# A laying of the sampling ghost.....

At the accepted risk of stirring up the Twitterers Of Twaddle from their slumber.....
-----OOOOO-----
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!
-----OOOOO-----
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.
-----OOOOO-----
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.
-----OOOOO-----
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?
-----OOOOO-----
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.

function
another
away
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Let me stop you right THERE.
The spikes are unity AREA not AMPLITUDE

Now ask yourself why the width is important. (Someone has told you this before. Try using Google.)
<Big K nonsense snipped as even Airy should see it isn't required.>
Chimera
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Under your alter-ego you have mistakenly assumed that the sampling spikes referred to are Unit Impulses, and your motivation to be insulting has merely made you look rather silly.
If you are really that bothered that people in other NG who are your technical superiors might criticise you, then perhaps you'd be better off not posting at all?

their
problems.
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I didn't see Brian's post about the sampling spikes but you still haven't answered my post about the same topic.
Sampling pulses ARE Unit Pulses. UNIT AREA. REPEAT UNIT AREA. If you don't understand that I suggest take up another line of study.
Now, I could go on about the nature of the Unit Pulse but I suspect you wouldn't understand it.
Chimera
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Ad homin wots it now? Let us just remember who is the socket puppet here, shall we, Gareth Alun Evans of Chippenham?
GW
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Plonk, again...

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Airy R. Bean wrote:

This is to my recollection the third time over the years you have raised the topic, and judging by the responses you have received, I am beginning to suspect that the 'problems' lie between your ears.
--
from
RVMJ
(dot) 99g (at) BTinternet (dot) com
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<snip>
the
their
Does not the 'hold' section of a DAC add back in an integrable component, explaining why the analog output has been restored? A 'hold' mechanism is central to all DACs.

Is not the digital sampling of the analog input actually a convolution of the two functions? Convolution can often be represented by multiplication, but it is distinctly different. <snip>
daestrom
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You'd be in one of those mathematical "trick" areas, like when you prove that 1 = 2, if you started to integrate something that was already zero.

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Judging by your maths I would assume this is standard practice for you.
Chimera
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Not much of a 'trick'. A simple 'hold' section is nothing more than a capacitor. And of course the voltage on a cap is well known to be proportional to the integral of the net input current.
The net input current isn't *always* zero. There exists a pulse with a width, however narrow, when the input is not zero. If you claim it is *always* zero, then you haven't applied any pulse and have thrown away all the information in the signal. No matter how narrow the pulse width, it is never zero, and always has *some* area (definition of limit in this sense is the width *approaches* zero but never reaches it).
daestrom
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If you have a mathematical function that has no width and exists only at discrete points then it is zero-integrable, with the exception of the contrivance of Dirac's Delta function.

is
is
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The "contrivance of Dirac's Delta function"? As we are talking about Dirac's Delta function does that not make it rather relevant?
Your problem is that you don't understand even the basics of what you are trying to study. Dirac's delta function has unit AREA and tends toward zero width. The fact that it is defined as an AREA _repeat AREA _ then it is intergrable. If not, then sampling would not work.
Do you have any technical qualifications in this field? Any at all?
Chimera
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Not to pile on here, but the fact is, if someone provides sampling data at something a bit more than twice the bandwidth of the initial waveform, it can be exactly reconstructed.
Is this one of those discussions were facts are inconvenient?
AJW
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1. The person masquerading as "Chimera" has latched onto the attributes of the Delta Function, which were never in dispute. However, taken with her infantile stance of name-calling, she is best ignored for the irrelevance that she undoubtedly is. Examples of her infantile stance may be seen below. 2. Whereas what you say is true, there is a problem with real sampling taken as points in that it yields samples that are zero-integrable and therefore impossible to determine the frequency spectrum for by conventional mathematical analysis. 3. The spectrum of real sampling changes dependant upon the width of the sampling pulse, and we are only interested in the value of the sampled waveform at the rising edge, and a spectrum that is independent of the sampling pulse width. 4. So, to represent that rising edge only, we treat our real samples as existing only at that edge 5. As such, those samples are isomorphic to the Unit Impulse but because of their small size are zero-integrable. 6. Therefore, in order to achieve an analysis, we replace the unity of one that is the characteristic of the real world with the unity of infinity which is the amplitude of the Unit Impulse. 7. This is a contrivance for modelling only, and it is important to stress that real sampling is NOT multiplying by a Unit Impulse, for this raises so many valid mathematical objections.
Facts are inconvenient? Only to those who have swallowed the religious lie that real sampling _IS_ multiplying by a comb of impulses rather then _IS MODELLED_ by such.

Dirac's
are
zero
can
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1. If the attributes of the unit impulse were never in dispute, why did you dispute them _AND_ insist that it had unit amplitude?
2. The unit impulse only approaches zero width, it never actually becomes zero. This concept is as old as Newton, surely you have come across it before?
3. Which part of the rising edge would that be? Even with your warped version of the unit pulse (before you stopped disputing its attributes) you should see that if it is only its edge that is significant then its amplitude and width would be irrelevant. What we are really interested in is the POWER in the waveform being sampled so the width can not be zero. Zero width zero power.
4. You might treat the samples as existing only at the edge but those who understand sampling know that it is invalid. See 3, above.
5. New word "isomorphic". Not applicable in this case but a new word for you. Maybe you can use it in Scrabble. If you will excuse the pun, don't stretch it to infinite amplitude......
6. Whoaaaa "the unity of infinity". A good title for a love song maybe but, in this context, nonsense.
7. Whoaa, you have suddenly come to the real world. A new experience for you, it would appear. Please stay awhile and have a look around. You will find there are people here that understand DSP.
You have never actually done any DSP, I assume? Never implemented a A/D converter so never analysed why you need a sample and hold? While you are in the real world I suggest you try doing so. A bit of reality helps everyone. Don't overdo it, it seems to have been awhile for you.
Chimera.
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Chimera wrote:

No, this is completely incorrect. The dirac delta function is just a convenient construct that simply encompasses the notion that the spectrum of the sampling process is flat. Similarly the notion of a bandlimited function is a convenient idealization. In reality there are lots of working sampling processes that don't meet these ideal conditions. In fact its fairly safe to say that you will never encounter a sampling process that meets these conditions perfectly.     In fact, a sampling process doesn't even have to come close to the idealized dirac pulse to work. For instance digital cameras work just fine.
-jim
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Dirac's
are
zero
convenient
sampling
convenient
don't
never
idealized
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.
--
73
Brian
G8OSN
www.g8osn.org.uk
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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.

point
sample
it
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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
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