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 ap-proximate in some sense a desired, usually an ideal, but not fea-sible frequency response of a filter. Among various approxima-tions, 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 ap-proximations, a maximally flat approximation and an equiripple approximation. We review here the polynomial equiripple ap-proximation, its highlights, applications, history, evolution and latest achievements including open questions.

I. Introduction

In 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.

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 his-tory, 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. Impor-tantly, a PERA is optimal in Chebyshev sense, i.e. the maximum deviation of the approximation of the speci-fied 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 num-ber of coefficients of its impulse response are minimal for any filter specification.

Abstract

The World of RipplesPavel Zahradnik and Miroslav Vlcek

Digital Object Identifier 10.1109/MCAS.2019.2909660

Date of publication: 20 May 2019

©IsTockPhoTo.coM/BAhAdIrTAnrIoVer

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Feature

60 Ieee cIrcUITs And sysTeMs MAgAZIne second QUArTer 2019

In a strict view, a true filter design consists of two essential steps. In the first step, the lowest filter or-der, which meets the filter specification, is obtained by an exact degree equation. Provided the filter or-der is acceptable for an anticipated application, the coefficients of the impulse response are evaluated in the second step. Equiripple FIR filters are usually de-signed numerically. There are two main drawbacks in a numerical filter design. In the first place, an exact de-gree 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 availabil-ity 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 approxi-mating polynomial of a particular filter type as it is almost impossible to deduce it.

III. Applications of Equiripple FIR FiltersEquiripple FIR filters based on PERAs represent a sort of a holy grail among FIR filters, especially in a real-time sig-nal 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], regard-less the filter is implemented in software or hardware, e.g. using digital signal and multi-core processors, field pro-grammable gate arrays, dedicated platforms etc. Narrow band equiripple FIR filters like notch filters, DC notch fil-ters, 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 frequen-cy receivers [9], DTMF applications [10], acoustic echo and reverberation cancellation [11], sound separation [12], im-provement 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 goniomet-ric functions. For illustration, an example of Chebyshev polynomial of first kind

( ) ( ( )) ,cos cosT w n w n T w1 1acosn # #~= = -^ h (1)

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 inter-val ,w 1 1! -6 @ by an equiripple polynomial of a speci-fied degree .n Polynomial ( )T wn has n roots and n 1- extremal values in the interval , .w 1 1! -6 @ 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 Vlcek is a full professor with the Dept. of Applied Mathematics, Faculty of Transportation Sciences, Czech Technical University in Prague, Czech Republic.

Figure 1. Pafnuty Lvovich chebyshev.

The origin of polynomial equiripple approximation can be bound in the pioneering work of P. L. Chebyshev.

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Chebyshev polynomials of first kind represent no ap-proximating polynomial of a useful filter because of no band selectivity, in a compound form, they are used in several PERAs which are discussed later. Further, a set of Chebyshev polynomials ( ), , , ...T w m n0 1m = forms a base, and, consequently, any polynomial ( )P wn of the degree n can be expressed besides its natural power series in w also in a form of an expansion into Cheby-shev polynomials

( ) ( ) ( ) ( ) .P w b m w a m T wnm

nm

m

n

m0 0

= == =

/ / (2)

An advantage of the Chebyshev representation of poly-nomials (2) is a dramatically smaller dynamic range of the coefficients ( )a m compared to the coefficients

( ),mb especially for polynomials of the degree greater than .n 40. It is also worth noting that a counter-part to ( ),T wn namely the Chebyshev polynomial of second kind

( )( )

( ) ( ),

sin arccossin arccos

U ww

n ww

11 1n # #=

+-^ h

6 @ (3)

represents no equiripple approximation (Fig. 3). Inter-estingly, the function ( )w U w1 n

2- is equirippled, howev-er, it is no longer a polynomial. A modification of ( ),U wn namely the antiderivate of its compound form is useful in the PERA of two constants in two disjoint intervals which is outlined in Sec. V.

Next milestone in the first period in the PER A emerged in 1877. Another celebrity in the approxima-tion theory, namely a student of P. L. Chebyshev and also a representative of the Sankt Peterburg School of Number Theory, E. I. Zolotarev (1847–1878), present-ed in [17] a generalization of the Chebyshev polyno-mial of first kind which represents a PERA of a single constant in two disjoint intervals. Unlike Chebyshev, Zolotarev’s interest were equiripple polynomial and rational approximations by itself. Zolotarev stated four approximation problems and the above men-tioned approximation in form of an equiripple polyno-mial ( , )Z w,p q \ is a solution of his first approximation problem, for details see pp. 1–26 in [17]. Polynomial

( , )Z w,p q \ is sketched in Fig. 5. By comparison of Fig. 2 bottom and Fig. 5, we can see that the third lobe from the left in Fig. 2 is pulled up in Fig. 5 forming a main lobe. This main lobe causes the selectivity of a related filter. Zolotarev expressed the solution of his first ap-proximation problem in terms of the Jacobi elliptic Eta function ( , )H z \ [18]

Figure 2. chebyshev polynomial ( )T w17 of an argument w (top) and of a transformed argument w (bottom).

–1–0.5

00.5

1

T17

(w)

–1–0.5

00.5

1

T17

(w)

acos(w)/π = ωT/π(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

w(a)

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

–20

–10

0

10

20

w

U17

(w

)

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

Figure 3. chebyshev polynomial ( ).U w17

Figure 4. egor Ivanovich Zolotarev.

Unlike Chebyshev, E. I. Zolotarev’s interest were equiripple polynomial and rational approximations by itself.

62 Ieee cIrcUITs And sysTeMs MAgAZIne second QUArTer 2019

( , )

( )

( ),

( ),

( ),

( ),

( )( ),

( ),

( ),

( ),.

Z w

H u np

H u np

H u np

H u np

TH u n

p

H u np

H u np

H u np

21

1 21

K

K

K

K

K

K

K

K

,p q

pn n

pn

\

\ \

\ \

\ \

\ \

\ \

\ \

\ \

\ \

=

=-

-

++

-

+

= -

-

++

+

-

-J

L

KKKK

J

L

KKKK

J

L

KKKK

``

``

``

``

N

P

OOOO

N

P

OOOO

N

P

OOOO

jj

jj

jj

jj

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

(4)

The real variable w in (4) is related to the standard complex variable z known in the z -transform. Specifi-cally, the variable w results from the modification of the Zhukovsky transform [19] and from its restriction to the unity circle, namely

.cosw z z T21 1

z e j T~= + =

= ~` j (5)

Integers p and q in (4) are related to the number of roots and ripples outside the main lobe, i.e. they control the position of the main lobe and thus the position of the narrow band in a related equiripple FIR filter. The real valued elliptic modulus \ controls the width and height of the main lobe, i.e. it controls the selectivity of the filter [20]. In fact, the elliptic modulus \ is a driving parameter of elliptic functions. Especially note that despite tran-scendental inner functions inside Chebyshev polynomi-al in (4), the outcome is still a polynomial of an unaltered degree .n Let us also note that a Zolotarev polynomial (4) reduces to a Chebyshev polynomial of first kind for elliptic modulus ,0\= i.e. ( , ) ( ) ( ).Z w T w0 1,p q

pp q= - + Un-

fortunately, Zolotarev provided no formulas for evaluat-ing coefficients of his polynomial ( , ).Z w,p q \ Useful for-mulas were introduced more than 120 years later [22]. In digital filters, Zolotarev polynomial forms the basis for a closed form design of narrow-band FIR filters [20],

[21], [23], [24]. Last but not least, it is also worth not-ing that the phenomenal publications [15], [17] in the first period and also the fundamental work [18] present besides text and formulas no graphs of functions. We can only guess if these pioneers of science ever saw the graphs of their functions.

V. The Second Period in the Polynomial Equiripple Approximation

Between 1877–1986, there was almost no noticeable ac-tivity available in the PERA. The paper [25] from 1928 of the creator of the Kharkov School of Mathematics, N. I. Achiezer, represents a review of Zolotarev results [17]. In 1934, E. Ja. Remes presented [26] a method, later named Remes exchange algorithm, which seeks for the coeffi-cients of equiripple polynomials numerically in an itera-tive way. It became later a basis of numerous computer algorithms in a numerical design of equiripple FIR filters including several Matlab functions, e.g. of firpm and firgr. However, the Remes approach and its applications repre-sents in fact no contribution in the true sense of the PERA as it provides no closed form solution of any approxima-tion problem in terms of an approximating equiripple polynomial. We attribute the long standstill between both periods in the PERA partly to the lack of ideas and partly to the complexity in solving equiripple approxima-tion problems, especially because of a non-trivial math-ematics involved. A renaissance and the second period in the PERA appeared with the advent of a wide availability of computing structures and their proliferation in digital filtering. An immaturity and gaps in the PERA as well as the desire for a robust design of optimal FIR filters spar-kled a new wave of activity in the PERA. In 1986, Chen and Parks presented [23] a design of narrow-band FIR fil-ters based on Zolotarev polynomial. Despite claiming to be an analytic design, it is not a true closed form design. It suffers from the lack of robust formulas for evaluating impulse response as well as from the lack of an explicit degree equation. A real enabler in the application of Zo-lotarev polynomials represents the work of Vlcek and Unbehauen [22] in 1999. They presented among others a simple, deterministic, fast and extremely robust closed form evaluation of coefficients of Zolotarev polynomial and of a corresponding impulse response as well as a corresponding degree equation. Further papers in the closed form design of equiripple narrow-band FIR filters including equiripple filter banks [27] and a cascade filter representation [28] are based on these results. Besides

0

2

4

12

10

8

6

Z6,

11 (w

, 0.7

5)

acos(w)/π 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5. Zolotarev polynomial ( , . )Z w 0 75,6 11 based on (4).

E. Ja. Remes presented a method that later became a basis of numerous computer algorithms for the design of equiripple FIR filters.

second QUArTer 2019 Ieee cIrcUITs And sysTeMs MAgAZIne 63

filter design, Zolotarev equiripple polynomials are useful also in other fields, e.g. in radar technology [29] and in digital processing of non-stationary signals [30].

In 2005 [31] and 2009 [32], Zahradnik and Vlcek in-troduced and perfected the PERA of a single constant in several regularly displaced intervals. It forms a basis for a closed form design of equiripple comb FIR filters (Fig. 6). Although a non-closed form design of equiripple comb FIR filters based on interleaving zero values in an impulse response of a prototype filter was known previ-ously, the approximating polynomial in terms of a com-pound Chebyshev polynomial

( )( )

,A w TT w

1s

r2

2

m

m=

-

-; E (6)

formulas for evaluating impulse response as well as an ex-act degree equation were presented for the first time in [31], [32]. The integers ,r s in (6) represent the numbers of particular ripples of the approximating polynomial and m controls the ripple size [32]. The relation of the nor-malized generating polynomial ( )A wo and of the mag-nitude frequency response ( )H e j T~ is apparent in Fig. 6. The normalized generating polynomial represents in fact

a mapping of the magnitude frequency response onto a half-cylinder in the z -plane which is illustrated in Fig. 7. An advantageous normalizing the approximating polyno-mial ( )A w results in ( )A wo with values within the range

( ) .A w0 1# #o This normalizing avoids the unwanted dis-continuities in the phase frequency response of a related filter which is demonstrated in Fig. 8 on the left.

In 2007, Zahradnik and Vlcek introduced [33] the PERA useful in a robust closed form design of an equiripple

00.10.20.30.40.5

1

0.60.70.80.9

Aν

(w)

00.10.20.30.40.5

1

0.60.70.80.9

H (e

jωT)

w–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1

ωT/π

(a)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6. normalized approximating polynomial ( )A wo for ,s 4= r 8= and .0 15m = based on (6) and a corresponding

magnitude frequency response.

01

0.10.20.3

0.8

0.40.50.6

0.6

0.70.80.9

0.4

1

0.20Im {z }

Re {z }ωT = π

ωT = 0

–1–0.8

–0.6–0.4–0.2

0.20.4

0.60.8

1

0

H (e

jωT)

Figure 7. ( )H ej T~ corresponding to ( )A wo from Fig. 6. A restriction to the unity circle z e j T= ~ in the complex z-half-plane is displayed.

00.20.40.60.8

1

–4

–2

0

2

4

00.20.40.60.8

1

–100–80–60–40–20

0

acos(w)/π

Aν

(w)

ϕ (e

jωT)

(rad

)

Aν

± (w

)ϕ

(ejω

T)

(rad

)

0 0.2 0.4 0.6 0.8 1acos(w)/π

0 0.2 0.4 0.6 0.8 1

ωT/π0 0.2 0.4 0.6 0.8 1

ωT/π0 0.2 0.4 0.6 0.8 1

Figure 8. Two types of normalizing ( )A w including corre-sponding phase frequency responses ( )e j T{ ~ . An advan-tageous normalizing ( ),A wo and a less useful normalizing

( )A w!o which is used e.g. in Matlab functions firpm and firgr.

Besides filter design, Zolotarev equiripple polynomials are useful also in other fields, e.g. in radar technology and in digital processing

of non-stationary signals.

64 Ieee cIrcUITs And sysTeMs MAgAZIne second QUArTer 2019

DC-notch FIR filter (Fig. 9) based on an approximat-ing polynomial

( ) ( )( )

A w TT w

1 2 1 11 1

n

n

m

m m= -

- +

+ - + (7)

where the real value 1$m controls the ripple size in the broad pass-band [33]. All of the above mentioned PERAs approximate a single constant in one or two in-tervals within , .w 1 1! -6 @ In the PERA of two constants in two disjoint intervals, the approximating polynomial is an antiderivative of a generating polynomial.

In 2009, Zahradnik and Vlcek presented [35] the PERA of two constants in two disjoint intervals of equal width. The approximating polynomial is based on the antiderivative of compound Chebyshev polynomials of second kind with quadratic inner argument

( )A w AUk

w k A Uk

w k w1

2 11

2 1 dn n2

2 2

1 1 2

2 2=

-- - +

-- -

- -ll

llc cm m#

(8)

where the real valued parameter kl controls the width of the main lobe of the symmetrical equripple approxi-mation. It is closely related to the elliptic modulus ,\ i.e. ( )/( )k k1 1 12\- = - +l l [35]. The approximating polynomial (8) represents the basis for a closed form design of equiripple half-band (HB) FIR filters [36], for illustration see Fig. 10. Note that the two individual polynomial components as well as their sum inside the integral (8) have a non-equiripple shape. The equirip-ple form is obtained by integration. A common prop-erty of all PERAs is that an integer number of ripples has to be accommodated in the interval , .w 1 1! -6 @ This fact may limit the practically attainable values of band edges.

VI. Recent DevelopmentsRecently, Zahradnik discovered the PERA of a very basic filter type, namely of an equiripple low-pass FIR filter based on the Zahradnik generating polynomi-al ( , )wZ ,p q \ [38]. In terms of approximation theory,

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

1

0.60.70.80.9

Aν

(w)

Figure 11. normalized approximating polynomial ( )A wo based on ( , )wZ ,p q \ for ,p 8= ,q 15= . ,0 4\ = .B 0 0585= and . .B 0 04741 =--

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

1

0.60.70.80.9

Aν

(w)

Figure 12. normalized approximating polynomial ( )A wo of an equiripple narrow multi band-pass FIr filter. A closed form solution is not available.

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

1

0.60.70.80.9

Aν(w

)

Figure 9. normalized approximating polynomial ( )A wo for n 37= and .1 0015m = based on (7).

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

1

0.60.70.80.9

Aν

(w)

Figure 10. normalized approximating polynomial ( )A wo for ,n 20= . ,k 0 0392=l .A 1 0853= and .A 0 95361 =- based on (8).

A common property of all PERAs is that an integer number of ripples has to be accommodated in the interval [–1, 1].

second QUArTer 2019 Ieee cIrcUITs And sysTeMs MAgAZIne 65

it represents a PERA of two distinct constants in two disjoint intervals within , ,w 1 1! -6 @ see Fig. 11. It also represents a step function in an equiripple sense. Note the absence of the Gibbs overshots. The Zahradnik ap-proximating polynomial ( , )wZ ,p q \ has the form

( , ) ( , )( , )

( )( )( ) ( , )

( )( )( ) ( , )

.

w w ww

Z w

B p q w w wZ w

B p q w w wZ w

w

2 11

21

1

d

dd

dd

d

Z Z, ,,

,

,

p q p qp q

m

pp q

m

pp q

22 1 2 1 2

1

11 1

1

,

,

p q

p q

1 1

\ \\

\

\

= =-

+

++ + -

-

++ -

-

+ +

++ +

-

+ +

c ^ h m=

G

# #

(9)

Similarly to (8), partial polynomials as well as their sum inside the integral (9) have a non-equiripple shape. Approximat-ing polynomial (9) reduces to its symmetric half-band form (8) for n p q2 2= = and ( )/( ).k 1 1 1 1 22 \\= - - + -l

VII. Open Problems in Polynomial Equiripple Approximation

Despite efforts in both periods of the PERA starting 1854, many PERA problems remain still unsolved. These include e.g. a narrow multi band-pass PERA (Fig. 12), a general band-pass PERA (Fig. 13), a general multi band-pass PERA (Fig. 14), their modifications with distinct values of ripples in particular bands, further two- (Fig. 15) and multi-dimensional PERAs of various filter types etc. Similarly to the already solved PERAs, it is a priori not known if a closed form solutions of these PERA problems do exist and if a potential solu-tion will be based on elliptic functions. However, the state-of-the-art and open problems demonstrate that the design of optimal equiripple FIR filters is so far not a closed chapter in the theory of digital filters as it may be wrongly seen. On contrary, PERAs repre-sent still a vivid, but slowly advancing scientific area. Based on our efforts so far, we believe that the un-solved PERAs, provided their solutions do exist, may be found empirically based on an prospective accu-

mulated experience rather than by a straightforward derivation. That is why we still expect a laborious and time consuming research.

Pavel Zahradnik was born in Melník, former Czechoslovakia. He received the M.Sc. and Ph.D. degrees in tele-communication engineering from the Czech Technical University in Prague in 1986 and 1991, respectively. In

1993–1994 he was a fellow of Swiss government at the Paul Scherrer Institut, Villigen, Switzerland, doing re-search in microwave tomography for a real-time lo-calization of tumors. In 1996–1997 he started his pio-neering work in polynomial equiripple approximations during his three Alexander-von-Humboldt fellow-ships at the Friedrich-Alexander-Universität, Erlan-gen-Nürnberg, Germany. Prof. Zahradnik has been working for over 30 years in the digital signal, image and video processing, in algorithms and their imple-mentation. In 2008 he became a full professor. His interests include also electronics, specialized pro-cessors, FPGA technology and optics. Prof. Zahrad-nik is a real guru in a closed-form design of equirip-ple FIR filters.

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

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0.60.70.80.9

Aν

(w)

Figure 13. normalized approximating polynomial ( )A wo of a general equiripple band-pass FIr filter. A closed form solu-tion is not available.

acos(w)/π0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

00.10.20.30.40.5

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0.60.70.80.9

Aν

(w)

Figure 14. normalized approximating polynomial ( )A wo of a general equiripple multi band-pass FIr filter. A closed form solution is not available.

01

0.2

0.8

0.40.6

0.6

0.8

0.4

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0.2 0.10.20.30.40.50.60.7

0.80.91

0 0

Aν(w

1,w

2)

acos(w1)/πacos(w2)/π

Figure 15. normalized approximating polynomial ( , )A w w1 2o of an equiripple two-dimensional narrow band-pass FIr filter. A closed form solution is not available.

66 Ieee cIrcUITs And sysTeMs MAgAZIne second QUArTer 2019

Miroslav Vlcek was born in Prague, the Czech Republic. He received the gradu-ate degree in theoretical physics from Charles University, Prague, in 1974, and the Ph.D. degree in communication engi-neering and the D.Sc. degree from the

Czech Technical University (CTU), Prague, in 1979 and 1994, respectively. From 1974 to 1993, he was with the De-partment of Circuit Theory, Faculty of Electrical Engineer-ing, CTU. Since 1995, he has been the Head of the Depart-ment of Applied Mathematics, Faculty of Transportation Sciences, CTU. He was the Alexander-von-Humboldt Fel-low at the Friedrich-Alexander-Universität Erlangen-Nürn-berg, Germany, in 1988, 1997, and 1999. In 2000 he became a full professor. He currently teaches courses in system theory and digital filter design. His scientific interests in-clude filter design and digital signal processing, and theo-ry of approximation and higher transcendental functions.

References[1] S. Samadi and A. Nishihara, “The world of flatness,” IEEE Circuits Syst. Mag., pp. 38–44, Third quarter 2007.[2] P. Zahradnik and M. Vlcek, “An analytical procedure for critical fre-quency tuning of FIR filters,” IEEE Trans. Circuits Syst. II, vol. 53, no. 1, pp. 72–76, Jan. 2006.[3] P. Zahradnik and M. Vlcek, “Notch filtering suitable for real time re-moval of power line interference,” Radioengineering, vol. 22, no. 1, pp. 186–193, 2013.[4] K. Aboutabikh, I. Haidar, and N. Aboukerdah, “Design and implemen-tation of a digital FIR notch filter for the ECG signals using FPGA,” Int. J. Adv. Res. Comput. Commun. Eng., vol. 5, no. 1, pp. 452–456, Jan. 2016.[5] M. S. Chavan, R. Agarvala, and M. D. Uplane, “Design and implemen-tation of digital FIR equiripple notch filter on ECG signal for removal of power line interference,” WSEAS Trans. Signal Process., vol. 4, no. 4, pp. 221–230, Apr. 2008.[6] K. Mandadi and B. K. Kumar, “Identification of inter-area oscillations using Zolotarev polynomial based filter bank with eigen realization al-gorithm,” IEEE Trans. Power Syst., vol. 31, no. 6, pp. 4650–4659, Nov. 2016.[7] V. Climente-Alarcon, J. A. Antonino-Daviu, M. Riera-Guasp, and M. Vlcek, “Induction motor diagnosis by advanced notch FIR filters and the wigner-ville distribution,” IEEE Trans. Ind. Electron., vol. 61, no. 8, pp. 4217–4227, Aug. 2014.[8] P. Zahradnik, B. Šimák, M. Vlcek, and T. Šadlák, “Very long band-pass FIR filters on the eight-core DSP TMS320C6678,” in Proc. 5th Euro-pean DSP Education and Research Conf. EDERC2012, Amsterdam, Neth-erlands, Sept. 2012, pp. 67–70.[9] P. Zahradnik, M. Vlcek, and B. Šimák, “DC-notch FIR filters for zero-IF receivers,” in Proc. 6th Int. Conf. Networking ICN’07, Sainte-Luce, Marti-nique, France, Paper No. 069, Apr. 2007.[10] P. Zahradnik, B. Šimák, and M. Vlcek, “Optimal FIR filters for DTMF applications,” in Proc. 12th Int. Conf. Networks ICN, Seville, Spain, Jan. 2013, pp. 163–168.[11] J. Benesty, D. R. Morgan, J. L. Hall, and M. M. Sondhi, “Stereophonic acoustic echo cancellation using nonlinear transformations and comb filtering,” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, Seattle, WA, USA, May 1998, pp. 3673–3676.[12] M. Gainza, R. Lawlor, and E. Coyle, “Harmonic sound source sepa-ration using FIR comb filters,” in Proc. 117th Audio Engineering Society AES Convention, San Francisco, CA, USA, Oct. 2004, pp. 1–7.[13] J. J. Chopade, N. P. Futane, and V. N. Shah, “FPGA implementation of comb filter for audibility enhancement of hearing impaired,” ICTACT J. Microelectron., vol. 3, no. 03, pp. 441–445, Oct. 2017.[14] M. A. Yakubu, N. C. Maddage, and P. K. Atrey, “Encrypted domain cloud-based speech noise reduction with comb filter,” in Proc. IEEE Int. Conf. Multimedia and Expo Workshops ICMEW, Seattle, WA, USA, July 2016, pp. 1–6.

[15] P. L. Chebyshev, “Théorie des mécanismes connus sous le nom de parallélogrammes,” Mém. Acad. Sci. Pétersb., vol. 7, pp. 539–568, 1854. in French. (Theory of mechanisms known as parallelograms).[16] B. N. Delone, The St. Petersburg school of number theory, 2005 (Transl.: R. Burns, Americal Mathematical Society).[17] E. I. Zolotarev, “Prilozhenie ellipticheskikh funkciy k voprosam o funkciyakh, naimenee i naibolee otklonyayushchikhsya ot nulya,” Za-piski S.-Peterburgskoy Akademii Nauk, vol. XXX, no. 5, pp. 1–59, 1877. in Russian. (Applications of elliptic functions to problems of functions deviating least and most from zero).[18] C. G. I. Iacobi, Fundamenta Nova Theoriae Functionum Ellipti-carum, Regiomonti Sumtibus Fratrum Borntræger, 1829, in Latin. (Fun-damental new theory of elliptic functions).[19] N. E. Zhukovsky, Über die Konturen der Tragflächen der Drachen-flieger, Zeitschrift für Flugtechnik und Motorluftschiffahrt, vol. 1, pp. 281–284, 1910; and vol. 3, pp. 81–86, 1912, in German. (About the contours of the wings of the hang gliders).[20] P. Zahradnik and M. Vlcek, “Fast analytical design algorithms for FIR notch filters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 3, pp. 608–623, Mar. 2004.[21] P. Zahradnik and M. Vlcek, “A computer program for designing notch FIR linear phase digital filters,” Int. Rev. Comput. Softw., vol. 7, no. 2, pp. 505–510, 2012.[22] M. Vlcek and R. Unbehauen, “Zolotarev polynomials and optimal FIR filters,” IEEE Trans. Signal Process., vol. 47, no. 3, pp. 717–730, Mar. 1999.[23] X. Chen and T. Parks, “Analytic design of optimal FIR narrow-band filters using Zolotarev polynomials,” IEEE Trans. Circuits Syst., vol. 33, no. 11, pp. 1065–1071, Nov. 1986.[24] P. Zahradnik, M. Šusta, M. Vlcek, and B. Šimák, “Degree of equi-ripple narrow bandpass FIR filter,” IEEE Trans. Circuits Syst. II, vol. 62, no. 8, pp. 771–775, Aug. 2015.[25] N. I. Achieser, “Über einige Funktionen, die in gegebenen Interval-len am wenigsten von Null abweichen,” Bull. Soc. Phys. Math. Kazan, vol. 3, pp. 1–69, 1928. in German. (About some functions which deviate least from zero in specified intervals).[26] E. J. Remes, “Sur la détermination des polynômes d’approximation de degré donnée,” Commun. Soc. Math. Kharkov, Ser. 4, vol. 10, pp. 41–63, 1934. in French. (On the determination of the approximation poly-nomials of given degree).[27] P. Zahradnik and M. Vlcek, “Perfect decomposition narrow-band FIR filter banks,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 59, no. 11, pp. 805–809, Aug. 2015.[28] P. Zahradnik, M. Šusta, B. Šimák, and M. Vlcek, “Cascade structure of narrow equiripple bandpass FIR filters,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 64, no. 4, pp. 407–411, Apr. 2017.[29] D. A. McNamara, “Direct synthesis of optimum difference patterns for discrete linear arrays using Zolotarev distributions,” IEE Microw. An-tennas Propag. Proc., vol. 140, no. 6, pp. 495–500, 1993.[30] J. Kubak, M. Vlcek, and P. Sovka, “Evaluation of computing sym-metrical Zolotarev polynomials of the first kind,” Radioengineering, vol. 26, no. 2, pp. 903–913, 2017.[31] P. Zahradnik and M. Vlcek, “Analytical design method for optimal equiripple comb FIR filters,” IEEE Trans. Circuits Syst. II, vol. 52, no. 2, pp. 112–115, Feb. 2005.[32] P. Zahradnik, M. Vlcek, and R. Unbehauen, “Design of optimal comb FIR filters-speed and robustness,” IEEE Signal Process. Lett., vol. 16, no. 6, pp. 465–468, June 2009.[33] P. Zahradnik and M. Vlcek, “Note on the design of an equiripple DC-notch FIR filter,” IEEE Trans. Circuits Syst. II, vol. 54, no. 2, pp. 196–199, Feb. 2007.[34] P. Zahradnik, M. Vlcek, and R. Unbehauen, “Almost equiripple FIR half-band filters,” IEEE Trans. Circuits Syst. I, vol. CAS-46, pp. 744–748, June 1999.[35] P. Zahradnik and M. Vlcek, “Equiripple approximation of half-band FIR filters,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 56, no. 12, pp. 941–945, 2009.[36] P. Zahradnik, Vlcek, B. Šimák, and M. Kopp, “Design of half-band FIR filters for signal compression,” Int. J. Adv. Telecommun., vol. 4, no. 3&4, pp. 240–248, 2011.[37] M. Vlcek and P. Zahradnik, “Almost equiripple low-pass FIR filters,” Circuits Syst. Signal Process., vol. 32, no. 2, pp. 743–757, 2013.[38] P. Zahradnik, “Approximation of equripple low-pass FIR filters,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 65, no. 4, pp. 526–530, Apr. 2018.