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Re: analysis of sound
Posted:
Jun 14, 1996 3:36 PM


In article <2givd2u5xb.fsf@pulsar.wku.edu>, Allen Adler <adler@pulsar.wku.edu> wrote:
...
>It seems to me that the following assertion is being made, at least >implicitly, when one speaks of time dependent amplitudes for a signal, >particularly a signal from a musical instrument or ensemble. It is >asserted that: (1) the signal f(t) has an expansion > sum a_n(t) exp(2 pi i n t)
maybe, maybe not. Of course, any signal can be written in this form: f(t) =: { f(t)exp(i 2 pi nt) } * exp( i 2 pi nt ),
for any "n". But, the author may not be trying to get you to think along these lines with the words "timedependent amplitudes". The bottom line is that "timedependent amplitude" is a concept and not a mathematically defined term. It means whatever the author wants it to mean, and sometimes the author is even kind enough to define his particular meaning.
> >as a series which looks like a Fourier series except that the >coefficients are time dependent. Since such an expansion is >clearly not uniquely determined by f, it is also asserted that: ^^^^^^^^^^^^^^^^^^^^^^^ key words here. In fact, not even close to being uniquely determined in any context. The Fourier transform itself doesn't uniquely pair functions with other functions. But, with fairly mild constraints on the classes of functions your talking about, then its does. More specifically, when you're talking about physical energy signals, any two functions which have the same Fourier transform represent the same physical signal. The main point here. You can't talk in a meaningful way about an expansion/transform without putting some restrictions on the a_n(t). >(2) There is a natural experimental apparatus which measures a_n(t) > for each n and t (e.g. a tuning fork tuned to the right frequency > and whose sympathetic vibrations are observed and measured).
with a clarification of the a_n(t), there are many ways to do the measurements. I don't think I'd use the terminology "natural" to describe one of them. You may be jumbling a couple of ideas here. One is the measurement/estimation apparatus/algorithm, another is the transform specification, and maybe what are called the natural modes of a vibrating system. Once the transform is specified, the natural way to get the "coefficients" of the transform is to, well, apply the transform. How you implement the transform is another problem.
As an aside, if I were studying or describing tuning forks, I might give serious consideration to using a Laplace transform rather than a Fourier transform. It'll get you a little closer to describing whats really happening inside the fork. The complex frequencies of the poles of the system can be described as sinusoids with timedependent amplitudes. >(3) There is an explicit transform which one can use to compute a > priori, starting with f, the coefficient a_n(t) measured by the > experimental apparatus. > >If such an apparatus, such a transform and such an expansion could >be given explicitly, I would find it a lot easier to accept the >intuitive descriptions of time dependent amplitudes as actually >representing what it is claimed they represent. > >Can this be done and, if so, where is it written down in all detail?
It can be, and it's written down in excruciating detail. Probably in a lot more detail than you want, and not quite in the way you like would like it phrased. I think what may be a good starting point here is a engineering textbook on linear systems theory, and another on digital signal processing.
So I don't leave hanging here, I'll show you one way that you can turn your expansion into a legitimate transform. Maybe this is even what you had in mind. Lets call it the Narrowband Boxcar Filter Bank transform.
First lets generalize the expansion slightly so we write, x(t) = sum(n = inf:inf) [ a_n(t) exp(i 2 pi n f0 t) ].
Here f0 is a fixed parameter of the transform, which we choose for convenience.
We'll use the notation F{x(t)} for the Fourier transform.
So we say, X(f) = F{x(t)}, to indicate a transform pair, and x(t) = Fi{X(f)} for the inverse.
Now we'll restrict the functions a_n(t) to be low frequency functions, so that
A_n(f) = 0 for f > f0/2, where A_n(f) = F{a_n(t)}.
Define the Boxcar function,
B(f) = 1, f < f0/2, = 0, f > f0/2.
Using the convolution theorem for the Fourier transform you can also write our transform as,
X(f) = sum(n = inf:inf)[ A_n( f  n*f0 ) ].
The rest is simple,
A_n(f) = B(f)X( f + n*f0 ), n = inf:inf,
and a_n(t) = Fi{A_n(t)}, n = inf:inf.
hope this helps, Jim



