I am trying to estimate the confidence limits for measurement of white
noise passed through a limited band filter.

In the first instance, can we consider the filter to be an ideal low
pass filter.

The noise voltage can be though of as a stream of instantaneous values
with Gaussian distribution, mean of zero, and standard deviation equal
to the RMS voltage.

If I take samples of this waveform, I should be able to calculate the
noise power (given the resistance). The noise power is proportional to
the variance of these samples, and the constant of proportionality is
1/R.

Shannon's Information Theory says to me that I need to sample the
waveform at least at double the highest frequency of any component
(the break point of the low pass filter).

It seems to me that what I am doing in statistical terms is taking a
limited set of samples and using it to estimate the population
variance (and hence the noise power in a resistor).

So, I can never be absolutely certain that my set of samples will give
the same variance as the population that I sampled.

I should expect that on repeated measurement of the same source, that
there will be variation, and that a component of that variation is the
chance selection of set of samples on which the estimate was based.

It seems reasonable to assume that taking more samples should give me
higher confidence that my estimate is closer to the real phenomena,
the population variance.

So, I am looking for a predictor of the relationship between sample
variance (the "measured" power) and population variance (the "actual"
power), number of samples, and confidence level.

The statistic Chi^2=(N-1)*S^2/sigma^2 (where S^2 is the sample
variance and sigma^2 is the population variance) seems a possible
solution. The distribution of Chi^2 is well known.

So I have plotted values for the confidence limits indicated by that
approach, the plot is at
http://www.vk1od.net/fsm/RmsConfidenceLimit01.gif . The x axis value
of number of samples relates to the minimum number of samples to
capture the information in the filtered output (in the sense of
Shannon), ie bandwidth*2.

It seems to me that this should also apply to a noise source that has
been passed through a bandpass filter (with a lower break point 0),
so long as the sampling rate is sufficient for the highest frequency,
but that the number of samples used for the graph lookup is
bandwidth*2.

I understand that there are other sources of error, this note is
focused on choice of appropriate number of samples (or integration
time) to manage the variation in the sampling process due to chance.

Am I on the right track?

Comments appreciated.

Owen

PS: This is not entirely off topic, I am measuring ambient noise from
a receiving system, and antenna performance assessment is the purpose.
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