# Random Access for Underwater Acoustic Cellular Systems
## 3 RAP Design
### Proposed RAP
#### 3.2.2 Proposed RAP 2
Figure 9a shows the case of \(\Delta\beta\
eq 0\). Figure 9b shows a special case when \(\Delta\beta=0\) and \(\Delta f=0\). In this case, the highest peak occurs at the correct timing (\(t=0\)) because the two RAPs are generated using the same LFM waveform. Figure 10 shows the CAFs of RAP 2 for different values of \(\beta_{b}\) and \(f_{b}\). Further, \(\beta_{c}=0.5\) kHz and \(f_{D}=0\) Hz. In Figure 10a, the analytical solution in (12) is plotted for three different values of \(\beta_{b}\), i.e., 0.55 kHz, 2 kHz, and \(-1\) kHz. As shown in this figure, the cross-correlation level decreases as \(\Delta\beta\) increases. In Figure 10b, the analytical solution in (14) is plotted for three different values of \(\Delta f\). As shown in this figure, there are two significant peaks other than the main peak at the origin. The two peaks occur when the two RAPs at \(f_{b}=350\) Hz (\(\Delta f=100\) Hz) and 450 Hz ((\(\Delta f=200\) Hz) are received and correlated with an RAP at 250 Hz. This may produce ambiguity in detecting the ID of the UE and time delay. The peak value and the corresponding time delay can be predicted using (15). The value of \(\Delta\beta\) should be carefully chosen to have small correlation among RAPs while supporting a large number of RAPs. In order to determine the number of possible RAPs generated by the LFM waveform, we will derive the upper bound of CAF of RAP 2. Here, we define \(\alpha_{C}\) and \(\alpha_{S}\) as the maximum values of cosine and sine Fresnel integrals, respectively. Subsequently, we can obtain the following relationships:\[ \begin{array}{ll}\text{i)}&-\alpha_{C}\leq C(t)\leq\alpha_{C},-\alpha_{S}\leq S (t)\leq\alpha_{S}\\ \text{ii)}&-2\alpha_{C}\leq C(t_{1})+C(t_{2})\leq 2\alpha_{C},-2\alpha_{S}\leq S (t_{1})+S(t_{2})\leq 2\alpha_{S}\\ \text{iii)}&0\leq C_{c,d}^{2}(t,f_{D})\leq 4\alpha_{C}^{2},0\leq S_{c,d}^{2}(t,f_{D}) \leq 4\alpha_{S}^{2}\\ \text{iv)}&0\leq C_{c,d}^{2}(t,f_{D})+S_{c,d}^{2}(t,f_{D})\leq 4\big{(} \alpha_{C}^{2}+\alpha_{S}^{2}\big{)}\\ \text{v)}&0\leq\frac{C_{c,d}^{2}(t,f_{D})+S_{c,d}^{2}(t,f_{D})}{\alpha_{c,d}^ {2}t^{2}}\leq\frac{4\big{(}\alpha_{C}^{2}+\alpha_{S}^{2}\big{)}}{\alpha_{c,d}^ {2}t^{2}}\\ \text{vi)}&0\leq\Big{|}A_{LFM}^{cd}(t,f_{D})\Big{|}\leq\frac{2}{\alpha_{c,d}^ {2}t}\sqrt{\alpha_{C}^{2}+\alpha_{S}^{2}}\end{array}  \tag{18}\]  
From (18), we can obtain the upper bound of the CAF of RAP 2 as follows:  
\[\Gamma(\tau,\Delta\beta)=2\sqrt{\frac{\alpha_{C}^{2}+\alpha_{S}^{2}}{\Delta \beta\tau\tau}},\ \Delta\beta>0 \tag{19}\]  
From (14), we can also obtain \(\alpha_{S}=0.8948\) and \(\alpha_{C}=0.9775\) by numerical evaluation of the Fresnel integral with a sampling rate of 4 kHz. As evident from (19), the upper bound value depends on neither the frequency shift \(f_{c}\) nor the Doppler shift \(f_{D}\), indicating that the values of \(f_{c}\) and \(f_{D}\) do not have a significant impact on the magnitude of CAF. The symbol duration (\(\tau\)) and separation value (\(\Delta\beta\)) between \(\beta_{c}\) and \(\beta_{d}\) are crucial. The larger the value of \(\Delta\beta\tau\), the smaller the value of the upper bound. From (11), we can observe that the range of frequency sweeping parameter is the union of the negative set, \(\mathbb{S}_{N}\), and positive set, \(\mathbb{S}_{P}\), i.e., \(\beta\in\{S_{N}\cup S_{P}\}\) where \(\mathbb{S}_{N}=[-B,-\beta_{0}]\) and \(\mathbb{S}_{P}=[\beta_{0},B]\). As discussed in the AF of RAP 2, \(\beta_{0}\) denotes the smallest value of \(\beta\) that provides a good AF in term of time shift and maximum magnitude value. Figure 11 shows the upper bound given in (19) and the maximum value of CAF of RAP 2 obtained through simulation. Here, \(\beta_{0}\) is set to 0.5 kHz and \(\beta_{d}(\beta_{c}+\Delta\beta)\) is a variable obtained by changing the value of \(\Delta\beta\). As shown in the figure, the upper bound value decreases with the increase of \(\tau\) or \(\Delta\beta\). The maximum value of CAF is slightly smaller than the upper bound and decreases with the increase of \(\tau\) or \(\Delta\beta\).