Biomedical Engineering Reference
In-Depth Information
and because signal measurements can often be interpreted more readily. The most widely
used counterparts for approximating and modeling biological signals are the
exponential
and
Fourier series.
The compact Fourier series is a close cousin of the standard Fourier series. This version
of the Fourier series is obtained by noting that the sum of sinusoids and cosines
can be rewritten by a single cosine term with the addition of a phase constant
a
m
compact
cos
m
o
o
t
þ
b
m
sin
m
o
o
t
¼
A
m
cos
ð
m
o
o
t
þ
f
m
Þ
, which leads to the compact form of the
Fourier series:
1
x
ð
t
Þ¼
A
0
2
þ
1
A
m
cos
ð
m
o
o
t
þ
f
m
Þ:
ð
11
:
4a
Þ
m
¼
The amplitude for each cosine,
A
m
, is related to the Fourier coefficients through
q
a
A
m
¼
2
m
þ
b
2
m
ð
11
:
4b
Þ
and the cosine phase is obtained from
a
m
and
b
m
as
:
b
m
a
m
tan
1
ð
11
:
4c
Þ
f
m
¼
EXAMPLE PROBLEM 11.7
Convert the standard Fourier series for the square pulse function of Example Problem 11.5 to
compact form and implement in MATLAB.
Solution
We first need to determine the magnitude,
A
m
, and phase, f
m
, for the compact Fourier series.
The magnitude is obtained as
q
5
q
a
j
sin
ð
m
p=
2
Þ
j
2
2
A
m
¼
2
m
þ
b
2
m
¼
ð
sinc
ð
m
p=
2
Þ
Þ
þ
ðÞ
0
¼
5
p
m
=
2
Since
1
0
m
¼
odd
j
sin
ð
m
p
=
2
Þ
j ¼
m
¼
even
we have
10
=
m
p
m
¼
odd
m
¼
even
:
A
m
¼
0
Unlike
a
m
or
b
m
in the standard Fourier series, note that
A
m
is strictly a positive quantity for
all
m
. The phase term is determined as
¼
0
p
for
m
¼
0, 1, 4, 5, 8, 9
...
b
m
a
m
0
tan
1
tan
1
f
m
¼
¼
5
sinc
ð
m
p=
2
Þ
m
¼
2, 3, 6, 7, 10, 11
...
