**5. Analysis of the** *TF* **Features Dispersion**

Several factors may influence the dispersion of the parameters' distributions described above and in the previous section. They concern both the stage of measurement and hardware configuration of the system as well as the subsequent stage of data processing and the selection of computational procedure parameters. It should be emphasized that one of the key factors affecting the smoothness of the characteristic transitions between successive measurement angles is the number of adopted angular steps. The step applied in the paper is relatively large, however, it still makes it possible to determine general trends of the characteristic. On the other hand, it can be noted that errors for individual measurement points have a significant impact on the local course of the characteristic. Another key factor affecting errors is the relatively large size of the transducer used, which means that the measured values should not be treated as spot, but measurements in a certain vicinity of the given point. Furthermore, the minimization of possible influence of random factors on the recorded signals was obtained by repeating the measurements sequence several times under the same conditions (excitation parameters, transducer orientation angle in relation to the sample axis) and then by implementing the procedure of data averaging. At the same time, the influence of alternating interfering components having a frequency constant over time (if appears) would result in constant energy level over the entire spectrogram period for a given frequency band, and thus could also be clearly identified on the time-frequency distributions. Some systematic error may also be the result of a small angular shift between the sample edges and the actual magnetization direction, but it is replicated for all orientations in very similar scale and has no major impact on values dispersion or on the achieved shape of the distributions, but possibly only on the angular shift of the characteristics. Another possible source of measurement errors may also be a small misalignment of the transducer's symmetry axis in relation to its rotation axis. This misalignment error would result in an asymmetrical angular relationship of the calculated *TF* parameters. However, it would not affect the assessment of the direction of the easy or hard magnetization axis, but would end in obtaining a different scale of values on both sides (assuming high repeatability of the angular magnetic properties of the steel the difference would be negligible) of the symmetry of *TF* features' characteristics. Nevertheless, it would also not results in dispersion variation between orientations of the *TF* parameters. Therefore, bearing in mind the above considerations, the assessment of the impact of the computational procedure configuration on the repeatability and dispersion of the results still remains. There are two key elements within this aspect. The first concerns the choice of window width used during the transformation of data into the *TF* domain. The second is related to the pre-transformation data preparation stage and refers to the adjustment of the digital high-pass filter (HPF) band *f* C. These parameters are crucial in the process of obtaining information.

The window size affects the resolution of time and frequency steps of the computed *TF* representation during the STFT transformation. For a narrow window, a high resolution in the time domain and a lower one in the frequency domain is achieved. In the opposite case, the wider the window is, the lower the resolution in time (larger time steps) and the higher in frequency is achieved. Therefore, the adjustment of the size is crucial to sensitivity for variations in analyzed spectrograms. As discussed in the introduction, anisotropy can be caused by a number of factors affecting the various stages of the reorganization of the domain structure during a single period of magnetization, and consequently appearing with different activity of the Barkhausen phenomenon in the time sequences. Thus, when analyzing individual factors, there is a need to obtain a high resolution over time, enabling unambiguous distinction between subperiods of increased Barkhausen phenomenon activity. Recently, the authors have considered the possibility of using *TF* analysis to distinguish time spans of MBN activity corresponding to various factors affecting anisotropy [24]. In that work, the key aspect was to maintain high resolution over time. Therefore, a relatively narrow window (guaranteeing steps Δ*T* = 128 μs and Δ*F* = 1.952 kHz) was used

during STFT transformation. However, the purpose of this work is to explore the possibility of conducting a quick assessment of the resultant anisotropy. The entire period of MBN activity is then analyzed, not its subperiods. Therefore, in this situation, a high resolution over time is not so crucial, and higher resolution in frequency may have a greater impact on the effectiveness of the proposed analysis.

The second factor that may affect the amount of dispersion of features in a given measurement group is the value of the lower band *f* <sup>C</sup> of the MBN signal frequency range. As can be noticed in Figure 4, the highest MBN activity is obtained within the lower frequency range of presented spectrograms *BN*TF\_S. This means that the selection of the lower band value can be decisive for the *TF* parameter distributions obtained during the analysis. In this range, interference from measuring instrumentation can affect the signal. In addition, distortions of a frequency close to the lower range of the hardware filter may also occur. Therefore, it is crucial to properly filter the signal prior further analysis. The cutoff frequency *f* <sup>C</sup> of the digital high-pass filter should be high enough to effectively cut off the effect of low-pass interference, but at the same time low enough not to lose information about the magnetic anisotropy of the material.

Finally, during the analysis of the impact of both factors on the spread of *TF* parameters, a series of calculations was carried out for four cases determined by two window sizes: 128 and 512 samples and two digital filter border frequencies: 0.5 kHz and 2 kHz. The choice of window width was made taking into account the resolution in time and frequency that they guarantee. In the first case, it is possible to obtain relatively small steps of Δ*T* = 128μs and large of Δ*F* = 1.952 kHz, while in the second one inversely, relatively large steps of Δ*T* = 512μs and small steps of Δ*F* = 488 Hz. In this way, it was possible to estimate the impact of the individual resolutions of the computational grid on the repeatability of the obtained distributions. When choosing the *f* <sup>C</sup> value, the cutoff frequency of the hardware filter (0.6 kHz) was taken into account, as well as the values of the frequency steps for both computational grids. Therefore, finally the analysis was carried out for *f* <sup>C</sup> less than 0.6 kHz and greater than 1.952 kHz, which allowed evaluation of the spread of *TF* parameters for two extreme settings.

In the first stage of the analysis the averaged spectrograms *BN*TF\_S were calculated for each out of 10 measurements made for a given angular orientation. Next, for each averaged spectrogram the *TF* parameters were calculated. Then, spreads of *TF* features values were determined in reference to their average values achieved from all measurements. Finally, the values obtained for individual orientation angles (individual subsets) were further averaged and presented in the form of a common bar representation. Figure 7 presents a bar graph illustrating the dispersion range expressed in percentage scale (*%CumRange* indicator) of selected *TF* parameters accumulating the results achieved for all transducer orientations. Each bar specifies a dispersion range received under different data processing conditions. It can be noted that in most cases the smallest range value is obtained for the *f* <sup>C</sup> of 2 kHz (first two columns of Figure 7). Considering the dispersion values of *BN*TF\_MEAN, one can notice that when the *f* <sup>C</sup> is 2 kHz the window size is not so significant, as the *%CumRange* values are comparable for both widths considered. However, the situation changes when the *f* <sup>C</sup> is equal to 0.5 kHz. In that case, a smaller *%CumRange* is achieved for wider window. This confirms the high importance of the low-frequency range of the spectrogram on the *TF* features distributions. For a window width of 128 samples, the frequency step Δ*F* is 1.952 kHz. As a result, the use of a cutoff frequency of 0.5 kHz causes that any undesirable components of the transformed *U*BN signal can be reflected in the lower range of the spectrogram band. Therefore, it can be concluded that the improvement of the robustness can be obtained when the *f* <sup>C</sup> is set to greater value than the first frequency step of the calculation grid, i.e., for *f* <sup>C</sup> = 2 kHz. On the other hand, in the case of window size of 512 sample the Δ*F* is equal to 488 Hz. Thus both considered cutoff frequencies are higher than the first frequency step of the spectrogram grid. Therefore, mostly similar results of the *%CumRange* are obtained for both *f* <sup>C</sup> values, and the distributions of *BN*TF\_MEAN are more convergent in those cases.

**Figure 7.** Bar graph of %CumRange indicator values of used features.

In the case of the spectral flatness parameter, smaller spreads were obtained using a smaller size of window. However, as in the case of the mean parameter, also in this case the use of a higher cut-off frequency significantly reduces the dispersion range. In the case of the *BN*TF\_CM parameter, convergent results were obtained to those obtained for the *BN*TF\_MEAN parameter. The smallest spread value was obtained for a window of 512 samples and *f* <sup>C</sup> of 2 kHz. However, it must be noted that the *BN*TF\_CM presents the highest stability over various computation parameters. The values of *%CumRange* obtained for feature *BN*TF\_SE are the smallest for window width 512 samples and *f* <sup>C</sup> = 2 KHz. Thus, the *BN*TF\_CM feature proves high repeatability and resistance to interfering factors.

In order to further examine the influence of the cutoff frequency on the information contained in the lower frequency ranges of spectrograms, one more analysis was performed. Its purpose was to determine the angular distributions (characteristics) of *TF* parameters as a function of successive frequency ranges of the spectrogram's computational grid. Graphic visualization of the calculation procedure is shown in Figure 8. For a given spectrogram, the entire time vector (single row) is considered, representing a single frequency range. Then the *TF* parameter value is calculated for this vector and the received value represents then a single cell of the resulting distribution presented in the form of the heat map.

Figure 9 presents two heat maps obtained for the *BN*TF\_MEAN parameter under two considered cutoff frequencies. In order to compare both conditions, the input spectrograms were computed using a window of 512 samples. Then the distributions were normalized to the (0–1) range and plotted in individual scales. If the lower frequency band value is used, the resulting distribution (top row) in the lower frequency ranges undergoes quite rapid changes of a rather random nature and not dependent on the angle of testing. Only for 90 and 270 degrees angles (along RD), in the higher frequency ranges, the characteristics present the angular relationship more visibly. In the second case, after applying a high-pass filter with a cutoff frequency of 2 kHz, a clear angular relationship in all compartments characterized by smooth and repeatable transitions between individual angles is obtained. Moreover, for angles that determine the axis of easy magnetization, large parameter values that at the same time cover a much wider frequency band are clearly visible. Attention should be paid to normalized ranges of values of both distributions as well. The unwanted components of low-frequency signals, not containing anisotropic information, reach much higher values. Therefore, they affect the sensitivity of the *TF* parameters and distort the crucial information.

**Figure 8.** Diagram of the computational procedure of TF feature angular distribution in successive frequency ranges.

**Figure 9.** Visualization of the BNTF\_MEAN angular distribution in successive frequency ranges of the spectrogram; left column is a perspective presentation of the results in the right column.

Summarizing, the results of the analysis confirmed the impact of the computational parameters selection on the achieved dispersion of *TF* features values in successive subgroups, which in turn affects the repeatability of the entire method. In addition, it should be emphasized that the choice of the digital filter cutoff frequency *f* <sup>C</sup> value is closely related to the window width value used during the STFT transformation. Therefore, in order to obtain the highest convergence of results and minimize the impact of external factors on parameter distribution, during the final analysis all results were (presented in previous chapter) achieved using the width of the window equal to 512 samples. This allowed the frequency resolution of spectrograms to be obtained enabling observation of even small variations of the MBN band, and at the same time to minimize the influence of the *f* <sup>C</sup> value on the results. Nevertheless, aiming at maximum reduction of the influence of external factors on the computed *TF* distributions, additionally during the signal-processing procedure a cutoff frequency of 2 kHz was used. With these values of calculation parameters, the percentage dispersion of *TF* features values practically do not exceed 2%.
