## 2 Principle and Theoretical Analysis
### Generation of NLFM Signals  
In \(\varphi\)-OTDR systems, the detection signal primarily consists of the backscattered light from the pulsed optical signal, forming a continuous signal that varies with distance. Excessive sidelobes can lead to crosstalk between Rayleigh signals at different locations [30]. This paper employs NLFM instead of LFM to achieve lower sidelobe levels and enhance the system's SNR. The theoretical foundation of NLFM signals is based on the phase progression principle [31]. NLFM signal models are diverse, and generating these signals is more complex compared to LFM signals, which lack precise signal design methods. In practice, various approximation methods are employed. LFM signals are typically represented as  
\[s(t)=u(t)e^{\big{[}2\pi f_{0}t}=\frac{1}{\sqrt{t_{p}}}rect(\frac{t}{\tau_{p}}) e^{\big{[}2\pi(f_{0}t+\frac{1}{2}\mu t^{2})} \tag{5}\]\[u(t)=\frac{1}{\sqrt{\tau_{p}}}rect(\frac{t}{\tau_{p}})e^{j2\pi(f_{0}t+\frac{1}{2} \mu t^{2})} \tag{6}\]  
Assuming the signal bandwidth is \(B\), then \(\mu=B/\tau_{p}\) represents the frequency modulation rate. By taking the derivative of the phase of the signal \(s(t)\), we obtain the instantaneous frequency of the signal:  
\[F(t)=\frac{1}{2\pi}\frac{d(\pi\mu t^{2}+2\pi f_{0}t)}{dt}=f_{0}+\mu t \tag{7}\]  
This paper employs the classical window function inversion method [32; 33] to generate NLFM signals by altering the duration of different frequency components within the signal while maintaining constant instantaneous power. Once a window function \(W\) is given, the group delay function \(T\) of the waveform can be approximately solved using the stationary phase principle. Then, the phase function is determined based on the instantaneous frequency. The specific process of signal generation is as follows:  
The NLFM signal is represented as:  
\[x(t)=a(t)\exp(j\theta(t)) \tag{8}\]  
In the equation, \(a(t)\) represents the signal's amplitude, which is typically a constant value, and \(\theta(t)\) represents the signal's phase function. Assuming the spectrum of \(x(t)\) is \(X(f)\), then the frequency response of its matched filter should be:  
\[H(f)=X^{*}(f)e^{-j\Theta(f)} \tag{9}\]  
In the equation, \"\" represents complex conjugation, \(\Theta(f)=2\pi ft_{0}\), where \(t_{0}\) is the moment at which the peak of the matched filter's output signal component is formed, and \(e^{-j\Theta(f)}\) is the linear phase factor. In this case, the frequency spectrum \(Y(f)\) of the matched filter's output \(y(t)\) is given by:  
\[Y(f)=\int_{-\infty}^{+\infty}y(t)\exp(-j2\pi ft)dt=X(f)H(f)=|X(f)|^{2} \tag{10}\]  
For a certain frequency-domain window function \(W(f)\), let:  
\[|X(f)|^{2}=W(f) \tag{11}\]  
According to the phase unwrapping principle, it is known that there is a relationship between the signal's spectrum and the frequency modulation rate, which is given by  
\[X(f)\propto\sqrt{\frac{1}{\theta^{\prime\prime}(t)}}=\sqrt{\frac{1}{\frac{df( t)}{dt}}}=\sqrt{\frac{dT(f)}{df}} \tag{12}\]  
In the equation, \(T(f)\) represents the group delay function of the signal, and \(f(t)\) is the frequency modulation function, which are inverse functions of each other. Assuming that the selected signal's spectrum after pulse compression is denoted as \(W(f)\), then \(W(f)\propto\frac{dT(f)}{df}\). Therefore, the group delay function can be obtained by integrating \(W(f)\) as follows:  
\[T(f)=K\int_{-\infty}^{f}W(v)dv \tag{13}\]In the equation, \(K\) is a constant coefficient. When it is required that the NLFM signal has a bandwidth of \(B\) and a duration of \(t_{p}\), we have:  
\[K=\frac{t_{p}}{\int_{-B/2}^{B/2}W(r)dv} \tag{14}\]  
Based on the relationship between \(T(f)\) and \(f(t)\), we can obtain:  
\[u(t)=\frac{1}{\sqrt{\tau_{p}}}rect(\frac{t}{\tau_{p}})e^{j2\pi(f_{0}t+\frac{1}{2} \mu t^{2})} \tag{6}\]0  
Therefore, the phase function of the signal is:  
\[u(t)=\frac{1}{\sqrt{\tau_{p}}}rect(\frac{t}{\tau_{p}})e^{j2\pi(f_{0}t+\frac{1}{2} \mu t^{2})} \tag{6}\]1  
At this point, the generation of the NLFM signal \(x(t)\) is complete.