Re: frequency resolution issue


"Rune Allnor" <allnor@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?
Why can we say that?

Others have already pointed out that the question here is *not* one of
resolution. Here's one of many possible quotes:

1. Accuracy is about conformity of a measured or calculated quantity to its
actual or true value.

2. Precision is about the variation in the measured or calculated quantities
AKA repeatability. Also used in discussion of word length in computers.

3. Resolution is the smallest change or increment in the measured quantity
that the measuring instrument can detect with certainty.

One might well ask what is the difference between precision and resolution?
It seems to me that they are very closely related but apply according to
context:

Resolution is an absolute measure of a process I should think. At least it
seems that's how we use it. In DSP we talk about resolution most often as
the sample interval in either time or frequency. This is usually a highly
precise measure unto itself.

Precision is more about outcomes given that there is underlying variation in
measurement of the same thing from measurement to measurement. So, a
computer with 8-bit word length or 8-bit resolution would yield precision
for each sample no better than 1 bit's worth of value in general. In the
worst case for low SNR there will be 1 bit of random variation (if the
measured value is very close to the bit threshold and there is at least some
noise - as there always will be). In the best case there will be no
variation and the resolution and precision will approach being equal. Then
there's the underlying thing being measured. What if its value is exactly
in the center of the LSB range? Then the measurements are very precise -
beyond the ability of the resolution. The problem is that one doesn't know
this and the terms blur. Not knowing, we have to assume that the value
measured may actually be +/- one LSB because there is at least some noise
and we don't know if the actual value is near the higher LSB threshold or is
near the lower LSB threshold.

If we see only 1 LSB variation in successive measurements of the same
quantity then we are probably justified saying that the likely value is
between bit thresholds and we might place the estimate according to the
number of 1's and 0's in the LSB.

Sampling in amplitude and sampling in time or sampling in frequency are all
examples of such quantization - thus discussion of resolution and precision.

In this case we are interested in making a precise measurement it appears
and may be somewhat hampered by the related resolutions (plural because each
method may have a different resolution). So, the question appears to be one
of precision.

The first thing to say I guess is that each outcome is an *estimate*. Do
estimates have an estimate independent of the signal? I don't think so.

For example, one might say that the Discrete Fourier Transform has a
resolution of 1/T where T is the temporal length of the sample. But, what
if a signal causes high, adjacent frequency samples in a "clump". Then,
what is the "precision" of the "estimate" if one insists on a single
frequency measure?

Then, what if one computes the average distance between zero crossings and
multiplies by 2 to get the "apparent period" = 1/"the apparent frequency"?
It's a different method and there's an SNR component in the definition of
"precision" here (not that there isn't in other methods of course). And,
there is still the issue of the signal character as with the FT.

And, so forth, going through all the methods one might use.....

I suppose one might say that all of the results were plotted and the
outcomes had a standard deviation or, in this case, an absolute range. Here
the absolute range is 0.055. Does that imply that the "precision" of: [the
signal and the methods combined] and other methods not used here have a
uniform distribution of outcomes that are distributed over a frequency range
of 0.055? No, you can't say that.

You might be a practical engineer and guess that there's a uniform
distribution of outcomes that's 2X this amount (just to pick a "safety
factor") or 0.11 but that would be a WAG for purposes of settling an
argument or for gross estimating purposes. But, an approach like this might
have real practical value.

Then there's the question of accuracy. It's fine to have good precision
(all the estimates seem to agree) but still have poor accuracy - as in the
case of a bias in the data.

I think we mean through observation that the standard deviation in the
outcomes is low. You have to decide how many standard deviations away from
the mean or the median define the precision of the measurement or aggregated
results of numerous measurement methods.
The standard deviation here is 0.00234 Hz.
The mean is 0.5049
The median is 0.50455
The distance from the median to the extreme measures is 0.00345.

So, it appears that the precision is pretty good.
As others have pointed out, calling this "resolution" is more about being
able to separate out one frequency from another.
If you applied the same 4 measures to another data set and the resulting
mean was 0.5040 with nearly the same standard deviation then is the new
measure of a "different" frequency? If you can't say then that's a
statement about resolution. If you believe it's justified to say, "yes,
it's a different frequency" then that's also a statement about resolution
because you are able to "resolve" two different frequencies.

But, if the question, as I believe it is, is:
How precise is your estimate? Then the answer is that it has a standard
deviation of 0.00234 Hz resulting from the methods used. That doesn't mean
that it's accurate.

Fred





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