Zero padding fftw

On Dec 8, 1:34&#4294967295;pm, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:
> Zero padding can make it look like you are getting more resolution,
> but you can't create resolution that isn't there in the first place.

This depends on the type of resolution you require.  If you want
to separate peaks, then zero padding won't buy you anything.
If you want to more accurately locate isolated peaks (separated
by say 3 bins or more), then interpolating by zero padding can
create more accurate resolution of these locations (assuming the
original signal was properly band-limited).

Experiment: take a band-limited signal with a single clear peak,
sample it above twice the highest frequency but with a sampling
grid that does not include the peak location.  You will then not
see the peak in the actual sample values.  Then resample the
signal by using a FFT plus zero padded IFFT as the interpolator.
The single peak location will now be resolved more accurately.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
 http://www.nicholson.com/rhn/dsp.html

Resolution is also a term using for vision.  Interpolation
does not create more information than is already present
in the data.  But it does allow one to "see" the existing
information in more detail more easily.

> IMHO. YMMV.
> --
> rhn A.T nicholson d.0.t C-o-M
> �http://www.nicholson.com/rhn/dsp.html

On Dec 9, 8:08&#4294967295;pm, illywhacker <illywac...@gmail.com> wrote:
> On Dec 9, 7:53&#4294967295;pm, Clay <c...@claysturner.com> wrote:
>
> > > That depends on your definition of "resolution". &#4294967295;Doing an FFT, zero
> > > padding, then doing an IFFT is, IMHO, a valid way to do sample rate
> > > conversion in batch applications.
>
> > I've actually read a few articles based on exactly this approach to
> > resampling. It is valid.
>
> > Clay
>
> It is interpolation. It is based on a model of the signal. It is valid
> to the extent that this model is valid.

Just like my Prof used to say when I was at Uni. Love it. :-)

>
> illywhacker;

On Dec 9, 9:07&#4294967295;pm, Clay <c...@claysturner.com> wrote:
> On Dec 9, 3:08&#4294967295;pm, illywhacker <illywac...@gmail.com> wrote:
>
> > On Dec 9, 7:53&#4294967295;pm, Clay <c...@claysturner.com> wrote:
>
> > > > That depends on your definition of "resolution". &#4294967295;Doing an FFT, zero
> > > > padding, then doing an IFFT is, IMHO, a valid way to do sample rate
> > > > conversion in batch applications.
>
> > > I've actually read a few articles based on exactly this approach to
> > > resampling. It is valid.
>
> > > Clay
>
> > It is interpolation. It is based on a model of the signal. It is valid
> > to the extent that this model is valid.
>
> > illywhacker;
>
> Yes, resampling is interpolation.

Right. So were your signal produced by a musical instrument, for
example, zero padding might not be the best thing to do, since the
existence of lower harmonics implies the existence of higher ones.

illywhacker;

On Dec 10, 6:55&#4294967295;am, illywhacker <illywac...@gmail.com> wrote:


> ...
.> > > It is interpolation. It is based on a model of the signal. It
is valid
.> > > to the extent that this model is valid.

> > > illywhacker;

.> > Yes, resampling is interpolation.

> Right. So were your signal produced by a musical instrument, for
> example, zero padding might not be the best thing to do, since the
> existence of lower harmonics implies the existence of higher ones.

> illywhacker;

Just as zero padding and filtering doesn't add any information, zero
padding and filtering needn't remove any information. The only "model"
that is relevant to the situation is whether the signal of interest
survived the anti-aliasing filtering required before the original
sampling.

Dale B. Dalrymple

On Dec 10, 4:21=A0pm, dbd <d...@ieee.org> wrote:
> On Dec 10, 6:55=A0am, illywhacker <illywac...@gmail.com> wrote:
>
> > ...
>
> .> > > It is interpolation. It is based on a model of the signal. It
> is valid
> .> > > to the extent that this model is valid.
>
> > > > illywhacker;
>
> .> > Yes, resampling is interpolation.
>
> > Right. So were your signal produced by a musical instrument, for
> > example, zero padding might not be the best thing to do, since the
> > existence of lower harmonics implies the existence of higher ones.
> > illywhacker;
>
> Just as zero padding and filtering doesn't add any information, zero
> padding and filtering needn't remove any information. The only "model"
> that is relevant to the situation is whether the signal of interest
> survived the anti-aliasing filtering required before the original
> sampling.

That model (i.e. that the signal 'survived' as you put it) is one
possible model that would lead to the interpolation method discussed
here under a squared error loss function. So would the model that the
signal is white noise, and indeed any other model in which the Fourier
components are independent with mean zero (this includes all
translation invariant Gaussian models).

illywhacker;

On Dec 10, 8:45=A0am, illywhacker <illywac...@gmail.com> wrote:
> On Dec 10, 4:21=A0pm, dbd <d...@ieee.org> wrote:
> ...
>
> That model (i.e. that the signal 'survived' as you put it) is one
> possible model that would lead to the interpolation method discussed
> here under a squared error loss function.

> So would the model that the
> signal is white noise, and indeed any other model in which the Fourier
> components are independent with mean zero (this includes all
> translation invariant Gaussian models).
>
> illywhacker;

White noise doesn't make sense as a model of the content of an anti-
aliased and sampled signal.

Dale B. Dalrymple

On Dec 11, 10:13=A0pm, dbd <d...@ieee.org> wrote:
> On Dec 10, 8:45=A0am, illywhacker <illywac...@gmail.com> wrote:
>
> > On Dec 10, 4:21=A0pm, dbd <d...@ieee.org> wrote:
> > ...
>
> > That model (i.e. that the signal 'survived' as you put it) is one
> > possible model that would lead to the interpolation method discussed
> > here under a squared error loss function.
> > So would the model that the
> > signal is white noise, and indeed any other model in which the Fourier
> > components are independent with mean zero (this includes all
> > translation invariant Gaussian models).
>
> > illywhacker;
>
> White noise doesn't make sense as a model of the content of an anti-
> aliased and sampled signal.
>
> Dale B. Dalrymple

Of course not. That is not what I said. When I say model, I am talking
about a model of the signal before is was anti-aliased and sampled.
Such a model is a prerequisite for signal reconstruction or
resampling, explicitly or not.

illywhacker;

On Dec 12, 5:24=A0am, illywhacker <illywac...@gmail.com> wrote:
> On Dec 11, 10:13=A0pm, dbd <d...@ieee.org> wrote:
>
>
>
> > On Dec 10, 8:45=A0am, illywhacker <illywac...@gmail.com> wrote:
>
> > White noise doesn't make sense as a model of the content of an anti-
> > aliased and sampled signal.
>
> > Dale B. Dalrymple
>
> Of course not. That is not what I said. When I say model, I am talking
> about a model of the signal before is was anti-aliased and sampled.
> Such a model is a prerequisite for signal reconstruction or
> resampling, explicitly or not.
>
> illywhacker;

In DSP we don't interpolate the signal as it was before anti-aliasing
and sampling so modeling the signal as it was before anti-aliasing and
sampling is irrelevant. If the signal of interest can be represented
by the anti-aliased and sampled data then the interpolation may be
valid whatever the signal source. If the signal can't be represented
by the anti-aliased sampled data the problem isn't the interpolation,
it's the choice to anti-alias and sample or the choice of how to anti-
alias and sample that's the problem.

Dale B. Dalrymple

On Dec 12, 6:30&#4294967295;pm, dbd <d...@ieee.org> wrote:
> On Dec 12, 5:24&#4294967295;am, illywhacker <illywac...@gmail.com> wrote:
>
>
>
>
>
> > On Dec 11, 10:13&#4294967295;pm, dbd <d...@ieee.org> wrote:
>
> > > On Dec 10, 8:45&#4294967295;am, illywhacker <illywac...@gmail.com> wrote:
>
> > > White noise doesn't make sense as a model of the content of an anti-
> > > aliased and sampled signal.
>
> > > Dale B. Dalrymple
>
> > Of course not. That is not what I said. When I say model, I am talking
> > about a model of the signal before is was anti-aliased and sampled.
> > Such a model is a prerequisite for signal reconstruction or
> > resampling, explicitly or not.
>
> > illywhacker;
>
> In DSP we don't interpolate the signal as it was before anti-aliasing
> and sampling so modeling the signal as it was before anti-aliasing and
> sampling is irrelevant. If the signal of interest can be represented
> by the anti-aliased and sampled data then the interpolation may be
> valid whatever the signal source. If the signal can't be represented
> by the anti-aliased sampled data the problem isn't the interpolation,
> it's the choice to anti-alias and sample or the choice of how to anti-
> alias and sample that's the problem.

Obviously you have never heard of superresolution. In any case, the
question is: what did the OP want? Or in other words, what are the
extra samples, samples of? The case you are discussing is that in
which:

1) only one signal in the range of the antialiasing is compatible with
the values of the given samples;

2) the resampled signal required can be derived from the antialiased
signal alone.

We can assume (1), if the initial system is well-designed. But (2)
renders the situation trivial, and it is hard to believe this is what
the OP wanted (remember he or she wants to increase the resolution).
If the resampled signal required is not a function of the antialiased
signal alone, then more is required.

illywhacker;