# Random Access for Underwater Acoustic Cellular Systems
## 3 RAP Design
### Proposed RAP
#### 3.2.2 Proposed RAP 2  
RAP 2 is generated using a Doppler-insensitive waveform, i.e., the LFM waveform. The information on the ID of a UE is mapped to the parameters of LFM waveform: frequency shift parameter, \(f\), and frequency sweeping parameter, \(\beta\). RAP 2 for UE \(c\) is generated in the continuous-time domain with a symbol duration, \(\tau\), as follows:  
\[ \begin{array}{l}x_{LFM}^{c}(t)=x_{\tau}(f_{c},\beta_{c},t)p_{\tau}(t)\\ -B\leq\beta_{c}\leq B,\ \frac{-B+|\beta_{c}|}{2}\leq f_{c}\leq\frac{B-|\beta_{c}|}{2} \end{array}  \tag{9}\]  
where \(B\) denotes the operational channel bandwidth (4 KHz in UL 0). The LFM waveform, \(x_{\tau}(f,\beta,t)\) and rectangular pulse, \(p_{\tau}(t)\), are defined as follows:  
\[x_{\tau}(f,\beta,t)=e^{j\pi(2f+\frac{\beta}{\tau}t^{2})},\ p_{\tau}(t)=\left\{ \begin{array}{ll}\frac{1}{\sqrt{\tau}},&-\frac{\tau}{2}\leq t\leq\frac{\tau} {2}\\ 0,&\text{otherwise}\end{array} \right. \tag{10}\]  
The AF of RAP 2 with the ID of UE, \(c\), is given as follows:  
\[ \begin{array}{lcl}A_{LFM}^{c}(t,f_{D})&=&\int\limits_{-\infty}^{\infty}x_{ LFM}^{c}(s)\big{(}x_{LFM}^{c}(s-t)\big{)}^{*}e^{j2\pi f_{D}s}ds\\ &=&\left\{\begin{array}{ll}\varphi_{\tau}^{*}\Big{(}\frac{2f_{c}+f_{D}}{2} \tau t\Big{)}\rho_{LFM}(t)\text{sinc}((f_{D}\tau+\beta t)\rho_{LFM}(t)),&-\tau \leq t<\tau\\ 0,&\text{otherwise}\end{array}\right.\end{array}  \tag{11}\]  
where \(\rho_{LFM}(t)=1-|t|/\tau\). The derivation of (11) is given in the Appendix. As evident from (11), the magnitude of \(A_{LFM}^{c}(t,f_{D})\) varies depending on \(\tau,f_{D}\) and \(\beta_{c}\)[16]. However, the parameter \(f_{c}\) does not affect the magnitude of AF. Further, we analyze the effect of AF in (11) on \(f_{D}\) and \(\beta_{c}\) with the symbol duration being fixed to 125 ms. Figure 8 shows the maximum magnitudes and corresponding time shifts of AF in \(\beta_{c}\) and \(f_{D}\) domains. Figure 8a,c show the maximum magnitudes and corresponding time shifts of AF in \(\beta_{c}\) domain for three different values of the Doppler shift, i.e., 0 Hz, 12 Hz and 30 Hz. As shown in these figures, there is no variation in both the magnitude (all 1 s) and time shift (all 0 s) when \(f_{D}\) = 0 Hz. However, magnitude and time shift vary when there is a Doppler shift. The degree of variation is significant especially at a small value of \(\beta_{c}\). It can be observed that, when \(\beta_{c}\) = 0 Hz and \(f_{D}\) = 30 Hz, the maximum value becomes smaller than 0.2 and the absolute value of time shift increases up to 480 samples. With a sampling rate of 4 kHz, the timing error of 480 samples corresponds to 180 m in the sound speed of 1500 m/s. In order to obtain a high detection probability of RAP with a small timing  
Figure 7: Maximum value of the CAF of RAP 1. (**a**) Doppler domain; (**b**) Distribution. error, the value of \(\beta_{c}\) should be selected such that high magnitude and small time shift of AF can be achieved. From these figures, the smallest value of \(\beta_{c}\) (denoted as \(\beta_{0}\)), i.e., 0.5 kHz, is selected and used in the following simulations. Figure 8b,d show the maximum value and corresponding time shifts of AF in \(f_{D}\) domain for three different values of \(\beta_{c}\), i.e., 0.5 kHz, 2 kHz and 4 kHz. As shown in Figure 8b, the maximum value decreases slowly in a triangular manner when \(\beta_{c}\) = 0.5 kHz, and fluctuates fast when \(\beta_{c}\) = {2, 4} kHz. However, the time shift varies in a reverse manner as shown in Figure 8d. The time shift in the case of \(\beta_{c}\) = 0.5 kHz increases faster than that in the cases of 2 kHz and 4 kHz. At the Doppler shift of 50 Hz, the time shift is 50, 12, and 6 samples when the value of \(\beta_{c}\) is 0.5 kHz, 2 kHz, and 4 kHz, respectively. In summary, RAP 2 is robust to Doppler shifts as long as the frequency sweeping parameter is set to be greater than \(\beta_{0}\). The time shift decreases in a Doppler environment as the frequency sweeping parameter increases.