1 Novel low-pass ﬁlter with adjustable parameters of exponential-type forgetting

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Novel low-pass ﬁlter with adjustable parameters

of exponential-type forgetting

Ivo Petr´

aˇ

s, Senior Member, IEEE

Abstract—In this paper, a novel form of Gaussian ﬁlter, the

Mittag-Lefﬂer ﬁlter, is presented. This new ﬁlter uses a Mittag-

Lefﬂer function in the probability density function. Such Mittag-

Lefﬂer distribution is used in the convolution kernel of the

ﬁlter. The ﬁlter has three parameters that may adjust the curve

shape due to the ﬁlter forgetting factor. Illustrative examples

present the main advantages of the proposed ﬁlter as compared

to classical Gaussian ﬁltering techniques. Some implementation

notes, together with the Matlab function, are also presented.

Index Terms—Exponential-type forgetting, Gaussian function,

Gaussian ﬁlter, Mittag-Lefﬂer function, Mittag-Lefﬂer ﬁlter.

I. INTRODUCTION

FILTERING is processing a signal whereby some un-

wanted components or properties are removed from the

signal or some aspects of the signal are suppressed. It often

means removing some frequencies or frequency bands from

the signal. However, we do not have to use ﬁlters exclusively

in the frequency domain, and certain frequency components

can be removed without having to act in the frequency domain.

Filters are widely used, for example, in signal processing in

electronics and telecommunications, in radars, control systems

sensors, as well as image processing, and computer graphics.

Various forms of ﬁlters are used, for instance, the Laplacian

ﬁlter [1], Bayesian ﬁlter [2], Gaussian ﬁlter [3], and so on.

This paper describes a new ﬁlter based on the famous

Gaussian ﬁlter. It is well known that this ﬁlter is often used

in many areas of signal and image processing for smoothing

and noise reduction, e.g., [4], [5], [6], [7]. It is a convolutional

ﬁlter that uses a Gaussian function as a convolution kernel and

mathematically adjusts the input signal by convolution with

a Gaussian function. In other words, a Gaussian ﬁlter is a ﬁlter

whose impulse response is a Gaussian function. Among other

things, these ﬁlters have the important property that they do

not overshoot at the input of the step function and, at the same

time, minimize the rise and fall time. This behavior is closely

related to the Gaussian ﬁlter having the minimum possible

group delay.

The main contributions of this paper are as follows:

•generalization of the Gaussian ﬁlter to the novel form,

based on the Mittag-Lefﬂer distribution function,

•suggestion of the implementation algorithm for the new

Mittag-Lefﬂer ﬁlter with adjustable forgetting parameters.

Manuscript received: October 3, 2022;

This work was supported in part by the Slovak Grant Agency for Science

under grant VEGA 1/0365/19, by the Slovak Research and Development

Agency under contracts No. APVV-14-0892 and No. APVV-18-0526, and

by Army Research Ofﬁce under grant No. W911NF-22-1-0264.

Ivo Petr´

aˇ

s is with the Faculty of BERG, Technical University of Koˇ

sice,

Nˇ

emcovej 3, 042 00, Koˇ

sice, Slovak Republic (e-mail: ivo.petras@tuke.sk).

The structure of this paper is as follows. Section I brieﬂy

describes the introduction to the problem. Section II presents

the essential mathematical tools. The main results are shown in

Section III. The illustrative examples are presented in Section

IV to demonstrate the beneﬁts of the proposed new ﬁlter.

Finally, some concluding remarks are given in Section V.

II. PRELIMINARIES

A. Gaussian function and Gaussian distribution

The Gaussian function, named after Johann Carl Friedrich

Gauss is a function that can be expressed in elemental form

f(x) = ae−(x−b)2

2c2,(1)

for arbitrary real parameters a,b, and c > 0.

Gaussian functions (1) are often used in statistics to repre-

sent the probability density function (PDF) of a normal shifted

distribution (a.k.a. Gauss distribution) for a real-valued random

variable with expected value (or mean) b=µand variance

c2=σ2. The general form of its PDF φ(x)is

φ(x;σ) = 1

σ√2πe−1

2(x−µ

σ)2,(2)

where the variable µ∈Ris the mean (or expectation) of the

distribution, while the positive variable σ∈Ris its standard

deviation. The variance of this Gauss distribution is then σ2.

The simplest case of the normal distribution is known as the

normal unit distribution or the standard normal distribution.

This is a particular case when σ= 1 and µ= 0. It means that

xhas variance, standard deviation of 1, and mean of 0.

Except for the mentioned utilization of the Gaussian func-

tion as PDF for normal distribution, we may use it in signal

processing to deﬁne Gaussian ﬁlters and image processing,

where a two-dimensional Gaussian ﬁlter is used for blurs.

Moreover, the exponential law is the classical approach

to studying the dynamics of systems, but there are many

systems where dynamics obey a faster or slower law than the

exponential law. In that case, the Mittag-Lefﬂer function can

best describe such anomalous dynamics changes [8].

B. Mittag-Lefﬂer function and Mittag-Lefﬂer distribution

The Mittag-Lefﬂer function Eα,β (z), named after Magnus

Gustaf Mittag-Lefﬂer, is a special function that depends on two

parameters, α, and β. It may be expressed by the following

series [9]:

Eα,β (z) =

∞

n=0

Γ(αn +β), α, β > 0, z ∈C,(3)

arXiv:2210.01195v1 [math.DS] 3 Oct 2022

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1Novellow-passlterwithadjustableparametersofexponential-typeforgettingIvoPetr´as,SeniorMember,IEEEAbstractInthispaper,anovelformofGaussianlter,theMittag-Leferlter,ispresented.ThisnewlterusesaMittag-Leferfunctionintheprobabilitydensityfunction.SuchMittag-Leferdistributionisusedintheconvoluti...

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