Digital Signal Processing Reference
In-Depth Information
The (1, 1) element of the product is given by
(1, 1) element = [3(2)
n
1
- 2(3)
n
1
][3(2)
n
2
- 2(3)
n
2
]
+ [-(2)
n
1
+ (3)
n
1
][6(2)
n
2
- 6(3)
n
2
] = 9(2)
n
1
+n
2
- 6(2)
n
1
(3)
n
2
- 6(2)
n
2
(3)
n
1
+ 4(3)
n
1
+n
2
- 6(2)
n
1
+n
2
+ 6(2)
n
1
(3)
n
2
+ 6(2)
n
2
(3)
n
1
- 6(3)
n
1
+n
2
.
Combining these terms yields
(1, 1) element = 3(2)
n
1
+n
2
- 2(3)
n
1
+n
2
,
(13.37)
which is the (1, 1) element of
≥[n
1
+ n
2
].
In a like manner, the other three elements of the
product matrices can be verified.
To illustrate the third property, (13.36), we assume that the property is true for this
example. Hence,
3(2)
n
- 2(3)
n
-(2)
n
+ (3)
n
≥[n]≥[-n] =
B
R
6(2)
n
- 6(3)
n
-2(2)
n
+ 3(3)
n
3(2)
-n
- 2(3)
-n
-(2)
-n
+ (3)
-n
*
B
R
=
I
.
6(2)
-n
- 6(3)
-n
-2(2)
-n
+ 3(3)
-n
As with the last property, we test only the (1, 1) element of the product. From (13.37), with
and
n
1
= n
n
2
=-n,
(1, 1) element = 3(2)
n-n
- 2(3)
n-n
= 3 - 2 = 1.
In a like manner, the other three elements of the product matrix can be verified.
■
In this section, three properties of the state transition matrix are developed.
The first property,
≥[0] =
I
,
is easily applied as a check of the calculation of a
state-transition matrix.
13.5
TRANSFER FUNCTIONS
A procedure was given in Section 13.2 for writing state equations of a system from
the transfer function. In this section, we investigate the calculation of transfer func-
tions from state equations.
The standard form of the state equations is given by
x
[n + 1] =
Ax
[n] +
B
u[n]
and
y[n] =
Cx
[n] + Du[n]
(13.38)
