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Content uploaded by Martin Holters
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Content may be subject to copyright.
PARAMETRIC HIGHERORDER SHELVING FILTERS
Martin Holters and Udo Z
¨
olzer
Department of Signal Processing and Communications, HelmutSchmidtUniversity
Holstenhofweg 85, 22043 Hamburg, Germany
phone: + (49) 40 65413468, fax: + (49) 40 65412822, email: {martin.holters,udo.zoelzer}@hsuhh.de
web: www.hsuhh.de/ant
ABSTRACT
The main characteristic of shelving ﬁlters, as commonly used
in audio equalization, is to amplify or attenuate a certain fre
quency band by a given gain. For parametric equalizers, a
ﬁlter structure is desirable that allows independent adjust
ment of the width and center frequency of the band, and the
gain. In this paper, we present a design for arbitraryorder
shelving ﬁlters and a suitable parametric digital ﬁlter struc
ture. A lowshelving ﬁlter design based on Butterworth ﬁl
ters is decomposed such that the gain can be easily adjusted.
Transformation to the digital domain is performed, keeping
gain and denormalized cutoff frequency independently con
trollable. Finally, we obtain band and highshelving ﬁlters
using a simple manipulation, providing the desired paramet
ric ﬁlter structure.
1. INTRODUCTION
For audio equalization, recursive shelving ﬁlters are com
monly employed. Their function is to amplify or attenuate
a speciﬁc frequency band, while leaving signal components
outside that band mainly unchanged. For equalizers which
can be tuned during operation, special parametric ﬁlter struc
tures are desirable which allow for easy and independent con
trol of the respective band’s width and center frequency and
the gain in that band.
For ﬁrstorder low and highshelving ﬁlters and second
order bandshelving ﬁlters, such designs based on an allpass
decompositions were presented in [1, 2]. In [3] a decom
position of secondorder low and highshelving ﬁlters and
fourthorder bandshelving ﬁlters was presented that allows
the gain to be adjusted without recomputation of the other
ﬁlter coefﬁcients, but where the band limits cannot be easily
adjusted.
In the following, we will ﬁrst derive arbitrary order low
shelving ﬁlters in the continuoustime domain based on a
Butterworth design. These designs will then be transformed
to the discretetime domain and a parametric realization will
be presented. Finally, by lowpass/highpass and lowpass/
bandpass transformations, highshelving and bandshelving
ﬁlters will be obtained.
2. CONTINUOUSTIME LOWSHELVING FILTERS
In [4] the transfer function of frequencynormalized ﬁrst and
secondorder lowshelving ﬁlters with gain g are given by
H
LS,1
(s) =
s + g
s + 1
(1)
and
H
LS,2
(s) =
s
2
+
√
2gs + g
s
2
+
√
2s + 1
, (2)
splane
σ
j
ω
−1
−g
(a) M = 1.
splane
σ
j
ω
1
√
g
(b) M = 2.
splane
σ
j
ω
1
6
√
g
(c) M = 6.
Figure 1: Pole/zero locations in the splane of lowshelving
ﬁlters with g = 2 for different ﬁlter orders M.
respectively. The pole and zero locations of these ﬁlters are
shown in Fig. 1 (a) and (b). These can be generalized to
ﬁlters of order M as
H
LS,M
(s) =
M
∏
m=1
s +
M
√
ge
j
α
m
s + e
j
α
m
,
α
m
=
1
2
−
2m −1
2M
π
(3)
of which (1) and (2) are special cases. Fig. 1 (c) gives an
example of the resulting pole and zero locations for M = 6.
2.1 Filter properties
The magnitude response of the ﬁlters according to (3) can be
found to be
H
LS,M
( j
ω
)
2
=
ω
2M
+ g
2
ω
2M
+ 1
,
of which examples are depicted in Fig. 2. As can be easily
seen,
H
LS,M
(0)
= g and
H
LS,M
(∞)
= 1, with a transitional
region around
ω
= 1. As desired, the slope in the transitional
region increases with the ﬁlter order M. At the normalized
cutoff frequency
ω
c
= 1, the magnitude response is
H
LS,M
( j
ω
c
)
2
=
g
2
+ 1
2
,
14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 48, 2006, copyright by EURASIP
ω
0.1
1
10 100
H
LS,M
( j
ω
)
2
0 dB
10 dB
20 dB
M = 1
M = 2
M = 6
Figure 2: Magnitude responses for continuoustime low
shelving ﬁlters of different orders M with gain g = 10
(20 dB).
ω
0.1
1
10 100
H
LS,6
( j
ω
)
2
−20 dB
−10 dB
0 dB
10 dB
20 dB
Figure 3: Magnitude responses for continuoustime low
shelving ﬁlters of order M = 6 and varying gain g.
i.e. approximately 3 dB below the maximum. It should be
noted that this yields an asymmetry between ampliﬁcation
(g > 1) and attenuation (g < 1), see Fig. 3.
By construction, the ﬁlters have minimumphase behav
ior. This can be exploited in several equalization applica
tions, for example when inputtooutput latency is a concern.
2.2 Parametric representation
To develop a ﬁlter structure later on where the parameters
gain, centerfrequency and bandwidth are decoupled and can
be adjusted independently, we will ﬁrst develop a represen
tation of the lowshelving ﬁlters where the gain can be easily
manipulated. We start by rewriting the complexvalued ﬁrst
order sections of (3),
H
(m)
LS,M
(s) =
s +
M
√
ge
j
α
m
s + e
j
α
m
,
as
H
(m)
LS,M
(s) = 1 +V
e
j
α
m
s + e
j
α
m
, V =
M
√
g −1.
Combining H
(m)
LS,M
and H
(M+1−m)
LS,M
, having complex con
jugate poles and zeros as e
j
α
m
= e
−j
α
M+1−m
, yields the para
metric realvalued secondorder section
¯
H
(m)
LS,M
(s) = H
(m)
LS,M
(s)·H
(M+1−m)
LS,M
(s)
= 1 + 2V
1 + c
m
s
s
2
+ 2c
m
s + 1
+V
2
1
s
2
+ 2c
m
s + 1
(4)
with c
m
= cos(
α
m
). If M is odd, (3) furthermore has one
realvalued ﬁrstorder section
H
(
M+1
2
)
LS,M
(s) =
s +
M
√
g
s + 1
= 1 +V
1
s + 1
. (5)
ω
0.1
1
10 100
H
LS,6
( j
ω
)
2
−20 dB
−10 dB
0 dB
10 dB
20 dB
Figure 4: Magnitude responses for continuoustime low
shelving ﬁlters of order M = 6 and varying gain g, alternative
design for gain symmetry.
Thus, the lowshelving ﬁlter of (3) can be rewritten in
terms of parametric realvalued ﬁrst and secondorder sec
tions according to (5) and (4) as
H
LS,M
= H
(
M+1
2
)
LS,M
(s)
 {z }
for odd M only
·
b
M
2
c
∏
m=1
¯
H
(m)
LS,M
(s).
2.3 Design alternatives
The ﬁlter deﬁned by (3) can in fact be understood as the
combination of a Butterworth lowpass with normalized cut
off frequency 1 and the inverse of a Butterworth lowpass
with cutoff frequency
M
√
g. In a similar manner, shelving ﬁl
ters based on Chebyshev and elliptic ﬁlters can be designed.
As expected, their magnitude responses have steeper rolloff,
but oscillation inside and/or outside the ampliﬁed/attenuated
band [5].
The asymmetry of the magnitude responses between am
pliﬁcation and attenuation (see Fig. 3) can be avoided by us
ing
H
ˆ
LS,M
(s) =
M
∏
m=1
s +
2M
√
ge
j
α
m
s +
1
2M
√
g
e
j
α
m
instead of (3), yielding the magnitude responses of Fig. 4. In
fact, for this design, taking the reciprocal gain is equivalent
to inverting the ﬁlter. However, this approach does not lend
itself to the decoupling as performed above, so we stay with
the design of (3).
3. DISCRETETIME REALIZATION
To obtain discretetime ﬁlters, we simply apply a bilinear
transformation [6]. In the same step, we also denormalize
the cutoff frequency, so that we arrive at the transformation
s =
1
K
1 −z
−1
1 + z
−1
, K = tan
Ω
B
2
, (6)
where Ω
B
denotes the ﬁlter bandwidth in radians per sample.
Substituting (6) in (4) and (5) gives
¯
H
(m)
LS,M
(z) =
1 + 2V K
K+c
m
+2Kz
−1
+(K−c
m
)z
−2
1+2Kc
m
+K
2
+(2K
2
−2)z
−1
+(1−2Kc
m
+K
2
)z
−2
+V
2
K
2
1+2z
−1
+z
−2
1+2Kc
m
+K
2
+(2K
2
−2)z
−1
+(1−2Kc
m
+K
2
)z
−2
(7)
14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 48, 2006, copyright by EURASIP
K
a
−1
0,m
z
−1
z
−1
−
−2c
m
K
2
−
K
−
−c
m
2
V
K V
Figure 5: Parametric realization of a secondorder section
¯
H
(m)
LS,M
(z).
a
z
−1
−
V
2
Figure 6: Parametric realization of a ﬁrstorder section
H
(
M+1
2
)
LS,M
(z).
and
H
(
M+1
2
)
LS,M
(z) = 1 +V K
1+z
−1
1+K+(K−1)z
−1
, (8)
respectively.
Realization of (7) with two parallel secondorder sys
tems, such that these subsystems only depend on K and
hence the cutoff frequency and are added to a directpath
using weights that only depend on V and hence the gain, is
straightforward. But as the two nontrivial summands of (7)
have the same denominator, they can share the feedback part
of the ﬁlter structure and only require distinct feedforward
parts. By further modifying the ﬁlter structure to reduce the
number of operations required, the realization of Fig. 5 can
be derived, where
a
−1
0,m
=
1
1 + 2Kc
m
+ K
2
.
The ﬁrstorder section of (8) actually is a standard low
shelving ﬁlter and can be realized with any of the known
approaches. For convenience, we reproduce the design pre
sented in [2] which uses the allpass decomposition
H
(
M+1
2
)
LS,M
(z) = 1 +
V
2
·
1 +
a + z
−1
1 + az
−1
, a =
K −1
K +1
yielding the realization shown in Fig. 6.
Ω
π
0 0.1 0.2 0.3 0.4
0.5 0.6
0.7 0.8
0.9
1
H
BS,M
(e
2
π
f / f
S
)
2
0 dB
10 dB
20 dB
Figure 7: Magnitude responses of bandshelving ﬁlters with
order M = 6, gain g = 10, bandwidth Ω
B
= 0.1
π
and center
frequencies Ω
0
= [0, 0.2
π
, 0.4
π
, 0.6
π
, 0.8
π
,
π
].
4. HIGH AND BANDSHELVING FILTERS
From the lowshelving ﬁlters, high and bandshelving ﬁlters
can easily be obtained as
H
HS,M
(z) = H
LS,M
(−z)
and
H
BS,M
(z) = H
LS,M
z
c
0
−z
1 −c
0
z
, c
0
= cos (Ω
0
),
where Ω
0
is the desired centerfrequency of the bandshelv
ing ﬁlter [7]. The resulting magnitude response is given by
H
BS,M
e
jΩ
2
=
(c
0
−cosΩ)
2M
+ (K sinΩ)
2M
g
2
(c
0
−cosΩ)
2M
+ (K sinΩ)
2M
,
as depicted in Fig. 7 for various center frequencies.
The center frequency here speciﬁes the frequency at
which the maximum gain is reached. This generally is not
the center of the ﬁlter’s active band in the sense of being the
arithmetic or geometric mean of the band edges. Instead,
tan
2
Ω
0
2
= tan
Ω
c,1
2
tan
Ω
c,2
2
where Ω
c,i
are the band edges where
H
BS,M
e
jΩ
c,i
2
=
g
2
+1
2
.
Especially, for Ω
0
→ 0, the bandshelving ﬁlter becomes
a lowshelving ﬁlter, and thus Ω
0
is at the lower end of the
active band, and likewise for Ω
0
→
π
, the ﬁlter becomes a
highshelving ﬁlter with Ω
0
at the upper end of the active
band.
To retain the decoupling of the ﬁlter parameters, no new
coefﬁcients depending on Ω
0
are computed, but instead, the
unit delays of Fig. 5 and 6 are replaced with an allpass as
z
−1
← A(z) = z
−1
c
0
−z
−1
1 −c
0
z
−1
. (9)
Note that (9) has the proper limits, i.e. A(z) = z
−1
for
Ω
0
= 0 and A(z) = −z
−1
for Ω
0
=
π
, so that the same im
plementation may be used for low, band and highshelving
ﬁlters, provided the realization of the frequency shifting all
pass is numerically stable for

c
0

= 1.
14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 48, 2006, copyright by EURASIP
f /Hz
100
1k
10k
200
2k
20k500 5k

H( f )

2
−5 dB
0 dB
5 dB
10 dB
(a) M = 1.
f /Hz
100
1k
10k
200
2k
20k500 5k

H( f )

2
−5 dB
0 dB
5 dB
10 dB
(b) M = 2.
f /Hz
100
1k
10k
200
2k
20k500 5k

H( f )

2
−5 dB
0 dB
5 dB
10 dB
(c) M = 6.
Figure 8: Design example results for different ﬁlter orders.
5. DESIGN EXAMPLE
Let us consider a digital parametric threeband equalizer op
erating at a sampling rate of 48 kHz. The equalizer is realized
as a cascade of shelving ﬁlters as presented in this paper. The
parameters of the three bands are set to
f
0,1
= 0 Hz, f
B,1
= 500 Hz, G
1
= 5 dB,
f
0,2
= 2 kHz, f
B,2
= 2 kHz, G
2
= 10 dB,
f
0,3
= 10 kHz, f
B,3
= 14 kHz, G
3
= −5 dB.
We shall compare realizations utilizing ﬁlter orders M =
1, 2 and 6. (Due the frequency shifting allpass, the effective
ﬁlter order of course is 2M.) The frequencydependent coef
ﬁcients execpt for the a
−1
0,m
are the same across ﬁlter orders
and can be computed to be
K
1
= 0.032737, c
0,1
= 1.00000,
K
2
= 0.131652, c
0,2
= 0.96593,
K
3
= 1.303225, c
0,3
= 0.25882.
Depending on the ﬁlter order, we furthermore ﬁnd the
gaindependent V
band,M
as
V
1,1
= 0.77828, V
1,2
= 0.33352, V
1,6
= 0.10069,
V
2,1
= 2.16228, V
2,2
= 0.77828, V
2,6
= 0.21153,
V
3,1
= −0.43766, V
3,2
= −0.25011, V
3,6
= −0.09148.
For the sake of brevity, the resulting values for the a
−1
0,m
are
omitted.
In Fig. 8, the resulting magnitude responses of the in
dividual shelving ﬁlters (dashed) and of their cascade (solid
line) are depicted. As can clearly be seen, the 0 dBvalley be
tween 500 Hz and 1 kHz can only be achieved for the higher
order ﬁlters. For M = 1, i.e. the traditional biquad case, in
the third band around 10 kHz, the desired gain of −5 dB is
not reached at all. This shows that higherorder shelving ﬁl
ters with their steep bandedges are a necessity for equalizers
that are used to model a speciﬁc magnitude response.
6. COMPUTATIONAL PERFORMANCE
While the parametric realization of the ﬁrstorder section in
Fig. 6 has the minimum number of two multipliers for two
coefﬁcients, the realization of the secondorder section in
Fig. 5 needs nine multipliers (not counting multiplication
by 2) — considerable more than a directform implementa
tion would need.
So the proposed structure is beneﬁcial only if the ﬁl
ter parameters are changed frequently, as recomputation of
the coefﬁcients is relatively cheap thanks to the decoupled
design. In particular, changing the bandwidth Ω
B
requires
one trigonometric function evaluation for the complete ﬁl
ter, which is unavoidable when the ﬁlter is constructed using
the bilinear transform, and M divisions to determine the a
−1
0,m
.
Changing the gain requires one exponentiation to determine
the gain per section, which also seems unavoidable.
The frequency shift to obtain bandshelving ﬁlters can
be efﬁciently realized with allpasses in onemultiplier form.
This also needs only one trigonometric function evalua
tion for the complete ﬁlter whenever the centerfrequency is
changed.
7. CONCLUSIONS
We have presented an approach to design minimumphase
shelving ﬁlters of arbitrary order. For this design, we
have derived a parametric ﬁlter structure in which center
frequency, bandwidth and gain of the shelving ﬁlter can be
adjusted independently and cheaply. This realization is suit
able for several equalization applications where the ﬁlter pa
rameters have to be updated frequently.
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¨
olzer and T. Boltze. Parametric digital ﬁlter struc
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New York, October 1995.
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¨
olzer. Parametric second and fourth
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¨
olzer. Digital Audio Signal Processing. John Wiley
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14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 48, 2006, copyright by EURASIP