Digital Signal Processing Reference
In-Depth Information
x
(
t
)
y
(
t
)
1/
a
2
b
2
1
1
a
1
b
1
2
2
a
0
b
0
(a)
x
(
t
)
y
(
t
)
1/
a
2
b
2
a
1
b
1
a
0
b
0
Figure 3.27
Direct form II realization of a
second-order system.
(b)
Using the procedure illustrated with this second-order system, we draw the simula-
tion diagrams for the direct form I and the direct form II in Figures 3.28 and 3.29,
respectively, for
It can be shown that at least
n
integrators are required to realize an
n
th-order
differential equation. If a realization has
n
integrators, the realization is
minimal
.
Otherwise, the realization is
nonminimal
. The direct form I is nonminimal, and the
direct form II is minimal.
m = n.
We now discuss some practical considerations. Suppose that, for example, the form
I simulation diagram of Figure 3.26(c) is constructed for a second-order mechanical
system and that is the position variable. Then velocity is and accelera-
tion is Generally, we must solve for these variables in a simulation, and
we obtain this information by requiring the outputs of two integrators to be these
y(t)
dy(t)/dt
d
2
y(t)/dt
2
.
