Name:
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Math 122 Lab 5: Fourier
Series
Introduction
Series
of functions like power series are very powerful, although sometimes difficult.
In class, we are using
What to Turn In
Turn
in this worksheet with everything filled in. This will be graded for accuracy
and understanding, so write good explanations when you are asked to explain
something. If you need more space, turn in additional pages typed or neatly
written.
Sound Waves
Sound and light are the most common forms of
electromagnetic waves in our lives. The sine and cosine function give very
accurate models of the waves that are found in nature. To start this lab, we’ll
look at a couple of sine graphs and interpret them as sound waves. First, graph
y = sinx on the TI-89 using the zoom trig option. Sketch the result here.
Next,
graph y = 2sinx.
Sketch the result and describe the difference in this graph and y = sinx. In this case, the 2 is known as the amplitude and affects how loud we perceive the sound to be.
Next,
graph y = sin2x.
Sketch the result and describe the difference in this graph and y = sinx. In this case, the 2 is known as the frequency and affects how high or low we perceive the pitch to be. This sound would be perceived as an octave higher
than the previous one. (When you double the frequency, you get the same note an
octave higher.)
Identify the amplitude and frequency of y = 4sin10x and y = 8sin20x. If these tones were played, describe as
completely as possible what differences you would perceive between the two.
Radio is based on changes to sine waves. For
example, if a station is broadcasting at frequency 16, the base signal is sin16x. If the station broadcasts a pure note
of frequency 2, then the combined signal is
y = sin2x sin16x. Graph this
function simultaneously with y = sin2x for −3 ≤ x ≤ 3 (show the result here) and
briefly explain where the term amplitude modulation (AM) comes from.
FM
radio is based on frequency modulation. In this case, if a station is broadcasting
at frequency 10 and the message signal is 2sinx, the combined signal is y
= cos(10x+2sinx). Graph this function here from 0 to 2π and briefly explain
why the term frequency modulation is used (look closely).
Fourier Series Basics
How
many different graphs can be made with combinations of sine functions? The
answer to this question is found in areas of mathematics known as Fourier
analysis and wavelets. In music synthesis, the question is exactly equivalent
to the following. How many different tones can be made by a machine (music
synthesizer) that generates many pure tones at different frequencies and
amplitudes? The answer to both questions is “as many as you want.” Our goal is
similar to
We
will start by rewriting f(x) = x
in terms of sinπx, sin2πx, sin3πx and so on. It can be shown that the following Fourier series
converges for x between −1 and
1.
As
with 
on the same graph with x
between −1 and 1. Show the result here. Comment on how close the graphs
are.
Graph
y = x and 
on the same graph with x
between −1 and 1. Show the result here. Comment on how close the graphs
are.
What
would you expect to happen as you graph more and more terms? At which values of
x does it appear that the
approximation is the worst?
Repeat
the last two graphs enlarging the x-range
to −3 to 3. What feature does the Fourier series have that the original
function does not? In other words, the Fourier series graph will always be
periodic (just as all sound and light waves are).
If
the above graphs were redone with all infinity of terms in the Fourier series,
sketch what you would expect the graph to look like from −3 to 3.
Engineers call this the “sawtooth wave.” Briefly explain why.
This
should give you the basic idea of Fourier series. It turns out that any
continuous function (or function with only jump discontinuities) can be exactly
written as a Fourier series. This means that a music synthesizer can mimic any
sound and that a digital processor can produce any image. To see how image
processing works, see page 829 for a picture being digitized. Then look at the
pictures shown below. The blurry images are transformed to the sharper image on
the right by taking more and more terms in the Fourier series. Notice in the
images that a “ghost” has been produced. This is a result of Fourier series
being less accurate at the edges than in the middle. Briefly describe what you
have seen in your graphs on the previous page that would correspond to this.
The
mathematics of Fourier series can be used to analyze any wave phenomenon. So,
the same mathematics is used to work on digital music as it is on digital
photography! Each instrument has a characteristic waveform (see the graph on
page 579 of a saxophone waveform). One task of a music synthesizer is to find
the right combination of pure tones (sines and cosines) to reproduce the timbre
of a given instrument. This is exactly the Fourier series problem. The language
of music focuses on what is called harmonic
content which is essentially a description of the size of Fourier
coefficients. For the Fourier series of x
found earlier, ignore the 2/π and minus signs and explain what is meant by
the harmonic content being 1/k.
The
harmonic content is graphed below to the left. To the right is the result of
boosting the treble on this wave. Briefly explain what is different about the
graph on the right. Given that high frequencies correspond to high pitches,
explain why the graph on the right shows a boost in treble and not in bass.
Take
the function y = (2/π) sinπx − (2/2π) sin2πx + (2/3π) sin3πx − (2/4π) sin4πx + (2/5π) sin5πx − (2/6π) sin6πx from before, modify some of the terms
to show a boost in bass and show the new function here. Graph the new function
and discuss whether the new waveform looks like the old. Do you think you could
hear the difference?
(I’ll
give a bonus point if you can name
the saxophone player on page 579 and the soccer player on page 829.)