Re: frequency resolution issue

On 26 Jul, 20:12, "Fred Marshall" <fmarshallx@xxxxxxxxxxxxxxxxxxxx>
wrote:
"Rune Allnor" <all...@xxxxxxxxxxxx> wrote in message

news:dcbb5ea6-bc6a-48cb-bf73-62f591b4ed6e@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx





On 26 Jul, 16:57, "biantai" <wrxd...@xxxxxxxxx> wrote:
These are good data. Whatever you might think of them, one seldom
sees data this clear and with so little noise.

As for the frequency estimates, I tried a number of methods:

1) FFT of full data set zero-padded to 12000 samples
2) FFT of samples 150:200 zero-padded to 12000 samples
3) Frequency estimator applied to full data set
4) Frequency estimator applied to samples 150:200.

The corresponding estimates for the frequency are:

1) 0.5025 Hz
2) 0.5052 Hz
3) 0.5080 Hz
4) 0.5039 Hz

All of these are well within the target accuracy of 0.01 Hz.

Rune

====
Hi Rune, I also got the frequency is 0.5078Hz. But How can we know the
resolution is in 0.01Hz?

Because the results vary by less than 0.01 Hz.

I agree with Rune but I wonder how that position is justified more formally?

The *quantitaive* justification is, as you point out, a matter of
testing the various methods and see how well they agree. Of course,
different method yeld different results either because they apply
different algorithms to the same data or because they use different
data selections as input to one algorithms. So all sorts of issues
regarding numerical stability or senistivity to noise become
important.

The difficult part is to defend the claim from a *qualitative*
point of view. All the methods mentioned above are based on the
temporal sinusoidal model

x(t) = A*exp(jwt)

where A and w are (possibly complex-valued) constants. Are these
assumptions met in the experiment?

Maybe they are and maybe they are not.

What the amplituda A is concerned, we have at least three domains
in the data frame:

1) Before the pulse reaches the sensor: A ~ 0
2) After the pulse reached the sensor but before
the reflection comes: A ~ a
3) After the reflection has arrived: A ~ (1+R)*a

where a is the excitation amplitude and R is the reflection
coefficient, both possibly complex-valued.

The frequency w may or may not be a constant. The waves
propagate very slowly in the horizontal direction, ~ 1-2 m/s,
and so the vertical particle displacement velocity vv might
become comparable to the wave velocity vh meaning that
the linear solution to the wave equation, which is based
on

vh >> vv

might not be valid. In addition, the frequency of the wave
might depend on physical parameters like fluid viscoity,
thickness of the ice sheet, water depth, all ow which
may or may not change along the propagated distance.

So the estimated accuracy is valid inasmuch as the
simple sinusoidal model with constant parameters is
valid for the propagating wave.

Rune
.

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