## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
The bound on the step-size is then reduced to

\[0<\mu_{\text{SeqLMS}}<\frac{2}{\text{max}\bigg{\{}\frac{1}{4}\bigg{[}\frac{L}{N} \ \pm\ \frac{\sin(\frac{L}{2}2\pi N/\ell_{0})}{\sin(2\pi N/\ell_{0})}\bigg{]}\bigg{\}} }=\frac{2}{\lambda_{\text{tot,max}}^{(N)}}.\] (A23)

Figures A2 and A3 show the size bound of the sequential PU LMS algorithm given by Equation (A23) for decimation factors \(N\) = 2 and \(N\) = 4, respectively. Results are given for \(L\) = 16, 32, 64, and 128. The normalized frequency of the input vector varies from 0 to 0.5. According to Figures A2 and A3, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one when the sequential PU LMS is used. of these critical frequencies, as well as the number and width of the notches, will be analyzed as a function of the sampling frequency \(F_{s}\), the length of the adaptive filter \(L\), and the decimation factor \(N\). According to Equations (A24) and (A25), with increasing decimation factor \(N\), the step-size can be multiplied by \(N\) and, as a result of that affordable compensation, the sequential PU LMS algorithm convergence is as fast as the full updates LMS algorithm as long as the undesired disturbance is free of components located at the notches of the gain in step-size. ## References

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