Inverse Fourier/Laplace transform of a periodic function?

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>
<end flames>
LOL

"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.

Androcles.

On a sunny day (Sun, 13 Aug 2006 14:40:53 GMT) it happened "Sorcerer"
<Headmaster@hogwarts.physics_a> wrote in
<VRGDg.40762$Ca.22585@fe1.news.blueyonder.co.uk>:

>
>"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.
>
>Androcles.

Oh, yes, I had a profi company do the layouts for the boards in IBM PC,
whole lot of phone calls from them, discussions about _where_ to put that
track.
But that was when we had no CAD ourselves (beginning of eighties).
And frequencies where in the MHz region.
These days you can get very highly specialized packages for very high
frequencies (several GHz, say rocket IO etc) that will make nice layout.
As for decoupling, I have heard SO many different stories over the years,
and I have _done_ wire wrap too in the old ages, use your common sense,
experience counts++++, it gets trikcy putting caps under a ball grid FPGA ..
and what caps.
Frequencies these days are indeed in the GHz range, we have entered the pico
second area, an antique wire wrap would even be too long as antenna.. ;-)

Anyways, what I am saying is, that _without_ those specialized layout
packages you will _NOT_ be able to someting like 1GHz boards correctly.
On the opther hand in the lab there was this person who did it .. and it
worked always first time, I have asked... but it is likely a gift.
Remember building a 10GHz link with his help...

It is all simulations, Androcles, the ciruits, the filters, the FPGA
logic, the layouts perhaps, before anything gets build.
Not adding a cap here or there later, although it may happen.

Problem I have is that these parts gets so small I can hardly see them.
And boards have so many layers you cannot figure out anything from looking
at the copper side :-)

For the generation that grows up now, all this stuff is normal, just do not
ask them to design a 10 tube BW TV.
Couple of million transistors .... is what they need now.

"Jan Panteltje" <pNaonStpealmtje@yahoo.com> wrote in message 
news:ebngvr$c5v$1@news.datemas.de...
| On a sunny day (Sun, 13 Aug 2006 14:40:53 GMT) it happened "Sorcerer"
| <Headmaster@hogwarts.physics_a> wrote in
| <VRGDg.40762$Ca.22585@fe1.news.blueyonder.co.uk>:
|
| >
| >"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.
| >
| >Androcles.
|
| Oh, yes, I had a profi company do the layouts for the boards in IBM PC,
| whole lot of phone calls from them, discussions about _where_ to put that
| track.
| But that was when we had no CAD ourselves (beginning of eighties).
| And frequencies where in the MHz region.
| These days you can get very highly specialized packages for very high
| frequencies (several GHz, say rocket IO etc) that will make nice layout.
| As for decoupling, I have heard SO many different stories over the years,
| and I have _done_ wire wrap too in the old ages, use your common sense,
| experience counts++++, it gets trikcy putting caps under a ball grid FPGA 
..
| and what caps.
| Frequencies these days are indeed in the GHz range, we have entered the 
pico
| second area, an antique wire wrap would even be too long as antenna.. ;-)
|
| Anyways, what I am saying is, that _without_ those specialized layout
| packages you will _NOT_ be able to someting like 1GHz boards correctly.
| On the opther hand in the lab there was this person who did it .. and it
| worked always first time, I have asked... but it is likely a gift.
| Remember building a 10GHz link with his help...
|
| It is all simulations, Androcles, the ciruits, the filters, the FPGA
| logic, the layouts perhaps, before anything gets build.
| Not adding a cap here or there later, although it may happen.
|
| Problem I have is that these parts gets so small I can hardly see them.
| And boards have so many layers you cannot figure out anything from looking
| at the copper side :-)
|
| For the generation that grows up now, all this stuff is normal, just do 
not
| ask them to design a 10 tube BW TV.
| Couple of million transistors .... is what they need now.
|
Yep... or an analogue computer. Thing is, to design something without
basic knowledge (or experience) of square wave behaviour will invariably
lead to disaster, so Lucy wants a firm grip on the mathematics.
But after all is said and done, it is very academic to say that we can
call any waveform an infinite set of sine waves, but you and I know
the only waves we are really interested in are sine, square and triangular
(for flyback) and  we are not about to create an infinite set of oscillators
and sum their outputs to create a square wave.
With the demise of the tube in favour of thin screen technology,
who needs flyback either? TV transmission is simply serialising the
data, but it doesn't have to be right to left, right to left, it can be
reversed on the next line.
 Without EHT and with data compression techniques it simply
isn't needed  anymore, so why keep it after the last few old TV's
have  burned out?
Androcles

glen herrmannsfeldt wrote:
> robert bristow-johnson wrote:
>
> > Lucy wrote:
>
> >>Yes, you can do that. But I am looking for a derivation which naturally
> >>goes from the integral from -inf to +inf, and then to the integral from
> >>-pi to +pi, and then to the sampling for the inversed function f(t), as
> >>you've said.
>
> > well, it depends on how rigorous you want to be.  the folks on sci.math
> > might blanch at the way we comp.dspers treat the Dirac impulse
> > function.
>
> Not to mention quantum mechanics, where it came from, as far as I know.

this is why, after we had another tiff here in comp.dsp a while ago
about the nature of the dirac delta, i decided to ask some hard core
physikers about it on sci.physics.research:

http://groups.google.com/group/sci.physics.research/browse_frm/thread/89bd3c294c5e5c47

essentially, what i got from the conversation is that a "function" can
only be defined by the strict mapping: what sends "x" to "f(x)".  it
doesn't have the right to be defined as a limit in the sense we
(engineers) do for the Dirac delta.  it isn't allowed to "remember" how
it may have been defined from a limit of real functions. it can only be
defined in terms of how it maps the real number "x" to another real
number "y" = f(x).

i.e. there are a lot (an infinite number) of possibilities for some
f(x) = 0 for |x|>0 and infinite (or undefined) for x=0.  for f(x) to be
a normal real function (as defined by the powers that be), we cannot
add to its definition any more than the function mapping (as stated
above).  we cannot add to its definition that it was a limit of some
other functions that all had an area of 1.  only the basic real-to-real
mapping is allowed.

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.  the
fact that it is not *physically* realizable was never the issue for me.
 we have concepts in mathematics, like infinities and imaginary
numbers, that do not exist in quantities of physical phenomena but they
conceptually exist in mathematical usage.

> There is a story about someone rewriting a QM book to remove all
> delta functions, because they didn't believe in them.
>
> Believe them or not, they do make many explanations much simpler.

that's what i think.  i still have a problem with the traditional
mathematicians "The Dirac-delta function is not a function, so you
can't do function-like things with it, such as define a Fourier series
for the Dirac comb."  and fairly recently at Wikipedia, in the talk
page of the Nyquist-Shannon Sampling Theorem, i got into the same
dispute with a German guy who felt it was completely fictitous to use
the identity below.  ("The Dirac comb is not really a function so you
can't have a Fourier Series for it.")

> > Lucy, can you accept the following identity?:
>
> >    +inf                     +inf
> >     SUM{ delta(t - n) }   =  SUM{ exp(i*2*pi*k*t) }
> >   n=-inf                    k=-inf
>
>
> It looks strange with k, which should be wave number or
> wave vector, but otherwise I agree.

for me, it's just a Fourier series that is easily determined.  "k" can
represent whatever you want.  for me it's "harmonic number".

r b-j

On a sunny day (Sun, 13 Aug 2006 17:01:29 GMT) it happened "Sorcerer"
<Headmaster@hogwarts.physics_a> wrote in
<JVIDg.89529$F8.78946@fe3.news.blueyonder.co.uk>:
 For the generation that grows up now, all this stuff is normal, just do 
>not
>| ask them to design a 10 tube BW TV.
>| Couple of million transistors .... is what they need now.
>|
>Yep... or an analogue computer. Thing is, to design something without
>basic knowledge (or experience) of square wave behaviour will invariably
>lead to disaster, so Lucy wants a firm grip on the mathematics.
>But after all is said and done, it is very academic to say that we can
>call any waveform an infinite set of sine waves, but you and I know
>the only waves we are really interested in are sine, square and triangular
>(for flyback) and  we are not about to create an infinite set of oscillators
>and sum their outputs to create a square wave.
>With the demise of the tube in favour of thin screen technology,
>who needs flyback either? TV transmission is simply serialising the
>data, but it doesn't have to be right to left, right to left, it can be
>reversed on the next line.
> Without EHT and with data compression techniques it simply
>isn't needed  anymore, so why keep it after the last few old TV's
>have  burned out?
>Androcles

Well the slogan Philips came with was 'lets go digital'.
Somebody in that same Philips once said: 'Why go digital if all you get
is more hum in the sound'.
Those times are of the past now, but:

It is interesting that these days sometimes to add 2 signals an AD, 
DSP or microprocessor, and AD are used.... Oh and the related filters :-)
and not simply 2 resistors.

My Creative SB live sound card resamples for the mixer and has more
distortion that you can clearly hear too, then an analog mixer...
I have build analog mixers as far back as i can remember, even in the
school days, that were better then that.

As far as waveforms are concerned, no, it gets more complex actually.
I have done design of EHT and deflection circuits for both tube and
transistor TV, it is not 'just a triangle'.
But in digital it gets a lot more complicated when you talk about resizing
say a 1920x1080 HDTV image to a 1024x768 monitor.
Those algorithms are really have math related, and set the display quality.

So, modern tech _can_ be better, but what is brought on the market often is
NOT.
LCD TVs still give a _lousy_ picture compared to a good CRT, yet CRT is
fading out.
They sell 300 Watt PC speaker boxes, in the old days 300 Watt would be RMS
and get you in trouble if you played it in the evening :-)
(I did).
 
Unfortunately consumers have no clue, and sales people neither, industry uses
that to sell ever new [not better] stuff.

mp3, well it sucks to be honest, more memory will now allow you to store
or copy the CDs in original wave format... problem solved.
So it will catch up, by the time all problems are fixed something else
will be on the shelves, some new technology, that will have its own problems,
_or_ it all fades in the dark ages that follow WW3 etc.. like much of the
knowledge of old civilisations was lost, and only now we find things like :
'How the Romans measured distance in their roads etc'.
Our stuff will be digged up: 'Look they already made circuits on silicon in
the twenty first century' (in 3035), all over again.

35 years ago we could land on the moon and return, NASA is working hard to
do _the same thing_.

Days of future.....

 

"Jan Panteltje" <pNaonStpealmtje@yahoo.com> wrote in message 
news:ebnt5c$5g3$1@news.datemas.de...
| On a sunny day (Sun, 13 Aug 2006 17:01:29 GMT) it happened "Sorcerer"
| <Headmaster@hogwarts.physics_a> wrote in
| <JVIDg.89529$F8.78946@fe3.news.blueyonder.co.uk>:
| For the generation that grows up now, all this stuff is normal, just do
| >not
| >| ask them to design a 10 tube BW TV.
| >| Couple of million transistors .... is what they need now.
| >|
| >Yep... or an analogue computer. Thing is, to design something without
| >basic knowledge (or experience) of square wave behaviour will invariably
| >lead to disaster, so Lucy wants a firm grip on the mathematics.
| >But after all is said and done, it is very academic to say that we can
| >call any waveform an infinite set of sine waves, but you and I know
| >the only waves we are really interested in are sine, square and 
triangular
| >(for flyback) and  we are not about to create an infinite set of 
oscillators
| >and sum their outputs to create a square wave.
| >With the demise of the tube in favour of thin screen technology,
| >who needs flyback either? TV transmission is simply serialising the
| >data, but it doesn't have to be right to left, right to left, it can be
| >reversed on the next line.
| > Without EHT and with data compression techniques it simply
| >isn't needed  anymore, so why keep it after the last few old TV's
| >have  burned out?
| >Androcles
|
| Well the slogan Philips came with was 'lets go digital'.
| Somebody in that same Philips once said: 'Why go digital if all you get
| is more hum in the sound'.
| Those times are of the past now, but:
|
| It is interesting that these days sometimes to add 2 signals an AD,
| DSP or microprocessor, and AD are used.... Oh and the related filters :-)
| and not simply 2 resistors.
|
| My Creative SB live sound card resamples for the mixer and has more
| distortion that you can clearly hear too, then an analog mixer...
| I have build analog mixers as far back as i can remember, even in the
| school days, that were better then that.
|
| As far as waveforms are concerned, no, it gets more complex actually.
| I have done design of EHT and deflection circuits for both tube and
| transistor TV, it is not 'just a triangle'.
| But in digital it gets a lot more complicated when you talk about resizing
| say a 1920x1080 HDTV image to a 1024x768 monitor.
| Those algorithms are really have math related, and set the display 
quality.
|
| So, modern tech _can_ be better, but what is brought on the market often 
is
| NOT.
| LCD TVs still give a _lousy_ picture compared to a good CRT, yet CRT is
| fading out.
| They sell 300 Watt PC speaker boxes, in the old days 300 Watt would be RMS
| and get you in trouble if you played it in the evening :-)
| (I did).
|
| Unfortunately consumers have no clue, and sales people neither, industry 
uses
| that to sell ever new [not better] stuff.
|
| mp3, well it sucks to be honest, more memory will now allow you to store
| or copy the CDs in original wave format... problem solved.
| So it will catch up, by the time all problems are fixed something else
| will be on the shelves, some new technology, that will have its own 
problems,
| _or_ it all fades in the dark ages that follow WW3 etc.. like much of the
| knowledge of old civilisations was lost, and only now we find things like 
:
| 'How the Romans measured distance in their roads etc'.
| Our stuff will be digged up: 'Look they already made circuits on silicon 
in
| the twenty first century' (in 3035), all over again.
|
| 35 years ago we could land on the moon and return, NASA is working hard to
| do _the same thing_.
|
| Days of future.....
|
The ultimate sound...  remember the "player piano" with a punched
paper roll that knocked out a tune automatically?

They are still business.
 http://mmd.foxtail.com/Exchange/


 Now, today, suppose we replaced the paper roll with a CD and
a frequency analyser and played a real piano from the recording...
What would vocal music sound like played by a piano?
Androcles

robert bristow-johnson wrote:
> glen herrmannsfeldt wrote:
> > robert bristow-johnson wrote:
> >
> > > Lucy wrote:
> >
> > >>Yes, you can do that. But I am looking for a derivation which naturally
> > >>goes from the integral from -inf to +inf, and then to the integral from
> > >>-pi to +pi, and then to the sampling for the inversed function f(t), as
> > >>you've said.
> >
> > > well, it depends on how rigorous you want to be.  the folks on sci.math
> > > might blanch at the way we comp.dspers treat the Dirac impulse
> > > function.
> >
> > Not to mention quantum mechanics, where it came from, as far as I know.
>
> this is why, after we had another tiff here in comp.dsp a while ago
> about the nature of the dirac delta, i decided to ask some hard core
> physikers about it on sci.physics.research:
>
> http://groups.google.com/group/sci.physics.research/browse_frm/thread/89bd3c294c5e5c47
>
> essentially, what i got from the conversation is that a "function" can
> only be defined by the strict mapping: what sends "x" to "f(x)".  it
> doesn't have the right to be defined as a limit in the sense we
> (engineers) do for the Dirac delta.  it isn't allowed to "remember" how
> it may have been defined from a limit of real functions. it can only be
> defined in terms of how it maps the real number "x" to another real
> number "y" = f(x).
>
> i.e. there are a lot (an infinite number) of possibilities for some
> f(x) = 0 for |x|>0 and infinite (or undefined) for x=0.  for f(x) to be
> a normal real function (as defined by the powers that be), we cannot
> add to its definition any more than the function mapping (as stated
> above).  we cannot add to its definition that it was a limit of some
> other functions that all had an area of 1.  only the basic real-to-real
> mapping is allowed.
>
> 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 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.

However, the integral of f(a,x) is defined for every a, and so
you can say "the integral of delta(x)" when what you really
mean is "the limit of the integral of f(a,x) as a->0". But that
isn't the same as the integral of the limit.

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.

That's why there's a necessity to build a rigorous theory of
what is meant by delta and its integrals (and derivatives too I guess).

                   - Randy

Randy Poe wrote:
> robert bristow-johnson wrote:

> > essentially, what i got from the conversation is that a "function" can
> > only be defined by the strict mapping: what sends "x" to "f(x)".  it
> > doesn't have the right to be defined as a limit in the sense we
> > (engineers) do for the Dirac delta.  it isn't allowed to "remember" how
> > it may have been defined from a limit of real functions. it can only be
> > defined in terms of how it maps the real number "x" to another real
> > number "y" = f(x).
> >
> > i.e. there are a lot (an infinite number) of possibilities for some
> > f(x) = 0 for |x|>0 and infinite (or undefined) for x=0.  for f(x) to be
> > a normal real function (as defined by the powers that be), we cannot
> > add to its definition any more than the function mapping (as stated
> > above).  we cannot add to its definition that it was a limit of some
> > other functions that all had an area of 1.  only the basic real-to-real
> > mapping is allowed.
> >
> > 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.

it is.  but you might be amazed at the quantity of electrical
engineering textbooks that do this.  practically any text on "Linear
System Theory" or "Signals and Systems" or the "Fundamentals of
Communications Systems" simply defines the Dirac delta function
(sometimes they call it the "unit impulse function") as this limit of
functions (what you are calling "f(a,x)" with a -> 0 but usually with a
"t" instead of an "x") and not mentioning anything about it being a
"distribution" or "not a function" or a "generalized function" or
whatever.

even graduate level EE texts (except, perhaps, Papoulos).  as if
electrical engineering students will never find the time to take a Real
Analysis course in the math department.  but i tell you, that very day
i asked the math prof about the Dirac delta when we learned Lebesgue
integration and that if f(x) = g(x) "almost everywhere", then the
integrals have to be the same.  (the problem, of course is that,
treating delta(x) like a function, f(x) = delta(x) is equal to g(x) = 0
everywhere except x=0 and their integrals are different.)

> 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.

and you can't even integrate it with the Lebesgue integral.  we EEs say
the integral is 1 but something that is zero almost everywhere has to
have an integral of 0.

> 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.
>
> However, the integral of f(a,x) is defined for every a, and so
> you can say "the integral of delta(x)" when what you really
> mean is "the limit of the integral of f(a,x) as a->0". But that
> isn't the same as the integral of the limit.

but we EEs treat it as if it is the same.

> 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.

well, what i do to hand-wave this problem away is treat the delta as
f(a,t) where "a" is not quite zero but smaller than anything i have to
worry about.  let "t" be time, then i treat the delta as f(a,t) with
a=Planck_time ~= 10^-44 second.  that's still a legitimate function,
for the sake of the hard core mathematicians, but it's behavior is so
close to how we think of the Dirac delta in *any* conceivable physical
situation, that the difference is not something to worry about.

nonetheless, what gets me into a "fighting" mood, is when i say that

        +inf                        +inf
    T * SUM{ delta(t - n*T) }   =   SUM{ exp(i*2*pi*k*t/T) }
       n=-inf                      k=-inf

and use it to, say, prove the Nyquist/Shannon Sampling & Reconstruction
Theorem or to derive the effect (on frequency response) of some
practical reconstruction like the Zero-Order Hold (because we can't
really be brick-wall filtering a string of weighted dirac impulse
functions because no real electronics will produce such a sequence of
impulses) and some math guy tells me i can't do that because "the Dirac
delta is not really a function and it is meaningless to express that
sequence of Dirac impulses as a Fourier Series".  when i hear that, i
do not let it slide.  about 20 kilograms of textbooks would have to be
tossed if we are not "allowed" to do that.

> That's why there's a necessity to build a rigorous theory of
> what is meant by delta and its integrals (and derivatives too I guess).

a necessity in the pure mathematics discipline, not in the engineering
(or, perhaps, physics) discipline.

r b-j

Randy Poe wrote:

   ...

> 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.
> 
> That's why there's a necessity to build a rigorous theory of
> what is meant by delta and its integrals (and derivatives too I guess).

And then, of course, we all need to wave out arms when call a "unit 
impulse" a single sample of amplitude 1.

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