## 2 Materials and Methods
### Sequential Partial Updates (PU) Applied to the Periodic LMS Algorithm  
The sequential partial updates (Seq. PU) LMS algorithm updates a subset of \(M/N\) coefficients per iteration, out of a total of \(M\) weights according to Equation (16). \[w_{l}(n+1)=\left\{ \begin{array}{l}w_{l}(n)+\mu x(n-l+1)e(n)\quad if\,\,(n-l +1)\text{mod}N=0\\ w_{l}(n)\end{array} \right. \tag{16}\]  
for \(1\leq l\leq M\), where \(w_{l}(n)\) represents the \(l\)-th coefficient of the filter, N the decimation factor of the PU strategy, \(\mu\) the step-size parameter of the algorithm, \(x(n)\) the input (often named regressor) signal, and \(e(n)\) the error signal. Thus, the computational complexity of the Seq. PU algorithm is reduced directly as N increases [30]. The Seq. PU strategy expressed by Equation (16) is summarized in Figure 1. Here, the weights to be updated at every cycle are determined, as well as the related samples of the regressor signal \(x(n)\). In this scheme, we assume that the first update is carried out at the first cycle, and the current value of the input signal is \(x(n)\). From Equation (16) and Figure 1, one can surmise that the current value given by \(x(n)\) is used to update the first \(N\) filter's taps during the upcoming \(N\) cycles. In general, in a full-update adaptive algorithm, it is necessary to renew the input vector at every cycle with a new sample of \(x(n)\). Nevertheless, according to Figure 1, the Seq. LMS adaptive algorithm makes use of every \(N\)-th position of the input vector. Hence, it is enough to obtain a new sample at just one out of \(N\) iterations. Figure 1: Scheme of the Seq. PU adaptive algorithm; here, we point out the coefficients to be updated at every cycle as well as the related samples of the regressor signal. Then, one can consider that the whole filter of \(M\) coefficients is made up of \(N\) logical subfilters of \((M/N)\) coefficients. These logical subfilters come from the process of sampling uniformly the weights of the original \(M\)-length filter with a sampling factor of \(N\) positions. To illustrate this idea, the weights of the first logical subfilter are marked with a circle in Figure 1. The weights that are placed at the same relative position in every logical subfilter are updated with the same value of the regressor signal \(x(n)\). This regressor signal is renewed just in 1 out of \(N\) cycles. Then, after \(N\) cycles, a new value of \(x(n)\) is sampled and taken to renew the first coefficient of each subfilter, whereas the oldest value of \(x(n)\) is shifted out of its active range. Summarizing, during \(N\) consecutive cycles, \(N\) (\(M/N\))-length logical subfilters are updated with the same regressor vector. This regressor signal is an \(N\)-decimated version of \(x(n)\). Therefore, we analyze the convergence of the \(M\)-length filter on the basis of the parallel convergence of \(N\) (\(M/N\))-length subfilters updated by an \(N\)-decimated input \(x(n)\). In order to use this PU strategy for the P-LMS algorithm, one should decimate the input signal and use a shorter version of the matrices involved in the updating process, because the active number of coefficients per iteration results in a filter of \(M^{\prime}\) coefficients. \[M^{\prime}=\frac{M}{N} \tag{17}\]  
If \(x(n)\) is the \(P\)-periodic signal, then the \(N\)-decimated signal \(x_{d}(n)\)  
\[x_{d}(n)=x(Nn) \tag{18}\]  
is also periodic, of period \(P^{\prime}\), with  
\[P^{\prime}=\left\{ \begin{array}{rl}&\frac{P}{N}if\ \frac{P}{N}\ \mbox{is integer}\\ &Pif\ \frac{P}{N}\ \mbox{is not integer}\end{array} \right. \tag{19}\]  
\(\mathbf{X}_{d}\) is a P\({}^{\prime}\)-by-\(M^{\prime}\) matrix, obtained by substituting in Equation (9) \(x(n)\) by its \(N\)-decimated version \(x_{d}(n)\), the period \(P\) by \(P^{\prime}\), and the number of coefficients \(M\) by \(M^{\prime}\).