## 5 The Adjustment Probability of Modulation Strategy  
When the user moves, the distance between the base station and user changes accordingly. Thus, the distance-dependent available frequency bandwidth varies and the modulation scheme should be adjusted too. Motivated by this, we design the modulation switch rules as shown in Table 2.  
According to the modulation switch rules, when the user moves, the probability that the SNR difference is not smaller than the threshold \(\gamma_{th}\), which is usually set to 0.5 dB in engineering practice, is calculated as follows: \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{\Delta}d\) & **SNR** & **Bandwidth** & **Adjustment** \\ \hline \multirow{4}{*}{\(>0\)} & \(\Delta\)SNR \(\geq\gamma_{th}\) & Arbitrary & Yes \\ \cline{2-4}  & \(\Delta\)SNR \(\leq\gamma_{th}\) & \(|B(d+\Delta d)-B(d)|\leq\min_{m,n}(\Delta f_{m,n}(d))\) & No \\ \cline{2-4}  & \(|B(d+\Delta d)-B(d)|\geq\min_{m,n}(\Delta f_{m,n}(d))\) & Yes \\ \hline \multirow{2}{*}{\(=0\)} & \(\Delta\)SNR \(\geq\gamma_{th}\) & Unchanged & Yes \\ \cline{2-4}  & \(\Delta\)SNR \(\leq\gamma_{th}\) & Unchanged & No \\ \hline \multirow{4}{*}{\(<0\)} & \(\Delta\)SNR \(\geq\gamma_{th}\) & Arbitrary & Yes \\ \cline{2-4}  & \(\Delta\)SNR \(\leq\gamma_{th}\) & \(|B(d+\Delta d)-B(d)|\leq\min_{m,n}(\Delta f_{m,n}(d))\) & No \\ \cline{1-1} \cline{2-4}  & \(|B(d+\Delta d)-B(d)|\geq\min_{m,n}(\Delta f_{m,n}(d))\) & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Switch Rules of Modulation Schemes. \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\] \[=\Pr(\left|-\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n}(\gamma_{m,n})}{-\ln(5 P_{b,m,n})N_{0}\Delta f_{m,n}(d)}\right.\] \[\qquad\qquad\left.-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+\Delta d)P_{ m,n}(\gamma_{m,n})}{-\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d+\Delta d)}\right|\geq\gamma_{th})\] \[=\Pr\left(\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n}(\gamma_{m,n})}{-\ln (5P_{b,m,n})N_{0}\Delta f_{m,n}(d)}\right. \tag{16}\] \[\qquad\qquad\left.-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+\Delta d)P_{ m,n}(\gamma_{m,n})}{-\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d+\Delta d)}\geq\gamma_{th}\right)\] \[\qquad\qquad\left.+\Pr\left(\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n}( \gamma_{m,n})}{-\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d)}\right.\right.\] \[\qquad\qquad\qquad\left.-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+ \Delta d)P_{m,n}(\gamma_{m,n})}{-\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d+\Delta d )}\leq-\gamma_{th}\right).\]  
By setting  
\[P_{1,prob} =\Pr\left(\frac{1.5\mu_{m,n}^{2}(t,d)P_{m,n}(\gamma_{m,n})}{-\ln(5 P_{b,m,n})N_{0}\Delta f_{m,n}(d)}\right.\] \[\qquad\left.-\frac{1.5\mu_{m,n}^{2}(t+\Delta t,d+\Delta d)P_{m,n} (\gamma_{m,n})}{-\ln(5P_{b,m,n})N_{0}\Delta f_{m,n}(d+\Delta d)}\geq\gamma_{ th}\right) \tag{17a}\] \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]0 \[P =\Pr(\left|\gamma(t)-\gamma(t+\Delta t)\right|\geq\gamma_{th})\]1  
we have \(P=P_{1,prob}+P_{2,prob}\). In Equation (16), the parameters \(h(t+\Delta t,d+\Delta d)\) and \(h(t,d)\) denote the channel power gain at the distance \(d+\Delta d\) and \(d\), respectively. The channel power gain \(h(t+\Delta t,d+\Delta d)=h(t,d)+\varepsilon\) with the channel error variable \(\varepsilon\sim N\big{(}0,\sigma_{n}^{2}\big{)}\), where \(\sigma_{n}^{2}\) is the variance of the error variable.