## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
The proposed method can be used in ANC systems to attenuate periodic disturbances, if the harmonics of the disturbance are not located at frequencies where the gain in step-size exhibits notches. The width and exact location of these notches depend on the system parameters. Figure 9: Computational complexities of different ANC algorithms in terms of the average number of multiplications per iteration. **Author Contributions:** Conceptualization, P.R.L. and G.P.-N.; methodology, P.R.L. and G.P.-N.; software, P.R.L., R.M.F. and F.A.M. ; validation, P.R.L. and G.P.-N.; formal analysis, P.R.L., R.M.F. ; investigation, P.R.L., R.M.F. ; resources, P.R.L., R.M.F. and F.A.M. ; data curation, P.R.L., R.M.F. and F.A.M. ; writing--original draft preparation, P.R.L. ; writing--review and editing, P.R.L. and G.P.-N.; visualization, R.M.F. and F.A.M. ; supervision, P.R.L. and G.P.-N.; project administration, P.R.L. and G.P.-N.; funding acquisition, P.R.L. and G.P.-N. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by \"Catedra Devoture de Tecnologia para el Ser Humano. Universidad de Zaragoza\". **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The data presented in this study are available on request from the corresponding author. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix A

The purpose of this Appendix A is to derive analytical expressions of the step-size bound and the gain in step-size (i.e., the factor by which the step-size parameter of an adaptive filter driven by a sequential PU algorithm can be increased to compensate for its intrinsic lower convergence rate due to PU) when the sequential PU LMS algorithm is used to update the coefficients of an adaptive FIR filter. The study is based on the eigenvalues of the autocorrelation matrix of the periodic filter input. To help the reader, we recall results that have been already presented in, namely, Equations (A7) and (A8), Equations (A14)-(A17), and Equations (A24) and (A25), because they allow us to determine in this paper:

(a) The dependence of the step-size bound of the adaptive algorithms on the length of the adaptive filter. The analysis includes not only the conventional full updates LMS algorithm, but also the sequential PU LMS algorithm with decimation factor \(N\),

(b) The gain in step-size of the sequential PU LMS algorithm. The traditional LMS algorithm analysis can be extended to the FxLMS algorithm, widely used in the field of ANC systems, under certain assumptions, namely the independence between the input signal and the filter weights, the slow convergence condition, and the exact off-line estimate of the secondary path [20], as long as the input signal corresponds to the reference signal filtered by the estimate of the secondary path. Thus, in the sequel we focus our attention on the LMS algorithm. The Appendix A is organized in four sub-sections. Firstly, one recalls the expression of the autocorrelation matrix of the input vector. Then, the analytical expression of the eigenvalues of a harmonic input signal is presented in Appendix A.2. It makes possible to express the step-size bound and the gain in step-size in and Appendices A.3 and A.4, respectively.