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This text provides an overview of ADev and Modified Allan Deviation (ModADev) as statistical tools for analyzing various types of noise, including white phase noise, flicker phase noise, and frequency noise. It explains how ADev and ModADev differ in their ability to measure different types of noise and offers insights into their limitations. Additionally, it explores the use of Hadamard variance and Total ADev for analyzing drift and long-term noise. The text also discusses the importance of time prediction and the calculation of TDev. Finally, it addresses issues related to unevenly spaced or missing data during the analysis.
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Modified Allan Deviation And a few other useful variances
ADev • Equation: second differences • Why try other statistics? • Limited in discriminating white from flicker phase noise • Does not work well with drifting frequency signals
White phase noise Flicker phase noise White frequency noise Flicker frequency noise Random walk Allan Variancefor 2pft >1 N.B. f is the high frequency cutoff.
Ideally for white phase Power dependence of ADev Log(sy(t)) White and flicker phase White frequency Random walk frequency Flicker frequency Log(t)
White Phase Noise 100 samples Red one : one sample Blue: average
ADEV of White Phase Noise Slope = -1 ! Not -3/2 ?!? ADev is not sensitive to white phase noise. It converts the short term noise into some sort of flicker phase noise.
Filcker Phase Noise One sample in red 100 samples average
ADev of Flicker Phase Noise Slope > -1 ADev does not measure flicker phase noise properly. There is an extra ln(t) factor.
White phase + white frequency noise: looks like white flicker phase noise The slope is slightly higher than -1, like the case of pure flicker phase noise. The sum of white phase and white frequency noises cannot be sorted out by ADev.
Mod ADev • Equation second differences of average phases • Reduces the phase noise over the interval tau considered by averaging • White and flicker average differently Let’s compare ADev and Mod Adev
MOD ADev ADev
MOD ADev You may be reluctant to calculate MOD ADev because of the double summation but there are tricks:
Practical way of calculating MOD Calculate the first D2 average for n Initialize accumulator One loop only Update overlapping average Update accumulator Calculate MODsy for nt0
Example of MOD Adev MOD Adev shows that t--1 noise does not really exist. There is t—3/2 and t—1/2 noise. ADev Note the lower values for each type of noise, except maybe random walk frequency MOD Adev
Is there more? Yes. • What about drift? • Hadamard variance will help solve it for us • What about long term? • TotalADev is another interesting solution • What about time prediction? • TDev is the proper answer.
Hadamard Using third difference gets rid of frequency drift. It can be used to compare with ADev in order to attempt to distinguish between random walk and frequency drift. Note the factor 6 at the denominator: since this is a second difference in frequency, you get 3!=6 gain of the original noise. Just to convince yourself, calculate the variance of a pure gaussian noise and its first, second and third differences.
Example of Hadamard on drifting clock ADev Hadamard Drift has been removed to reveal the underlying noise
Total variance • To improve the estimation for time intervals close to the time of measurements, a technique similar to what is assumed in FFT can be used. • Wrapping the data from the end to the beginning of the time series smoothes the results for long time intervals.
How to realize Total ADev t t Very useful for long t 1-Duplicate data Total ADev ADev 2-shift data to align ends 3-Run ADev from t=0 to t=T/2 starting within the interval 0-T
TDev • Do we need more? • Frequency stability is not all • We need to determine where will be the clock after some time. • Using the MOD ADev, you get it
Work in time : TDev Easy to compute once you can compute MODADev You can always use not too bad substitute for rough evaluation:
Example of TDev From these ADev and MOD ADev below you get this TDev and sx(t) on the right sx(t) ADev TDev MOD ADev
Before concluding • What if data is not equally spaced? • If intervals do not vary by more than 5%, it does not matter at all. We will see later. • What if data is missing? • Just use the time tag to skip the data in your statistics. No real need to interpolate.
Not equally spaced data Note that close to the main diagonal where the t1 = t2 there is a very slow change in the actual value of ADev ADev Flicker frequency White/flicker phase ADev log(t2) log(t2) log(t1) log(t1) Random walk frequency White frequency ADev ADev log(t2) log(t2) log(t1) log(t1)
Beware of long time measurements When you calculate ADev or any other over long time intervals, there may be some changes in the conditions of measurements
Closer look at the moving ADev The one second is fluctuating with time, due to some interference of some kind. Average value over the whole period is meaningless.
Degrees of freedom For all types of noise the distribution is nearly 2 within a few percent. We can safely assume for the purpose of calculating the degrees of freedom that it is 2 distributed. In the case of white phase noise: The degrees of freedom associated with ADev, ModADev, etc. can be estimated to be roughly the time span T-nt divide by t, where n is one for ADev and 2 for ModADev. In the case of Total Dev, n=0. Ndof=(T-n*t)/t, n=0,1 or 2 for ADev, ModADev or TotalADev For other type of noise, there is a reduction in the number of degrees of freedom going as low as 67% for white frequency noise and large degrees of freedom. Note that for Ndof cannot be lower than 1. Use this number for calculating the coverage factor.
Degrees of freedom For all types of noise the distribution is nearly 2 within a few percent. We can safely assume for the purpose of calculating the degrees of freedom that it is 2 distributed. In the case of white phase noise: The degrees of freedom associated with ADev, ModADev, etc. can be estimated to be roughly the time span T-nt divide by t, where n is one for ADev and 2 for ModADev. In the case of Total Dev, n=0. Ndof=(T-n*t)/t, n=0,1 or 2 for ADev, ModADev or TotalADev For other type of noise, there is a reduction in the number of degrees of freedom going as low as 67% for white frequency noise and large degrees of freedom. Note that for Ndof cannot be lower than 1. Use this number for calculating the coverage factor.
conclusion • Again and again, use all the tools available to characterize your data. • Look at your data from different point of view. • All those tools are there to help you sortout the various problems of timekeeping.
beware • TDev should be used carefully for prediction over interval different than the one estimated. • Better use the structure factors • A little bit of it follows
Generalized Variance in the average frequency transfer 3 phase 2 B A 4 1 time
Standard uncertainty In theory you need to measure those h coeficients
Useful results Knowing , we want with the following form: where each type of noise is calculated independently.
What fA-B are • AB structure factors which are directly linked with the measured Allan deviation. • Can be used for any linear combination of results in the calibration interval. • For end-point average frequency, results simple enough for hand-calculator use. • Provided similar equipment (same high cutoff frequency) is used, no dependence on passband.
White and flicker phase noise white frequency noise flicker frequency noise random walk Ratio of uy2a to Adeva2for 2pft >1
Time prediction reference Clock A
Timescale work • Prediction of uncertainty of time after interval tB • following calibration over interval tA • equivalent to prediction of frequency uncertainty on tB after calibration on tA with no dead time.
