On Sun, 9 Jul 2006 21:33:27 +0100, "Reg Edwards"
wrote:
I am designing the instrument. I am exploring the number of samples
required to reduce the effect of chance on the measurement result
(in
respect of the sampling issue) to an acceptable figure.
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The first thing to do is calibrate the instrument against a standard
noise source. Immediately, the uncertainty in the standard is
transferred to the instrument - plus some more uncertainty due to the
manner in which the standard and instrument are associated.
Reg, I think you have missed my point. Because of the random nature of
white noise, an attempt to measure the noise source by sampling the
noise for a short period introduces an error due to the sampling
process. That sampling error is related to the quantity of
"information" gathered by the sampling process, ie the length of
"integration" or number of samples.
The issue is not about absolute calibration, it is about one source of
error in measuring a white noise source, and quantification of bounds
on that error to a level of confidence.
Does the instrument read in watts, decibels, or what?
The second thing to do is to be verbally and numerically more precise
about "to reduce the effect of chance on the measurement result to an
acceptable figure."
I am sorry if that is wordy, but I think it is precise in expressing
the problem.
To give a specific application, suppose that I want to do an receiver
system performance test by comparing noise from one cosmic noise
source with quiet sky, and I expect the variation with my G/T to be
0.5dB.
At the outset you should define the acceptable figure. What effects?
In what units is the acceptable figure?
The acceptable figure will depend on the application, I am trying to
understand the principle.
It is then not a difficult matter to decide the number of
measurements, by taking samples, to give a predetermined level of
confidence in the average or mean. But I have the feeling you are
over-flogging the issue. You don't really have a problem.
So, coming back to the application above, I note that successive
measurements of the same white noise source passed through a limited
bandwidth filter have variation from measurement to measurement, and
that variation is related to the length of time that length of
"integration" time or number of samples used for each measurement.
In trying to understand this relationship, I explored the use of the
Chi-square distribution as discussed in my initial posting.
In looking for more information on that relationship, I found Dicke
being quoted with an estimate of the sensitivity of a radiometer as
the minimum detectable signal being the one in which the mean
deflection of the output indicator is equal to the standard deviation
of the fluctuations about the mean deflection of the indicator. He is
quoted as saying:
mean(delta-T)= (Beta * Tn) /( delta-v * t)^0.5
where delta-T is the minimum detectable signal; Beta is a constant of
proportionality that depends on the receiver and is usually in the
range 1 to 2; Tn is the receiving system noise temperature; delta-v is
the pre-detection receiver bandwidth; and t is the post detection
integration time constant.
(I do not have a derivation of Dicke's formula.)
This suggests that an estimate of the error (in dB) due to the
sampling process is 10*log(1+Beta /( delta-v * t)^0.5).
I have plotted the above expression at Beta=2 over the plots that I
did based on the Chi-square distribution, they are at
http://www.vk1od.net/fsm/RmsConfidenceLimit03.gif . You will see that
the Dicke (Beta=2) line follows (ie it pretty much obscures by
overwriting) my Chi-square based 95% confidence line. It appears that
the two methods arrive at similar answers.
Dicke's Beta seems to be determined empiracally. Varying Beta has the
same effect as changing the confidence level in my Chi-square based
estimator.
Owen
PS: Still remains relevant to antennas, I am measuring the performance
of a receiver system, which includes the antenna and alll noise
sources.
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