## 2 Theoretical Background  
### Time-Frequency Domain Reflectometry  
The TFDR process consists of three steps: (1) First, we designed the optimal incident signal, which is applied to the cable, and (2) the reflected signal is then measured. (3) Then, we calculated the time-frequency cross-correlation (TFCC) between the reference signal and the reflected signal [9].  
The TFDR uses a Gaussian envelope with a linear chirp signal as a reference signal, which can be expressed as follows [9]:  
\[s(t)=exp\left(\frac{-(t-t_{0})^{2}}{2\tau_{0}^{2}}\right)exp\left(j\frac{1}{2} \dot{\omega}_{0}(t-t_{0})^{2}+j\omega_{0}(t-t_{0})\right) \tag{1}\]  
where \(\tau_{0}\) is the effective time duration, \(t_{0}\) is the center time, \(\omega_{0}\) is the center frequency of the chirp signal, and \(\dot{\omega}_{0}\) is the frequency increment by time.  
The reflected signals are attenuated and distorted over the entire frequency band during propagation. Therefore, the attenuation and distortion of the signal vary depending on cable length and insulation. Alternatively, through designing adequate time and frequency ranges of the reference signal according to the frequency characteristics of the cable, a less distorted and attenuated reflected signal can be obtained, which improves the accuracy and resolution of the results. In addition, reducing overlapping reflections by designing the reference signal leads to an improved diagnostic performance. The designing process of the optimal reference signal is described in [20].  
Then, the time-frequency distribution similarity between the incident signal and the reflected signal was calculated via the normalized TFCC [9]. Consequently, the impedance discontinuity points in the cable was detected and localized according to the TFCC value.  
Therefore, this allows the specific detection of the reflected signal which has the same time-frequency signature as the incident signal. The normalized TFCC is expressed as follows:  
\[C_{sr}(t)=\frac{\iint\left[R_{s}(\tau,\omega)R_{r}(\tau-t,\omega)\right]d\tau d \omega}{\sqrt{\iint\left[R_{s}(\tau,\omega)\right]^{2}d\tau d\omega}\sqrt{ \iint\left[R_{r}(\tau-t,\omega)\right]^{2}d\tau d\omega}} \tag{2}\]  
where \(R_{s}(t,\omega)\) and \(R_{r}(t,\omega)\) are the RID-Rihaczek distributions of the incident signal and the reflected signal, respectively [21,22]. The TFCC are normalized by the energy of the incident signal and the reflected signal in the time-frequency domain. The TFCC results are focused on the detection of all possible impedance discontinuity points which results from various causes including the fault, damaged equipment, loose connectors, and unterminated cables.  
Figure 1 shows examples of the TFDR and step-pulse TDR measurement signals, respectively. The cable length is 20 m, and an incipient fault (insulation fray) is located at 8.5 m of the cable. In Figure 1a,d, the cable start and end points can be easily detected, however, the incipient fault at 8.5 m is indistinguishable from noise when using the TDR. On the other hand, the TFDR calculates the cross-correlation between the reference signal and the reflected signal. Therefore, the fault was successfully detected through using the time-frequency similarity of the signal despite its low energy as shown in Figure 1c. Hence, compared to the TDR method, the TFDR is more adequate for detecting fine signals of the fault as shown in Figure 1c,d.