## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
\begin{table}
\begin{tabular}{c c c c c} \hline
**Task\(\backslash\)Algorithm** & **FxLMS** & **Mod FxLMS** & **Mod Fx H LMS** & **G\(\bullet\)-Mod Fx H Seq LMS** \\ \hline Computing output of & - & \(L\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta}\ \beta\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta}\ \beta\) \\   the slave filter & & & & \\ \hline Filtering reference with & \(L_{\mathsf{s}}\) & \(L_{\mathsf{s}}\) & \(L_{\mathsf{s}}\) & \(L_{\mathsf{s}}\) \\ \(\widehat{S}(z)\) & & & & \\ \hline Filtering output of the & - & \(L_{\mathsf{s}}\) & \(L_{\mathsf{s}}\) & \(L_{\mathsf{s}}\) \\ slave filter with \(\widehat{S}(z)\) & & & & \\ \hline Computing output of & \(L\) & \(L\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta}\ \beta\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta}\ \beta\) \\   the adaptive filter & & & & \\ \hline Updates of the & \(L+1\) & \(L+1\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta^{\prime}}\ (\beta+1)\) & \(\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta^{\prime}}\ \left(\frac{\delta}{N}+1\right)\) \\ \hline \# Total Mutilications & \(2L+L_{\mathsf{s}}+1\) & \(3L+2L_{\mathsf{s}}+1\) & \(\left[\sum\limits_{l=1}^{\mathsf{a}}\frac{L}{\beta}\ (3\beta+1)\right]+2L_{\mathsf{s}}\) & \(\left[\sum\limits_{l=1}^{\mathsf{a}}\frac{l}{\beta}\ \left(2\beta+\frac{\delta}{N}+1\right)\right]+2L_{\mathsf{s}}\) \\ \end{tabular}
\end{table} 
Table 1: Comparison of computational complexities of active noise control (ANC) algorithms, in terms of the average number of multiplications per task and iteration, in a single channel implementation of an ANC system. Figure 8: Compared performance of FxLMS algorithm, Mod FxLMS algorithm and G\(\bullet\)—Mod Fx H Seq LMS, (**a**) \(N=G\mu=1\) and (**b**) \(N=G\mu=4\). Only the sequential PU algorithm depends on the decimation factor \(N\). The \(L=625\) taps are organized in 25 subfilters of \(\beta=25\) weights. The computational complexity for the decimation factors \(N=1\) and \(N=4\) are marked with a circle over the curve. According to Figure 9, it is not necessary to use a very large decimation factor to reduce the computational complexity of the proposed algorithm to the order of that of the conventional FxLMS. If the decimation factor \(N\) is set to 4, the computational complexity of the modified Fx hierarchical LMS algorithm with sequential PU shows an evident reduction with regards to the full updates (\(N=1\)) algorithm. In our example, setting the decimation factor \(N=4\) leads to a good compromise between computational complexity and convergence rate. In addition, it is well suited with the disturbance PSD. More generally, we suggest the practitioner to take into account these above features to select the parameters of our approach. ## 5 Conclusions

This work presents a contribution to the selection of the step-size used in the modified Fx hierarchical LMS adaptive algorithm with sequential PU. The periodic-input signal case is studied, and it is verified that, under certain conditions, the stability range of the step-size is increased compared to the full updates modified Fx hierarchical LMS. The algorithm we propose is based on the sequential PU of the coefficients of a hierarchical filter and on the controlled increase in the step-size of the adaptive algorithm. To boost up the convergence rate of the approach, we suggest combining the modified FxLMS and the hierarchical LMS algorithm. Nevertheless, in so doing, the increase in the computational complexity turns out to be the main drawback. Therefore, to reduce the number of operations required per cycle, we propose to use the sequential PU algorithm. Finally, the existence of an affordable increase in step-size--or gain in step-size--allows the global control strategy to compensate for the lack of adaptation of most of the coefficients, even when the number of operations per iteration is significantly reduced due to PU. The proposed algorithm has the advantage of increasing the convergence rate of the well-known FxLMS algorithm, with a moderate increase in the computational complexity.