Having estimated the undesired noise \(\widetilde{d}(n)\), the secondary path and the adaptive filter are swapped in the updates path. Then, the error signal of the adaptive algorithm is calculated as the difference between the estimated noise and the output of the adaptive filter. Hence, the behavior of the system is similar  
Figure 1: Single-channel active noise control system using the FxLMS algorithm. (**a**) Physical arrangement of the electro-acoustic elements. (**b**) Equivalent block diagram. to that of the conventional LMS algorithm. Nevertheless, the secondary path impulse response is assumed to be accurately estimated (A frequency domain analysis about the behaviour of the Mod FxLMS algorithm in the presence of secondary path modelling errors is proposed in [19]). The convergence rate of the Mod FxLMS algorithm is increased at the cost of an additional computational complexity, which turns out to be the main drawback of the approach. A trade-off has hence to be found between convergence rate and computational complexity [20]. Reduced-complexity implementations of the Mod FxLMS have been proposed in [21; 22], but the convergence speed can still be improved. In [23] a new delay-less frequency-domain ANC algorithm is proposed. The proposal not only removes the delay in the weight adaptation (as the modified filter-x scheme implies) but also removes the delay in the signal path. The proposed strategy exhibits lower computational complexity than other state-of-the-art frequency-domain FxLMS algorithms [24]. In this paper, we propose to combine the Mod FxLMS structure and the hierarchical LMS (HLMS) algorithm, initially developed in the field of channel equalization by [PERSON] [25]. One may find a performance analysis of the HLMS algorithm in [26]. In [27], the mean-squared error in a two-level HLMS algorithm is analyzed; in this example, the HLMS is used as a predictive strategy that can significantly speed up the convergence rate during the initial stage of the algorithm. In the HLMS adaptive algorithm, the filter coefficients are organized into a hierarchy of subfilters of shorter length distributed in \(\alpha\) levels (details on the hierarchical arrangement of subfilters are provided in Section 2). The output signals of the subfilters at level \((l-1)\) are the input signals of the subfilters placed at the next level \(l\), with \(l\) varying from \(1\) to \(\alpha\). Then, the number of subfilters per level is divided from level \((l-1)\) to level \(l\) by a factor given by the length of the subfilters at level \((l-1)\). At the last level of the hierarchy, namely level \(\alpha\), there is only one subfilter. Since the subfilters have shorter length than a conventional FIR filter, they can converge faster, as recalled in the Appendix A.3 of the Appendix A. However, the computational complexity associated to this multi-level structure is higher than that of the conventional LMS algorithm. To address the above problem, we suggest using partial updates (PU) of the adaptive filter coefficients. A widely used PU algorithm is the sequential PU LMS algorithm with decimation factor \(N\)[28]. This algorithm updates a subset of size \(L/N\), out of \(L\) coefficients--\(w_{j}(n)\), \(1\leq j\leq L\)--per iteration of a conventional \(L\)-length FIR filter according to  
\[w_{j}(n+1)=\left\{ \begin{array}{cc}w_{j}(n)+\mu&x(n-j+1)\ e(n)&\text{if }(n-j+1)\text{ mod }N\ =\ 0\\ w_{j}(n)&\text{otherwise}\end{array} \right. \tag{1}\]  
where \(\mu\) is the step-size of the algorithm, \(x(n)\) the input signal, and \(e(n)\) the error. Nevertheless, the higher the decimation factor \(N\) is, the lower the convergence rate will be. In [29], we have shown that, in the context of a conventional adaptive FIR filter, this lower convergence rate can be compensated, under the assumption of a periodic input signal, by an affordable increase in the step-size \(\mu\).