Digital Signal Processing Reference
In-Depth Information
8.1.
Consider the
RL
circuit of Figure P8.1. The circuit input is the voltage
v
i
(t).
(a)
Write the state equations for the circuit, with the state variable equal to the current
and the output equal to the resistance voltage
(b)
Write the state equations for the circuit, with both the state variable and the output
equal to v
R
(t).
i(t)
v
R
(t).
L
i
(
t
)
v
i
(
t
)
v
R
(
t
)
R
Figure P8.1
8.2.
In Figure P8.1, replace the resistor with a capacitor of
C
farads and voltage
v
c
(t).
The
input voltage is
v
i
(t).
(a)
Write the state equations for the circuit, with the state variables equal to the cur-
rent the capacitor voltage and the output equal to
(b)
Write the state equations of the circuit, with the state variables equal to
i(t),
v
C
(t),
v
C
(t).
i(t)
and
v
C
(t),
and the output equal to
i(t).
8.3.
Find a set of state equations for each of the systems described by the following differ-
ential equations:
y
#
(t) + ay(t) = bu(t)
(a)
(b)
(c)
(d)
(e)
y
#
(t) - 2y(t) = 4u(t)
$
(t) + 5y
#
(t) + 6y(t) = 9u(t)
6
$
(t) + 3y
#
(t) + y(t) = 4u(t)
#
#
1
(t) + 3y
1
(t) + 6y
1
(t) - 9y
2
(t) = 3u
1
(t) - u
2
(t)
y
#
2
(t) + 2y
2
(t) + 4y
1
(t) = u
1
(t) - 4u
2
(t)
y
#
1
(t) + 2y
1
(t) + 4y
2
(t) = 5u
1
(t) - u
2
(t)
(f)
y
$
(t) - 3y
2
#
(t) + 6y
2
(t) - y
1
(t) = 3u
1
(t) + 2u
2
(t)
8.4.
(a)
Draw a simulation diagram for the system described by the transfer function
Y(s)
U(s)
= H(s) =
6
s + 4
.
(b)
Write the state equations for the simulation diagram of part (a).
(c)
Give the system differential equation for the simulation diagram of part (a).
(d)
Repeat parts (a), (b), and (c) for the transfer function
Y(s)
U(s)
= H(s) =
50s
+ 3s + 1
.
s
2
