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Content uploaded by Martin Holters
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All content in this area was uploaded by Martin Holters
Content may be subject to copyright.
PARAMETRIC HIGHER-ORDER SHELVING FILTERS
Martin Holters and Udo Z
¨
olzer
Department of Signal Processing and Communications, Helmut-Schmidt-University
Holstenhofweg 85, 22043 Hamburg, Germany
phone: + (49) 40 6541-3468, fax: + (49) 40 6541-2822, email: {martin.holters,udo.zoelzer}@hsu-hh.de
web: www.hsu-hh.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 arbitrary-order
shelving filters and a suitable parametric digital filter struc-
ture. A low-shelving 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 cut-off frequency independently con-
trollable. Finally, we obtain band- and high-shelving 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 first-order low- and high-shelving filters and second-
order band-shelving filters, such designs based on an all-pass
decompositions were presented in [1, 2]. In [3] a decom-
position of second-order low- and high-shelving filters and
fourth-order band-shelving 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 continuous-time domain based on a
Butterworth design. These designs will then be transformed
to the discrete-time domain and a parametric realization will
be presented. Finally, by low-pass/high-pass and low-pass/
band-pass transformations, high-shelving and band-shelving
filters will be obtained.
2. CONTINUOUS-TIME LOW-SHELVING FILTERS
In [4] the transfer function of frequency-normalized first- and
second-order low-shelving 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)
s-plane
σ
j
ω
−1
−g
(a) M = 1.
s-plane
σ
j
ω
1
√
g
(b) M = 2.
s-plane
σ
j
ω
1
6
√
g
(c) M = 6.
Figure 1: Pole/zero locations in the s-plane of low-shelving
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
cut-off 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 4-8, 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 continuous-time 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 continuous-time 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 minimum-phase behav-
ior. This can be exploited in several equalization applica-
tions, for example when input-to-output latency is a concern.
2.2 Parametric representation
To develop a filter structure later on where the parameters
gain, center-frequency and bandwidth are decoupled and can
be adjusted independently, we will first develop a represen-
tation of the low-shelving filters where the gain can be easily
manipulated. We start by rewriting the complex-valued 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 real-valued second-order 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
real-valued first-order 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 continuous-time low-
shelving filters of order M = 6 and varying gain g, alternative
design for gain symmetry.
Thus, the low-shelving filter of (3) can be rewritten in
terms of parametric real-valued first- and second-order 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 low-pass with normalized cut-
off frequency 1 and the inverse of a Butterworth low-pass
with cut-off 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 roll-off,
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. DISCRETE-TIME REALIZATION
To obtain discrete-time filters, we simply apply a bilinear
transformation [6]. In the same step, we also denormalize
the cut-off 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 4-8, 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 second-order section
¯
H
(m)
LS,M
(z).
a
z
−1
−
V
2
Figure 6: Parametric realization of a first-order 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 second-order sys-
tems, such that these sub-systems only depend on K and
hence the cut-off frequency and are added to a direct-path
using weights that only depend on V and hence the gain, is
straight-forward. But as the two non-trivial summands of (7)
have the same denominator, they can share the feed-back part
of the filter structure and only require distinct feed-forward
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 first-order 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 all-pass 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 band-shelving 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 BAND-SHELVING FILTERS
From the low-shelving filters, high- and band-shelving 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 center-frequency of the band-shelv-
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 band-shelving filter becomes
a low-shelving filter, and thus Ω
0
is at the lower end of the
active band, and likewise for Ω
0
→
π
, the filter becomes a
high-shelving 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 all-pass 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 high-shelving
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 4-8, 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 three-band 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 all-pass, the effective
filter order of course is 2M.) The frequency-dependent 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
gain-dependent 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 dB-valley 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 higher-order shelving fil-
ters with their steep band-edges are a necessity for equalizers
that are used to model a specific magnitude response.
6. COMPUTATIONAL PERFORMANCE
While the parametric realization of the first-order section in
Fig. 6 has the minimum number of two multipliers for two
coefficients, the realization of the second-order section in
Fig. 5 needs nine multipliers (not counting multiplication
by 2) — considerable more than a direct-form 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 band-shelving filters can
be efficiently realized with all-passes in one-multiplier form.
This also needs only one trigonometric function evalua-
tion for the complete filter whenever the center-frequency is
changed.
7. CONCLUSIONS
We have presented an approach to design minimum-phase
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.
REFERENCES
[1] P. A. Regalia and S. K. Mitra. Tunable digital fre-
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[2] U. Z
¨
olzer and T. Boltze. Parametric digital filter struc-
tures. In Proc. 99th AES Convention, Preprint No. 4099,
New York, October 1995.
[3] F. Keiler and U. Z
¨
olzer. Parametric second- and fourth-
order shelving filters for audio applications. In Proc.
IEEE 6th Workshop on Multimedia Signal Processing,
pages 231–234, Siena, Italy, September 2004.
[4] U. Z
¨
olzer. Digital Audio Signal Processing. John Wiley
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14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP
