**3. BN Signal Analysis**

The basic unit of information is a single cycle of changes in magnetization. The cycle is presented in detail in Figure 3. Two halves of such a full cycle can be distinguished: one in which the magnetization current intensity diminishes (marked as *I*) and the other in which the current intensity rises (marked as *I* ).

**Figure 3.** Detailed description of a single cycle of changes in magnetization.

To obtain information from a single cycle of changes in magnetization, the signal is processed in different ways. The processing method produces values of individual properties of the Barkhausen noise. For a multiple number of cycles of changes in magnetization, a multiple number is obtained of values of the quantity for which a statistical analysis has to be carried out. The analysis gives the average value of the quantity and the value of standard deviation being a variability measure of the BN signal analyzed properties.

The measured signal of the Barkhausen noise is a set of voltage pulses, among which so-called events can be distinguished. It is assumed that for a set value of threshold voltage *Ug*, the event occurrence is determined based on three subsequent signal samples with a value exceeding the threshold voltage, where the value of the second signal is higher than the value of the first and third (cf. Figure 4). Detected events are summed up, which gives the distribution of the total number of events—*NoETOT*—depending on threshold voltage *Ug* (cf. the example presented in Figure 5). The threshold voltage-dependent *NoETOT* distribution is obtained for each cycle of changes in magnetization. In the case of a multiple number of cycles of changes in magnetization, averaged *NoETOT* distributions are determined together with statistical features, such as standard deviation.

**Figure 4.** Method of event determination above the set value of threshold voltage *Ug.*

**Figure 5.** Averaged distribution of the total number of events (*NoETOT*) with standard deviation values (marked in red).

The relation between the number of events (*NoETOT*) and the corresponding values of hardness were analyzed for individual values of discrimination voltage *Ugj* using a linear regression model. The analysis resulted in the value of determination coefficient *R*<sup>2</sup> for each value of discrimination voltage *Ugj* . The value reflects the quality of the goodness of fit of the linear model developed based on input data. A value of determination coefficient *R*<sup>2</sup> equal or higher than 0.8 proves a good fit of the linear regression model.

Additionally, the resolution coefficient (*RECOF.*) was defined. The coefficient makes it possible to find the discrimination voltage value for which the best resolution is obtained, where the resolution is defined for a given value of discrimination voltage *Ugj* as the quotient of the difference between the maximum and the minimum value of the obtained total number of events (*NoETOTmax*(*Ugj* ) and *NoETOTmin*(*Ugj* ), respectively) and the sum of the standard deviation values *SDi*(*Ugj* ) corresponding to individual points of the hardness measurement.

$$RE(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}}) = \frac{\text{No}E\_{TOT\max}(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}}) - \text{No}E\_{TOT\min}(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}})}{\sum\_{i}^{n} SD\_{i}(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}})} \tag{1}$$

To normalize the resolution coefficient (*RE*) results with the determination coefficient (*R*2) varying in interval <0,1>, the normalized coefficient of resolution (*RECOF.*) was introduced. The coefficient is defined as:

$$RE\_{\text{CFF.}}(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}}) = \frac{RE(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}})}{\max(RE(\mathcal{U}\_{\mathcal{S}\_{\mathcal{I}}}) }. \tag{2}$$

The best correlation occurs for the discrimination voltage for which maximum values of *R*<sup>2</sup> and *RECOF* are obtained.
