# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 5 Bistatic Radar Receiver Synchronization and Ambiguity Function
### Cross-Ambiguity Functions  
Once the bistatic radar receiver is synchronized and the transmitted waveform is reconstructed, \(s_{rec}(t)\) is used to correlate with the echo from the target \(r_{tgt}(t)\). This echo is a time-delayed and Doppler-shifted version of the transmitted waveform. That is  
\[r_{tgt}(t)=s(t-\tau)exp^{|2\pi[f_{c}+f_{D}]t}. \tag{9}\]  
From Figure 8, let the range between the transmitter and the target be \(R_{T}\), the range between the target and the receiver be \(R_{R}\). The bistatic configuration where \(L<R_{R}\) is called the cosine region, otherwise it is called the receiver-centered region. If \(\theta_{R}\) is the receiver look angle, then the relationships between the time-delay \(\tau\), range between the target-receiver \(R_{R}\) and Doppler frequency \(f_{D}\), and radial velocity \(V\) of the target are, respectively, given as [61]  
\[\left(\tau=\frac{R_{R}+\sqrt{R_{R}^{2}+L^{2}+2R_{R}Lsin\theta_{R}}}{c};f_{D}= \frac{2f_{c}}{c}V\sqrt{\frac{1}{2}+\frac{R_{R}+Lsin\theta_{R}}{2\sqrt{R_{R}^ {2}+L^{2}+2R_{R}Lsin\theta_{R}}}}\right) \tag{10}\]  
where \(c\) is the speed of light. To detect a target, the bistatic radar parameters to be investigated are the estimates of time delay \(\widehat{\tau}\) and the Doppler frequency \(\widehat{f}_{D}\). Similar to Equation (10), they are expressed as  
\[\left(\widehat{\tau}=\frac{\widehat{R}_{R}+\sqrt{\widehat{R}_{R}^{2}+L^{2}+2 \widehat{R}_{R}Lsin\theta_{R}}}{c};\widehat{f}_{D}=\frac{2f_{c}}{c}\widehat{ V}\sqrt{\frac{1}{2}+\frac{\widehat{R}_{R}+Lsin\theta_{R}}{2\sqrt{\widehat{R} _{R}^{2}+L^{2}+2\widehat{R}_{R}Lsin\theta_{R}}}}\right) \tag{11}\]  
When the target echo is processed through the matched filter [62], the estimates \(\widehat{\tau}\) and \(\widehat{f}_{D}\) can be determined via the cross-ambiguity function \(\chi\big{(}\widehat{\tau}-\tau,\widehat{f}_{D}-f_{D}\big{)}\), which is expressed as Equation (12). However, due to the nonlinear relationship between  
Figure 10: Auto–correlation (solid line plot) between \(\{s(t),s(t)\}\) and cross–correlation between \(\{s(t),s_{rec}(t)\}\) for (**a**) the Lorenz CBFM waveform (blue color plot) and (**b**) the Burke–Shaw CBFM waveform (red color plot).  
\(\{\widehat{\tau},\ \widehat{R}_{R}\}\) and \(\{\widehat{f}_{D},\ \widehat{V}\}\), the cross-ambiguity function of a bistatic radar is plotted on the range-velocity plane.  
\[\chi\Big{(}\widehat{\tau}-\tau,\widehat{f}_{D}-f_{D}\Big{)}=\int\limits_{- \infty}^{\infty}s(t-\tau)s_{rec}^{*}(t-\widehat{\tau})exp^{[-\beta\pi(f_{D}- \widehat{f}_{D})t]}dt. \tag{12}\]  
For the simulations, we considered a point target moving with a velocity \(V_{tgt}=100\) m/s, \(L=0.5\) Km, \(R_{R}=1\) Km, and \(\theta_{R}=+30^{\circ}\). When the received echo is driven through the matched filter with a response \(s_{rec}(t)\), a peak is expected at \(R_{R}\) and \(V_{tgt}\). Figure 11a,b shows the cross-ambiguity surface of the Lorenz CBFM and the Burke-Shaw CBFM waveforms, respectively, which is computed between the \(\{s_{rec}(t),r_{tgt}(t)\}\).  
For the Lorenz CBFM waveform, the sidelobes on the velocity plane decorrelate very quickly. On the range plane, the first SLL occurs at 0.136 \((-17.3\) dB). The subsequent sidelobes disappear at a faster rate. Similarly, for the Burke-Shaw CBFM waveform, the first range SLL occurs at 0.081 \((-21.93\) dB). If the digital information is embedded onto the chirp signal, the corresponding ambiguity surface resembles a shape of the mountain ridge with range-Doppler coupling [14; 15]. In contrast, the cross-ambiguity surface of the CBFM waveform closely resembles a thumb-tack ambiguity function with a significant peak occurring at \(R_{R}=1\) Km and \(V=100\) m/s.