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.
 
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