## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
Therefore, at the second level, there is one subfilter of \(\beta\) = 25 taps. The performance of the proposed algorithm is tested for different values of the decimation factor \(N\), namely \(N\) = 2, 3, and 4. Figure 7 shows the theoretically predicted gain in step-size, defined by Equation (4), over the frequency band of interest for \(N\) = 2, 3, and 4. Figure 6: Theoretically predicted gain in step-size vs. simulation results of the G\({}_{\mu}\)—Mod Fx H Seq LMS algorithm in a modelled ANC system. The frequency response of the gain in step-size depends on the decimation factor and exhibits \(N-1\) notches, located at \(\left\{iF_{s}/2N\right\}_{i=1,\ldots,N-1}\). Therefore, the PSD of the undesired disturbance has to be taken into account to select the decimation factor. Figure 8a,b compares the learning curves of the FxLMS algorithm, the Modified FxLMS algorithm and the G\(\mu\)--Mod Fx H Seq LMS, for \(N=G\mu=1\) and 4, respectively. The learning curves are averaged over 500 realizations. To carry out a fair comparison of algorithms, we have adjusted the step sizes to make the FxLMS and the Modified FxLMS converge to the same residual noise level that the proposed algorithm reaches. Given Figure 8, it can be affirmed that the Modified FxLMS algorithm (blue) converges faster than the standard FxLMS (black), and when a hierarchical filter is included (red), the convergence rate is further boosted. Although the hierarchical filter accelerates the convergence rate of the modified FxLMS (this was the reason to include a hierarchical structure), it presents an inherent drawback: including the hierarchical filter implies an increase in the computational complexity. To minimize this negative effect, we included partial updates of the coefficients with a decimation factor \(N\). In so doing, the convergence rate is reduced proportionally to \(N\), unless gain in step size is applied. By increasing the step size in a factor \(G\mu=N\), the convergence rate and the residual error of our approach are approximately the same when \(N\) is modified. As the reference signal given by Equation (5) does not include harmonics in the vicinity of the notches presented in the gain in step size when \(N=4\) (see Figure 7), full gain in step-size can be applied, and performance of the G\(\mu\) -Mod Fx H Seq LMS algorithm in both cases--(a) full updates, \(N=G\mu=1\) and (b) partial updates \(N=G\mu=4\)--are similar in terms of convergence rate and residual error, with a significant reduction in the computational cost when partial updates are implemented. As far as the computational complexity is concerned, let us assume for the sake of simplicity that the length of every subfilter is \(\beta\), regardless of its position in the hierarchy. Therefore \(L=\beta^{\alpha}\). In addition, we recall that \(Ls\) is the length of the off-line estimate of the secondary path. Figure 7: Theoretically predicted step-size gain of the G\(\mu\)—Mod Fx H Seq PU LMS over the frequency band of interest—from 0 to 800 Hz—for \(N=2\), 3, and 4. \(L=625\) taps are organized in 25 subfilters of \(\beta=25\) weights. According to Table 1, the modified version of the FxLMS requires \(L+Ls\) extra multiplications with regard to the computational complexity of the conventional FxLMS algorithm. When the hierarchical filter is used, the computational complexity is further increased because of the multi-level arrangement. Finally, our algorithm has the advantage of reducing the computational complexity thanks to the PU strategy. Figure 9 compares the computational complexities of different ANC algorithms when the decimation factor \(N\) varies from \(1\) to \(50\).