Digital Signal Processing Reference
In-Depth Information
and from (13.20), the output vector is given by
y
[n] =
Cx
[n] +
Du
[n]
n- 1
=
C
≥[n]
x
[0] +
a
C
≥[n - 1 - k]
Bu
[k] +
Du
[n].
(13.24)
k= 0
Note that the summation in (13.24) is a convolution sum. This is not surprising, be-
cause the output signal
y
[
n
] for an LTI system is expressed as a convolution sum in
(10.13).
The complete solution of the state equations is given in Equation (13.24).
However, in practice, we generally do not solve for as a function of
n
. Instead,
we calculate the solution of state equations recursively on a digital computer, using
the state equations. An example is now given to illustrate the recursive nature of the
solution.
≥[n]
Recursive solution of state equations
EXAMPLE 13.4
Consider the discrete-time system with the transfer function
b
1
z + b
2
z
2
+ a
1
z + a
2
.
z + 3
z
2
- 5z + 6
H(z) =
=
From (13.18), we write the state equations
01
-65
0
1
B
R
B
R
x
[n + 1] =
x
[n] +
u[n];
y[n] = [3
1]
x
[n].
2]
T
Assume that the initial state is
x[0] = [2
and that the input signal is a unit step function,
such that
u[n] = 1
for
n G 0.
We obtain the recursive solution by evaluating the state equa-
tions first for
n = 1,
next for
n = 2,
then for
n = 3,
and so on:
01
-65
0
1
01
-65
2
2
0
1
2
-1
x
[1] =
B
R
x
[0] +
B
R
u[0] =
B
RB
R
+
B
R
(1) =
B
R
;
2
-1
y[1] = [3
1]
B
R
= 5
and
01
-65
2
-1
0
1
-1
-16
B
RB
R
B
R
B
R
x
[2] =
+
=
;
-1
-16
B
R
y[2] = [3
1]
=-19.
