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Lab 6: Fourier Analysis

A steady musical tone from an instrument or a voice has, in most cases, quite a complicated wave shape. The oscillations repeat themselves f times a second, where f is called the fundamental frequency. We have learned that f is related to the pitch of the tone. Tones played on different instruments sound different — musicians say that the tones have different timbre or different tone color.

How does one describe wave shape? In Fourier Analysis we represent the complex wave shape as a sum of sine waves (or a sum of “partials”), each of a different amplitude. If the wave shape is periodic, the frequencies of the partials are multiples of the fundamental frequency and are called the “harmonics” of the tone being played. If the frequency of the musical tone is, for example, 200 Hz, the fundamental (also called the “first harmonic”) has a frequency of 200 Hz; the second harmonic (also called the first overtone) has a frequency of 400 Hz; the third harmonic (or second overtone) has a frequency of 600 Hz; and so on. Many musical instruments, including voices, have ten or more overtones.

A Fourier Analyzer is a device that tells us how much of the various overtones are present in the sound that is being analyzed, i.e. it calculates and displays a graph of the amplitude and the frequency of the various harmonics. Expressed in popular terms, the Fourier Analyzer gives the “voice print” or the “sound spectrum” of any periodic wave shape you feed into it.

Fourier Synthesis

Our Fourier synthesizer produces a fundamental mode of a given frequency and higher harmonics. The amplitude and phase of each of these waves can be adjusted. Extra features of the synthesizer are in the right column.

Study the regions on the Fourier synthesizer. An oscilloscope display at the top allows you to inspect the synthesized wave shape. Below that, the amplitude of each harmonic can be adjusted between 0 and 1 and the phases can range from -180° to +180° shift. To listen to changes in the tone quality, you use a small speaker or headphones.

1. Two Sine Waves of the Same Frequency

You will notice that the Fourier synthesizer has the ability to save a waveform and to show the current waveform (red) along with the saved waveform (blue) and a superposition of the two (green). This gives us the opportunity to study the wave shape that results when two waves are added — the questions of superposition. The simplest case adds two sine waves of the same frequency but different phase and different amplitude.

A point to remember when you are adding two sine waves of the same frequency is that the result of the superposition will depend on the relative phase of the two components being added. If they are in phase, the resultant sine wave will have large amplitude (the maximal resultant amplitude we can get). If the two superposed sine wave are out of phase, though, the resultant will have smaller amplitude. Exactly how much smaller depends upon exactly how much out of phase the waves are.

2. Building a Square Wave from Sine Waves

The next part of the game is to build a square wave by adding harmonics. Look at it as a puzzle.

3. Does One Hear Phase?

Checking the sound box allows you to hear the waves as you change their properties. Using this, answer the following questions.

This experiment shows why the Fourier spectrum is more useful to specify the tone “color” than the wave shape itself.

Fourier Analysis

1. Fourier Analysis of Sine Waves

In this part of the lab, we will analyze preset functions and also the signal picked up by a microphone when you sing a steady tone. To learn how to use the equipment, the preset functions are more convenient than your voice because they produce steady output whose frequency we can set accurately. We can also use the Fourier synthesizer to produce complex waveforms.

2. Fourier Spectrum of the Square Wave

3. Fourier Analysis of Your Voice

This is done with a different program. Download this applet and run it on your computer. Be sure a microphone is connected to the input of your computer.