# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 5 Bistatic Radar Receiver Synchronization and Ambiguity Function  
### Synchronizing the Bistatic Radar Receiver  
The bistatic radar shown in Figure 8 has the transmitter and receiver at different locations separated by a considerable distance called baseline (\(L\)). Compared to a monostatic radar, some advantages of using bistatic configuration are passive sensing, detection of low radar cross-sectional (RCS) targets, and avoiding jammer systems, etc. Despite these advantages, bistatic radars are not commonly employed due to their geometry and the problem associated with synchronization of the receiver and transmitter, which is necessary to obtain range-Doppler information of the target. In [55,56], tools such as global positioning systems (GPS), crystal oscillators, low-cost quartz GPS-disciplined oscillators, satellites, and wired communication channels, etc. are used to synchronize the receiver and the transmitter. Furthermore, due to the geometry limitations of bistatic radars, unique signal  
Figure 6: Encoded information and errors \(|e_{1}(t)|\) and \(|e_{2}(t)|\) for (**a**) the Lorenz CBFM waveform and (**b**) Burke-Shaw CBFM waveform. (Illustration purpose only. For practical reasons, a single CBFM pulse transmits only one bit of information.) Figure 7: Bit error rate (\(BER\)) for BPSK, Lorenz CBFM and Burke-Shaw CBFM waveforms, and BRPSK as a function of the ratio of energy per bit to noise spectral density (\(E_{b}/N_{0}\)). processing methods are required to obtain high-resolution imagery of targets. Instead, chaotic oscillators offer a cost-effective way of synchronizing bistatic radar receivers [57; 58], as well as generating high-resolution radar imagery [59]. To synchronize and detect the target, we consider two antennas at the receiver. One is dedicated to face the transmitter, and the other is used to track the target. We assume that the transmitter and the receiver are in line-of-sight (LOS) and are on non-moving platforms. A direct synchronization approach is utilized at the bistatic radar receiver [60], as shown in Figure 9. Here, the synchronization is achieved directly by receiving a signal (through the dedicated antenna), demodulating it, and synchronizing the local oscillator to achieve the ranging and detection. Hence, the term direct synchronization. Usually, the signal transmission takes place using a separate channel such as a land-line cable, communication link, or a satellite. Instead, we propose using the transmitted CBFM waveform to achieve synchronization while providing excellent range resolution. The waveform incident on the dedicated antenna is demodulated using the PLL (with the same specifications) mentioned above. The scaled version of the output of PLL, i.e., \(d(t)=\zeta p(t)\) is used to drive the local RCOs. Depending on the bit encoded at the transmitter, the RCO that is tuned with respective control parameters will be synchronized. That is, when a bit (\(-1\)) is encoded, the RCO\({}_{1}\) is synchronized, and when a bit (\(+1\)) is encoded, the RCO\({}_{2}\) is synchronized. The comparator (given in Equation (6) or part of Figure 5) is used to determine which RCO is synchronized with the least error level. The normalized version of the corresponding RCO's \(\widehat{x}(t)\), i.e., \(\widehat{v}_{i}(t)=\widehat{x}(t)/\zeta_{r}\) forms the instantaneous frequency to reconstruct the CBFM waveform. Here, \(\zeta_{r}\) is maximum of \(|\widehat{x}(t)|\). The reconstructed CBFM waveform \(s_{rec}(t)\) given in Equation (8) serves as a reference waveform for the matched filter. \[s_{rec}(t)=Aexp\left(j2\pi f_{c}t+j2\pi K\int\limits_{0}^{t}\widehat{v}_{i}( \alpha)d\alpha\right). \tag{8}\]  
The effectiveness of the reconstruction approach is characterized using correlation analysis. In Figure 10a, the auto-correlation between \(\{s(t),s(t)\}\) of the Lorenz CBFM is plotted using a solid line (black color) plot while the cross-correlation between \(\{s(t),s_{rec}(t)\}\)  
Figure 8: Bistatic radar configuration. Figure 9: Architecture for the bistatic radar receiver.