Reply by glen herrmannsfeldt●January 10, 20072007-01-10
jddaviswy wrote:
(snip)
> I don't think that anybody else has mentioned this so I will put my two
> cents in here. The kind of resolution that people are talking about
> when they say that you can't get any more by zero padding is the
> ability to resolve two very close tones. If you want to do that, you
> need a longer record in the first place. This is the same as using a
> larger lens to be able to image objects that are close together, with a
> small lens the points (think stars) will be blured to some radius, with
> a larger lens they will be blurred to some smaller radius.
And then there is deconvolution to undo some kinds of resolution loss.
There is a very interesting book by Peter Jansson called
"Deconvolution of Images and Spectra." Non-linear algorithms
take into account that some signals can't go negative (intensity)
and some have a maximum, too.
-- glen
Reply by jddaviswy●January 8, 20072007-01-08
Ron N. wrote:
> That's the common meaning in optics, but not necessarily what
> all people are talking about here. A common usage, especially
> in newbie questions, it related to the accuracy of measuring a
> single windowed tone, not the separation of multiple tones (unless
> you consider the noise floor a "tone"). That's the cause of some
> of the confusion; people are asking and answering different
> questions.
Point well taken, I've had this confusing conversation with many
co-workers, fellow students, etc. What I wanted to get across was that
when people say "you can't get more resolution by zero padding" this is
the definition of resolution that they are using, and not the
interpolation related concept used when trying to find the location of
a peak, etc. I have found it to be a very confusing usage in the past
myself.
-John
Reply by Ron N.●January 8, 20072007-01-08
jddaviswy wrote:
> j...@gmail.com wrote:
> >....
> > Correct. The interpolated DFT outputs don't have any new information,
> > but you have greater resolution in the sense that the frequency bins
> > are more finely spaced, not any other way. It's just a semantic issue
> > on whether or not you want to call that greater "resolution."
> > ...
>
> I don't think that anybody else has mentioned this so I will put my two
> cents in here. The kind of resolution that people are talking about
> when they say that you can't get any more by zero padding is the
> ability to resolve two very close tones.
That's the common meaning in optics, but not necessarily what
all people are talking about here. A common usage, especially
in newbie questions, it related to the accuracy of measuring a
single windowed tone, not the separation of multiple tones (unless
you consider the noise floor a "tone"). That's the cause of some
of the confusion; people are asking and answering different
questions.
IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
Reply by Jerry Avins●January 8, 20072007-01-08
jddaviswy wrote:
> j...@gmail.com wrote:
>> ....
>> Correct. The interpolated DFT outputs don't have any new information,
>> but you have greater resolution in the sense that the frequency bins
>> are more finely spaced, not any other way. It's just a semantic issue
>> on whether or not you want to call that greater "resolution."
>> ...
>
> I don't think that anybody else has mentioned this so I will put my two
> cents in here. The kind of resolution that people are talking about
> when they say that you can't get any more by zero padding is the
> ability to resolve two very close tones. If you want to do that, you
> need a longer record in the first place. This is the same as using a
> larger lens to be able to image objects that are close together, with a
> small lens the points (think stars) will be blured to some radius, with
> a larger lens they will be blurred to some smaller radius.
>
> For more information on this refer to Hayes Statistical Digital Signal
> Processing and Modeling pp 402-403, or any Spectral Estimation book
> (Stoica, Kay, Marple).
John,
Your likening the resolution of a DFT to the resolution of a lens is a
fine analogy. With a 2D DFT, it would be exact. The blur you mentioned
is called an Airy disc, which has a sinc shape along a line through its
center. Just as tapered windows suppress the side lobes at the cost of
broadening the main lobe, so can graded neutral-density filters apodize
a lens's image. "Apodize" means "without feet"; the "toes" formed by the
extension of the sinc outside the main lobe. The word has taken on a
more general meaning in applied mathematics, and I'm surprised that it
isn't more generally used in signal processing.
http://en.wikipedia.org/wiki/Apodizing
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by jddaviswy●January 8, 20072007-01-08
j...@gmail.com wrote:
>....
> Correct. The interpolated DFT outputs don't have any new information,
> but you have greater resolution in the sense that the frequency bins
> are more finely spaced, not any other way. It's just a semantic issue
> on whether or not you want to call that greater "resolution."
> ...
I don't think that anybody else has mentioned this so I will put my two
cents in here. The kind of resolution that people are talking about
when they say that you can't get any more by zero padding is the
ability to resolve two very close tones. If you want to do that, you
need a longer record in the first place. This is the same as using a
larger lens to be able to image objects that are close together, with a
small lens the points (think stars) will be blured to some radius, with
a larger lens they will be blurred to some smaller radius.
For more information on this refer to Hayes Statistical Digital Signal
Processing and Modeling pp 402-403, or any Spectral Estimation book
(Stoica, Kay, Marple).
-John
Reply by ●January 7, 20072007-01-07
Jerry Avins wrote:
> cincydsp@gmail.com wrote:
> > John182 wrote:
> >> thanks.
> >>
> >> Now can someone tell me if zero padding affects the frequency resoluti=
on?
> >>
> >> i read that it does not?
> >> but i can't undersatnd this because when we zero pad we increase our
> >> number of samples and our frequency resolution is totally dependent on=
the
> >> number of samples therfore it has to change.
> >>
> >> thanks
> >
> > Zero-padding a finite-length time-domain signal will increase the
> > number of frequency bins output if you perform a DFT. Since
> > zero-padding doesn't affect the sample rate, the greater number of bins
> > still span the same frequency range as before. So, the frequencies that
> > you get are more finely-spaced; this can be interpreted to mean you
> > have greater resolution.
>
> Such an interpretation would be wrong. What you get is mere
> interpolation, not new data. The interpolation might as well have been
> done with a French curve or draftsman's spline.
>
> Jerry
> --
> Engineering is the art of making what you want from things you can get.
> =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=
=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF
Correct. The interpolated DFT outputs don't have any new information,
but you have greater resolution in the sense that the frequency bins
are more finely spaced, not any other way. It's just a semantic issue
on whether or not you want to call that greater "resolution."
By the sampling theorem, as long as you sample a bandlimited signal
under the Nyquist rate, you know everything you need to know to
perfectly reconstruct the signal (provided you could construct the
filters or some other implementation needed to do so, which we know is
impossible). Therefore, at that point, you've done the best you can do;
you can zero-pad all you want to get more points at the output of your
DFT, but you don't get any new information, because you already have
all the information there is on the continuous-time signal.
Jason
Reply by Ron N.●January 7, 20072007-01-07
John182 wrote:
> Now can someone tell me if zero padding affects the frequency resolution?
If you think interpolation algorithms (as opposed to just picking
the largest bin) affect frequency resolution, then zero padding
affects frequency resolution is a similar manner. Zero padding a
much larger fft is a computationally expensive way of creating a
large number of interpolated points. But usually you only want
only one interpolated maxima, and this can be done with a more
local interpolation (a table corrected parabolic interpolation was
posted a few weeks ago) at a much lower computation cost than doing
a whole and longer fft.
IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
Reply by Jerry Avins●January 6, 20072007-01-06
cincydsp@gmail.com wrote:
> John182 wrote:
>> thanks.
>>
>> Now can someone tell me if zero padding affects the frequency resolution?
>>
>> i read that it does not?
>> but i can't undersatnd this because when we zero pad we increase our
>> number of samples and our frequency resolution is totally dependent on the
>> number of samples therfore it has to change.
>>
>> thanks
>
> Zero-padding a finite-length time-domain signal will increase the
> number of frequency bins output if you perform a DFT. Since
> zero-padding doesn't affect the sample rate, the greater number of bins
> still span the same frequency range as before. So, the frequencies that
> you get are more finely-spaced; this can be interpreted to mean you
> have greater resolution.
Such an interpretation would be wrong. What you get is mere
interpolation, not new data. The interpolation might as well have been
done with a French curve or draftsman's spline.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by glen herrmannsfeldt●January 6, 20072007-01-06
John182 wrote:
> Now can someone tell me if zero padding affects the frequency resolution?
> i read that it does not?
> but i can't undersatnd this because when we zero pad we increase our
> number of samples and our frequency resolution is totally dependent on the
> number of samples therfore it has to change.
To increase the resolution in a real sense you need more information.
Say you take a discrete frequency spectrum (such as an FFT output),
transform to the time domain (all the information is still there),
zero out the last half of the points (some information is now gone).
If you transform back, you will still have as many frequency
points, but half of the information is gone.
For a different point of view, say I measure the lengths of some
objects with a ruler to 1mm resolution. Now I write down the
numbers with 10 digits after the decimal point and give them to you.
(Maybe using a random number generator.)
You now have what looks like high resolution measurements, but
there is no more information than the 1mm that I actually measured.
-- glen
Reply by John182●January 5, 20072007-01-05
thanks.
Now can someone tell me if zero padding affects the frequency resolution?
i read that it does not?
but i can't undersatnd this because when we zero pad we increase our
number of samples and our frequency resolution is totally dependent on the
number of samples therfore it has to change.
thanks