| Author |
Message |
Pierian Spring
Guest
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 Posted:
Sat Dec 17, 2005 9:15 am Post subject:
A FAQ for Xmas |
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What better than a season of goodwill for laying to rest a few
grmelins, or for those pretentious "experts" to admit that they
don't know the answers?.....
Frequently Asked Questions (F.A.Q.).....
1. (A Frequently Added Quotation, F.A.Q., is appended below.)
Assuming that we were able to generate a Diracian, and then produce
a comb of them by delays and by superposition, there wouldn't be a
factor
of "T" in such superposition, so where does yours come from?
Where did the factor of "T" come from in the "sampling function" in
your opening lines?
Consider a 16-bit ADC capable of 100 M Samples per sec. In the first
instance we'll use it to sample a geophysical signal of bandwidth
limited to 300 HZ and sample at 1 kHz, with suitable analogue
instrumentation to match the input signal to the full range of
the ADC.
If we keep the circuit the same, but now sample
at 65.536 MHZ, your claimed factor of "T" will result in the
16 bit range being compressed down to one bit, because the new
sampling frequency is 2^(16) faster than the old. We know this
doesn't happen - there will be more samples, but they'll still
be of the same magnitude and over the same full-scale 16-bit range.
So, where does the factor of "T" come from?
2. Where have your Diracian impulses come from?
The Diracian has some interesting properties - zero width,
area of unity, infinite sum of all possible cosines, a height
which is not discussed but which appears to be greater in magnitude
than any voltage appearing in your circuits. In the
systems that you deal with, what experimental evidence do you
have that there are pulses with the attributes of Diracians upon
which to base your theory?
The Diracian, or Unit Impulse is a very good mathematical tool
to analyse the response of systems once a mathematical model of
those systems had been produced.
It is, however, a poor mathematical claim to make that such
impulses are found to be part of a system when neither the
area nor the magnitude of such impulses are found anywhere in such
systems.
To those who ask, "Who cares? I get good results." , I suggest that
their approach is unscientific and compares to the religious
loonies who sacrifice goats and virgins to stop the Sun falling
out of the sky and justify the continuing practice by the Sun
remaining in the sky.
So.....is the world of DSP a world of scientific men, or is
it a world of snake-oil charlatans and of religious loonies?
Where do these Diracian impulses come from?
| Quote: | robert bristow-johnson wrote:
x(t)*q(t) = T*SUM{x[k]*d(t-k*T)} .------.
x(t)--->(*)------------------------------------->| H(f) |---> x(t)
^ '------'
|
| +inf
'------- q(t) = T * SUM{ d(t - k*T) }
k=-inf
where: d(t) = 'dirac' impulse function
and T = 1/Fs = sampling period
Fs = sampling frequency
+inf
q(t) = T * SUM{ d(t - k*T) } is the "sampling function", is periodic
k=-inf with period T, and can be expressed as a
Fourier series. It turns out that ALL of
the Fourier coefficients are equal to 1. |
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Jerry Avins
Guest
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| Posted:
Sat Dec 17, 2005 5:15 pm Post subject:
Re: A FAQ for Xmas |
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Pierian Spring wrote:
...
Taste not or drink deep.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Guest
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| Posted:
Sun Dec 18, 2005 1:15 am Post subject:
Re: A FAQ for Xmas |
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Rather than responding with your habitual infantile
remarks, why not answer the question? (Because you
don't know the answer and prefer to make a sneering
fool of yourself; certainly if your childish posting history is
anything to go by?)
Jerry Avins wrote:
| Quote: | Pierian Spring wrote:
...
Taste not or drink deep.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
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Bevan Weiss
Guest
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| Posted:
Sun Dec 18, 2005 1:15 am Post subject:
Re: A FAQ for Xmas |
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Of all three current posts in this thread, Jerry's was by far the most
enlightened.
You post this same long message over and over, and never give any
context for the snippet you present from RBJ.
The comb of impulse functions is simply used to mathematically model a
sampling process. I'm not sure why you struggle so much to understand this.
As for the snippet from RBJ, it's similar to any such sampling block
shown in a text book, though with intrinsic oddities that offhanded
explanations tend to include.
I'd personally not use x(t) to represent the sampled function, i'd use
x[t] instead, where t is now a quantized value, perhaps even illustrate
that it is no longer strictly time referred by using n instead, hence
the output would be x[n] (or x[k] to keep with RBJ's notation).
Also the factor of T is erroneous for basic sampling, and hence makes me
wonder what the snippet was originally referring to. Put the context
back in!
radio.ham@lycos.co.uk wrote:
| Quote: | Rather than responding with your habitual infantile
remarks, why not answer the question? (Because you
don't know the answer and prefer to make a sneering
fool of yourself; certainly if your childish posting history is
anything to go by?)
Jerry Avins wrote:
Pierian Spring wrote:
...
Taste not or drink deep.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Tim Wescott
Guest
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| Posted:
Sun Dec 18, 2005 1:15 am Post subject:
Re: A FAQ for Xmas |
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radio.ham@lycos.co.uk wrote:
| Quote: | Rather than responding with your habitual infantile
remarks, why not answer the question? (Because you
don't know the answer and prefer to make a sneering
fool of yourself; certainly if your childish posting history is
anything to go by?)
Jerry Avins wrote:
Pierian Spring wrote:
...
Taste not or drink deep.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
|
Would you please stick to one email address? I know they're all fake,
but when you dodge around like that it's tiresome to keep my killfiles
updated.
Thanks in advance.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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John Monro
Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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Pierian Spring wrote:
(snip)
| Quote: |
2. Where have your Diracian impulses come from?
The Diracian has some interesting properties - zero width,
area of unity, infinite sum of all possible cosines, a height
which is not discussed but which appears to be greater in magnitude
than any voltage appearing in your circuits. In the
systems that you deal with, what experimental evidence do you
have that there are pulses with the attributes of Diracians upon
which to base your theory?
|
A simple example:
Take a LP filter consisting of one resistor R and one capacitor C.
If we apply a short pulse of voltage E to the input, the output rises by
delta_v = (1/RC)E.delta_t
This is a good approximation if the pulse width (delta_t) is much
shorter than the time constant (1/RC) and becomes a better approximation
as delta_t approaches 0.
Having delta_t equal to 0 brings up mathematical difficulties, so we
recognise that the important thing here is not E or delta_t alone, but
the product E.delta_t, which is of course an impulse. If the product is
1.0 then it is called a unit impulse.
As long as we know the value of the impulse, and as long as delta_t is
short, the precise value of E and delta_t do not matter because the
response of the system will be the same.
You ask: "In the systems that you deal with, what experimental evidence
do you have that there are pulses with the attributes of Diracians upon
which to base your theory?"
The experimental evidence is that when the systems that I deal with are
tested with an impulse, the response depends on the product E.delta_t.
As long as delta_t is short enough, the response is independent of its
precise length. In fact, the voltage output control of the function
generator or its pulse-width control can be, within limits, used
interchangeably as an 'impulse' control and this affects only the
amplitude of the output of the system and not any other aspect of the
system response.
I acknowledge that I am not the first in this newsgroup to point out all
this, admittedly with arguments that are far more elegant than mine, but
thought it may be of interest expressed in this particular way.
Regards,
John |
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Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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I seek a sound mathematical basis for that model,
a search that arises from curiosity and a desire
for the truth.
There is no sound basis forthcoming, either from
the "experts" in this NG or from the various textbooks
that I encounter.
The impulses do not exist in any of the systems with
which we deal - either in amplitude or in width - so why
claim that they are a true model?
Other aspects of engineering model the parameters being
measured with numerical values that are true representations
of what is being measured - so what is the sound derivation
to claim that the infinitely large (amplitude) and the infinitesimally
small (time) are true models?
In an application which is inherently mathematical, the lack
of a sound mathematical basis for the model - the analogy -
the paradigm - is appalling.
Those of us who aspire to be engineers know the mathematical
basis of the work that we do. it is the technician who is given
a lump of mathematics and who accepts it blindly.
So, aspiring to be an engineer and not a technician, I see something
that is a glaring error - a model that is not representative of that
which is being modelled - and so I query it.
I find it disturbing that a number of those who would be the experts
in this NG resort to infantile outbursts when faced with these
queries. Their attitude presents a dismal picture of acadaemic
excellence, as does your own perception of Avins' infantile stance.
Bevan Weiss wrote:
| Quote: | The comb of impulse functions is simply used to mathematically model a
sampling process. I'm not sure why you struggle so much to understand this. |
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Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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There is nothing fake about my email addresses or my
Internet address. I do not respond with infantile outbursts
as you do. I seek truth. I am always direct and honest
in my approach.
If you dislike the way that my pseudonym changes in order to
protect my family from attacks, then your complaint lies
with those malicious persons who deliberately reveal
my identity when I change; I suggest that you take it up with them.
If you dislike truth, if you dislike being hauled over the coals
for repeated infantile outbursts, by all means run away
and hide in a kill-file. My remarks are not directed, in any case,
at the infantile bigot (if there are any such in this NG)
Tim Wescott wrote:
| Quote: | radio.ham@lycos.co.uk wrote:
Rather than responding with your habitual infantile
remarks, why not answer the question? (Because you
don't know the answer and prefer to make a sneering
fool of yourself; certainly if your childish posting history is
anything to go by?)
Jerry Avins wrote:
Pierian Spring wrote:
...
Taste not or drink deep.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ
Would you please stick to one email address? I know they're all fake,
but when you dodge around like that it's tiresome to keep my killfiles
updated.
Thanks in advance.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com |
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Bevan Weiss
Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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radio.ham@lycos.co.uk wrote:
| Quote: | I seek a sound mathematical basis for that model,
a search that arises from curiosity and a desire
for the truth.
There is no sound basis forthcoming, either from
the "experts" in this NG or from the various textbooks
that I encounter.
The impulses do not exist in any of the systems with
which we deal - either in amplitude or in width - so why
claim that they are a true model?
Other aspects of engineering model the parameters being
measured with numerical values that are true representations
of what is being measured - so what is the sound derivation
to claim that the infinitely large (amplitude) and the infinitesimally
small (time) are true models?
In an application which is inherently mathematical, the lack
of a sound mathematical basis for the model - the analogy -
the paradigm - is appalling.
Those of us who aspire to be engineers know the mathematical
basis of the work that we do. it is the technician who is given
a lump of mathematics and who accepts it blindly.
So, aspiring to be an engineer and not a technician, I see something
that is a glaring error - a model that is not representative of that
which is being modelled - and so I query it.
I find it disturbing that a number of those who would be the experts
in this NG resort to infantile outbursts when faced with these
queries. Their attitude presents a dismal picture of acadaemic
excellence, as does your own perception of Avins' infantile stance.
Bevan Weiss wrote:
The comb of impulse functions is simply used to mathematically model a
sampling process. I'm not sure why you struggle so much to understand this.
|
If you want to use a more complicated model then by all means do so.
The simple impulse comb is not the only model, it is the zero order
model. If you want to take into account some width of the sampling
function then do so, however it doesn't provide much more information
than is provided by the simple impulse comb and so for almost all
situations merely requires additional mathematical complexity for
limited model accuracy.
I'm not sure why you're looking for an impulse to exist in physical
reality. That has nothing at all to do with modeling a physical
situation. If you have a simple RC leaky integrator we model this as a
transfer function that consists of an ideal integrator and some leakage
factor, clearly neither component is strictly an integrator and neither
provides pure leakage. The capacitor has some inductance, some
resistance, as well as capacitance, none of which are strictly constant
(they are almost all frequency/voltage/time/temperature/humidity
dependent) and yet we nearly always ignore these aspects of reality.
Why do we ignore them? Because the model is close enough to
representing the true nature of the circuit that those details are for
the most part not important.
Sure in a sampling process there's no infinite amplitude impulse,
however there is some function that extracts just the value of the input
signal at some discrete time instants, much like the impulse function
does. And having an already existing mathematical function that
performs just such an operation (the dirac delta) we use that. It's not
a perfect representation, but for most aspects it's close enough.
If you really want to be an engineer and not a technician, then you have
to realize that a model is required for this sampling process. And
until *you* or someone else comes up with a better standard model, the
current standard model is the one to use. You just have to understand
the limitations of it.
You may say that it wouldn't truly be understanding the mathematical
basis of the work to just use the impulse model, however you just take
as faith that the basic capacitor model (ie perfect capacitor) is true
for a standard capacitor that you get from the shelf, what's the difference? |
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Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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John,
Thank you for taking the trouble to respond, but I am afraid
that I am familiar with everything that you say and have been
for the last 34 years (33 since graduating in Electronics).
The essence of providing impulses to analyse systems is that the
impulse
need have a continuous frequency spectrum over the range of
frequencies to which a system responds and needs no others.
Other frequencies that appear will not produce a response and this
is how a large family of pulses can all act as impulses, irrespective
of their shape.
It is not true, however, your claim that provided that the volt/time
product of energy is unity that systems will respond identically - it
is only
true if the energy product IN THE PART OF THE SPECTRUM TO
WHICH A SYSTEM RESPONDS remains constant. If we were to
provide a pulse with a unity volt/time product in which 99% of the
energy were outside the frequency response of the system, then
the response of the system will be different in amplitude to a
pulse with a unity volt/time product where the energy lies totally
withing the frequency response of the system.
It is this difference in amplitude that gives rise to my protests.
In our sampling, we take an infinitesimally small width but
only an amplitude which is of the order of the analogue signal,
and therefore the volt/time product of this is zero and it is zero
integrable using the calculus. The volt/time product is an
infinite number of orders of magnitude less than unity, so would you
justify claiming that
it is a scaled Unit Impulse?
John Monro wrote:
| Quote: | Having delta_t equal to 0 brings up mathematical difficulties, so we
recognise that the important thing here is not E or delta_t alone, but
the product E.delta_t, which is of course an impulse. If the product is
1.0 then it is called a unit impulse.
As long as we know the value of the impulse, and as long as delta_t is
short, the precise value of E and delta_t do not matter because the
response of the system will be the same. |
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Guest
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| Posted:
Sun Dec 18, 2005 9:15 am Post subject:
Re: A FAQ for Xmas |
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Not only is it not perfect, but it exhibits a glaring
mathematical error.
The use of a Diracian to extract the value of a function at
one point, sifting, requires an integration operation, and when
transformed, that integration must appear in the analysis as
a factor (in the Laplacian case) of 1/s.
In the claimed use of an impulse comb (Shah?) to sample,
it is claimed that the extraction of value at certain instants
is achieved by multiplication alone; and the critical integration
that provides the sifting operation magically vanishes.
Bevan Weiss wrote:
| Quote: | Sure in a sampling process there's no infinite amplitude impulse,
however there is some function that extracts just the value of the input
signal at some discrete time instants, much like the impulse function
does. And having an already existing mathematical function that
performs just such an operation (the dirac delta) we use that. It's not
a perfect representation, but for most aspects it's close enough. |
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Stan Pawlukiewicz
Guest
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| Posted:
Mon Dec 19, 2005 5:16 pm Post subject:
Re: A FAQ for Xmas |
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radio.ham@lycos.co.uk wrote:
| Quote: | I seek a sound mathematical basis for that model,
a search that arises from curiosity and a desire
for the truth.
|
Keep on posting the same stuff. The truth is most don't care.
| Quote: |
There is no sound basis forthcoming, either from
the "experts" in this NG or from the various textbooks
that I encounter.
|
I gave you a reference to Stark and Tutuer. Go encounter that.
| Quote: |
The impulses do not exist in any of the systems with
which we deal - either in amplitude or in width - so why
claim that they are a true model?
Other aspects of engineering model the parameters being
measured with numerical values that are true representations
of what is being measured - so what is the sound derivation
to claim that the infinitely large (amplitude) and the infinitesimally
small (time) are true models?
In an application which is inherently mathematical, the lack
of a sound mathematical basis for the model - the analogy -
the paradigm - is appalling.
Those of us who aspire to be engineers know the mathematical
basis of the work that we do. it is the technician who is given
a lump of mathematics and who accepts it blindly.
So, aspiring to be an engineer and not a technician, I see something
that is a glaring error - a model that is not representative of that
which is being modelled - and so I query it.
I find it disturbing that a number of those who would be the experts
in this NG resort to infantile outbursts when faced with these
queries. Their attitude presents a dismal picture of acadaemic
excellence, as does your own perception of Avins' infantile stance.
Bevan Weiss wrote:
The comb of impulse functions is simply used to mathematically model a
sampling process. I'm not sure why you struggle so much to understand this.
|
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robert bristow-johnson
Guest
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| Posted:
Tue Dec 20, 2005 1:16 am Post subject:
Re: A FAQ for Xmas |
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Bevan Weiss wrote:
| Quote: | radio.ham@lycos.co.uk wrote:
....
Bevan Weiss wrote:
The comb of impulse functions is simply used to mathematically model a
sampling process. I'm not sure why you struggle so much to understand this.
|
i presume "radio ham", being someone name "Gareth" a.k.a. "Beanie, is
not struggling to understands anything. he knows what is going on. he
knows everything.
| Quote: | I seek a sound mathematical basis for that model,
a search that arises from curiosity and a desire
for the truth.
|
of course it's just a desire for the truth. that's why Beanie simply
shuns all attention directed toward him.
| Quote: | If you want to use a more complicated model then by all means do so.
The simple impulse comb is not the only model, it is the zero order [hold]
model. If you want to take into account some width of the sampling
function then do so, however it doesn't provide much more information
than is provided by the simple impulse comb and so for almost all
situations merely requires additional mathematical complexity for
limited model accuracy.
|
the zero-order hold, is a D/A or reconstruction issue. from a POV of
the sampling theorem, it makes no difference if the method of the A/D
was such that required a sample & hold to precede the A/D (like
successive approximation or the old dual slope A/D). that does not
matter. what matters is that the number coming out of the A/D (going
into the D/A) is proportional to the input voltage (or whatever is the
physical signal) at some instance of time within some tolerance of
quantization error. so we have a sequence of numbers, x[n], that are
equal to (discounting scaling and error) to the input, x(t), at equally
spaced times x(n*T). attaching these sample values to a string of
dirac impulse is part of a hypothetical mathematical model to show that
(given the bandwidth restriction of Nyquist) sufficient information is
left in the samples that the original x(t) can be reconstructed
somehow. (it might require an acausal LPF, but that's just another
nasty detail.)
| Quote: | If you dislike the way that my pseudonym changes in order to
protect my family from attacks,
|
this tactic works pretty good. my organized crime contacts in the UK
have been unable to track down "radio ham's" address (so they could
attack his family, of course). are you living with your parents,
radio? because they couldn't find any Mr. "Ham" with first name
"Radio" in the public directories.
| Quote: | then your complaint lies
with those malicious persons who deliberately reveal
my identity when I change;
|
yeah, those malicious identifiers.
| Quote: | I suggest that you take it up with them.
If you dislike truth,
|
yeah, that evil truth...
| Quote: | if you dislike being hauled over the coals
|
.... and those loathsome coals.
| Quote: | for repeated infantile outbursts, by all means run away
and hide in a kill-file.
|
it doesn't work so well if other people respond to the troll.
r b-j |
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Howard Long
Guest
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| Posted:
Tue Dec 20, 2005 12:01 pm Post subject:
Re: A FAQ for Xmas |
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Folks
| Quote: | In our sampling, we take an infinitesimally small width but
only an amplitude which is of the order of the analogue signal,
and therefore the volt/time product of this is zero and it is zero
integrable using the calculus. The volt/time product is an
infinite number of orders of magnitude less than unity, so would you
justify claiming that
it is a scaled Unit Impulse?
|
Maybe I'm being way to simplistic here, but isn't this just a basic calculus
issue? Although t tends to zero, v tends to infinity. With only these two
terms you are left without a full definition: you need to state that v*t
tends to 1 as t tends to zero. This limit is part of the definition of the
Dirac Delta function. They teach the introductory calculus concept of limits
at school at about the age of 16.
Although it is an abstract concept in that we can never physically attain
infinite amplitude with zero width, as we approach closer and closer to that
experimentally, we can see that the unit impulse model fits closer and
closer.
If there is another method you'd prefer to use, then I am sure that's fine,
and I'm sure it would be very useful and interesting to compare your method
and results with the current more generally accepted perceived wisdom.
Cheers, Howard |
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robert bristow-johnson
Guest
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| Posted:
Wed Dec 21, 2005 12:29 am Post subject:
Re: A FAQ for Xmas |
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Beanie regurjitates:
| Quote: | In our sampling, we take an infinitesimally small width but
only an amplitude which is of the order of the analogue signal,
and therefore the volt/time product of this is zero and it is zero
integrable using the calculus. The volt/time product is an
infinite number of orders of magnitude less than unity, so would you
justify claiming that it is a scaled Unit Impulse?
|
Howard Long wrote:
| Quote: | Folks
Maybe I'm being way to simplistic here, but isn't this just a basic calculus
issue? Although t tends to zero, v tends to infinity. With only these two
terms you are left without a full definition: you need to state that v*t
tends to 1 as t tends to zero. This limit is part of the definition of the
Dirac Delta function. They teach the introductory calculus concept of limits
at school at about the age of 16.
Although it is an abstract concept in that we can never physically attain
infinite amplitude with zero width, as we approach closer and closer to that
experimentally, we can see that the unit impulse model fits closer and
closer.
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there is not just the physical impossibility of a real infinite height
impulse, but the different mathematical understanding of the dirac
delta function that engineers have vs. what the pure mathematicians
say. there is a theorem in real analysis (where we learn about
Lebesgue integration) that says that for a function that is zero
"almost everywhere" (that is everywhere but a countable number of
points) that the integral of such a function is zero. but we engineers
look at the dirac delta as a function that is zero everywhere but at
t=0, yet its integral (over all t) is one. so something is not
connecting there. the math guyz like to call the dirac delta a
"distribution" (not a "function") and really just define it as a
notation that when you put it alongside ("multiply") a legit function
and integral, that construction is "functional" that maps that function
f(t) to a number which is f(0) (or f(a) for delta(t-a)). for the math
guys, the dirac delta has little meaning outside of an integral.
Beanie is just trolling, but he wraps himself in this issue to draw
attention. i was hoping that ignoring the troll would make him go
away, but it's not happening. i'm still not removing him from my
killfile, though.
i don't particularly like the pure mathematicians view of the dirac
delta, but i do not dispute their correctness about it (when we define
a function, it is a mapping of a number to another number and that
mapping doesn't really get to "remember" the limit of how it got
defined, and without the nascent dirac impulse - those in the limit -
there is no way to represent that finite area in an infinitely thin
pulse). they way i deal with in conceptually is, for the purpose of
*any* real or practical or engineering context, i just define the dirac
impulse as one of those nascent approximations to it (like one
femto-second wide and the reciprocal of that high). it will make
absolutely no numerical difference for any practical situation and then
i am allowed to have a bunch of dirac impulses outside of the integral
that i can manipulate as if they were some "normal" functions.
r b-j |
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