Digital Signal Processing Reference
In-Depth Information
Step 2
The exponential CTFS coefficients
D
n
of the periodic signal
x
T
(
t
) are
obtained from Eq. (5.63) as
=
1
T
0
D
n
X
(
ω
)
with
T
0
=
2
π
and
ω
0
=
1
.
ω=
n
ω
0
The above expression simplifies to
=
3
n
2
3
n
π
n
π
2
D
n
2
sinc
=
sin
.
5.9 LTIC systems a
nalysis using CTFT
In Chapters 2 and 3, we showed that an LTIC system can be modeled either
by a linear, constant-coefficient differential equation or by its impulse response
h
(
t
). A third representation for an LTIC system is obtained by taking the CTFT
of the impulse response:
CTFT
←−−→
h
(
t
)
H
(
ω
)
.
The CTFT
H
(
ω
) is referred to as the
Fourier transfer function
of the LTIC
system and provides meaningful insights into the behavior of the system. The
impulse response relates the output response
y
(
t
) of an LTIC system to its input
x
(
t
) using
y
(
t
)
=
h
(
t
)
∗
x
(
t
)
.
Calculating the CTFT of both sides of the equation, we obtain
Y
(
ω
)
=
H
(
ω
)
X
(
ω
)
,
(5.64)
where
Y
(
ω
) and
X
(
ω
) are the respective CTFTs of the output response
y
(
t
) and
the input signal
x
(
t
). Equation (5.64) provides an alternative definition for the
transfer function as the ratio of the CTFT of the output response and the CTFT
of the input signal. Mathematically, the transfer function
H
(
ω
)isgivenby
H
(
ω
)
=
Y
(
ω
)
X
(
ω
)
.
(5.65)
5.9.1 Transfer function of an LTIC system
It was mentioned in Section 3.1 that, for an LTIC system, the relationship
between the applied input
x
(
t
) and output
y
(
t
) can be described using a constant-
coefficient differential equation of the following form:
n
m
a
k
d
k
x
d
t
k
b
k
d
k
x
=
d
t
k
.
(5.66)
k
=
0
k
=
0
From the time-differentiation property of the CTFT, we know that
d
n
x
d
t
n
CTFT
←−−→
(j
ω
)
n
X
(
ω
)
.
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