## 2 Working Models
### Colpitts Chaotic Synchronization
More precisely, the Rx model is given by:  
\[\left\{ \begin{array}{c}C_{1}\frac{dV_{CE,RX}(t)}{dt}=I_{L_{L}RX}(t)-I_{C_{-} TX}(t))\\ C_{2}\frac{dV_{BE,RX}(t)}{dt}=-\frac{V_{FE}+V_{BE,RX}(t)}{RE_{EE}}-I_{L_{-}RX}(t)-I_{ B_{-}RX}(t)\\ L\frac{dI_{L,RX}(t)}{dt}=V_{CC}-V_{CE\_RX}(t)+V_{BE\_TX}(t)-I_{L_{-}RX}(t)R_{L} \end{array} \right., \tag{3}\]  
Note that the Rx's piecewise linear function is identical to (2), and therefore omitted for clarity purposes. Using (1) and (3) we plot the simulated results for the Colpitts time series, synchronized as depicted in Figure 4. We observe that after an initial transient due to the initial condition mismatch both oscillators are synchronized. The Tx and Rx are shown in blue and orange, respectively. Next, we plot the error against time as illustrated in Figure 5, which shows that the coupling mechanism leads to perfect chaotic synchronization for an ideal channel. This is represented by a time series that stabilizes to zero after a transient period. Finally, we plot both attractors together as shown in Figure 6 and observe that both overlap after the initial transient. The suggested coupling leads to a robust form of identical chaotic synchronization. We show in the following section that this is suitable for future implementations of chaotic inclusion techniques with Colpit  
Figure 4: Colpitts time series synchronized. We finish this subsection with a brief discussion regarding the frequency of oscillation in chaotic models, in relation to time normalization mechanisms. Recalling that chaotic oscillators are composed of an infinite number of frequencies, it is tacitly admitted that a chaotic signal has a practical finite frequency bandwidth in which the most dominant frequency components are contained. Also, it is important to highlight that the frequency concept is borrowed from harmonic oscillation theory and that it does not neatly fit in the chaotic concept. The Colpitts chaotic bandwidth, for the model described in (1) and (2), is in the order of 100 kHz. One can see this by analyzing the spectral composition of the chaotic signal. Figure 5: Error plots. Figure 6: Colpitts attractors for Tx and Rx synchronized. V\({}_{\text{BE}}\)(t). In order to increase this frequency, we apply a time normalization procedure. This consists in replacing \(d/dt\) with \(d/d\tau\) where \(\tau=t/T\), with \(T\) being the normalized time factor in the equation model. Then, with time normalization, the normalized frequency \(\omega_{\text{norm}}\) can be gradually shifted from 100 kHz to the tens of MHz and above, allowing us to progressively observe the multipath effect on chaos synchronization. In the analog domain, the chaotic Colpitts time constant can be manipulated using the different discrete components to achieve higher frequencies (e.g., [30] for chaotic radar applications). Finally, since the model described in (1) and (2) is in continuous time; for numerical simulation purposes we employed a fixed step solver (Runge-Kutta, ode4) with a step size of \(\Delta t=10^{-7}\) s, which is scaled with \(T\). This is effectively a discretization procedure from the analog to the digital domain.