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 (white). 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.

• Click the Clear button on the right-hand side of the applet window, then raise the first harmonic to an amplitude of 0.1, and then click the Save Waveform button near the bottom.
  
• Make sure the checkboxes for Show Saved Waveform and Show Superposition are both checked.
  
• Vary the amplitude and phase of the current waveform. Listen to the sound.
• What conclusion do you reach about the wave shape when two sine waves of the same frequency are added?
• Can you get the two waves to cancel one another?
• What are the two conditions necessary for cancellation?
  

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.

• Uncheck the checkboxes for Show Saved Waveform and Show Superposition to display only the current wave on the oscilloscope. Start with the fundamental (Harmonic 1). Rather than just playing with the many different harmonics in the hope of making a square wave by luck, it is better to draw a big square wave in your notebook and next draw in the fundamental sine wave, such that it resembles the square wave as closely as possible.
• Next ask yourself what higher harmonic should be added to get closer to the square wave — would the second or the third harmonic do a better job, and how should their phases be adjusted compared to the fundamental? One thing to keep in mind is that the square wave is mirror symmetric about an axis (can you point it out?), and the waves you use to build up the square wave should be mirror symmetric too.
• Try it out! The oscilloscope pictures below might give you some clues:
• 
• Use the mouse to adjust levels first to draw a rough square wave. Then, positioning the mouse over the amplitude you would like to adjust, use the up/down arrows (hold down the shift/ctrl keys for finer adjustments) to tune the components.
• Once you've created a decent square wave, use the table of the components to try to figure out the pattern of harmonics that creates a square wave.

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.

• Add two or more sine waves (for instance, harmonics 1, 2, and 3) of similar amplitudes.
• Change the phase of one of them (note the phase doesn't actually change until you release the mouse button).
  • Does the wave shape change?
  • Does the sound you hear change?
  • Does the Fourier spectrum of the tone change

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.

• Click the Sine wave function button in the right column.
• Look at the Fourier spectrum of the sine wave (amplitudes section below the waveform). Does it look like it should? (Note: Since it is a simple sine wave, there has to be just one component in the Fourier spectrum.) Move the cursor over the dot to get the frequency.  Read off the frequency and the amplitude of this component.

2. Fourier Spectrum of the Square Wave

• Now, click the Square wave function button. The Fourier spectrum of this square wave is displayed.
• From the screen, measure the amplitude and frequency of each harmonic and write them down in a table in your notebook. Can you observe some regularity in the amplitudes and frequencies of the harmonics? Recall what you did in the first part of the lab, where you generated the square wave with the help of the synthesizer.

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.

• Let's test the new applet--go back to the Fourier Synthesizer, click the Sine wave function like you did a few minutes ago, and hold the earphones up to the microphone.
• Do you see a single peak on the spectrum of the new applet?  Is it at the correct frequency?
  
• Sing a steady tone into the microphone, for instance a tone like “aah” in father. Watch the signal in the upper half of the screen and the Fourier spectrum of the tone in the lower half. Since it is hard to sustain the sound for the length of time needed to measure the spectrum, you can make use of another feature of the software. By clicking anywhere in the Fourier analysis window, you can freeze or release the waveform and frequency spectrum. You can then make measurements at your own pace.
  
• If you right-click (ctrl-click with one button mice), a popup menu appears that allows you to set the frequency maximum of the Fourier analyzer to 2000 Hz. Using the cursor, try to figure out the fundamental frequency of your voice. (Note: The fundamental is not always the first peak in the Fourier spectrum, nor is it always the highest! The oral cavity might amplify some overtones more than it does the fundamental. The fundamental frequency is determined by the rate of oscillations in your vocal cords, but only those overtones that are amplified by the oral cavity produce audible sound. Therefore, use the fact that if the harmonics are all multiple of the fundamental, they have to be equally spaced, with the spacing in frequency between them being equal to the fundamental! Remember, we use vocal cords, that is vocal strings, to produce sound.)
• A group of nearby frequencies is called a frequency range.  In the last step, you right-clicked to set the frequency range of the analyzer to be 0-2000 Hz. 
  
• The frequency range amplified by your oral cavity is called a “formant.” In what frequency range is your formant when you sing “aah?”
• Change the pitch of the “aah” and observe the change in the Fourier spectrum. Does the fundamental frequency change? Does the formant region change? To more easily observe the difference between two spectra, you can choose to save a waveform from the popup menu.
• Ask your TA to explain (once again) what timbre is!
• A more elaborate study of your voice, such as the analysis of different vowels, like “eeh” or “ooh,” can be used to find out what patterns of overtones makes one vowel different from another.