Digital Signal Processing Reference
In-Depth Information
3.17
A sinusoidal signal
x
(
t
)
=
A
sin(
ω
0
t
+ θ
) is applied at the input of an
LTIC system with real-valued impulse response
h
(
t
). By expressing the
sinusoidal signal as the imaginary term of a complex exponential, i.e. as
A
e
j(
ω
0
+
t
)
j
A
sin(
ω
0
t
+ θ
)
=
Im
,
A
∈ℜ,
show that the output of the LTIC system is given by
y
(
t
)
=
A
H
(
ω
0
)
sin(
ω
0
t
+ θ +
arg(
H
(
ω
0
))
,
where
H
(
ω
) is the Fourier transform of the impulse response
h
(
t
)as
defined in Problem 3.16.
Hint: If
h
(
t
) is real and
x
(
t
)
→
y
(
t
), then Im
x
(
t
)
→
Im
y
(
t
)
.
3.18
Given that the LTIC system produces the output
y
(
t
)
=
5 cos(2
π
t
) when
the signal
x
(
t
)
=−
3 sin(2
π
t
+ π/
4) is applied at its input, derive the
value of the tranfer function
H
(
ω
)at
ω =
2
π
. Hint: Use the result derived
in Problem 3.17.
3.19
(a) Compute the solutions of the differential equations given in P3.2 for
duration 0
≤
t
≤
20 using M
ATLAB
. (b) Compare the computed solution
with the analytical solution obtained in P3.2.
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