Digital Signal Processing Reference
In-Depth Information
y
(
t
)
=
i
(
t
)
R
. Substituting the value of
i
(
t
)
=
y
(
t
)
/
R
into Eq. (3.3) yields
d
2
y
d
t
2
L
R
+
d
y
d
t
RC
y
(
t
)
=
d
x
1
+
d
t
,
(3.7)
which is a linear, second-order, constant-coefficient, differential equation mod-
eling the relationship between the input voltage
x
(
t
) and the output voltage
y
(
t
)
measured across resistor
R
.
A more compact representation for Eq. (3.1) is obtained by denoting the differ-
entiation operator d
/
d
t
by D:
D
n
y
+
a
n
−
1
D
n
−
1
y
++
a
1
D
y
+
a
0
y
(
t
)
=
b
m
D
m
y
+
b
m
−
1
D
m
−
1
y
++
b
1
D
y
+
b
0
x
(
t
)
.
By treating D as a differential operator, we obtain
(
D
n
+
a
n
−
1
D
n
−
1
++
a
1
D
+
a
0
)
y
(
t
)
Q(D)
=
(
b
m
D
m
+
b
m
−
1
D
m
−
1
++
b
1
D
+
b
0
)
x
(
t
)
,
(3.8)
P(D)
or
Q(D)
y
(
t
)
=
P(Q)
x
(
t
)
,
(3.9)
where Q(D) is the
n
th-order differential operator, P(D) is the
m
th-order differen-
tial operator, and the
a
i
and
b
i
are constants. Equation (3.9) is used extensively
to describe an LTIC system.
To compute the output of an LTIC system for a given input, we must solve the
constant-coefficient differential equation, Eq. (3.9). If the reader has little or no
background in differential equations, it will be helpful to read through Appendix
C before continuing. Appendix C reviews the direct method for solving linear,
constant-coefficient differential equations and can be used as a quick look-up
of the theory of differential equations. In the material that follows, it is assumed
that the reader has adequate background in solving linear, constant-coefficient
differential equations.
From the theory of differential equations, we know that output
y
(
t
) for
Eq. (3.9) can be expressed as a sum of two components:
y
(
t
)
=
y
z
i
(
t
)
zero
-
input response
+
y
z
s
(
t
)
zero
-
state response
,
(3.10)
where
y
zi
(
t
)isthe
zero-input response
of the system and
y
zs
(
t
)isthe
zero-
state response
of the system. Note that the zero-input component
y
zi
(
t
)isthe
response produced by the system because of the initial conditions (and not due
to any external input), and hence
y
zi
(
t
) is also known as the natural response
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