# Application of Dynamic Time Warping (DTW) in Comparing MRT Signals of Steel Ropes
## 2 Materials and Methods
### MRT Methods
#### 2.1.1 Numerical Model of Search Coils
The nonlinearity of the LF1 and LF2 coils means that the fundamental frequencies of the source of the local magnetic anomaly and their harmonics and the frequencies of the modulation products are expected in the signal spectrum. As the density of defects in the rope increases, the spectrum will be further modified by the effect of averaging the spectra of individual defects, considering the distance between defects. Discrete signals\(LF1(k)\) and signals \(LF2(k)\) recorded in the flaw detector are described by the signal model (Equation (14)):  
\[\text{LFi}(k)=signal_{\text{LFi}}(k)+noise_{\text{LFi}}(k) \tag{14}\]  
where  
* \(k\) is the discrete time;
* \(signal_{\text{LFi}}\) is the voltage component at the coil output generated by the technical condition of the wire rope LFi;
* \(noise_{\text{LFi}}\) is the voltage component at the coil output generated by other sources of magnetic field heterogeneity LFi. A measurement head equipped with two LF sensing coils LF1 and LF2, positioned at different radii R\({}_{1}\) and R\({}_{2}\), enables quantitative analysis of rope defects (anomaly sources) and localization of their position relative to the axis of the measurement head. This process takes into account a magnetic anomaly model, the amplitudes of the magnetic anomalies recorded in the LF1 and LF2 channels, and calibration data from the measurement system, as described by Equations (15) and (16) [38]. \[\frac{\Delta r}{\Delta R}=\frac{\left(k\cdot\frac{A_{1}}{A_{2}}\right)^{1/m}}{ 1-\left(k\frac{A_{2}}{A_{1}}\right)^{1/m}} \tag{15}\]  
\[Z_{c}(\omega_{s})=|Z_{c}|(\omega_{s})e^{i\varphi(\omega_{s})}=R(\omega_{s})+jX( \omega_{s})=R_{c}+j\omega_{s}L_{c}-j\frac{1}{\omega_{s}C_{c}} \tag{7}\]0  
where  
\(\Delta r=R_{1}-r\) is the distance from the anomaly source to the inner sensing coil LF1; \(\Delta R=R_{2}-R_{1}\) is the fundamental signal that depends only on the flaw parameters and is independent of the flaw location and the geometry of the search coils;  
\(A_{1}=k_{1}\frac{A_{0}}{(R_{1}-r)^{m}}\) is the amplitude of the magnetic anomaly recorded by the LF1 coil;  
\(A_{2}=k_{2}\frac{A_{0}}{(R_{2}-r)^{m}}\) is the amplitude of the magnetic anomaly recorded by the LF2 coil. Reliable detection of rope defects and meeting the criterion of defect symptoms according to EN 12927:2019 requires minimizing the noise level, including the following components:  
* Signal harmonics that reproduce the structural features of the rope and the measuring head, and in the current data analysis algorithms, MRT methods are considered structural noise;
* Color noise being the resultant of quantization noise, thermal noise of the measurement path, and external low-frequency electromagnetic interference. For this purpose, various non-standardized algorithms for the analysis of MRT measurement data in the time, frequency, rope scanning distance, order, and modulation domains are used around the world [39; 40; 41; 42], which, together with the design features of the measuring head, often constitute the manufacturer's know-how. As a result, the user of the MRT test track is not able to fully use the diagnostic potential of the existing testing equipment and software and optimally adapt the diagnostic criteria to new types of steel ropes, especially polymer-impregnated steel ropes. Such a situation is one of the premises for the occurrence of type II errors in MRT diagnoses and occasional cases of emergencies due to breaking of steel ropes. The spectral complexity of the LF1 and LF2 signals makes these signals preferable for numerical synchronization of data from periodic technical inspections of the rope, which will ensure the objective comparison of test results recommended by the ISO 4309 standard and minimize the impact of the human factor on the probability of diagnosis. One of the possible methods for the automatic synchronization of time series with a similar spectrum is the DTW algorithm--Dynamic Time Warping.