# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 2 Materials and Methods
### Algorithm DTW  
Time-series synchronization can provide a better understanding of the interaction mechanisms between two different signals or systems [49]. Synchronization analysis between spatially separated features is a fundamental procedure in visual pattern recog  
Figure 7: Representation of rope defect in the LMA signal [48]. nition [50]. The goal of synchronization measurement and analysis is to characterize the relationship between two systems (e.g., current and reference), which may be identical, non-identical, or different. As coupling strength increases, systems may experience phase synchronization (PS), generalized synchronization (GS), lag synchronization (LS), or complete synchronization (CS) [51]. Synchronization is most often estimated using coherence and spectral properties of the signals. However, this approach is not suitable for characterizing signals that are non-stationary and is limited to capturing linear dependencies between time series. Because of the analysis of the MRT method, it was found that the synchronization of time series from periodic MRT tests of wire ropes should be carried out based on LF1 and LF2 signals from search coils, which for each channel represent the MISO (Multi-Input, Single-Output) class system. Elongation of the wire rope (structural, operational, final degradation phase) changes the position of the points mapping the same fragment of the rope in the LF signal on the horizontal axis, both in the TIMER and ENCODER modes (Figure 8). As a result, during rope operation, the spectral content of the harmonic components in the LF1 and LF2 signals, as well as the colored noise \(\epsilon\)(k), changes. However, the following remain constant:  
* The proportions between characteristic frequencies resulting from the rope's pitch and its structural features (such as the number of outer, inner, and core strands);
* The main component of the colored noise, which reflects dynamic phenomena in the rope system, the noise of the measurement channel, and the level of electromagnetic interference near the measurement head. \[ \begin{array}{l}LFi_{ref}(k)=\sum\limits_{j}a_{j}(k)\cdot sin\big{(}\omega_{ j}\cdot k+\varphi_{k}\big{)}+\varepsilon_{i}(k)\\ LFi_{b}(k)=\sum\limits_{j}\big{(}a_{j}(k)+\Delta a_{j}(k)\big{)}\cdot sin \big{(}\big{(}\omega_{j}-\Delta\omega_{j}\big{)}\cdot k+\varphi_{k}+\Delta \varphi_{k}\big{)}+\varepsilon_{i}(k)+\Delta\varepsilon_{i}(k)\end{array}  \tag{22}\]  
Figure 8: Illustrative (not to scale) effect of rope elongation \(\Delta\)x(t) between successive MRT inspections. Elongation causes a shift in the position of a given rope segment and changes in rope diameter; in polymer-impregnated ropes, it also alters the spacing between strand layers. where I = 1,2..k is the discrete time of signal recording; \(\omega=2\pi\)f is the angular frequency of the harmonic signal component; \(\varphi_{\rm k}\) is the initial phase of the signal; \(\Delta\epsilon_{\rm i}\) is the change in noise characteristics relative to the reference signal. The spectral features of the above signals, including information about the type of colored noise, are described by power spectral density (PSD) functions, which are normalized to a constant frequency interval width, most commonly, 1 Hz. The power spectral density is denoted by S(f) and is given by  
\[S(f)=\lim_{\Delta f\to 0}\frac{|X(f)|^{2}}{\Delta f} \tag{23}\]  
For a discrete time signal \(x[n]\), there is a relationship between the power spectral density \(S_{x}(f)\) and autocorrelation function \(R_{x}[m]\), which can be given as follows:  
\[\therefore S_{x}\Big{(}e^{2\pi f}\Big{)}\longleftrightarrow\sum\
olimits_{m=- \infty}^{+\infty}R_{x}[m]e^{-12\pi fm} \tag{24}\]  
For long ropes, e.g., those used in a mining hoisting machine, the total elongation of the wire rope may exceed many times the pitch of the rope, and the phase shift of the current signal \(LF_{b}\) relative to the reference signal \(LF_{ref}\) (e.g., the first recording) will increase with the length of the rope scan.