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I have attached all the lab files. You can do this lab in excel.Fourier Synthesis 1. Notice that this report only requires the excel file. 2. You need to start the lab by doing the sample given in Figure 5. 3. Refer to the uncertainty lab (lab 1) to review how to generate the X values. Please
remember that the x-values must be in radians and equally divided. 4. For the two exercises you need 257 x values, starting with zero. 5. I have included in the website two PDF files that show the derivation of the Fourier
Transform for the Square and the Sawtooth waves. 6. Notice that in both cases the an coefficient is zero. Therefore, we will only be using the
sine function part of the Fourier series. 7. Refer to the analysis lab to review how to include the amplitude and the harmonic
numbers (frequency) in the sine function equation 8. For the square wave the values of n are odd. Therefore, the harmonic numbers will be
odd values. Example n = 1, 3, 5,…
9. The function for the square wave is equals to 1/n * sin(n*x) 10. For the Sawtooth wave the values of harmonic numbers, n, are 1, 2, 3… 11. The function for the Sawtooth function is 1/n * sin (n*x) 12. Do at least 20 harmonics for each function. 13. The amplitude of each function is equal to 1/n. 14. Your last column should have the sum of the sine values for each harmonic.
15. You will graph the sum column in the y-axis and the x values in the x-axis. 16. Select line as your type chartFourier Synthesis
Introduction: We will take a simple waveform, break it into its components using a form of Fourier analysis, and try
to recreate it with Fourier synthesis. Fourier synthesis is a wave addition algorithm. Wave addition is also called
superposition of waves, a simple addition of instantaneous amplitudes.
Theory: Any complex waveform can be constructed from the sum of sine and cosine waves with the appropriated
amplitudes and frequencies. This summation, called a Fourier series, looks like this:
Equation 1
where f(x) is a periodic function {f(x) = f(x + 2L)}. Either x or t can be used as the variable. If you use x then L is the
half length of the wave; likewise, if your horizontal axis is time, T takes the place of L and is called the period. The
term ao/2 is an offset AKA bias, that is, a constant which shifts the waveform up or down the y axis. The function f(x)
defines the position of a point on the wave in space with 2L being the wavelength. The harmonics are n multiples (n =
1, 2, 3…) of the fundamental frequency for wavelength 2L. The coefficients an and bn are the amplitudes of each
harmonic wave, given by the following integrals:
Equation 2
Equation 3
If one has an explicit function (e.g. f(x) = sin(2x)) to analyze, it is a fairly simply task to take the integrals, find the
harmonic amplitudes, and using a math program recreate the original f(x). There is also a way to combine an and bn
into a single An called the harmonic strength; it employs a phase angle, but we won’t need that today.
However, this semester you won’t be taking waves apart: you’ll be putting them together. The assembly or synthesis of
a wave is done discreetly by adding the instantaneous amplitudes, that is, the f(x) at a fixed interval i along the waves
being summed.
Start with the fundamental (n=1) and sum from there. This technique is well-suited to how spreadsheets work: each
cell contains a formula that is the instantaneous amplitude at that instant with harmonic strength An and harmonic n.
Wave addition is extant throughout engineering and physics, from optics to digital circuitry to seismic analysis. Two
standard waveforms used in audio to simulate acoustic instruments are these:
Figure 1: Sawtooth
Figure 2: Square
These waves are the starting point for mimicking stringed instruments (sawtooth) and clarinets (square). Of course,
the process doesn’t end there: the envelope (time-dependent amplitude modulation) of the overall amplitude as well as
the harmonic amplitude envelope play a critical role.
Figure 3: Spectral Envelope
This wave-addition technique doesn’t produce the best results, but it is light on storage requirements compared with
actually sampling the original wave for playback.
To see what combination of waves produces certain standard waveforms
Procedure:
You will be making three waves, so let’s have each wave on a separate Excel sheet (tabs at the bottom). Double click
on the tab and you can name it appropriately.
Figure 4
1. First I want to see a practice sine wave. I know you’ve made Excel sine waves twice before in this course but to do
this exercise efficiently you should set things up thusly:
Figure 5
a. Obviously this is for a single wave; your lab will add a series of single waves, sines and cosines, at different
amplitudes and frequencies.
b. See how there is one cell above the sine function that will affect its amplitude and another that will affect its
frequency?
2. For your practice wave you may have 13 indices as in Figure 5, but for your actual synthesized waves I want to
see 257 for smoothness.
a. Construct your x values so that the range from 0 to 2 pi, both here and later on.
b. Construct your sine function so that by changing the value in ONE cell the height of the wave is adjusted.
c. Construct your sine function so that by changing the value in ONE cell the frequency of the wave is adjusted.
d. Place the small chart of your wave next to your calculations.
e. When you have an adjustable sine wave think of it as one of the twenty harmonics needed for the synthesis;
now move on to 3.
3. Construct a separate column for each harmonic’s sine and cosine and then sum them; do NOT put the harmonic
sum into one formula.
a. Your first column is the fundamental, n = 1 and a1 = 1. This is called normalizing; let all your other
amplitudes be < 1. b. You need not have every harmonic beyond n = 1: for instance, you might have n = 3, n = 6, n = 9, etc. It depends on what shape you are trying to emulate. c. You can get close, but not perfect reproductions. If the sawtooth wave is reversed, technically called a ramp wave, that will be sufficient. 5. Graph both functions. Search MathWorld Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Calculus and Analysis > Series > Fourier Series >
Interactive Entries > Interactive Demonstrations >
fourier series—sawtooth wave
Fourier Series­­Sawtooth Wave
THINGS TO TRY:
fourier series—sawtooth wave
Foundations of Mathematics
{25, 35, 10, 17, 29, 14, 21, 31}
Geometry
eigenvectors {{1,0,0},{0,0,1},
{0,1,0}}
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Approximation of
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David von Seggern
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Consider a string of length
plucked at the right end and fixed at the left. The functional form of this configuration is
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The components of the Fourier series are therefore given by
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(8)
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The Fourier series is therefore given by
(10)
(11)
(12)
Fourier Series, Fourier Series­­Square Wave, Fourier Series­­Triangle Wave, Sawtooth Wave
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 762­763, 1985.
CITE THIS AS:
Weisstein, Eric W. “Fourier Series­­Sawtooth Wave.” From MathWorld­­A Wolfram Web Resource.
http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html
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Fourier series square wave (2*pi*10*x)
Fourier Series­­Square Wave
THINGS TO TRY:
Fourier series square wave
(2*pi*10*x)
Foundations of Mathematics
Geometry
representations square wave(x)
History and Terminology
sum_(k=0)^infinity sin(2(1+2 k)
pi x)/(1+2 k)
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Consider a square wave
of length
. Over the range
, this can be written as
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, the function is odd, so
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The Fourier series is therefore
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Fourier Series, Fourier Series­­Sawtooth Wave, Fourier Series­­Triangle Wave, Gibbs Phenomenon, Square Wave
CITE THIS AS:
Weisstein, Eric W. “Fourier Series­­Square Wave.” From MathWorld­­A Wolfram Web Resource.
http://mathworld.wolfram.com/FourierSeriesSquareWave.html
Wolfram Web Resources
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technical.
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across science, mathematics,
finance, social sciences, and more.
Computerbasedmath.org »
Join the initiative for modernizing math
education.
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Solve integrals with Wolfram|Alpha.
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Walk through homework problems step­
by­step from beginning to end. Hints help
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Fourier Synthesis
1. Notice that this report only requires the excel file.
2. You need to start the lab by doing the sample given in Figure 5.
3. Refer to the uncertainty lab (lab 1) to review how to generate the X values. Please
remember that the x-values must be in radians and equally divided.
4. For the two exercises you need 257 x values, starting with zero.
5. I have included in the website two PDF files that show the derivation of the Fourier
Transform for the Square and the Sawtooth waves.
6. Notice that in both cases the an coefficient is zero. Therefore, we will only be using the
sine function part of the Fourier series.
7. Refer to the analysis lab to review how to include the amplitude and the harmonic
numbers (frequency) in the sine function equation
8. For the square wave the values of n are odd. Therefore, the harmonic numbers will be
odd values. Example n = 1, 3, 5,…
9. The function for the square wave is equals to 1/n * sin(n*x)
10. For the Sawtooth wave the values of harmonic numbers, n, are 1, 2, 3…
11. The function for the Sawtooth function is 1/n * sin (n*x)
12. Do at least 20 harmonics for each function.
13. The amplitude of each function is equal to 1/n.
14. Your last column should have the sum of the sine values for each harmonic.
15. You will graph the sum column in the y-axis and the x values in the x-axis.
16. Select line as your type chart.

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