[time-nuts] ADEV vs. OADEV

[Ulrich Bangert]

Magnus,

I am aware that you know a lot about these things. Nevertheless I
believe you are starting a most dangerous discussion in the sense that
you put some terms into question of which I believed that they have well
been established. For that reason let me test where we agree and where
not:

Mr. Allan decided that for his new statistical measure the summation
shall run over

square(y(i+1)-y(i))

for frequency data and over

square(x(i+2)-2*x(i+1)+(xi))

for phase data. Both in contrast to the standard deviation where the
summation runs over squares of distances from the mean. This new
variance was called "Allan variance" and its square root "Allan
deviation" to honor Mr. Allan for his work. This variance/deviation has
a certain "overlapping aspect" since a single y(i) or x(i) appears in
multiple terms of the summation. Agreed?

Both terms require that the elements with subsequent indices are spaced
apart at the "Tau" for wich the computation shall be done. Considered a
number of phase measurements spaced 1 s apart then the computation will
run over 

square(x(i+2)-2*x(i+1)+(xi))

for Tau = 1 s. If you are going to compute for Tau = 2 s from the SAME
data set you will have to use the "original" samples

square(x(5)-2*x(3)+x(1)) 

for the first summand and

square(x(7)-2*x(5)+x(3))

for the second summand and

square(x(9)-2*x(7)+x(5))

for the third summand and so on. All indices are incremented by two
between neighbour summands because the next summand is 2 s (or two
original samples) apart from the current summand. Agreed?

As we notice the summation leaves out a number of summands where the
elements are also spaced 2 s apart, for example

square(x(6)-2*x(4)+x(2))

or

square(x(8)-2*x(6)+x(4))

If we use these additional terms in the summation the number of summands
increases a lot and improves the confidence interval of the estimation,
even though the added summands are NOT completely statistical
independend from the original ones and therefore this measure shall be
clearly distincted from the original Allan variance/deviation. The
summation over the original terms plus the added terms delivers the
"Overlapping Allan variance/deviation" in conjunction with a suitable
normation factor. Agreed?

Best regards
Ulrich 

> -----Ursprüngliche Nachricht-----
> Von: time-nuts-bounces at febo.com 
> [mailto:time-nuts-bounces at febo.com] Im Auftrag von Magnus Danielson
> Gesendet: Mittwoch, 21. Januar 2009 09:33
> An: Discussion of precise time and frequency measurement
> Betreff: [time-nuts] ADEV vs. OADEV
> 
> 
> Hi!
> 
> I have been quite surprised to see the abbreviation OADEV appear. I 
> assume that this means Overlapped Allan Deviation, but this 
> is confusing 
> since the Allan Deviation estimates already is overlapping. 
> However, I 
> have seen that some use a non-overlapping estimator, but this type of 
> estimator has an unwanted filtering effect and should not be used.
> 
> If a distinction between these ADEV estimators should be 
> used, then the 
> standard (overlapping) estimator should continue to be called 
> ADEV and 
> the non-overlapping (back-to-back) could be called NOADEV ór 
> whatever...
> 
> I have done a fair amount of digging around many sources 
> around ADEV and 
> friends so I think I got it right.
> 
> Unless someone can give a meaningful explanation and I really 
> expect a 
> good article detailing the difference and benefits...
> 
> Cheers,
> Magnus
> 
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