## 2 Principle and Theoretical Analysis  
### System Principles  
The schematic diagram of the \(\varphi\)-OTDR based on OFDM-NLFM time-domain pulse modulation is shown in Figure 1. It utilizes an OFDM-NLFM time-domain signal as the pulse modulation signal. This sequence consists of \(N\) pulses, and \(N\) detection pulses are multiplexed within one sequence period \(T\). The pulse repetition period is \(T/N\), where the sequence period \(T\) is greater than the maximum round-trip time of light for detection pulses through the long-distance sensing fiber. The value of \(N\) depends on the maximum response frequency to be achieved and the ratio determined by the sensing distance for sequence repetition frequency. The pulse width of each detection pulse is \(\tau_{p}(\tau_{p}<T/N)\), where \(\tau_{p}\) determines the energy of the detection pulse light, and this energy cannot exceed the threshold of fiber nonlinearity effects. The detection pulse is an NLFM signal with a modulation bandwidth of \(B_{S}\), which dictates the system's spatial resolution. The frequencies of multiple detection pulses are orthogonal to each other. Additionally, \(f_{0}\) represents the central frequency of the detection pulse electrical signal, and the central frequency of each detection pulse varies with a step size of \(\Delta f\). Through OFDM, the input signal is divided into multiple parallel signals, with each modulated on independent orthogonal subcarriers. This effectively utilizes the available spectrum and overcomes the bandwidth limitations in long-distance sensing. NLFM, with its non-linear variation of instantaneous frequency over time, enhances the sidelobe suppression ratio, reduces interference between adjacent signals, and improves the system's SNR [24; 25; 26; 27; 28]. The system combines the advantages of OFDM and NLFM, simultaneously enhancing spatial resolution, frequency response range, and sensing distance. The probing light is modulated into chirp pulses, which can be represented as:  
\[s(t)=rect(t/\tau_{p})exp\{j2\pi f_{a}t+j\pi kt^{2}\} \tag{1}\]  
where \(rect(t/\tau_{p})\) is a rectangular window function within the time range \(t\in\left[0,\tau_{p}\right]\), \(\tau_{p}\) is the pulse width, \(f_{a}\) is the initial frequency of the chirp pulse signal, and \(k\) is the frequency chirp rate [29]. After coherent detection and photodetection, the photocurrent signal is transformed into a complex signal through the Hilbert transformation in the digital domain. To maximize  
Figure 1: Schematic diagram of \(\varphi\)-OTDR based on OFDM-NLFM. the SNR, a matched filter should be used for data processing. In the digital domain, the matched filter \(s^{*}(-t)\) is generated to process \(i(t)\), resulting in the output signal:  
\[r_{c}(t)=i(t)\otimes s^{*}(-t)=h(t)\otimes R(t) \tag{2}\]  
Here, \(\otimes\) denotes convolution, and \(r_{c}(t)\) represents the trace line of the synthesized reflectivity of the test fiber. \(R(t)\) can be considered as the target detection pulse. Through the matched filtering process, the original chirped frequency-modulated pulse \(s(t)\) is compressed into \(R(t)\). \(R(t)\) is a function similar to sine and can be represented as:  
\[R(t)=s(t)\otimes s^{*}(-t)=rect(\frac{t}{2\tau_{p}})\frac{sin[\tau kt(\tau_{p} -\big{|}t\big{|})]}{\tau kt}exp\{-\big{/}2\pi\left(f_{a}+\frac{k\tau_{p}}{2} \right)t\} \tag{3}\]  
The system's spatial resolution \(R(t)\) is determined by the full width at half maximum (FWHM) of the main lobe and is inversely proportional to the chirp frequency range, which is independent of the pulse width. This breaks the mutual constraint between spatial resolution and sensing distance. Therefore, by selecting appropriate chirped modulation pulses, higher spatial resolution can be achieved at longer sensing distances. The matched filter is the optimal linear filter for white noise signals.