DFT point = decimated filter output?

Hi,

I am struggling with some FFT interpretation.

To me it looks that each sample of an FFT can be considered as the output of 
a process consiting of downconversion, fitlering ,decimating.
e.g .the DC bin output is calculated by:
- downconverting the signal by 0 (hence doing nothing)
- applying a very simple filter that has N taps equal to 1 (for an FFT of 
size N).
- decimating the filtered signal by N (hence aliasing some content in the 
due to far from perfect filter)

Is this an OK way of thinking? This makes things like leakage, windowing 
somewhat easier to grab for me.

I am struggling a bit with the fact that after decimation we only keep 1 
sample, so we can hardly speak of decimating a signal to another "signal". 
Or can this be happily neglected because the DFT assumes the original signal 
has a period of N samples, hence after decimating we would actually still 
have a series of identical sample points?

Thanks,

NL

On Apr 9, 3:47 pm, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> I am struggling with some FFT interpretation.
>
> To me it looks that each sample of an FFT can be considered as the output of
> a process consiting of downconversion, fitlering ,decimating.
> e.g .the DC bin output is calculated by:
> - downconverting the signal by 0 (hence doing nothing)
> - applying a very simple filter that has N taps equal to 1 (for an FFT of
> size N).
> - decimating the filtered signal by N (hence aliasing some content in the
> due to far from perfect filter)
>
> Is this an OK way of thinking? This makes things like leakage, windowing
> somewhat easier to grab for me.
>
> I am struggling a bit with the fact that after decimation we only keep 1
> sample, so we can hardly speak of decimating a signal to another "signal".
> Or can this be happily neglected because the DFT assumes the original signal
> has a period of N samples, hence after decimating we would actually still
> have a series of identical sample points?

Everything you said is completely correct (I think!).

There is indeed no problem with keeping just one sample; that is what
happens if you decimate a length-N signal by N.  But yes, in view of
the implied periodicity of the DFT, then really what we have is an
infinite sequence of identically-valued sample points, just as you
said.


--
Oli

On Apr 9, 9:47 am, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> Hi,
>
> I am struggling with some FFT interpretation.
>
> To me it looks that each sample of an FFT can be considered as the output of
> a process consiting of downconversion, fitlering ,decimating.
> e.g .the DC bin output is calculated by:
> - downconverting the signal by 0 (hence doing nothing)
> - applying a very simple filter that has N taps equal to 1 (for an FFT of
> size N).
> - decimating the filtered signal by N (hence aliasing some content in the
> due to far from perfect filter)
>
> Is this an OK way of thinking? This makes things like leakage, windowing
> somewhat easier to grab for me.
>

The best way to look at the DFT is to consider it as a projection.
It's right there in the formula:

  X_k = \sum_{n=0}^{N-1} x[n] \exp(-j 2\pi n k / \Omega).

Write it out as inner product X_k = < x[n], f_k[n] >.  Now,
what is the correct f_k[n] ?

Remember that the DFT is for periodic, discrete-time signals,
so taking a finite segment of a signal and taking the DFT
of that segment can be a two-edged sword!

Julius

On Apr 9, 6:47 am, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> I am struggling with some FFT interpretation.
>
> To me it looks that each sample of an FFT can be considered as the output of
> a process consiting of downconversion, fitlering ,decimating.
> e.g .the DC bin output is calculated by:
> - downconverting the signal by 0 (hence doing nothing)
> - applying a very simple filter that has N taps equal to 1 (for an FFT of
> size N).
> - decimating the filtered signal by N (hence aliasing some content in the
> due to far from perfect filter)
>
> Is this an OK way of thinking? This makes things like leakage, windowing
> somewhat easier to grab for me.
>
> I am struggling a bit with the fact that after decimation we only keep 1
> sample, so we can hardly speak of decimating a signal to another "signal".

Each DFT result sample not only has a value, but an
associated bin number as well as a transform length.
So the information content of the "signal" is not
only the complex sample value, but knowledge about
the down-conversion frequency and the filter width
implied by the bin number and transform length, and
perhaps even phase information from the absolute
location of the transform window with respect to
some external reference time or point, etc.

Also note that the number of sample points needed to
reconstruct certain types of signals or functions is
not related to the max frequency content, but to the
max bandwidth of the signal.  Given a narrower and
narrower max bandwidth, it seems reasonable for the
number of samples required to represent the signal
to decrease, eventually to just one, the output of
one complex down-conversion followed by an averaging
low pass filter, per each "bin" of a DFT.


IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M

On 9 Apr, 16:47, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> Hi,
>
> I am struggling with some FFT interpretation.
>
> To me it looks that each sample of an FFT can be considered as the output of
> a process consiting of downconversion, fitlering ,decimating.
> e.g .the DC bin output is calculated by:
> - downconverting the signal by 0 (hence doing nothing)
> - applying a very simple filter that has N taps equal to 1 (for an FFT of
> size N).
> - decimating the filtered signal by N (hence aliasing some content in the
> due to far from perfect filter)
>
> Is this an OK way of thinking?

I don't know. If you thinks it helps, then stick with it for now.
The danger with elaborate "physical" analogies is that at some
point they slow you down and obfuscate more than they enlighten.

As long as you are aware of this, and is both willing and capable
of letting the analogy go when the time comes, then your approach
is OK.

> This makes things like leakage, windowing
> somewhat easier to grab for me.
>
> I am struggling a bit with the fact that after decimation we only keep 1
> sample, so we can hardly speak of decimating a signal to another "signal".

I don't know if this is correct or not, but if you find yourself
struggling over many more questions like this, you should consider
very carefully whether your approach to understanding the FFT
causes more trouble than it helps.

Rune

On Apr 9, 7:47 am, "NewLine" <umts_remove_this_and_t...@skynet.be>
wrote:
> Hi,
>
> I am struggling with some FFT interpretation.
>
> To me it looks that each sample of an FFT can be considered as the output of
> a process consiting of downconversion, fitlering ,decimating.
> e.g .the DC bin output is calculated by:
> - downconverting the signal by 0 (hence doing nothing)
> - applying a very simple filter that has N taps equal to 1 (for an FFT of
> size N).
> - decimating the filtered signal by N (hence aliasing some content in the
> due to far from perfect filter)
>
> Is this an OK way of thinking? This makes things like leakage, windowing
> somewhat easier to grab for me.
>
> I am struggling a bit with the fact that after decimation we only keep 1
> sample, so we can hardly speak of decimating a signal to another "signal".
> Or can this be happily neglected because the DFT assumes the original signal
> has a period of N samples, hence after decimating we would actually still
> have a series of identical sample points?
>
> Thanks,
>
> NL


There are many alternate interpretations of the DFT that correspond to
different applications of the DFT calculation. Discussion of the DFT
is often confused when different speakers have different
interpretations. Some speakers don't even acknowledge the existance of
other interpretations. So, you have to listen carefully.

Some people interpret the DFT output as samples of the Fourier
transform of the input. That is valid for certain signal types. Such
people have said in this thread: "the DFT is for periodic, discrete-
time signals", which is true in this interpretation.

Another interpretation of the DFT output as a set of filters as you
have suggested. This interpretation is valid as well, given its own
assumptions. The best discussion I know of on this interpretation is
in "Handbook of Digital Signal Processing: Engineering Applications"
edited by Douglas Elliot. Chapter 8 by fred harris discusses this and
some other time domain signal processing interpretations of the DFT
algorithm.

Some people only 'get' one interpretation and can find discussion of
other interpretations confusing. Some people practice more than one
interpretation which can be confusing if you don't catch the changes
from one to another in their discussion.

Be sure to 'listen' carefully and evaluate the interpretation that
underlays the input you are getting from each poster.

Dale B. Dalrymple
http://dbdimages.com