[time-nuts] Frequency Stability of An Individual Oscillator: Negative Values?
Magnus Danielson
magnus at rubidium.dyndns.org
Thu Apr 22 17:46:52 EDT 2010
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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