## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
The \(\beta/N\)-length logical subfilters have a slower convergence rate than that of the \(\beta/N\)-length logical subfilters (in Appendix A.3 of the Appendix A, it is proved that the larger an adaptive filter, the smaller the bound on its step-size). As there is at least one \(\beta/N\)-length logical subfilter per iteration at the first level, the convergence rate of a \(\beta/N\)-length logical subfilter determines the convergence of the hierarchy of the subfilters. For the sake of simplicity to obtain the dependence of the gain in step-size on the length \(\beta\) and on the decimation factor \(N\), let a single tone of normalized frequency \(f_{0}\) be the input signal of the hierarchical structure. Thus, the gain in step-size is given by

\[G_{\mu}(1,f_{0},\beta,\ N)\ =\ \frac{\max\Bigl{\{}\frac{1}{4}\Bigl{[}\beta\ \pm\ \frac{\sin(\beta 2\pi f_{0})}{\sin(2\pi f_{0})}\Bigr{]}\Bigr{\}}}{\max \biggl{\{}\frac{1}{4}\Biggl{[}\Bigl{[}\ \beta\ \pm\ \frac{\sin(\beta\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! The output of the primary path is then disturbed by additive zero-mean white Gaussian noise \(\eta(n)\), with a signal-to-noise ratio SNR of 27 dB. The decimation factor \(N\) of the PU strategy is set to 3. To obtain a previous coarse estimate of the experimental upper bound \(\widetilde{\mu}_{max}\), the step-size is increased until divergence appears. At that stage, one focuses more carefully on the neighborhood of that value by making the step-size varying with an increment equal to \(10^{-3}\widetilde{\mu}_{max}\). The comparison between the simulated and the experimental results is shown in Figure 6.

According to Figure 6, the predicted gain in step-size is in good agreement with the experimental results obtained by simulations. It should be noted that in both cases, the gain in step-size corresponds to the decimation factor \(N\), namely 3, for most of the frequencies apart from the notches that appear at 1333.3 Hz and 2666.6 Hz. Moreover, the experiment confirms that the global convergence of the hierarchical controller is determined by the convergence of the 24-length subfilter at the first level. Figure 4: Primary path. Figure 5: Secondary path. ### Comparative Study between Different ANC Algorithms

In this part, we compare the G\({}_{\mu}\)--Mod Fx H Seq LMS algorithm with three approaches:

1. Standard FxLMS, with a \(L\)-length FIR adaptive filter [4];
2. Mod FxLMS with a \(L\)-length FIR adaptive filter [12;13];
3. Mod H FxLMS with \(L\) coefficients at the first level of the hierarchical filter. The computational complexity and the convergence rate are the two criteria we take into account. As in the previous sub-section, the \(1\times 1\times 1\) arrangement is chosen for the ANC system. Nevertheless, in this second example, the selected reference signal \(x(n)\) is a multi-tone signal defined as follows

\[ \begin{split} x(n)=\cos\bigl{(}2\pi\tfrac{90}{F5}n\bigr{)}+5\cos \bigl{(}2\pi\tfrac{100}{F5}n\bigr{)}+3\,\cos\bigl{(}2\pi\tfrac{110}{F5}n\bigr{)} +2\,\cos\bigl{(}2\pi\tfrac{300}{F5}n\bigr{)}+\,3\cos\bigl{(}2\pi\tfrac{320}{F5} n\bigr{)}\\ +\cos\bigl{(}2\pi\tfrac{340}{F5}n\bigr{)}+2\cos\bigl{(}2\pi\tfrac{650 }{F5}n\bigr{)}+5\cos\bigl{(}2\pi\tfrac{665}{F5}n\bigr{)}+4\,\cos\bigl{(}2\pi \tfrac{680}{F5}n\bigr{)}\end{split}  \tag{5}\]

where the sampling frequency \(F_{S}\) is set to 1600 samples/s. With such an input, the conventional FxLMS algorithm shows difficulties to rapidly cancel the undesired noise. Primary and secondary paths are the 25 th order IIR filters shown in Figures 4 and 5, respectively. The output of the primary path is disturbed by an additive zero-mean white Gaussian noise \(\eta(n)\) leading to a SNR = 50 dB. The off-line estimate of the secondary path is carried out by an adaptive FIR filter of 128 coefficients. To obtain a feasible DSP-board implementation, \(L\) is set to 625 taps. When the hierarchy is used, these taps are organized in 25 subfilters of \(\beta\) = 25 weights.