At the accepted risk of stirring up the Twitterers Of Twaddle from their slumber.....
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The conversion from the continuous time world of analogue to the discrete time world of digital causes some mathematical problems.
The action of sampling is to multiply an incoming analogue waveform by a series of unity amplitude spikes (ideally of zero width) to yield, still in the analogue world, a set of samples of the incoming waveform, again of zero width.
How do we represent these samples mathematically? In particular, how do we analyse their frequency spectra?
The calculus available to us in the analogue world holds these samples to be zero-integrable and therefore their application into any integral transform (Fourier, Laplace, etc) will yield a zero spectrum.
However, we know that this is not true of the systems into which this sampling takes place, for after processing of these samples and re-application back to the analogue world through a DAC, the whole of the works of Mozart re-appear in all their glory.
So, what is the nature of these samples, and how shall they be represented? We have a mathematical model, integral calculus, that is failing us because of a zero-integrable result in an area where the results (Mozart, above) are far from zero!
The approach taken, (and we are free to adopt any approach that may appeal to us to model an aspect of engineering for which conventional mathematics has failed us) is to model the sampled spikes as though they are a multiple of the Analogue Unit Impulse (not to be confused with the Discrete Unit Sample, more of that later). We need to be very cautious when we do this, because if we claim that the action of our sampling is to multiply the incoming waveform by a Unit Impulse, there are many, many valid mathematical objections that can be raised against such a claim, not the least of which is the lack of the order of infinity in a sample that is only a volt, or so, high.
So, the model (and it is only a model and not a REAL representation) that we take is to say that the unit of unity that was the continuous analogue world has become the unit of infinity in the discrete analogue world. This resolves our problem of zero-integrability, the properties of the Unit Impulse, Dirac's Delta function, being well established.
It may help to think of some fiddle factor that has caused this huge multiplication of size. I had previously referred to a non-specific, "Big-K", although what I am implying now would, in fact, be the reciprocal of what was previously posited as "Big-K". Such a fiddle factor would be justified because even if though the scale of our analogue samples in unity and not infinity, the shape of our sampled pulses is isomorphic with the Unit Impulse. We must, however, remember to remove this fiddle factor, this Big-K, this reciprocal of Big-K; and we do this when applying the outputs of our DSP to the DAC from which comes out our Mozart once again in signals that are of the order of unity..... The transfer function, or impulse response, of the combined effect of the DAC and the sample-and-hold is taken to be a single pulse of unity value(existing till the time of the next conversion from the output of our DSP). So, in response to a voltage spike of no width and of unity height, the sample-and-hold responds with its pulse lasting as long as the sample time. Now, in reality we have a spike of one volt high which we have been analysing as though it was infinity volts high. How do we mentally model this removal of Big-K?
Think of this......if a network responds with an output that is it impulse response, then what it was triggered with must have been an impulse!
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IT IS IMPORTANT TO REMEMBER THAT THE ACTION OF SAMPLING IS NOT TO MULTIPLY THE INCOMING ANALOGUE WAVEFORM BY A COMB OF UNIT IMPULSES AND THEIR DELAYED SIBLINGS; IF THAT WERE TO BE CLAIMED THEN IT COULD BE EASILY KNOCKED DOWN.
The action of sampling is only _REPRESENTED_ by such a multiplication and only then to resolve the specific problem of zero-integrability.
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In response to the OP.....
On the whole, it seems to me that despite your undoubted standing in the DSP community, you were one of those who swallowed the plausible story about sampling, and because you understood well at that point all that you had needed to know about the Delta function up to that time, that you, as did so many others, accepted statements made about the Delta function without objection and without query.
It is only now, much later on, that you have allowed yourself to be drawn into a challenge that was not aimed at you personally, and I suspect that you cannot accept that there is a flaw in your underlying knowledge; causing you to react with Freudian Rationalisation by interjecting a rather silly adaptation of the integral calculus, as quoted from you below.
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This has, however, been an interesting series of threads over the last few years. It is not an area in which I am yet professionally involved, and with so many other things to occupy my time, the ugly head got reared again whenever I drifted back round to the topic.
Based upon the explanation given above, I am satisfied. However, I also remain satisfied that all the objections and protests that I raised against the glib throw-away-lines of the text-books were also correct!
It is interesting that in response to my claim that the text-books were dubious was that so many of you responded by indignantly requoting the text-book position and thereby contributing nothing to the discussion. More Freudian Rationalisation, perhaps?
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I leave it to the Twitterers Of Twaddle to now exhibit their playground habituation. No doubt there will be an indignant mass fart of, "I told you so" when, in fact, they told nothing of the kind.