FAQ - Perhaps there are real professionals here today rather than uppity technicians?

Frequently Asked Questions (F.A.Q.).....

1. (A Frequently Added Quotation, F.A.Q., is appended below.)
Assuming that we were able to generate a Diracian, and then produce
a comb of them by delays and by superposition, there wouldn't be a
factor
of "T" in such superposition, so where does yours come from?

Where did the factor of "T" come from in the "sampling function" in
your opening lines?

Consider a 16-bit ADC capable of 100 M Samples per sec. In the first
instance we'll use it to sample a geophysical signal of bandwidth
limited to 300 HZ and sample at 1 kHz, with suitable analogue
instrumentation to match the input signal to the full range of
the ADC.

If we keep the circuit the same, but now sample
at 65.536 MHZ, your claimed factor of "T" will result in the
16 bit range being compressed down to one bit, because the new
sampling frequency is 2^(16) faster than the old. We know this
doesn't happen - there will be more samples, but they'll still
be of the same magnitude and over the same full-scale 16-bit range.

So, where does the factor of "T" come from?

2. Where have your Diracian impulses come from?

The Diracian has some interesting properties - zero width,
area of unity, infinite sum of all possible cosines, a height
which is not discussed but which appears to be greater in magnitude
than any voltage appearing in your circuits. In the
systems that you deal with, what experimental evidence do you
have that there are pulses with the attributes of Diracians upon
which to base your theory?

The Diracian, or Unit Impulse is a very good mathematical tool
to analyse the response of systems once a mathematical model of
those systems had been produced.

It is, however, a poor mathematical claim to make that such
impulses are found to be part of a system when neither the
area nor the magnitude of such impulses are found anywhere in such
systems.

To those who ask, "Who cares? I get good results." , I suggest that
their approach is unscientific and compares to the religious
loonies who sacrifice goats and virgins to stop the Sun falling
out of the sky and justify the continuing practice by the Sun
remaining in the sky.

So.....is the world of DSP a world of scientific men, or is
it a world of snake-oil charlatans and of religious loonies?

Where do these Diracian impulses come from?

> robert bristow-johnson wrote:
>
>                   x(t)*q(t) = T*SUM{x[k]*d(t-k*T)}   .------.
>     x(t)--->(*)------------------------------------->| H(f) |---> x(t)
>              ^                                       '------'
>              |
>              |                  +inf
>              '------- q(t) = T * SUM{ d(t - k*T) }
>                                k=-inf
>
>
> where:   d(t) = 'dirac' impulse function
>   and       T = 1/Fs = sampling period
>            Fs = sampling frequency
>
>
>                +inf
>     q(t) = T * SUM{ d(t - k*T) }  is the "sampling function", is periodic
>               k=-inf              with period T, and can be expressed as a
>                                   Fourier series.  It turns out that ALL of
>                                   the Fourier coefficients are equal to 1.

I think a dirac function is useful to develop math of sampling theory.
To model a realistic stuation, we convolve a dirac function with a
function that approximates to the real sampling function (which is
likely to be some kind of average over a short time, perhaps weighted).
The math supports this quite easily, but is not so easy to do, and in
most cases the rsult is found ot be not much different. So we tend to
work with the easy math rather than always using the hard stuff.
However of course in many cases the non-dirac nature of sampling
functions is important (for instance in image sensors which need to be
quite large and so are averagers): in these cases we do take account of
the sampling function (its effect in images is to attenuate certain
spatial frequencies, which is easily demonstrated in practice).

Chris
=====================
Chris Bore
BORES Signal Processing
www.bores.com

Polymath wrote:
> Frequently Asked Questions (F.A.Q.).....
>
> 1. (A Frequently Added Quotation, F.A.Q., is appended below.)
> Assuming that we were able to generate a Diracian, and then produce
> a comb of them by delays and by superposition, there wouldn't be a
> factor
> of "T" in such superposition, so where does yours come from?
>
> Where did the factor of "T" come from in the "sampling function" in
> your opening lines?
>
> Consider a 16-bit ADC capable of 100 M Samples per sec. In the first
> instance we'll use it to sample a geophysical signal of bandwidth
> limited to 300 HZ and sample at 1 kHz, with suitable analogue
> instrumentation to match the input signal to the full range of
> the ADC.
>
> If we keep the circuit the same, but now sample
> at 65.536 MHZ, your claimed factor of "T" will result in the
> 16 bit range being compressed down to one bit, because the new
> sampling frequency is 2^(16) faster than the old. We know this
> doesn't happen - there will be more samples, but they'll still
> be of the same magnitude and over the same full-scale 16-bit range.
> to 1.


So what you are asking is why does oversampling improve the resolution.

Consider that the quantizing noise is distributed over the entire
Nyquist bandwidth.  When you raise the sampling rate, the Q noise is
spread over a wider frequency range so the noise density is reduced and
the noise in a given bandwidth of interest  is reduced.

The quantizing noise being distributed over the full Nyquist bandwidth
depends on an assumpition of wideband random noise present at the input
A/D input that excedds 1 LSB.  This assumption can be met by the
application of dither.

Mark

chris_bore@yahoo.co.uk wrote:
> 
> I think a dirac function is useful to develop math of sampling theory.
> To model a realistic stuation, we convolve a dirac function with a
> function that approximates to the real sampling function (which is
> likely to be some kind of average over a short time, perhaps weighted).
> The math supports this quite easily, but is not so easy to do, and in
> most cases the rsult is found ot be not much different. So we tend to
> work with the easy math rather than always using the hard stuff.
> However of course in many cases the non-dirac nature of sampling
> functions is important (for instance in image sensors which need to be
> quite large and so are averagers): 

This is nonsense. The primary problem with an image sensor is the signal
being measured is not bandlimited. If the signal were in fact properly
bandlimited then one could simply derive the signal magnitude at the 
exact center of each sensor in the array. In order to do that one must
assume that each sensor is sized and located with perfect exactness.
That is, of course, another real limitation to realizing the perfect
sampling that the theory predicts. The width of the sensor is not.

-jim


in these cases we do take account of
> the sampling function (its effect in images is to attenuate certain
> spatial frequencies, which is easily demonstrated in practice).
> 
> Chris
> =====================
> Chris Bore
> BORES Signal Processing
> www.bores.com

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(Pseudonym changed from "Polymath", BTW)

1. How is it useful if it does not represent the engineering
values being sampled?

2. Convolving with a Diracian is no more than multiplying
by unity in the frequency domain - not a lot of use to anyone.

3. Thanks anyway.

chris_bore@yahoo.co.uk wrote:
> I think a dirac function is useful to develop math of sampling theory.
> To model a realistic stuation, we convolve a dirac function with a
> function that approximates to the real sampling function (which is
> likely to be some kind of average over a short time, perhaps weighted).
> The math supports this quite easily, but is not so easy to do, and in
> most cases the rsult is found ot be not much different. So we tend to
> work with the easy math rather than always using the hard stuff.
> However of course in many cases the non-dirac nature of sampling
> functions is important (for instance in image sensors which need to be
> quite large and so are averagers): in these cases we do take account of
> the sampling function (its effect in images is to attenuate certain
> spatial frequencies, which is easily demonstrated in practice).
>
> Chris
> =====================
> Chris Bore
> BORES Signal Processing
> www.bores.com

No. What I am asking is, "Where did the factor of "T" come from in the
"sampling function" in ... opening lines?"

Mark wrote:
> Polymath wrote:
> >
> > Where did the factor of "T" come from in the "sampling function" in
> > your opening lines?
> >
>
> So what you are asking is why does oversampling improve the resolution.
>

Dominus Dominorum wrote:

> 2. Convolving with a Diracian is no more than multiplying
> by unity in the frequency domain 

Precisely.

- not a lot of use to anyone.

Maybe. But considerably more useful than attempting to multiply by
anything else.

-jim

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>The primary problem with an image sensor is the signal
>being measured is not bandlimited. If the signal were in fact properly
>bandlimited then one could simply derive the signal magnitude at the
>exact center of each sensor in the array. In order to do that one must
>assume that each sensor is sized and located with perfect exactness.
>That is, of course, another real limitation to realizing the perfect
>sampling that the theory predicts. The width of the sensor is not.

Are you saying that if the optical image were band-limited and we knew
the exact size and sampling function of the sensor then we could
reconstruct the signal at the center of the sensor?
If so, then I agree with you but I do not see how that makes the
statement that the sampling function is important be 'nonsense'.

Also, surely there are many problems with image sensors apart from the
fact that optical images are not bandlimited?
Such as noise, light gathering power, that lead to a desire for large
sensors, and sometimes microlenses - these mean the effective sampling
function is not easy to assess perfectly.

On the question of band limiting, an optical signal can be bandlimited
with optical low pass filters - again, within the limitations of the
technology but just as characterizable as antialias filters in any
other domain.

Chris
=====================
Chris Bore
BORES Signal Processing
www.bores.com

Polymath wrote:
> Consider a 16-bit ADC capable of 100 M Samples per sec. In the first
> instance we'll use it to sample a geophysical signal of bandwidth
> limited to 300 HZ and sample at 1 kHz, with suitable analogue
> instrumentation to match the input signal to the full range of
> the ADC.
>
> If we keep the circuit the same, but now sample
> at 65.536 MHZ, your claimed factor of "T" will result in the
> 16 bit range being compressed down to one bit, because the new
> sampling frequency is 2^(16) faster than the old.

Do you assume the dimensions represented by the 16 bits of data
word are the same in both cases and independant of sample rate?

Why?

Would a vector of samples of the geophysical signal stand on
its own and be useful if the sample rate were completely unknown
and undeduceable from just the vector contents?


-- rhn

chris_bore@yahoo.co.uk wrote:
> 
> >The primary problem with an image sensor is the signal
> >being measured is not bandlimited. If the signal were in fact properly
> >bandlimited then one could simply derive the signal magnitude at the
> >exact center of each sensor in the array. In order to do that one must
> >assume that each sensor is sized and located with perfect exactness.
> >That is, of course, another real limitation to realizing the perfect
> >sampling that the theory predicts. The width of the sensor is not.
> 
> Are you saying that if the optical image were band-limited and we knew
> the exact size and sampling function of the sensor then we could
> reconstruct the signal at the center of the sensor?

We could but if we are talking ordinary photography we don't because we
don't want to. Digital cameras can do a pretty good job of creating
life-like images. For the signal they are intended to capture (that
which the mind's eye would see) they are faithful. That is, the
frequency response for the target signal is pretty flat. To change its
"non-dirac nature" as you called it to more resemble a narrow pulse
would degrade the result. Not because of issues with "noise, light
gathering power" as you postulate but because it would change the
frequency content of the signal.

-jim

> If so, then I agree with you but I do not see how that makes the
> statement that the sampling function is important be 'nonsense'.



> 
> Also, surely there are many problems with image sensors apart from the
> fact that optical images are not bandlimited?
> Such as noise, light gathering power, that lead to a desire for large
> sensors, and sometimes microlenses - these mean the effective sampling
> function is not easy to assess perfectly.
> 
> On the question of band limiting, an optical signal can be bandlimited
> with optical low pass filters - again, within the limitations of the
> technology but just as characterizable as antialias filters in any
> other domain.
> 
> Chris
> =====================
> Chris Bore
> BORES Signal Processing
> www.bores.com

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