Re: minimum cycles for fft, and limits of filling short sample out with zeros

glen herrmannsfeldt <gah@xxxxxxxxxxxxxxxx> writes:

> Stan Pawlukiewicz wrote:
>
> (snip)
>
>> The problem with this, is that in quantum mechanics the location of
>> the particle is governed by a probability density which is related
>> to the observation. In signal processing, the transform broadens
>> but the location of peak governs the frequency measurement. The
>> frequency content of a truncated signal increases but its not a
>> probability density, i.e. the true frequency isn't a random
>> parameter governed by chance, like a quantum particle.
>
> Say I have a signal that in frequency space has a broad peak,
> maybe not so peak shaped at all. What do I call the frequency
> of that signal? Somewhere within the peak, but where?

That is an excellent question, Glen. At least it sure got me thinking.

I think I see the "time-frequency" uncertainty thing now and
understand at least partially why it doesn't necessarily apply to some
types of frequency measurements.

This may seem really obvious, but if you look at the frequency domain
from the purely mathematical POV, you can only have a *perfect*
sinusoid if that sinusoid extends infinitely in time. If you have only
an "observation" of that sinusoid, which can be modeled by multiplying
the infinite sinusoid by a rectangular window with a width
corresponding to the observation period, then that infinite-thin peak
in the frequency domain spreads out. Why? Cause you don't know what
that sinusoid is "doing" outside the window.

Now, those facts are perfectly correct WHEN ANALYZING A SIGNAL OF INFINITE
EXTENT. However, we are free to perform an *instantaneous* frequency
measurement by, e.g., measuring the change in phase of the analytic signal
from one sample to the next and dividing by the sample interval. Thus even
thought that "sine" wave may be doing some really funky things outside the
observation interval, we can still measure the INSTANTANEOUS frequency and
that frequency can be measured PERFECT ACCURATELY if there is NO NOISE.

Now if there IS noise, you can't measure even the INSTANTANEOUS frequency
accurately and must do some averaging or whatever to get the accuracy up,
so introducing noise requires that we measure the signal longer to get an
accurate estimate.

Does that make sense?
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