A laying of the sampling ghost.....

"Airy R. Bean" wrote:


What's not so? Your post appears in response to mine, but you quote some one else.

Standard analysis of what? Is f(t) the notes Mozart scored? or is it the sound waves coming out of the orchestra? or is it the current in the microphone? at what point does f(t) become f(t)?

what circuit? who said anything about circuits.
-jim

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No - it's posted in response to Mr.Reay.

open,
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is
that
will
point
sample
it
Sanity at last.
Chimera
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I'm responding here in that my newsreader is not showing the post quoted below:
Chimera wrote:

Sane maybe, but not a very good grasp of the sampling process. By the reasoning you guys are using the Weather Service is going about it all wrong for sampling rainfall. Instead of recording the rainfall in the gauge once a day and then emptying it. what they should be doing is just sticking the gauge out the window for a second at the same time each day. Mr Bean would go even further and not even allow them to retract their arm before noting the measurement, but instead insist that they record the measurement as soon as the arm is fully extended.     There's no integration involved - the contents of the rain are just recorded once a day. The sampling theorem tells you that there exists a continuos function that can be derived that interpolates these samples. It doesn't tell you whether the rain gauge is in a desert or tropical island, or if the gauge is in inches or millimeters. Those things will affect what the continuous function looks like. A crack in the bottom of the rain gauge or placing the rain gauge where the water runs off a roof will also give you a different function f(t). You can't possibly imagine that the dirac function is a mapping that ecompasses all these variable and yields f(t) ~ that pure function "rainfall".
-jim
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below:
open,
reasoning
sampling
then
window
not even

insist
recorded once

function that

whether the

or
like. A

water
possibly
variable
For the rain gauge, this equates to measuring the rain fall for on day, the sampling period. From that you can determine no more than how much it rained on one day (the sample time, which equates to the width of the sample pulse).
For the unit pulse sampling a waveform, a single pulse sampling the waveform once, all you determine is the value of wave form being sampled over the sample pulse width. You cannot determine anything about the nature of the waveform at other times.
In the case of the rain gauge, the water collect is the rain that fell in the sample period (assuming there are no error sources). In the sample pulse case, it is the power of the sampled waveform averaged over the pulse width.
Also, no single sample process is immune from the error due to things link leaky gauges etc. It is possible to account for some error sources but can we get Airy to understand the basics first.
Sampling theory (Nyquist) tells use that we must sample waveform at twice its frequency of repetition, if we are to reconstructed or, as is more appropriate in this analogy, be able to predict a value at some future time.
Airy's problem hasn't extended that far yet, he still can't grasp the way a single sample works. The width of the Dirac pulse is finite AND its area 1. If it is repeated, like a comb as he refers to, you get a series of samples, each sample being the width of the sampling pulse and repeated at the samplying frequency.
Chimera
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Chimera wrote:
...

The case is unusual in that the sampling window -- "aperture time" -- and the sampling period are the same.

So? If you have a train of samples and know that the signal is bandlimited, you can tell a lot about the signal as a whole.

Commonly, "sample period" refers to the time from one sample and the next. For classical sampling, the aperture time is kept short enough so that the bandlimited signal being sampled can't change as much as 1 LSB. Many ADCs are slower than that and so need a sample-and-hold front end. A rain gauge integrates the rainfall from dump to dump.

Barmbrain understands all this at least well enough to keep you trying vainly to enlighten him. When you play this theme out, he'll amuse himself by taunting you with another. I bet he pulls the legs of creepy things as another form of amusement.

Sampling theory is also based on the taking of a sample being an instantaneous process. Even when the data collector is an integrator -- rain gauge, CCD imager -- acquiring the signal is associated with a particular instant.
...
Jerry
--
Engineering is the art of making what you want from things you can get.

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Jerry Avins wrote:

No there is no aperture time - you just made that up:). You imagine that you are measuring the 'Rain function'. That's OK. But if you are going to do that then you can't use the dirac function as a model for the sample process.     You're going to have to substitute a fat pulse into the model and if you do that then your spectrum for the sampling process will no longer be flat. And if the spectrum is no longer flat then when its convolved with the spectrum represented by the samples you will not have the same spectrum that the samples represent. And when you reconstruct the signal with a linear combination of sinusoids that the spectrum represents it will no longer pass thru the sample points. As I said that's all well and good, but then there's no point arguing about the dirac function after you've discarded it.     But that's not what sampling theory says the samples represent. The samples are what they are - that's all. You're simply measuring the samples in a rain gauge - how they got there is outside the scope of the samples. Your personal knowledge of the whether makes you imagine all sorts of functions beyond the scope of the samples.     Here's another way of looking at. The rain gauge is an anti-alias filter. If it has a tiny leak and you don't empty it manually, its an even better anti-alias filter. Is it good sampling practice to take into account the effects of an anti-alias filter when attempting to reconstruct the signal from the samples.
-jim
PS. I did read the rest of your post and concur.

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jim wrote:

...
I think you missed my point. If daily rain gauge samples aren't recorded at the same time each day, they have less value than if collection were strictly regular. Now since there is a limit to how quickly rain can accumulate, a few seconds one way or the other is immaterial -- that much jitter doesn't matter. Nor do a few seconds to complete the measurement -- the aperture time. You are perfectly correct in pointing out that how the gauge got filled is beyond the scope of the sampling process. Nevertheless, it the samples are called "daily rainfall", the gauge had better accumulate its contents throughout the 24 hours preceding the measurement.
Jerry
--
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below:
open,
reasoning
sampling
then
window
Not so. The two methods you mention for rainfall are a *sampling* technique, and a *measurement* technique. If you have the resources to measure every item (in this case rate of rainfall over time), then 100% measurement is the way to go. But if you don't have the resources to sample continuously, you can sometimes get a very good approximation by sampling at appropriate intervals and infering the continuous function based on the sample points. To get an *accurate* reproduction you would need to sample at least twice the frequency of interest. Considering that rain storms typically come and go in the course of hours or faster, 'the Weather Service' would have to sample more often than once a day.
And such a sample would be the *rate* of rainfall, not the total so far. To find the total, one would have to *assume* the rate is changing between samples by some known function f(t). Then one could integrate the total rainfall with some degree of accuracy.
But since the weather is unpredictable, you may have to sample the rate of rainfall quite often and then just *assume* the rate changes linearly between samples.

insist
recorded once

function that

whether the

or
like. A

water
possibly
variable
Because all those things imply that the rate of rainfall is a rather complex f(t), to get a reasonable accuracy, one would have to estimate the highest frequency component and sample at least twice this rate. No simple matter. But the beauty of such a rain gauge is it never needs emptying :-)
If you empty a conventional gauge once a day, how do you calculate the average rainfall for a month? Easy, you have ~30 samples of rainfall with each sample measuring inches-of-rain per day. You just have a sample width that is very close to the sample period (minus a second or two to empty it each day).
Some weather observers empty the gauge more often than once a day. The more often one empties it, the more the situation approaches a digital sample of the volume/time and you have a rain-rate sample. The 'gauge' is just a long-time-constant 'hold' device in a sample and hold system.
daestrom
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Whoever is the lady masquerading as, "Chimera", she seems to have become obsessed by the wrong end of the stick.
I have never disputed the facts below that she attributes to the Delta function. I have held these to be true for years, but it is their very truth that means that using a comb of Delta Functions can only be an idealised model of sampling and never the actuality of sampling.
Still, right from her very first appearance a couple of months ago, "Chimera"'s motivation seems to have been to vent her spleen at me rather than to introduce any technical enlightenment. The google record speaks! Her aggressive jeering from the sidelines makes her irrelevant and she is best ignored.

Dirac's
are
zero
.....[All of jim's remarks snipped!].....
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You have disputed the nature of the Delta function, you claimed the amplitude was unity whereas it is the _AREA_ which is unity.
Have you ever actually had any formal education in maths or engineering? If so, can I suggest you ask for a refund.
Chimera
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Dirac's
are
zero
convenient
sampling
convenient
don't
never
idealized
Your link to the 'real world' is very apt and valid for the most part. But, as Dr Reay has pointed out, sampling does not take place on zero time. If it did, no energy would be transferred, which is clearly not tenable.
The _AREA_ of the unit pulse is relevant and, in the real world, equates to the sample time.
Chimera
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Ah... Your problem is you think the pulse has zero width. It is *not* exactly zero. It can approach zero and get as close as you are physically able to make it and/or want. But it *never* reaches zero. If it did, then there would be no pulse at all.
The area under the pulse is what is important. If the pulse width is fixed at any arbitrary size greater than zero, the pulse is definitely intergable.
daestrom
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But it isn't my problem, nor anyone else's. It is a fact of the action of sampling. We are interested in the amplitude of the sampled function at the rising edge only.
If we consider the rising edge to be the point of sample, then we become independent of the width of the sampling pulse in whatever circuit you are using.
As such, we wish to analyse a single point. That we wish to analyse through an integrable transform gives us a difficulty because such points are zero-integrable. The solution is to model using the infinity that is represented by the Unit Impulse's amplitude as our unity.
It is misleading to state that the area of the pulse is of interest, because there is no part of the calculus that supports the instantaneous multiplying of one function by the area of another. This last ruse is one of the religious icons of those who state that sampling _IS_ the effect of multiplying the incoming waveform by a comb of Delta Functions rather than stating that it is only _MODELLED_ by such. It is a religious icon because it is a matter of blind faith and not a matter of the underlying calculus. Like all religions, you need to contrive more and more weird and ridiculous explanations to explain away your initial assumption which is wrong.

then
fixed
intergable.
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OK, but if you sample the pulse during its rise time, how can you predict its final amplitude?
--
;>)
73 de Frank Turner-Smith G3VKI - mine's a pint.
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It is only the convolution of the two functions once you have reached the frequency domain. The objections that I am raising to the plausible standard explanation prevent you from reaching the frequency domain.

multiplication,
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