**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 a_{o}/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 a_{n} and b_{n} 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 a_{n} and b_{n} into a single A_{n} 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. The animation below approximates the process:

Animation 1

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 *A _{n}* and harmonic

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 create an easily adjustable spreadsheet to add twenty waves
- To see what combination of waves produces certain standard waveforms

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

- 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

- Obviously this is for a single wave; your lab will add a series of single waves, sines and cosines, at different amplitudes and frequencies.
- See how there is one cell above the sine function that will affect its amplitude and another that will affect its frequency?
- 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.
- Construct your x values so that the range from 0 to 2 pi, both here and later on.
- Construct your sine function so that by changing the value in ONE cell the height of the wave is adjusted.
- Construct your sine function so that by changing the value in ONE cell the frequency of the wave is adjusted.
- Place the small chart of your wave next to your calculations.
- 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.

- Construct a separate column for each harmonic's sine and cosine and then sum them; do NOT put the harmonic sum into one formula.
- Use your adjustable spreadsheet to synthesize a square wave and a sawtooth wave.
- Your first column is the fundamental, n = 1 and a
_{1}= 1. This is called normalizing; let all your other amplitudes be < 1. - 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.
- You can get close, but not perfect reproductions. If the sawtooth wave is reversed, technically called a ramp wave, that will be sufficient.
- Make a graph of the
*amplitude*vs*harmonic number*for each waveform. Note, this is not a waveform but a column graph similar but not identical to this one:

Figure 6