IEEE Circuits and Systems Magazine - Q2 2019 - 64

A common property of all PERAs is that an integer number of ripples
has to be accommodated in the interval [-1, 1].
DC-notch FIR filter (Fig. 9) based on an approximating polynomial
A (w) = 1 -

Tn (mw + m - 1) + 1
Tn (2m - 1) + 1

(7)

where the real value m $ 1 controls the ripple size in
the broad pass-band [33]. All of the above mentioned
PERAs approximate a single constant in one or two intervals within w ! 6-1, 1@ . In the PERA of two constants
in two disjoint intervals, the approximating polynomial
is an antiderivative of a generating polynomial.
In 2009, Zahradnik and Vlcˇek 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

#

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

2
2
l2
l2
AU n c 2w - 1 -2 k m + A -1 U n -1 c 2w - 1 -2 k m dw
1 - kl
1 - kl
(8)

0

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

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

1

Aν (w)

Figure 10. normalized approximating polynomial A o (w) for
n = 20, k l = 0.0392, A = 1.0853 and A -1 = 0.9536 based on (8).

64

Ieee cIrcUITs And sysTeMs MAgAZIne

0

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

1

Figure 11. normalized approximating polynomial A o (w)
based on Z p, q (w, \) for p = 8, q = 15, \ = 0.4, B = 0.0585
and B -1 =-0.0474.

Aν (w)

0

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

1

Figure 9. normalized approximating polynomial A o (w) for
n = 37 and m = 1.0015 based on (7).

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

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

Aν (w)

Aν (w)

A (w) =

where the real valued parameter kl controls the width
of the main lobe of the symmetrical equripple approximation. It is closely related to the elliptic modulus \,
i.e. 1 - \ 2 = (1- kl ) / (1 + kl ) [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 equiripple form is obtained by integration. A common property of all PERAs is that an integer number of ripples
has to be accommodated in the interval w ! 6-1, 1@ .
This fact may limit the practically attainable values of
band edges.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
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 12. normalized approximating polynomial A o (w) of
an equiripple narrow multi band-pass FIr filter. A closed form
solution is not available.

second QUArTer 2019
IEEE Circuits and Systems Magazine - Q2 2019

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