Image Processing Reference
In-Depth Information
e
j
ω
)
(
G
G
∞
ω
π
0
Fig. 1. The
H
∞
norm
e
j
ω
)
∞
is the maximum gain of the frequency response
G
(
G
.
By the packed notation, the following formulae are often used in this chapter:
⎡
⎣
⎤
⎦
,
A
1
B
1
C
1
D
1
A
2
B
2
C
2
D
2
A
2
0
B
2
B
1
C
2
A
1
B
1
D
2
D
1
C
2
C
1
D
1
D
2
×
=
⎡
⎣
⎤
⎦
.
A
1
B
1
C
1
D
1
A
2
B
2
C
2
D
2
A
1
0
B
1
±
±
=
B
2
C
1
C
2
D
1
±
0
A
2
D
2
Next, we define
stability
of linear systems. The state-space system
G
in (1) is said to be
stable
λ
1
,...,
λ
n
of the matrix
A
lie in the open unit circle
D
=
{
∈
C
|
| <
}
if the eigenvalues
z
:
z
1
.
Assume that the transfer function
G
(
z
)
is irreducible. Then
G
is stable if and only if the poles
of the transfer function
G
. To compute the eigenvalues of
A
in MATLAB, use the
command
eig(A)
, and for the poles of
G
(
z
)
lie in
D
(
z
)
use
pole(Gz)
.
The
H
∞
norm
of a stable transfer function
G
(
z
)
is defined by
e
j
ω
)
∞
:
=
(
G
max
ω∈
[
G
.
π
]
0,
e
j
ω
)
This is the maximum gain of the frequency response
G
as shown in Fig. 1. The
MATLAB code to compute the
H
∞
norm of a transfer function is given as follows:
(
of
G
>> z=tf('z',1);
>> Gz=(z-1)/(z^2-0.5
*
z);
>> norm(Gz,inf)
ans =
1.3333
This result shows that for the stable transfer function
−
z
1
(
)=
G
z
0.5
z
,
z
2
−
the
H
∞
norm is given by
G
∞
≈
1.3333.