## 3 Probabilistic Topology Control Algorithm
### Communication Network Architecture of SWARMs Project
#### 3.1.3 The Sweep-Spread Carrier Model
\right., \tag{5}\]  
where \(t_{c}=\Big{\{}\frac{t}{T_{sw}}\Big{\}}T_{sw}\) is the cycle time defined in Equation (2), and \(\tau_{ci}=\Big{\{}\frac{\tau_{i}}{T_{sw}}\Big{\}}T_{sw}\) is a fractional part of time delay related to sweep duration \(T_{sw}\). Thus, every delayed arrival represented in the second member of Equation (3) can be rewritten as:  
\[x(t-\tau_{i})=\left\{ \begin{array}{l}s(t-\tau_{i})\cdot e^{j(\omega_{L}\{t _{c}-\tau_{ci}\}+m(t_{c}-\tau_{ci})^{2})},\quad t_{c}\geq\tau_{ci}\\ s(t-\tau_{i})\cdot e^{j(\omega_{L}\{T_{sw}+t_{c}-\tau_{ci}\}+m(T_{sw}+t_{c}- \tau_{ci})^{2})},\quad t_{c}<\tau_{ci}\end{array} \right., \tag{6}\]  
After transformation of Equation (6), each delayed arrival can be written as:  
\[x(t-\tau_{i})=s(t-\tau_{i})\cdot e^{j(\omega_{L}t_{c}+mt_{c}^{2})}\cdot e^{j( -\Delta\omega_{i}t_{c}+\phi_{i})}, \tag{7}\]  
where \(\Delta\omega_{i}=\left\{ \begin{array}{l}2m\tau_{ci},\quad t_{c}\geq\tau_{ci }\\ -2m(T_{sw}-\tau_{ci}),\quad t_{c}<\tau_{ci}\end{array} \right.\) is the frequency deviation of the \(i\)-th multipath arrival caused by delay \(\tau_{i}\), and \(\phi_{i}=\left\{ \begin{array}{l}(m\tau_{ci}-\omega_{L})\tau_{ci},\quad t_{c }\geq\tau_{ci}\\ (\omega_{L}+\omega_{L}-2m\tau_{ci})\frac{T_{sw}-\tau_{ci}}{2},\quad t_{c}<\tau _{ci}\end{array} \right.\) is the phase of the \(i\)-th multipath arrival. The term with \(i\) = 0 in Equation (3) represents an attenuated version of the original signal, and the other term is the multipath diversity of its delayed, attenuated and frequency shifted reproductions. The most important feature of Equation (7) is that at any instant all the interfering multipath arrivals have different frequencies spaced by \(\Delta\omega_{i}\) from each other. Signal with Sweep Spread Carrier under Doppler Spreading  
The same can be shown for time-varying channels. The sweep spread carrier under Doppler spreading can be expressed as [31]:  
\[x(t-\tau_{i})=s(t-\tau_{i})\cdot e^{j(\omega_{L}\{\frac{t-\tau_{i}}{T_{sw}}\}T _{sw}+m(\{\frac{t-\tau_{i}}{T_{sw}}\}T_{sw})^{2})}\cdot e^{j\omega_{d}^{j}(t- \tau_{i})}, \tag{8}\]  
where \(\omega_{d}^{i}\) is the Doppler frequency encountered in \(i\)-th propagation path, which reflects the influence of Doppler effection on the received signal. The last exponent in Equation (8) can reflect time-varying phase/frequency. In this case, the \(\omega_{d}^{j}\) is characterized with a time dependent function specific for \(i\)-th path induced. Equation (8) demonstrates that Doppler shifts belonging to different paths will not be coupled while the \(\omega_{d}^{j}\) stays within certain borders; so a maximum value \(\omega_{d\text{max}}^{j}\) of the time-varying bandwidth enlargement \(\omega_{d}^{j}\) does not extend a half of frequency separation space between respective multipath arrivals (e.g., \(\omega_{d\text{max}}^{j}<\frac{\omega_{d}^{j}}{2}\)). In this case, every arrival stays within a definite frequency range and does not influence another frequency bands; no inter-modulation between differently varying Doppler terms belonging to different propagation paths takes place.