People are generally comfortable thinking about functions of time. For example a signal might be described as x(t), where "t" is time. This is
referred to as the
"time domain." However, it is often useful to think of signals and systems
in the "frequency domain" where frequency, instead of time, is the independent
variable, e.g., X(f) where "f" is frequency. This brings us to the concept of Fourier Analysis. The
next several paragraphs try to describe why Fourier Analysis is important.
Motivation
What follows are some examples that show how a function can be made up of a
sum of sinusoidal functions. These are given to show that it is plausible
that general waveforms can be considered as a sum of sines and cosines.
The next section will describe exactly how we determine how we determine which
sinusoids need to added together to form a particular function. Included are
Also included are a few examples that show, in a very basic way, a couple of
applications of Fourier Theory, thought the number of applications and the ways
that Fourier Theory is used are many.
Average + 1st harmonic
up to 3rd harmonic
...5th harmonic
...7th
...21st
If you click the leftmost button, you will see four functions: 1) a square
wave (solid blue line), 2) the average value of the square wave, 3) a sine wave with a period
of two seconds (dash-dot blue), or frequency = ½ Hz (solid magenta), and 4) the sum of the last three
functions (solid red). This last function is the a very crude approximation to the
square wave. Note: the sine wave is the same frequency as the
square wave; we call this the 1st (or fundamental) harmonic.
If you click the second button another (smaller) sine wave is added to the
picture with a frequency of 3/2 Hz (this is three times as fast as the
square wave (and the original sine wave); we call this the 3rd harmonic). If you look at the sum of the
average plus the two sine waves, you see that we get an even better
approximation to the original square wave.
Notes:
As you add sinusoids waves of increasingly higher frequency, the
approximation gets better and better.
The addition of higher frequencies better approximates the rapid
changes, or details, (i.e., the discontinuity) of the original function
(in this case, the square wave).
On either side of the discontinuity there is a small overshoot
(called "Gibb's phenomenon" or "Gibb's overshoot"). This overshoot
is always present in the Fourier representation of a signal with a
discontinuity and has the same height for any number of harmonics
greater than 1. The magnitude of the overshoot varies with the number of harmonics used but quickly converges to an amplitude of about 9% of the
magnitude of the discontinuity for a square wave. In this case the discontinuity has
an amplitude of 1 unit, so the Gibb's overshoot is about 0.09.
Though the height of the overshoot is finite (at about 9%) as we add more
harmonics, note that the width decreases, so the area of the
overshoot (and hence the energy) decreases. As the area goes to
zero, its effect in most systems of interest to engineers also goes to
zero.
Because of the symmetry of the waveform, only odd harmonics (1, 3,
5, ...) are needed to approximate the function. The reasons for
this are discussed elsewhere.
The rightmost button shows the sum of all harmonics up to the 21st
harmonic, but not all of the individual sinusoids are explicitly shown on the plot.
In particular harmonics between 7 and 21 are not shown.
Example: The Triangle Wave as a Sum of Sinusoids
Average + 1st harmonic
up to 3rd harmonic
...5th harmonic
...7th
Note:
As you add sine waves of increasingly higher frequency, the
approximation gets better and better.
The addition of higher frequencies better approximates the details, (i.e., the
change in slope) in the original function.
The amplitudes of the harmonics for this example drop off much more
rapidly than those for the square wave (we prove this later). As
an example, compare the amplitude of the 3rd harmonic in both
cases. Conceptually, this occurs because the triangle wave looks
much more like the 1st harmonic, so the contribution of the higher harmonics
is less. Even with only the 1st through 7th harmonics we have a
very good approximation to the original function.
There is no discontinuity, so no Gibb's overshoot.
As before, only odd harmonics (1, 3, 5, ...) are needed to
approximate the function.
Example: The Sawtooth as a Sum of Sinusoids
Average + 1st harmonic
up to 3rd harmonic
...5th harmonic
...7th
...21st
Notes:
As you add sine waves of increasingly higher frequency, the
approximation gets better and better.
The addition of higher frequencies better approximates the rapid
changes, or details, (i.e., the discontinuity) in the original function.
Because of the discontinuity we see Gibb's phenomenon.
As before, only odd harmonics (1, 3, 5, ...) are needed to
approximate the function.
The rightmost button shows the sum of all harmonics up to the 21st
harmonic, but not all of the individual sinusoids are explicitly
included in the plot.
In particular harmonics between 7 and 21 are not shown.
Example: The Electrocardiogram as a Sum of Sinusoids
Average + 1st harmonic
up to 2nd harmonic
...3rd harmonic
...4th
...10th
...20th
Notes:
As you add sine waves of increasingly higher frequency, the
approximation gets better and better.
The addition of higher frequencies better approximates the rapid
changes, or details, in the original function.
Because the waveform lacks symmetry, both even and odd harmonics are
needed to approximate the function.
The two rightmost button shows the sum of all harmonics up to the
10th and 20th
harmonics, but not all of the individual sinusoids are explicitly
included in the plot.
A Fourier Approach to Audio Signals
Fourier analysis plays a key role in the study of signals. For example
consider the function of time shown at the left below (the vertical axis is
arbitrary). If you play the
signal (controls are below the image) you will hear the word "hello." The
signal is quite complicated as you can see by a detail of the image (from 0.30 to 0.35 seconds is
shown) shown in the upper right. It is possible, using Fourier Analysis
to find the magnitude of sinusoidal functions in the signal. This is shown
in the lower right where the amplitude of the signal as a function of frequency
("frequency domain") is shown. Again, the units of the vertical
axis can be considered to be arbitrary - the reason for the large scale (and
correspondingly small signal) is to match the scale for
the example after this
one.
Example: Saying "Hello"
Now consider the signal shown below. The time domain
representation appears to be even more complicated than the one above. If you play it, you will
find that there is a lot of noise in the signal. Looking at the detail, you see that the signal is quite
complicated to describe as a function of time. However, if you look at the
frequency domain representation (lower right) the problem becomes immediately
apparent. There are large signals near 300 and 700 Hz. The distortion is easily
seen in both frequency and time, but only the frequency domain clearly shows the
nature of the distortion.
Example: Saying "Hello" with Noise Added
Fourier Theory Applied to Systems Analysis
Now consider the example of a car moving a long a road. This model of
a car is very simple - a mass (m), connected to the road through a suspension
system (spring (k) and dashpot, or friction (b)). The road surface is
irregular, but we can find the frequencies of the sinusoids that add up to
describe the surface using Fourier Analysis.
We can also find the frequencies at which the suspension might be resonant using
sinusoidal steady state analysis, or the Bode
plot. If certain frequencies are generally present, we would design
are suspension so that it damps those frequencies.
The same
principle could apply to the design of buildings. If we are trying to
design a building that is resistant to earthquakes, we could find the range of
frequencies that are present in the shaking of the ground during an earthquake,
and ensure that are building is insensitive to those frequencies.
Now consider the example of a car moving a long a road. This model of
a car is very simple - a mass (m), connected to the road through a suspension
system (spring (k) and dashpot, or friction (b)). The road surface is
irregular, but we can find the frequencies of the sinusoids that add up to
describe the surface using Fourier Analysis.
We can also find the frequencies at which the suspension might be resonant using
sinusoidal steady state analysis, or the