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Elements of Electrical Engineering
Dr. Fred J. Taylor, Professor
Lesson Title: Fourier Series
Lesson Number: 5 (Sections 3.53.8)
Introduction
In Chapter 3, it was established that a continuoustime signal x(t), having a fundamental period
T
0
, has a
Fourier series signal representation given by:
(
29
(
29
∑
∞
∞
=
=
k
kt
T
j
k
e
a
t
x
)
/
2
(
0
π
(Synthesis Equation)
1.
It was also established that this representation is valid only for periodic signals.
The Fourier
coefficients a
k
, found in Equation 1, are computed using the following production rules:
(
29
(
29
(
29
(DC)
harmonic
k
;
1
(DC)
harmonic
0
;
1
th
/
2
0
0
th
0
0
0
0
0
0
dt
e
t
x
T
a
dt
t
x
T
a
kt
T
j
T
k
T

∫
∫
=
=
(Analysis Equation)
2.
or
(
29
(
29
(
29
(DC)
harmonic
k
;
1
(DC)
harmonic
0
;
1
th
/
2
2
/
2
/
0
th
2
/
2
/
0
0
0
0
0
0
0
dt
e
t
x
T
a
dt
t
x
T
a
kt
T
j
T
T
k
T
T



∫
∫
=
=
(Analysis Equation)
3.
The resulting Fourier series representation of x(t) is seen to be defined in terms of (in general)
complex coefficients a
k
and complex basis functions v
k
(t)=e
j2
π
kt/T
0
, k
∈
{
∞
,
∞
}.
The coefficient a
k
corresponds to the complex amplitude and phase of the k
th
harmonic.
In addition it was stated
that the basis functions are orthogonal.
The graphical display of the Fourier coefficients defines
the signal’s spectrum.
Example
Zeromean Gaussian noise is added to a sinusoidal signal to define a signal x(t).
Technically,
because of the noise, the signal x(t) is aperiodic.
A mathematician would not proceed any
further, knowing that a Fourier series solution does not exists.
However, using a computer the
Fourier series, defined by Equations 1 and 2, can be machine calculated and displayed.
The
resulting display defines the signal’s approximate spectrum (engineers will always take a good
approximation that can be easily obtained (using a computer) over nothing at all.
The results
are shown in Figure 1.
Fourier Series
1
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Dr. Fred J. Taylor, Professor
x(t)=cos(
ϖ
0
t)+n(t)
Magnitude spectrum

ϖ
0
0
ϖ
0
Phase spectrum

ϖ
0
0
ϖ
0
Real spectrum

ϖ
0
0
ϖ
0
Imaginary spectrum

ϖ
0
0
ϖ
0
Figure 1:
Spectrum of a cosine in noise.
An alternative form of the Fourier series is called the trigonometric Fourier series which is
defined below (synthesis equation followed by analysis equation):
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
dt
t
n
t
x
T
b
dt
t
n
t
x
T
a
value
DC
or
average
dt
t
x
T
a
T
f
f
t
n
b
t
n
a
a
t
x
T
n
T
n
T
n
n
n
n
0
0
0
0
0
0
0
0
0
0
1
0
1
0
sin
2
;
cos
2
)
(
1
/
1
;
2
;
)
sin
)
cos
ϖ
π
∫
∫
∫
∑
∑
=
=
=
=
=
+
+
=
∞
=
2
π

π
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 Electrical Engineering

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