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A Sound Mathematical Basis For Sampling - Lesson 3

Started by Airy R. Bean November 14, 2004
A Sound Mathematical Basis For Sampling - Lesson 3
--------------------------------------------------

Good Morning, once again, Boys and Girls!

I'm sorry that I got called away yesterday; SWMBO,
indeed, MBO!

Today I'll derive for you the mathematics of sampling,
based on an analysis of the circuits that we actually use,
rather than on some dubious mathematics (I will show later
why it is dubious) that most authors use, and which they
seem to crib from each other without checking! (Perhaps
the unsoundness of their mathematics is why they emphasize
so much the importance to them of "hands on" experience?
If so, that would seem to render them as technicians
rather than engineers, would it not? A disturbing
thought when so many or our life-support systems
depend on the results that the former pupils (that
these experts have "taught") produce!)

-----ooooo-----

Take the Unit Step function, U(t) which has the spectrum
of 1/s from our Laplace knowledge, together with another
Unit Step that has been delayed for a time T. (If we were
to be discussing land-line telephony where samples are taken
8000 times per second, then T would be 125 uSecs). That second
Unit Step is represented by U(t - T) and has a spectrum
of (1/s) * e^(-sT).

(I will cover in my notes at the end why a delayed signal
is a function of (t - T), and also the effect of a delay
causing the spectrum to be multiplied by e^(-sT). In respect
of this latter, remember that e^(-sT) is a complex number
that represents a phase shift of all constituent parts of
the spectra)

Now, if we take the first Unit Step, (which you remember rises
at zero and continues thereafter to eternity) and subtract
from it the second, we will get a single short pulse lasting
for T, which will be 125 uSecs in the land-line telephony case.

We are relying on the Principle Of Linear Superposition here,
in order to determine the spectrum of two signals added together.

Our combined signal, creating the single short pulse,
is given by U(t) - U(t - T) and its spectrum is
given by 1/s  - (1/s) * e^(-sT) which comes down
to ...

            (1 - e^(-sT)) / s

-----ooooo-----

Now, let us create a second pulse which starts as the first
one finishes, and then goes on for a further 125 uSecs.

We can create this with a Unit Step that is delayed by T
and then subtract from it another Unit Step that has been
delayed by 2T...
                 U(t - T) - U(t - 2T)

This second pulse will clearly have the same frequency content
as the first, but with a different set of phases so that the
infinite spectral components of the Laplace Transform now
add up constructively at a different point in time to give the
delayed pulse, and cancel out every where else. (Remember that the
effect on the frequency spectrum of a delay of T is to multiply
the spectrum by e^(-sT), a complex number having a modulus of 1
thus retaining the amplitude of the spectrum, but shifting the
phase of all components linearly)

The spectrum of our second pulse is given by...

             (e^(-sT) - e^(-s2T))/s

which simplifies to ...
            e^(-sT) * (1 - e^(-sT)) / s

This is the same as the spectrum of the first pulse except that
all the phases have been modified by the delay factor e(-sT)!

-----ooooo-----

In the same way, a third such delayed pulse would be ...
                 U(t - 2T) - U(t - 3T)

with a spectrum of ....

             (e^(-s2T) - e^(-s3T))/s

which simplifies to ...
            e^(-s2T) * (1 - e^(-sT)) / s

and so on, and so on, ad infinitum.

-----ooooo-----

Our real-world sampling circuits take a sample of the input
circuit, and hold it steady so that it can be digitised. Then,
at the sample interval (Yes, our T of 125 uSecs above, you're
getting ahead of me, well done!) take another sample, then
another, and another and so on, and so on.

What we end up with is a series of pulses of differing heights
determined by the value of the input signal at the start
of the sampling interval.

-----ooooo-----

If our analogue input signal could be termed f(t), then
we end up with a series of samples, f(0), f(T), f(2T), f(3T)
and so on and so on.

-----ooooo-----

Now, each sample looks like the pulses that we created just now,
but with each one of an amplitude governed by the input waveform
at the start of the pulse ....

We end up with the pulse sequence ...

        f(0) * [ U(t) - U(t - T) ]
        +
            f(T) * [ U(t - T) - U(t - 2T) ]
            +
                f(2T) * [ U(t - 2T) - U(t - 3T) ]
                +
                    f(3T) * [ U(t - 3T) - U(t - 4T) ]
                    +
                        ...

with the resultant frequency spectrum given by ...

        f(0) * [ (1 - e^(-sT)) / s ]
        +
            f(T) * [ (e^(-sT) - e^(-s2T))/s ]
            +
                f(2T) * [ (e^(-s2T) - e^(-s3T))/s ]
                +
                    f(3T) * [ (e^(-s3T) - e^(-s4T))/s ]
                    +
                        ...
Quite a mouthful!

-----ooooo-----

We now take out the common term (1 - e^(-sT)) / s , which,
as you will remember from above, is the Laplace Transform
of the first pulse, and we get ....

    [ (1 - e^(-sT)) / s ] * [ f(0)
                              +
                                  f(T) * e^(-sT)
                                  +
                                      f(2T) * e^(-s2T)
                                      +
                                          f(3T) * e^(-s3T)
                                          +
                                              ... ]

Still quite a mouthful for the output of our sampler ready
to be presented to the input of our DSP's!!

----oooo----

We do two things to reduce the seeming complexity of the
above expression.

1. In the frequency domain, you will remember that multiplication
is commutative, A * B giving the same result as B * A, and so
we take the factor [ (1 - e^(-sT)) / s ], (what might be thought
of as what gives us the pulse shapes), and ignore it until the
output stages of our DSP. This leaves us with a sequence ...

    [ f(0)
      +
          f(T) * e^(-sT)
          +
              f(2T) * e^(-s2T)
              +
                  f(3T) * e^(-s3T)
                  +
                      ... ]


IT IS MOST IMPORTANT NOT TO TAKE THE ABOVE COEFFICIENTS AS THOUGH
THEY ARE A SIGNAL SEQUENCE THAT HAS ARRIVED FROM SOMEWHERE ELSE.

THEY ARE MERELY A GROUP OF COEFFICIENTS THAT HAVE BEEN TEMPORARILY
DISCONNECTED FROM THEIR SHAPING FORMULA [ (1 - e^(-sT)) / s ].

Remember from yesterday's lecture that the sampling circuit is,
as far as Laplacian analysis goes, a SIGNAL GENERATOR that cannot
be analysed any deeper!

2. We make a parameter substition using the relationship ...

              z = e^(sT)

    so that e^(-sT) becomes z^(-1) or 1/z

so giving us ...

    [ f(0) + f(T) * z^(-1)
          +
              f(2T) * z^(-2)
              +
                  f(3T) * z^(-3)
                  +
                      ... ]
-----ooooo-----

So, we have arrived with the same mathematical expression
for a sampled input sequence as other authors, but we have
not had to appeal to Dirac's Delta Function and neither
have we had to invent mythical properties for that function
in order to justify the claims made for that approach!

-----ooooo-----

In my next lecture, I will attack the misuse of Dirac's Delta
Function as it is usually presented by other authors and
I will show the fallacies that they propagate - they are
good examples of the  faulty memes described by Richard Dawkins!


"Airy R. Bean" <me@privacy.net> wrote in message
news:2vpj4gF2ojquvU1@uni-berlin.de...
> A Sound Mathematical Basis For Sampling - Lesson 3 > -------------------------------------------------- > > > So, we have arrived with the same mathematical expression > for a sampled input sequence as other authors, but we have > not had to appeal to Dirac's Delta Function and neither > have we had to invent mythical properties for that function > in order to justify the claims made for that approach! > > -----ooooo----- > > In my next lecture, I will attack the misuse of Dirac's Delta > Function as it is usually presented by other authors and > I will show the fallacies that they propagate - they are > good examples of the faulty memes described by Richard Dawkins! > >
Yes it looks right.However I am not sure you are saying anything new.We already have results for sampling with finite width pulses.Also,when you assume the signal exist at f(0),f(T),f(2T) etc you are back to the impulse idea!Nice try though. Tom
Sampling by finite-width pulses is what happens in real circuits
and it is the way of the engineer to analyse the circuits that he
is actually using.

Please illustrate your claim that f(0) etc are impulses.

f(0) is a constant function which when graphed is a
horizontal straight line, and not an impulse function which
would be graphed a vertical straight line.

"Number 6" <No6@distant.island.nz> wrote in message
news:1100678258.651165@ftpsrv1...
> Yes it looks right.However I am not sure you are saying anything new.We > already have results for sampling with finite width pulses.Also,when you > assume the signal exist at f(0),f(T),f(2T) etc you are back to the impulse > idea!Nice try though.
Insofar as I have presented an analysis of sampling by real circuits
which did not have an appeal to the writings of P.A.M.Dirac, and
presents practical results that are tied in to previous knowledge
of the world of Fourier and Laplace, why do others feel the need
to introduce the semi-religious mysticism of sampling using
Dirac's Delta Function? How is it at all relevant to the practising
engineer as opposed to the acadaemic mathematician?

That mysticism leads to ever-more arcane explanations involving
distributions and generalised functions (ref. M.Lighthill) that lead the
reader further and further away from the task in hand, which is
understanding the frequency responses of our DSP's.

I suggest, once again, that it is religious mania - ridiculous claims
have been made which need more and more weird explanations
to sustain those claims.

Most readers, I believe (Argumentum Ad Populum?) shy away from
those Diracian explanations and rapidly skip over that part of their
studies.
When those readers reach positions of authority in the DSP World, they
cannot accept their lack of knowledge in what is a very fundamental
aspect of DSP and respond emotionally. That is my experience of them
in this NG today, yesterday and over the past few months.

The Google record of this NG shows that I, in my turn, found difficulty
with the Diracian approach, but being convinced that the fault lay with
me as the earnest student I sought solid explanations and none were
forthcoming. All I encountered were emotional tirades which seemed to
owe more to religious belief than to sound mathematical principles.

The situation is now that I know the practicalities behind sampling (see
my lesson 3). I also understand in full all the claims - false claims - made
by the Diracian school. I understand them, but I do not agree with them.

"Airy R. Bean" <me@privacy.net> wrote in message
news:300onkF2qar23U7@uni-berlin.de...
> Sampling by finite-width pulses is what happens in real circuits > and it is the way of the engineer to analyse the circuits that he > is actually using. > > Please illustrate your claim that f(0) etc are impulses. > > f(0) is a constant function which when graphed is a > horizontal straight line, and not an impulse function which > would be graphed a vertical straight line. > > "Number 6" <No6@distant.island.nz> wrote in message > news:1100678258.651165@ftpsrv1... > > Yes it looks right.However I am not sure you are saying anything new.We > > already have results for sampling with finite width pulses.Also,when you > > assume the signal exist at f(0),f(T),f(2T) etc you are back to the
impulse
> > idea!Nice try though. > >
There rests the matter.

I shall not now proceed with an attack on the Diracian
approach - my time as a lecturer should have reminded me how
much typing is needed to support even only a few minutes of lecturing
and I am very busy at the moment - hence the non-appearance
of Lesson 4 yesterday.

All the points that I would have raised have, in any case, been covered
by my many questions and scepticisms over the years and may be found
by a search of the Google record.

"Airy R. Bean" <me@privacy.net> wrote in message
news:300sifF2q4pjtU1@uni-berlin.de...
> Insofar as I have presented an analysis of sampling by real circuits > which did not have an appeal to the writings of P.A.M.Dirac, and > presents practical results that are tied in to previous knowledge > of the world of Fourier and Laplace, why do others feel the need > to introduce the semi-religious mysticism of sampling using > Dirac's Delta Function? How is it at all relevant to the practising > engineer as opposed to the acadaemic mathematician? > > That mysticism leads to ever-more arcane explanations involving > distributions and generalised functions (ref. M.Lighthill) that lead the > reader further and further away from the task in hand, which is > understanding the frequency responses of our DSP's. > > I suggest, once again, that it is religious mania - ridiculous claims > have been made which need more and more weird explanations > to sustain those claims. > > Most readers, I believe (Argumentum Ad Populum?) shy away from > those Diracian explanations and rapidly skip over that part of their > studies. > When those readers reach positions of authority in the DSP World, they > cannot accept their lack of knowledge in what is a very fundamental > aspect of DSP and respond emotionally. That is my experience of them > in this NG today, yesterday and over the past few months. > > The Google record of this NG shows that I, in my turn, found difficulty > with the Diracian approach, but being convinced that the fault lay with > me as the earnest student I sought solid explanations and none were > forthcoming. All I encountered were emotional tirades which seemed to > owe more to religious belief than to sound mathematical principles. > > The situation is now that I know the practicalities behind sampling (see > my lesson 3). I also understand in full all the claims - false claims -
made
> by the Diracian school. I understand them, but I do not agree with them. > > "Airy R. Bean" <me@privacy.net> wrote in message > news:300onkF2qar23U7@uni-berlin.de... > > Sampling by finite-width pulses is what happens in real circuits > > and it is the way of the engineer to analyse the circuits that he > > is actually using. > > > > Please illustrate your claim that f(0) etc are impulses. > > > > f(0) is a constant function which when graphed is a > > horizontal straight line, and not an impulse function which > > would be graphed a vertical straight line. > > > > "Number 6" <No6@distant.island.nz> wrote in message > > news:1100678258.651165@ftpsrv1... > > > Yes it looks right.However I am not sure you are saying anything
new.We
> > > already have results for sampling with finite width pulses.Also,when
you
> > > assume the signal exist at f(0),f(T),f(2T) etc you are back to the > impulse > > > idea!Nice try though. > > > > > >
"Airy R. Bean" <me@privacy.net> wrote in message
news:300t07F2qbuurU1@uni-berlin.de...
> There rests the matter. > > I shall not now proceed with an attack on the Diracian > approach - my time as a lecturer should have reminded me how > much typing is needed to support even only a few minutes of lecturing > and I am very busy at the moment - hence the non-appearance > of Lesson 4 yesterday. > > All the points that I would have raised have, in any case, been covered > by my many questions and scepticisms over the years and may be found > by a search of the Google record. >
Big K ?
"Airy R. Bean" <me@privacy.net> wrote in message
news:300t07F2qbuurU1@uni-berlin.de...
> There rests the matter. > > I shall not now proceed with an attack on the Diracian > approach - my time as a lecturer
Oh? When was that? What subject? What institution? Pre or Post Westinghouse?

Airy R. Bean wrote:
> Insofar as I have presented an analysis of sampling by real circuits > which did not have an appeal to the writings of P.A.M.Dirac, and > presents practical results that are tied in to previous knowledge > of the world of Fourier and Laplace, why do others feel the need > to introduce the semi-religious mysticism of sampling using > Dirac's Delta Function? How is it at all relevant to the practising > engineer as opposed to the acadaemic mathematician?
Because, as you have shown, it is equvalent to your more complicated approach and is more intuitive from a pedagogic perspective. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein