## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size  
In this section, we propose the G\({}_{\mu}\)-Mod Fx H Seq LMS algorithm by combining the Mod FxLMS, the HLMS, and the sequential PU LMS with gain in step-size. Our goal is to derive an ANC adaptive algorithm with a faster convergence rate than the conventional FxLMS with a similar computational complexity. Figure 2 shows the block diagram of the proposed algorithm. According to Figure 2, the filtered reference \(x(n)\) is the input of the control adaptive filter whereas the reference signal \(x(n)\) is filtered by a slave filter, which is a copy of the control adaptive filter. By cascading the slave filter and the estimat  
Figure 2: Block diagram of the modified filtered-x hierarchical LMS algorithm with sequential partial updates. \(\widetilde{S}(z)\), and then by passing the reference signal \(x(n)\) through the resulting filter, one can obtain an estimation \(\widetilde{y}(n)\) of the antinoise and hence an estimation \(\widetilde{d}(n)\) of the undesired noise. Inaccuracy of the secondary path estimate and its effects are discussed in [12; 13; 14] in the context of filtered x LMS algorithms. At that stage, the output \(z(n)\) of the adaptive control filter is directly subtracted from the estimated noise \(\widetilde{d}(n)\) to provide the numerical error \(e_{n}(n)\). One may find in [30] a complete review of techniques of estimation of the error signal using signal processing algorithms. As this error is used to update the adaptive control filter, the limitations imposed on the step-size \(\mu\) for the standard version of the FxLMS algorithm are now overcome. Let us now focus our attention on the hierarchical filter. Given the number \(L\) of taps at the first level of the hierarchy, the number of subfilters at the \(l\)th level is given by  
\[N_{l}=\frac{L}{\prod\limits_{r=1}^{l}\beta_{r}}=\frac{\prod\limits_{r=1}^{ \alpha}\beta_{r}}{\prod\limits_{r=1}^{l}\beta_{r}}=\prod\limits_{r=1+1}^{ \alpha}\beta_{r} \tag{2}\]  
where \(\beta_{l}\) denotes (with this notation, we implicitly assume that the subfilters at the same level have the same number of taps and this number may vary from one level to another) the number of weights of a subfilter at level \(l\), varying from \(1\) to \(\alpha\). As the subfilter length may vary from one level to another, the step-size bound of every subfilter can be different. In the sequel, the coefficients of the \(i\)th hierarchically arranged subfilter impulse response at the \(l\)th level are denoted as  
\[\mathbf{w}_{i}^{l}(n)=\left[w_{i1}^{l}(n)\ w_{i2}^{l}(n)\ \cdots\ w_{i\beta_{l}}^{l}(n)\right],\quad 1 \leq l\leq\alpha,\ 1\leq i\leq\ N_{l} \tag{3}\]  
where \(w_{ij}^{l}\) denotes the weight for the \(j\)th tap of the \(i\)th subfilter at the \(l\)th level. In addition, \(z_{ij}^{l}\) and \(y_{ij}^{l}\), respectively, denote the input signals of the the \(j\)th tap of the \(i\)th subfilter at the \(l\)th level of the adaptive and the slave hierarchical filters. The outputs of the adaptive and the slave hierarchical filters, respectively denoted as \(z(n)\) and \(y(n)\), are given by the last loop of the multilevel filtering, that is, \(z(n)=z_{11}^{\alpha+1}(n)\) and \(y(n)=y_{11}^{\alpha+1}(n)\). The error signal of the \(i\)th subfilter at the \(l\)th level is denoted as \(\varepsilon_{i}^{l}\). These error signals are obtained by subtracting the output of every subfilter from the estimated noise \(\widetilde{d}(n)\). It should be noted that the necessity of using the estimated noise \(\widetilde{d}(n)\) to update the subfilters placed at the intermediate levels of the hierarchy is already solved as we use the Mod FixLMS version of the ANC algorithm. Figure 3 shows the architecture of a 2-level hierarchical filter. In this example, the number of subfilters at levels \(1\) and \(2\) are \(N_{1}=L/\beta\) and \(N_{2}=1\), respectively. The number of coefficients of every subfilter at levels \(1\) and \(2\) are \(\beta\) and \(L/\beta\), respectively.