[time-nuts] Hadamard variance

Magnus Danielson magnus at rubidium.dyndns.org
Wed Apr 8 23:43:29 UTC 2009

Tom Van Baak skrev:
>> According to Wikipedia, this is insensitive to drift and so seems like
>> a better tool for measuring oscillators, like ocxo. I don't think I've
>> seen this being used by anyone on the list and wonder why?
> Hi Steve,
> I use it sometimes when I need to. But note that in most cases
> you do NOT want to ignore drift. If you measure an OCXO for
> the purpose of using it in a clock or appliance or radio or test
> equipment you really do want to know if it has drift or not.

True, but using ADEV is not a good method to bring it into the equation.

ADEV is meant to measure frequency stability resulting from phase noise 
of various slopes. It is not meant to give a measure of drift components 
which is best estimates through other measures.

> ADEV will show this, while HDEV will not. So you have to be careful
> about using statistics that deliberately and quietly ignore effects
> that may be important to your application.

You need to understand what the various measures gives you. The ADEV and 
its various estimators is usually not drift compensating. The drift will 
limits the ADEV estimators ability to show noise slopes, but if drift 
compensations is performed then ADEV estimators typically just continues 
the slopes further down below the drift rate floor.

HDEV is another approach that gives very similar properties.

Drift and other higher orders of frequency shift, i.e. shifts in 
frequency that produces DC shifts needs to be combined and the best way 
of combing them is to estimate phase, frequency and drift rate errors 
and build their error over time along with the TDEV over the same time.

This time error formulation have been provided in several sources, such 
as the NIST Special Publication 1065.

The TDEV measure (a variant of the MDEV measure) is an estimator for 
time stability rather than frequency stability. Again it is only meant 
to be used for estimating the contribution from noise processes, not 
linear time drift components.

Drift is a noise component when estimating ADEV, MDEV and TDEV, so it 
needs to be removed if sufficiently large in amplitude. ADEV without 
drift compensation is not a good combined measure for noise and drift.

> On the other hand, if you are choosing an OCXO to be used
> in a smart GPSDO which you know has internal adaptive drift
> rate calculation and compensation then, yes, HDEV would be
> a more appropriate statistic than ADEV.
> But before you run off and use HDEV for everything note that
> the other practice that is far more common -- simply remove
> frequency drift from the raw data before computing an ADEV
> on the residuals. If you look at plots in professional journals you
> will often find comments to the effect that phase, frequency, or
> drift offsets have been added or removed prior to making said
> phase, frequency, or stability plots.

The normal ADEV estimators is insensitive to phase offset and frequency 
offset in the data series. Phase and Frequency deviations such as jumps, 
spikes needs to be canceled or avoided. Frequency drift such as that of 
a frequency slope will affect ADEV estimators, since the core of ADEV 
estimators is a second difference, squaring and averaging and a 
frequency ramp will produce a static value after the second difference.
HDEV uses a third difference, so that is why the drift insensitivity 
comes from, but that is still just a theoretical things since many times 
the drift is not a pure linear frequency ramp but more complex, so some 
of it will leak in, but first degree cancelation can occur.
Estimating the drift rate is straightforward thought, just use the 
second difference of the time data and average, then subtract that out 
of the second differences before squaring and averaging.

Infact, the assumption that drift can be canceled is part of building 
the ADEV theory already in Allans original paper in 1966. It's hidden in 
the early equations if you look carefully.

Regardless of which measure one uses, one should spend some time to read 
up on the peculiarities of them as one can easily confuse oneself with 
freely compare results from different stability measures, different 
estimators of these and then know what form of noise sources may obscure 
the data.

Spending quality time with something like NIST SP1065 is adviceable.
Using simulated noise and real measurements and processing them through 
the various variants can be good, as they react differently to different 
noise slopes and also to non-noise error sources. Length of measurement 
list vs. longest tau trusted, effects of repetitive signals etc. etc. 
all comes into it. Some measures depend on measurement bandwidth where 
as newer measures can cancel that through a combination of algorithmic 
change in bandwidth and t0 vs lowest tau analyzed margins.

So while HDEV is indeed useful, don't expect it to be a drop-in 
replacement for ADEV but with drift compensation.


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