# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 4 Communication Receiver
\[\int\limits_{nT_{b}}^{(n+1)T  
To assess the performance of a communication waveform, the bit error rate (\(BER\)) as the function of the ratio of energy per bit to noise spectral density (\(E_{b}/N_{0}\)) is used. Figure 7 shows the BER plots for the bipolar shift keying (BPSK), chaos-based FM waveforms using the Lorenz and Burke-Shaw systems. A total of 100,000 signal realizations are used to compute the BER values. For higher noise, the Lorenz CBFM yields high BER values. In contrast, the Burke-Shaw CBFM waveform performs well with BER values close to the theoretical BER values of the BPSK waveform. However, when the noise levels are less, i.e., for \(E_{b}/N_{0}>4\), the BER curve of the Lorenz CBFM waveform closely follows the theoretical BER curve of the BPSK waveform. Consequently, there is a tradeoff of transmission between these two CBFM waveforms depending on the noise in communication channels.  
To show the effectiveness of our proposed communications waveform, we compared our results with state-of-the-art binary reduced phase shift keying (\(BRPSK\)) as presented in [14; 15]. The BER of the BRPSK is given as  
\[P_{b}=Q\left(\sqrt{\frac{2E_{b}sin^{2}(\phi)}{N_{0}}}\right) \tag{7}\]  
where, phases \(-\phi\) and \(\phi\) represent the binary data (two constellation points). If \(\phi=90^{\circ}\), the BRPSK waveform is essentially the BPSK waveform. The essence of the BRPSK waveform is to have a minor phase difference such that the constellation points have the minimum distance between them. For comparison purposes, we considered \(|\phi|=60^{\circ}\) and simulated the BER against the \(E_{b}/N_{0}\). It can be observed that the BER curves of the CBFM waveforms closely follow the BRPSK that is embedded in the traditional radar chirp waveform. In particular, for low noise, the Lorenz CBFM waveform performs better compared to the BRPSK, whereas the Burke-Shaw CBFM performs well in a high noise environment. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**System** & \multicolumn{2}{c}{**Differential Equations of RCOs**} & \multicolumn{2}{c}{**Conditional Lyapunov Exponents**} \\ \hline \multirow{6}{*}{Lorenz} & \multirow{3}{*}{\(\text{RCO}_{1}\)} & \(\hat{x}_{r_{1}}=\gamma_{L}[10(y_{r_{1}}-x_{r_{1}})]\) & \(\lambda_{\text{C}_{1}}=-1.7814\) \\  & & \(\hat{y}_{r_{1}}=\gamma_{L}[d(t)(\rho-z_{r_{1}})-y_{r_{1}}]\) & \(\lambda_{\text{C}_{2}}=-1.8862\) \\  & & \(\hat{z}_{r_{1}}=\gamma_{L}[d(t)y_{r_{1}}-\frac{8}{2}z_{r_{1}}]\) & \\ \cline{2-3}  & \multirow{3}{*}{\(\text{RCO}_{2}\)} & \(\hat{x}_{r_{2}}=\gamma_{L}[14(y_{r_{2}}-x_{r_{2}})]\) & \(\lambda_{\text{C}_{1}}=-2.1332\) \\  & & \(\hat{y}_{r_{2}}=\gamma_{L}[d(t)(\rho-z_{r_{2}})-y_{r_{2}}]\) & \(\lambda_{\text{C}_{2}}=-2.1353\) \\  & & \(\hat{z}_{r_{2}}=\gamma_{L}[d(t)y_{r_{2}}-(\frac{8}{3}+0.6)z_{r_{2}}]\) & \\ \hline \multirow{6}{*}{Burke–Shaw} & \multirow{3}{*}{\(\text{RCO}_{1}\)} & \(\hat{x}_{r_{1}}=\gamma_{B}[-10(x_{r_{1}}+y_{r_{1}})]\) & \(\lambda_{\text{C}_{1}}=-0.4906\) \\  & & \(y_{r_{1}}=\gamma_{B}[-10d(t)z_{r_{1}}-y_{r_{1}}]\) & \(\lambda_{\text{C}_{2}}=-0.5102\) \\ \cline{1-1}  & & \(\hat{z}_{r_{1}}=\gamma_{B}[10d(t)y_{r_{1}}+13]\) & \\ \cline{1-1} \cline{2-3}  & \multirow{3}{*}{\(\text{RCO}_{2}\)} & \(\hat{x}_{r_{2}}=\gamma_{B}[-11(x_{r_{2}}+y_{r_{2}})]\) & \(\lambda_{\text{C}_{1}}=-0.4933\) \\  & & \(\hat{y}_{r_{2}}=\gamma_{B}[-11d(t)z_{r_{2}}-y_{r_{2}}]\) & \(\lambda_{\text{C}_{2}}=-0.5076\) \\ \cline{1-1}  & & \(\hat{z}_{r_{2}}=\gamma_{B}[11d(t)y_{r_{2}}+16]\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamical equations for the response oscillator and its corresponding conditional-Lyapunov exponents demonstrating the stability of the response chaotic oscillators. Typically the radar waveforms are transmitted at a very high power in the magnitudes of 1 MW. In contrast, a typical communication waveform is transmitted in the order of KW. Therefore, the proposed communication receiver will have a significant signal-to-noise ratio compared to the traditional communication receivers.