## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
The main drawback of the HLMS is the high computational complexity inherently associated to its multi-level structure. Sequential PU of the coefficients of the hierarchical filter are used to reduce the computational complexity. PU are applied to every coefficient at every level, of the hierarchical organization of taps, from the first tap of the first subfilter to the last tap of the last subfilter. For instance, in Figure 3, the shadowed coefficients of the hierarchical filter are the \(N\)-equally-spaced taps that have to be updated at a given time \(n\). At the following iterations of the updating process, namely \(n+1\), \(n+2\),..., \(n+N-1\), the next subsets of equally spaced coefficients of the hierarchical filter are updated. Due to PU, the algorithm suffers from a reduction in convergence rate as \(N\) increases. Then, by using the gain in step-size, the slower convergence rate of the sequential PU adaptive algorithm can be compensated. The strategy hence gives the same performance as that of the full updates algorithm in terms of convergence rate, but with lower computational complexity. In previous works [29], this strategy is analyzed in the context of a conventional adaptive FIR filter. The G\({}_{\mu}\)--Mod Fx H Seq LMS deals with periodic disturbances. These periodic noises, such as engine noise, are very often the subject of cancellation in ANC applications. This is due to two reasons. First, these disturbances are the most annoying and, second, it is usually easy to find a good reference signal to cancel them. In the Algorithm 1, the G\({}_{\mu}\)--Mod Fx H Seq LMS algorithm is given:  
Figure 3: Two-level hierarchical filter. ``` for\(i=1\) to # iterations /* MAIN LOOP*/ \(\textbf{y}_{\forall i}^{1}\;y_{j}(n)=\textbf{x}(n)\) /* SLAVEHERARCHICAL FILTER*/ for\(l=1\) to \(\alpha\) for\(i=1\) to \(\prod\limits_{r\;=\;l+1}^{\alpha}\delta_{r}\) /* Computing the output of every subfilter */ \(y_{p\;q}^{l+1}(n)=\textbf{w}_{i}^{l}\;\mathcal{T}(n)\;\textbf{y}_{i}^{l}\; \Big{|}_{p=\left[\frac{l}{\beta_{l}}\right],\;q=i-\left[\frac{|i}{\beta_{l}} \right]\delta_{l}}\) end of for (i) end of for (\(l\)) \(\to y(n)=y_{1}^{\alpha+1}(n)\) /* END OF SLAVEHERARCHICAL FILTER*/ /* Computing the antinostic signal where */ \(\alpha\;\tilde{y}(n)=\left[\tilde{y}_{1}\;\;\tilde{y}_{2}\;\;\;\ldots\;\;\; \tilde{y}_{L}\right]^{\textsf{T}}\)and */ \(\alpha\;\tilde{y}_{1}^{\alpha+1}(n)=\) /* Measured error \(e_{m}(n)=d(n)-y^{\prime}(n)\)*/ /* Computing the estimated noise */ /* Filtering the referenoc\(\textbf{x}(n)=[x(n)\;\;\;\ldots\;\;x(n-L_{\textsf{s}}+1))]^{\textsf{T}}\)/* First level of adaptive hierarchical filter is filled with */ * ADAPTIVEHERARCHICAL FILTER*/ for\(l=1\) to \(\alpha\) for\(i=1\) to \(\prod\limits_{r\;=\;l+1}^{\alpha}\delta_{r}\) for\(i=1\) to \(\prod\limits_{r\;=\;l+1}^{\alpha}\delta_{r}\) /* Computing the output of every subfilter */ \(z_{p\;q}^{l+1}(n)=\textbf{w}_{i}^{l}\;\mathcal{T}(n)



namely the independence theory between the reference signal and the filter weights, and the slow convergence condition (The reader is referred to [29] for more information on the assumption of independence theory and the slow convergence condition. Despite the fact that such assumptions might be initially questionable when dealing with periodic inputs, we confirm in [29] the feasibility of assuming both conditions in the analysis of a FIR-based Fx sequential PU LMS strategy to attenuate periodic disturbances). When using a more complex filtering structure [25] based on hierarchically arranged subfilters \(\mathbf{w}_{i}^{l}(n)\), \(1\leq l\leq\alpha,\ 1\leq i\leq N\), the overall convergence of the hierarchical structure is assumed to be constrained by the hierarchical arranged subfilter \(\mathbf{w}_{slow}(n)\) that converges with the slowest convergence rate. By applying to \(\mathbf{w}_{slow}(n)\) the convergence analysis for a conventional FIR filter recalled in the Appendix A, one can obtain the analytical expression of the gain in step size of the G\({}_{\mu}\)--Mod Fx H Seq LMS algorithm.