On 04/22/2010 11:07 PM, Kyle Wesson wrote:
Hello,
I am working to determine the Allan variance of an individual
oscillator from a series of three paired measurements as described in
the paper by Gray and Allan "A Method for Estimating the Frequency
Stability of An Individual Oscillator" (NIST, 1974,
tf.nist.gov/general/pdf/57.pdf). In this report they make reference to
the statistical uncertainty of the measurement due to ensemble noise
and potential clock phase correlation which can potentially make the
Allan variance for an individual oscillator have a negative value.
They write:
"If the noise level of the oscillator being measured is low enough,
and the scatter high enough, equation (4) may occasionally give a
negative value for the variance."
This is rather an effect of imperfect measurements than real world.
My question is: how should I treat negative variance values in this
case? For example, if my data set were to produce an individual
oscillator Allan variance with a value of -5e-12, should I convert
this value to 0 (ie. the closest valid sigma value to the number since
0<= sigma< inf ), take the absolute value of the result (ie. turn
-5e-12 to +5e-12), or drop the result from my estimate of individual
oscillator frequency stability altogether?
You variance can't be negative. It's the sum of squares of real values,
so it can't be negative.
If the oscillators in a so called three-cornered hat has the variances
of sigma_1^2, sigma_2^2 and sigma_3^2 then the measurements between them
becomes
sigma_12^2 = sigma_1^2 +sigma_2^2
sigma_13^2 = sigma_1^2 +sigma_3^2
sigma_23^2 = sigma_2^2 +sigma_3^2
This is just the sum of noise energies. They are never negative either.
However, imprecision in measurements will yield negative solutions from
the above equation system. This is a good indication that you should
have less difference between the different sources.
Is there another method that will produce estimates of individual
oscillators from an ensemble approach but assures non-negative output
variances?
Better noise references, better rig, better counters, whatever is your
limit at that range of tau.
Cheers,
Magnus
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