## 1 Introduction  
High-strength low-alloyed steels (HSLAs) are well-known materials with a very good ratio between high yield and ultimate strength and low cost. These materials are frequently used in the automotive, petrochemical, and civil industries for heavily loaded parts [1, 2]. Their application also enables a reduction in the weight of construction for improved corrosion resistance [3]. The low content of alloying elements with respect to their hardening is compensated by thermo-mechanical treatment and a small addition of Nb, Ti, and/or V. The sheets of HSLAs are usually rolled at elevated temperatures (above the austenitization temperature), followed by rapid cooling. The final strength of HSLAs is a function of hot rolling conditions and the superimposed rate of consecutive cooling [2; 3; 4; 5]. Furthermore, Nb, Ti, and V can play a certain role in microstructure refinement, their precipitates, and dissolution in the matrix [3; 5; 6; 7]. The microstructure of thermo-mechanically treated low-alloyed steels of lower strength is mostly composed of refined ferrite as soon as the refined ferrite matrix becomes a mixture of ferrite + bainite or bainite + martensite. However, increasing strength comes at the expense of reduced toughness [1; 8], and the residual stresses are also altered [8]. Components and structures made of HSLAs are very often of vital importance with respect to the reliability of the system and their long-term use. Their functionality is a function of the initial state, design, load manner, etc. Their non-destructive monitoring in different applications is essential. Magnetic Barkhausen noise (MBN) is obtained during the magnetization of ferromagnetic bodies in an altered magnetic field. Domain walls (DWs) are pinned in their positions due to the presence of lattice imperfections, and their sudden motion in the form of irreversible jumps occurs when the external magnetic field reaches the pinning strength of the pinning sites [9; 10]. Electromagnetic or acoustic pulses produced by these DWs during their irreversible motion can be detected on the free surface. MBN is sensitive to the microstructure due to interference between DWs and all lattice imperfections, especially dislocation tangles [11; 12], precipitates [13; 14], grain boundaries [15; 16], hard-ferromagnetic or/and non-ferromagnetic phases [17], etc. Furthermore, preferential crystallography and the corresponding magnetic alignment (anisotropy) affect the angular distribution of MBN [18]. Finally, certain contributions of the stress state to MBN should be considered due to the positive magnetostriction and biaxial anisotropy of Fe alloys [19; 20]. Consecutive bursts of MBN are usually very similar when the surface is magnetized through the hysteresis loop. Two consecutive bursts represent the ascending and descending routes of a magnetic field. However, these bursts may be different in some cases. During hard milling, such an asymmetry can be found due to the magneto-coupling between the hard white layer and the underlying soft heat-affected zone [21]. This effect was also observed in an amorphous bilayer ribbon (Fe73.5 Nb3 Si13.5B9 Cu1/Fe74.5 Nb3 Si13.5B9) consisting of two layers of different magnetic hardness [22]. Similar behavior was reported with respect to plastic deformation under the uniaxial tensile test when this asymmetry was produced as a result of heterogeneity in dislocation density among neighboring grains and the associated differences in magnetic hardness of mutually coupled grains [23]. The lack of structure homogeneity is nowadays an important aspect of component functionality, as reported in many studies. Coarse grains can result in deformation heterogeneity under some conditions [24]. In some conditions, this heterogeneity can result in premature failure [25]. However, in some cases, structural heterogeneity can improve mechanical properties, especially in additively manufactured wire arc stainless steel [26].