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Inverse Fourier/Laplace transform of a periodic function?

Started by Lucy August 11, 2006
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. �����������������������������������������������������������������������
jim wrote:
> > Jerry Avins wrote: > > >> That too. Just don't tell a kid who doesn't know better that Mr. Gibbs >> will calm down if he gets enough terms. > > I don't get it why wouldn't you tell someone that. There are an infinite > number of ways you can weight the sinusoids to approximate a square wave > some will have more ripple some have less. The more terms you have the > closer you can get to an ideal square wave. That's essentially what fir > filter design is all about. You're not going to tell that kid if he > designs a half-band filter he is going to have to live with 18% (or > whatever the amount you have claimed) ripple are you?
To that's why I mentioned windows. You can also apply the sigma factor. But in the series we were discussing, a[sin(x) + sin(3x)/3 + sin(5x)/5 ...], the amplitude of the ringing is undiminished as terms are added. I didn't say that you can't suppress Gibbs ringing. I said you can't do it be adding terms. A basic phenomenon bay be confusing you. All of the infinite ways to suppressing ringing also soften the transition. That can be made steeper again by adding terms. The added terms don't suppress the ringing, but they do compensate for the slope degradation that suppression methods impose. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Sorcerer wrote:
> "Jan Panteltje" <pNaonStpealmtje@yahoo.com> wrote in message > news:ebn0rc$hqf$1@news.datemas.de... > | On a sunny day (Sat, 12 Aug 2006 18:44:35 -0500) it happened jim > | <"sjedgingN0sp"@m@mwt.net> wrote in > <1155425653_14911@sp6iad.superfeed.net>: > | > | > > | > > | >Jerry Avins wrote: > | > > | > > | >> > | >> That too. Just don't tell a kid who doesn't know better that Mr. Gibbs > | >> will calm down if he gets enough terms. > | > > | >I don't get it why wouldn't you tell someone that. There are an infinite > | >number of ways you can weight the sinusoids to approximate a square wave > | >some will have more ripple some have less. The more terms you have the > | >closer you can get to an ideal square wave. That's essentially what fir > | >filter design is all about. You're not going to tell that kid if he > | >designs a half-band filter he is going to have to live with 18% (or > | >whatever the amount you have claimed) ripple are you? > | > | Here is a nice free windows FIR filer design program. > | http://www.mediatronix.com/FIRTool.htm > | It allows you to play with parameters and shows the result. > | Supports some stuff, I am posting from sci.physics, you dsp guys probably > | have this.... made nice video filters with it in Xilinx FPGA. > | > | As for 'adding Farads' no, I disagree, that is tinkering, if you are > really > | good you know how to make a stable design. > | ;-) > | <insert flames> > > Ok... In theory, theory is the same as practise. In practise, they are > different. > > | <end flames> > | LOL > | > > I can recall when I was working with wire-wrap boards and a > decoupling capacitor was added adjacent to each logic device (chip) > at the supply pins. DEC PDP 11's had a wirewrap back plane as well. > The schematic showed all the capacitors connected in parallel so > they were wired in parallel, wire links from one to the next, AFTER > the wiring between devices had been installed, making a rat's nest > on top of a rat's nest. Electrically, the caps were nowhere close > to the chips, even though they were physically adjacent. Not surprisingly, > the board did not work as expected. > No matter how good you are or how good your prototype is, when > it comes into production you'll be thwarted by the manufacturing > dept. a year later when the designer has moved on to his next project, > or perhaps found another job. > Even a PCB layout will be carried out by a technician at a CAD > station, there will always be corrections needed that you call tinkering.
A colleague designed a CATV amplifier in which as much copper as possible was devoted to ground plane; it oscillated anyway. the ground copper formed a loop around the periphery of the board, as well as finger-like extensions to the interior when possible. I suggested cutting the loop to see how that effected stability. It still oscillated, but with an entirely different waveform. We bridged the gap with two free-standing turns of wire formed over a 3/16" rod, and all became stable. The schematic dutifully indicated an inductor grounded at both ends. 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;
In sci.physics, Jerry Avins
<jya@ieee.org>
 wrote
on Mon, 14 Aug 2006 00:40:32 -0400
<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
A square wave can be represented as sum (i=1,+oo) (sin( (2*i+1) * f * t / Pi) * (1/(2*i + 1))) where f is the desired frequency of the wave. If one fiddles with the primary (the i = 1 term) one gets a saggy wave. If one fiddles with the high frequencies one can probably get overshoot. -- #191, ewill3@earthlink.net Windows Vista. Because it's time to refresh your hardware. Trust us.

Jerry Avins wrote:

> > To that's why I mentioned windows. You can also apply the sigma factor. > But in the series we were discussing, a[sin(x) + sin(3x)/3 + sin(5x)/5 > ...],
Don't have a clue what windows or sigma have to do with it. You were responding to the statement: "After all is said and done a square wave is just a set of sine waves. For example....... ............ You can see the signal is "ringing" because insufficient terms were used" This statement was correct. One could add more terms (sinusoids) and reduce the ringing.
>the amplitude of the ringing is undiminished as terms are added. I > didn't say that you can't suppress Gibbs ringing. I said you can't do it > be adding terms.
But that's incorrect. You can in fact reduce the ringing by adding more terms if you are free to add whatever terms you want which no one ever said you weren't free to do. If you are going to correct someone for not being clear you ought to at least make some attempt to be clear yourself.
> > A basic phenomenon bay be confusing you. All of the infinite ways to > suppressing ringing also soften the transition. That can be made steeper > again by adding terms. The added terms don't suppress the ringing, but > they do compensate for the slope degradation that suppression methods > impose.
No I'm afraid it is you that is confused. The series that you are using 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 selected points then you can certainly do much better than that. The fact is, given the particular finite series of terms that the original poster (which you challenged) had presented, you can certainly add one or more terms which will result in making the waveform shape closer to a square-wave with less ripple. 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. -jim ----== 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 =----
"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.


"Jerry Avins" <jya@ieee.org> wrote in message 
news:8cOdnYgxY8uxnn3ZnZ2dnUVZ_uydnZ2d@rcn.net...
| Sorcerer wrote:
| > "Jan Panteltje" <pNaonStpealmtje@yahoo.com> wrote in message
| > news:ebn0rc$hqf$1@news.datemas.de...
| > | On a sunny day (Sat, 12 Aug 2006 18:44:35 -0500) it happened jim
| > | <"sjedgingN0sp"@m@mwt.net> wrote in
| > <1155425653_14911@sp6iad.superfeed.net>:
| > |
| > | >
| > | >
| > | >Jerry Avins wrote:
| > | >
| > | >
| > | >>
| > | >> That too. Just don't tell a kid who doesn't know better that Mr. 
Gibbs
| > | >> will calm down if he gets enough terms.
| > | >
| > | >I don't get it why wouldn't you tell someone that. There are an 
infinite
| > | >number of ways you can weight the sinusoids to approximate a square 
wave
| > | >some will have more ripple some have less. The more terms you have 
the
| > | >closer you can get to an ideal square wave. That's essentially what 
fir
| > | >filter design is all about. You're not going to tell that kid if he
| > | >designs a half-band filter he is going to have to live with 18% (or
| > | >whatever the amount you have claimed) ripple are you?
| > |
| > | Here is a nice free windows FIR filer design program.
| > | http://www.mediatronix.com/FIRTool.htm
| > | It allows you to play with parameters and shows the result.
| > | Supports some stuff, I am posting from sci.physics, you dsp guys 
probably
| > | have this.... made nice video filters with it in Xilinx FPGA.
| > |
| > | As for 'adding Farads' no, I disagree, that is tinkering, if you are
| > really
| > | good you know how to make a stable design.
| > | ;-)
| > | <insert flames>
| >
| > Ok... In theory, theory is the same as practise. In practise, they are
| > different.
| >
| > | <end flames>
| > | LOL
| > |
| >
| > I can recall when I was working with wire-wrap boards and a
| > decoupling capacitor was added adjacent to each logic device (chip)
| > at the supply pins. DEC PDP 11's had a wirewrap back plane as well.
| > The schematic showed all the capacitors connected in parallel so
| > they were wired in parallel, wire links from one to the next, AFTER
| > the wiring between devices had been installed, making a rat's nest
| > on top of a rat's nest. Electrically, the caps were nowhere close
| > to the chips, even though they were physically adjacent. Not 
surprisingly,
| > the board did not work as expected.
| > No matter how good you are or how good your prototype is, when
| > it comes into production you'll be thwarted by the manufacturing
| > dept. a year later when the designer has moved on to his next project,
| > or perhaps found another job.
| > Even a PCB layout will be carried out by a technician at a CAD
| > station, there will always be corrections needed that you call 
tinkering.
|
| A colleague designed a CATV amplifier in which as much copper as
| possible was devoted to ground plane; it oscillated anyway. the ground
| copper formed a loop around the periphery of the board, as well as
| finger-like extensions to the interior when possible. I suggested
| cutting the loop to see how that effected stability. It still
| oscillated, but with an entirely different waveform. We bridged the gap
| with two free-standing turns of wire formed over a 3/16" rod, and all
| became stable. The schematic dutifully indicated an inductor grounded at
| both ends.

Yep. Woe betide anyone attempting to build the amplifier from the
schematic alone, he needs the identical PCB.
Androcles.


|
| 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; 


The Ghost In The Machine wrote:
> In sci.physics, Jerry Avins > <jya@ieee.org> > wrote > on Mon, 14 Aug 2006 00:40:32 -0400 > <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 > > A square wave can be represented as > > sum (i=1,+oo) (sin( (2*i+1) * f * t / Pi) * (1/(2*i + 1))) > > where f is the desired frequency of the wave. If one fiddles > with the primary (the i = 1 term) one gets a saggy wave. > If one fiddles with the high frequencies one can probably get overshoot.
I'm rather poor at fiddling. Can you put a little rosin on the bow, so we get a better grip on things? If you eliminate the i = 1 term altogether, the lowest frequency of the composite is a third of the inverse of the period. What of it? You get overshoot at the corners no matter how few or many of the terms you choose to include. When you limit the count to 1, the "corners" merge halfway between and add, giving an "overshoot" of 4/pi. Revisit http://www.sosmath.com/fourier/fourier3/gibbs.html or plot a few waveforms instead of theorizing. 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;
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. 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;
jim wrote:
> > Jerry Avins wrote: > >> To that's why I mentioned windows. You can also apply the sigma factor. >> But in the series we were discussing, a[sin(x) + sin(3x)/3 + sin(5x)/5 >> ...], > > Don't have a clue what windows or sigma have to do with it. You were > responding to the statement: > > "After all is said and done a square wave is just a set of sine waves. > For example....... > ............ > You can see the signal is "ringing" because insufficient terms were > used" > > This statement was correct. One could add more terms (sinusoids) and > reduce the ringing.
Strictly, I suppose one needs to define "reduce". More terms will not reduce the peak amplitude of the ringing. More terms will reduce the period of the ringing and the fraction of the "flat" substantially affected by it. I took "reduce" to apply to amplitude. What's your meaning?
>> the amplitude of the ringing is undiminished as terms are added. I >> didn't say that you can't suppress Gibbs ringing. I said you can't do it >> be adding terms. > > But that's incorrect.
That is precisely correct, if by "terms", you mean terms in the Fourier series, which was the clear (to me) implication of the discussion.
> You can in fact reduce the ringing by adding more terms if you are free > to add whatever terms you want which no one ever said you weren't free > to do. If you are going to correct someone for not being clear you ought > 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 served 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 steeper >> again by adding terms. The added terms don't suppress the ringing, but >> they do compensate for the slope degradation that suppression methods >> impose. > > No I'm afraid it is you that is confused. The series that you are using > 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 selected > 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 reduce the ringing either. It has practically none at any count.
> The fact is, given the particular finite series of terms that the > original poster (which you challenged) had presented, you can certainly > 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. &#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;