Jim Lux wrote in
:
Owen Duffy wrote:
...
Just following through on the '4 digit' issue...
I have done two series of 250 measurements of audio noise voltage
from a SSB receiver using two different digital multimeters, the 9932
is a modern digital multimeter that is NOT true RMS responding, and
the 506 is a modern digital multimeter that is true RMS responding
with bandwidth adequate to cover the receiver output response.
From observation with a stopwatch, I estimate that the 9932 updates 3
times per second, and the 506 updates 2 times per second. The
integration times are probably .33 and .5 seconds respectively.
I have measured the receiver equivalent noise bandwidth and it is
1600Hz.
95% of 250 readings were within 0.41dB for the 9932 and 0.31dB for
the 506. These observations reconcile well with my Chi-squared based
estimate of the uncertainty that I referred to in an earlier post.
As for the number of digits, they are both 4 digit multimeters which
doesn't mean a lot. They were used to measure 200mV with 1mV
resolution, so the representational error is 0.04dB.
Gotta be a bit careful there, because quantization error has a uniform
distribution, so the variance is 1/12 of the span. This is different
than the (presumably) normally distributed actual measurands.
Ok, point taken. I think more correctly, the maximum error would be 20
*log(1+1/200/2) or 0.0217dB.
The expected error due to representation in three digits does not account
for the variation in measurements.
When giving an uncertainty (sampling error), one should also say
whether it's a one sigma, two sigma, or 3 sigma number. *Standard
uncertainty* is 1 sigma... *expanded uncertainty*, often given as a
+/- number is usually the 95% percent confidence interval, which, for
normal distributions, is 2 sigma
Whilst it might be reasonable to assume that the combined error in
measurement of a high S/N sine wave voltage might be normally
distributed, and that might also be true of measurement of noise voltage
in some circumstances, I propose that measurement of noise power in
narrow bandwidth with short integration times is distributed as Chi-
squared and the number of samples becomes relevant in determining the
number of degrees of freedom for the distribution. For this reason, I
have talked about a confidence level rather than sigma (which is more
applicable to normally distributed data).
Just for interest, in the case of the 9932 measurement set:
Average=0.201, sigma=0.0046, 1sigma based uncertainty estimate=0.20dB,
2sigma based uncertainty estimate=0.41dB, 3sigma based uncertainty
estimate=0.62dB.
given your statistics above, you would be giving the expanded
uncertainty as 0.41dB
I stated it as 95% of obs within 0.41, I should have said 95% of obs
within +/-0.41, I was explicit about the implied confidence, the 95%
doesn't equate to either the 1sigma or 3sigma confidence, it is very
close to the 2sigma confidence (95.45%), and it is at the high confidence
end of the scale.
By the way, unless your device actually directly measures dB (e.g. it
has a log detector) or the errors are inherently ratios, it's probably
better to give the value in a linear scale (milliwatts?) with the
uncertainty in the same units. That gets you around the "ratio"
problem where log(1+delta) -log(1-delta)
I understand what you mean in your last sentence.
I did record the voltage, and converted the values to dB for analysis.
The interval was calculated by taking the average of the 2.5 percentile
and 97.5 percentile, which is an approximation, but as such small values
is pretty close.
I have converted results to dB to make it easier to see the relevance of
the error or uncertainty, but in so doing, another (small in this case)
error is introduced.
http://physics.nist.gov/cuu/Uncertainty/index.html has the simple
explanation, and the technical note (TN1297) , and references to the
ISO Guide to Expression of Uncertainty in Measurment (aka the GUM)
In terms of the above, I am proposing that measurements of narrowband
noise with short integration time is not strictly normally distributed,
and an estimate of its uncertainty to a given confidence level can be
obtained from the Chi-square distribution.
One could not estimate the results of the test from knowledge of the
instrument accuracy (inherent and representational error) alone.
I think the experiment supports the proposition that digital multimeters
with typically short integration times do not deliver high resolution
measurement of narrow band (eg SSB telephony) noise.
The error due to the number of
digits in this downscale three digit application is insignificant
compared to the sampling error of 0.4dB and 0.3dB.
Graphically, the distributions are shown at
http://www.vk1od.net/nfm/temp.gif .
Different meters with different integration times, and different
receivers with different noise bandwidth will result in different
outcomes, but I argue that the uncertainty is predictable.
Indeed, it is.
Thanks, appreciate the comments.
Owen