## 2 Principle and Signal Processing Method
### Short-Time Fractional Fourier Transform  
Like the conventional OTDR, DLFM-OTDR also aims to obtain the attenuation characteristics of FUT by detecting the intensity variation of the backscattered signal with the round-trip time. The probe signal used is an intensity-modulated optical pulse whose modulation signal is the electrical digital LFM signal. Therefore, considering how to effectively detect the digital LFM signal in the backscatter signal is very important. The Fourier series decomposition is performed on Equation (2), which can be rewritten as  
\[DS(t)=rect(\frac{t}{T})[\cos(2\pi f_{0}t+\pi Kt^{2})-\frac{1}{3}\cos(6\pi f_{0 }t+3\pi Kt^{2})+\frac{1}{5}\cos(10\pi f_{0}t+5\pi Kt^{2})+\cdots]. \tag{7}\]  
Hence, \(DS(t)\) can be considered as a linear superposition of various LFM signals, with different FM slopes and start frequencies, which are referred to as primary, tertiary and quinary LFM signals in sequence. Therefore, we only need to effectively extract the LFM signal from the backscattered one. Figure 2a,b show, respectively, the time-domain waveform and time-frequency distribution of an LFM signal, which has a large time width and bandwidth and is non-stationary, having the frequency changing with time. Thus, the LFM signal can be processed and extracted neither in the time nor frequency domains. The fractional Fourier transform (FrFT) is a generalized Fourier transform that breaks through the traditional way of analysis only in the time, frequency, or time-frequency joint domains [25; 26]. It allows the signal analysis in the fractional domain between the time and frequency domains, and all signal characteristics from both domains can be observed [27]. Like the Fourier transform, FrFT is also a base representation method, whose basis function is a set of LFM signals. As shown in Figure 2c, when the FM slope of the selected fractional Fourier domain decomposition is equal to that of the LFM signal, the latter will have an energy aggregation effect on the selected basis [27]. Similar to a narrow bandpass filter in the frequency domain, filtering is performed with a spike in the fractional domain, and with a properly selected bandwidth, the signal energy can be preserved while filtering out most of the noise energy. Therefore, FrFT is very effective in extracting LFM signals and filtering noise. The effect of filtering can be measured by the signal-to-noise ratio improvement factor IF, which is defined as the ratio of signal-to-noise ratio (SNR) before and after filtering [28],  
\[IF=\frac{SNR_{out}}{SNR_{in}}. \tag{8}\]  
If \(N\) is defined as the length of the normalized signal sample, the input SNR can be defined as [29]  
\[SNR_{in}=\frac{E[\sum\limits_{0}^{N-1}|s(n)|^{2}]}{N\sigma_{n}^{2}}. \tag{9}\]  
Figure 2: T-section of an LFM signal (time normalizes with respect to T). (**a**) Time-domain waveform and (**b**) time–frequency distribution; (**c**) fractional Fourier transform when the FM slope of the decomposition basis is equal to that of the LFM signal. We use a bandpass filter to filter the signal in the fractional domain, and then the output SNR can be estimated as  
\[SNR_{out}\approx\frac{E\big{|}\sum\limits_{0}^{N-1}\big{|}s(n)^{2}\big{|}\big{|}} {\Delta n\sigma_{n}^{2}}, \tag{10}\]  
where \(\Delta n\) is the bandwidth of the narrow bandpass filter on the fractional domain. At this point, the improvement in signal-to-noise ratio is  
\[IF=SNR_{out}/SNR_{in}\approx N/\Delta n. \tag{11}\]  
The improvement in SNR is proportional to the length of the signal sample and inversely proportional to the bandwidth of the narrow bandpass filter on the fractional domain. DLFM-OTDR requires the detection of not only the probe pulse intensity but also its changes over time. STFrFT, a windowed FrFT, is used to detect the strength of an LFM signal of local sections of a signal as it changes over time [23; 24]. Therefore, we choose STFrFT as the signal processing method for our proposed DLFM-OTDR.