Simplified Computation of l
2
-Sensitivity for 1-D and a Class of
2-D State-Space Digital Filters Considering 0 and ±1 Elements
Yoichi Hinamoto
1
and Akimitsu Doi
2
1
Department of Electronical and Computer Engineering, Kagawa National College of Technology
Takamatsu, Kagawa 761-8058, Japan
2
Department of Computer Science, Hiroshima Institute of Technology, Hiroshima 731-5193, Japan
Keywords:
Coefficient Sensitivity Analysis, L
2
-Sensitivity Measures, Improved l
2
-Sensitivity, State-space Digital Filters,
a Class of 2-D State-space Digital Filters, Dual Systems, Fornasini-Marchesini Second Local State-space
Model, 0 and ±1 Elements.
Abstract:
A simplified method of computing an improved l
2
-sensitivity measure is developed for state-space digital
filters by reducing the number of the Lyapunov equations, and it is expanded into a class of two-dimensional
(2-D) state-space digital filters. First, a conventional improved l
2
-sensitivity for state-space digital filters is
reviewed and simplified to two novel forms so that the number of the Lyapunov equations is reduced. Next, the
resulting mehod is expanded into a class of 2-D state-space digital filters. Finally, two numerical examples are
presented to evaluate more precise (improved) l
2
-sensitivity measures for 1-D and a class of 2-D state-space
digital filters by employing the proposed methods.
1 INTRODUCTION
In the case when a state-space model is realized from
a given transfer function and implemented with a fi-
nite binary representation, the truncation or rounding
of the coefficients is required to meet the finite-word-
length (FWL) constraints. As a result, unacceptable
degradation of the characteristics of a recursive dig-
ital filter may be caused, and a stable recursive dig-
ital filter may be changed to an unstable one. This
motivates the study of coefficient sensitivity analysis
and its minimization problem. Several methods have
been explored to evaluate the coefficient sensitivity of
a state-space digital filter and to minimize the coeffi-
cient sensitivity (Thiele, 1984; Thiele, 1986; Iwatsuki
et al., 1989; Li and Gevers, 1990; Li et al., 1992; Yan
and Moore, 1992; Li and Gevers, 1992; Gevers and
Li, 1993; Xiao, 1997; Hinamoto et al., 2005; Yamaki
et al., 2006). The analysis and minimization problems
of l
2
-sensitivity have also been considered for two-
dimensional (2-D) state-space digital filters (Kawa-
mata et al., 1987; Hinamoto et al., 1990; Li, 1997;
Li, 1998; Hinamoto et al., 2002; Hinamoto et al.,
2006; Yamaki et al., 2007). Some of them evaluate
the coefficient sensitivity by using a mixture of l
1
/l
2
norms (Thiele, 1984; Thiele, 1986; Iwatsuki et al.,
1989; Li and Gevers, 1990; Li et al., 1992; Kawa-
mata et al., 1987; Hinamoto et al., 1990), while the
others rely on the use of a pure l
2
norm (Yan and
Moore, 1992; Li and Gevers, 1992; Gevers and Li,
1993; Xiao, 1997; Hinamoto et al., 2005; Li, 1997;
Li, 1998; Hinamoto et al., 2002; Hinamoto et al.,
2006; Yamaki et al., 2006; Yamaki et al., 2007). It
is noted that the l
2
-sensitivity measure is more nat-
ural and reasonable than the l
1
/l
2
mixed sensitivity
measure. In (Xiao, 1997), an improved l
2
-sensitivity
measure has been presented to evaluate l
2
-sensitivity
more precisely when the state-space model contains
0 and ±1 coefficients. In (Hinamoto and Doi, 2012),
simple l
2
-sensitivity measures have been explored for
evaluating the l
2
-sensitivity of canonical forms in 1-
D and 2-D separable-denominator state-space digital
filters.
In this paper, a simplified method of comput-
ing an improved l
2
-sensitivity measure for state-space
digital filters is developed by reducing the number
of the Lyapunov equations. The resulting method
is expanded into a class of two-dimensional (2-
D) state-space digital filters reported in (Hinamoto,
2001). This class of 2-D state-space digital filters
can be viewed as a dual system of the Fornasini-
Marchesini second local state-space model (Fornasini
and Marchesini, 1978). First, a conventional im-
proved l
2
-sensitivity for state-space digital filters in
53
(Xiao, 1997) is reviewed and simplified to two novel
forms so that the number of the Lyapunov equations is
reduced. The novel contribution exists in two alterna-
tive formulations for the Lyapunov equations where
two independent variables are replaced by a single
independent variable. This enables one to reduce
the amount of computations considerably. Further-
more, the possible coefficient values equal to 0 and
±1 are considered as special cases. Then the simpli-
fied method of computing the improved l
2
-sensitivity
measure for 1-D filter is expanded into a class of 2-D
state-space digital filters. The analysis is carried out
more precisely than that in (Hinamoto et al., 2006)
by taking into account 0 and ±1 elements in the 2-D
state-space digital filter. Finally, two numerical exam-
ples are presented to demonstrate the validity and ef-
fectiveness of simplified methods for computing more
precise (improved) l
2
-sensitivity measures in 1-D and
a class of 2-D state-space digital filters.
2 REVIEW OF IMPROVED
l
2
-SENSITIVITY FOR
STATE-SPACE DIGITAL
FILTERS
Consider a stable, controllable and observable state-
space digital filter (A,b,c)
n
described by
x(k + 1) = Ax(k) +bu(k)
y(k) = cx(k)
(1a)
where x(k) is an n × 1 state-variable vector, u(k) is a
single input, y(k) is a single output, and
A =
a
11
a
12
··· a
1n
a
21
a
22
··· a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
··· a
nn
, b =
b
1
b
2
.
.
.
b
n
c =
c
1
c
2
··· c
n
.
(1b)
The transfer function of the filter in (1a) can be ex-
pressed as
H(z) = c(zI
n
− A)
−1
b. (2)
The l
2
-sensitivity measure for the filter in (1a) is de-
fined as (Yan and Moore, 1992)
S =
n
∑
k=1
n
∑
l=1
1
2π j
I
|z|=1
∂H(z)
∂a
kl
2
dz
z
+
n
∑
k=1
1
2π j
I
|z|=1
∂H(z)
∂b
k
2
dz
z
+
n
∑
l=1
1
2π j
I
|z|=1
∂H(z)
∂c
l
2
dz
z
(3a)
where
∂H(z)
∂a
kl
= G(z)e
k
e
T
l
F(z)
∂H(z)
∂b
k
= G(z)e
k
,
∂H(z)
∂c
l
= e
T
l
F(z)
(3b)
with
F(z) = (zI
n
−A)
−1
b, G(z) = c(zI
n
−A)
−1
. (3c)
It is noted that coefficients 0 and ±1 can be real-
ized precisely in the implementation of FWL digital
systems. Therefore, system’s l
2
-sensitivity is not af-
fected by these coefficients.
By taking this situation into account, the individ-
ual sensitivities for the elements of coefficient matri-
ces A, b and c should be changed to (Xiao, 1997)
∂H(z)
∂a
kl
= G(z)e
k
e
T
l
F(z)φ
kl
∂H(z)
∂b
k
= G(z)e
k
ϕ
k
,
∂H(z)
∂c
l
= e
T
l
F(z)ψ
l
(4a)
where
φ
kl
=
1 for a
kl
6= 0,±1
0 for a
kl
= 0,±1
ϕ
k
=
1 for b
k
6= 0,±1
0 for b
k
= 0,±1
ψ
l
=
1 for c
l
6= 0,±1
0 for c
l
= 0,±1
.
(4b)
Lemma : The improved l
2
-sensitivity measure for
a state-space model (A,b,c)
n
in (1a) is presented by
(Xiao, 1997)
S
I
=
n
∑
k=1
n
∑
l=1
φ
kl
[ c 0 ] R(k,l)
c
T
0
+
n
∑
k=1
ϕ
k
W
kk
+
n
∑
l=1
ψ
l
K
ll
(5a)
where R(k,l), K
ll
((l,l)th entry of K) and W
kk
((k, k)th
entry of W) are obtained by solving the Lyapunov
equations
R(k, l) =
A e
k
e
T
l
0 A
R(k, l)
A e
k
e
T
l
0 A
T
+
"
0 0
0 bb
T
#
K = AKA
T
+ bb
T
W = A
T
WA + c
T
c
(5b)
for k = 1, 2, · · · , n and l = 1,2,··· ,n.
SIGMAP2013-InternationalConferenceonSignalProcessingandMultimediaApplications
54
In the following two theorems, it is shown that the
above improved l
2
-sensitivity measure in (5a) can be
modified to two novel forms so that the number of the
Lyapunov equations is reduced.
Theorem 1 : The improved l
2
-sensitivity measure
in (5a) is changed to the form
S
0
I
=
n
∑
k=1
n
∑
l=1
φ
kl
[ e
T
l
0 ] M(k)
"
e
l
0
#
+
n
∑
k=1
ϕ
k
W
kk
+
n
∑
l=1
ψ
l
K
ll
(6a)
where M(k) is obtained by solving the Lyapunov
equation
M(k) =
A bc
0 A
M(k)
A bc
0 A
T
+
0 0
0 e
k
e
T
k
(6b)
for k = 1, 2, · · · , n.
Proof : It is noted that
F(z)G(z) = (zI
n
− A)
−1
bc(zI
n
− A)
−1
=
I
n
0
zI
2n
−
A bc
0 A
−1
0
I
n
.
(7)
Defining Φ(z) = F(z)G(z), from (4a) and (7) it fol-
lows that
1
2π j
I
|z|=1
∂H(z)
∂a
kl
2
dz
z
= φ
kl
e
T
l
1
2π j
I
|z|=1
Φ(z)e
k
e
T
k
Φ
T
(z
−1
)
dz
z
e
l
= φ
kl
[ e
T
l
0 ] M(k)
e
l
0
(8a)
where
M(k) =
∞
∑
p=0
A bc
0 A
p
0
e
k
0
e
k
T
A
T
0
(bc)
T
A
T
p
(8b)
which yields the Lyapunov equation in (6b). This
completes the proof of Theorem 1.
Theorem 2 : The improved l
2
-sensitivity measure
in (5a) can be written as
S
00
I
=
n
∑
k=1
n
∑
l=1
φ
kl
[ 0 e
T
k
] N(l)
0
e
k
+
n
∑
k=1
ϕ
k
W
kk
+
n
∑
l=1
ψ
l
K
ll
(9a)
where N(l) is obtained by solving the Lyapunov equa-
tion
N(l) =
A bc
0 A
T
N(l)
A bc
0 A
+
e
l
e
T
l
0
0 0
(9b)
for l = 1, 2, · · · , n.
Proof : From (4a) and (7) it follows that
1
2π j
I
|z|=1
∂H(z)
∂a
kl
2
dz
z
= φ
kl
e
T
k
1
2π j
I
|z|=1
Φ
T
(z
−1
)e
l
e
T
l
Φ(z)
dz
z
e
k
= φ
kl
[ 0 e
T
k
] N(l)
0
e
k
(10a)
where
N(l) =
∞
∑
p=0
A
T
0
(bc)
T
A
T
p
e
l
0
e
l
0
T
A bc
0 A
p
(10b)
which yields the Lyapunov equation in (9b). This
completes the proof of Theorem 2.
It is noted that the novel contribution in this paper
exists in two alternative formulations in (6b) and (9b)
for the Lyapunov equations where two independent
variables (k, l) in (5b) are replaced by a single inde-
pendent variable either k or l. This makes it possible
to reduce the amount of computations considerably.
3 MORE PRECISE
l
2
-SENSITIVITY FOR A CLASS
OF 2-D STATE-SPACE DIGITAL
FILTERS
Consider a stable, locally controllable and lo-
cally observable 2-D local state-space model
(A
1
,A
2
,b,c
1
,c
2
,d)
n
defined for a class of 2-D IIR
digital filters (Hinamoto, 2001)
x(i + 1, j + 1)
y(i, j)
=
A
1
A
2
c
1
c
2
x(i, j + 1)
x(i + 1, j)
+
b
d
u(i, j) (11a)
where x(i, j) is an n × 1 local state vector, u(i, j) is a
single input, y(i, j) is a single output, and
A
i
=
a
i11
a
i12
··· a
i1n
a
i21
a
i22
··· a
i2n
.
.
.
.
.
.
.
.
.
.
.
.
a
in1
a
in2
··· a
inn
, b =
b
1
b
2
.
.
.
b
n
c
i
=
c
i1
c
i2
··· c
in
for i = 1, 2.
(11b)
SimplifiedComputationofl2-Sensitivityfor1-DandaClassof2-DState-SpaceDigitalFiltersConsidering0and+-1
Elements
55
The transfer function of (11a) is given by (Hinamoto,
2001)
H(z
1
,z
2
) = (z
−1
1
c
1
+ z
−1
2
c
2
)
·
I
n
− z
−1
1
A
1
− z
−1
2
A
2
−1
b + d.
(12)
A more precise (an improved) l
2
-sensitivity measure
for the local state-space model in (11a) can be defined
as
M =
n
∑
k=1
n
∑
l=1
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂a
1kl
2
dz
1
dz
2
z
1
z
2
+
n
∑
k=1
n
∑
l=1
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂a
2kl
2
dz
1
dz
2
z
1
z
2
+
n
∑
k=1
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂b
k
2
dz
1
dz
2
z
1
z
2
+
n
∑
l=1
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂c
1l
2
dz
1
dz
2
z
1
z
2
+
n
∑
l=1
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂c
2l
2
dz
1
dz
2
z
1
z
2
(13a)
where
∂H(z
1
,z
2
)
∂a
ikl
= z
−1
i
G(z
1
,z
2
)e
k
e
T
l
F(z
1
,z
2
)φ
ikl
∂H(z
1
,z
2
)
∂b
k
= G(z
1
,z
2
)e
k
ϕ
k
∂H(z
1
,z
2
)
∂c
il
= z
−1
i
e
T
l
F(z
1
,z
2
)ψ
il
for i = 1,2
(13b)
with
F(z
1
,z
2
) =
I
n
− z
−1
1
A
1
− z
−1
2
A
2
−1
b
G(z
1
,z
2
) = (z
−1
1
c
1
+ z
−1
2
c
2
)
·
I
n
− z
−1
1
A
1
− z
−1
2
A
2
−1
φ
ikl
=
1 for a
ikl
6= 0,±1
0 for a
ikl
= 0,±1
ϕ
k
=
1 for b
k
6= 0,±1
0 for b
k
= 0,±1
ψ
il
=
1 for c
il
6= 0,±1
0 for c
il
= 0,±1
(13c)
for i = 1, 2.
It is noted that unlike those in (Hinamoto et al.,
2006), the individual sensitivities in (13b) are taken
into account 0 and ±1 elements in the 2-D local state-
space model of (11a) to evaluate the l
2
-sensitivity
more precisely.
By substituting (13b) into (13a), a more precise l
2
-
sensitivity measure for a 2-D state-space digital filter
in (11a) is derived.
Theorem 3 : The more precise l
2
-sensitivity mea-
sure for the 2-D filter in (11a) can be computed from
either of
M
I
=
n
∑
k=1
n
∑
l=1
φ
1kl
e
T
k
M
l
e
k
+
n
∑
k=1
n
∑
l=1
φ
2kl
e
T
k
M
l
e
k
+
n
∑
k=1
ϕ
k
W
kk
+
n
∑
l=1
ψ
1l
K
ll
+
n
∑
l=1
ψ
2l
K
ll
(14a)
M
0
I
=
n
∑
k=1
n
∑
l=1
φ
1kl
e
T
l
N
k
e
l
+
n
∑
k=1
n
∑
l=1
φ
2kl
e
T
l
N
k
e
l
+
n
∑
k=1
ϕ
k
W
kk
+
n
∑
l=1
ψ
1l
K
ll
+
n
∑
l=1
ψ
2l
K
ll
(14b)
Proof : It follows from (13b) and (13c) that
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂a
ikl
2
dz
1
dz
2
z
1
z
2
= φ
ikl
e
T
k
M
l
e
k
= φ
ikl
e
T
l
N
k
e
l
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂b
k
2
dz
1
dz
2
z
1
z
2
= ϕ
k
e
T
k
We
k
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
∂H(z
1
,z
2
)
∂c
il
2
dz
1
dz
2
z
1
z
2
= ψ
il
e
T
l
Ke
l
for i = 1,2
(15a)
where
M
l
=
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
[F(z
1
,z
2
)G(z
1
,z
2
)]
T
e
l
·e
T
l
F(z
−1
1
,z
−1
2
)G(z
−1
1
,z
−1
2
)
dz
1
dz
2
z
1
z
2
N
k
=
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
F(z
1
,z
2
)G(z
1
,z
2
)e
k
·e
T
k
[F(z
−1
1
,z
−1
2
)G(z
−1
1
,z
−1
2
)]
T
dz
1
dz
2
z
1
z
2
K=
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
F(z
1
,z
2
)F
T
(z
−1
1
,z
−1
2
)
dz
1
dz
2
z
1
z
2
W=
1
(2π j)
2
I
|z
1
|=1
I
|z
2
|=1
G
T
(z
1
,z
2
)G(z
−1
1
,z
−1
2
)
dz
1
dz
2
z
1
z
2
.
(15b)
Matrices M
l
, N
k
, K and W are the 2-D Gramians
SIGMAP2013-InternationalConferenceonSignalProcessingandMultimediaApplications
56
which can be derived from
M
l
=
∞
∑
i=0
∞
∑
j=0
H
T
(i, j)e
l
e
T
l
H(i, j)
N
k
=
∞
∑
i=0
∞
∑
j=0
H(i, j)e
k
e
T
k
H
T
(i, j)
K =
∞
∑
i=0
∞
∑
j=0
f(i, j) f
T
(i, j)
W =
∞
∑
i=0
∞
∑
j=0
g
T
(i, j) g(i, j)
(16a)
where
f(i, j) = A
(i, j)
b
g(i, j) = c
1
A
(i−1, j)
+ c
2
A
(i, j−1)
A
(0,0)
= I
n
, A
(i, j)
= 0 for i < 0 , j < 0
A
(i, j)
= A
1
A
(i−1, j)
+ A
2
A
(i, j−1)
= A
(i−1, j)
A
1
+ A
(i, j−1)
A
2
for (i, j) > (0,0)
H(i, j) =
∑∑
(0,0)≤(k,r)<(i, j)
f(k, r)g(i − k, j − r)
(16b)
with the partial ordering for integer pairs (i, j) defined
in (Roessor, 1975).
4 NUMERICAL EXAMPLES
Example 1: Let a state-space digital filter (A, b, c)
3
in
(1a) be specified by (Xiao, 1997)
A =
0 1 0
0 0 1
0.1732 −1.0227 1.8155
, b =
0
0
1
c =
0.1174 −0.3818 0.2984
From (6a) and (9a), the improved l
2
-sensitivity
was found to be
S
0
I
= 240.433072, S
00
I
= 240.433072
which essentially coincide with S
I
= 240.43 in (Xiao,
1997). The optimal state-space digital filter with the
minimum l
2
-sensitivity can be constructed as (Yan
and Moore, 1992),(Xiao, 1997)
A
o
=
0.6883 −0.2234 −0.0297
−0.2234 0.5394 −0.1394
−0.0297 −0.1394 0.5879
b
o
=
0.5183
0.1718
0.0158
c
o
=
0.5183 0.1718 0.0158
For this optimal realization, from (6a) and (9a), the
improved l
2
-sensitivity was found to be
S
0
I
= 2.458368, S
00
I
= 2.458368
which essentially coincide with S
I
= 2.4579 in (Xiao,
1997). In fact, S
I
= 2.458368 was derived from (5a).
Example 2: Consider a local state-space model in
(11a) specified by (Hinamoto et al., 2006)
A
1
=
0 1 0 0
0 0 1 0
0 0 0 1
−0.0041 0.0801 −0.4246 1.0446
A
2
=
−0.2261 1.6143 0.1005 −0.0072
−0.4059 1.6104 −0.6062 0.2458
−0.3096 1.0234 −0.4532 0.3867
−0.1447 0.4387 −0.3102 0.5629
b =
0 0 0 1
T
c
1
=
−0.0145 0.0123 0.0205 0.0476
c
2
=
0.01190 0.0235 −0.0064 0.0209
.
In (16a), the Gramians M
l
, N
k
, K, and W were
computed over (0,0) ≤ (i, j) ≤ (100, 100). From
(14a) and (14b), the more precise l
2
-sensitivity was
found to be
M
I
= 1.426236 × 10
5
, M
0
I
= 1.426236 × 10
5
which are smaller than the value of the l
2
-sensitivity
measure: 1.579936 × 10
5
reported in (Hinamoto
et al., 2006) since coefficients 0 and ±1 were not
taken into account in (Hinamoto et al., 2006).
The optimal 2-D state-space digital filter structure
that minimizes the l
2
-sensitivity subject to l
2
-scaling
constraints was constructed as (Hinamoto et al., 2006)
A
o
1
=
0.31822 0.36329 −0.21491 −0.14635
−0.01438 0.13734 0.56514 −0.07384
−0.08182 −0.07710 0.16782 0.18698
−0.01344 0.07553 −0.03754 0.42122
A
o
2
=
0.53774 −0.04378 0.15039 0.21219
0.08849 0.38433 0.01571 0.00945
−0.22483 0.36107 0.11917 −0.07048
−0.09396 −0.04963 0.14221 0.45275
b
o
=
−0.38108
−0.36371
−0.77857
0.53703
c
o
1
=
−0.19351 −0.07547 0.06028 −0.01237
c
o
2
=
0.15022 0.38727 0.40379 0.99327
For this optimal realization that does not contain 0 and
±1 elements, from (14a) and (14b) the more precise
l
2
-sensitivity was found to be
M
I
= 372.778156, M
0
I
= 372.778156
which essentially coincide with the value of an l
2
-
sensitivity measure: 372.776303 in (Hinamoto et al.,
2006).
SimplifiedComputationofl2-Sensitivityfor1-DandaClassof2-DState-SpaceDigitalFiltersConsidering0and+-1
Elements
57
5 CONCLUSIONS
This paper has developed a simplified method of com-
puting an improved l
2
-sensitivity measure for state-
space digital filters by reducing the number of the
Lyapunov equations. The simplified method has also
been expanded into a class of 2-D state-space digital
filters. First, a conventional improved l
2
-sensitivity
for state-space digital filters has been reviewed and
its computation method has been simplified with two
novel forms such that the number of the Lyapunov
equations is reduced. Next, the resulting method has
been applied to a class of 2-D state-space digital fil-
ters. This has been done more precisely by taking into
account 0 and ±1 elements in the filter. Finally, two
numerical examples have been presented to explain
the validity and effectiveness of simplified methods
of computing more precise (improved) l
2
-sensitivity
measures for 1-D as well as a class of 2-D state-space
digital filters.
The simplified method has also been investigated
for computing a more precise l
2
-sensitivity measure in
2-D state-space digital filters described by the Roes-
sor model (Roessor, 1975) and the results will appear
elsewhere.
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