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?
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? ************************
David C. Ullrich
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?
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?
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?
[*Exactly* the same questions that he asked many moons ago, and
which have been answered completely in these groups, with both the
theoretical and practical rammifications thoroughly covered.]
You have provided no coherent or correct refutation of any of the
responses that were provided in the previous discussions.
Rather than asking the same questions again, I suggest that you use
groups.google.org to research the answers already provided, and post
followup questions if you feel that specific clarification is necessary.
Going back to your original post is unlikely to be more helpful another
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
in article firstname.lastname@example.org, Airy R. Bean at email@example.com
wrote on 12/9/04 7:29 AM:
What is truly impossible is explaining things to Airy-head. I usually avoid
dissing people, but Airy is such an inviting target.
] Rather than asking the same questions again, I suggest that you use
] groups.google.org to research the answers already provided, and post
] followup questions if you feel that specific clarification is
] necessary.Going back to your original post is unlikely to be more
] helpful another time around.
Andrew, you're assuming that understanding is the point of the
exercise. I don't think it is.
Why not cite some of those responses and show
how they answered the questions?
The answer is, that they did not, and merely repeated
parrot-fashion (or religionist fashion if you prefer) what
could be read from the text books. As I referred to such
textbook context initially, then those responses were meaningless,
other than, perhaps, to serve as an ego-trip for the posters.
First of all, shame on you for lowering the tone
by introducing the behavioural standards of the CBer.
That you are a CBer is indicated by your failing to realise
that Ham Radio is a technical pursuit, where interest in
technical development is the essence. DSP is now an increasing
part of the techiques developed by _REAL_ Radio Hams and
therefore any discussion relating to the understanding of DSP
is entirely relevant to Ham Radio.
(You're a CBer, so you won't have a clue about what the
above means, and so I append a short article to assist you.)
What is Ham Radio?
Ham Radio is a technical pursuit for those who
are interested in the science of radio wave
propagation and who are also interested in the
way that their radios function. It has a long-standing
tradition of providing a source of engineers who
are born naturals.
Ham Radio awakens in its aficionados a whole-life
fascination with all things technical and gives
an all-abiding curiosity to improve one's scientific
knowledge. It's a great swimming pool, please dive in!
This excitement causes a wish to share the experience
with ones fellow man, and shows itself in the
gentlemanly traditions of Ham Radio.
Radio Hams are qualified to design, build and then
operate their own pieces of equipment. They do this
with gusto, and also repair and modify their own
The excitement that drives a Radio Ham starts with
relatively simple technologies at first, perhaps making
his own Wimshurst machine and primary cells. Small pieces
of test equipment follow, possibly multimeters and signal
generators. Then comes receivers and transmitters. It is with
the latter that communication with like-minded technically
motivated people takes off. The scope for technical
development grows with the years
and now encompasses DSP and DDS. There is also a great deal
of excitement in the areas of computer programming to
be learnt and applied.
The technical excitement motivates Radio Hams to compete
with each other to determine who has designed and manufactured
the best-quality station. This competitiveness is found in DXing,
competitions and fox-hunts.
However, beware! A Ham Radio licence is such a
desirable thing to have that there are large
numbers of people who wish to be thought of
as Radio Hams when, in fact, they are nothing
of the kind! Usually such people are a
variation of the CB Radio hobbyist; they buy their
radios off the shelf and send them back to be
repaired; they are not interested in technical discussion
and sneer at those who are; they have no idea how
their radios work inside and have no wish to find out;
they are free with rather silly personal insults;
they have not satisfied any technical qualification
and their licences prevent the use of
These CB types engage in the competitive activities
with their Cheque-Book-purchased off-the-shelf radios
in a forlorn effort to prove that they are Radio Hams.
No _REAL_ Radio Hams are deceived by such people!
If you think that's bad, how about the fact that signals are required to last
for infinitely long in order to be correctly represented by their Fourier
For a signal to be accurately represented by its Fourier transform, it must be
periodic. For a signal s(t) to be periodic with period P, it must be true that
s(t) = s(t + n * P) for all integral value of n (all values, including plus and
So, how does anyone justify using Fourier transforms when these are clearly only
valid for signals which have always been around and which will last forever!!
PS: One professor who tried to brainwash me into accepting these Fourier
transforms tried a light-hearted remark along the way that t = -innfinite was
clearly going back too far - it would be enough to be gack to 1819 where H C
Oersted discovered electrodynamics. Ol' HC went on the found the university I
attended, but that doesn't make such rubbish slipshod pseudo-math any more
There's no difficulty there because the infinite spectrum
of sinusoids cancel out to zero everywhere else
apart from the place at which they add constructively
to produce the pulse being analysed.
That they are zero everywhere else, including the
extremes of bipolar infinities means that the infinities
do not feature.