# Synchronization of Monostatic Radar Using a Time-Delayed Chaos-Based FM Waveform
## 6 Chaotic Signal-Based on FM Modulation
Thus, the time-delay chaos source (transmitter system) parameter values have been carefully chosen as follows: \(a=1,\ b=3\), and with parameters (\(\tau=0.6\) ms, \(k=1500\)), and the initial condition is set to \(x_{m}(0)=0.5\). Now, the effect value of the loop gain \((G)\) on the recovered signal \(\hat{x}_{m}(t)\) from the PLL can be seen in Figure 12a-d, where the right \(G\) value must be carefully chosen for the recovered signal \(\hat{x}_{m}(t)\). As a result, it is apparent that when the loop gain value becomes 4.6, it has a significant impact on the recovering signal. The resulting signal \(\hat{x}_{m}(t)\) closely resembles the original signal \(x_{m}(t)\). \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]1 _Cont._  
On the other hand, once the PLL locks on the original signal, we will examine the effect of the various noise values on the \(S_{t}(t)\) over the wireless channel, assuming the sampling frequency, \(f_{s}~{}=~{}5\) GHz, and echo delay (\(\tau_{d}\)) = 0.2 ns. Upon these assumptions, Figures 13 and 14 show the limited continuous baseband signal \(x_{m}(t)\) with bandwidth (BW) = 1 MHz and the output signal \(S_{t}(t)\) from the VCO with a carrier frequency \(f_{c}\) = 0.3 GHz, loop gain \(G\) = 4.6, and \(K_{vco}~{}=~{}75\) MHz, respectively. Note that the effect of the time delay in Figure 13, for instance \(\tau~{}=~{}0.6\) ms, shows more chaos dynamics generated in the transmitted FM signal as compared to the counterpart signal in case of \(\tau~{}=~{}0.5\) ms, depicted in \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]2 These FM signals are generated using the chaotic attractors obtained in Figure 8a,b. Furthermore, the correlation performance of the proposed mono-static radar system is also estimated under various values of signal-to-noise ratio (SNR) when VHF (very high frequency) chaotic signal based on FM is considered. Under the effect of noise, the PLL is locked on the phase of the incoming signal \(s_{t}(t)\), and the recovered signal \(\hat{x}_{m}(t)\) disappears when the noise becomes below the value of SNR = \(-10\) dB, as shown in Figure 15e,f. Meanwhile, a clear peak in cross-correlation performance can provide the desired information about the target, as shown in Figure 15a, when SNR = \(20\) dB. \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]3 Impact of the loop gain on the recovered signal \(\hat{x}_{m}(t)\) in the PLL (**a**) \(G~{}=~{}1\), (**b**) \(G~{}=~{}3\), (**c**) \(G~{}=~{}4\), (**d**) \(G~{}=~{}4.6\). \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]4 _Cont._  
\[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]5 Monostatic continuous chaotic radar system for \(\tau=0.5\) ms and \(x(0)=0.5\), \(a=1\), \(b=3\), \(k=1500\). (**a**) \(x_{m}(t)\): the limited chaotic master signal (**b**) \(s_{t}(t)\): the transmitted FM signal. \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]6 Monostatic continuous chaotic radar system for \(\tau~{}=~{}0.6\) ms and \(x(0)~{}=~{}0.5\), \(a~{}=~{}1\), \(b~{}=~{}3\), \(k=1500\). (**a**) \(x_{m}(t)\): the limited chaotic master signal (**b**) \(s_{t}(t)\): the transmitted FM signal. \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]7 The correlation performance of monostatic continuous chaotic radar system for various SNR, (**a**) \(\mathrm{SNR}\ =\ 20\ \mathrm{dB}\), (**b**) \(\mathrm{SNR}\ =\ 10\ \mathrm{dB}\), (**c**) \(\mathrm{SNR}\ =\ 0\ \mathrm{dB}\) (**d**) \(\mathrm{SNR}\ =\ -10\ \mathrm{dB}\), no signal: (**e**) \(\mathrm{SNR}\ =\ -15\ \mathrm{dB}\), (**f**) \(\mathrm{SNR}\ =\ -20\ \mathrm{dB}\).