# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 4 Communication Receiver  
To decode the information, the self-synchronization scheme of chaotic systems is utilized [52; 53]. Consider two identical chaotic systems: a driver and a response system (an  
Figure 4: Block diagram for the joint mono and bistatic RadComm transmitter. Figure 3: Illustration of the joint radar-communication systems. RCO). These two systems are synchronized by driving (forcing) the response system using a chaotic signal generated from the driver. This type of synchronization is only possible if both the systems are identical in the sense that they have the same dynamical equations and control parameters. If the control parameters are mismatched, the error between the driver and response does not approach zero asymptotically. Furthermore, for a given driving function, the RCO is stable if the conditional Lyapunov exponents are negative [53]. Figure 5 shows the communication receiver that consists of a phase-locked loop (PLL) and a nonlinear detector. The received CBFM waveform is driven through the PLL, which consists of an infinite impulse response lowpass Butterworth third-order filter to eliminate higher harmonics, a mixer, and a local voltage-controlled oscillator that is arbitrarily initialized. The output of PLL \(p(t)\) approximates the instantaneous frequency \(v_{i}(t)\) of the transmitted waveform, i.e., \(p(t)\approx v_{i}(t)\)[54]. The scaled version of \(p(t)\), i.e., \(d(t)=\zeta p(t)\) acts as a driving function to synchronize the RCOs that are tuned to two sets of control parameters. The first RCO is tuned to the control parameters with respect to bit (\(-1\)) and the second RCO to the bit (+1). The corresponding dynamical equations of the Lorenz and Burke-Shaw RCOs are given in Table 2. The conditional Lyapunov exponents of RCOs for both sets of control parameters are negative, making them stable response systems. In this context, the combination of two RCOs, summers and a comparator, is called the nonlinear detector. The two RCOs will become synchronized to \(d(t)\) according to the control parameters associated with the transmitted signal. When a bit (\(-1\)) is encoded, the control parameters of the Lorenz system at the transmitter are \(\{10,\ 28,\ 8/3\}\) and hence RCO\({}_{1}\) will be synchronized to \(d(t)\) with the error \(|e_{1}(t)|=|d(t)-x_{r_{1}}(t)|\) approaching zero asymptotically. Due to parameter mismatch, RCO\({}_{2}\) will not be synchronized causing a significant amplitude in error \(|e_{2}(t)|=|d(t)-x_{r_{2}}(t)|\). Similarly, when a bit (+1) is encoded, the control parameters of the Lorenz system at the transmitter are \(\{14,\ 28,\ 8/3+0.6\}\). In this case, RCO\({}_{2}\) will synchronize with a very low error \(|e_{2}(t)|\) while RCO\({}_{1}\) will be out of sync with high error amplitudes. Figure 5(a) shows an illustration of errors \(|e_{1}(t)|\) and \(|e_{2}(t)|\) when information is encoded onto the Lorenz system. Similar results can be obtained for the Burke-Shaw system, as shown in Figure 5(b). For illustration purposes, we concatenated several encoded bits as a single waveform. However, considering the bistatic radar system, we considered that one CBFM waveform pulse could transmit only one bit of information. A data rate of 0.2 Mbits/s is used for our simulations. To decode the digital information, a nonlinear detection scheme is used. The comparator is used to compare the error between RCO\({}_{1}\), RCO\({}_{2}\) and \(d(t)\) to obtain \(|e_{1}(t)|\) and \(|e_{2}(t)|\) and decode the bits. It is mathematically expressed as Equation (6), where, \(T_{b}=5\)\(\upmu\)s is the duration of bit, \(n=0,\ 1,\ 2,\ \dots,\ N\) is the n\({}^{\text{th}}\) bit and \(N\) is the total number of encoded bits.