Digital Signal Processing Reference
In-Depth Information
as shown. We then write the equations for the input signals to the delays of
Figure 13.2:
x
1
[n + 1] = x
2
[n];
x
2
[n + 1] = x
3
[n];
o
(13.16)
x
N- 1
[n + 1] = x
N
[n];
Á
-a
2
x
N- 1
[n] - a
1
x
N
[n] + u[n].
x
N
[n + 1] =-a
N
x
1
[n] - a
N- 1
x
2
[n] -
From Figure 13.2, the equation for the output signal is
y[n] = (b
N
- a
N
b
0
)x
1
[n] + (b
N- 1
- a
N- 1
b
0
)x
2
[n]
Á
+
+ (b
1
- a
1
b
0
)x
N
[n] + b
0
u[n].
(13.17)
We now write (13.16) and (13.17) as matrix equations:
Á
0
1
0
0
0
0
0
o
0
1
Á
0
0
1
0
0
x
[n + 1] =
E
o
o
o
o
o
U
x
[n] +
E
U
u[n];
(13.18)
Á
0
0
0
0
1
Á
-a
N
-a
N- 1
-a
N- 2
-a
2
-a
1
y
[n] = [(b
N
- a
N
b
0
)(b
N- 1
- a
N- 1
b
0
)
Á
(b
1
- a
1
b
0
)]
x
[n] + b
0
u[n].
Note that the state equations can be written directly from the transfer function
(13.13) or from the difference equation (13.15), since the coefficients and are
given in these two equations. The intermediate step of drawing the simulation dia-
gram is not necessary. An example is now given.
a
i
b
i
State equations from a transfer function
EXAMPLE 13.2
Suppose that we have a system with the transfer function
b
0
z
2
+
b
1
z
+
b
2
z
2
+ a
1
z + a
2
2z
2
+ 3z + 1.5
z
2
- 1.1z + 0.8
H(z) =
=
.
We write the state equations directly from (13.18):
0
1
0
1
x
[n + 1] =
B
R
x
[n] +
B
R
u[n];
-0.8
1.1
y[n] = [(1.5 - 0.8 * 2)(3 + 1.1 * 2)]
x
[n] + 2u[n]
= [-0.1
5.2]x[n] + 2u[n].
(13.19)
