**3. The Time-Frequency Based Procedure for Evaluation of Easy Magnetization Axis**

Figure 3 presents the successive stages of the procedure for evaluation of magnetic anisotropy based on the *TF* representation of the measured MBN signals along with exemplary MBN signals received for three different transducer's orientations. The early step is to perform the initial signal processing and determine the *TF* representation.

First, in order to minimize low-frequency instrumentation disturbances, the digital signal filtration was performed by the Butterworth high pass filter (with a cutoff frequency *f* <sup>C</sup> of 2 kHz). Then the measurements obtained were divided into half periods of magnetization, each corresponding to the individual MBN burst. In the next step, for each individual burst, transformation of the *U*BN (time representation) to the *TF* domain was performed. For this purpose, the STFT transformation was used, which enables a homogeneous division of the computational grid in the *TF* space to be obtained and, in consequence, the dynamics of the Barkhausen phenomenon to be observed in detail [24]. The Kaiser type of the computational window having a length of 512 samples along with the overlapping technique with a rate of 0.75 was used during the STFT transformation. The given above parameters of the window resulted in a computational time step Δ*T* of 512 μs and frequency step Δ*F* of 488 Hz, what allowed to precisely observe the changes in dynamics of the MBN activity for various measurement angles. A more detailed analysis of the impact of the window width and the value of the filter's cutoff frequency *f* <sup>C</sup> on the quality of the acquired angular characteristics and the possibility of assessing the examined dependence is presented in the following sections of this paper. Finally, as a result, the complex *TF* representations *S*BN(*t*, *f*) of *U*BN voltage signals were obtained. In next step, the spectrograms expressed as |*S*BN(*t*, *f*)| <sup>2</sup> were calculated.

Multiple measurements for a single angular orientation enabled implementation of a smoothing procedure of the results obtained. Its purpose was to average all the spectrograms achieved for a single angular orientation of the transducer. Selected averaged spectrograms *BN*STFT\_S calculated for the measurements acquired within the half range of the transducer rotation (with respect to the TD from 0◦ to 180◦) are shown in Figure 4. There is a noticeable difference between spectrograms for subsequent angles in the activity of the measured MBN. The highest activity was obtained for orientation of the transducer in accordance with the rolling direction of steel and along its easy magnetization axis (Figure 4, α = 90◦ i.e., RD). In this case, the highest energy level is visible practically throughout the entire MBN period, excluding the time span in which the magnetizing field changes its direction (around a time of 25 ms, see Figure 3b). At the same time, the frequency band occupied by the highest activity level of MBN is also the widest. Moreover, it can be seen that along with the gradual rotation of the transducer into the transverse direction to the RD (TD direction: α = 0◦ and α = 180◦), a gradual delay of the beginning of the MBN activity area (arises more slowly) as well as the general decrease in its area can be seen. The bandwidth of the MBN activity is also noticeably reduced. On all the distributions presented, regardless of the angle of measurement, one can notice two distinct areas of MBN activity, one at the beginning of the period around 15 ms and the other at its end around 40 ms. In addition, a third area of activity is strongly visible at the RD angle. This follows the observations presented by other researchers. Considering this, those activity regions could be associated with the aforementioned (three) processes of nucleation of reverse domains, 180◦ and 90◦ DWs motion and the growth of the MBN activity for RD could be explained in reference to the large quantity of 180◦ DWs [2,25,28,30,31,33]. Despite clearly visible variation of the MBN activity in spectrograms, for proper assessment it is necessary to carry out a detailed quantitative analysis enabling the quantification of observed relationships.

**Figure 3.** Time-frequency (TF) calculation procedure: (**a**) diagram of the procedure, (**b**) exemplary UBN signals for orientation α = 0◦ (transverse direction, TD), α = 45◦, and α = 90◦ (rolling direction, RD) vs. time (left column) and magnetic field (right column—presented in accordance to descending half of magnetizing period).

**Figure 4.** View of spectrogram activity in angle range between 0–180 degrees.

The magnetic anisotropy of the material ends up with differences in the magnetic properties occurring for different test angles. Based on the spectrograms, it can be noticed that these changes in properties are then reflected in the course and the intensity of the observed Barkhausen phenomenon. In consequence, the dynamics of the energy distribution, concentration, centroid shift in time and frequency, and the degree of order/disorder or scope of changes of the MBN *TF* representation may be affected. Therefore, in order to quantify information expressing the angular variations of magnetic characteristics, a multi-parameter extraction was used to define vectors of *TF* features for each test angle. As a result, the set of several parameters carried by the *TF* representation were calculated from the *BN*STFT\_S. The first group refers to some statistical properties, including various forms of mean values (i.e., arithmetic, geometric, etc.), centroid, variance or standard deviation, skewness or kurtosis. The next subset are the features describing of the shape of the *TF* spectrogram and its energy distribution or entropy, thus allowing the dynamics of variance, uniformity of distribution or the degree of disorder of the MBN spectral content to be assessed. The last, but no less important part of the *TF* features vector is parameters that indicate different characteristics values of the *BN*TF\_S, such as: symmetry, center shift in the *t* or *f* axis, flatness, homogeneity or monotonicity, etc. The definition and properties of all proposed *TF* features and the used calculation procedure have been introduced earlier and discussed in detail in [32]. The results of the angular distributions of selected features will be presented in the following chapter.
