City University of Hong Kong
Department of Electronic Engineering
EE3210
Lab 2: Introduction to Complex Exponentials
Prelab:
Read the Background section.
Verification:
The Warm-Up section must be completed during your assigned lab time. The steps
marked
Instructor Verification
must also be signed off during the lab time. When you have
completed a step that requires verification, simply demonstrate the step to the instructor.
Lab Report:
It is only necessary to turn in a report on the Experiment section with graphs and
explanations.
1.
Background
Recalled that
sinusoids
may
be expressed in the form:
(
)
{
}
t
f
j
j
o
o
e
Ae
t
f
A
t
x
π
φ
φ
π
2
Re
2
cos
)
(
=
+
=
.
Now consider
the sum
of cosine waves
given
by
(
∑
=
+
=
N
k
k
k
t
f
A
t
x
1
0
2
cos
)
(
φ
π
)
.
(1)
Note that
x
(
t
)
is
the sum of
N
cosine waves whose
frequencies are all equal to
f
0
. Although
this
formula
is expressed
by
using trigonometric
identities, it
can also
be
concisely expressed as the
sum
of
complex exponentials. By
using
complex exponential representation of cosine functions we
have
{
}
(
)
s
s
t
f
j
s
t
f
j
N
k
k
N
k
t
f
j
k
t
f
A
e
Z
e
Z
e
Z
t
x
o
o
o
φ
π
π
π
π
+
=
=
⎭
⎬
⎫
⎩
⎨
⎧
⎟
⎠
⎞
⎜
⎝
⎛
=
⎭
⎬
⎫
⎩
⎨
⎧
=
∑
∑
=
=
0
2
2
1
1
2
2
cos
Re
Re
Re
)
(
where
k
j
k
k
e
A
Z
φ
=
, and
.
s
j
s
N
k
k
s
e
A
Z
Z
φ
=
=
∑
=
1
We see that the
signal
x
(
t
)
itself is a single
sinusoid and it
is also periodic with period
T
0
=
1 /
f
0
.
Harmonic function
is an important
extension of sinusoidal
function.
For
harmonic function,
x
(
t
) is
the sum of
N
cosine waves whose frequencies
f
k
are
multiples of one single frequency
f
0
:
Frequency)
(Harmonic
0
kf
f
k
=

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