How did...?

How did Fourier know in the first place that periodic vibrations were
comprised sine waves?

--Bhooshan
		
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The same way that Walsh knew they were made of square waves.
cheers, Syms.
"bhooshaniyer" <bhooshaniyer@gmail.com> wrote in message 
news:crSdnfsbmNIf4czfRVn-3g@giganews.com...
>
> How did Fourier know in the first place that periodic vibrations were
> comprised sine waves?
>
> --Bhooshan
>
> This message was sent using the Comp.DSP web interface on
> www.DSPRelated.com 

Hi--

>The same way that Walsh knew they were made of square waves.
>cheers, Syms.

Gee,Thanks! But let me confess that I was looking forward to a more
historical perspective to this stream of thought...

--Bhooshan
		
This message was sent using the Comp.DSP web interface on
www.DSPRelated.com

bhooshaniyer wrote:

> How did Fourier know in the first place that periodic vibrations were
> comprised sine waves?

 x" = -w^2 x

Ciao,

Peter K.

in article 1112638173.176761.54290@l41g2000cwc.googlegroups.com, Peter K. at
p.kootsookos@iolfree.ie wrote on 04/04/2005 14:09:

>> How did Fourier know in the first place that periodic vibrations were
>> comprised sine waves?
> 
> x" = -w^2 x

yeah, i think that's the answer.  sinusoids and exponentials have this nice
property that all the derivatives are sinusoids or exponentials.  that's
also why they are related to each other via Euler.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

bhooshaniyer wrote:
> How did Fourier know in the first place that periodic vibrations were
> comprised sine waves?
> 
> --Bhooshan
> 	

see http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html

as well as other links from
http://www.google.com/search?hl=en&lr=&q=%22Fourier++series%22+history+heat

Symon wrote:
> The same way that Walsh knew they were made of square waves.
> cheers, Syms.

And that Daubechie knew they were made of...  Well, whatever 
those recursive things are that bear her name.  :-)

BTW, wasn't it Harr that did the step function decomposition?

This is actually a rather important thing because many 
people actually do believe that signals are comprised of 
lots of sin waves.  Really hard these days to get past that 
misconception.

I believe that the fact that all of the discrete forms of 
these decompositions have a unifying theory belies Robert's 
belief that the Fourier one is special in somehow extending 
the knowledge of the argument beyond the bounds of the 
calculation.

I think it is the operations that he talks about, 
abbreviated convolution and the shift theorem, for example, 
which superimpose a periodic view.  Doesn't the shift 
theorem arise simply from a basis transformation which 
assumes periodicity of the basis?  It must be qualified with 
the conditional, "If the signal is periodic then..."  That 
periodicity is assumed nowhere in the calculation of the DFT 
itself.


Bob
-- 

"Things should be described as simply as possible, but no 
simpler."

                                              A. Einstein

Bob Cain wrote:

> That periodicity is assumed nowhere in the calculation of the DFT itself.

The periodicity is not assumed. It turns up on its own in any 
decomposition to continuous, non-time-limited functions.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Bob Cain wrote:

> I believe that the fact that all of the discrete forms of
> these decompositions have a unifying theory belies Robert's
> belief that the Fourier one is special in somehow extending
> the knowledge of the argument beyond the bounds of the
> calculation.

Except that the exponential/sinusoidal basis are the solution to
unforced LCCDEs (linear, constant coefficient difference [differential]
equations)... which is the basis for all filter theory.  The others are
not.

I do agree that the other bases are just as valid (and certainly have
sensible uses).

Ciao,

Peter K.

in article d31qvh02k41@enews3.newsguy.com, Bob Cain at
arcane@arcanemethods.com wrote on 04/06/2005 19:22:

> I believe that the fact that all of the discrete forms of
> these decompositions have a unifying theory belies Robert's
> belief that the Fourier one is special in somehow extending
> the knowledge of the argument beyond the bounds of the
> calculation.

i'm glad you too are reverting to the crutch of anthopomorhizing this
mindless mathematical operators.

one data is truncated and thrown away, no mortal knows what it was or could
have been with any certainty.  i'm not saying that the DFT "knew" the data
was periodically extended outside the data set given it.  i'm just saying it
"assumes" that it is.  that might be wrong for the DFT to do that, but
that's what it does.  and to forget that might get you into trouble.

this is not because if any issue regarding decompositions in general, it is
one of a specific decomposition to a bunch of periodic basis functions.
that example i gave Andor about the linear mapping that maps the N samples
to N coefficients of an (N-1)th order polynomial does not share this
property of periodically extension.  perhaps it would have a "polynomial
extension".  i dunno.  if it did, it would have to make use of that when
operations such as shifting are applied, but i haven't lookd into it.

i agree with Peter K. and whoever he was agreeing with that "... the other
bases are just as valid (and certainly have sensible uses).

> I think it is the operations that he talks about,
> abbreviated convolution and the shift theorem, for example,
> which superimpose a periodic view. Doesn't the shift
> theorem arise simply from a basis transformation which
> assumes periodicity of the basis?

i think so (if i understand your meaning).  it wouldn't be true, as stated,
if the basis were not periodic.  perhaps, with another transform (like
above) there is a shifting theorem that utilizes the extension of the basis
beyond the original domain of the input data, but again i dunno.

> It must be qualified with  the conditional, "If the signal is periodic
> then..."

No, Bob!  that's the point!  even if the original signal was not periodic
with period N, the DFT made it that way when you passed it those N samples
and you better get used to it!  it's kinda like a window function has made
all the data outside the window equal to zero, even if it wasn't before the
window was applied.

> That  periodicity is assumed nowhere in the calculation of the DFT itself.

and that is where we most fundamentally disagree.  when, in the context of
the DFT, we say that the spectrum X[k] and Y[k] are related by a factor of
exp(j*2*pi*d*k/N) (whether "d" as an integer or not), the DFT *is* assuming
that periodicity.  if "d" happens to be an integer, it is saying that y[n] =
x[n+d], but if you don't also understand that the DFT has made that
assumption, you have to say the clumsier y[n] = x[mod(n+d,N)] which i say is
tantamount to periodically extending x[n].

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."