Decimation vs. Collapsing and their spectral effects

Hello all,

I am struggling with what happens to the time axis in a sampled
sequence when it is collapsed by adding every two samples together,
with or wihout averaging.
I am working in the spatial domain but believe the principles should
be the same as with time varying signals.

I understand decimation and how it effectively resamples at a lower
sampling frequency and has the potential for inducing aliasing if the
sequence is not first band limited. I understand this because it is
obvious that in decimation you are just using a different sampling
period and resampling. But I am not sure what actually takes place in
collapsing.

What is tripping me up is that it seems like when adding two samples
together the sample is not instantaneous but rather the integral over
the duration of the two samples used. I have been working with matlab
to demonstrate to myself what happens to a sampled sequence with
repeated interpolations and collapsings. When I collapse a sequence
say of N=64 to N=32 I have combined each pair of the original samples
and thus must have doubled the time/space of each sample. The collapse
acts like a lowpass filter and attenuates the high frequencies but it
is not directly analagous I don't think because the bin width has been
doubled. With a FIR filter the bin width doesn't change.

It is difficult for me to frame the actual question and I apologize
for that. I want to be able to understand the theoretical
underpinnings of just what happens during a collapse. We do this all
the time in medical imaging and I want to be able to explain it with
DSP.

Any insights greatly appreciated.

-Martin

Martin J. Stumpf wrote:

>Hello all,
>
>I am struggling with what happens to the time axis in a sampled
>sequence when it is collapsed by adding every two samples together,
>with or wihout averaging.
>I am working in the spatial domain but believe the principles should
>be the same as with time varying signals.
>
>I understand decimation and how it effectively resamples at a lower
>sampling frequency and has the potential for inducing aliasing if the
>sequence is not first band limited. I understand this because it is
>obvious that in decimation you are just using a different sampling
>period and resampling. But I am not sure what actually takes place in
>collapsing.
>
>What is tripping me up is that it seems like when adding two samples
>together the sample is not instantaneous but rather the integral over
>the duration of the two samples used. I have been working with matlab
>to demonstrate to myself what happens to a sampled sequence with
>repeated interpolations and collapsings. When I collapse a sequence
>say of N=64 to N=32 I have combined each pair of the original samples
>and thus must have doubled the time/space of each sample. The collapse
>acts like a lowpass filter and attenuates the high frequencies but it
>is not directly analagous I don't think because the bin width has been
>doubled. With a FIR filter the bin width doesn't change.
>
>It is difficult for me to frame the actual question and I apologize
>for that. I want to be able to understand the theoretical
>underpinnings of just what happens during a collapse. We do this all
>the time in medical imaging and I want to be able to explain it with
>DSP.
>
>Any insights greatly appreciated.
>
>-Martin
>  
>

Hello Martin :  it looks as though you are redefining the spatial sampling
interval and doing some sort of averaging.   If the original space between
your samples was dphi your new samples are spaced 2*dphi and offset from the
original samples by dphi/2  (either +/- depending on which edge you start to
collapse from).  By adding them together it's just like taking the
averaqge*2.   You must have already had these thoughts though so what are
you really worried about?

Mike

"christophe grimault" <christophe.grimault@novagrid.com> wrote in message
news:4150497D.5040402@novagrid.com...
>
>
> Martin J. Stumpf wrote:
>
> >Hello all,
> >
> >I am struggling with what happens to the time axis in a sampled
> >sequence when it is collapsed by adding every two samples together,
> >with or wihout averaging.
> >I am working in the spatial domain but believe the principles should
> >be the same as with time varying signals.
> >
> >I understand decimation and how it effectively resamples at a lower
> >sampling frequency and has the potential for inducing aliasing if the
> >sequence is not first band limited. I understand this because it is
> >obvious that in decimation you are just using a different sampling
> >period and resampling. But I am not sure what actually takes place in
> >collapsing.
> >
> >What is tripping me up is that it seems like when adding two samples
> >together the sample is not instantaneous but rather the integral over
> >the duration of the two samples used. I have been working with matlab
> >to demonstrate to myself what happens to a sampled sequence with
> >repeated interpolations and collapsings. When I collapse a sequence
> >say of N=64 to N=32 I have combined each pair of the original samples
> >and thus must have doubled the time/space of each sample. The collapse
> >acts like a lowpass filter and attenuates the high frequencies but it
> >is not directly analagous I don't think because the bin width has been
> >doubled. With a FIR filter the bin width doesn't change.
> >
> >It is difficult for me to frame the actual question and I apologize
> >for that. I want to be able to understand the theoretical
> >underpinnings of just what happens during a collapse. We do this all
> >the time in medical imaging and I want to be able to explain it with
> >DSP.
> >
> >Any insights greatly appreciated.
> >
> >-Martin
> >
> >
>

Another way of looking at it is as a 1/2 sample shift due to using a
trapezoidal interpolator then decimation by two.  Why don't you just throw
every other sample away instead?

Best of Luck - Mike

"Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message
news:cipll6$ru2$1@titan.btinternet.com...
> Hello Martin :  it looks as though you are redefining the spatial sampling
> interval and doing some sort of averaging.   If the original space between
> your samples was dphi your new samples are spaced 2*dphi and offset from
the
> original samples by dphi/2  (either +/- depending on which edge you start
to
> collapse from).  By adding them together it's just like taking the
> averaqge*2.   You must have already had these thoughts though so what are
> you really worried about?
>
> Mike
>
> "christophe grimault" <christophe.grimault@novagrid.com> wrote in message
> news:4150497D.5040402@novagrid.com...
> >
> >
> > Martin J. Stumpf wrote:
> >
> > >Hello all,
> > >
> > >I am struggling with what happens to the time axis in a sampled
> > >sequence when it is collapsed by adding every two samples together,
> > >with or wihout averaging.
> > >I am working in the spatial domain but believe the principles should
> > >be the same as with time varying signals.
> > >
> > >I understand decimation and how it effectively resamples at a lower
> > >sampling frequency and has the potential for inducing aliasing if the
> > >sequence is not first band limited. I understand this because it is
> > >obvious that in decimation you are just using a different sampling
> > >period and resampling. But I am not sure what actually takes place in
> > >collapsing.
> > >
> > >What is tripping me up is that it seems like when adding two samples
> > >together the sample is not instantaneous but rather the integral over
> > >the duration of the two samples used. I have been working with matlab
> > >to demonstrate to myself what happens to a sampled sequence with
> > >repeated interpolations and collapsings. When I collapse a sequence
> > >say of N=64 to N=32 I have combined each pair of the original samples
> > >and thus must have doubled the time/space of each sample. The collapse
> > >acts like a lowpass filter and attenuates the high frequencies but it
> > >is not directly analagous I don't think because the bin width has been
> > >doubled. With a FIR filter the bin width doesn't change.
> > >
> > >It is difficult for me to frame the actual question and I apologize
> > >for that. I want to be able to understand the theoretical
> > >underpinnings of just what happens during a collapse. We do this all
> > >the time in medical imaging and I want to be able to explain it with
> > >DSP.
> > >
> > >Any insights greatly appreciated.
> > >
> > >-Martin
> > >
> > >
> >
>
>

Thanks for your response Mike,

I am collapseing the samples on purpose to double the pixel size of
the image.  This is necessary because I am looking at the effects of
different projection bin sizes on iterative image reconstruction from
projections. I know what you are saying in your first response and
appreciate you saying "What are you worried about?". I am worried that
when we collapse an image by two say to change 1mm pixels into 2mm
pixels that we are also changing the frequencies in that image.
ie. The noise will be attenuated just by virtue of the collapsing. My
goal is to be able to describe or predict the outcome of changing the
pixel sizes using DSP theory. Another more direct way of asking the
question is this: Is collapsing in this manner considered resampling?
And if it is can I predict the effect on the resulting power spectrum?
I guess what I'm 'worried about' is whether or not I am changing the
spatial resolution in the image by collapsing. I guess in the time
domain the analogy would be that by adding together every two samples
one would be doubling the time per sample and therefore affecting the
time resolution.

Thank you very much for your time

-Martin

In article <cipmaf$c8c$1@hercules.btinternet.com>, Mike Yarwood wrote:
> Another way of looking at it is as a 1/2 sample shift due to using a
> trapezoidal interpolator then decimation by two.  Why don't you just throw
> every other sample away instead?
> 
> Best of Luck - Mike
> 
> "Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message
> news:cipll6$ru2$1@titan.btinternet.com...
>> Hello Martin :  it looks as though you are redefining the spatial sampling
>> interval and doing some sort of averaging.   If the original space between
>> your samples was dphi your new samples are spaced 2*dphi and offset from
> the
>> original samples by dphi/2  (either +/- depending on which edge you start
> to
>> collapse from).  By adding them together it's just like taking the
>> averaqge*2.   You must have already had these thoughts though so what are
>> you really worried about?
>>
>> Mike
>>
>> "christophe grimault" <christophe.grimault@novagrid.com> wrote in message
>> news:4150497D.5040402@novagrid.com...
>> >
>> >
>> > Martin J. Stumpf wrote:
>> >
>> > >Hello all,
>> > >
>> > >I am struggling with what happens to the time axis in a sampled
>> > >sequence when it is collapsed by adding every two samples together,
>> > >with or wihout averaging.
>> > >I am working in the spatial domain but believe the principles should
>> > >be the same as with time varying signals.
>> > >
>> > >I understand decimation and how it effectively resamples at a lower
>> > >sampling frequency and has the potential for inducing aliasing if the
>> > >sequence is not first band limited. I understand this because it is
>> > >obvious that in decimation you are just using a different sampling
>> > >period and resampling. But I am not sure what actually takes place in
>> > >collapsing.
>> > >
>> > >What is tripping me up is that it seems like when adding two samples
>> > >together the sample is not instantaneous but rather the integral over
>> > >the duration of the two samples used. I have been working with matlab
>> > >to demonstrate to myself what happens to a sampled sequence with
>> > >repeated interpolations and collapsings. When I collapse a sequence
>> > >say of N=64 to N=32 I have combined each pair of the original samples
>> > >and thus must have doubled the time/space of each sample. The collapse
>> > >acts like a lowpass filter and attenuates the high frequencies but it
>> > >is not directly analagous I don't think because the bin width has been
>> > >doubled. With a FIR filter the bin width doesn't change.
>> > >
>> > >It is difficult for me to frame the actual question and I apologize
>> > >for that. I want to be able to understand the theoretical
>> > >underpinnings of just what happens during a collapse. We do this all
>> > >the time in medical imaging and I want to be able to explain it with
>> > >DSP.
>> > >
>> > >Any insights greatly appreciated.
>> > >
>> > >-Martin
>> > >
>> > >
>> >
>>
>>
> 
> 

This is a pretty easy case to analyze from a DSP perspective.   You're
applying an averaging filter that is essentially a rectangular FIR
with an impulse response length of two samples, in other words, a FIR
with coefficients h = 1,1.   In addition to applying this small FIR,
you're also decimating 2:1.

The FIR filter you've described doesn't have a very good frequency
response, but it is a low pass filter of sorts, with a sinx/x response
with the first null at fs/2.  This means that their will be a
non-negligible (depending on your application, of course) amount of
aliasing due to the sidelobes, especially the first sidelobes since
they're only -13dB peak from the passband peak.

So it's a very simple, more-or-less brute force 2:1 decimating filter.
It lacks some elegance and isn't very high performance, but it's
awfully simple to compute.

I'd opine that this isn't a very good thing to do from a DSP
perspective, but since what you're really doing is image processing it
might be suitable for whatever it is that you're trying to do (which I
don't completely understand).

Hope that helps a little, anyway.

On Tue, 21 Sep 2004 16:58:04 +0000 (UTC), "Martin J. Stumpf"
<mjs@neuron.rad.jhmi.edu> wrote:

>Thanks for your response Mike,
>
>I am collapseing the samples on purpose to double the pixel size of
>the image.  This is necessary because I am looking at the effects of
>different projection bin sizes on iterative image reconstruction from
>projections. I know what you are saying in your first response and
>appreciate you saying "What are you worried about?". I am worried that
>when we collapse an image by two say to change 1mm pixels into 2mm
>pixels that we are also changing the frequencies in that image.
>ie. The noise will be attenuated just by virtue of the collapsing. My
>goal is to be able to describe or predict the outcome of changing the
>pixel sizes using DSP theory. Another more direct way of asking the
>question is this: Is collapsing in this manner considered resampling?
>And if it is can I predict the effect on the resulting power spectrum?
>I guess what I'm 'worried about' is whether or not I am changing the
>spatial resolution in the image by collapsing. I guess in the time
>domain the analogy would be that by adding together every two samples
>one would be doubling the time per sample and therefore affecting the
>time resolution.
>
>Thank you very much for your time
>
>-Martin
>
>In article <cipmaf$c8c$1@hercules.btinternet.com>, Mike Yarwood wrote:
>> Another way of looking at it is as a 1/2 sample shift due to using a
>> trapezoidal interpolator then decimation by two.  Why don't you just throw
>> every other sample away instead?
>> 
>> Best of Luck - Mike
>> 
>> "Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message
>> news:cipll6$ru2$1@titan.btinternet.com...
>>> Hello Martin :  it looks as though you are redefining the spatial sampling
>>> interval and doing some sort of averaging.   If the original space between
>>> your samples was dphi your new samples are spaced 2*dphi and offset from
>> the
>>> original samples by dphi/2  (either +/- depending on which edge you start
>> to
>>> collapse from).  By adding them together it's just like taking the
>>> averaqge*2.   You must have already had these thoughts though so what are
>>> you really worried about?
>>>
>>> Mike
>>>
>>> "christophe grimault" <christophe.grimault@novagrid.com> wrote in message
>>> news:4150497D.5040402@novagrid.com...
>>> >
>>> >
>>> > Martin J. Stumpf wrote:
>>> >
>>> > >Hello all,
>>> > >
>>> > >I am struggling with what happens to the time axis in a sampled
>>> > >sequence when it is collapsed by adding every two samples together,
>>> > >with or wihout averaging.
>>> > >I am working in the spatial domain but believe the principles should
>>> > >be the same as with time varying signals.
>>> > >
>>> > >I understand decimation and how it effectively resamples at a lower
>>> > >sampling frequency and has the potential for inducing aliasing if the
>>> > >sequence is not first band limited. I understand this because it is
>>> > >obvious that in decimation you are just using a different sampling
>>> > >period and resampling. But I am not sure what actually takes place in
>>> > >collapsing.
>>> > >
>>> > >What is tripping me up is that it seems like when adding two samples
>>> > >together the sample is not instantaneous but rather the integral over
>>> > >the duration of the two samples used. I have been working with matlab
>>> > >to demonstrate to myself what happens to a sampled sequence with
>>> > >repeated interpolations and collapsings. When I collapse a sequence
>>> > >say of N=64 to N=32 I have combined each pair of the original samples
>>> > >and thus must have doubled the time/space of each sample. The collapse
>>> > >acts like a lowpass filter and attenuates the high frequencies but it
>>> > >is not directly analagous I don't think because the bin width has been
>>> > >doubled. With a FIR filter the bin width doesn't change.
>>> > >
>>> > >It is difficult for me to frame the actual question and I apologize
>>> > >for that. I want to be able to understand the theoretical
>>> > >underpinnings of just what happens during a collapse. We do this all
>>> > >the time in medical imaging and I want to be able to explain it with
>>> > >DSP.
>>> > >
>>> > >Any insights greatly appreciated.
>>> > >
>>> > >-Martin
>>> > >
>>> > >
>>> >
>>>
>>>
>> 
>> 

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

"Martin J. Stumpf" wrote:
> 
> Hello all,
> 
> I am struggling with what happens to the time axis in a sampled
> sequence when it is collapsed by adding every two samples together,
> with or wihout averaging.
> I am working in the spatial domain but believe the principles should
> be the same as with time varying signals.
> 
> I understand decimation and how it effectively resamples at a lower
> sampling frequency and has the potential for inducing aliasing if the
> sequence is not first band limited. I understand this because it is
> obvious that in decimation you are just using a different sampling
> period and resampling. But I am not sure what actually takes place in
> collapsing.
> 
> What is tripping me up is that it seems like when adding two samples
> together the sample is not instantaneous but rather the integral over
> the duration of the two samples used. I have been working with matlab
> to demonstrate to myself what happens to a sampled sequence with
> repeated interpolations and collapsings. When I collapse a sequence
> say of N=64 to N=32 I have combined each pair of the original samples
> and thus must have doubled the time/space of each sample. The collapse
> acts like a lowpass filter and attenuates the high frequencies but it
> is not directly analagous I don't think because the bin width has been
> doubled. With a FIR filter the bin width doesn't change.
> 
> It is difficult for me to frame the actual question and I apologize
> for that. I want to be able to understand the theoretical
> underpinnings of just what happens during a collapse. We do this all
> the time in medical imaging and I want to be able to explain it with
> DSP.

Hi Martin
	The frequency response of your filter [1,1] is just cos function. To
see how aliasing works with this filter take a piece of paper and plot
cos(x) with x=0 at the left edge of the paper and x=pi/2 on the right
edge. Now while the ink is still wet on the paper fold it in half
exactly in the middle so that the wet ink on the right imprint onto the
left side of the paper. That ink that transfered from the left side of
the plot to the right represents the aliasing that occurs using this
filter. That is to say, when you decimate by 2 the frequencies from the
top half of the spectrum (the right side of your plot) are folded into
the bottom half of the spectrum (the left side).  
	Now if you could create a filter that had a frequency response so that
the plot was one on the left side of your paper and zero on the right
side you would have a "ideal brick wall filter". And then when you
decimate no aliasing would occur. But alas, such a filter is not
possible. It is possible to make a filter that is a lot closer to the
ideal than what you are using. But your filter is better than no filter
since as you can see with the frequency response of your filter a lot of
the upper half of the spectrum is removed and much of the lower half is
preserved.

-jim


-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
http://www.newsfeeds.com - The #1 Newsgroup Service in the World!
-----==  Over 100,000 Newsgroups - 19 Different Servers! =-----

jim wrote:

   ...

> Hi Martin
> 	The frequency response of your filter [1,1] is just cos function. 

   ...

That seems very strange to me; I would have thought it is a sinc. Can 
you explain?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Sorry, if I distract a little from the OPs main issue, 
but it's very close to my problem, therefore I would like to 
challenge you a little more...

Eric Jacobsen wrote:

> This is a pretty easy case to analyze from a DSP perspective.  
> You're applying an averaging filter that is essentially a
> rectangular FIR with an impulse response length of two samples, in
> other words, a FIR
> with coefficients h = 1,1.   In addition to applying this small
> FIR, you're also decimating 2:1.

I do the same, because my decimation rate is variable (and not 
predictable). (See my other post "Decimation/interpolation filters 
issue when sample rate ratio is variable" for further details)
 
My FIR has coefficients h=1,1,...,1  with a variable number of 
elements. So I'm decimating by N(t):1.

> 
> The FIR filter you've described doesn't have a very good frequency
> response, but it is a low pass filter of sorts, with a sinx/x
> response
> with the first null at fs/2.  This means that their will be a
> non-negligible (depending on your application, of course) amount
> of aliasing due to the sidelobes, especially the first sidelobes
> since they're only -13dB peak from the passband peak.
> 
> So it's a very simple, more-or-less brute force 2:1 decimating
> filter. It lacks some elegance and isn't very high performance,
> but it's awfully simple to compute.

Can you imagine a better version of anti-aliasing filter when 
decimation ratio is variable as N(t):1 ?

> 
> I'd opine that this isn't a very good thing to do from a DSP
> perspective, 
So do I, and indeed, I have a very bad feeling, especially,
because I'm doing it in a measurement application where aliasing 
might be a critical effect.

> but since what you're really doing is image
> processing it might be suitable for whatever it is that you're
> trying to do (which I don't completely understand).
> 

> Hope that helps a little, anyway.

Helps me to see, that others deal with similar issues.
Gives hope, that there's a better approach already...

Bernhard

Hi again Martin,
    As Eric Jacobsen said, one collapse is equivalent to a two tap filter
with weights=1 (or a two sample boxcar averager) followed by decimation by
two. This is also the same (within a scaling factor) as trapezoidal
interpolation followed by discarding every other interpolated sample. When
you itterate the collapse you double the length of your boxcar filter and
halve the number of outputs that you keep (so collapsing three times in a
row is like convolving your sample sequence with a sequence of 8 'ones' then
keeping only every eighth o/p from the convolution). It's all very well
talking about how high frequency components that may be present in your scan
will only be imperfectly suppressed by the averageing/summing part of your
collapser and will alias back into the collapsed image but how does this
affect your perception of the image?  Do you have a set of criteria for
determining what is significant and what constitutes a perceptible
impairment in quality? As far as I can tell, regardless of the aliasing,
your collapsing gives good preservation of contrast, maintains the average
'grey scale in an area covered by a few pixels, is pretty simple and
straightforward , could be applied to 2D rectangular rasters of  sample
points without worrying about row/column order etc. etc.  do you have any
more sophisticated criteria in mind?

Best of Luck - Mike


"Eric Jacobsen" <eric.jacobsen@ieee.org> wrote in message
news:4150a405.526659078@news.west.cox.net...
> This is a pretty easy case to analyze from a DSP perspective.   You're
> applying an averaging filter that is essentially a rectangular FIR
> with an impulse response length of two samples, in other words, a FIR
> with coefficients h = 1,1.   In addition to applying this small FIR,
> you're also decimating 2:1.
>
> The FIR filter you've described doesn't have a very good frequency
> response, but it is a low pass filter of sorts, with a sinx/x response
> with the first null at fs/2.  This means that their will be a
> non-negligible (depending on your application, of course) amount of
> aliasing due to the sidelobes, especially the first sidelobes since
> they're only -13dB peak from the passband peak.
>
> So it's a very simple, more-or-less brute force 2:1 decimating filter.
> It lacks some elegance and isn't very high performance, but it's
> awfully simple to compute.
>
> I'd opine that this isn't a very good thing to do from a DSP
> perspective, but since what you're really doing is image processing it
> might be suitable for whatever it is that you're trying to do (which I
> don't completely understand).
>
> Hope that helps a little, anyway.
>
> On Tue, 21 Sep 2004 16:58:04 +0000 (UTC), "Martin J. Stumpf"
> <mjs@neuron.rad.jhmi.edu> wrote:
>
> >Thanks for your response Mike,
> >
> >I am collapseing the samples on purpose to double the pixel size of
> >the image.  This is necessary because I am looking at the effects of
> >different projection bin sizes on iterative image reconstruction from
> >projections. I know what you are saying in your first response and
> >appreciate you saying "What are you worried about?". I am worried that
> >when we collapse an image by two say to change 1mm pixels into 2mm
> >pixels that we are also changing the frequencies in that image.
> >ie. The noise will be attenuated just by virtue of the collapsing. My
> >goal is to be able to describe or predict the outcome of changing the
> >pixel sizes using DSP theory. Another more direct way of asking the
> >question is this: Is collapsing in this manner considered resampling?
> >And if it is can I predict the effect on the resulting power spectrum?
> >I guess what I'm 'worried about' is whether or not I am changing the
> >spatial resolution in the image by collapsing. I guess in the time
> >domain the analogy would be that by adding together every two samples
> >one would be doubling the time per sample and therefore affecting the
> >time resolution.
> >
> >Thank you very much for your time
> >
> >-Martin
> >
> >In article <cipmaf$c8c$1@hercules.btinternet.com>, Mike Yarwood wrote:
> >> Another way of looking at it is as a 1/2 sample shift due to using a
> >> trapezoidal interpolator then decimation by two.  Why don't you just
throw
> >> every other sample away instead?
> >>
> >> Best of Luck - Mike
> >>
> >> "Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message
> >> news:cipll6$ru2$1@titan.btinternet.com...
> >>> Hello Martin :  it looks as though you are redefining the spatial
sampling
> >>> interval and doing some sort of averaging.   If the original space
between
> >>> your samples was dphi your new samples are spaced 2*dphi and offset
from
> >> the
> >>> original samples by dphi/2  (either +/- depending on which edge you
start
> >> to
> >>> collapse from).  By adding them together it's just like taking the
> >>> averaqge*2.   You must have already had these thoughts though so what
are
> >>> you really worried about?
> >>>
> >>> Mike
> >>>
> >>> "christophe grimault" <christophe.grimault@novagrid.com> wrote in
message
> >>> news:4150497D.5040402@novagrid.com...
> >>> >
> >>> >
> >>> > Martin J. Stumpf wrote:
> >>> >
> >>> > >Hello all,
> >>> > >
> >>> > >I am struggling with what happens to the time axis in a sampled
> >>> > >sequence when it is collapsed by adding every two samples together,
> >>> > >with or wihout averaging.
> >>> > >I am working in the spatial domain but believe the principles
should
> >>> > >be the same as with time varying signals.
> >>> > >
> >>> > >I understand decimation and how it effectively resamples at a lower
> >>> > >sampling frequency and has the potential for inducing aliasing if
the
> >>> > >sequence is not first band limited. I understand this because it is
> >>> > >obvious that in decimation you are just using a different sampling
> >>> > >period and resampling. But I am not sure what actually takes place
in
> >>> > >collapsing.
> >>> > >
> >>> > >What is tripping me up is that it seems like when adding two
samples
> >>> > >together the sample is not instantaneous but rather the integral
over
> >>> > >the duration of the two samples used. I have been working with
matlab
> >>> > >to demonstrate to myself what happens to a sampled sequence with
> >>> > >repeated interpolations and collapsings. When I collapse a sequence
> >>> > >say of N=64 to N=32 I have combined each pair of the original
samples
> >>> > >and thus must have doubled the time/space of each sample. The
collapse
> >>> > >acts like a lowpass filter and attenuates the high frequencies but
it
> >>> > >is not directly analagous I don't think because the bin width has
been
> >>> > >doubled. With a FIR filter the bin width doesn't change.
> >>> > >
> >>> > >It is difficult for me to frame the actual question and I apologize
> >>> > >for that. I want to be able to understand the theoretical
> >>> > >underpinnings of just what happens during a collapse. We do this
all
> >>> > >the time in medical imaging and I want to be able to explain it
with
> >>> > >DSP.
> >>> > >
> >>> > >Any insights greatly appreciated.
> >>> > >
> >>> > >-Martin
> >>> > >
> >>> > >
> >>> >
> >>>
> >>>
> >>
> >>
>
> Eric Jacobsen
> Minister of Algorithms, Intel Corp.
> My opinions may not be Intel's opinions.
> http://www.ericjacobsen.org