On Sat, 08 Jul 2006 17:10:34 -0700, Richard Clark
wrote:

On Sat, 08 Jul 2006 22:08:20 GMT, Owen Duffy wrote:


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.

Hi Owen,

This will possibly be your greatest source of error, the clipping of
the spectrum. In Fourier Analysis, the operation is called
"windowing" and there are a world of window shapes that offer either
excellent frequency resolution at the cost of amplitude accuracy, or
the t'other way 'round. Insofar as the window shape, this deviates
from an "ideal" filter response, but then an "ideal" filter response
(infinite skirt) does not guarantee accuracy.

Blackman and Tukey in their seminal work, "The Measurement of Power
Spectra" (1958) assert that
"a realistic white noise spectrum must be effectively
band-limited by an asymptotic falloff at least as fast as 1/f²."

Consider the discussion at:
http://www.lds-group.com/docs/site_d...%20Windows.pdf




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).

That's Nyquist sampling rate at slightly more than double than Fmax.
Shannon predicts the bit error rate for a signal to noise ratio.

Ok.


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).

Seeing that the RMS voltage is applied fully to the resistance,
shouldn't that be signal + noise power in a resistor? The noise in
this sense only describes the deviation from the distributions' shape.

In this case, the KTB noise due to the loads own resistance is so
small as to be insignificant and not require a correct to be applied.


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.

When I've done brute force noise reduction through ever increasing
samples, it always appeared to follow a square law relationship.

I expected that, and the radioastronomy folk seem to work on that
basis from Dicke's work, if I understand it correctly.


Am I on the right track?

What are you using as a source of noise?

Noise from the real world which I understand is not exactly white, but
I figure that if I understand the behavior from a white noise point of
view, the answer will be very close for noise that resembles white
noise.

The noise is audio output from an SSB receiver (operating below AGC
gain compression threshold) that, if you like, is acting as a linear
downconverter with a narrow pass band filter. Typically, the passband
is 300-2400Hz. The sampling is done in a PC sound card at a rate of
11kHz. The application here is using these samples to synthesise a
"true RMS voltmeter".

The question is how many samples, or how long an integration time at a
given bandwidth, is required to reduce the likely contribution of
chance to the sampling process below, say 0.1dB, at a confidence level
of, say 90% (in a two tailed test).

Thanks for your response Richard.

Owen
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