# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 3 Generation of CBFM Waveforms for Joint Radar-Communications
In all the cases, the largest Lyapunov exponent is positive, indicating that the system is always chaotic despite changing the selected control parameter values. Note that despite changing the control parameters by a fraction of \(\pm 1\) (as the function of time), the \(\lambda\)s are identical. An identical \(\lambda\) value indicates that the characteristics of the system do not change drastically, and this can be observed from the time-series plot shown in Figure 2.  
With this chaos-shift keying approach, the output of a chaotic system inherently has information encoded in it without altering the bandwidth of the chaotic signal. Figure 2 shows the normalized time series plots of \(x(t)\) for both the Lorenz and Burke-Shaw chaotic systems. The systems are simulated using the Runge-Kutta fourth-order method with a  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**System** & \ \begin{tabular}{c} **Differential Equations** \\ **at the Transmitter** \\ \end{tabular}  & **Control Parameters** & \begin{tabular}{c} **Largest** \\ **Lyapunov Exponents** \\ \end{tabular}  \\ \hline Lorenz & \(\dot{y}_{t}=\gamma_{L}[x_{t}(\rho-z_{t})-y_{t}]\) & \(\rho=28\) & \(\lambda_{(+1)}=0.9335\) \\  & \(\dot{z}_{t}=\gamma_{L}[x_{t}y_{t}-\beta_{i}z_{t}]\) & \(\beta_{(-1)}=\frac{8}{3},\ \beta_{(+1)}=\frac{8}{3}+0.6\) & \\ \hline [PERSON]–[PERSON] & \(\dot{x}_{t}=\gammasampling interval of \(1\)\(ns\). Both the time series plots consist of information but without revealing what bits are encoded. The scaled version of \(x(t)\) acts as an input to the voltage-controlled oscillator (VCO). The scaling factor \(\zeta\) equal to the absolute maximum value of the \(x(t)\) is necessary to avoid spectral aliasing. Alternately, to reduce the additional hardware components, this scaling process can be achieved using the amplitude control approach, as shown in [51]. \[v_{i}(t)=\frac{x(t)}{\zeta}. \tag{4}\]  
The above input voltage \(v_{i}(t)\) controls the frequency oscillations of the VCO output, thus generating the CBFM waveform which, is expressed as  
\[s(t)=A\ exp\left(j2\pi f_{c}t+j2\pi K\int\limits_{0}^{t}v_{i}(\alpha)d\alpha \right). \tag{5}\]  
where \(A=1\), is the amplitude to obtain a constant envelope, \(f_{c}=300\) MHz is the carrier frequency, and \(K\) is the modulation index adjusted to obtain a bandwidth of \(150\) MHz. This particular bandwidth is chosen in order to obtain the range-resolution \(\Delta r=\frac{\zeta}{2P}=1\) m and considering the demodulation limits of the phase-locked loop. Notice that the generated FM waveform has a fractional bandwidth \(\frac{\beta}{f_{c}}=0.5\), thereby not satisfying the narrow band criteria where the fractional bandwidth should be less than \(0.1\). It should be noted that the frequencies of the chaotic systems are typically limited based on the circuitry. Unlike the work shown in [41], we modulate the chaotic signal onto the frequency carrier to generate the CBFM waveform. The spectral width of the generated CBFM waveform can be controlled using voltage-controlled oscillators to achieve the described bandwidths. Consequently, the spectrum of the CBFM waveform is independent of the circuit components of chaotic oscillators. Figure 3 shows an illustration of the joint radar-communication system. The generated CBFM waveform transmitted from location A is used for both sensing and communications. Communication is possible via ground-to-air (communicating with aircraft) or ground-to-ground (a device on the ground). The radar sensing is possible either in monostatic  
Figure 2: Instantaneous frequency (\(v_{i}(t)\) is the scaled version of \(x(t)\)) of the CBFM waveform using (**a**) Lorenz and (**b**) Burke–Shaw chaotic systems (zoom in plots). The solid line indicates the encoded information. The time-series plot does not have any discontinuities when digital information is encoded. mode in which the transmitter and receiver are collocated or in the bistatic mode where the transmitter and receiver are separated.