# Random Access for Underwater Acoustic Cellular Systems
## 3 RAP Design
### Proposed RAP
#### 3.2.1 Proposed RAP 1
Significant side peaks occur at \(|n|=\{100,200,300,400\}\) when the Doppler shift is 12 Hz (star). When \(f_{D}=0\) Hz, the magnitudes of side peaks are smaller than 0.2. However, when \(f_{D}=12\) Hz, the magnitude at the correct timing is reduced to approximately 0.2 and the side peaks become larger, leading to RAP detection with a large timing error. Figure 6b shows the behavior of AF evaluated at three different time indices, \(n=\{0,-100,-200\}\), when the Doppler shift varies. As shown in this figure, \(\left|A_{ZC}^{c}(0,\varepsilon)\right|\), \(\left|A_{ZC}^{c}(-100,\varepsilon)\right|\) and \(\left|A_{ZC}^{c}(-200,\varepsilon)\right|\) have sinc shapes decreasing from the maximum values of 1, 0.8, and 0.6, respectively. The largest magnitudes of \(\left|A_{ZC}^{c}(-100,\varepsilon)\right|\) and \(\left|A_{ZC}^{c}(-200,\varepsilon)\right|\) occur at the first and second nulling points in \(\left|A_{ZC}^{c}(0,\varepsilon)\right|\), i.e., at the Doppler shifts of 8 Hz and 18 Hz, respectively. The cross-over point between \(\left|A_{ZC}^{c}(0,\varepsilon)\right|\) and \(\left|A_{ZC}^{c}(-100,\varepsilon)\right|\) occurs near the Doppler shift of 4.13 Hz, with the magnitude of 0.61. Notably, the magnitudes of other side peaks including the peak at the correct timing are less than 0.2 when the Doppler shift is 4.13 Hz. These simulation results confirm that RAP 1 is sensitive to Doppler shift; the magnitude of AF at the correct timing is reduced and side peaks increase in a Doppler environment. From this result, it can  
Figure 6: AF of RAP 1. **(a)** AF vs. time delay; **(b)** AF vs. Doppler shift. also be concluded that the approach widely used in LTE-based systems to increase the number of RAPs, i.e., cyclic-shifting a ZC sequence, is not suitable for carrying the IDs of different UEs in underwater acoustic cellular systems owing to high auto-correlation in RAP 1 in a Doppler environment. Subsequently, in order to analyze the cross-correlation properties of the proposed RAPs with the different IDs of UEs under the effect of time delay and Doppler shift, the definition of AF is extended to CAF in this paper. The discrete-time CAF of RAP 1 with two different root indices, \(u_{c}\) and \(u_{d}\), is defined as:  
\[ \begin{array}{ll}A_{ZC}^{c,d}(n,\varepsilon)&=\frac{1}{N_{ZC}}\sum\limits_{m =-\infty}^{\infty}x_{ZC}^{c}(m)\Big{(}x_{ZC}^{d}(m-n)\Big{)}^{*}e^{j\frac{2\pi n }{N_{ZC}}m}\\ &=\frac{1}{N_{ZC}}\sum\limits_{m=-\infty}^{\infty}x_{u_{c}}(m)x_{u_{d}}^{*}(m -n)q(m)q(m-n)e^{j\frac{2\pi n}{N_{ZC}}m}\\ &=\frac{1}{N_{ZC}}\left\{\begin{array}{ll}x_{u_{d}}^{*}(-n)\sum\limits_{m= 0}^{N_{ZC}+n-1}x_{u_{cd}}(m)e^{j\frac{2\pi n}{N_{ZC}}(nu_{d}+\varepsilon)m},& \qquad-N_{ZC}<n\leq 0\\ x_{u_{c}}(n)e^{j\frac{2\pi n}{N_{ZC}}nN_{ZC}-1-n}x_{u_{cd}}(m)e^{j\frac{2\pi}{ N_{ZC}}(nu_{c}+\varepsilon)m},&\qquad 0<n<N_{ZC}\end{array}\right.\end{array}  \tag{6}\]  
where \(u_{cd}=u_{c}-u_{d}\). Unlike the AF of RAP 1 in (5), a closed-form expression of (6) cannot be derived for the general values of Doppler shift and \(n\). However, when \(\varepsilon\) is an integer value (\(\varepsilon_{l}\)), the CAF of RAP 1 can be derived as follows:  
\[ \begin{array}{ll}A_{ZC}^{c,d}(0,\varepsilon_{l})&=\frac{1}{N_{ZC}}\sum \limits_{m=0}^{N_{ZC}-1}x_{u_{cd}}(m)e^{j\frac{2\pi n}{N_{ZC}}m}\\ &=x_{u_{cd}}^{*}\Big{(}u_{cd}^{-1}\varepsilon_{l}\Big{)}\frac{1}{N_{ZC}}\sum \limits_{m=0}^{N_{ZC}-1}x_{u_{cd}}\Big{(}m+u_{cd}^{-1}\varepsilon_{l}\Big{)} \\ &=x_{u_{cd}}^{*}\Big{(}u_{cd}^{-1}\varepsilon_{l}\Big{)}\frac{1}{N_{ZC}}\sum \limits_{m=0}^{N_{ZC}-1}x_{u_{cd}}(m)\\ &=x_{u_{cd}}^{*}\Big{(}u_{cd}^{-1}\varepsilon_{l}\Big{)}e^{-j\frac{2\pi u_{cd} q\theta^{2}}{N_{ZC}}}\left\{\begin{array}{ll}G(\eta(u_{c}-u_{d}),N_{ZC}),&u_{c}>u_{ d}\\ G^{*}(\eta(u_{d}-u_{c}),N_{ZC}),&u_{c}<u_{d}\end{array}\right.\end{array}  \tag{7}\]  
where \(u_{cd}^{-1}\) denotes the multiplicative inverse of \(u_{cd}\). Further, \(\eta=(N_{ZC}+1)/2\) and \(\theta=(N_{ZC}-1)/2\).