Digital Signal Processing Reference
In-Depth Information
to as the transfer function of the LTIC system and is given by
∞
−
j
ω
t
d
t
.
H
(
ω
)
=
h
(
t
)e
(4.73)
−∞
Step 2
Using the principle of superposition, the overall output
y
(
t
) by adding
individual outputs
y
n
(
t
)isgivenby
n
=−∞
y
n
(
t
)
∞
y
(
t
)
=
(4.74)
or
n
=−∞
D
n
e
j
n
ω
0
t
H
(
ω
)
∞
y
(
t
)
=
ω=
n
ω
0
.
(4.75)
Step 3
Based on Eq. (4.75), it is clear that the response
y
(
t
) of an LTIC system
to a periodic input
x
(
t
) is also periodic with the same fundamental period as
x
(
t
). In addition, the exponential CTFS coefficients
E
n
of the output
y
(
t
) are
related to the CTFS coefficients
D
n
of the periodic input signal
x
(
t
) by the
following relationship
E
n
=
D
n
H
(
ω
)
ω=
n
ω
0
.
(4.76)
Example 4.25
Calculate the exponential CTFS coefficients of the output
y
(
t
) if the square
wave
x
(
t
) illustrated in Fig. 4.14 is applied as the input to an LTIC system with
impulse response
h
(
t
)
=
exp(
−
2
t
)
u
(
t
).
Solution
The exponential CTFS coefficients of the square wave
x
(
t
) shown in Fig. 4.14(a)
are given by (see Example 4.14)
=
1
n
4
D
n
4
sinc
,
for
−∞ <
n
< ∞.
The transfer function
H
(
ω
) of the LTIC is given by
∞
∞
1
−
j
ω
t
d
t
−
(2
+
j
ω
)
t
d
t
H
(
ω
)
=
h
(
t
)e
=
e
=
j
ω
)
.
(4.77)
(2
+
−∞
0
For
ω
0
=
1 radian/s, the exponential CTFS coefficients of the output
y
(
t
) are
given by
=
1
n
4
1
=
sinc(
n
/
4)
8
E
n
=
D
n
H
(
ω
)
4
sinc
,
(4.78)
ω=
n
(2
+
j
n
)
+
j4
n
and the output
y
(
t
)isgivenby
n
=−∞
E
n
e
j
n
ω
0
t
∞
∞
sinc(
n
/
4)
8
+
j4
n
e
j
nt
.
y
(
t
)
=
=
(4.79)
n
=−∞
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