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.

Task:

To create an easily adjustable spreadsheet to add twenty waves

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.

4. Use your adjustable spreadsheet to synthesize a square wave and a sawtooth wave.

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.
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fourier series—sawtooth wave

Fourier SeriesSawtooth Wave

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fourier series—sawtooth wave

Foundations of Mathematics

{25, 35, 10, 17, 29, 14, 21, 31}

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The Fourier series is therefore given by

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(11)

(12)

SEE ALSO:

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.

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Fourier series square wave

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representations square wave(x)

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sum_(k=0)^infinity sin(2(1+2 k)

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SEE ALSO:

Fourier Series, Fourier SeriesSawtooth Wave, Fourier SeriesTriangle Wave, Gibbs Phenomenon, Square Wave

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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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