## 4 Discussion  
First of all, it is important to note the fact that active noise control of periodic disturbances is a common task that many control systems have to deal with. As already said, narrowband active noise control is a current-day topic of great importance because the attenuation of noises from engines, compressors, turbines, fans, or propellers is a problem worldwide addressed by many researchers.  
The gain in step-size is a concept addressed in previous works in the context of the sequential PU LMS algorithm leading to the convergence of a standard FIR filter. The main idea behind this topic is that the inherent reduction of convergence speed due to partial updates can be compensated for by increasing the step-size parameter \(\mu\), because the \(\mu\) bound seems to be increased by a factor of up to the decimating factor \(N\). Nevertheless, the gain in step-size is not a constant of value \(N\), but exhibits notches whose number, width, and location can be predicted and, consequently, avoided.  
Then, we have considered the alternative matrix-based approach of the LMS formulation proposed by [PERSON] et al. [9], that we refer to as periodic LMS (P-LMA). In order to reduce its computational complexity by a decimating factor \(N\), sequential PU have been  
Figure 14: Learning curves comparison. Version 2 of the experiment: desired signal \(d_{2}(n)\) and reference signal \(x_{2}(n)\). (a) LMS, (b) LMS with Seq PU; \(N=2\) and \(G_{\mu}=2\), (c) P-LMS, (d) P-LMS with Seq. PU; \(N=2\) and \(G_{\mu}=2\). Average of 50 runs. The performance measure used is the instantaneous squared error in logarithmic scale (dB).  
proposed in the framework of the P-LMS algorithm. As expected, PU reduce operations per cycle but, as a drawback, the convergence rate slows down proportionally by \(N\).  
On this point, we have looked into the gain in step-size when sequential PU are considered in the P-LMS. We conducted a study of the theoretical framework of the adaptive strategy as well as several experiments to conclude that the gain in step-size does not exhibit notches. The frequency-dependent shape of the gain in step-size of the P-LMS tends to \(N^{2}\) for high frequencies but shows a lower value in the low frequency range. Nevertheless, there is no evidence of the presence of the notches that were visible when the standard (and not the periodic LMS) was considered.  
To sum up, research results show that a reduction in the computational costs associated with PU is not achieved at the expense of a reduction of the convergence rate. This statement is justified by the existence of a gain in step-size, when PU are considered, and, more importantly, thanks to the matrix-based formulation of the LMS algorithm that we have referred to as P-LMS. This gain in step-size does not show notches in the frequency domain as it does in the case of a standard LMS formulation but just a poor behavior in the low frequency range. Thus, paying attention to the low frequency components, one may take full advantage of the application of the step-size gain.  
Conceptualization, P.R.L. and R.M.F.; methodology, P.R.L. and G.P.-N.; software, P.R.L., R.M.F., and F.A.M.; validation, P.R.L. and G.P.-N.; formal analysis, P.R.L. and R.M.F.; investigation, P.R.L. and R.M.F.; resources, P.R.L., R.M.F., and F.A.M.; data curation, P.R.L., R.M.F., and F.A.M.; writing--original draft preparation, P.R.L.; writing--review and editing, P.R.L. and G.P.-N.; visualization, R.M.F. and F.A.M.; supervision, P.R.L. and G.P.-N.; project administration, P.R.L. and G.P.-N.; funding acquisition, P.R.L. and G.P.-N. All authors have read and agreed to the published version of the manuscript. This research received no external funding.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.