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.
Highlights
The least mean-square (LMS) [1,2,3] is an adaptive algorithm where a simplification of the gradient vector computation is carried out by means of an appropriate modification of the goal function
In order to figure out the dependence of gain in step-size on (a) the frequency, (b) the decimating factor N, and (c) the length of the filter M given by Equation (29), we carried out two experiments described in the upcoming subsubsections
Because we want to deal with periodic signals, we have considered a range of values for the period P from 512 down to 8 samples, corresponding, respectively, to analog frequencies F0 from 15,6 to 1000 Hz
Summary
The least mean-square (LMS) [1,2,3] is an adaptive algorithm where a simplification of the gradient vector computation is carried out by means of an appropriate modification of the goal function. Very different fields of knowledge such as underwater communications [4] or ultrawide bandwidth systems [5] make use of the LMS algorithm to optimize an objective function by iteratively minimizing the error signal. The periodic nature of the reference ( referred to as regressor or input) signal x(n) and the training signal d(n) allows us to use the matricial approach of the LMS algorithm proposed by Parra et al [9], periodic least-mean-square (P-LMS) algorithm, in the following. The referenced paper is based on a previous work [10] where a matrix-based approach is proposed to analyze the stability of adaptive algorithms
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