## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
\tag{100}\]

Therefore, the largest eigenvalue \(\lambda_{k,max}^{(N)}\ (kf_{0})\) of each matrix \(\mathbf{R}_{k}^{(N)}\) can be expressed as follows

\[\lambda_{k,\max}^{(N)}(kf_{0})\ =\ \max\left\{\frac{1}{4}\left[\frac{L}{N}\ \pm\ \frac{\sin \Bigl{(}\frac{L}{N}2\pi kNf_{0}\Bigr{)}}{\sin(2\pi kNf_{0})}\right]\right\}. \tag{101}\]

Considering the triangle inequality, the largest eigenvalue \(\lambda_{tot,max}^{(N)}\) of the (\(L\)/\(N\)) \(\times\) (\(L\)/\(N\)) matrix \(\mathbf{R}^{(N)}\ =\ \sum\limits_{k=1}^{K}C_{k}^{2}\ \mathbf{R}_{k}^{(N)}\) is bounded by

\[\lambda_{\text{tot,max}}^{(N)}\ \leq\ \sum\limits_{k\ =\ 1}^{K}C_{k}^{2}\lambda_{k,\max}^{(N)}( kf_{0})\ =\ \sum\limits_{k\ =\ 1}^{K}C_{k}^{2}\max\left\{\frac{1}{4}\left[\frac{L}{N}\ \pm\ \frac{\sin \Bigl{(}\frac{L}{N}2\pi kNf_{0}\Bigr{)}}{\sin(2\pi kNf_{0})}\right]\right\}. \tag{102}\]

It should be noticed that for \(N\) = 1 the sequential PU LMS algorithm reduces to the conventional full updates LMS algorithm and Equations (101) and (102) reduce to Equations (100) and (101), respectively. ### Effect of the Length of the Filter on the Step-Size Bound

Full updates LMS algorithm

Let the input signal of the LMS algorithm be the periodic signal given by Equation (100). The convergence in mean of the weights of the filter is guaranteed if the step-size satisfies the inequality [33]

\[0<\mu_{LMS}<\frac{2}{\lambda_{\text{tot,max}}}. \tag{103}\]

Thus, combining Equations (101) and (103), we obtain a more restrictive bound on the step-size that ensures convergence in mean. \[0<\mu_{LMS}<\frac{2}{\sum\limits_{k\ =\ 1}^{K}C_{k}^{2}\text{max}\Bigl{\{} \frac{1}{4}\Bigl{[}L\ \pm\ \frac{\sin(12\pi kf_{0})}{\sin(2\pi kf_{0})}\Bigr{]}\Bigr{\}}}. \tag{104}\]

Hence, the bound on \(\mu_{LMS}\) depends on the frequency \(f_{0}\), the length of the filter \(L\), and the weights \(C_{k}\) of the input signal. To simplify the graphical representation of the bound, the input signal of a \(L\)-length adaptive filter updated by the conventional LMS algorithm is defined by the following single tone of normalized frequency \(f_{0}\). \[x(n)\ =\ \cos(2\pi f_{0}n\ +\ \varphi)\,. \tag{105}\]Due to Equations (A19) and (A20), the bound on the step-size is then given by

\[0<\mu_{LMS}<\frac{2}{\max\left\{\frac{1}{4}\left[L\ \pm\ \frac{\sin(L2\pi f_{0})}{\sin(2\pi f _{0})}\right]\right\}}=\frac{2}{\lambda_{\text{tot,max}}}.\] (A21)

Figure 11 shows the size bound of the conventional LMS algorithm for \(L=16\), \(32\), \(64\), and \(128\). The normalized frequency of the input vector varies from \(0\) to \(0.5\). According to Figure 11, and considering that a large step-size guarantees fast convergence rate, we conclude that a shorter filter can converge faster than a large one. (b) Sequential PU LMS algorithm. Let the \(L\times 1\) input vector of the sequential PU LMS algorithm be given by the \(N\)-decimated version \(\mathbf{x}^{(N)}(n)\) of the periodic signal \(\mathbf{x}(n)\) expressed by Equation (A9). A similar analysis as the one carried out in the previous section for the conventional LMS algorithm yields a more restrictive bound on the step-size that ensures convergence in mean for the case of the sequential PU LMS algorithm. \[0<\mu_{SeqLMS}<\frac{2}{\sum\limits_{k=1}^{K}C_{k}^{2}\text{max}\left\{\frac{1 }{4}\left[\frac{L\ \pm\ \frac{\sin(k/2\pi kNf_{0})}{\sin(2\pi kNf_{0})}\right]}\right\}}<\frac{2}{ \lambda_{\text{tot,max}}^{(N)}}.\] (A22)

This bound on \(\mu_{SeqLMS}\) depends on the frequency \(f_{0}\), the length of the filter \(L\), the weights of the input signal \(C_{k}\), and the decimation factor \(N\). As we did in the previous section to simplify the graphical representation of the bound, we reduce the number of harmonics of the input signal to \(K=1\). In so doing, the \(L\times 1\) input vector of the sequential PU LMS algorithm is given by the \(N\)-decimated version \(\mathbf{x}^{(N)}(n)\) of a sinusoidal signal defined in Equation (A20).