IEEE Circuits and Systems Magazine - Q2 2019 - 62

E. Ja. Remes presented a method that later became a basis of numerous
computer algorithms for the design of equiripple FIR filters.
Z p, q (w, \) =
RJ
Nn J
N-nV
p
p
S
W
(-1) p SK H ` u + n K (\), \ j O K H ` u + n K (\), \ j O W
K
O +K
O
=
2 SK
p
p
W
S H ` u - n K (\), \ j O K H ` u - n K (\), \ j O W
L
L
P
P
T
X
VN
J R
p
p
S
W
`
`
j
j
H
u
K
H
u
K
+
(
\
),
\
(
\
),
\
K
O
n
n
WO . (4)
= (-1) p Tn K 1 S
+
2
p
p
S
W
K S H ` u - K (\), \ j H ` u + K (\), \ j WO
n
n
L T
XP
The real variable w in (4) is related to the standard
complex variable z known in the z -transform. Specifically, the variable w results from the modification of the
Zhukovsky transform [19] and from its restriction to the
unity circle, namely
w = 1 `z + 1 j
= cos ~T.
2
z z = e j~T

(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 transcendental inner functions inside Chebyshev polynomial 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 p, q (w, 0) = (-1) p Tp +q (w). Unfortunately, Zolotarev provided no formulas for evaluating coefficients of his polynomial Z p, q (w, \). Useful formulas 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],

Z6,11 (w, 0.75)

12
10
8
6
4
2
0
0

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

1

Figure 5. Zolotarev polynomial Z 6, 11 (w, 0.75) based on (4).

62

Ieee cIrcUITs And sysTeMs MAgAZIne

[21], [23], [24]. Last but not least, it is also worth noting 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 activity 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 coefficients of equiripple polynomials numerically in an iterative 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 represents in fact no contribution in the true sense of the PERA
as it provides no closed form solution of any approximation 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 approximation problems, especially because of a non-trivial mathematics 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 sparkled a new wave of activity in the PERA. In 1986, Chen
and Parks presented [23] a design of narrow-band FIR filters 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 Zolotarev polynomials represents the work of Vlcˇek 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
second QUArTer 2019
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