Digital Signal Processing Reference
In-Depth Information
and, hence,
≥[0] =
I
,
(13.34)
where
I
is the identity matrix. This property can be used in the verification of the
calculation of
Next, from (13.22),
≥[n].
≥[n] =
A
n
.
Thus, the second property is seen to be
≥[n
1
+ n
2
] =
A
n
1
+n
2
=
A
n
1
A
n
2
=≥[n
1
]≥[n
2
].
(13.35)
The third property is derived from the relationships
≥[-n] =
A
-n
= [
A
n
]
-1
=≥
-1
[n].
Consequently,
≥
-1
[n] =≥[-n].
(13.36)
In summary, the three properties of the state transition matrix are given by the
following equations:
[eq(13.34)]
[eq(13.35)]
[eq(13.36)]
≥[0] =
I
;
≥[n
1
+ n
2
] =≥[n
1
]≥[n
2
];
≥
-1
[n] =≥[-n].
An example illustrating these properties is now given.
Illustration of properties of a state transition matrix
EXAMPLE 13.7
We use the state transition matrix from Example 13.6 to illustrate the three properties:
3(2)
n
- 2(3)
n
-(2)
n
+ (3)
n
B
R
≥[n] =
.
6(2)
n
- 6(3)
n
-2(2)
n
+ 3(3)
n
From (13.34), the first property is satisfied:
3(2)
0
- 2(3)
0
-(2)
0
+ (3)
0
10
01
B
R
B
R
≥[0] =
=
=
I
.
6(2)
0
- 6(3)
0
-2(2)
0
+ 3(3)
0
The second property, (13.35), yields
3(2)
n
1
- 2(3)
n
1
-(2)
n
1
+ (3)
n
1
≥[n
1
]≥[n
2
] =
B
R
6(2)
n
1
- 6(3)
n
1
-2(2)
n
1
+ 3(3)
n
1
3(2)
n
2
- 2(3)
n
2
-(2)
n
2
+ (3)
n
2
*
B
R
.
6(2)
n
2
- 6(3)
n
2
-2(2)
n
2
+ 3(3)
n
2
