## 2 Principle and Theoretical Analysis
### Generation of OFDM-NLFM Signals  
When a pulse consisting of \(M_{t}\) individual frequency-modulated pulses enters the sensing optical fiber, where each individual pulse has a duration of \(\tau_{p}\) and a frequency modulation bandwidth of \(B_{s}\), and the frequencies of adjacent individual pulses are separated by \(\Delta f\), with the initial pulse having a central frequency of \(f_{0}\), the central frequency of the signal for the \(m\) pulse is represented as \(f_{m}=f_{0}+(m-1)\Delta f\). The bandwidth of all pulse signals is denoted as \(B=B_{s}+(m-1)\Delta f\).  
Assuming the \(m\) waveform's pulse-compressed output spectrum is represented by the combined window spectrum \(W_{m}(f)\), then  
\[W_{m}(f)=\frac{\sum\limits_{i=1}^{N}a_{i}W_{i}(f)}{\sum\limits_{i=1}^{N}a_{i}} \tag{17}\]  
According to Equation (12), the group delay \(T_{m}(f)\) of the \(m\) pulse signal can be obtained by integrating Equation (16).  
The frequency modulation function of the \(m\) pulse signal can be derived from the group delay as follows:  
\[f_{m}(t)=T_{m}^{-1}(f) \tag{18}\]  
Similarly, integrating Equation (17) yields the phase function:  
\[\theta_{m}(t)=2\pi\int_{-\infty}^{t}f_{m}(\tau)d\tau \tag{19}\]  
Once the phase function is determined, we can obtain the signal for the \(m\) instance of OFDM-NIFM as follows:  
\[S_{m}(t)=\exp(j\theta_{m}(t)) \tag{20}\]  
Combining the OFDM-NLFM \(\varphi\)-OTDR system with the basic system is essentially the same. The only difference lies in the driving signal generated by the arbitrary waveform generator. In the basic \(\varphi\)-OTDR system, the arbitrary waveform generator produces linear sweep pulses with exactly the same sweep range. Constrained by the length of the optical fiber, the time interval \(T\) between each pulse must be greater than the total round-trip time of light in the fiber. However, in the combined OFDM-NLFM \(\varphi\)-OTDR system, the driving signal generated by the arbitrary waveform generator is as shown in Figure 3.  
Figure 3: Spectrum of OFDM-NLFM signal.  
The repetition period of the pulse combination remains \(T\), but within each pulse combination, there are \(N\) individual swept-frequency pulses with a time spacing of only \(T/N\). By employing a frequency reuse algorithm, each pulse can effectively perform one detection cycle. This enhancement increases the system's sampling rate for vibrations by a factor of \(N\) and also expands the vibration frequency response bandwidth by \(N\) times.  
Although the time interval between two pulses is merely \(T/N\), when viewed in the frequency domain, the frequencies of adjacent pulses do not interfere with each other. Consequently, the Raman signals generated by these adjacent pulses in the frequency domain are also free from mutual interference, thus allowing them to be separated using a frequency-domain multiplexing algorithm.  
The spectrum diagram of the OFDM-NLFM signal is shown in Figure 3. When \(\Delta f=B_{s}\), the spectra of each signal are precisely separate and arranged sequentially, with each occupying a distinct channel. This separation significantly reduces inter-signal interference, simultaneously enhancing spectrum utilization and favoring matched filtering.