## 2 Materials and Methods
### Gain in Step-Size of Standard Version of the Sequential PU LMS Algorithm
\[\lambda_{k,max}^{(N)}(kf_{0})=max\left\{\frac{1}{4}\Bigg{[}\frac{M}{N}\pm\frac {\sin\!\left(\frac{M}{N}2\pi kNf_{0}\right)}{\sin(2\pi kNf_{0})}\Bigg{]}\right\} \tag{36}\]  
Considering the triangle inequality, the largest eigenvalue \(\lambda_{tot,max}^{(N)}\) of the \((M/N)\times(M/N)\) matrix \(\mathbf{R}^{(N)}=\sum_{k=1}^{K}C_{k}^{2}\mathbf{R}_{k}^{(N)}\) is bounded by  
\[\lambda_{tot,max}^{(N)}\leq\sum_{k=1}^{K}C_{k}^{2}\,\lambda_{k,max}^{(N)}(kf_ {0})=\sum_{k=1}^{K}C_{k}^{2}max\left\{\frac{1}{4}\Bigg{[}\frac{M}{N}\pm\frac{ \sin\!\left(\frac{M}{N}2\pi kNf_{0}\right)}{\sin(2\pi kNf_{0})}\Bigg{]}\right\} \tag{37}\]  
It should be noticed that for \(N=1\) the sequential PU LMS algorithm reduces to the conventional full-update LMS algorithm and Equations (36) and (37) reduce to Equations (34) and (35), respectively. The quotient between the limits of the step-sizes \((\mu)\) in two different cases (\(N>1\), sequential PU LMS and \(N=1\), conventional LMS), defines the step-size gain \(G\mu\) as the factor by which one can multiply the step-size meeting convergence requirements [29]. \[G_{\mu}(K,f_{0},M,N)=\frac{bound\{\mu_{SeqPL_{\rm LMS}}\}}{bound\{\mu_{LMS}\}}= \frac{\frac{2}{max\left\{\lambda_{k,max}^{(N)}\right\}}}{\frac{2}{max\left\{ \lambda_{tot,max}^{(N)}\right\}}}=\frac{\sum_{k=1}^{K}C_{k}^{2}\lambda_{k,max}( kf_{0})}{\sum_{k=1}^{K}C_{k}^{2}\lambda_{k,max}^{(N)}(kf_{0})}\frac{\sum_{k=1}^{K}C_{ k}^{2}\text{max}\Big{\{}\frac{1}{4}\Big{[}M\pm\frac{\sin\!\left(\frac{M}{N}2\pi kNf_{0} \right)}\Big{]}\Big{\}}}{\sum_{k=1}^{K}C_{k}^{2}\text{max}\Big{\{}\frac{1}{4} \Big{[}M\pm\frac{\sin\!\left(\frac{M}{N}2\pi kNf_{0}\right)}\Big{]}\Big{\}}}. \tag{38}\]  
The gain in step-size given by Equation (38) depends on the length of the filter \(M\) and on the decimation factor \(N\). In order to visualize this double dependence, we set to \(K=1\) the number of harmonics of the input signal. In so doing, the gain in step-size yields  
\[G_{\mu}(1,f_{0},M,N)=\frac{bound\{\mu_{SeqPL_{\rm LMS}}\}}{bound\{\mu_{LMS}\}}= \frac{max\Big{\{}\frac{1}{4}\Big{[}M\pm\frac{\sin\!\left(M2\pi f_{0}\right)}{ \sin\!\left(2\pi f_{0}\right)}\Big{]}\Big{\}}}{max\bigg{\{}\frac{1}{4}\Big{[} \frac{M}{N}\pm\frac{\sin\!\left(\frac{M}{N}2\pi Nf_{0}\right)}{\sin\!\left(2\pi Nf _{0}\right)}\Big{]}\bigg{\}}} \tag{39}\]  
The step-size gain for a pure tone when different decimation factors \(N\) and different filter lengths \(M\) are considered, is shown in Figures 2 and 3[29]. According to Figures 2 and 3, one can infer that the step-size can be increased by a factor up to \(N\) if certain frequencies are not present in the input signal. These frequencies are those that exhibit notches in the gain in step-size. The location of these critical frequencies, as well as the width and the number of the notches, can be analyzed as a function of the length of the adaptive filter \(M\), the decimating factor \(N\), and the sampling rate \(F_{s}\). Thus, the step-size parameter can be multiplied by a factor of \(N\) in order to ensure that the sequential PULMS algorithm convergence rate is as fast as the full-update LMS algorithm. To afford that increase in convergence rate, the undesired disturbance must be free of significant components placed at frequency notches in the step-size gain. Figure 3: Gain in step-size with filter lengths, \(M\) = 8, 32, and 128 for a single tone. The decimating factor is set to \(N\) = 2. Figure 2: Gain in step-size for decimation factors, \(N\) = 1, 2, 4, and 8 for a single tone. The number of coefficients of the filter is set to \(M\) = 256.