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

Bob Camp lists at rtty.us
Thu Apr 22 18:17:23 EDT 2010

```Hi

There are normally two cases where you get negative variances out of the three corner hat:

1) One negative, two roughly equal (the common outcome).
2) Two negative, one positive (rare)

In the first case, one normally uses the data for the two positive results and drops the third. In the second case, you go back to the pair wise data and see what's actually going on.

Bob

On Apr 22, 2010, at 5:46 PM, Magnus Danielson wrote:

> 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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