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Inverse Fourier/Laplace transform of a periodic function?

Started by Lucy August 11, 2006

Jerry Avins wrote:
 I took "reduce" to apply to amplitude. What's your meaning?
> =
Yes, reduce means lower amplitude from peak to peak. =
> =
> That is precisely correct, if by "terms", you mean terms in the Fourier=
> series, which was the clear (to me) implication of the discussion.
Yes, obviously I mean sinusoids which have cycles that are some integer multiple of the square-wave cycle. Anything else would obviously mot produce the desired periodic waveform. They also all have to have phase agreement to constructively contribute to a square wave shape. = The issue at hand is adding one more such component to an already existing summed series (there was a link given to a picture of an example). You get to choose the amplitude of the additional component. If you choose well it will reduce the ringing. You could if you wanted select such an additional component that would increase the ringing also - if that is what you desired. = -jim
> =
> > You can in fact reduce the ringing by adding more terms if you are fr=
ee
> > to add whatever terms you want which no one ever said you weren't fre=
e
> > to do. If you are going to correct someone for not being clear you ou=
ght
> > to at least make some attempt to be clear yourself. > =
> Dragging in elements outside the realm of discourse, especially after > the fact, amounts to pulling a fast one. I'm not interested in such > discussions; moreover, my heads-up to the unwary must surely have serve=
d
> its purpose by now. > =
> >> A basic phenomenon may be confusing you. All of the infinite ways to=
> >> suppressing ringing also soften the transition. That can be made ste=
eper
> >> again by adding terms. The added terms don't suppress the ringing, b=
ut
> >> they do compensate for the slope degradation that suppression method=
s
> >> impose. > > > > No I'm afraid it is you that is confused. The series that you are usi=
ng
> > is not the closest approximation to a square wave by any reasonable > > metric of the meaning of the word "closest". The series that you are > > talking about is the one that interpolates a square wave. That is, it=
> > coincides with a square wave at certain sample points. If the goal is=
to
> > come up with a set of sinusoids that add up to a waveform that is > > everywhere closer to a square wave and not just close at a few select=
ed
> > points then you can certainly do much better than that. > =
> The series I assumed is the Fourier series; the series that was being > discussed. Adding terms to the sigma-approximation series doesn't reduc=
e
> the ringing either. It has practically none at any count.
> =
> > The fact is, given the particular finite series of terms that t=
he
> > original poster (which you challenged) had presented, you can certain=
ly
> > add one or more terms which will result in making the waveform shape > > closer to a square-wave with less ripple. > =
> What does "less ripple" mean to you? If it means "peak amplitude of the=
> ripple", you're wrong. > =
> > You can easily do this yourself manually by making a reasonable=
guess
> > for the coefficient of an additional in phase sinusoid and seeing > > whether it moves the shape closer to the ideal square shape or not. > =
> If it's so easy, show us. > =
> Jerry > -- > Engineering is the art of making what you want from things you can get.=
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF ----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==---- http://www.newsfeeds.com The #1 Newsgroup Service in the World! 120,000+ Newsgroups ----= East and West-Coast Server Farms - Total Privacy via Encryption =----
jim wrote:
> > Jerry Avins wrote: > I took "reduce" to apply to amplitude. What's your meaning? > > Yes, reduce means lower amplitude from peak to peak. > >> That is precisely correct, if by "terms", you mean terms in the Fourier >> series, which was the clear (to me) implication of the discussion. > > Yes, obviously I mean sinusoids which have cycles that are some integer > multiple of the square-wave cycle. Anything else would obviously mot > produce the desired periodic waveform. They also all have to have phase > agreement to constructively contribute to a square wave shape. > The issue at hand is adding one more such component to an already > existing summed series (there was a link given to a picture of an > example). You get to choose the amplitude of the additional component. > If you choose well it will reduce the ringing. You could if you wanted > select such an additional component that would increase the ringing also > - if that is what you desired.
The amplitude of the nth harmonic of the Fourier series of a square wave of amplitude +/-1 is (4/pi)*1/(2n-1). You don't get to choose. Perhaps you mean some other series? If so, then in the context of a discussion such as this one, it is incumbent on you to say so. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
"Jerry Avins" <jya@ieee.org> wrote in message 
news:Bq-dnaxcGZ4zBH3ZnZ2dnUVZ_qGdnZ2d@rcn.net...
| Sorcerer wrote:
| > "Jerry Avins" <jya@ieee.org> wrote in message
| > news:rKydnX3pheBZY0LZnZ2dnUVZ_tCdnZ2d@rcn.net...
| > | Sorcerer wrote:
| > |
| > |   ...
| > |
| > | > Incidentally, overshoot is more voltage than was supplied.
| > | >  http://tinyurl.com/n67uy
| > |
| > | How does adding up odd harmonics of diminishing amplitude apply a 
voltage?
| > |
| > | Jerry
| > | -- 
| > | Engineering is the art of making what you want from things you can 
get.
| > | 
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
| >
| > That's the art of engineering, making what you want from things you can 
get.
| > Androcles.
|
| It's a poor engineer who can't distinguish a voltage from a
| trigonometric series.

It's a moronic troll that behaves as you do. Have a nice flame, then fuck 
off.
http://www.androcles01.pwp.blueyonder.co.uk/flame.gif

*plonk*
Androcles.





Jerry Avins wrote:

> =
> The amplitude of the nth harmonic of the Fourier series of a square wav=
e
> of amplitude +/-1 is (4/pi)*1/(2n-1). You don't get to choose. =
Oh really? well sorry to have to inform you, but I do get to choose. And if you ever decide you also get to choose the weighting of the components you can come up with much better approximations to a square wave than what you've got now. The original statement which you took offense at made it quite clear that the particular example given was just one example of a set of sine waves that can approximate a square wave. -jim
>Perhaps > you mean some other series? If so, then in the context of a discussion > such as this one, it is incumbent on you to say so. > =
> Jerry > -- > Engineering is the art of making what you want from things you can get.=
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF ----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==---- http://www.newsfeeds.com The #1 Newsgroup Service in the World! 120,000+ Newsgroups ----= East and West-Coast Server Farms - Total Privacy via Encryption =----
jim wrote:
> > Jerry Avins wrote: > >> The amplitude of the nth harmonic of the Fourier series of a square wave >> of amplitude +/-1 is (4/pi)*1/(2n-1). You don't get to choose. > > > Oh really? well sorry to have to inform you, but I do get to choose. And > if you ever decide you also get to choose the weighting of the > components you can come up with much better approximations to a square > wave than what you've got now.
There are many series that approximate a square wave, but only one Fourier series. If you choose a sigma-approximated series, there is no substantial Gibbs ringing for additional terms to reduce.
> The original statement which you took > offense at made it quite clear that the particular example given was > just one example of a set of sine waves that can approximate a square > wave.
I was not and am not offended. By initial remark was a warning to the unwary. Enough of this. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;

Jerry Avins wrote:
> =
> jim wrote: > > > > Jerry Avins wrote: > > > >> The amplitude of the nth harmonic of the Fourier series of a square =
wave
> >> of amplitude +/-1 is (4/pi)*1/(2n-1). You don't get to choose. > > > > > > Oh really? well sorry to have to inform you, but I do get to choose. =
And
> > if you ever decide you also get to choose the weighting of the > > components you can come up with much better approximations to a squar=
e
> > wave than what you've got now. > =
> There are many series that approximate a square wave, but only one > Fourier series. =
Nobody had mentioned fourier series until it was pointed out your remarks in regards to a sine series approximation of a square wave were incorrect. And now you say there is only one Fourier series. But there is more than one fourier series for a square-wave. Just shift the square wave in time and you get a completely different fourier series, So you're still wrong even after changing the subject. -jim =
>If you choose a sigma-approximated series, there is no > substantial Gibbs ringing for additional terms to reduce. > =
> > The original statement which you took > > offense at made it quite clear that the particular example given was > > just one example of a set of sine waves that can approximate a square=
> > wave. > =
> I was not and am not offended. By initial remark was a warning to the > unwary. > =
> Enough of this. > =
> Jerry > -- > Engineering is the art of making what you want from things you can get.=
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF ----== Posted via Newsfeeds.Com - Unlimited-Unrestricted-Secure Usenet News==---- http://www.newsfeeds.com The #1 Newsgroup Service in the World! 120,000+ Newsgroups ----= East and West-Coast Server Farms - Total Privacy via Encryption =----
Randy Poe wrote:

> robert bristow-johnson wrote:
(snip)
>>it *still* seems to me a semantic. whether we call it a "function" or >>a "distribution" or a "farg", it is what it is. we evidently can >>integrate it and get a non-zero real number, whatever we call it.
> Well, no you can't, and I think that's the main issue.
> The "integral" of the delta-"function" is not an ordinary > Riemann integral. You can't obtain it as the limit of a > sequence of Riemann sums.
You can't always do the exchange of the order of integration that is required, either, but that is also assumed. As I recently wrote in a completely unrelated discussion (related to shielded ethernet cable) much of physics is knowing which approximations to make, and when.
> You could define the delta-function delta(x) as the limit > of a sequence of functions f(a,x) as a->0 (gaussians for > instance), except that the limit of f(a,0) doesn't exist either. > It diverges.
They get much worse. I won't claim to understand it, but if you consider Feynman's explanation of Quantum Electrodynamics, it is much worse than just the value of one limit. To start, consider the energy of an electron of radius zero in its own electric field. (snip)
> When I'm dealing with delta's, I just tell myself "it's all OK, > this stuff doesn't really have meaning away from the limit > process and the integration, and so I'm just writing a short-hand > for those things." But that kind of assumption can sometimes > get you into trouble.
The trick is to know when the assumptions are valid and when they aren't. -- glen