# Random Access for Underwater Acoustic Cellular Systems
## Appendix  
Some useful properties, which will be used to simplify the algebraic equations related to the LFM waveform, are given as follows:  
\[ \begin{array}{l}x_{\tau}(f,\beta,t)=e^{-j\pi\beta\tau(f/\beta)^{2}}e^{j\text{ sgn}(\beta)(at+b)^{2}},\ \beta\
eq 0\\ x_{\tau}^{\tau}(f,\beta,t)=x_{\tau}(-f,-\beta,t)\\ x_{\tau}(f,\beta,s-t)=x_{\tau}(-f,\beta,t)x_{\tau}(f,\beta,s)\varphi_{\tau}( \beta st)\\ x_{\tau}(f_{1},\beta_{1},t)x_{\tau}(f_{2},\beta_{2},t)=x_{\tau}(f_{1}+f_{2}, \beta_{1}+\beta_{2},t)\end{array}  \tag{1}\]  
where \(\text{sgn}(\beta)\) denotes the signum function with the values of \(\{-1,0,1\}\) for negative, zero, and positive values of \(\beta\), respectively. Furthermore:  
\[\varphi_{\tau}(t)=e^{-j\frac{2\pi}{\tau}t},\ a=\sqrt{\frac{|\beta|\pi}{\tau}}, \ b=\sqrt{|\beta|\pi\pi}\frac{f}{\beta} \tag{2}\]  
Additional integral functions, which will be used to simplify the algebraic equations in this sub-section, are given as follows:  
\[ \begin{array}{l}\int_{t_{1}}^{t_{2}}\sin(as+b)^{2}ds=\frac{S(at_{2}+b)-S(at_ {1}+b)}{a}\\ \int_{t_{1}}^{t_{2}}\cos(as+b)^{2}ds=\frac{C(at_{2}+b)-C(at_{1}+b)}{a}\end{array}  \tag{3}\]  
where \(S(t)=\sqrt{\pi/2}S_{n}(\sqrt{2/\pi}t)\) and \(C(t)=\sqrt{\pi/2}C_{n}(\sqrt{2/\pi}t)\) denote the sine and cosine Fresnel integrals, respectively. The normalized forms of Fresnel integrals are given by:  
\[S_{n}(t)=\int_{0}^{t}\sin\left(\frac{\pi s^{2}}{2}\right)ds,\quad C_{n}(t)= \int_{0}^{t}\cos\left(\frac{\pi s^{2}}{2}\right)ds \tag{4}\]  
The AF and CAF of RAP 2 are derived using (1) and (3) as follows:  
\[\begin{array}{lcl}A_{LFM}^{c}(t,f_{D})&=&\int\limits_{-\infty}^{\infty}x_{ \tau}(f_{c},\beta_{c},s)x_{\tau}^{\tau}(f_{c},\beta_{c},s-t)e^{j2\pi f_{D}s}p_ {\tau}(s)p_{\tau}(s-t)ds\\ &=&\frac{x_{\tau}(f_{c}-\beta_{c},t)}{\tau}\left\{ \begin{array}{l}\int \limits_{-\tau/2+t}^{\tau/2+t}e^{j\frac{2\pi(\beta_{c}t+f_{D}\tau)}{\tau}}ds, \ -\tau\leq t<0\\ \int\limits_{-\tau/2+t}^{\tau/2+t}e^{j\frac{2\pi(\beta_{c}t+f_{D}\tau)}{\tau} s}ds,\ 0\leq t<\tau\end{array} \right.\\ A_{LFM}^{c}(t,f_{D})&=&\int\limits_{-\infty}^{\infty}x_{\tau}(f_{c},\beta_{c},s )x_{\tau}^{\tau}(f_{d},\beta_{d},s-t)e^{j2\pi f_{D}s}p_{\tau}(s)p_{\tau}(s-t) ds\\ &=&\frac{x_{\tau}(f_{d}-\beta_{d},t)}{\tau}\int\limits_{-\infty}^{\infty}x_{ \tau}\left(\Delta f+\frac{\beta_{d}t}{\tau}+f_{D},\Delta\beta,s\right)* _[PERSON] (2008)_ _[PERSON] Design and capacity analysis of cellular-type underwater acoustic network._ IEEE J. Ocean. Eng._**2008**, _33_, 171-181. [CrossRef]
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