IEEE Circuits and Systems Magazine - Q2 2019 - 59

Feature

The World of Ripples
Pavel Zahradnik and Miroslav Vlcˇek
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Abstract
Approximation techniques are useful in numerous engineering
and non-engineering tasks. In electrical engineering and in the
filter design in particular, approximation techniques allow to approximate in some sense a desired, usually an ideal, but not feasible frequency response of a filter. Among various approximations, polynomial approximations are very useful. The design of
feasible linear digital filters with finite impulse response is based
on them. There emerge two unique types among polynomial approximations, a maximally flat approximation and an equiripple
approximation. We review here the polynomial equiripple approximation, its highlights, applications, history, evolution and
latest achievements including open questions.

I. Introduction
n the paper "The World of Flatness" [1], selected aspects
of a polynomial maximally flat approximation (PMFA)
were outlined. The PMFA can be seen as a limit case of
a polynomial equiripple approximation (PERA) for the
ripple size approaching to zero. While it is easy to obtain a
PMFA as a limit value of a PERA, it is, in general, impossible
to reverse this process in order to de-limit a maximally flat
approximation for obtaining an equiripple approximation.
Our overview presented here complements the paper [1]
by covering selected aspects of the PERA.

Digital Object Identifier 10.1109/MCAS.2019.2909660
Date of publication: 20 May 2019

second QUArTer 2019

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We can see two essential periods in the history of
the PERA which are separated by an almost complete
standstill lasting longer than one century. Similarly to
the evolution in the PMFA, the initial motivation in the
PERA was not related to frequency filters. On the other
hand, the evolution in the PERA in its second period is
strongly motivated by a robust closed form design of
digital filters. In next sections, we outline the optimality
and advantages of the PERA, useful applications, its history, recent results and open problems.
II. Highlights of a Polynomial Equiripple
Approximation
It is emphasized that "Exactness = Flatness" in the
PMFA [1]. By analogy, we can point out to the fact
that "Optimality = Equal ripples" in the PERA. Importantly, a PERA is optimal in Chebyshev sense, i.e. the
maximum deviation of the approximation of the specified curve, called maximum approximation error, in
interval(s) of interest is minimal. Each non-equiripple
approximation is worse than the PERA in terms of the
maximum deviation. Specifically, in equiripple finite
impulse response (FIR) filters, which are based on
PERAs, the degree of an approximating polynomial,
the filter order and the filter length in terms of the number of coefficients of its impulse response are minimal
for any filter specification.
Ieee cIrcUITs And sysTeMs MAgAZIne

IEEE Circuits and Systems Magazine - Q2 2019

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