## 2 Principle and Signal Processing Method
### Short-Time Fractional Fourier Transform
In practice, the procedure for computing STFrFT involves dividing a longer time signal into shorter segments of equal length and successively computing the FrFT separately on each of them. To clarify the analysis, we first consider how to extract the primary LFM component \(x(t)\). The signal \(x(t)\) is represented in the complex number to facilitate subsequent mathematical derivation, and is expressed as  
\[x(t)=rect(\frac{t}{T})\exp[j(2\pi f_{0}t+\pi Kt^{2})]. \tag{12}\]  
After \(x(t)\) is processed by STFrFT, the output signal can be expressed as [22]  
\[ \begin{array}{rl}STFRFT_{x}^{\alpha}(t,u)=&\frac{1}{\sqrt{2\pi}}\exp[-(j/2)t ^{2}\cot\alpha]\cdot\int_{-\infty}^{+\infty}x(\tau)\exp[(j/2)\tau^{2}\cot\alpha ]\\ &\cdot h^{*}(t-\tau)\exp[-j(t-\tau)u\csc\alpha]d\tau,\end{array}  \tag{13}\]  
where \(\alpha\) is the FrFT order, -cot\(\alpha\) is the FM slope of the fractional Fourier domain decomposition, \(u\) is an independent variable in the fractional domain, \(h(t)\) is the STFrFT window, and the superscript \"\(\ast\)\" indicates the conjugate operation. As an example, we choose the simplest rectangular window for STFrFT, expressed as  
\[h(t)=rect(\frac{t}{T}), \tag{14}\]  
where \(T\) is the window length, which is selected as long as the length of the LFM probe pulse. To achieve energy aggregation of the primary LFM components in the fractional domain, the FM slope of the selected fractional Fourier domain decomposition must be equal to the FM slope of \(x(t)\), as follows:  
\[-\cot\alpha=2\pi K, \tag{15}\]  
while the relationship between the FrFT independent variables and the Fourier transform independent variables is determined by  
\[f=\frac{u\csc\alpha}{2\pi}. \tag{16}\]  
By substituting Equations (13-16) into Equation (12) and taking into account the relationship between the frequency f of \(x(t)\) and the time t,\(f=Kt+f_{0}\), we can simplify Equation (12) as  
\[ \begin{array}{rl}STFRFT_{s}^{\alpha}(t)&=\frac{A_{0}}{\sqrt{2\pi}}\exp(-j\pi K t ^{2})\cdot\int_{-\infty}^{+\infty}rect(\frac{t}{T})rect(\frac{t-\tau}{T}) \exp[2j\pi(K\tau+f_{0})t]d\tau\\ &=\frac{TA_{0}}{\sqrt{2\pi}}rect(\frac{t}{2T})\frac{\sin\pi K(T-|t|)t}{\pi K \tau t}\exp(2j\pi f_{0}t).\end{array}  \tag{17}\]When \(t\leqq T\), the envelope of (17) can be approximated as a sinc function, denoted by \(Sa(\cdot)\), as follows:  
\[\mathrm{E}(t)\approx\frac{TA_{0}}{\sqrt{2\pi}}rect(\frac{t}{2T})Sa(\pi KTt). \tag{18}\]  
Finally, Equation (6) is substituted into Equation (18) and only the primary LFM components are considered. According to Equation (7), the output expression \(y(t)\) of the signal \(i_{s}\) (\(t\)) after STFrFT is  
\[y(t)=\frac{TP_{0}R_{d}}{\sqrt{2\pi}}rect(\frac{t}{2T})Sa(\pi KTt)\otimes G(t). \tag{19}\]  
According to Equations (7) and (19), for other higher-order LFM components, the output result would only differ in the intensity and width of the reflection peak. According to the characteristics of the sinc function, the 3-dB width or full width at half maximum (FWHM) of the mainlobe of the sinc function is  
\[\mathrm{BW}=2\pi t\frac{1}{\pi KTt}=\frac{1}{KT}. \tag{20}\]  
The results of the STFrFT numerical calculation for a section of the LFM pulse based on intensity modulation are shown in Figure 3; the original LFM pulse is compressed into a sinc-shaped narrow pulse by STFrFT. According to Equation (20), the 3 dB width or full width at half maximum (FWHM) of the mainlobe of the sinc function is equal to \(1/KT\), as shown in Figure 3b. In theory, the DLFM-OTDR spatial resolution is determined by the probe pulse propagation distance in FUT within half time length of the mainlobe FWHM, which can be expressed as  
\[\mathrm{R}=\frac{c}{2 nKT}=\frac{c}{2 nB}, \tag{21}\]  
where \(c\) is the speed of light propagation in vacuum, \(n\) is the refractive index of FUT, and \(B=KT\) is the sweep frequency range of the LFM pulse. Equation (21) shows that the spatial resolution of DLFM-OTDR is determined by the sweep frequency range \(B\) and not by the pulse width of the probe pulse.