IEEE Circuits and Systems Magazine - Q2 2019 - 60

The origin of polynomial equiripple approximation can be bound
in the pioneering work of P. L. Chebyshev.
In a strict view, a true filter design consists of two
essential steps. In the first step, the lowest filter order, which meets the filter specification, is obtained
by an exact degree equation. Provided the filter order is acceptable for an anticipated application, the
coefficients of the impulse response are evaluated in
the second step. Equiripple FIR filters are usually designed numerically. There are two main drawbacks in
a numerical filter design. In the first place, an exact degree formula is, by principle, not available. Secondly, a
numerical evaluation of the coefficients of the impulse
response is a numerical fragile task, especially in case
of long filters. The drawbacks of a numerical design
stand in contrast to a closed form design of equiripple
FIR filters based on PERAs. Thus, additional highlights
of a true filter design based on PERAs is the availability of an exact degree equation as well as of formulas
for a deterministic, fast and robust evaluation of the
impulse response of a filter. It is also worth noting that
the true unique art in the PERA is to find the approximating polynomial of a particular filter type as it is
almost impossible to deduce it.
III. Applications of Equiripple FIR Filters
Equiripple FIR filters based on PERAs represent a sort of
a holy grail among FIR filters, especially in a real-time signal filtering. It profoundly profits from the minimum filter
length for a specified filter selectivity. Further, formulas for

a deterministic, fast and robust evaluation of the impulse
response as well as formulas for fast tuning [2] the filter
selectivity are appreciated in adaptive filtering [3], regardless the filter is implemented in software or hardware, e.g.
using digital signal and multi-core processors, field programmable gate arrays, dedicated platforms etc. Narrow
band equiripple FIR filters like notch filters, DC notch filters, narrow band-pass filters and comb filters are applied
in a rich portfolio of areas, e.g. in attenuation of unwanted
signals, typically of power line frequencies in weak signals
[3]-[5], damping oscillations in power systems [6], power
system diagnostics [7], separation of narrow band signals
in direct sampling receivers [8], zero intermediate frequency receivers [9], DTMF applications [10], acoustic echo and
reverberation cancellation [11], sound separation [12], improvement of audibility [13] and speech denoising [14] to
name few of them.
IV. The First Period in the Polynomial
Equiripple Approximation
The first period in the PERA is represented by three
decades in the second half of 19th century. The origin
of the PERA can be seen essentially in the pioneering
work [15] of P. L. Chebyshev (1821-1894) presented in
1854, creator of the Sankt Peterburg School of Number
Theory [16]. Chebyshev was motivated by some tasks
of practical interest for obtaining optimal precision in
the design and production of mechanical components,
specifically in parallel linearly moving mechanisms like
rods in steam engines, so called parallelograms [15]. In
terms of approximation theory, Chebyshev introduced
the PERA of a single constant in a single interval in form
of his famous polynomial which is based on goniometric functions. For illustration, an example of Chebyshev
polynomial of first kind
Tn (w) = cos (n acos (w)) = cos ^n~T h, - 1 # w # 1 (1)

Figure 1. Pafnuty Lvovich chebyshev.

is displayed for n = 17 in Fig. 2. The symbol T stands
in (1) for a normalized sampling period. It is apparent,
that a constant is approximated optimally in the interval w ! 6-1, 1@ by an equiripple polynomial of a specified degree n. Polynomial Tn (w) has n roots and n - 1
extremal values in the interval w ! 6-1, 1@ . Despite

Pavel Zahradnik is a full professor with the Dept. of Telecommunication Engineering, Faculty of Electrical Engineering, Czech Technical University in
Prague, Czech Republic. Miroslav Vlcˇek is a full professor with the Dept. of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical
University in Prague, Czech Republic.

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IEEE Circuits and Systems Magazine - Q2 2019

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