Re: Wave decompositions

On Sat, 18 Oct 2008 06:00:14 EDT, riderofgiraffes
<mathforum.org_am@xxxxxxxxxxxxxx> wrote:

I'm trying to figure out whether a wave ...
some physical basis ... has some essential
"canonical" decomposition into sinusoid
waves
(snip)
Any periodic function (with various generous
assumptions that are covered by your "physical basis"
comment) is uniquely decomposable into a linear
combination of sin and cos functions by using Fourier.

This says that any "real, physical wave" really
is (in some sense) made up of sine waves (of some
amplitude and phase).

Wrong! or at least not quite right.

Some time ago:
On Tue, 06 Feb 2007 01:47:55 GMT, andy everett <vze2q...@xxxxxxxxxxx>
wrote:
http://groups.google.com/group/sci.math/msg/6f38ae8ab0baed5a?dmode=source

If a saw tooth wave can be represented by an alternating sum of sin
waves can a sin wave be represented by some sum of saw tooth waves?

Lee Rudolph wrote:

Now that I know the answer is yes it is easy to sketch the alternating
sum of the first couple of "triangle" waves with wavelength L, L/3, L/5,
... such a sum starts to "get smooth"

Thanks for an amusing challenge. You seem to have found your answer
but it took me a while to re-orient myself to waveform harmonics to
see the answer. Its a lot like division. The first order sawtooth
has sub-harmonics of period 1/2, 1/3, 1/4 etc. Use a second sawtooth
of 1/2 amplitude to suppress the second harmonic. Keeping track of
the new harmonics added to the result of this subtraction, just keep
subtracting the next remaining subharmonic until your sum starts to
"get smooth"

Netting this out--any waveform can be decomposed into any number of
orthogonal components. As to whether there is one unique
decomposition--that is one excellent question.

There is the Wold Decomposition Theorem. This theorem states that
any real-world process can be decomposed into a deterministic
component and a noise process. "The noise process can be modeled as
the output of a linear filter, excited at its input by a white noise
signal."

That is the closest relevent wisdom I am aware of. If seriously
pursued, I would start by finding a proof of Wold's theorem and see if
it can be adapted to the purpose.




.

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