Biomedical Engineering Reference
In-Depth Information
If the amplitudes and frequencies are chosen appropriately, the trigonometric signals
add constructively, thus recreating an arbitrary periodic signal. This is akin to combining
prime colors in precise ratios to recreate an arbitrary color and shade. RGB are the building
blocks for more elaborate colors, much as sinusoids of different frequencies serve as the
building blocks for more complex signals. All of these elements (the color and the required
proportions; the frequencies and their amplitudes) have to be precisely adjusted to achieve
a desired result. For example, a first-order approximation of the square wave is achieved by
fitting the square wave to a single sinusoid of appropriate frequency and amplitude. Suc-
cessive improvements in the approximation are obtained by adding higher-frequency sinu-
soid components, or
, to the first-order approximation. If this procedure is
repeated indefinitely, it is possible to approximate the square wave signal with infinite
accuracy.
The Fourier series summarizes this result as
harmonics
1
x
ð
t
Þ¼
a
0
þ
1
a
m
ð
cos
m
o
o
t
þ
b
m
sin
m
o
o
t
Þ
ð
11
:
3a
Þ
m
¼
where
x
(
t
) is the periodic signal to be approximated, o
0
¼
2p
=
T
is the fundamental frequency
of
x
(
t
) in units of radians/s, and the coefficients
a
m
and
b
m
determine the amplitude of each
cosine and sine term at a specified frequency o
m
¼
m
o
0
. Equation (11.3a) tells us that the peri-
odic signal,
), is precisely replicated by summing an infinite number of sinusoids. The fre-
quencies of the sinusoid functions always occur at integer multiples of o
o
x
(
t
and are referred to
as “harmonics” of the fundamental frequency. If we know the coefficients
a
m
and
b
m
for each
of the corresponding sine or cosine terms, we can completely recover the signal
x
(
t
) by evalu-
ating the Fourier series. How do we determine
for an arbitrary signal?
The coefficients of the Fourier series correspond to the amplitude of each sine and cosine.
These are determined as
a
m
and
b
m
ð
T
x
ð
t
Þ
dt
1
T
a
0
¼
ð
11
:
3b
Þ
ð
T
x
ð
t
Þ
2
T
a
m
¼
cos
ð
m
o
o
t
Þ
dt
ð
11
:
3c
Þ
ð
T
x
ð
t
Þ
2
T
b
m
¼
sin
ð
m
o
o
t
Þ
dt
ð
11
:
3d
Þ
where the integrals are evaluated over a single period, T, of the waveform.
EXAMPLE PROBLEM 11.6
Find the trigonometric Fourier series of the square wave signal shown in Figure 11.7A, and
implement the result in MATLAB for the first ten components. Plot the time waveform and the
Fourier coefficients.
Continued
