Impossible sampling theory!

A number of texts suggest that sampling can be modelled by multiplying the incoming waveform by a comb of Diracian Delta Functions.
How can this be?
1. The samples that you get are measured in the order of single volts whereas the Diracian is infinitely tall. Surely, if something of the order of unity were to be multiplied by something of the order of infinity, the result would be of the order of infinity?
How do you account for the difference? Do you have some internal mental model where there is an invisible constant, "Big K", perhaps, to account for the difference in scaling?
2. The area of the sampled pulse is very much less than unity, the volts being ooo unity and the time being typically ooo usecs.
How do you handle this mentally when the area of the Diracian is unity?
How do you come to terms with the attributes of your claimed model being orders of magnitude different from the signals of the real world?
3. If you are one of those who claim that the sampled signal is a short spike of zero width, then it is zero-integrable and not analysable by any process involving Laplace Transforms.
How do you overcome the problem that your sampled signals are not representable in the way that you claim?
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wrote:

Huh. A second ago you denied ever saying that every text on the planet is wrong. Can you name a _relevant_ text, that being a text that discusses this issue, that denies what you just said?

No, it's not infinitely tall. It's not a function.

But it's true that after you do that sampling what's left is no longer a function (at least not a function defined on R).

How do you account for the fact that you ask us all these silly questions, even though you're determined to pay no attention if anyone tries to explain?
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David C. Ullrich
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And I haven't said it below. Are you a troll?
And what I denied ever saying was that every textBOOK was wrong.
You'ew changing your goalposts by the minute.

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Why would there be any texts that deny that "A number of texts suggest that sampling can be modelled by multiplying the incoming waveform by a comb of Diracian Delta Functions"?
What texts do you suggest would discuss other texts in that manner?

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How tall do you think it is for the multiplication to take effect in a real circuit?

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Airy R. Bean wrote:

A real sample-and-hold circuit has a finite width and a height of 1.
And the Fourier theory in actual DSP is based on discrete Fourier series, which involves integrals over finite time windows, not the continuous Fourier transform which involves integrals over all time.
And the DFT of a comb function of height 1 is another comb function of finite height. But you knew all that already, didn't you?
- Randy
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how old are you beanie?
you cause a lot of problems. thats my observation
interesting indeed. whats in it for you? do you learn from this group? do you provide input that others appreciate?
dr. x
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Unheard of for Gareth to post anything helpful.
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: Unheard of for Gareth to post anything helpful. : : 5.6 on the cluckometer
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Airy R. Bean wrote:

One of my professors implied that the Dirac delta wasn't mathematically rigorous but provides correct results so it's used nonetheless.
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It can be mathematically rigorous if you were to use the curve borrowed from the Normal Distribution of statistics, but it cannot produce the correct results for the simple reason that the pulses obtained are several orders of magnitude different from the pulses in real circuits.
(And if you regard the pulses produced in real circuits as existing only at a point, then those pulses are not analysable)
In all aspects of engineering, the numbers that you analyse are the physical values that arise in your equipment. I wonder how others come to terms with the fact that the numbers produced by the claim that sampling is the multiplication by a comb of Diracian are simply far, far too large?

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Airy R. Bean wrote:

Good thing they don't claim that, but include an integrator in the idealized sampling circuit model.
- Randy
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On Thu, 09 Dec 2004 10:11:07 -0700, Kevin Neilson

There's no problem with the mathematics involved in the delta function. You don't see the actual math in typical undergraduate courses - the exposition in a typical differential equations book is certainly far from rigorous. That's just because the rigorous explanation is not going to be accessible to that audience, not a problem with the delta function itself.
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David C. Ullrich
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David C. Ullrich wrote:

Can you recommend a good website or book?
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On Thu, 09 Dec 2004 15:11:54 -0800, Tim Wescott

If you want the whole story you need to learn some "real analysis" first (measure theory, topological vector spaces, etc). There are many places you can find the theory of distributions worked out in detail - the two that are standard texts where I come from would be Folland "Real Analysis" (or maybe it's "Real Analysis and Applications" or something) and Rudin "Functional Analysis".
Website? Hmm, google...
It seems that wikipedia
http://en.wikipedia.org/wiki/Distribution
discusses the topic, although I doubt that there's a complete exposition of the theory there, in an "encyclopedia" they must have just statements of the main results.
The description of
http://www.math.ku.dk/~grubb/distcon.pdf
on google sounds like it might be what you want, but that actual pdf is just a table of contents. I didn't see how to find the actual notes on the site, but maybe you can if you hunt around.
Otoh I wouldn't be surprised if there is no web site that actually contains the whole story.
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David C. Ullrich wrote:

I have "Intermediate Real Analysis" by Emanuel Fischer, but if it discusses those topics it does so by entirely different names -- I rather suspect that it leaves off where your other texts start.
So far any time I've felt a need for rigor around the delta function (distribution, whatever) I've just constructed some real function with area one that's either parameterized by height (or width), found my result, then taken the limit as the parameter goes to infinity (or zero). It's probably not entirely kosher, but it's served my purposes.
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On Fri, 10 Dec 2004 08:30:58 -0800, Tim Wescott

I don't know that book but based on the title that seems likely - "real analysis" covers a lot of ground.

Can't say for sure without seeing exactly what you've done, but that could very well be just fine (although it's not _really_ ok unless you can explain why...). For example:
Say f_n(t) = n for 0 < t < 1/n, 0 for other t. Then f_n -> delta "in the sense of distributions" as n -> infinity. What convergence "in the sense of distributions" means is that if g is an infinitely differentiable function then
(*) int f_n g -> int delta g = g(0) as n -> infinity.
(Here int is the integral from -infinity to infinity.)
If all you're doing is things that look like (*) then the things you're doing are ok.
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But it doesn't provide correct results!

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Does the definition whose URL you quoted (and how much better a form of debate if you present your own argument rather than seeking to send your correspondents off somewhere else?) describe how the attributes of area and amplitude of the Diracian can possibly be representative of real-world sampling pulses with "plenty of rigor"?

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wrote:

For some reason you appear frightened of researching information, such as following a url, and integrating the knowledge gained with what you already know in order to take things into new territory. But this is how science advances, and the study for a degree of PhD requires that the current position be adequately researched as a prerequisite to moving on to one's particular research topic. This is called the Literature Survey, and it is a fundamental part of the PhD. Fail to perform this adequately, and your PhD is doomed.
As you will not follow urls, I append a short article to help you, and as you are prone to ISP failures, I may repost this from time-to-time.
Note particularly the paragraph headed "RESEARCH METHODS".
For the information of other readers, the url is http://www.itacs.uow.edu.au/research/postgrad/comments.shtml
GENERAL COMMENTS ON RESEARCH
PREAMBLE: Research, by its very nature, is a step into the unknown and therefore open-ended; there are no guarantees. As such your supervisor(s) will not know the answer to your research questions (research is not the same as coursework). This step is usually guided by the results of previous researchers in the field. Such previous work "sets the scene"/points you in the right direction/tells you where to look. Steady, methodical and persistent effort on your part is then necessary to reach your research goal, often employing the scientific/experimental method(s) (e.g. hypothesis testing). Of itself, this might not be sufficient; genuine insight, serendipity and unexpected "connections" from seemingly unrelated areas are often necessary. These can neither be anticipated nor manifested at will. Many scientific breakthroughs come from the most unexpected sources.
RESEARCH METHODS: In order to (a) become familiar with your chosen area of research, and (b) to ensure you don't "reinvent the wheel" and commence working on a topic which has been previously researched, it is essential to become familiar with the published literature in the field. A good way of doing this is to write your own literature survey/review article, perhaps even presenting a seminar/conference paper on your findings. This helps you not only to familiarise yourself with previous work, but also to highlight what has yet to be done/what problems remain to be solved in your chosen field. It also helps to identify areas in which you are perhaps weak and need to learn and/or improve your skills. The first six months of a 3-year PhD programme should be devoted to a literature survey; the second six months to replicating previous work. By the end of the first year, it should become clear as to how the earlier work can be extended/improved, thus enabling a detailed research proposal to be formulated. Naturally, the remaining two years are spent in following these ideas (and periodically backtracking and revising your research plan in the light of your findings).
NOTE: For Research Masters (and undergraduate Honours), it is quite valid to work on a topic which has been researched previously, but from a different perspective/extending it in some manner. For a PhD, an original contribution to knowledge is required - establishing what has been done previously and identifying a substantial problem to tackle is even more critical here. Successfully applying new/different (and better) techniques to problems previously solved by other means is still a valid approach for a PhD however. In order to conduct a literature survey, you will need to hone your library skills, specifically: (i) how to track down survey papers/introductory books, (ii) developing the art of quickly reading and evaluating abstracts (at least - entire papers if appropriate), (iii) identification of the classic references in the field, and subsequently tracking them down (in hard copy form, either within the UoW Library, or via Inter-Library Loans), (iv) use of the UoW on-line Library resources, as well as more general searching of the World Wide Web, & (v) the ability to critically evaluate what's been done previously. In short, who are the key researchers in the field? What are the seminal works/books/survey papers? What are the most important journals in your chosen area?
NOTE: It is very important to keep abreast of the latest developments in the field, especially if someone publishes what you are currently working on. If this happens, you may need to take a significant change of direction with your work. Thus periodic updates of your literature survey will be necessary during the course of your study.
--
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