## 2 Modified FX Hierarchical Sequential PU LMS Algorithm with Gain in Step-Size
### Autocorrelation Matrix of the Decimated Input Vector of the Sequential PU LMS Algorithm

Let be the \(L\)-length LMS-filter vector response updated as follows

\[\mathbf{w}(n+1)\ =\ \mathbf{w}(n)\ +\ \mu\ e(n)\ \mathbf{x}(n)\] (A1)

where \(\mu\) is the step-size of the adaptive algorithm, \(e(n)\) the error signal between the output of the filter and the desired response, and \(\mathbf{x}(n)=[x(n)\ x(n-1)\ \ldots\ x(n-L+1))]^{T}\) the \(L\times 1\) input vector of the LMS algorithm, whose Toeplitz autocorrelation matrix is given by

\[\mathbf{R}\ =\ E\Big{[}\mathbf{x}(n)\mathbf{x}^{T}(n)\Big{]}.\] (A2)



### Eigenvalues of the Autocorrelation Matrix of a Periodic Signal Consisting of \(K\) Harmonics

Let us assume that the input signal \(x(n)\) of an adaptive filter is defined as follows

\[x(n)\ =\ \sum_{k\ =\ 1}^{K}C_{k}\cos(2\pi kf_{0}n\ +\ \varphi_{k}) \tag{10}\]

where \(f_{0}\) is the normalized fundamental frequency, \(\{\phi_{k}\}_{k=1,\ldots,K}\) the initial random phases mutually independent and uniformly distributed from 0 to \(2\pi\) and \(\{C_{k}\}_{k=1,\ldots,K}\) the amplitudes of the harmonics. Given Equation (10), the autocorrelation function of input signal \(x(n)\) can be expressed as

\[r_{xx}(\tau)\ =\ \sum_{k\ =\ 1}^{K}\frac{C_{k}^{2}}{2}\cos(2\pi kf_{0}\tau). \tag{11}\]

Therefore, the autocorrelation matrix of the input vector \(x(n)\) can be expressed as the sum of \(K\) matrices \(\mathbf{R}_{k}\) of size \(L\times L\) as follows

\[\mathbf{R}\ =\ \sum_{k=1}^{K}C_{k}^{2}\ \mathbf{R}_{k} \tag{12}\]

where

\[\mathbf{R}_{k}\ =\ \frac{1}{2}\left\{ \begin{array}{ccccc}1&\cos(2\pi kf_{0})& \cdots&\cdots&\cos[2\pi k(L-1)f]\\ \cos(2\pi kf_{0})&1&\ddots&&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&\ddots&\cos(2\pi kf_{0})\\ \cos[2\pi k(L-1)f]&\cdots&\cdots&\cos(2\pi kf_{0})&1\end{array} \right\}. \tag{13}\]

The largest eigenvalue \(\lambda_{k,max}(kf_{0})\) of each matrix \(\mathbf{R}_{k}\) is given by [31]

\[\lambda_{k,\max}(kf_{0})\ =\ \max\biggl{\{}\frac{1}{4}\biggl{[}L\ \pm\ \frac{\sin(L2\pi kf_{0})}{\sin(2\pi kf_{0})}\biggr{]}\biggr{\}} \tag{14}\]

where the subscript \(k\) refers to the index of the submatrix \(\mathbf{R}_{k}\). According to the triangle inequality [32], appendix E, the largest eigenvalue of a sum of matrices is bounded by the sum of the largest eigenvalues of each of its components. Therefore, the largest eigenvalue \(\lambda_{tot,max}\) of \(\mathbf{R}\) is bounded by

\[\lambda_{\text{tot,max}}\ \leq\ \sum_{k\ =\ 1}^{K}C_{k}^{2}\lambda_{k,\max}(kf_{0} )\ =\ \sum_{k\ =\ 1}^{K}C_{k}^{2}\text{max}\biggl{\{}\frac{1}{4}\biggl{[}L\ \pm\ \frac{\sin(L2\pi kf_{0})}{\sin(2\pi kf_{0})}\biggr{]}\biggr{\}} \tag{15}\]

where the subscript tot refers to the whole autocorrelation matrix \(\mathbf{R}\). As far as the sequential PU LMS algorithm with decimation factor \(N\) is concerned, the convergence condition of the whole filter might be translated to the parallel convergence of \(N\) logical subfilters of length \(L/N\) updated by a \(N\)-decimated input signal \(\mathbf{x}^{(N)}(n)\)[29]. Adjusting the above approach to the case of sequential PU LMS, where the size of the autocorrelation matrix is \(L/N\) and the sampling frequency is divided by \(N\), we deal with \(K\) matrices \(\mathbf{R}_{k}^{(N)}\) of size \((L/N)\ \times(L/N)\) defined by\[\mathbf{R}_{k}^{(N)}\ =\ \frac{1}{2}\left\{ \begin{array}{ccccc}1&\cos(2\pi Nkf_{0})& \cdots&\cdots&\cos\Bigl{[}2\pi Nk\Bigl{(}\frac{L}{N}-1\Bigr{)}f\Bigr{]}\\ \cos(2\pi Nkf_{0})&1&\ddots&&\vdots\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&\ddots&\cos(2\pi Nkf_{0})\\ \cos\Bigl{[}2\pi Nk\Bigl{(}\frac{L}{N}-1\Bigr{)}f\Bigr{]}&\cdots&\cdots&\cos( 2\pi Nkf_{0})&1\end{array} \right\}\,.