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
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?
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?
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?
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
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?
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.
David C. Ullrich
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
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
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.
David C. Ullrich
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.
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.
David C. Ullrich
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
and amplitude of the Diracian can possibly be representative of
real-world sampling pulses with "plenty of rigor"?
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
GENERAL COMMENTS ON RESEARCH
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.
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
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).
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?
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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