Signals and Systems
Lecture 8

A Fourier series is an expansion of a
periodic function
f
(
t
)
in terms of an infinite sum
of
cosines
and
sines
Introduction
1
0
)
sin
cos
(
2
)
(
n
n
n
t
n
b
t
n
a
a
t
f
Fourier series

In other words, any
periodic function
can be
resolved as a summation of
constant
value and
cosine
and
sine
functions:
1
0
)
sin
cos
(
2
)
(
n
n
n
t
n
b
t
n
a
a
t
f
)
sin
cos
(
1
1
t
b
t
a
2
0
a
)
2
sin
2
cos
(
2
2
t
b
t
a
)
3
sin
3
cos
(
3
3
t
b
t
a

The computation and study of Fourier series is
known as
harmonic analysis
and is extremely
useful as a way to
break up
an arbitrary periodic
function
into a set of simple terms
that can be
plugged in,
solved individually
, and
then
recombined to obtain the solution to the
original problem
or an approximation to it to
whatever accuracy is desired or practical.

=
+
+
+
+
+
…
Periodic Function
2
0
a
t
a
cos
1
t
a
2
cos
2
t
b
sin
1
t
b
2
sin
2
f(t)
t

1
0
)
sin
cos
(
2
)
(
n
n
n
t
n
b
t
n
a
a
t
f
where
T
dt
t
f
T
a
0
0
)
(
2
frequency
l
Fundementa
2
T
T
n
tdt
n
t
f
T
a
0
cos
)
(
2
T
n
tdt
n
t
f
T
b
0
sin
)
(
2
*we can also use the integrals limit
.
2
/
2
/
T
T

Example 1
Determine the Fourier series representation of the
following waveform.

Solution
First, determine the period & describe the one period
of the function:
T
= 2
2
1
,
0
1
0
,
1
)
(
t
t
t
f
)
(
)
2
(
t
f
t
f

Then, obtain the coefficients
a
0
,
a
n
and
b
n
:
1
0
1
0
1
)
(
2
2
)
(
2
2
1
1
0
2
0
0
0
dt
dt
dt
t
f
dt
t
f
T
a
T
Or, since
y
=
f
(
t
)

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- Spring '18
- Fourier Series, Sin, Cos