ESE 351-01, Fall 2017
Homework Set #11, due Nov. 21
11 Problems
In problems 1-6, use known transform pairs and properties (tables), results from the
Laplace transform, or Euler’s formula. You should
not
evaluate the Fourier
Transform integral, or its inverse.
11.1
. Find the Fourier Transform of the function graphed below.
11.2
. Find the Fourier Transform.
f t
( )
=
sin 2
π
t
−
2
(
)
⎡
⎣
⎤
⎦
t
−
2
11.3
. Find the Fourier Transform. (Hint: Duality.)
f t
( )
=
2
α
α
2
+
t
2
,
α
>
0
11.4
. The figures below depict functions of time,
f
(
t
).
(a) Write the Fourier Transform of the function in (a).
(b) Write the Fourier Transform of the function in (b). (Hint: It is the function in (a)
multiplied by
t
.) Write your answer so that no sine, cosine, or sinc is involved. (Euler’s
formula to the rescue.)

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ESE 351-01, Fall 2017
Homework Set #11, due Nov. 21
11.5.
Based on Mukai problem 11.13.7. The Fourier Transform of
f
(
t
) is given by.
F
ω
(
)
=
4
25
−
ω
2
+
i
ω
6
Find the Fourier Transform,
G
(
ω
), of the given
g
(
t
). (You do not have to make the
denominator all real.)
a
( )
g t
( )
=
f
3
t
(
)
b
( )
g t
( )
=
f t
−
10
(
)
c
( )
g t
( )
=
f
−
3
t
(
)
d
( )
g t
( )
=
f t
( )
cos4
t
e
( )
g t
( )
=
f t
( )
sin4
t
f
(
)
g t
( )
=
d
dt
f t
( )
11.6
. Mukai 11.13.14.
11.7.
Mukai 11.13.16(a). Use Parseval’s theorem to evaluate
sinc
2
t
( )
dt
−∞
∞
∫
.
For problems 8-11, you will need these key results regarding Fourier transforms,
linear systems (input
u
, output
y
) and Power Spectral Density:
S
f
ω
(
)
PSD
!"#
=
F
R
f
τ
( )
{
}
,
where R
f
τ
( )
=
Autocorrelation function of f t
( )
S
y
ω
(
)
=
H i
ω
(
)
2
S
u
ω
(
)
11.8
(a) Sketch the real part of the Fourier transform of
r
(
t
). (Hint: It’s all real.)
r t
( )
=
sin
t
π
t
1
+
cos4
t
(
)
(b) Find and sketch the squared gain, |
H
(
i
ω
)|
2
, of the system below.
!!
y
+
4
!
y
+
8
y
=
8
u
(c) Suppose that
r
(
t
) from part (a) is the autocorrelation function of the input
u
(
t
) in part
(b). Sketch the Power Spectral Density of
y
(
t
) in that case. (I don’t know if such an
autocorrelation function is possible, but assume that it is.)