Realization of Variable Band-Pass/Band-Stop IIR
Digital Filters Using Gramian-Preserving Frequency
Transformation
Shunsuke Koshita, Keita Miyoshi, Masahide Abe and Masayuki Kawamata
Department of Electronic Engineering, Graduate School of Engineering, Tohoku University
6-6-05, Aoba, Aramaki, Aoba-ku, Sendai, 980-8579 Japan
Email: kosita@mk.ecei.tohoku.ac.jp
Abstract— This paper proposes variable band-pass/band-stop
infinite impulse response (IIR) state-space digital filters with
high tuning accuracy and reduced computational complexity.
We derive our proposed variable filters from the Gramianpreserving frequency transformation based on the normalized
lattice structure of second-order all-pass functions. This approach gives a simple algorithm for control of the frequency
characteristics of our proposed variable filters. In addition, we
show that our proposed variable filters have smaller number
of additions and multiplications than general state-space filters.
Furthermore, it is shown that our proposed variable filters have
the same controllability/observability Gramians as those of the
prototype filters, regardless of the value of the tuning parameter.
This property shows the high accuracy of our proposed filters
with respect to quantization effects for all tunable frequency
characteristics.
that the controllability/observability Gramians of our proposed
variable filters are invariant under the change of frequency
characteristics. This property shows that our proposed variable
filters maintain high tuning accuracy with respect to quantization effects for all tunable frequency characteristics. In
addition, we prove that the algorithm for tuning frequency
characteristics can be implemented in a simple form without
expensive tasks such as matrix inversion. We also prove that
our proposed variable filters consist of smaller number of
nonzero coefficients than general state-space digital filters.
These facts show the effectiveness of our proposed method.
I. I NTRODUCTION
Let Hd (z) be the transfer function of a variable digital filter.
This transfer function is obtained by the following frequency
transformation [1]
Variable band-pass and band-stop digital filters play important roles in many signal processing applications such as
telecommunications, acoustic signal processing and control
systems. Many techniques have been proposed for design of
variable band-pass/band-stop digital filters, and one of the
well-known design methods is to apply the theory of frequency
transformation [1]. This method enables independent tuning
of the bandwidth and the center frequency of variable bandpass/band-stop filters.
The aim of this paper is to develop an efficient realization method of variable band-pass/band-stop filters that have
high tuning accuracy with respect to quantization effects. We
attempt to achieve this goal by applying state-space representation to realization of variable digital filters because the
state-space representation is known to be a powerful tool for
synthesis of high-accuracy digital filters [2]–[5]. Variable lowpass digital filters based on the state-space representation are
proposed in [6], and this method can be also extended to
variable high-pass state-space filters. However, realization of
variable band-pass/band-stop digital filters in state-space form
has been an open problem.
In this paper we present a simple, closed-form description
of variable band-pass/band-stop infinite impulse response (IIR)
digital filters in state-space form. We derive our proposed
variable filters from the Gramian-preserving frequency transformation given by our previous work [7], [8]. It is proved
978-1-4244-5309-2/10/$26.00 ©2010 IEEE
II. P RELIMINARIES
A. Frequency Transformation
Hd (z) = Hp (z)|z−1 ←T (z)
(1)
where Hp (z) and T (z) are the transfer functions of a prototype
low-pass filter and an appropriate all-pass filter. In the case of
design of variable band-pass/band-stop filters, T (z) is given
by a second-order all-pass function.
In this paper we focus on the specific case that the bandwidth of variable band-pass/band-stop filters is the same as that
of prototype low-pass filters, i.e. we assume that the variable
band-pass/band-stop filters to be discussed here have the tunable center frequency and fixed bandwidth. The variable bandpass filters with fixed bandwidth are given by the following
specific low-pass-band-pass (LP-BP) transformation:
Hd (z)
TBP (z)
= Hp (z)|z−1 ←TBP (z)
z −1 − ξBP
= −z −1
1 − ξBP z −1
(2)
where ξBP = cos ωBP with |ξBP | < 1 and ωBP is the desired
center frequency of the passband. Similarly, the variable bandstop filters with fixed bandwidth are given by the specific
low-pass-band-stop (LP-BS) transformation using the all-pass
function TBS (z) defined as
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TBS (z) = z −1
z −1 − ξBS
1 − ξBS z −1
(3)
where ξBS = cos ωBS , |ξBS | < 1 and ωBS is the center
frequency of the stopband.
It is well-known that variable band-pass/band-stop filters
with tunable bandwidth can be obtained by applying the above
specific transformations to a variable low-pass filter with a
single-parameter tunable cutoff frequency given by the lowpass-low-pass (LP-LP) transformation.
B. Gramian-Preserving Frequency Transformation
Here we introduce the Gramian-preserving frequency transformation [7], [8], which plays a central role in derivation of
our main results. Consider the state-space representation of an
N -th order prototype filter Hp (z) as
xp (n + 1)
y(n)
= Ap xp (n) + bp u(n)
= cp xp (n) + dp u(n)
(4)
(5)
where u(n), y(n) and xp (n) are the scalar input, the scalar
output and the state vector of size N ×1, and Ap , bp , cp and dp
are real-valued coefficient matrices with appropriate size. The
coefficients (Ap , bp , cp , dp ) and the transfer function Hp (z)
are related as
Hp (z) = dp + cp (zI N − Ap )−1 bp
(6)
where I N is the N × N identity matrix. In this paper, the
system (Ap , bp , cp , dp ) is assumed to be a minimal realization
of Hp (z), i.e. the system (Ap , bp , cp , dp ) is controllable and
observable.
For the system (Ap , bp , cp , dp ), the solutions K p and W p
to the following Lyapunov equations are called the controllability Gramian and the observability Gramian, respectively:
which are respectively denoted by K d and W d , are given in
terms of K p and W p as
K d = I M ⊗ K p, W d = I M ⊗ W p.
(12)
This theorem shows that (8)–(11) gives the state-space representation of Hd (z) obtained by the frequency transformation
(1), and that the controllability/observability Gramians of this
state-space system are the same as those of the prototype statespace filter (Ap , bp , cp , dp ) with multiplicity M .
III. M AIN R ESULTS
We derive our main results by applying the Gramianpreserving frequency transformation to realization of variable
band-pass/band-stop filters.
First, we consider the realization of variable band-pass
filters. In this case the required all-pass function for the LP-BP
transformation is given by (2). Using the procedure given by
[7], [8], we obtain the appropriate state-space representation
γ
of TBP (z) with the normalized lattice structure as
β,
, δ)
(α,
⎞
⎛
2
0
1 − ξBP
ξ
BP
β
α
2
⎠ . (13)
=⎝
1 − ξBP
0
−ξBP
δ
γ
0
−1
0
Substitution of (13) into (8)–(11) yields
2 A
ξBP I N
− 1 − ξBP
p
Ad =
2 I
1 − ξBP
ξBP Ap
N
T
2
bd =
1 − ξBP bp −ξBP bp
01×N −cp
cd =
dd
= dp
(14)
(15)
(16)
(17)
where 01×N is the zero matrix of size 1 × N . This description
W p = ATp W p Ap + cTp cp . (7) is the state-space representation of our proposed variable bandpass filters. It follows from Theorem 1 that the controllability
The Gramians K p and W p are symmetric and positive Gramian K d and the observability Gramian W d of the above
definite because the system (Ap , bp , cp , dp ) is assumed to realization (Ad , bd , cd , dd ) are given as
be asymptotically stable, controllable and observable. These
Kp
W p 0N ×N
0N ×N
Gramians are essential to realization of high-accuracy digital K d =
,Wd =
. (18)
0N ×N
Kp
0N ×N W p
filters with respect to quantization effects [2]–[5].
We now introduce the Gramian-preserving frequency transRemark 1: Since the tuning parameter
ξBP is defined as
2
formation by the following theorem.
1 − ξBP
as sin ωBP .
cos ωBP , we can rewrite the term
Theorem 1 ( [7], [8]): Consider the state-space representa- Therefore, another description of our proposed variable bandpass filters can be obtained in terms of the tuning parameter
tion (Ad , bd , cd , dd ) described by
ωBP as follows:
γ ) ⊗ [Ap (I N − δA
p )−1 ] (8)
Ad = α̃ ⊗ I N + (β
cos ωBP I N − sin ωBP Ap
−1
A
=
(19)
d
(9)
bd = β ⊗ [(I N − δAp ) bp ]
sin ωBP I N
cos ωBP Ap
p )−1 ]
T
⊗ [cp (I N − δA
(10)
cd = γ
sin ωBP bp − cos ωBP bp
(20)
bd =
−1
p (I N − δA
p ) bp
(11)
dd = dp + δc
01×N −cp
(21)
cd =
where ⊗ denotes the Kronecker product for matrices [9],
dd = dp .
(22)
γ
is the state-space representation of an M β,
, δ)
Here we discuss the significance of our proposed variable
and (α,
th order allpass function T (z) that is realized as the cascaded band-pass filters described by (14)–(17). Comparing (14)–(17)
normalized lattice structure. Then, the above representation with the original version of the Gramian-preserving frequency
(Ad , bd , cd , dd ) satisfies the transfer function Hd (z) that transformation (8)–(11), we see that the original description
is given by the frequency transformation (1). In addition, (8)–(11) requires computation of an inverse matrix, whereas
the controllability/observability Gramians of (Ad , bd , cd , dd ), the expression of (14)–(17) does not include the inverse matrix.
K p = Ap K p ATp + bp bTp ,
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The reason for this is that the constant term in the numerator
of TBP (z) given by (2) is zero, which results in δ = 0 and
forces the inverse matrix in (8)–(11) to be the identity matrix.
This fact leads to significant reduction of system complexity
of our proposed variable filters. Furthermore, our proposed
variable filters have smaller number of nonzero coefficients
than general state-space digital filters: our proposed variable
filters have 2(N 2 − N ) zero elements in Ad and N zero
elements in cd . This reduction comes from the fact that the allpass function TBP (z) is constructed as the normalized lattice
γ
β,
, δ)
structure and that its state-space representation (α,
consists of sparser matrices than general state-space descriptions, as shown in (13).
The significance of our proposed method can be also
pointed out from the viewpoint of quantization effects. As
mentioned previously, realization of high-accuracy digital filters with respect to quantization effects is closely related to
the controllability/observability Gramians: such high-accuracy
digital filters can be obtained by constructing the controllability/observability Gramians of the filters appropriately [2]–[5].
Now, noting that the controllability/observability Gramians of
our proposed variable band-pass filters satisfy (18) for any
value of the tuning parameter ξBP , we see that our proposed
variable filters retain the same performance with respect to
quantization effects as that of the prototype filter. Therefore,
by constructing a prototype state-space filter in such a manner
that its Gramians exhibit a good performance with respect to
quantization effects, we can force our proposed variable filters
to have the same performance as that of the prototype filter
regardless of the change of frequency characteristics.
We next present the realization of variable band-stop filters.
γ
of (3) with the
β,
, δ)
The state-space representation (α,
normalized lattice structure is given by
⎞
⎛
2
0
1 − ξBS
ξBS
β
α
2
⎠,
(23)
=⎝
1 − ξBS
0
−ξBS
δ
γ
0
1
0
which can be easily obtained by only changing the sign of γ
in (13). Thus, the proposed variable band-stop filters are also
easily obtained as
2 A
1 − ξBS
p
ξBS I N
Ad =
(24)
2 I
1 − ξBS
−ξBS Ap
N
T
2 b
bd =
(25)
1 − ξBS
p −ξBS bp
01×N cp
(26)
cd =
dd
= dp
or equivalently
Ad
=
bd
=
cd
dd
(27)
cos ωBS I N
sin ωBS I N
sin ωBS bp
01×N cp
=
= dp .
sin ωBS Ap
− cos ωBS Ap
T
− cos ωBS bp
(28)
(29)
(30)
(31)
IV. N UMERICAL E XAMPLE
This section gives a numerical example to demonstrate
the utility of our proposed method from the viewpoint of
coefficient quantization effects. It is also possible to show the
utility of our proposed method from other aspects such as
roundoff noise, dynamic ranges and limit cycles.
The prototype filter used here is the fourth-order elliptic
low-pass filter with the following transfer function:
Hp (z) =
0.0101−0.0362z −1 +0.0524z −2 −0.0362z −3 +0.0101z −4
1−3.7895z −1 +5.4142z −2 −3.4553z −3 +0.8310z −4
. (32)
The peak-to-peak ripple, the minimum stopband attenuation
and the passband-edge frequency of this filter are 0.5 dB, 40
dB and 0.05π rad, respectively.
In our proposed method, we construct the state-space representation of this prototype filter as
⎛
⎞
0.9838 −0.1007 −0.0165 −0.0171
⎜ 0.1007
0.9582 −0.1029 −0.0273 ⎟
⎟
Ap = ⎜
⎝ −0.0165 0.1029
0.9336 −0.1015 ⎠
0.0171 −0.0273 0.1015
0.9139
T
0.1490 −0.1953 0.1669 −0.0995
bp =
0.1490 0.1953 0.1669 0.0995
cp =
dp
=
0.0101.
(33)
The controllability/observability Gramians of this realization
are calculated as
K p = W p = diag(0.8850, 0.6124, 0.2761, 0.0817),
(34)
which shows that this realization is the balanced realization
[10] and exhibits high-accuracy with respect to quantization
effects [6]. We obtain the desired variable band-pass filter
by substituting (33) into (14)–(17). Note that the resultant
variable band-pass filter has eighth-order because the LP-BP
transformation makes use of second-order all-pass functions.
Figures 1(a), (b), (c) and (d) show the magnitude responses
of our proposed variable filter for ξBP = −0.8, −0.4, 0.5
and 0.9, respectively. For comparison purpose, the magnitude
responses in the case of the cascaded direct form are also
shown here, and all the coefficients of these two variable
filters are quantized to 10 fractional bits. From Figs. 1(a), (b),
(c) and (d) we know that our proposed variable filter shows
very good agreement with the ideal magnitude responses
for all ξBP . This result confirms that, our proposed variable
filter exhibits high accuracy for all tunable characteristics by
constructing the state-space representation of the prototype
filter appropriately with respect to the Gramians. On the other
hand, the magnitude responses of the cascaded direct form are
degraded in all cases and the degradation is extremely large
for ξBP = 0.9. As is well-known, direct form digital filters
are very sensitive to quantization effects. In addition, since
variable digital filters with direct form do not take into account
the controllability/observability Gramians, the performance of
the direct form with respect to quantization effects highly
depends on the frequency characteristics. These facts show
the utility of our proposed method.
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30
30
Ideal
Proposed
Cascaded direct form
20
10
Magnitude [dB]
10
Magnitude [dB]
Ideal
Proposed
Cascaded direct form
20
0
−10
−20
0
−10
−20
−30
−30
−40
−40
−50
0
0.2
0.4
0.6
Normalized frequency
0.8
−50
0
1
0.2
(a)
30
1
Ideal
Proposed
Cascaded direct form
20
10
Magnitude [dB]
10
Magnitude [dB]
0.8
(b)
30
Ideal
Proposed
Cascaded direct form
20
0.4
0.6
Normalized frequency
0
−10
−20
0
−10
−20
−30
−30
−40
−40
−50
0
0.2
0.4
0.6
Normalized frequency
0.8
−50
0
1
(c)
0.2
0.4
0.6
Normalized frequency
0.8
1
(d)
Fig. 1. Magnitude responses of the eighth-order variable band-pass digital filters: (a) Responses for ξBP = −0.8. (b) Responses for ξBP = −0.4. (c)
Responses for ξBP = 0.5. (d) Responses for ξBP = 0.9.
V. C ONCLUSION
This paper has proposed a new method of realization of
high-accuracy variable band-pass/band-stop digital filters by
using the Gramian-preserving frequency transformation. It
has been proved that our proposed variable filters can be
implemented in a simple form without complicated tasks such
as matrix inversion. In addition, our proposed variable filters
consist of smaller number of nonzero coefficients than general
state-space filters, which shows the efficiency of our proposed
method. Moreover, we have proved that our proposed variable
filters have the same controllability/observability Gramians as
those of the prototype filter for any value of the tuning parameter. Therefore, by constructing the state-space representation
of the prototype filter in such a manner that its Gramians
show high performance with respect to quantization effects,
we can force our proposed variable filters to possess the same
high performance as that of the prototype filter for all tunable
frequency characteristics.
In this paper, we have restricted ourselves to the special
case that the bandwidth of the variable band-pass/band-stop
filters is fixed and only the center frequency is tuned. However,
we can obtain variable filters with tunable bandwidth by
combining the proposed method with a method of realization
of variable low-pass state-space filters such as [6].
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