# Chaos Based Frequency Modulation for Joint Monostatic and Bistatic Radar-Communication Systems
## 5 Bistatic Radar Receiver Synchronization and Ambiguity Function
### Cross-Ambiguity Functions in the Presence of Noise  
To assess the proposed joint bistatic RadComm system, we further computed the cross-ambiguity functions in the presence of noise. The received signal from the transmitter is typically corrupted with noise that can be modeled as bandlimited additive white Gaussian noise (BL-AWGN) [62]. That is  
\[r(t)=s(t)+\eta(t). \tag{15}\]  
The same procedure as illustrated in Figure 9 is used to reconstruct the CBFM waveform at the bistatic radar receiver. Once the CBFM waveform is reconstructed in the  
Figure 12: Signature analysis of the complex target obtained using (**a**) the Lorenz CBFM waveform and (**b**) the Burke–Shaw CBFM waveform.  
presence of noise, it can be used to correlate with the target echo, which is also assumed to be corrupted with BL-AWGN. That is  
\[r_{tgt}(t)=s(t-\tau)exp^{\beta\pi[f_{c}+f_{D}]t}+\eta(t). \tag{16}\]  
Figure 13a shows the cross-ambiguity functions between \(\{r(t),r_{tgt}\}\) for the Lorenz CBFM waveform with an SNR of 3 dB. Figure 13b is for the Burke-Shaw CBFM waveform. In both cases, the noise floor tends to rise, and the energy from the mainlobe peak spills over to the adjacent sidelobes. While the sidelobes on the velocity plane were not affected, the range sidelobes for the Lorenz CBFM and the Burke-Shaw CBFM waveforms were raised to \(-11.5\) and \(-13.5\) dB, respectively. The sidelobes adjacent to the mainlobe could be further reduced using standard Chebyshev windows.  
The bistatic radar synchronization using chaos and reconstruction of the FM waveform is susceptible to very high noise levels. It is because the self-synchronization of the chaotic system is sensitive to parameter mismatch and noise [53]. Furthermore, since the bistatic radar receiver employs the self-synchronization scheme, the reconstruction of the FM waveform suffers in the presence of noise, especially if SNR is less than 3 dB.  
The quality of the simulated cross-ambiguity functions in the presence of noise can be evaluated using the sidelobe volume function [67] given in Equation (17). For an ideal thumbtack ambiguity function, \(V_{SL}\) should be equal to zero. Any positive value indicates the presence of sidelobes and an increase in noise-floor. It may not provide information about the value of SL, but it conveys the quality of the ambiguity function by computing the overall sidelobes adjacent to the prominent peak.  
\[V_{SL}=\int\limits_{f}\int\limits_{t}|\chi|^{2}dtdf-\int\limits_{-\Delta f}^{ \Delta f}\int\limits_{-\Delta t}^{\Delta t}|\chi|^{2}dtdf. \tag{17}\]  
Figure 14 shows the sidelobe volume of the CBFM waveforms in bistatic configuration plotted against the SNR. As the noise increases, there is a clear indication of the growth of SLL. Overall the Burke-Shaw CBFM waveform outperforms the Lorenz CBFM waveform. For the Lorenz CBFM waveform, as indicated in the gold color square, the reference SLL of \(-13.33\) dB occurs for \(SNR=5\) dB, while for the Burke-Shaw CBFM waveform, it appears for \(SNR=2\) dB (indicated in the green square). A detailed study on the bistatic radar  
Figure 13: The cross–ambiguity surface obtained using **(a)** the Lorenz CBFM waveform and **(b)** the Burke–Shaw CBFM waveform considering the \(SNR=3\) dB.  
synchronization using the CBFM waveform and its high-resolution capabilities is presented in [59].  
The Burke-Shaw CBFM waveform has better performance in terms of sidelobe volume because of its apparent random behavior compared to the Lorenz CBFM waveform. It can be observed by looking at the time series plot, as shown in Figure 2, as well as the higher value of the largest Lyapunov exponent, indicating a more chaotic nature of the waveform.