## 1 Introduction  
Domain wall (DW) motion under the alteration of a magnetic field near the coercivity usually occurs in the form of irreversible and discontinuous jumps due to interference of DWs with lattice imperfections [1, 2, 3]. These jumps produce acoustic as well as electromagnetic pulses which can be detected on the free surface in the form of bursts, known as Barkhausen noise [1, 2, 3]. Electromagnetic components of these pulses are referred to as magnetic Barkhausen noise (MBN). The main reason for discontinuous DW motion can be viewed in their interference with the pinning sites such as dislocation tangles [4, 5], grain boundaries [6], surface irregularities [7], precipitates [8], and non-ferromagnetic phases [9], etc. For this reason, MBN signals contain information about the interfering microstructure features as well as the stress state [10, 11, 12, 13]. Plastic straining usually increases dislocation density [12, 14], alters grain size and the corresponding grain boundaries, and initiates phase transformations [15], etc. For this reason, MBN is widely employed for monitoring of bodies exposed to the variable degrees of plastic strains under the various regimes of loading (uniaxial, biaxial, bending, etc.) [12, 16, 17]. Sensitivity of MBN against dislocation density and phase transformations has been investigated and explained in a satisfactory manner. On the other hand, such aspects as the alteration of magnetic anisotropy or/and the asymmetry in MBN bursts have been reported [12; 18], but their origin has not been satisfactorily discussed and explained yet. It is considered that the alterations of magnetic anisotropy are due to realignment of DWs due to the preferential crystallographic orientation [18; 19; 20; 21]. On the other hand, remarkable asymmetry in the consecutive MBN bursts in many cases is due to residual magnetism. This effect diminishes when the surface is demagnetised. However, despite demagnetisation, in some cases, this asymmetry remains in the components as a result of magnetic coupling between the phases of the different magnetic (and the corresponding mechanical) hardness. [PERSON] et al. [22] investigated monolithic amorphous bilayer ribbons (Fe\({}_{73.5}\)Nb\({}_{3}\)Si\({}_{13.5}\)B\({}_{9}\)Cu\({}_{1}\)/Fe\({}_{74.5}\)Nb\({}_{3}\)Si\({}_{13.5}\)B\({}_{9}\)) consisting of two layers of different magnetic hardness. The authors separated the individual contributions of each layer into the global magnetic behaviour and analysed the magneto-coupling between them. The findings of this study match well with the magnetic behaviour of the hard-milled surface when the magnetic coupling can be found between the surface hard white layer and the subsurface soft heat-affected region [18]. The marked MBN asymmetry was also reported during uniaxial plastic straining [12]. It was found that this asymmetry is sensitive to the magnetic anisotropy and the degree of plastic straining. However, its origin has not been investigated and explained yet. Remarkable asymmetry in the consecutive MBN bursts can be detected as a results of insufficient samples demagnetization. However, this asymmetry can be found in some cases after components processing despite their demagnetization due to heterogeneity in magnetic hardness of the neighboring regions. This asymmetry remarkably increases standard deviations of MBN expressed in term of the effective (rms) values as well as the positions of MBN envelopes and the corresponding Peak Positions. This aspect limits reliability of MBN technique employed for monitoring of components in the real industry. For these reasons, the asymmetry in the consecutive MBN bursts should be investigated, explained, and considered when a suitable concept for monitoring of real components is proposed. Therefore, this study provides a deeper insight and explanation of the MBN asymmetry after the uniaxial plastic straining and the superimposing magnetic anisotropy.