minimum cycles for fft, and limits of filling short sample out with zeros

glen herrmannsfeldt wrote:
> Rune Allnor wrote:
>
> > glen herrmannsfeldt wrote:
>
> (snip)
>
> >>Now consider sodium at higher temperature and pressure.
> >>The lines are broadened due to doppler shift and collisions,
> >>and there is also reabsorption by other atoms.
>
> > Now I am confused. Are you capable of studying each line in
> > its own right? Or do you just see a fuzzy wnergy distribution
> > with respect to frequency?
>
> Well, the question had to do with being able to measure
> frequency exactly when there is no noise.  I was trying to
> find real world examples that, even without noise, one would
> not be able to measure exactly.  You can study each line in
> its own right if you can build a filter to separate them.
> That filter is then part of the system.  It can be considered
> noiseless, but otherwise should have properties that real
> filters have.
>
> > If you are not able to study single absorption/emission lines,
> > then I would say you are dealing with a power spectrum density
> > estimation problem.
>
> So, in the noiseless case can you do an exact power
> spectral density measurement?
>
> > If you can see single lines and want to find their frequency
> > as exact as possible, then you are dealing with a frequency
> > estimation problem.
>
> The claim was exact.  I am looking for reasons why it
> won't be exact.
>
> > If you want to see if one line has split in two when you
> > modify the material, you are dealing with a line resolution
> > problem.
>
> Well, sodium already has two lines without any modifying.

I don't understand the basis for your post.

If you know everything about an experiment (like with
sodium) there is no reason for making a measurement.
If you state a model case wher you have access to the
model parameters, there is no reason to make a measurement.

What I am discussing, are real-world contraints where one
aims to learn something about the process being measured,
and where one have to deal with noise and system
imperfections.

The first factors to consider (there can be far more)
is the finite-duration measurement in the presence of
noise. In that case, the resolution principle applies
(if one aims to find if there is one or two lines
present by mean of the DFT) or the Cramer Rao limit
applies, if one uses a parametric approach.

Whatever one tries to do, it is crucial to have a clear
appreciation of the problem one tries to solve. What
method will work an what will not, depend on the
question one seks the answer for.

Rune

Rune Allnor wrote:

(snip originally regarding noiseless measurements)

>>Well, sodium already has two lines without any modifying.

> I don't understand the basis for your post.

> If you know everything about an experiment (like with
> sodium) there is no reason for making a measurement.
> If you state a model case wher you have access to the
> model parameters, there is no reason to make a measurement.

Some of the discussion had to do with the ability to measure
the frequency of a single sine wave, in the absence of
noise, with only three points.  The case with two sines
was also discussed.  I was using the sodium doublet as a
two sine case at least a little bit real.

> What I am discussing, are real-world contraints where one
> aims to learn something about the process being measured,
> and where one have to deal with noise and system
> imperfections.

I was trying to add a little real world into an idealized
problem.  One can never have an infinite duration sine in
a real problem, so it must have finite duration.  I was then
trying to understand the measurement of a finite duration
noiseless sine or small number of sine source.  To understand
whether the finite duration limits the ability to measure the
frequency even without any noise.

> The first factors to consider (there can be far more)
> is the finite-duration measurement in the presence of
> noise. In that case, the resolution principle applies
> (if one aims to find if there is one or two lines
> present by mean of the DFT) or the Cramer Rao limit
> applies, if one uses a parametric approach.

First I wanted to consider a finite duration noiseless
source.  To add more realism, though still without noise,
I didn't allow a perfect rectangle envelope, as that
would require excessive bandwidth.  (Even putting
the edges on sine zero crossings still requires a
discontinuous, and unrealistic, derivative.)

> Whatever one tries to do, it is crucial to have a clear
> appreciation of the problem one tries to solve. What
> method will work an what will not, depend on the
> question one seks the answer for.

That is true, and it wasn't so clear here.  I wanted
to add realism to a measurement of a noiseless source,
but how much realism is needed?  How about detector
noise?  Mixer noise for a heterodyne detector?
There are many possibilities.

-- glen