Digital Signal Processing Reference
In-Depth Information
t
2
0
t
1
t
2
t
1
t
Figure 8.7
Time axis.
x
(t
2
+ t
1
) =≥(t
2
)
x
(t
1
) =≥(t
2
)≥(t
1
)
x
(0).
Figure 8.7 shows the points on the time axis. Also, from (8.39), with
t = t
1
+ t
2
,
x
(t
2
+ t
1
) =≥(t
2
+ t
1
)
x
(0).
From the last two equations, we see the second property:
≥(t
1
+ t
2
) =≥(t
1
)≥(t
2
).
(8.42)
We can derive the third property from the second property by letting
t
1
= t
and
t
2
=-t.
Then, in (8.42),
≥(t - t) =≥(t)≥(-t) =≥(0) =
I
,
from (8.40). Thus,
≥
-1
(t) =≥(-t).
(8.43)
≥
-1
(t)
It can be shown that always exists for
t
finite [7].
In summary, the three properties of the state transition matrix are given by
[eq(8.40)]
[eq(8.42)]
[eq(8.43)]
≥(0) =
I
;
≥(t
1
+ t
2
) =≥(t
1
)≥(t
2
);
≥
-1
(t) =≥(-t).
An example illustrating these properties will now be given.
Illustrations of properties of the state transition matrix
EXAMPLE 8.9
We use the state transition matrix from Example 8.4 to illustrate the three properties:
2e
-t
- e
-2t
e
-t
- e
-2t
B
R
≥(t) =
.
-2e
-t
+ 2e
-2t
-e
-t
+ 2e
-2t
From (8.40),
2e
0
- e
0
e
0
- e
0
10
01
B
R
B
R
≥(0) =
=
=
I
,
-2e
0
+ 2e
0
-e
0
+ 2e
0
