Fourier
Analysis  introduction from a practical point of view
In
calculus or mathematics, Fourier
analysis is a topic
which grew from the study of Fourier
series. This field began with the study of
the way periodic or general functions might be represented by summations of
simpler trigonometric functions (sine
or cosine series).
Fourier analysis has
many applications in science, physics, digital signal processing,
imaging,
numerical analysis, acoustics... and in many other areas.
We are going to comment
on some electronic applications, where
there are two emphasized domains: analysis in time domain, to find
the voltages
or currents at specific moments, and the analysis in the frequency domain, to
study (in order to possibly manipulate) the frequencies present in a
given
signal.
Fourier series
This figure shows a sine wave with a peak value V_{p}
and a period T.
When AC analysis is
presented in circuit theory courses, they
usually focus on the sine wave because it helps understand many
effects. Based
on those effects or behaviors of electronic components we can start the
study
of nonsinusoidal signals.
Periodic waves
The shapes below (a halfsine, a sawtooth and a square wave) complete
their basic
form over a period T, and each
cycle
is an exact copy of the first period or cycle. This kind of waveform
with
repetitive cycles is called a periodic wave with period T.
The addition of sine waves with appropriate
amplitudes and
phases can represent triangular waves, or a different combination of
sine series
and waves can produce a sawtooth signal, and another different
combination may produce
a halfwave signal. In other words, any
periodic wave is a superposition of
sine waves.
The sine waves are harmonically related, which means that
the frequencies are harmonics (or multiples) of a fundamental (lowest)
frequency.
Given a periodic wave, you can find the period T
when measured with an oscilloscope. The reciprocal of T
is equal to the fundamental frequency.
In formulas:
The fundamental frequency
f_{1}
= 1 / T
The second harmonic has a frequency f_{2}
= 2f_{1}
The third harmonic has a frequency f_{3}
= 3f_{1}
In
general, the n^{th}
harmonic has a frequency
f_{n}
= nf_{1}
Formulas for the Fourier series
Expressed in words, the figure above means:
periodic wave = DC component
+
first harmonic +
second harmonic +
... +
n^{th}
harmonic
In mathematical terms:
v = V_{0}
+ V_{1}
sin(wt + Ф_{1})
+ V_{2} sin(2wt + Ф_{2}) +
... + V_{n} sin(nwt + Ф_{n})
This famous equation is called the Fourier
series, and it means that a periodic
wave is a superposition of harmonically related sine waves.
Voltage v
is the
value of the periodic wave at any instant, this value can be calculated
by
adding the DC component and the instant values of all the the
harmonics. The
first term in the Fourier series is V_{0},
which is a constant and represents the DC component. The coefficients V_{1}, V_{2}...
V_{n}
are the peak values of the harmonics. The angular frequency w is equal to 2πf,
so you can
see that each term in the Fourier series represents the next higher
harmonic.
Harmonic waves theoretically continue to infinity. However,
five to ten terms are often enough to represent a periodic signal
within an
accuracy of 5%. With the right combination of magnitudes, angles and
phases, it is
possible to reproduce any periodic waveform.
Now, let’s do a brief experiment with our numerical software. You can
see how to produce a square
wave, here.
We
can implement this code:
%
Define and plot one cycle of a square wave
x = 0 :
.01 : 2*pi;
y =
square(x);
plot(x,y)
axis([min(x)
max(x) min(y).5 max(y)+.5])
%
Plot f1 and f3 and add them up
figure
y1 =
sin(x);
y2 =
.2*sin(3*x);
plot(x,y1,'b', x,y2,'b:', x,
y1+y2, 'g', 'linewidth', 2)
axis([min(x)
max(x) min(y).5 max(y)+.5])
We get two figures:
The figure on the left is
a plain square wave. We want to
find an equivalent of that periodic wave by utilizing only sines.
The
figure on the right represents our first attempt to do it. We use the
first and
third harmonics of the original wave, we add them up and show
the results.
We can clearly see that the green line is getting closer to a square
wave. In
order to produce a better approximation, we’d need to adjust our amplitudes and
include more terms in the series.
In
the next article, we’re going to explore three very
common periodic waves and the specific Fourier Series to produce them.
Reference:
Malvino, A. P.; Electronic Principles; 2nd. edition; McGrawHill Co.,
1979.
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