6
F
OURIER
S
ERIES
A
PPROXIMATION
U
SING
O
RDINARY
L
EAST
S
QUARES
6.1
O
BJECTIVES
The objective of this lab is to familiarize you with the basics of the Fourier series analysis using
artificial signals.
6.2
I
NTRODUCTION
The Fourier series analysis is one of the foundations of signal processing. The assumption here is
that the signal has been present from –inf to + inf. The basic principle is that assume that any
signal can be written as a sum of sine and cosine waves with coefficients. In formula form:
()
0
11
cos
sin
nn
f
ta
a
n
t
b
n
∞∞
==
=+
+
∑∑
t
(1.)
The
term is simply the mean value of the signal (type help mean). Also, the cos and sin terms
can be estimated separately using the OLS method. From the previous lab, we know that the
parameter vector can be estimated based on the regressor matrix
A
and the data vector
0
a
y
as
follows:
( )
1
TT
AA Ay
θ
−
=
(2)
If we substitute the following model
1
cos
n
n
f
∞
=
=
∑
n
t
(3.)
into the OLS estimator equation we get a coefficient (parameter) vector
and a regressor
matrix as follows:
1
2
3
4
5
a
a
a
a
a
⎛⎞
⎜⎟
⎝⎠
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1111
cos
cos2
cos3
cos4
cos5
cos
mmmm
tttt
A
⎛⎞
⎜⎟
=
⎝⎠
1
m
t
t
The combination
becomes a unity matrix, and therefore the OLS estimator can be
written as follows.
()
1
T
AA
−
2
T
Ay
N
θ
=
(5.)
Where
N
is the number of data points.
Now the coefficient estimation boils down to taking the
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 Spring '10
 TonyGrift
 Fourier Series, basis functions

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