I don't think there is a disconnect between Chimera's version and the point you make. The feature that is common is the idea that sampling takes a finite time which is, in itself, and integration process. Think of a sample and hold- it integrates the signal over the time the sample gate is open, it doesn't take a sample in zero time.
As Chimera and daestrom point out, the area of the unit pulse is significant.
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 m> > > If you have a mathematical function that has no width
.....[All of jim's remarks snipped!].....
Not so. The accepted analyses of sampling use only the value of f(T) and not the integrated sum over the interval f(T) - f(T+dt). There is no integration of the input function f(t) over the sampling period in the standard analyses.
Whatever the width of the sampling pulse in your circuit, the value of the input function at the rising edge only is used.
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
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
Accepted by you only, to those of us with real experience your analysis is laughable.
Your comment re the rising edge shows a lack of understanding that I have never seen the like of.
If only the edge matters, why did Dirac specify the area as unity and not some measure of the rise time of the unit impulse. It would then be the zero rise time pulse not the unit pulse.
Chimera
Sanity at last.
Chimera
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
OK, but if you sample the pulse during its rise time, how can you predict its final amplitude?
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
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
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.
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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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.
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Jerry
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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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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