[time-nuts] Frequency Stability of An Individual Oscillator: Negative Values?

[Magnus Danielson]

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