d)}{1.5P_{1,prob,n}(\gamma_{m,n})}}\) and \(\sqrt{\frac{-\gamma_{th}\ln\left(5P_{1,prob,n}\right)N_{0}\Delta f_{m,n}(d+ \Delta d)}{1.5P_{1,prob,n}(\gamma_{m,n})}}<\varepsilon<+\infty\). Therefore, \(P_{2,prob}\) is calculated as  
\[