## 2 Materials and Methods
### Gain in Step-Size of Standard Version of the Sequential PU LMS Algorithm  
In previous works [29] we have carried out a similar analysis on the basis of the eigenvalues of the autocorrelation matrix of the input signal. As we see, the shape of the gain in step-size has nothing to do in both cases (LMS vs. P-LMS). Eigenvalues of the Autocorrelation Matrix of a P-Periodic Signal Composed of K Pure Harmonics  
Let us assume that the input signal \(x(n)\) of an adaptive filter is defined as follows. \[x(n)=\sum_{k=1}^{K}C_{k}cos(2\pi kf_{0}n+\varphi_{k}) \tag{30}\]  
where \(f_{0}\) is the fundamental frequency normalized by the sampling rate and \(\{\phi_{k}\}_{k=1,\ldots,K}\) are the initial random phases. Phases are uniformly distributed from 0 to \(2\pi\) radians and mutually independent. Finally, \(\{C_{k}\}_{k=1,\ldots,K}\) are the amplitudes of the harmonics. The autocorrelation function of input signal \(x(n)\) can be expressed as  
\[r_{xx}(\tau)=\sum_{k=1}^{K}\frac{C_{k}^{2}}{2}\text{cos}(2\pi kf_{0}\tau) \tag{31}\]  
Therefore, the autocorrelation matrix of the input vector \(x(n)\) can be expressed as the sum of \(K\) matrices \(\mathbf{R}_{k}\) of size \(M\times M\) as follows. \[\mathbf{R}=\sum_{k=1}^{K}C_{k}^{2}\mathbf{R}_{k} \tag{32}\]  
where  
\[\mathbf{R}_{k}=\frac{1}{2}\left\{ \begin{array}{ccccc}1&\cos(2\pi kf_{0})&\cdots &\cdots&\cos[2\pi k(M-1)f]\\ \cos(2\pi kf_{0})&1&\ddots&&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&\ddots&\cos(2\pi kf_{0})\\ \cos[2\pi k(M-1)f]&\cdots&\cdots&\cos(2\pi kf_{0})&1\end{array} \right\} \tag{33}\]  
The largest eigenvalue \(\lambda_{k,max}(kf_{0})\) of each matrix \(\mathbf{R}_{k}\) is given by [33]  
\[\lambda_{k,max}(kf_{0})=max\left\{\frac{1}{4}\bigg{[}M\pm\frac{\sin(M2\pi kf_{ 0})}{\sin(2\pi kf_{0})}\bigg{]}\right\} \tag{34}\]  
where the subscript \(k\) refers to the index of the submatrix \(\mathbf{R}_{k}\). Let us consider a matrix made up of a sum of matrices of the same dimensions. According to the triangle inequality [34], appendix E, we have that the sum of the largesteigenvalues of every matrix from the sum of matrices establishes the bound for the largest eigenvalue of the matrix defined as a sum of matrices. As a result of that, the bound of the largest eigenvalue of \(\mathbf{R}\), which we refer to as \(\lambda_{tot,max}\), is limited according to  
\[\lambda_{tot,max}\leq\sum_{k=1}^{K}C_{k}^{2}\lambda_{k,max}(kf_{0})=\ \sum_{k=1}^{K}C_{k}^{2}max\bigg{\{}\frac{1}{4}\bigg{[}M\pm\frac{\sin(M2\pi kf_ {0})}{\sin(2\pi kf_{0})}\bigg{]}\bigg{\}} \tag{35}\]  
where the subscript _tot_ is used to refer to the autocorrelation matrix \(\mathbf{R}\)[29]. As far as the Seq. PU LMS adaptive algorithm is concerned, the requirements for the convergence that the entire filter has to meet can be translated to the joint convergence of \(N\) logical (\(M\)/\(N\))-length subfilters updated by an \(N\)-decimated input signal \(\mathbf{x}^{(N)}(n)\)[29]. Adjusting the above approach to the case of sequential PU LMS, where the length of the autocorrelation matrix is \(M/N\) and the sampling rate is divided by \(N\), we deal with \(K\) matrices \(\mathbf{R}_{k}^{(N)}\) of dimension \((M/N)\times(M/N)\) obtained by substituting in Equation (33) the number of coefficients \(M\) by \(M/N\). Thus, the largest eigenvalue \(\lambda_{k,max}^{(N)}\) (\(kf_{0}\)) of each matrix \(\mathbf{R}_{k}^{(N)}\) can be expressed as follows.