COSC 4214: Digital Communications
Assignment # 1: Signals and Spectra
Due Date: Quiz will be held on September 28, 2006
1.
Determine the energy spectral density
G
x
(
f
) of a square pulse
⎩
⎨
⎧
≤
≤
−
=
elsewhere.
0
1
)
(
2
2
T
T
t
t
x
Also, calculate the normalized energy
E
x
in the pulse.
2.
Determine which of the following functions satisfy the properties of autocorrelation functions.
a.
⎩
⎨
⎧
≤
τ
≤
−
=
τ
elsewhere.
0
1
1
1
)
(
x
b.
)
2
sin(
)
(
)
(
0
τ
π
+
τ
δ
=
τ
f
x
.
c.
).
exp(
)
(
τ
=
τ
x
d.
⎩
⎨
⎧
≤
τ
≤
−
τ
−
=
τ
elsewhere.
0
1
1
1
)
(
x
3.
Determine which of the following functions satisfy the properties of power spectral density functions.
a.
).
2
(
cos
)
(
)
(
2
f
f
f
X
π
+
δ
=
b.
).
10
(
10
)
(
−
δ
+
=
f
f
X
c.
()
10
2
exp
)
(
−
π
−
=
f
f
X
.
d.
( )
( )
10
2
exp
)
(
2
−
π
−
=
f
f
X
.
4.
Consider a random process given by
( )
φ
+
π
=
t
f
A
t
x
0
2
cos
)
(
, where
A
and
are constants and
φ
is a random
variable that is uniformly distributed over (0, 2
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