| Author |
Message |
David C. Ullrich
Guest
|
 Posted:
Sat Dec 11, 2004 6:47 pm Post subject:
Re: Impossible sampling theory! |
|
|
On Fri, 10 Dec 2004 08:30:58 -0800, Tim Wescott
<tim@wescottnospamdesign.com> wrote:
| Quote: | David C. Ullrich wrote:
Can you recommend a good website or book?
If you want the whole story you need to learn some
"real analysis" first (measure theory, topological
vector spaces, etc). There are many places you can
find the theory of distributions worked out in
detail - the two that are standard texts where I
come from would be Folland "Real Analysis" (or
maybe it's "Real Analysis and Applications" or
something) and Rudin "Functional Analysis".
Website? Hmm, google...
It seems that wikipedia
http://en.wikipedia.org/wiki/Distribution
discusses the topic, although I doubt that
there's a complete exposition of the theory
there, in an "encyclopedia" they must have
just statements of the main results.
The description of
http://www.math.ku.dk/~grubb/distcon.pdf
on google sounds like it might be what
you want, but that actual pdf is just a
table of contents. I didn't see how to find
the actual notes on the site, but maybe you
can if you hunt around.
Otoh I wouldn't be surprised if there is no
web site that actually contains the whole story.
************************
David C. Ullrich
I have "Intermediate Real Analysis" by Emanuel Fischer, but if it
discusses those topics it does so by entirely different names -- I
rather suspect that it leaves off where your other texts start.
|
I don't know that book but based on the title that seems likely -
"real analysis" covers a lot of ground.
| Quote: | So far any time I've felt a need for rigor around the delta function
(distribution, whatever) I've just constructed some real function with
area one that's either parameterized by height (or width), found my
result, then taken the limit as the parameter goes to infinity (or
zero). It's probably not entirely kosher, but it's served my purposes.
|
Can't say for sure without seeing exactly what you've done, but
that could very well be just fine (although it's not _really_ ok
unless you can explain why...). For example:
Say f_n(t) = n for 0 < t < 1/n, 0 for other t. Then f_n -> delta
"in the sense of distributions" as n -> infinity. What convergence
"in the sense of distributions" means is that if g is an
infinitely differentiable function then
(*) int f_n g -> int delta g = g(0) as n -> infinity.
(Here int is the integral from -infinity to infinity.)
If all you're doing is things that look like (*) then
the things you're doing are ok.
************************
David C. Ullrich |
|
| Back to top |
|
 |
David C. Ullrich
Guest
|
| Posted:
Sat Dec 11, 2004 6:51 pm Post subject:
Re: Impossible sampling theory! |
|
|
On Sat, 11 Dec 2004 10:38:41 -0000, "Airy R. Bean" <me@privacy.net>
wrote:
| Quote: | "Dracaena"?? "Diracian"!!! The perils of using a
spell checker mechanically!
|
Coming from you that's very funny. Hint: there's
no such word as "Diracian". Even though you use
the word all the time. (Ok, it appears that
there is such a word. But it's not a noun referring
to the delta function, execpt in your posts.)
| Quote: | "Airy R. Bean" <me@privacy.net> wrote in message
news:3200rtF3fjpeeU11@individual.net...
Not disputed at all.
However, the sampling waveforms in practice
are not represented by an area of 1.
If anyone wishes to claim that their sampled
signals are represented in some way by the Diracian,
then they must mentally model an invisible scaling factor to
bring the magnitudes of their sampled waveforms
into the order of magnitude of the attributes of the
Dracaena.
|
************************
David C. Ullrich |
|
| Back to top |
|
|
David Kastrup
Guest
|
| Posted:
Sat Dec 11, 2004 9:22 pm Post subject:
Re: Impossible sampling theory! |
|
|
"Airy R. Bean" <me@privacy.net> writes:
| Quote: | Not disputed at all.
However, the sampling waveforms in practice
are not represented by an area of 1.
If anyone wishes to claim that their sampled
signals are represented in some way by the Diracian,
then they must mentally model an invisible scaling factor to
bring the magnitudes of their sampled waveforms
into the order of magnitude of the attributes of the
Dracaena.
I enquired as to how others came to terms with this
blatant discrepancy.
|
Well, if you want to work with the Fourier transform of a sampled
signal, the integral over the sampled function must be non-zero, and
you just have isolated points with content. Something has to give,
and it turned out that it was easiest to sacrifice strict point-wise
defined functions since they don't survive integration and
differention unmolested, anyway. And if one wants to have an entity
that will do the right things under integral transforms, it is
probably easiest to define it by its behavior for integration in the
first place: distributions. Stieltjes integrals might be related,
though.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum |
|
| Back to top |
|
|
Tachyon
Guest
|
| Posted:
Sun Dec 12, 2004 6:03 am Post subject:
Re: Impossible sampling theory! |
|
|
On 2004-12-11, Airy R. Bean <me@privacy.net> wrote:
| Quote: | As indeed do I. All mathematics when applied to
engineering is a tool or a model.
Certainly, we use it as a tool in the Analogue
Signal Processing when we consider the Impulse
Response of our networks.
But in the case of sampling, we are not exciting
our circuit designs with a tool signal - we generate
a specific sampling waveform whose attributes
are not those of the Diracian, and therefore the
derivation is erroneous.
|
The purpose of using the Dirac integral is to capture a
level at an instant in time. In real-life sampling, the average,
or something close to it, over a finite time is captured.
When the bandwidth of the signal is small
relevant to the reciprocal of the time interval, then the
signal isn't changing much in that time interval. It
is like DC. Averaging it over the time interval and averaging
it over 1/10 the interval give the same results. So would
averaging it over an infinitely small time interval.
| Quote: | However, the Diracian is not a model of signal
that appears in real-world sampling circuits.
|
Something very close does. It produces nearly the same
result the dirac delta function would.
| Quote: | All other mathematics that the electronic engineer
encounters have values that represent those
measurable from the circuits under consideration.
|
Well a lot of people can get away without ever
thinking about how a real a/d converter works
because they're buying working modules. But a
real designer would have to understand it. The
relavent design parameters are the bandwidth
of the signal in the stopband, and the convergence
time of the sample and hold. The closer
that s&h works to a dirac, the better the sample
quality will be.
I'm sure it's pretty trivial in the audio range,
but digitizing satellite data etc is probably
a lot harder.
| Quote: | "ETS" <emale80919@yahoo.com> wrote in message
news:ADkud.35028$Mu3.2413177@twister.southeast.rr.com...
What about this, do you believe the dirac impulse is a real function or
not? I look at it as a mathematical tool, that is all.
|
--
different MP3 every day! http://gweep.net/~shifty/snackmaster
. . . . . . . . ... . . . . . .
"Anything moving in the zone, even a three-year- | Niente
old, needs to be killed." -Captain R. | shifty@gweep.net |
|
| Back to top |
|
|
Tachyon
Guest
|
| Posted:
Sun Dec 12, 2004 6:17 am Post subject:
Re: Impossible sampling theory! |
|
|
On 2004-12-10, ETS <emale80919@yahoo.com> wrote:
| Quote: | You can't take a dirac in pieces, you have to take the amplitude (oo)
and the width 0 for a dirac to work.
|
i don't believe the dirac has width 0. a quick web search shows
mixed results:
Non-Zero
--------
" a very tall narrow spike of height 1/(c*pi) and width c "
(http://people.ccmr.cornell.edu/~muchomas/ 8.04/Lecs/lec_quant_states/node8.html)
"as a normalized Gaussian function (normal distribution) in the limit of zero width."
(http://planetmath.org/encyclopedia/DeltaFunction.html)
"can also be defined as a normalized Gaussian function in the limit of zero width." (http://rkb.home.cern.ch/rkb/AN16pp/node57.html)
" should not be considered to be an infinitely high spike of zero width "
(www.ph.ed.ac.uk/~wjh/teaching/Fourier/delta-page.pdf)
" This function is one that is infinitesimally narrow, infinitely tall, "
(http://cnx.rice.edu/content/m10059/latest/)
Zero
----
"Very loosely we may
say that the delta function is a spike of infinite height, zero width "
(personal.rhul.ac.uk/UKAP/ 042/PH2130/PH2130_files/delta.pdf)
" One way is to think of it as a narrow spike having infinite height and zero width "
(http://www.engr.udayton.edu/faculty/jloomis/ece561/notes/intro/signals.html)
--
different MP3 every day! http://gweep.net/~shifty/snackmaster
. . . . . . . . ... . . . . . .
"Anything moving in the zone, even a three-year- | Niente
old, needs to be killed." -Captain R. | shifty@gweep.net |
|
| Back to top |
|
|
Guest
|
| Posted:
Mon Dec 13, 2004 6:59 am Post subject:
Re: Impossible sampling theory! |
|
|
As for Dirac delta function, it should not be treated as "function";
it is "distribution" or "generalized function". The following link
is with a collection of links about that:
http://www.ScienceOxygen.com/math/547.html
I believed even Dirac himself did not understand the
true meaning of the "Dirac Function" when he defined it.
Basically, it belongs to the realm of "Functional Analysis"
in the field of Math Analysis.
Why the "generalized function" is defined? Basically, it
is trying to extend to concept of "Differentiability". For
example, we learned that some continuous functions are
not "differentiable" in Calculus. The concept of
"generalized function" is to find the derivative of
some functions such that the solution does not have to
be "function", but a "generalized function".
For example, the derivative of "step function" is not
everywhere-differentiable according what we learned in the
course of Calculus. But in the context of "generalized function",
the derivative exists -- known as Delta Function.
Since the derivative of a function is not necessarily
as a function in board sense, we should not expect that
it has values at any points like function.
Why the concept of "generalized function" or "distribution"
is important ? It is originated from the following needs:
1. finding the solutions of some Differential equations or
PDEs
2. giving the solutions for the limit of some mathematical
problems
3. serving as the ideal cases for some physical problems
Thus, when you said "impossible sampling theory", it is correct
in the context of Calculus. But in the context of "functional analysis"
( more advanced math than Calculus ), it makes sense. But it should not
be treated as "classical way".
Hopefully, this would help. If you are interested in that, you can
look into the literature from some advanced math ( although it is not
straightforward ...)
Tachyon wrote:
| Quote: | On 2004-12-10, ETS <emale80919@yahoo.com> wrote:
You can't take a dirac in pieces, you have to take the amplitude
(oo)
and the width 0 for a dirac to work.
i don't believe the dirac has width 0. a quick web search shows
mixed results:
Non-Zero
--------
" a very tall narrow spike of height 1/(c*pi) and width c "
(http://people.ccmr.cornell.edu/~muchomas/
8.04/Lecs/lec_quant_states/node8.html)
"as a normalized Gaussian function (normal distribution) in the limit
of zero width."
(http://planetmath.org/encyclopedia/DeltaFunction.html)
"can also be defined as a normalized Gaussian function in the limit
of zero width." (http://rkb.home.cern.ch/rkb/AN16pp/node57.html)
" should not be considered to be an infinitely high spike of zero
width "
(www.ph.ed.ac.uk/~wjh/teaching/Fourier/delta-page.pdf)
" This function is one that is infinitesimally narrow, infinitely
tall, "
(http://cnx.rice.edu/content/m10059/latest/)
Zero
----
"Very loosely we may
say that the delta function is a spike of infinite height, zero width
"
(personal.rhul.ac.uk/UKAP/ 042/PH2130/PH2130_files/delta.pdf)
" One way is to think of it as a narrow spike having infinite height
and zero width "
(http://www.engr.udayton.edu/faculty/jloomis/ece561/notes/intro/signals.html)
--
different MP3 every day!
http://gweep.net/~shifty/snackmaster
. . . . . . . . ... . . . .
.. .
"Anything moving in the zone, even a three-year- | Niente
old, needs to be killed." -Captain R. | shifty@gweep.net |
|
|
| Back to top |
|
|
ETS
Guest
|
| Posted:
Mon Dec 13, 2004 6:59 am Post subject:
Re: Impossible sampling theory! |
|
|
Airy R. Bean wrote:
| Quote: | I was referring to the LPF following the DAC on the
output - a bit late to apply an anti-aliasing filter there!
"ETS" <emale80919@yahoo.com> wrote in message
news:ADkud.35028$Mu3.2413177@twister.southeast.rr.com...
Not sure about you second paragraph there, the spectral droop is a
funtion of the sample aperture. With a dirac, zero aperture means no
droop. Real world sampled signal do experience this because no one has
built a diracian sampler. The LPF is used for antialiasing, not for
spectral droop.
:-), that's an anti-alias filter after the DAC as well, but I thought we |
were talking about sampling an analog signal.
Anywho....
If you are that hung up on using the dirac as a sampling waveform, you
can always work out the math for a more realistic sampling waveform. It
still works and you can get away from bothering with the dirac. Just
work it through with flat top sampling. That way you have a waveform
that is definitely producible in the real world and a mathematical
description of it.
Take care |
|
| Back to top |
|
|
ETS
Guest
|
| Posted:
Mon Dec 13, 2004 6:59 am Post subject:
Re: Impossible sampling theory! |
|
|
Tachyon wrote:
| Quote: | On 2004-12-10, ETS <emale80919@yahoo.com> wrote:
You can't take a dirac in pieces, you have to take the amplitude (oo)
and the width 0 for a dirac to work.
i don't believe the dirac has width 0. a quick web search shows
mixed results:
Non-Zero
--------
" a very tall narrow spike of height 1/(c*pi) and width c "
(http://people.ccmr.cornell.edu/~muchomas/ 8.04/Lecs/lec_quant_states/node8.html)
"as a normalized Gaussian function (normal distribution) in the limit of zero width."
(http://planetmath.org/encyclopedia/DeltaFunction.html)
"can also be defined as a normalized Gaussian function in the limit of zero width." (http://rkb.home.cern.ch/rkb/AN16pp/node57.html)
" should not be considered to be an infinitely high spike of zero width "
(www.ph.ed.ac.uk/~wjh/teaching/Fourier/delta-page.pdf)
" This function is one that is infinitesimally narrow, infinitely tall, "
(http://cnx.rice.edu/content/m10059/latest/)
Zero
----
"Very loosely we may
say that the delta function is a spike of infinite height, zero width "
(personal.rhul.ac.uk/UKAP/ 042/PH2130/PH2130_files/delta.pdf)
" One way is to think of it as a narrow spike having infinite height and zero width "
(http://www.engr.udayton.edu/faculty/jloomis/ece561/notes/intro/signals.html)
|
All the math I dealt with defined the dirac as something like 1/2*e as
e approaches 0. So, in the limit, the height approaches infinity and
the width approaches 0. If the width did not approach 0, then the
integral of that would have to be infinite. |
|
| Back to top |
|
|
Rune Allnor
Guest
|
| Posted:
Mon Dec 13, 2004 1:05 pm Post subject:
Re: Impossible sampling theory! |
|
|
Airy R. Bean wrote:
| Quote: | A number of texts suggest that sampling can be modelled
by multiplying the incoming waveform by a comb of
Diracian Delta Functions.
How can this be?
|
Taken as a definitive statement, the above is plain wrong.
However, it is very easy to mis-interpret the sampling theory
along such lines as suggested above. It only takes a little bit
of formalism to demonstrate the pitfall. One needs to be aware
of one very tiny snag with the standard formalism of functions.
The Dirac delta function D(t) is defined to have the properties
int D(t) dt = 1 [1.a]
A int D(t) dt = A [1.b]
int x(tau)D(t-tau) dtau = x(t) [1.c]
where the integrals are taken from -infinite to infinite.
[1.c] above represents the sampling of the function x(t)
to extract the instantaneous value of x(t) at time tau.
Hence, the notation is ambiguous in that the x()'s in
[1.c] represent different types of 'quantities'. Inside
the integral, x(tau) is a function defined on the interval
-infinite < tau < infinite, while the x(t) at the right-hand
side represents the scalar value at time t.
Eq [1] shows us how to extract the instantaneous value
of a function (i.e. sample a function), what remains is to
find a formal way to order the samples in time. One convenient
way is to represent the sampled series as a continuous function
x'(t) that is formed as
x'(t) = sum c_n D(nT-t) [2]
where the summation is over n such that -infinite < n < infinite.
The coefficients c_n represent the instantaneous amplitude of
x(t) at time nT, T being the sampling interval, and is found
according to [1.c] as
c_n = integral x(tau) D(nT-tau) dtau = x(nT). [3]
Insert this back into [2] to find
x'(t) = sum x(nT)D(nT-t). [4]
Having reached [4], it is easy to mis-read the contents of the
equation and draw the conclusion that it "suggests", as you said,
that "sampling can be modelled by multiplying the incoming waveform
by a comb of Diracian[sic!] Delta Functions."
Communicating the statement correctly requires two things:
- The author must write it out correctly, and not over-simplify
anything.
- The reader must be aware that there is a difference between
the *function* x(t) and the *instantaneous value* of x(t) at
time t. This subtle difference does not come through in the
standard notation of functions, except for through the
context.
If both authors and readers are aware of such subtle details,
the sampling theorem ought not to be a problem.
Rune |
|
| Back to top |
|
|
David Kirkland
Guest
|
| Posted:
Mon Dec 13, 2004 7:41 pm Post subject:
Re: Impossible sampling theory! |
|
|
ETS wrote:
| Quote: | Tachyon wrote:
On 2004-12-10, ETS <emale80919@yahoo.com> wrote:
You can't take a dirac in pieces, you have to take the amplitude (oo)
and the width 0 for a dirac to work.
i don't believe the dirac has width 0. a quick web search shows
mixed results:
Non-Zero
--------
" a very tall narrow spike of height 1/(c*pi) and width c "
(http://people.ccmr.cornell.edu/~muchomas/
8.04/Lecs/lec_quant_states/node8.html)
"as a normalized Gaussian function (normal distribution) in the limit
of zero width."
(http://planetmath.org/encyclopedia/DeltaFunction.html)
"can also be defined as a normalized Gaussian function in the limit of
zero width." (http://rkb.home.cern.ch/rkb/AN16pp/node57.html)
" should not be considered to be an infinitely high spike of zero width "
(www.ph.ed.ac.uk/~wjh/teaching/Fourier/delta-page.pdf)
" This function is one that is infinitesimally narrow, infinitely tall, "
(http://cnx.rice.edu/content/m10059/latest/)
Zero
----
"Very loosely we may
say that the delta function is a spike of infinite height, zero width "
(personal.rhul.ac.uk/UKAP/ 042/PH2130/PH2130_files/delta.pdf)
" One way is to think of it as a narrow spike having infinite height
and zero width "
(http://www.engr.udayton.edu/faculty/jloomis/ece561/notes/intro/signals.html)
All the math I dealt with defined the dirac as something like 1/2*e as
e approaches 0. So, in the limit, the height approaches infinity and
the width approaches 0. If the width did not approach 0, then the
integral of that would have to be infinite.
|
The dirac delta can be be seen as the limit of a family of functions
e.g. rectangular, exponential, etc. In analyzing the limit it is
sometimes easier to visualize using one set of functions over another.
Sampling is easier to visualize using a rectangular function, while
handling derivatives and such is easier with a smoother function like
the exponential.
Cheers,
Dave |
|
| Back to top |
|
|
Phil Scott
Guest
|
| Posted:
Sat Dec 25, 2004 5:55 am Post subject:
Re: Impossible sampling theory! |
|
|
"Airy R. Bean" <me@privacy.net> wrote in message
news:31r9h0F3basdsU1@individual.net...
| Quote: | A number of texts suggest that sampling can be modelled
by multiplying the incoming waveform by a comb of
Diracian Delta Functions.
How can this be?
|
Texts and thier authors are often wrong in one aspect or
another as you approach absolute limits... as with most
instruments they read most accurately in the mid range. The
usual electricians meter of course wouldnt read a tenth of a
volt with any accuracy and he doesnt need it to... it wouldnt
read a trillionth of a volt at all..
A meter that would read a trillionth of a volt would be toast
at 480 volts.. its not practical to design a meter for both
uses... the same applies with mathematics or any science.
Techiques used to measure spin of subatomic particles are not
useful in measuring items the size of molecules,
To trash one or the other as worthless because it doesnt
address both is not good logic nor good argument.
so its a matter of whats most applicable for a given range.
In your case you are at an extreme limit for any kind of
workability for that solution to be the best choice....you
need to pick, or invent another approach.
| Quote: |
1. The samples that you get are measured in the order
of single volts whereas the Diracian is infinitely tall.
Surely,
if something of the order of unity were to be multiplied by
something of the order of infinity, the result would
be of the order of infinity?
How do you account for the difference? Do you have
some internal mental model where there is an invisible
constant,
"Big K", perhaps, to account for the difference in scaling?
2. The area of the sampled pulse is very much less than
unity,
the volts being ooo unity and the time being typically ooo
usecs.
How do you handle this mentally when the area of the
Diracian
is unity?
How do you come to terms with the attributes of your claimed
model
being orders of magnitude different from the signals of the
real world?
3. If you are one of those who claim that the sampled signal
is a short
spike of zero width, then it is zero-integrable and not
analysable by
any process involving Laplace Transforms.
|
If a person chose to apply that notion he wouldnt use zero
in the calculation obviously, but instead a wild guess on the
duration, something real close to zero, but not zero... or an
actual number based on what is discoverable regarding the
signal... nanoseconds or whatever.
If you are any sort of mathematician you know that the use of
imaginary, or trial numbers is quite common and accepted...you
can plot a curve using these imaginaries and from that can see
the actual trend line...and extrapolate from there in many but
not of course all cases...especially as you approach quantum
levels... then it all changes, often unpredicatably.
We know only a faint trace of what there is to know in the
world... maybe 0.00001% at best... and much of that, in a
million years will be shown to poorly founded.
If it were me, I would then examine the results to see if
they made any sense...and I would also use various other
means...in the end arriving at the best number the project
warranted...narrowing it down as the function shaped up on a
graph. Absolute pure perfect accuracy though is almost never
attainable for reasons you are no doubt familiar with.
If you see some person writing a texts that seems to
indicate that his his approach is somehow workable in all
cases and situations to near perfect accuracy... then all you
have to do is what you have done...point out the loopholes in
his logic... and do not use his approach in those
situation...find another.
If every author used 15 pages of text to point out the
unworkable ranges of his solution texts would be too large to
be useful...so in a text referring to one aspect of level of
application you will see time tested formula's for that
particular range of applications.... not some exotic other,
inapplicable range of course.
You are spendiing time focusing on the inapplicable range...
thats either legitimate or insane, depending on your intent.
If you are looking to trash people for believing texts..well
you wont have to look far or long for targets... its human
nature to believe whats in a text...and as any student of
history knows, that is virtually always in error at some level
of later advance, very often 100% in error.
Spending your life searching for those who believe without
thinking... is a waste of your time...beyond satisfying
yourself that one must always look deeper...depending on his
need for accuracy or truth in the matter.
| Quote: |
How do you overcome the problem that your sampled signals
are
not representable in the way that you claim?
|
One as always does the best he personally can...for some thats
not very good..... others have many options...and take a broad
view, and integrate the results of several calculations and
observations, and trials with some common sense to achieve at
the least, a workable solution.
I find graphical solutions work well in many areas. If by use
of mathematics, or direct observation, you can develop a
verifiable curve on either side of your particular
situation...then the points inbetween in almost all cases will
fall on the curve.
Its not magic.
Phil Scott
|
|
| Back to top |
|
|
Airy R. Bean
Guest
|
| Posted:
Mon Dec 27, 2004 9:10 pm Post subject:
Re: Impossible sampling theory! |
|
|
But it doesn't provide correct results!
"Kevin Neilson" <kevin_neilson@removethiscomcast.net> wrote in message
news:cpa0vb$quo1@xco-news.xilinx.com...
| Quote: | Airy R. Bean wrote:
A number of texts suggest that sampling can be modelled
by multiplying the incoming waveform by a comb of
Diracian Delta Functions.
How can this be?
1. The samples that you get are measured in the order
of single volts whereas the Diracian is infinitely tall. Surely,
if something of the order of unity were to be multiplied by
something of the order of infinity, the result would
be of the order of infinity?
One of my professors implied that the Dirac delta wasn't mathematically
rigorous but provides correct results so it's used nonetheless. |
|
|
| Back to top |
|
|
U-CDK_CHARLES\\Charles
Guest
|
| Posted:
Mon Dec 27, 2004 9:51 pm Post subject:
Re: Impossible sampling theory! |
|
|
On Mon, 27 Dec 2004 16:10:32 -0000, Airy R. Bean <me@privacy.net> wrote:
| Quote: | But it doesn't provide correct results!
"Kevin Neilson" <kevin_neilson@removethiscomcast.net> wrote in message
news:cpa0vb$quo1@xco-news.xilinx.com...
Airy R. Bean wrote:
A number of texts suggest that sampling can be modelled
by multiplying the incoming waveform by a comb of
Diracian Delta Functions.
How can this be?
1. The samples that you get are measured in the order
of single volts whereas the Diracian is infinitely tall. Surely,
if something of the order of unity were to be multiplied by
something of the order of infinity, the result would
be of the order of infinity?
One of my professors implied that the Dirac delta wasn't mathematically
rigorous but provides correct results so it's used nonetheless.
|
Mathworld says that "In engineering contexts, the functional nature of
the delta function is sometimes suppressed."
Their definition seems to have plenty of rigor.
http://mathworld.wolfram.com/DeltaFunction.html |
|
| Back to top |
|
|
Airy R. Bean
Guest
|
| Posted:
Tue Dec 28, 2004 4:48 pm Post subject:
Re: Impossible sampling theory! |
|
|
Does the definition whose URL you quoted (and how much better
a form of debate if you present your own argument rather than seeking
to send your correspondents off somewhere else?) describe how the attributes
of area
and amplitude of the Diracian can possibly be representative of
real-world sampling pulses with "plenty of rigor"?
"U-CDK_CHARLES\Charles" <"Charles Krug"@aol.com> wrote in message
news:I4Xzd.15035$vF5.2684@trndny07...
| Quote: | On Mon, 27 Dec 2004 16:10:32 -0000, Airy R. Bean <me@privacy.net> wrote:
But it doesn't provide correct results!
"Kevin Neilson" <kevin_neilson@removethiscomcast.net> wrote in message
news:cpa0vb$quo1@xco-news.xilinx.com...
Airy R. Bean wrote:
A number of texts suggest that sampling can be modelled
by multiplying the incoming waveform by a comb of
Diracian Delta Functions.
How can this be?
1. The samples that you get are measured in the order
of single volts whereas the Diracian is infinitely tall. Surely,
if something of the order of unity were to be multiplied by
something of the order of infinity, the result would
be of the order of infinity?
One of my professors implied that the Dirac delta wasn't mathematically
rigorous but provides correct results so it's used nonetheless.
Mathworld says that "In engineering contexts, the functional nature of
the delta function is sometimes suppressed."
Their definition seems to have plenty of rigor. |
|
|
| Back to top |
|
|
Spike
Guest
|
| Posted:
Tue Dec 28, 2004 10:57 pm Post subject:
Re: Impossible sampling theory! |
|
|
On Tue, 28 Dec 2004 11:48:13 -0000, "Airy R. Bean" <me@privacy.net>
wrote:
| Quote: | Does the definition whose URL you quoted (and how much better
a form of debate if you present your own argument rather than seeking
to send your correspondents off somewhere else?) describe how.....
|
For some reason you appear frightened of researching information, such
as following a url, and integrating the knowledge gained with what you
already know in order to take things into new territory. But this is
how science advances, and the study for a degree of PhD requires that
the current position be adequately researched as a prerequisite to
moving on to one's particular research topic. This is called the
Literature Survey, and it is a fundamental part of the PhD. Fail to
perform this adequately, and your PhD is doomed.
As you will not follow urls, I append a short article to help you, and
as you are prone to ISP failures, I may repost this from time-to-time.
Note particularly the paragraph headed "RESEARCH METHODS".
For the information of other readers, the url is
http://www.itacs.uow.edu.au/research/postgrad/comments.shtml
GENERAL COMMENTS ON RESEARCH
PREAMBLE:
Research, by its very nature, is a step into the unknown and therefore
open-ended; there are no guarantees. As such your supervisor(s) will
not know the answer to your research questions (research is not the
same as coursework). This step is usually guided by the results of
previous researchers in the field. Such previous work "sets the
scene"/points you in the right direction/tells you where to look.
Steady, methodical and persistent effort on your part is then
necessary to reach your research goal, often employing the
scientific/experimental method(s) (e.g. hypothesis testing). Of
itself, this might not be sufficient; genuine insight, serendipity and
unexpected "connections" from seemingly unrelated areas are often
necessary. These can neither be anticipated nor manifested at will.
Many scientific breakthroughs come from the most unexpected sources.
RESEARCH METHODS:
In order to (a) become familiar with your chosen area of research, and
(b) to ensure you don't "reinvent the wheel" and commence working on a
topic which has been previously researched, it is essential to become
familiar with the published literature in the field. A good way of
doing this is to write your own literature survey/review article,
perhaps even presenting a seminar/conference paper on your findings.
This helps you not only to familiarise yourself with previous work,
but also to highlight what has yet to be done/what problems remain to
be solved in your chosen field. It also helps to identify areas in
which you are perhaps weak and need to learn and/or improve your
skills.
The first six months of a 3-year PhD programme should be devoted to a
literature survey; the second six months to replicating previous work.
By the end of the first year, it should become clear as to how the
earlier work can be extended/improved, thus enabling a detailed
research proposal to be formulated. Naturally, the remaining two years
are spent in following these ideas (and periodically backtracking and
revising your research plan in the light of your findings).
NOTE:
For Research Masters (and undergraduate Honours), it is quite valid to
work on a topic which has been researched previously, but from a
different perspective/extending it in some manner. For a PhD, an
original contribution to knowledge is required - establishing what has
been done previously and identifying a substantial problem to tackle
is even more critical here. Successfully applying new/different (and
better) techniques to problems previously solved by other means is
still a valid approach for a PhD however.
In order to conduct a literature survey, you will need to hone your
library skills, specifically: (i) how to track down survey
papers/introductory books, (ii) developing the art of quickly reading
and evaluating abstracts (at least - entire papers if appropriate),
(iii) identification of the classic references in the field, and
subsequently tracking them down (in hard copy form, either within the
UoW Library, or via Inter-Library Loans), (iv) use of the UoW on-line
Library resources, as well as more general searching of the World Wide
Web, & (v) the ability to critically evaluate what's been done
previously. In short, who are the key researchers in the field? What
are the seminal works/books/survey papers? What are the most important
journals in your chosen area?
NOTE:
It is very important to keep abreast of the latest developments in the
field, especially if someone publishes what you are currently working
on. If this happens, you may need to take a significant change of
direction with your work. Thus periodic updates of your literature
survey will be necessary during the course of your study.
--
from
Aero Spike |
|
| Back to top |
|
|
|
|
|
|
FAQ
Memberlist
Register
Profile
Log in to check your private messages
Log in