Re: FFT Resolution bandwidth


"dbd" <dbd@xxxxxxxx> wrote in message
news:62fb458f-3fce-41f1-b0f3-19295765c4a0@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


You still suggest no definition of 'resolution' or why it should not
relate to frequency estimation.
In analog swept spectrum analyzers the verb 'to resolve' meant to see
two peaks separated by minimun in the ploted response. The plotted
positions of the peaks were frequency estimates.
There is a related term 'resolution bandwidth' that is an attempt to
characterize 'resolution'.
How this fits in a digital world has been discussed here before:
http://groups.google.com/group/comp.dsp/browse_thread/thread/6ae040dd9657edcd/06006678e3be9d10?hl=en&lnk=gst&q=resolution+bandwidth#06006678e3be9d10
Dale B. Dalrymple
http://dbdimages.com

Dale,

Yes, you're right, I didn't. That's because I believe the common definition
in DSP is first: the frequency sample interval. And second, if the time
function is zero-padded then the frequency sample interval *as if* the zero
padding wasn't there. The second version is often overlooked or not
understood.

I think it's that simple.

Adding issues of frequency estimation just makes it more convolved.

I'm not sure I agree completely with: "meant to see two peaks separated by a
minimum in the ploted response." This implies that resolved peaks have to
be separated by fully 2x the resolution so a full minimum may be seen.
After all, a full minimum requires using up one of the frequency samples.

Actually I think that any minimum is adequate between peaks for an analog
system to see a doublet. There are 3 conditions with two adjacent
sinusoids: 1) where the output shows a single peak that's relatively broad;
2) where the output shows two peaks with an amount of intervening minimum;
3) where the output shows two peaks with a minimum somewhere around the
noise floor.

The question of resolution is different than the question of frequency
estimation. The sampling theorem says that you can perfectly reconstruct a
sampled time signal if .....(as you well know).

In theory this means reconstructing with a sinc. As long as the summation
of sincs passes through the sample values exactly then one can say that the
samples are resolved (in time and amplitude). So, the same must apply if
one samples in frequency. As long as the summation of reconstructing sincs
(or Dirichlets) passes through the sample values exactly then one can say
that the samples are resolved. If the samples happen to occur because of a
doublet then the doublet is "resolved" in that context - but it may not be
detectable as a doublet unless the separation is greater than the
resolution. And, as above, it will definitely be detected if the doublet
spacing is 2x the resolution. There are a few cases at the "edges" of this.

Just so I'm not totally off base, I Googled on "resolution" and found these
definitions:

"The act or process of separating or reducing something into its constituent
parts"
(Sampling does that - you could say that it's separating the signal into its
constituent sincs).

"The fineness of detail that can be distinguished in an image, as on a vido
display terminal"
(The transition from one value to another unambiguously does that - which is
roughly half the distance between two peaks separated by a zero).

In this context there's a factor of 2 difference between resolving two
values and separating two peaks.

Fred


.

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