## 3 Probabilistic Topology Control Algorithm
### Communication Network Architecture of SWARMs Project
#### 3.1.3 The Sweep-Spread Carrier Model  
The S2 CR module shown in Table 2 is built upon the sweep-spread carrier (S2C) technology [31]. In the following, we will introduce this technology under multipath environment and Doppler spreading environment briefly. Digital Signal with Sweep Spread Carrier  
Assuming that the sweep spread carrier (S2-carrier) consists of a succession of sweeps with frequency variation from \(\omega_{L}\) to \(\omega_{H}\) within a time interval \(T_{sw}\), and all the sweeps will be uniformly produced in a linear manner with rapid frequency variation following each other successively without any gap between them. Then the S2-carrier can be expressed as:  
\[c(t)=A_{c}e^{j(\omega_{L}(t-\lfloor\frac{t}{T_{sw}}\rfloor T_{sw})+m(t- \lfloor\frac{t}{T_{sw}}\rfloor T_{sw})^{2})}, \tag{1}\]  
where \(A_{c}\) is the amplitude; \(m=\frac{(\omega_{H}-\omega_{L})}{2T_{sw}}\) is a coefficient denoting the frequency variation rate; \(\omega_{L}\) and \(\omega_{H}\) denote the lowest and highest angular frequencies, respectively; \(T_{sw}\) is the sweep duration; the term \(\left\lfloor\frac{t}{T_{sw}}\right\rfloor\) denotes the operand for truncating the value to the nearest least integer, which is defined as:  
\[\left(t-\left\lfloor\frac{t}{T_{sw}}\right\rfloor T_{sw}\right)=\left\{\frac{ t}{T_{sw}}\right\}T_{sw}, \tag{2}\]  
Equation (2) can be interpreted in Equation (1) as an actual cycle time with the cycle duration \(T_{sw}\). Signal with Sweep Spread Carrier under Multipath Channel  
Based on the conclusions in Equations (1) and (2), the signal with S2C in a multipath channel can be calculated. Let the symbol \(s(t)\) be phase encoded data. The symbols are modulated onto the S2C, which is \(x(t)=s(t)\cdot c(t)\). The signal is transmitted over a dispersive underwater channel. The part of the model which represents the water medium consists of a number of delay elements \(\tau_{i}\) which denote the time intervals between two successive multipath arrivals, and a number of multiplication elements \(V_{i}\) which takes possible attenuations on interfering multipath arrivals into account. If both \(c(t)\) and \(s(t)\) have unit amplitudes, and every coefficient \(V_{i}\) and delay element \(\tau_{i}\) remain constant during the entire transmission time, then after propagation along different paths in an underwater medium, the signals received by a receiver can be calculated as:  
\[y(t)=V_{0}x(t)+\sum_{i}V_{i}x(t-\tau_{i})+n(t), \tag{3}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Items** & **Parameters** \\ \hline Operating depth & 200 m to 2000 m depending on the housing material; 6000 m for Ti \\ \hline Operating range & 3500 m \\ \hline Frequency band & 18–34 kHz \\ \hline Transducer beam pattern & Horizontally omnidirectional \\ \hline Acoustic connection & (1) Burst data mode: up to 13.9 kbit/s in good channel conditions; 2.2–3.2 kbit/s in complex channel condition shallow water; (2) instant message mode: 1 kbit/s \\ \hline Bit error rate & Less than \(10^{-10}\) \\ \hline Power & Stand-by mode: 2.5 mW; listen mode: 5–285 mW; receive mode: \(\leq\)1.3 W; transmission mode: 2.8 W (1000 m), 8 W (2000 m), 35 W (3500 m), 80 W (max available) \\ \hline Power supply & External 24 VDC; internal rechargeable battery \\ \hline Dimensions & diameter 110 mm; total length 265 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the S2 CR communication module. where \(x(t)\) is defined as above, and \(x(t-\tau_{i})\) can be expressed as:  
\[x(t-\tau_{i})=s(t-\tau_{i})\cdot e^{j(\omega_{L}\{\frac{t-\tau_{i}}{T_{sw}}\}T_{ sw}+m(\{\frac{t-\tau_{i}}{T_{sw}}\}T_{sw})^{2})}, \tag{4}\]  
where \(n(t)\) is the white noise. It is evident that:  
\[\bigg{\{}\frac{t-\tau_{i}}{T_{sw}}\bigg{\}}T_{sw}=\left\{ \begin{array}{l}t_ {c}-\tau_{ci},\quad t_{c}\geq\tau_{ci}\\ T_{sw}+t_{c}-\tau_{ci},\quad t_{c}<\tau_{ci}\end{array}