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Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
C.
Role of the Amplitude and Phase Spectra
In general, the Fourier series of a periodic signal
( )
x t
decomposes
()
x t
into an infinite series of sinusoids. An
important question is the following:
How many of these sinusoids are “really” needed to
reconstruct the original signal
x t
?
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
¾
In general, the signal
x t
can be reconstructed
“perfectly” from all of the Fourier series’ harmonic
component sinusoids (an infinite number of them).
¾
However, sometimes, we only need a small number
(much smaller than the infinite number in the series)
to reconstruct the signal
x t
with “reasonable quality”.
¾
By increasing the number of harmonic sinusoid terms
(from the Fourier series), our approximation of ()
x t
should be improved.
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
Example
The Fourier series of the pulse train function shown is,
6
4
2
0
2
4
6
0.2
0
0.2
0.4
0.6
0.8
1
t
x(t)
6
4
2
0
2
4
6
0.2
0
0.2
0.4
0.6
0.8
1
t
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
This signal, with a period
0
T
π
=
and
0
0.5
a
=
, has an
even symmetry and hence all of the sine amplitudes are
zeros:
0
n
b
=
. The cosine amplitudes are:
02
,
4
,
6
2
1, 5, 9,
1
1
2
3,7,11,
n
n
n
n
an
n
=
⎧
⎪
⎪
⎪
=+ ⋅
=
⎨
⎪
⎪
−⋅
=
⎪
⎩
"
"
"
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Copyright © 20052009 – Hayder Radha
Hence:
12
1
1
1
( )
cos
cos3
cos5
cos7
23
5
7
xt
t
t
t
t
π
⎛
⎞
=+
−
+
−
+
⎜
⎟
⎝
⎠
…
We will now reconstruct it by starting with an
approximation that uses only the lowest (zero) frequency
component (which is the DC component),
0
1
2
a
=
:
6
4
2
0
2
4
6
0
0.5
1
t
x
n
(t)
6
4
2
0
2
4
6
0
0.5
1
t
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
This is a “coarse” approximation of
()
x t
by its mean.
Now we add the lowest frequency component, the 1
st
harmonic, i.e.
cos
2
t
+
.
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
Adding successive harmonics yields closer approx.
If we use the 3
rd
harmonic,
1
cos
cos3
tt
⎛
⎞
+−
⎜
⎟
⎝
⎠
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
And the 5
th
, 7
th
and 9
th
harmonics …
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
6
4
2
0
2
4
6
0
0.5
1
t
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
In the above example, the role of additional (higher
frequency) harmonic terms diminish at the rate
1
n
.
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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
 Signal Processing

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