HW #3: Using Spectra

One of the most important "tools" in the repertoire of an audio
specialist is the use of the Fourier transform to find the 
spectrum of a sound. The goal in this assignment is to 
learn as much as possible about a pair of bell strikes.

You may use any software at your command to carry out this
assignment (it will need to be able to manipulate sound .wav 
files, take their spectrum and draw spectrograms).
For concreteness, I will state the problem in terms 
of using Matlab to do the calculations.
If you are hazy on the FFT, the DFT, and related topics,
you may find the following useful:

http://www.cae.wisc.edu/~sethares/DFTchapter.pdf

especially the section 7.1.2 called "Using the DFT."
Here will find Matlab routines that mimic much of what is needed
in this assignment.

Download the sound file

http://www.cae.wisc.edu/~sethares/chinesebells.wav

Listen to it (either in Matlab using the "sound" or "play" command
or in your favorite audio player. You hear the same bell
begin struck twice, once on its front and once on its side.

Here is a transcript of a radio program talking about the
history of the bells

http://www.uh.edu/engines/epi175.htm

Your goal in this assignment is to carry out a complete
analysis of the two bell sounds.
In your analysis, please consider the following:

What are the most prominant partials (modes, or sine wave components) of
each sound.  How do these change over the duration of the sound?

How are the most prominent partials of the first strike
related to those of the second?

Are the sounds harmonic, that is, are the partials in
a harmonic series? Is there any pattern that you notice 
in the location of the partials?

What is the pitch of each sound? Explain how you determined the
pitches.

Explain how the pitch is (or is not)  related to the frequency of
the various partials.

How does the start of each sound compare to the latter portions.  Is there a
"steady state"?  Is there more than one "steady state"?  Is the ending
abrupt or drawn out?  Why is this so?

What portion of the energy is in the main modes, as compared to the "rest"
of the sound?  Be sure to describe the techniques/criteria you use to
distinguish the "partials" from the "remainder"?  What is the physical
meaning of these partials?  The physical meaning of the "remainder"?

Does the phase information in the FFT tell you anything?  What?

What are the sources of error?  How have you attempted to minimize these,
given the constraints imposed by the data?  Is the data "noisy", and what
does this mean (i.e., what are the likely sources of "noise")?

What does your ear tell you about the sound?  How
does this complement the more formal analysis?

As your analysis proceeds, be sure to explain how you got your results (how
was the sound partitioned, what window functions were used, the length of
the FFTs, what the accuracy of the results is, etc.).