###### Abstract  
Partial updates (PU) of adaptive filters have been successfully applied in different contexts to lower the computational costs of many control systems. In a PU adaptive algorithm, only a fraction of the coefficients is updated per iteration. Particularly, this idea has been proved as a valid strategy in the active control of periodic noise consisting of a sum of harmonics. The convergence analysis carried out here is based on the periodic nature of the input signal, which makes it possible to formulate the adaptive process with a matrix-based approach, the periodic least-mean-square (P-LMS) algorithm In this paper, we obtain the upper bound that limits the step-size parameter of the sequential PU P-LMS algorithm and compare it to the bound of the full-update P-LMS algorithm. Thus, the limiting value for the step-size parameter is expressed in terms of the step-size gain of the PU algorithm. This gain in step-size is the quotient between the upper bounds ensuring convergence in the following two scenarios: first, when PU are carried out and, second, when every coefficient is updated during every cycle. This step-size gain gives the factor by which the step-size can be multiplied so as to compensate for the convergence speed reduction of the sequential PU algorithm, which is an inherently slower strategy. Results are compared with previous results based on the standard sequential PU LMS formulation. Frequency-dependent notches in the step-size gain are not present with the matrix-based formulation of the P-LMS. Simulated results confirm the expected behavior. active noise control; adaptive signal processing; periodic input signal; sequential partial updates; gain in step-size +
Footnote †: journal: Applied Sciences  
15 June 2021  
**Problems:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. secondary sound sources is arranged so as to destructively interfere with the acoustic noise from the primary source. The basic idea behind ANC systems has been reported for the last decades [11; 12; 13; 14], and it still is a research field with continuous dissemination of interesting results [15; 16; 17; 18; 19]. ANC systems' efficiency has been demonstrated by many researchers who have published complete references on the topic [20; 21; 22]. Active techniques, therefore, are generally considered a valid strategy of overcoming passive systems' performance in certain cases. Many applications of ANC have to deal with periodic disturbances consisting of several harmonics [23; 24]. The reader might take into account that narrowband active noise control systems may attenuate a great variety of noises, e.g., those generated by compressors, turbines, engines, or fans. In [25], [PERSON] et al. proposed a robust adaptive strategy for the rejection of periodic components of a disturbance and analyzed its performance and stability properties. One may find in the literature numerous examples of adaptive algorithms derived from the conventional LMS aimed at improving their performance by dealing with the step-size parameter. For instance, [PERSON] et al. proposed [26] a variable step-size LMS strategy that achieves a fast convergence rate and low mis-adjustment by improvement of the updating stage. Besides, system identification is a common task in many application fields. If systems have to be estimated using sparse computation due to computational complexity constraints, partial updates (PU) of the coefficients turn out to be an optimal option. Thus, algorithms exploiting sparseness in the coefficient update domain, where most of the weights are small and only a few are relevant, are often based on variations of the LMS adaptive algorithm [27; 28]. In this sense, gain in step-size [29] is a parameter defined in the context of periodic primary noise attenuation when PU of the weights of the filter are used to lower the computational cost of an adaptive algorithm [30; 31; 32].