This preview shows pages 1–4. Sign up to view the full content.
Introduction to Signal Processing
1
Copyright © 20052009 – Hayder Radha
VI.
Frequency Domain Analysis of
ContinuousTime Periodic Signals:
The Fourier Series
In this part of the course, we focus on the representation
of
periodic signals
in terms of sums of sinusoids or
exponentials.
Introduction to Signal Processing
2
Copyright © 20052009 – Hayder Radha
A.
Periodic Signal Representation by
Trigonometric Fourier Series
A periodic signal
()
x t
with period
0
T
has the property,
0
(
)
x
tT x
t
+=
for all
t
.
The smallest value of
0
T
for which this condition holds is
the
fundamental period
of
x t
.
Introduction to Signal Processing
3
Copyright © 20052009 – Hayder Radha
0
500
1000
1500
2000
1
0.5
0
0.5
1
1.5
t
x(t)
T
0
The area under
x t
over any interval of duration
0
T
is
always the same. For any real numbers
a
and
b
;
Introduction to Signal Processing
4
Copyright © 20052009 – Hayder Radha
00
aT
bT
ab
xtd
t
t
++
=
∫∫
For example, for a periodic signal, we have the following
equivalent integrals:
0
0
0
/2
3
/4
0/
2/
4
3
/
4
TT
T
T
T
t
t
t
t
−−−
===
=
∫
∫
"
The
frequency
in
cycles per second
or
Hertz
for a signal
with period
0
T
is denoted by
0
f
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Introduction to Signal Processing
5
Copyright © 20052009 – Hayder Radha
Then
00
2
f
ω
π
=
is the frequency in
radians per second
.
So,
0
12
T
f
==
and
2
T
=
A fundamental question regarding periodic signals is the
following:
can we represent an arbitrary periodic signal
using a linear combination of “template” periodic
signals?
We begin by looking at such linear combination.
Introduction to Signal Processing
6
Copyright © 20052009 – Hayder Radha
Consider a signal
()
x t
that is a linear combination of
sinusoidal functions (sines and cosines) with a
fundamental (radian) frequency
0
and all its
harmonics
(
0
n
, where
1, 2, ,
n
=
"
):
0
1
cos
s
in
nn
n
x
ta
a
n
t
b
n
t
∞
=
=+
+
∑
Note that the above equation includes the zeroth
harmonics which is represented by the term
0
a
.
Introduction to Signal Processing
7
Copyright © 20052009 – Hayder Radha
This type of series is known as the
Fourier Series
(FS).
An important set of questions regarding the FS
representation of the signal
( )
x t
are the following:
¾
Is
( )
x t
periodic over all values of
t
?
¾
Is it periodic for all values of the
a
and
b
parameters?
We address these questions by examining if the condition
0
(
)
x
tT x
t
+=
is true or not.
Introduction to Signal Processing
8
Copyright © 20052009 – Hayder Radha
0
0
1
0
0
0
0
0
1
0
1
0
1
c
o
s
s
i
n
cos(
)
sin(
)
cos(
2
)
sin(
2
)
cos
sin
n
n
n
n
x
tT a
a
n tT b
n tT
aa
n
t
n
T
b
n
t
n
T
n
t
n
b
n
t
n
n
t
b
n
t
xt
ωω
ωπ
∞
=
∞
=
∞
=
∞
=
+
++
+
+
+
+
+
+
+
+
=
∑
∑
∑
∑
Introduction to Signal Processing
9
Copyright © 20052009 – Hayder Radha
Therefore,
any
linear combination of sinusoidal functions
(sines and cosines) with (any) harmonic frequencies,
00
0
0,
,2
,
,
,
f
fk
f
……
and any amplitudes,
012
12
,,, ,,
aaa
bb
produces a periodic signal of frequency
0
f
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.
 Spring '08
 STAFF
 Frequency, Signal Processing

Click to edit the document details