# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 3 Generation of CBFM Waveforms for Joint Radar-Communications  
An N-dimensional chaotic system is mathematically given by a set of differential equations [42] such as  
\[\frac{d\mathbf{X}}{dt}=f(\mathbf{X}) \tag{1}\]  
where \(f\) is a nonlinear function, and \(\mathbf{X}\) is a vector of N-chaotic state variables. These systems are governed by control parameters that dictate the periodic, quasi-periodic and chaotic nature of the system [43]. For illustration purposes, we considered three-dimensional Lorenz [44] and Burke-Shaw [45] systems that are given in Table 1. Many other chaotic systems, such as [PERSON]'s circuit (multi-scroll) [46], can yield similar results. It should be noted that the performance of the communication receiver depends on the time it takes to self-synchronize. Lorenz, Burke-Shaw, and Chua systems self-synchronize relatively faster than other oscillators such as the Rossler system. Varying time constants of chaotic systems [47] and linear methodologies such as frequency scaling [48] can improve the communication and bistatic radar performance. However, these implementations are beyond the scope of this work. Instead, the digital information is encoded using the chaos-shift keying approach, where the control parameters of the system are continuously varied in proportion to the digital information \(i(t)\)[24; 25; 49]. Here, \(i(t)\) takes on values of \(-1\) and \(+1\), i.e., \(i(t)\), a data stream that could be thought of as a pseudo-random sequence of antipodal bits. Figure 1 shows the bifurcation plot of the (a) Lorenz and (b) Burke-Shaw systems. The y-axis is the maxima of the x-state variable (\(X_{n}\)). For the Lorenz system, the bifurcation parameter is considered to be \(\beta\). When \(\rho=28\) and \(\sigma=10\), the chaotic region is from \(0.72<\beta<3.46\). Similarly, when \(\rho=28\) and \(\sigma=14\), the chaotic region is \(0.97<\beta<3.315\). For the Burke-Shaw system, the bifurcation parameter is \(V\). When \(U=10\), the chaotic region occurs between \(5<V<28\), and for \(U=11\), the chaotic region is \(5<V<30\). In both cases, with change in \(\sigma,\beta,U,\) and \(V\), the dynamics of the bifurcation plot are not easily altered. For the Lorenz system, the control parameter \(\rho=28\) is constant, while the other two control parameters \(\sigma\) and \(\rho\) are a function of time given as  
Figure 1: Bifurcation plot of (**a**) the Lorenz system with varying parameter \(\beta\) and (**b**) the Burke–Shaw system with varying parameter \(V\). \[\sigma_{i}(t) =12+2i(t) \tag{2}\] \[\beta_{i}(t) =\left(\frac{8}{3}+0.3\right)+0.3i(t).\]  
when a bit \(-1\) is encoded, \(\sigma_{(-1)}=10\), and \(\beta_{(-1)}=8/3\). When a bit \(+1\) is encoded, \(\sigma_{(+1)}=14\), and \(\beta_{(+1)}=8/3+0.6\). For the remainder of the paper, the subscripts \((-1)\) and \((+1)\) indicate that encoded bits are \(-1\) and \(+1\), respectively. Similarly, for the Burke-Shaw system, the control parameters \(U\) and \(V\) are varied as  
\[U_{i}(t) =10.5+0.5i(t) \tag{3}\] \[V_{i}(t) =14.5+1.5i(t).\]  
Hence, \(U_{(-1)}=10\), \(V_{(-1)}=13\) and \(U_{(+1)}=11\), \(V_{(+1)}=16\). The selection of the control parameters given in Equations (2) and (3) are based on the chaotic region, as shown in Figure 1. In addition to bifurcation analysis, to verify that the system is chaotic for the selected control parameters, we computed the largest Lyapunov exponent (\(\lambda\)) using the QR decomposition method [50]. A positive value of \(\lambda\) indicates that the system quickly diverges from its original orbit with an infinitesimal change in the initial conditions, making the system chaotic. Values of the largest Lyapunov exponents \(\lambda_{(-1)}\) and \(\lambda_{(+1)}\) for the Lorenz and Burke-Shaw systems are summarized in Table 1. The values \(\gamma_{L}\) and \(\gamma_{B}\) are the time constants used to compress or dilate the chaotic system. These particular values are chosen to obtain the identical bandwidth for the Lorenz-based CBFM and Burke-Shaw-based CBFM waveforms.