**4. Angular Distribution Results of Time-Frequency (TF) Features**

Figure 5 presents results of four selected parameters obtained from the *TF* representation: spectral flatness *BN*TF\_SF, concentration measures *BN*TF\_CM, spectral entropy *BN*TF\_SE and mean value *BN*TF\_MEAN of the spectrogram. The chosen parameters refer to all three groups described above, allowing generalization of the *TF* characteristic. The definitions of the presented parameters were shown in Table 1. Before presentation all features were normalized, to eliminate the isotropic part of information (presented in 0–1 scale).


**Table 1.** Definition of selected features based on spectrogram *BN*TF\_S of *TF* representation *S*BN(*t*, *f*).

All feature distributions were approximated using a piecewise fitted curve to facilitate the analysis and interpretation of the obtained angular characteristic. It should be emphasized that obviously interpreting the distributions obtained is not the same in each case and the minimum values of some presented features do not have to refer to the minimum values of others, and the same for maximums. First, two of the selected parameters relate to the assessment of the energy concentration of the analyzed spectrogram. The *BN*TF\_SF parameter specifies the ratio of the geometric to the arithmetic mean and obtains higher values when the distribution is homogeneous (e.g., random). In reference to the presented *TF* spectrograms, this parameter takes the highest values for the hard magnetization axis. As the transducer orientation approaches to 90 degree angle (easy magnetization axis), value of the parameter begins to decrease, which indicates a growth of MBN activity (energy) area. Confirmation of

this observation is found in the distribution of the second parameter, the *BN*TF\_CM, which takes higher values in case of the evenly distributed energy over the entire *TF* plane. However, at the same time, the parameter is not sensitive to small quantities. Thus when the level of MBN activity is growing in all spectral bands in general, the parameter value is reaching higher values as well. According to the received angular distribution, the *BN*TF\_CM value increases as the angle increases within the 0–90 degrees range. This parameter achieves the largest value in the direction of the easy magnetization axis, which is consistent with the observation for the angular distribution of the *BN*TF\_SF parameter. The spectral entropy allows the rate of disorder of the spectrogram to be assessed. It can be noticed that *BN*TF\_SE assumes the highest value for relatively wide orientation range around TD (α = 0◦ and α = 180◦), at the same time showing a sudden decrease of value for the angle close to RD (α = 90◦ and α = 270◦). This can be understood as leading the spectral distribution *BN*TF\_S to a higher degree of order and to accommodate the existing energy states for easy magnetization direction. The last presented parameter, *BN*TF\_MEAN, relates to the statistical quantity. The angular characteristic of the feature is well correlated with the course of the *BN*TF\_CM. This confirms the earlier observations regarding the increase in value levels practically throughout the whole spectrogram space for angles consistent with the RD direction or close to it. All obtained distributions allow to draw similar conclusions. One can see a general indication of the directions of high and low activity of MBN, which are also consistent with the direction of, respectively, easy and hard magnetization. In reference to Figure 4, for the RD angle, one can observe a clear increase in MBN activity, expressed by a global increase in its level (reflecting among others, in *BN*TF\_MEAN and *BN*TF\_SF values). At the same time, the greatest energy values accumulate within the three mentioned sub-periods (affecting among others, the *BN*TF\_CM), with a growing difference between areas with low and high MBN activity (accommodation of existing energy states depicted by among others, *BN*TF\_SE). Moreover, this increase in activity is most visible on the presented spectrograms near the central part of the entire MBN signal period. From extensive research reported in the literature, RMA has a particular impact on activity in this area [25,30,31]. This would indicate the greatest impact of RMA on the distribution of resultant anisotropy and the occurrence of the easy magnetization axis. The information contained in the proposed *TF* features can be validated by classical MBN features. Figure 6 presents the distribution of two frequently utilized MBN parameters, i.e., number of events *BN*<sup>N</sup> and energy *BN*EN. The distributions obtained confirm the directional properties of the tested steel, showing a clear increase in energy for the direction consistent with RD. An increase in activity for the RD direction was also observed in other publications [26,28], where authors explained the higher MBN activity (determined by the higher amplitude of the MBN envelope and RMS or energy values) by a much larger number of 180◦ domain walls in this direction. Similar conclusions were also presented in a number of other works [25,27,30,31], under consideration of only a middle part of the MBN signal associated with the movement of the 180◦ DWs and related to the influence of RMA. Based on the obtained angular characteristics (Figures 5 and 6) one can notice good agreement with the presented *TF*-based results. Recently the authors presented detailed comparison of the various features obtained from the time, frequency and time-frequency domain [32]. A good correlation between the spectral flatness and number of events, and between concentration measure and energy as well was reported. The presented results in Figures 5 and 6 confirmed the previous observations. The observed increase for RD in activity also translates into energy carried by MBN, while the decreasing number of events can be explained by the increase in the MBN phenomenon intensity and the superposition of smaller impulses into a larger cluster. This confirms the observations made for the parameters of the *TF* characteristics. Higher pulse values obtained for the RD direction compared to the TD direction cause a significant increase in energy value, but also a noticeable increase in the bandwidth occupied by the areas of highest activity, which (considering the scale of both factors) finally affects an increase in the value of the *BN*TF\_CM parameter. Furthermore, the increase in the difference between the energy states of the highest activity areas and the rest of the spectrogram determines the decrease in the *BN*TF\_SF parameter value. In addition, the overlapping of MBN events and accumulation of energy lead to a decrease in

the parameter *BN*TF\_SE. Considering all aspects, the results obtained indicated the possibility of using this proposed method based on the *TF* representation of full-period for the analysis of the resultant anisotropy in SiFe steel.

**Figure 5.** View of calculated parameters for 2 kHz and 512 window size: Markers represents the features values obtained for selected test angles; solid line refers the piecewise fitting result; all results are normalized with respect to maximum value.

**Figure 6.** Results of the angular distributions of classical magnetic Barkhausen noise (MBN) parameters, that is the number of events BNN and the energy BNEN derived from time domain representation: markers represents the features values obtained for selected test angles; solid line refers the piecewise fitting result; all results are normalized with respect to maximum value.

Finally, despite the sparse measurements (22.5◦ angular step) and the occurrence in some cases of non-compliance of measuring points with angular characteristics, which also affects the achieved approximations, the RD and TD axes can in almost all cases be uniquely identified by the location of maximum or minimum values of individual parameters.

However, due to the observed discrepancies, it becomes justified to verify the repeatability of the results obtained. As a significant number of MBN bursts were registered each time (10 measurements, 10 magnetization periods each) the assessment of the divergence of individual *TF* features obtained for subsequent measurement angles can be performed. Therefore, in the next stage of the work, the

impact of possible causes affecting the range of the dispersion of parameter values obtained for a single angular orientation of the transducer was analyzed.
