Digital Signal Processing Reference
In-Depth Information
of the system. For example, the initial conditions may include charges stored
in a capacitor or energy stored in a mechanical spring. The zero-input response
y
zi
(
t
) is evaluated by solving a homogeneous equation obtained by setting the
input signal
x
(
t
)
=
0 in Eq. (3.9). For Eq. (3.9), the homogeneous equation is
given by
Q(D)
y
(
t
)
=
0
.
The zero-state response
y
zs
(
t
) arises due to the input signal and does not depend
on the initial conditions of the system. In calculating the zero-state response,
the initial conditions of the system are assumed to be zero. The zero-state
response is also referred to as the
forced response
of the system since the zero-
state response is forced by the input signal. For most stable LTIC systems, the
zero-input response decays to zero as
t
→∞
since the energy stored in the
system decays over time and eventually becomes zero. The zero-state response,
therefore, defines the steady state value of the output.
Example 3.2
Consider the RLC series circuit shown in Fig. 3.1. Assume that the inductance
L
=
0 H (i.e. the inductor does not exist in the circuit), resistance
R
=
5
,
and capacitance
C
=
1
/
20 F. Determine the output signal
y
(
t
) when the input
voltage is given by
x
(
t
)
=
sin(2
t
) and the initial voltage
y
(0
−
)
=
2 V across
the resistor.
Solution
Substituting
L
=
0,
R
=
5, and
C
=
1
/
20 in Eq. (3.7) yields
d
y
d
t
+
4
y
(
t
)
=
d
x
d
t
=
2 cos(2
t
)
.
(3.11)
Zero-input response of the system
Using the procedure outlined in Appendix
C, we determine the characteristic equation for Eq. (3.11) as
(
s
+
4)
=
0
,
which has a root at
s
=−
4. The zero-input response of Eq. (3.11) is given by
−
4
t
,
zero input response
y
zi
(
t
)
=
A
e
where
A
is a constant. The value of
A
is obtained from the initial condition
y
(0
)
=
2 V in the above equation yields
A
=
2.
The zero-input response is given by
y
zi
(
t
)
=
2e
−
)
=
2 V. Substituting
y
(0
−
−
4
t
.
Zero-state response of the system
The zero-state response is calculated by
solving Eq. (3.11) with a zero initial condition,
y
(0
−
)
=
0. The homogeneous
component of the zero-state response of Eq. (3.11) is similar to the zero input
response and is given by
y
(h)
zs
−
4
t
,
(
t
)
=
C
e
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