## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
Let us now give the criteria to recognize the slowest subfilter \(\mathbf{w}_{slow}(n)\) of the arrangement. The convergence conditions of \(\mathbf{w}_{i}^{l}(n)\), \(1\leq l\leq\alpha,\ 1\leq i\leq N\), depends on its length and on its input signal. The larger the subfilter is, the smaller the maximum step-size is. Therefore, the larger the subfilter is, the slower the convergence will be (See Appendix A.3 of the Appendix A, where the dependence of the step-size bound on the length of a filter is derived). As far as the input signal is concerned, [PERSON] [25] states that the influence of the input on the convergence is related to the level of the subfilter in the hierarchy. The eigenvalue spread of the input-signal autocorrelation matrix becomes smaller from level \(l\) to level \(l+1\) because the hierarchical structure tends to average the eigenvalues of the related input-signal autocorrelation matrix. Then, assuming that the number of taps \(\beta\) of every subfilter is the same, regardless the position in the hierarchy, the bottle-neck in the convergence process is located at the subfilters of the first level. Nevertheless, in our approach, every hierarchically arranged subfilter, and more particularly \(\mathbf{w}_{slow}(n)\), is updated by the sequential PU algorithm. Therefore, we have to consider that the logical subfilter (we consider a logical subfilter as the set of \(N\)-equally-spaced taps of a filter updated at every iteration of the updating process according to a sequential LMS algorithm with decimation factor \(N\) (see Appendix A)) is formed by the subset of \(\beta\)/\(N\) coefficients of the \(\beta\)-length subfilter \(\mathbf{w}_{slow}(n)\). These \(\beta\)/\(N\) coefficients are updated in one iteration of the sequential LMS algorithm with decimation factor \(N\). Therefore, the convergence condition of the hierarchical arrangement is established on the basis of the joint convergence of the \(N\) logical subfilters into which the slowest subfilter \(\mathbf{w}_{slow}(n)\) is decomposed. Having determined the element that limits the convergence rate, we derive the gain in step-size for the hierarchical structure in the next section. ### Gain in Step-Size of the G\({}_{\mu}\)--Mod Fx H Seq LMS Algorithm

Results derived in the Appendix A for a conventional FIR filter (theoretical derivation of the gain in step-size for a conventional adaptive FIR filter whose coefficients are partially updated according to the sequential LMS algorithm can be found in the Appendix A.4 of the Appendix), are extended to the \(\beta\)-length slowest subfilter of the hierarchy \(\mathbf{w}_{slow}(n)\), when sequential PU with decimation factor \(N\) are applied to the hierarchical filter. The role of this slowest subfilter can be played by any of the subfilters located at the first level of the hierarchy. To obtain more easily the factor by which the step-size parameter \(\mu\) of the proposed algorithm can be increased with regard to the step-size of the full updates approach, we impose the use of the same value for \(\mu\) for every subfilter of the hierarchy. Then, we consider the bound on the step-size of the hierarchical filter as the maximum value of \(\mu\) that ensures convergence in all the subfilters. Since \(\beta\) is shorter than the total number \(L\) of taps at the first level of the hierarchy (\(\beta=\sqrt[4]{L}\) typically), and, provided that the decimation factor \(N>1\), the number \(\beta\)/\(N\) of subfilter coefficients that are effectively updated per iteration is small. When \(\beta\)/\(N\) is not integer, this ratio must be rounded to the nearest integer either towards zero, \(\beta\)/\(N\), or towards infinity, \(\beta\)/\(N\). In that case, the \(\beta\)-length hierarchical subfilter is decomposed into \(\beta-N\beta/N\) logical subfilters of length \(\beta/N\), and \(N-\beta+N\beta/N\) logical subfilters of length \(\beta/N\).