# Synchronization of Monostatic Radar Using a Time-Delayed Chaos-Based FM Waveform
## 6 Chaotic Signal-Based on FM Modulation  
After we generated the chaotic attractor from the time-delay chaotic system in Section 5, we could use it in the FM modulation. Thus Figures 10 and 11 show the chaotic signal's modulation and demodulation structures, respectively. The voltage control oscillator (VCO) is used to modulate the chaotic signal and generate continuous FM signals. The low-pass filter (LPF) receives the output signal from the time-delay chaos source block (the master signal \(x(t)\)) and assuming the cutoff frequency of LPF (\(f_{cut}=1\) MHz), first-order, and with the Butterworth method). Then, the limited chaotic master signal (\(x_{m}(t)\)) is regarded as an input signal to the VCO during the modulation process. Then, the output signal from the VCO becomes a transmitted signal (\(S_{t}(t)\)) through wireless channel [4; 35]. \[S_{t}(t)=A\cos[2\pi f_{c}t+2\pi\ K_{vco}\int_{0}^{t}x_{m}(t)dt] \tag{5}\]  
where the carrier frequency is \(f_{c}\) and the sensitivity of the VCO in Hz/V is \(K_{vco}\), and  
\[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]  
where \(f_{min}\) and \(f_{max}\) are the minimum and maximum frequency, while \(v_{min}\) and \(v_{max}\) denote the minimum and maximum voltages of \(x_{m}(t)\), respectively. The difference between maximum and minimum frequencies has represented the bandwidth (\(BW=f_{max}-f_{min}\)) of the modulated signal and the instantaneous frequency deviation described in the term \(K_{vco}\ \ x_{m}(t)\). The nonlinear frequency relation is between the input of the master signal to VCO and the transmitted signal (\(S_{t}(t)\)), as noted in Equation (5). The received signal \(S_{t}(t)\) is demodulated by the analog Phase-Locked Loop (PLL), as shown in Figure 11.  
Figure 9: The power spectrum frequency of the output signal from Lur’e system with time delay \(\tau=0.6\) ms and sampling frequency \(f_{\rm s}=4\) MHz. As shown in Figure 11, a PLL consists of three components: a VCO, a low pass filter (LPF), and a mixer. Here, the LPF is built with \(f_{cut}=150\) MHz, first order, Butterworth method. The phase detector (PD) acts as a mixer to compare the phase of the received signal \(\hat{S}_{t}(t)\) with the phase of the VCO. The output signal of PD represents the error voltage (\(m_{e}(t)\)), which is filtered by the loop LPF. However, this signal is amplified by the loop gain \(G\) and is assumed to produce \(\hat{\vartheta}(t)\) that later feeds the VCO. Subsequently, the output signal from the VCO (\(v(t)\)) is multiplied by the received signal \(\hat{S}_{t}(t)\) to obtain \(m_{e}(t)\). Then, the output of the VCO is described as [4; 35; 36]:  
\[v(t)=V_{r}\sin(\theta_{v}) \tag{7}\]  
where,  
\[\theta_{v}=2\pi f_{e}t+2\pi\ K_{\mathrm{VCO}}\int_{0}^{t}\hat{\vartheta}(t) \mathrm{dt} \tag{8}\]  
\[m_{e}=\hat{S}_{t}*v(t) \tag{9}\]  
and the voltage amplitude of the VCO output signal is \(V_{r}\). Then, the output of the LPF can be described by  
\[\hat{\vartheta}(t)=\frac{1}{2}\sin(\theta_{t}-\theta_{v}) \tag{10}\]  
where the phase of \(S_{t}(t)\) is \(\theta_{t}\). The phase difference (\(\theta_{t}-\theta_{v}\)) will equal zero if the PLL is locked on the phase of the incoming signal \(\hat{S}_{t}(t)\), and the nonlinear (sinusoidal) may be discarded; therefore, the Equation (10) can be rewritten as follows:  
\[\hat{\vartheta}(t)=G(\theta_{t}-\theta_{v}) \tag{11}\]  
Later, the output from PLL (represented by \(\hat{\vartheta}(t)\)) becomes as an input to another LPF (which is chosen with \(f_{cut}=1\) MHz, 3 rd order, and Butterworth method) to recover \(\hat{x}_{m}(t)\). Then, the performance of the proposed continuous chaotic radar system can be measured by  
Figure 11: The structure of signal demodulation using PLL. \[K_{vco}=\frac{f_{max}-f_{min}}{v_{max}-v_{min}}=\frac{BW}{v_{max}-v_{min}} \tag{6}\]0 The block diagram of a Time-delayed Chaotic-FM Radar System. the final output of the correlated over the noisy wireless channel. The correlation qualities are used to assess the chaotic radar's performance.