Re: Interpolation

On Fri, 04 Apr 2008 08:23:53 -0600, jim <"sjedgingN0sp"@m@xxxxxxx>
wrote:



Eric Jacobsen wrote:

On Thu, 03 Apr 2008 12:13:23 -0600, jim <"sjedgingN0sp"@m@xxxxxxx>
wrote:

Eric Jacobsen wrote:


A signal which has energy only in the unity-gain passband of a filter,
regardless of how much bandwidth reduction the filter has done, will
still preserve the input samples in the output if it isn't
time-shifting the samples.

That is an absurd position. If the samples are preserved the filter has
done no bandwidth reduction at all. Besides absurd, it is also irrelevant.
The case that was asked to be considered was when the band reduction
filter does change the samples.

How is it absurd that a signal in the unity gain passband of a
frequency selective filter is preserved at the ouput? If it *isn't*
preserved it's a problem, I'd think.

Your whole position is absurd because it has no substance. You have stated
no clear definition of interpolation or any of the other terminology. The
average politician can address the issues more straight forward than you
can. You just dance around and say nothing.

Here is what you said "A signal which has energy only in the unity-gain
passband of a filter, regardless of how much bandwidth reduction the
filter has done, will still preserve the input samples in the output.

That statement is absurd. Of course it will preserve the input sample as
it has done absolutely no bandwidth reduction at all. If I move your data
from one hard drive to another the samples will be (hopefully) preserved
also. Does that mean there has been interpolation? Or what? Or is my
example just as meaningless as your example?

I think what that means is that you still don't understand my example
or my point.

You could simply resample your sinusoid without doing any filtering at
all. If you have a sinusoid at a frequency below Fs/4 you can simply toss
out every other sample in order to down sample by 2. So what? Are you
saying doing that is interpolation? No of course you are not because you
are not saying anything at all.

Well, you're not getting my points, that's for certain.

And you're completely missing the point, despite me trying to make it
as clearly as I can multiple times. It *is* relevant, because it
illustrates that the process is separable. The separation of the
bandwidth reducing filter and the resampler illustrates that the point
of whether or not the input samples are preserved at the output or not
is not a strong one when arguing whether or not the resampler is an
"interpolator" or not.

But that is not correct. Previously in this thread I wrote a definition
for interpolation.

"The process called interpolation is simply to find new points on a
continuous curve that passes thru the input samples."

Your response:

"I've said as much a few times here, and was perhaps the first one in
this thread to do so."

So let's suppose that is the definition. Now if you are doing a
downsampling process and if that downsampling involves reducing the
bandwidth - the new samples will not lie on the curve that passes thru the
original samples. As a consequence downsampling (in many case) cannot be
considered to be interpolating the input samples. It doesn't meet the
definition you agreed to.

You've already evidently agreed that it does:

___________ You (jim) wrote on April 1st:

Jerry Avins wrote:


I don't care what you call it. I think it's clear that the filter
changes the samples whose location in time matches the original samples.

Your previous post said it wasn't so clear.


After the filtering is done, there is no further change brought about by
the downsampling process. If that's not the case. please let me know.

Yes, that was pretty much all there was to the original statement ->
The
AA filter changes the samples.

-jim
_____________________

Or were you not agreeing with Jerry? He was making a very similar
point to mine but from a different, but also very valid, perspective.
The filtering and downsampling processes are separable, so the
examination of whether or not "interpolation" is taking place in the
downsampling can be assessed separately from the "bandwidth reduction"
that the filter is doing. I'm making the point that one can also take
the perspective that there's an easy example case where the decimating
filter, even with "anti-alias" bandwidth reduction, preserves the
input samples. This addresses the "purist" point of view that the
samples shouldn't change, so it's not hard to argue that
"interpolation" is happening in a decimating filter.

The nice thing there is that whether or not you take the perspective
that it matters that the input samples are preserved, it's still
straightforward to show that the processes are separable and
"interpolation" is taking place. If one still wants to argue that
when the output samples don't match the input (in other words, when
there's energy in the stop band), that "interpolation" is no longer
happening, it suggests that the definition is data dependent. I
think it just adds more confusion when the name of an algorithm is
data dependent.

I'm not sure what part of that you don't understand. From my
perspective it fully addresses the point, so if you think I'm just
"dancing around saying nothing" it may be due to a problem on your
end.

Now if you want to talk about implementation details of downsampling and
say the process can be conceptualized as one process with several
sub-processes and a particular sub-process meets the definition of
interpolation - That is fine. Saying that would be actually saying
something meaningful. It also makes it clear that you don't regard the
entire process as nothing but interpolation.

That's what I've been getting at all along. Maybe you're finally
catching up.

But instead you are reciting over and over again about filtering a
sinusoid with a filter that has a gain of one at the sinusoid's frequency.
That example says nothing. It says nothing meaningful about downsampling
or interpolation. It doesn't even suggest that "process is separable" as
you claim, if anything that example obscures the fact that the sub-process
is separable.

No, it demonstrates that the same filter can, with a perfectly valid
input signal, produce an example case where the input samples are the
same as the output samples. Since that seems to be the condition that
you feel is necessary for the term "interpolation" to apply (even
though you apparently agreed when Jerry said otherwise), it
demonstrates that even in that case one can see that the subsequent
resampling process is, exactly, by the description that you gave,
interpolation of the input samples.

Now look at upsampling. If the task one wishes to undertake is upsampling
(when you choose to) you can always make the output samples lie on the
(interpolating) curve. So clearly there is a difference between upsampling
and downsampling with respect to the definition of interpolation that you
agreed to. You claim there is no difference, but you offer nothing in
support of your position.

-jim

No, you've either ignored or not understood the support of the
position.

It's possible that you just can't or won't see it, I don't know. Your
responses to the example have been that it's "irrelevant" or that I'm
"saying nothing", rather than any sort of solid technical rebuttal.
Neither response does anything to support your case in my view.

Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org
.

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