zero padding avoids aliasing

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.
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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

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

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