A simple-to-use sound file writer
Alf P. Steinbach
alfps at start.no
Sat Jan 16 02:33:59 CET 2010
* Steve Holden:
>
> For the record, yes, summing any waveforms that can be represented as
> Fourier Series will necessarily result in another Fourier series, since
> any linear combination of Fourier series must itself, be a Fourier
> series, and therefore the representation of the sum of the summed waveforms.
As it is I do not know whether the above represents what I've written, or might
perhaps be /misprepresenting/ the conclusions of this thread.
If I knew a lot more about Fourier series (it's been a long time since college!
lots forgotten) I might agree or disagree with the above as it applies to
constructing a sine wave from square waves.
I just note that representing a sine wave as square waves, forming it from
square waves, in the practical sense works (demonstrated here), even though it's
totally impractical :-), but just in answer to your original question.
And I note that in the mathematematical sense when n goes to infinity and
vanishingly thin pulses result from sums of square waves, like *impulse* waves,
then one is over in some regime where it is not at all clear to me that it is
valid to talk about Fourier series any more. Perhaps it's valid if the term
"Fourier series" does not necessarily imply sum of sine waves. I don't know.
I'm guessing that applying the Fourier series view for that is like actually
dividing by zero to maintain the running product of a collection of numbers when
an instance of 0 is removed from the collection.
It's no practical problem to maintain a running product (for programmers it's
interesting to note that polar representation complex numbers can do the job),
and it's well-defined also mathematically, with any /reasonable/ approach. But
the simple-minded direct way, just dividing the current product by the number
removed from the collection, is then invalid. And some people might take that
limited applicability of the direct simple way as evidence that it's impossible
to remove numbers, failing to see any of the trivial practical solutions. ;-)
Cheers,
- Alf
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