A METHOD FOR ESTIMATING LOCAL EXTREMA OF DIGITAL SIGNALS BASED ON INTERPOLATION ANALOGS OF THE FEJÉR OPERATOR

Abstract

In a number of problems, it becomes necessary to find local extrema of a function that describes a certain process or phenomenon over a specific interval of its argument. This task becomes particularly relevant in the context of signal processing. However, as is often the case in signal processing, the analyzed signal may be presented either as a sequence of discrete samples or as a function that is too complex for analytical determination of its local extrema, which typically complicates solving the problem.

An overview of existing optimization and signal processing methods reveals that one of the most common approaches to solving this problem is to tabulate the function that represents the signal and analyze the resulting sequence of samples. If the signal is already presented in digital form, the process is usually limited to the second step. However, this method is unreliable due to its strong dependence on the sampling density and the number of samples. For this reason, signal approximation using Lagrange interpolation polynomials is sometimes suggested. Nevertheless, this approach also has limitations, as interpolation polynomials such as those of Lagrange type possess certain mathematical properties that may lead to the appearance of so-called fictitious extrema, potentially resulting in inaccurate conclusions.

As an alternative to classical interpolation polynomials in such cases, approaches based on Fourier analysis are sometimes proposed. One of the most well-studied tools in the context of signal approximation is the class of interpolation analogs of operators generated by linear summation methods of Fourier series. As shown by previous research, some of these interpolation polynomials allow for high-accuracy signal approximation. However, their use in locating the local extrema of functions describing signals has received relatively little attention. Therefore, the aim of this work is to investigate this aspect using one of the oldest and most well-known interpolation analogs of operators generated by summation of Fourier series — namely, the interpolation analogs of the Fejér operator.