# Random Access for Underwater Acoustic Cellular Systems
## 3 RAP Design
### Proposed RAP
#### 3.2.2 Proposed RAP 2
Subsequently, to analyze the cross-correlation properties of the proposed RAP 2 with different IDs of UEs under the effect of time delay and Doppler shift, the CAF of RAP 2 with IDs \(c\) and \(d\) is defined as:  
\[ \begin{array}{ll}A_{LFM}^{c,d}(t,f_{D})&=\int\limits_{-\infty}^{\infty}x_{ LFM}^{c}(s)\Big{(}x_{LFM}^{d}(s-t)\Big{)}^{*}e^{|2\pi f_{D}s}ds\\ &=\Big{(}\frac{\mathbb{C}_{c,d}(t,f_{D})}{a_{c,d}\tau}+j\text{sgn}(\Delta\beta )\frac{S_{c,d}(t,f_{D})}{a_{c,d}\tau}\Big{)}x_{\tau}(f_{d},-\beta_{d},t)e^{- j\pi\Delta\beta\tau(\frac{Af_{\tau}+\beta_{d}\tau+f_{D}\tau}{\Delta\beta\tau})} ^{2}\end{array}  \tag{12}\]  
Figure 8: Maximum value of AF of RAP 2. (**a**) Maximum magnitude in \(\beta_{c}\) domain; (**b**) Maximum magnitude in\(f_{D}\) domain; (**c**) Time shift in \(\beta_{c}\) domain; (**d**) Time shift in\(f_{D}\) domain. where \(-\tau\leq t\leq\tau\), \(\Delta f=f_{c}-f_{d},\ \Delta\beta=\beta_{c}-\beta_{d},\ \Delta\beta\
eq 0\), and  
\[\begin{array}{l}S_{c,d}(t,f_{D})=\left\{ \begin{array}{l}S(a_{c,d}s_{t}+b_{c, d}(t,f_{D}))+S(a_{c,d}s_{0}-b_{c,d}(t,f_{D})),-\tau\leq t<0\\ S(a_{c,d}s_{0}+b_{c,d}(t,f_{D}))+S(a_{c,d}s_{t}-b_{c,d}(t,f_{D})),\ 0\leq t<\tau\\ C_{c,d}(t,f_{D})=\left\{\begin{array}{l}C(a_{c,d}s_{t}+b_{c,d}(t,f_{D}))+C(a_ {c,d}s_{0}-b_{c,d}(t,f_{D})),-\tau\leq t<0\\ C(a_{c,d}s_{0}+b_{c,d}(t,f_{D}))+C(a_{c,d}s_{t}-b_{c,d}(t,f_{D})),\ 0\leq t<\tau\\ \end{array}\right.\\ s_{t}=\frac{\tau}{2}-|t|,a_{c,d}=\sqrt{\frac{|\Delta\beta|\pi}{\tau}},b_{c,d}( t,f_{D})=\left(\frac{\Delta\tau+\beta_{d}t+f_{D}\tau}{\Delta\beta\tau}\right) \sqrt{|\Delta\beta|\tau\pi}\end{array}  \tag{13}\]  
The derivation of (12) is given in the appendix. When \(\beta_{c}=\beta_{d}\), the CAF of RAP 2 can be derived as:  
\[ \begin{array}{ll}A_{LFM}^{c,d}(t,f_{D})&=\frac{x_{\tau}(f_{c}-\beta_{c}t)}{ \tau}\int\limits_{-\infty}^{\infty}\varphi_{\tau}^{*}(\Delta f\tau s+\beta_{c} ts+f_{D}\tau s)p_{\tau}(s)p_{\tau}(s-t)ds\\ &=\varphi_{\tau}\Big{(}-\frac{f_{c}+\Delta f+f_{D}}{2}\tau t\Big{)}\rho_{\tau }(t)\text{sinc}((\Delta f\tau+f_{D}\tau+\beta_{c}t)\rho_{\tau}(t))\end{array}  \tag{14}\]  
Since (14) can be viewed as a shifted version of a sinc function, the time shift, \(t_{S}^{c,d}\), and the corresponding peak value, \(\widetilde{\rho}_{LFM}^{c,d}\), are expressed as follows:  
\[t_{S}^{c,d}=-\frac{\tau}{\beta_{c}}(\Delta f+f_{D}),\widetilde{\rho}_{LFM}^{c,d}=\rho_{LFM}\Big{(}t_{S}^{c,d}\Big{)} \tag{15}\]  
when \(\beta_{c}\) is fixed, the time shift increases proportionally with the values of \(f_{c},f_{d}\), and \(f_{D}\). Moreover, the peak value decreases as the time shift increases. Further, (15) signifies that the time shift should be larger than the symbol duration to avoid time ambiguity, which may lead to false detection, i.e. :  
\[\frac{\left|t_{S}^{c,d}\right|}{\tau}\geq 1\Leftrightarrow|\Delta f+f_{D}| \geq|\beta_{c}| \tag{16}\]  
From (9) and (16), we can obtain the condition of \(\Delta f\) that does not produce time ambiguity, assuming that \(\Delta f\) is much larger than \(f_{D}\), as follows:  
\[\beta_{c}\leq\Delta f\leq\frac{B-\beta_{c}}{2},\ \beta_{c}>0 \tag{17}\]  
The condition in (17) is satisfied for all the values of \(\beta_{c}\) less than \(B/3\). If \(\beta_{c}\) is larger than \(B/3\), only one RAP can be generated with \(\beta_{c}\). Otherwise, time ambiguity will occur. For example, when \(\Delta f=\beta_{c}=0.5\) kHz, we can generate seven RAPs with \(|f_{c}|=\{0,\ 0.5,\ 1,\ 1.5\}\) kHz from a single value of the frequency sweeping parameter. In this case, the frequency of RAP after sweeping stays within the channel bandwidth and time ambiguity does not occur. In Figure 9, the analytical solutions in (12) and (14) are compared with the simulation results when \(f_{D}=0\) Hz. The analytical solution in (12) is obtained by numerical evaluation of Fresnel integral whereas the simulation results are obtained through a direct cross-correlation operation between the two LFM waveforms. As shown in this figure, the analytical solutions are consistent with the simulation results (almost indistinguishable).