Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.02
Intro to EECS II
Spring 2008
Homework #1: Complex numbers, Fourier Series, Fourier Transform
Issued: February 8, 2008
Due: February 15, 2008
Copyright
©
2007 by M.H. Perrott and C.G. Sodini
1.
For the following exercises on complex numbers, assume
j
=−
1
.
a.
Write simplified expressions for
jj
,
jjj
,and
jjjj
.
b.
Draw a labeled plot of vector
z
where
12
zj
=
+
and the xaxis and yaxis correspond to the real and imaginary components,
respectively. Be sure to include labels for the magnitude and phase (in
degrees
as
opposed to
radians
) of
z
.
c.
Calculate the magnitude,
K
,
and phase (in
radians
),
Φ
,
of
z
,where
j
zK
e
j
Φ
=
=+
d.
Calculate the real component,
a
, and imaginary component,
b
, for
4/
6
6
j
aj
b e
π
+=
2.
For the following exercises on Fourier Series, use only the complex exponential form of the
Fourier Series:
x
(
t
)
=
ˆ
X
n
e
jn
ω
o
t
n
=−∞
∞
∑
ˆ
X
n
=
1
T
x
(
t
)
e
−
jn
o
t
dt
t
o
t
o
+
T
∫
a.
Calculate the Fourier Series
ˆ
X
n
for
( ) ( )
() s
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 Spring '08
 Terman
 Computer Science, Electrical Engineering, Fourier Series, Complex number, Fourier Series Fourier Transform

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