Allan variance
Encyclopedia
The Allan variance also known as two-sample variance, is a measure of frequency stability in clock
s, oscillators and amplifier
s. It is named after David W. Allan. It is expressed mathematically as
The Allan deviation (ADEV) is the square root of Allan variance. It is also known as sigma-tau, and is expressed mathematically as
The M-sample variance is a measure of frequency stability using M samples, time T between measures and observation time
. M-sample is expressed as
The Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability, i.e. the stability in frequency. See also the section entitled "Interpretation of value" below.
There are also different adaptations or alterations of Allan variance. Notably the modified Allan variance
MAVAR or MVAR, the total variance, and the Hadamard variance. There also exist time stability variants such as time deviation
TDEV or time variance
TVAR. Allan variance and its variants have proved useful outside of the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, but a derivative will be.
The M-sample variance is of historic importance as well as important background but has essentially been replaced by its special case of 2-sample variance with
now being called Allan variance. It remains important since it allows dead time
in measurements and bias functions allows conversion into Allan variance values.
s and atomic clock
s it was found that they did not have a phase noise
consisting only of white noise
, but also of white frequency noise and flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation
as the estimator will not converge. The noise is thus being said to be divergent. Efforts in analysing the stability provided both the theoretical analysis as well as practical measurements.
An important side-consequence of having these types of noises was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources, specify a needed quality and be able to get it from a supplier, essentially all forms of scientific and commercial uses was limited to dedicated measurements which hopefully would capture the need for that application.
To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not completely resolve all noise forms, it provided means to separate noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure.
An early concern was related to time and frequency measurement instruments which had a dead time
between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero dead time counters removed the need, but the bias analysis tools have proved useful.
Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation
, only low 
values would be affected while higher values would be unaffected. The change of 
is done by letting it be an integer multiple 
of the measurement timebase 
.
The physics of crystal oscillator
s was analyzed by D. B. Leeson which is now referred to as the Leeson effect. The feedback in the oscillator will make the white noise
and flicker noise
of the feedback amplifier and crystal become the power-law noises of
white frequency noise and 
flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time error samples. This mechanics of the feedback oscillators was unknown when the work on oscillator stability started but was presented by Leeson at the same time as the statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect see modern phase noise literature.
average of the squares of the differences between successive readings of the frequency deviation
sampled over the sampling period. The Allan variance depends on the time period used between samples: therefore it is a function of the sample period, commonly denoted as τ, as well as the distribution being measured, and is displayed as a graph rather than a single number. A low Allan variance is a characteristic of a clock with good stability over the measured period.
Allan deviation is widely used for plots (conveniently in log-log
format) and presentation of numbers. It is preferred as it gives the relative amplitude stability, allowing ease of comparison with other sources of errors.
An Allan deviation of 1.3×10−9 at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations a second apart with a relative root mean square
(RMS) value of 1.3×10−9. For a 10-MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed then the time deviation
variants should be consulted and used.
The 
-sample variance is defined (here in a modernized notation form) as
or with average fractional frequency time series
where
is the number of frequency samples used in variance, 
is the time between each frequency sample and 
is the time-length of each frequency estimate.
An important aspect is that
-sample variance model counter dead-time by letting the time 
be different from that of 
.
which is conveniently expressed as
where
is the observation period, 
is the nth fractional frequency average over the observation time 
.
The samples are taken with no dead-time between them, which is achieved by letting
and variance
, the Allan deviation is defined as the square root of the Allan variance.
The oscillator is assumed to have the nominal frequency of vn being the nominal number of cycles per second or Hertz (Hz), corresponding to the nominal angular frequency
as related in
Removing the nominal phase ramp the total phase can be divided as
For measured values a time error series TE(t) is defined from the reference time function TREF(t) as
where the average is taken over observation time τ, the y(t) is the fractional frequency error at time t and τ is the observation time.
Since y(t) is the derivate of x(t) we can without loss of generality rewrite it as
, integrating over infinite time. Real world situation does not allow for such time-series, in which case a statistical estimator
needs to be used in its place. A number of different estimators will be presented and discussed.
The relation between the number of fractional frequency samples and time error series is fixed in the relationship
where T is the time between measurements. For Allan variance, the time being used has T set to the observation time τ.
The time error sample series let N denote the number of samples (x0 ...xN-1) in the series. The traditional convention uses index 1 through N.
which gives
For the Allan variance assumption of T being τ it becomes
The average fractional frequency sample series let M denote the number of samples (
) in the series. The traditional convention uses index 1 through M.
As a short-hand is average fractional frequency often written without the average bar over it. This is however formally incorrect as the fractional frequency and average fractional frequency is two different functions. A measurement instrument able to produce frequency estimates with no dead-time will actually deliver a frequency average time series which only needs to be converted into average fractional frequency and may then be used directly.
or for the time series
These formulas however only provides the calculation of τ = τ0 case, wishing any other τ and a new time-series needs to be provided.
or for the time series
These estimators have a significant drawback in that they will drop a significant amount of sample data as only 1/n of the available samples is being used.
or for the time series
The overlapping estimators have far superior performance over the non-overlapping estimators as n rises and the time-series is of moderate length. The overlapped estimators have been accepted as the preferred Allan variance estimators in IEEE, ITU-T and ETSI standards for comparable measurements such as needed for telecommunication qualification.
and similarly for Allan deviation to time deviation
. The modified Allan variance measure is a frequency stability measure, just as the Allan variance.
. The confidence interval depends on the number of observations in the sample series, the dominant noise type, and the estimator being used. The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values. For variable-τ estimators, the τ0 multiple n is also a variable.
can be established using chi-squared distribution by using the distribution of the sample variance :
where s2 is the sample variance of our estimate, σ2 is the true variance value, d.f. is the degrees of freedom for the estimator and χ2 is the degrees of freedom for a certain probability. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality:
which after rearrangement for the true variance becomes:
represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on N and n has been empirically found to be:
As found in and in modern forms.
The Allan variance is unable to distinguish between WPM and FPM, but is able to resolve the other power-law noise types. In order to distinguish WPM and FPM, the modified Allan variance
need to be employed.
The above formulas assume that
and thus that the bandwidth of the observation time is much lower than the instruments bandwidth. When this condition is not met, all noise forms depend on the instruments bandwidth.
where
or frequency modulation of the form
into the Allan variance of the form
can be significantly simplified by providing a mapping between α and μ. A mapping between α and Kα is also presented for convenience:
The mapping is taken from.
Thus, linear drift will contribute to output result. When measuring a real system, the linear drift or other drift mechanism may need to be estimated and removed from the time-series prior to calculating the Allan variance.
where
Replacing the time series of
with the Fourier transformed variant 
the Allan variance can be expressed in the frequency domain as
Thus the transfer function for Allan variance is
These bias functions is not sufficient for handling the bias resulting from concatenating M samples to the Mτ0 observation time over the MT0 with has the dead-time distributed among the M measurement blocks rather than in the end of the measurement. This rendered the need for the B3 bias.
The bias functions is evaluated for a particular µ value, so the α-µ mapping needs to be done for the dominant noise form as found using noise identification. Alternatively as proposed in and elaborated in the µ value of the dominant noise form may be inferred from the measurements using the bias functions.
where
The bias function becomes after analysis
where
The bias function becomes after analysis
where
The B3 bias function is useful to adjust non-overlapping and overlapping variable τ estimator values based on dead-time measurements of observation time τ0 and time between observations T0 to normal dead-time estimates.
The bias function becomes after analysis (for the N = 2 case)
where
The bias function becomes after analysis
where
Similarly, for concatenated measurements using M sections, the logical extension becomes
. As can be seen in the power-law noise formulas, the white and flicker noise modulations both depends on the upper corner frequency
(these systems is assumed to be low-pass filtered only). Considering the frequency filter property it can be clearly seen that low-frequency noise has greater impact on the result. For relatively flat phase modulation noise types (e.g. WPM and FPM), the filtering has relevance, where as for noise types with greater slope the upper frequency limit becomes of less importance, assuming that the measurement system bandwidth is wide relative the 
as given by
When this assumption is not met, the effective bandwidth
needs to be notated alongside the measurement. The interested should consult NBS TN394.
If however one adjust the bandwidth of the estimator by using integer multiples of the sample time
then the system bandwidth impact can be reduced to insignificant levels. For telecommunication needs, such methods have been required in order to ensure comparability of measurements and allow some freedom for vendors to do different implementations. The ITU-T Rec. G.813 for the TDEV measurement.
It can be recommended that the first
multiples be ignored such that the majority of the detected noise is well within the passband of the measurement systems bandwidth.
Further developments on the Allan variance was performed to let the hardware bandwidth be reduced by software means. This development of a software bandwidth allowed for addressing the remaining noise and the method is now referred to modified Allan variance
. This bandwidth reduction technique should not be confused with the enhanced variant of modified Allan variance
which also changes a smoothing filter bandwidth.
during which the signal is not being observed. Such dead time introduces systematic measurement biases, which needs to be compensated for in order to get proper results. For such measurement systems will the time T denote the time between the adjacent start events (and thus measurements) while
denote the time-base length, i.e. the nominal length between the start and stop event of any measurement.
Dead time effects on measurements have such an impact on the produced result that much study of the field have been done in order to quantify its properties properly. The introduction of zero dead-time counters removed the need for this analysis. A zero dead-time counter has the property that the stop-event of one measurement is also being used as the start-event of the following event. Such counters creates a series of event and time timestamp pairs, one for each channel spaced by the time-base. Such measurements have also proved useful in order forms of time-series analysis.
Measurements being performed with dead time can be corrected using the bias function B1, B2 and B3. Thus, dead time as such is not prohibiting the access to the Allan variance, but it makes it more problematic. The dead time must be known such that the time between samples T can be established.
The effect may be that the estimated value may be much smaller or much greater than the real value, which may lead to false conclusions of the result.
It is recommended that the confidence interval is plotted along with the data, such that the reader of the plot is able to be aware of the statistical uncertainty of the values.
It is recommended that the length of the sample sequence, i.e. the number of samples N is kept high to ensure that confidence interval is small over the τ-range of interest.
It is recommended that the τ-range as swept by the τ0 multiplier n is limited in the upper end relative N such that the read of the plot is not being confused by highly unstable estimator values.
It is recommended that estimators providing better degrees of freedom values be used in replacement of the Allan variance estimators or as complementing them where they outperform the Allan variance estimators. Among those the Total variance and Theo variance estimators should be considered.
In order to provide evenly spaced measurements will the reference clock be divided down to form the measurement rate, triggering the time-interval counter (ARM input). This rate can be 1 Hz (using the 1 PPS
output of a reference clock) but other rates like 10 Hz and 100 Hz can also be used. The speed of which the time-interval counter can complete the measurement, output the result and prepare itself for the next arm will limit the trigger frequency.
A computer is then useful to record the series of time-differences being observed.
The Allan variance can then be calculated using the estimators given, and for practical purposes the overlapping estimator should be used due to its superior use of data over the non-overlapping estimator. Other estimators such as Total or Theo variance estimators could also be used if bias corrections is applied such that they provide Allan variance compatible results.
To form the classical plots, the Allan deviation (square root of Allan variance) is plotted in log-log format against the observation interval tau.
The NASA-IEEE Symposium on Short-Term Stability in November 1964 brings together many fields and uses of short and long term stability with papers from many different contributors. The articles and panel discussions is interesting in that they concur on the existence of the frequency flicker noise and the wish for achieving a common definition for short and long term stability (even if the conference name only reflect the short-term stability intention).
The IEEE proceedings on Frequency Stability 1966 included a number of important papers including those of David Allan, James A. Barnes, L. S. Cutler and C. L. Searle and D. B. Leeson. These papers helped shape the field.
The classical M-sample variance of frequency was analysed by David Allan in along with a initial bias function. This paper tackles the issues of dead-time between measurements and analyses the case of M frequency samples (called N in the paper) and variance estimators. It provides the now standard α to µ mapping. It clearly builds on James Barnes work as detailed in his article in the same issue. The initial bias functions introduced assumes no dead-time, but the formulas presented includes dead-time calculations. The bias function assumes the use of the 2-sample variance as a base-case, since any other variants of M may be chosen and values may be transferred via the 2-sample variance to any other variance for of arbitrary M. Thus, the 2-sample variance was only implicitly used and not clearly stated as the preference even if the tools where provided. It however laid the foundation for using the 2-sample variance as the base case of comparison among other variants of the M-sample variance. The 2-sample variance case is a special case of the M-sample variance which produces an average of the frequency derivative.
The work on bias functions was significantly extended by James Barnes in in which the modern B1 and B2 bias functions was introduced. Curiously enough it refers to the M-sample variance as "Allan variance" while referencing to. With these modern bias functions full conversion among M-sample variance measures of variating M, T and τ values could used, by conversion through the 2-sample variance.
James Barnes and David Allan further extended the bias functions with the B3 function in to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead time in between.
The IEEE Technical Committee on Frequency and Time within the IEEE Group on Instrumentation & Measurements provided a summary of the field in 1970 published as NBS Technical Notice 394. This paper could be considered first in a line of more educational and practical papers aiding the fellow engineers in grasping the field. In this paper the 2-sample variance with T = τ is being the recommended measurement and it is referred to as Allan variance (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and to get comparable measurements, it is essentially the least common denominator with the aid of the bias functions B1 and B2.
An improved method for using sample statistics for frequency counters in frequency estimation or variance estimation was proposed by J.J. Snyder. The trick to get more effective degrees of freedom out of the available dataset was to use overlapping observation periods. This provides a square-root n improvement. It was included into the overlapping Allan variance estimator introduced in. The variable τ software processing was also included in. This development improved the classical Allan variance estimators as well as providing a direct inspiration going into the work on modified Allan variance
.
The confidence interval and degrees of freedom analysis, along with the established estimators was presented in.
The first meaningful summary is the NBS Technical Note 394 "Characterization of Frequency Stability". This is the product of the Technical Committee on Frequency and Time of the IEEE Group on Instrumentation & Measurement. It gives the first overview of the field, stating the problems, defining the basic supporting definitions and getting into Allan variance, the bias functions B1 and B2, the conversion of time-domain measures. This is useful as it is among the first references to tabulate the Allan variance for the five basic noise types.
A classical reference is the NBS Monograph 140 from 1974, which in chapter 8 has "Statistics of Time and Frequency Data Analysis". This is the extended variant of NBS Technical Note 394 and adds essentially in measurement techniques and practical processing of values.
An important addition will be the Properties of signal sources and measurement methods. It covers the effective use of data, confidence intervals, effective degree of freedom as well as introducing the overlapping Allan variance estimator. It is a highly recommended reading for those topics.
The IEEE standard 1139 Standard definitions of Physical Quantities for Fundamental Frequency and Time Metrology is beyond that of a standard a comprehensive reference and educational resource.
A modern book aimed towards telecommunication is Stefano Bregni "Synchronisation of Digital Telecommunication Networks". This summarises not only the field but also much of his research in the field up to that point. It aims to include both classical measures as well as telecommunication specific measures such as MTIE. It is a handy companion when looking at telecommunication standard related measurements.
The NIST Special Publication 1065 "Handbook of Frequency Stability Analysis" of W.J. Riley is a recommended reading for anyone wanting to pursue the field. It is rich of references and also covers a wide range of measures, biases and related functions that a modern analyst should have available. Further it describes the overall processing needed for a modern tool.
s, atomic clock
s and frequency-stabilized laser
s over a period of a second or more. Short term stability (under a second) is typically expressed as phase noise
. The Allan variance is also used to characterize the bias stability of gyroscopes, including fiber optic gyroscopes and MEMS
gyroscopes.
Clock
A clock is an instrument used to indicate, keep, and co-ordinate time. The word clock is derived ultimately from the Celtic words clagan and clocca meaning "bell". A silent instrument missing such a mechanism has traditionally been known as a timepiece...
s, oscillators and amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
s. It is named after David W. Allan. It is expressed mathematically as
The Allan deviation (ADEV) is the square root of Allan variance. It is also known as sigma-tau, and is expressed mathematically as
The M-sample variance is a measure of frequency stability using M samples, time T between measures and observation time
. M-sample is expressed asThe Allan variance is intended to estimate stability due to noise processes and not that of systematic errors or imperfections such as frequency drift or temperature effects. The Allan variance and Allan deviation describe frequency stability, i.e. the stability in frequency. See also the section entitled "Interpretation of value" below.
There are also different adaptations or alterations of Allan variance. Notably the modified Allan variance
Modified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
MAVAR or MVAR, the total variance, and the Hadamard variance. There also exist time stability variants such as time deviation
Time deviation
TDEV is a metric often used to determine an aspect of the quality of timing signals in telecommunication applications and is a statistical analysis of the phase stability of a signal over a given period of time...
TDEV or time variance
Time deviation
TDEV is a metric often used to determine an aspect of the quality of timing signals in telecommunication applications and is a statistical analysis of the phase stability of a signal over a given period of time...
TVAR. Allan variance and its variants have proved useful outside of the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, but a derivative will be.
The M-sample variance is of historic importance as well as important background but has essentially been replaced by its special case of 2-sample variance with
now being called Allan variance. It remains important since it allows dead timeDead time
For detection systems that record discrete events, such as particle and nuclear detectors, the dead time is the time after each event during which the system is not able to record another event....
in measurements and bias functions allows conversion into Allan variance values.
Background
When investigating the stability of crystal oscillatorCrystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...
s and atomic clock
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...
s it was found that they did not have a phase noise
Phase noise
Phase noise is the frequency domain representation of rapid, short-term, random fluctuations in the phase of a waveform, caused by time domain instabilities...
consisting only of white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
, but also of white frequency noise and flicker frequency noise. These noise forms become a challenge for traditional statistical tools such as standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
as the estimator will not converge. The noise is thus being said to be divergent. Efforts in analysing the stability provided both the theoretical analysis as well as practical measurements.
An important side-consequence of having these types of noises was that, since the various methods of measurements did not agree with each other, the key aspect of repeatability of a measurement could not be achieved. This limits the possibility to compare sources, specify a needed quality and be able to get it from a supplier, essentially all forms of scientific and commercial uses was limited to dedicated measurements which hopefully would capture the need for that application.
To address these problems, David Allan introduced the M-sample variance and (indirectly) the two-sample variance. While the two-sample variance did not completely resolve all noise forms, it provided means to separate noise-forms for time-series of phase or frequency measurements between two or more oscillators. Allan provided a method to convert between any M-sample variance to any N-sample variance via the common 2-sample variance, thus making all M-sample variances comparable. The conversion mechanism also proved that M-sample variance does not converge for large M, thus making them less useful. IEEE later identified the 2-sample variance as the preferred measure.
An early concern was related to time and frequency measurement instruments which had a dead time
Dead time
For detection systems that record discrete events, such as particle and nuclear detectors, the dead time is the time after each event during which the system is not able to record another event....
between measurements. Such a series of measurements did not form a continuous observation of the signal and thus introduced a systematic bias into the measurement. Great care was spent in estimating these biases. The introduction of zero dead time counters removed the need, but the bias analysis tools have proved useful.
Another early aspect of concern was related to how the bandwidth of the measurement instrument would influence the measurement, such that it needed to be noted. It was later found that by algorithmically changing the observation
, only low values would be affected while higher values would be unaffected. The change of is done by letting it be an integer multiple of the measurement timebase .The physics of crystal oscillator
Crystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...
s was analyzed by D. B. Leeson which is now referred to as the Leeson effect. The feedback in the oscillator will make the white noise
White noise
White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...
and flicker noise
Flicker noise
Flicker noise is a type of electronic noise with a 1/ƒ, or pink power density spectrum. It is therefore often referred to as 1/ƒ noise or pink noise, though these terms have wider definitions...
of the feedback amplifier and crystal become the power-law noises of
white frequency noise and flicker frequency noise respectively. These noise forms have the effect that the standard variance estimator does not converge when processing time error samples. This mechanics of the feedback oscillators was unknown when the work on oscillator stability started but was presented by Leeson at the same time as the statistical tools was made available by David W. Allan. For a more thorough presentation on the Leeson effect see modern phase noise literature.Interpretation of value
Allan variance is defined as one half of the timeTime
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
average of the squares of the differences between successive readings of the frequency deviation
Frequency deviation
Frequency deviation is used in FM radio to describe the maximum instantaneous difference between an FM modulated frequency and the nominal carrier frequency...
sampled over the sampling period. The Allan variance depends on the time period used between samples: therefore it is a function of the sample period, commonly denoted as τ, as well as the distribution being measured, and is displayed as a graph rather than a single number. A low Allan variance is a characteristic of a clock with good stability over the measured period.
Allan deviation is widely used for plots (conveniently in log-log
Log-log graph
In science and engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes...
format) and presentation of numbers. It is preferred as it gives the relative amplitude stability, allowing ease of comparison with other sources of errors.
An Allan deviation of 1.3×10−9 at observation time 1 s (i.e. τ = 1 s) should be interpreted as there being an instability in frequency between two observations a second apart with a relative root mean square
Root mean square
In mathematics, the root mean square , also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids...
(RMS) value of 1.3×10−9. For a 10-MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed then the time deviation
Time deviation
TDEV is a metric often used to determine an aspect of the quality of timing signals in telecommunication applications and is a statistical analysis of the phase stability of a signal over a given period of time...
variants should be consulted and used.
-sample variance
The -sample variance is defined (here in a modernized notation form) asor with average fractional frequency time series
where
is the number of frequency samples used in variance, is the time between each frequency sample and is the time-length of each frequency estimate.An important aspect is that
-sample variance model counter dead-time by letting the time be different from that of .Allan variance
The Allan variance is defined aswhich is conveniently expressed as
where
is the observation period, is the nth fractional frequency average over the observation time .The samples are taken with no dead-time between them, which is achieved by letting
Allan deviation
Just as with standard deviationStandard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
, the Allan deviation is defined as the square root of the Allan variance.
Oscillator model
The oscillator being analysed is assumed to follow the basic model of
The oscillator is assumed to have the nominal frequency of vn being the nominal number of cycles per second or Hertz (Hz), corresponding to the nominal angular frequency
as related in
Removing the nominal phase ramp the total phase can be divided as
Time function
The time function T(t) is the progress of time for the oscillator as defined in
Time error
The time error function x(t) is the difference between expected nominal time and expected normal time
For measured values a time error series TE(t) is defined from the reference time function TREF(t) as
Frequency function
The frequency function v(t) is the frequency over time defined as
Fractional frequency
The fractional frequency y(t) is the normalized delta from the nominal frequency vn, thusAverage fractional frequency
The average fractional frequency is defined aswhere the average is taken over observation time τ, the y(t) is the fractional frequency error at time t and τ is the observation time.
Since y(t) is the derivate of x(t) we can without loss of generality rewrite it as
Estimators
The definition is based on the statistical expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, integrating over infinite time. Real world situation does not allow for such time-series, in which case a statistical estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
needs to be used in its place. A number of different estimators will be presented and discussed.
Conventions
- The number of frequency samples in a fractional frequency series is denoted with M.
- The number of time error samples in a time error series is denoted with N.
The relation between the number of fractional frequency samples and time error series is fixed in the relationship
- For time error sample series, xi denotes the i;th sample of the continuous time function x(t) as given by
where T is the time between measurements. For Allan variance, the time being used has T set to the observation time τ.
The time error sample series let N denote the number of samples (x0 ...xN-1) in the series. The traditional convention uses index 1 through N.
- For average fractional frequency sample series,
denotes the ith sample of the average continuous fractional frequency function y(t) as given by
which gives
For the Allan variance assumption of T being τ it becomes
The average fractional frequency sample series let M denote the number of samples (
) in the series. The traditional convention uses index 1 through M.As a short-hand is average fractional frequency often written without the average bar over it. This is however formally incorrect as the fractional frequency and average fractional frequency is two different functions. A measurement instrument able to produce frequency estimates with no dead-time will actually deliver a frequency average time series which only needs to be converted into average fractional frequency and may then be used directly.
- It is further a convention to let τ denote the nominal time-difference between adjacent phase or frequency samples. A time series taken for one time-difference τ0 can be used to generate Allan variance for any τ being an integer multiple of τ0 in which case τ = nτ0 is being used, and n becomes a variable for the estimator.
- The time between measurements is denoted with T, which is the sum of observation time τ and dead-time.
Fixed τ estimators
A first simple estimator would be to directly translate the definition intoor for the time series
These formulas however only provides the calculation of τ = τ0 case, wishing any other τ and a new time-series needs to be provided.
Non-overlapped variable τ estimators
If taking the time-series and skipping past n − 1 samples a new (shorter) time-series would occur with τ0 as the time between the adjacent samples, for which the Allan variance could be calculated with the simple estimators. These could be modified to introduce the new variable n such that no new time-series would have to be generated, but rather the original time series could be reused for various values of n. The estimators becomesor for the time series
These estimators have a significant drawback in that they will drop a significant amount of sample data as only 1/n of the available samples is being used.
Overlapped variable τ estimators
A technique presented by J.J. Snyder provided an improved tool, as measurements was overlapped in n overlapped series out of the original series. The overlapping Allan variance estimator was introduced in. This can be shown to be equivalent to averaging the time or normalized frequency samples in blocks of n samples prior to processing. The resulting predictors becomesor for the time series
The overlapping estimators have far superior performance over the non-overlapping estimators as n rises and the time-series is of moderate length. The overlapped estimators have been accepted as the preferred Allan variance estimators in IEEE, ITU-T and ETSI standards for comparable measurements such as needed for telecommunication qualification.
Time stability estimators
The Allan variance and Allan deviation provides the frequency stability variance and deviation. The time stability variants can be provided by using frequency to time scaling from Allan variance to time varianceTime deviation
TDEV is a metric often used to determine an aspect of the quality of timing signals in telecommunication applications and is a statistical analysis of the phase stability of a signal over a given period of time...
and similarly for Allan deviation to time deviation
Time deviation
TDEV is a metric often used to determine an aspect of the quality of timing signals in telecommunication applications and is a statistical analysis of the phase stability of a signal over a given period of time...
Modified estimators
In order to address the inability to separate white phase modulation from flicker phase modulation using traditional Allan variance estimators an algorithmic filtering to reduce the bandwidth by n. This filtering provides a modification to the definition and estimators and is now identifies as a separate class of variance called modified Allan varianceModified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
. The modified Allan variance measure is a frequency stability measure, just as the Allan variance.
Other estimators
Further developments have produced improved estimation methods for the same stability measure, the variance/deviation of frequency, but these are known by separate names such as the Hadamard variance, modified Hadamard variance, the total variance, modified total variance and the Theo variance. These distinguish themselves in better use of statistics for improved confidence bounds or ability to handle linear frequency drift.Confidence intervals and equivalent degrees of freedom
Statistical estimators will calculate an estimated value on the sample series used. The estimates may deviate from the true value and the range of values which for some probability will contain the true value is referred to as the confidence intervalConfidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
. The confidence interval depends on the number of observations in the sample series, the dominant noise type, and the estimator being used. The width is also dependent on the statistical certainty for which the confidence interval values forms a bounded range, thus the statistical certainty that the true value is within that range of values. For variable-τ estimators, the τ0 multiple n is also a variable.
Confidence interval
The confidence intervalConfidence interval
In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...
can be established using chi-squared distribution by using the distribution of the sample variance :
where s2 is the sample variance of our estimate, σ2 is the true variance value, d.f. is the degrees of freedom for the estimator and χ2 is the degrees of freedom for a certain probability. For a 90% probability, covering the range from the 5% to the 95% range on the probability curve, the upper and lower limits can be found using the inequality:
which after rearrangement for the true variance becomes:
Effective degrees of freedom
The degrees of freedomDegrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
represents the number of free variables capable of contributing to the estimate. Depending on the estimator and noise type, the effective degrees of freedom varies. Estimator formulas depending on N and n has been empirically found to be:
| Noise type | degrees of freedom |
| white phase modulation (WPM) |  |
| flicker phase modulation (FPM) |  |
| white frequency modulation (WFM) |  |
| flicker frequency modulation (FFM) |  |
| random walk frequency modulation (RWFM) |  |
Power-law noise
The Allan variance will treat various power-law noise types differently, conveniently allowing them to be identified and their strength estimated. As a convention, the measurement system width (high corner frequency) is denoted fH.| Power-law noise type | Phase noise slope | Frequency noise slope | Power coefficient | Phase noise | Allan variance |
| white phase modulation (WPM) |  |
 |
 |
 |
 |
| flicker phase modulation (FPM) |  |
 |
 |
 |
 |
| white frequency modulation (WFM) |  |
 |
 |
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 |
| flicker frequency modulation (FFM) |  |
 |
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 |
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| random walk frequency modulation (RWFM) |  |
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 |
As found in and in modern forms.
The Allan variance is unable to distinguish between WPM and FPM, but is able to resolve the other power-law noise types. In order to distinguish WPM and FPM, the modified Allan variance
Modified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
need to be employed.
The above formulas assume that
and thus that the bandwidth of the observation time is much lower than the instruments bandwidth. When this condition is not met, all noise forms depend on the instruments bandwidth.
α-μ mapping
The detailed mapping of a phase modulation of the formwhere
or frequency modulation of the form
into the Allan variance of the form
can be significantly simplified by providing a mapping between α and μ. A mapping between α and Kα is also presented for convenience:
| α | β | μ | Kα |
| -2 | -4 | 1 |  |
| -1 | -3 | 0 |  |
| 0 | -2 | -1 |  |
| 1 | -1 | -2 |  |
| 2 | 0 | -2 |  |
The mapping is taken from.
Linear response
While Allan variance is intended to be used to distinguish noise forms, it will depend on some but not all linear responses to time. They are given in the table:| Linear effect | time response | frequency response | Allan variance | Allan deviation |
|---|---|---|---|---|
| phase offset |  |
 |
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 |
| frequency offset |  |
 |
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| linear drift |  |
 |
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 |
Thus, linear drift will contribute to output result. When measuring a real system, the linear drift or other drift mechanism may need to be estimated and removed from the time-series prior to calculating the Allan variance.
Time and frequency filter properties
In analysing the properties of Allan variance and friends, it has proven useful to consider the filter properties on the normalize frequency. Starting with the definition for Allan variance forwhere
Replacing the time series of
with the Fourier transformed variant the Allan variance can be expressed in the frequency domain asThus the transfer function for Allan variance is
Bias functions
The M-sample variance, and the defined special case Allan variance, will experience systematic bias depending on different number of samples M and different relationship between T and τ. In order address these biases the bias-functions B1 and B2 has been defined and allows for conversion between different M and T values.These bias functions is not sufficient for handling the bias resulting from concatenating M samples to the Mτ0 observation time over the MT0 with has the dead-time distributed among the M measurement blocks rather than in the end of the measurement. This rendered the need for the B3 bias.
The bias functions is evaluated for a particular µ value, so the α-µ mapping needs to be done for the dominant noise form as found using noise identification. Alternatively as proposed in and elaborated in the µ value of the dominant noise form may be inferred from the measurements using the bias functions.
B1 bias function
The B1 bias function relates the M-sample variance with the 2-sample variance (Allan variance), keeping the time between measurements T and time for each measurements τ constant, and is defined aswhere
The bias function becomes after analysis
B2 bias function
The B2 bias function relates the 2-sample variance for sample time T with the 2-sample variance (Allan variance), keeping the number of samples N = 2 and the observation time τ constant, and is definedwhere
The bias function becomes after analysis
B3 bias function
The B3 bias function relates the 2-sample variance for sample time MT0 and observation time Mτ0 with the 2-sample variance (Allan variance) and is defined aswhere
The B3 bias function is useful to adjust non-overlapping and overlapping variable τ estimator values based on dead-time measurements of observation time τ0 and time between observations T0 to normal dead-time estimates.
The bias function becomes after analysis (for the N = 2 case)
where
τ bias function
While formally not formulated, it has been indirectly inferred as a consequence of the α-µ mapping. When comparing two Allan variance measure for different τ assuming same dominant noise in the form of same µ coefficient, a bias can be defined asThe bias function becomes after analysis
Conversion between values
In order to convert from one set of measurements to another the B1, B2 and τ bias functions can be assembled. First the B1 function converts the (N1, T1, τ1) value into (2, T1, τ1), from which the B2 function converts into a (2, τ1, τ1) value, thus the Allan variance at τ1. The Allan variance measure can be converted using the τ bias function from τ1 to τ2, from which then the (2, T2, τ2) using B2 and then finally using B1 into the (N2, T2, τ2) variance. The complete conversion becomeswhere
Similarly, for concatenated measurements using M sections, the logical extension becomes
Measurement issues
When making measurements to calculate Allan variance or Allan deviation a number of issues may cause the measurements to degenerate. Covered here is the effects specific to Allan variance, where results would be biased.Measurement bandwidth limits
A measurement system is expected to have a bandwidth at or below that of the Nyquist rate as described within the Shannon–Hartley theoremShannon–Hartley theorem
In information theory, the Shannon–Hartley theorem tells the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise. It is an application of the noisy channel coding theorem to the archetypal case of a continuous-time...
. As can be seen in the power-law noise formulas, the white and flicker noise modulations both depends on the upper corner frequency
(these systems is assumed to be low-pass filtered only). Considering the frequency filter property it can be clearly seen that low-frequency noise has greater impact on the result. For relatively flat phase modulation noise types (e.g. WPM and FPM), the filtering has relevance, where as for noise types with greater slope the upper frequency limit becomes of less importance, assuming that the measurement system bandwidth is wide relative the as given byWhen this assumption is not met, the effective bandwidth
needs to be notated alongside the measurement. The interested should consult NBS TN394.If however one adjust the bandwidth of the estimator by using integer multiples of the sample time
then the system bandwidth impact can be reduced to insignificant levels. For telecommunication needs, such methods have been required in order to ensure comparability of measurements and allow some freedom for vendors to do different implementations. The ITU-T Rec. G.813 for the TDEV measurement.It can be recommended that the first
multiples be ignored such that the majority of the detected noise is well within the passband of the measurement systems bandwidth.Further developments on the Allan variance was performed to let the hardware bandwidth be reduced by software means. This development of a software bandwidth allowed for addressing the remaining noise and the method is now referred to modified Allan variance
Modified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
. This bandwidth reduction technique should not be confused with the enhanced variant of modified Allan variance
Modified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
which also changes a smoothing filter bandwidth.
Dead time in measurements
Many measurement instruments of time and frequency have the stages of arming time, time-base time, processing time and may then re-trigger the arming. The arming time is from the time the arming is triggered to when the start event occurs on the start channel. The time-base then ensures that minimum amount of time goes prior to accepting an event on the stop channel as the stop event. The number of events and time elapsed between the start event and stop event is recorded and presented during the processing time. When the processing occurs (also known as the dwell time) the instrument is usually unable to do another measurement. After the processing has occurred, an instrument in continuous mode triggers the arm circuit again. The time between the stop event and the following start event becomes dead timeDead time
For detection systems that record discrete events, such as particle and nuclear detectors, the dead time is the time after each event during which the system is not able to record another event....
during which the signal is not being observed. Such dead time introduces systematic measurement biases, which needs to be compensated for in order to get proper results. For such measurement systems will the time T denote the time between the adjacent start events (and thus measurements) while
denote the time-base length, i.e. the nominal length between the start and stop event of any measurement.Dead time effects on measurements have such an impact on the produced result that much study of the field have been done in order to quantify its properties properly. The introduction of zero dead-time counters removed the need for this analysis. A zero dead-time counter has the property that the stop-event of one measurement is also being used as the start-event of the following event. Such counters creates a series of event and time timestamp pairs, one for each channel spaced by the time-base. Such measurements have also proved useful in order forms of time-series analysis.
Measurements being performed with dead time can be corrected using the bias function B1, B2 and B3. Thus, dead time as such is not prohibiting the access to the Allan variance, but it makes it more problematic. The dead time must be known such that the time between samples T can be established.
Measurement length and effective use of samples
Studying the effect on the confidence intervals that the length N of the sample series have, and the effect of the variable τ parameter n the confidence intervals may become very large since the effective degree of freedom may become small for some combination of N and n for the dominant noise-form (for that τ).The effect may be that the estimated value may be much smaller or much greater than the real value, which may lead to false conclusions of the result.
It is recommended that the confidence interval is plotted along with the data, such that the reader of the plot is able to be aware of the statistical uncertainty of the values.
It is recommended that the length of the sample sequence, i.e. the number of samples N is kept high to ensure that confidence interval is small over the τ-range of interest.
It is recommended that the τ-range as swept by the τ0 multiplier n is limited in the upper end relative N such that the read of the plot is not being confused by highly unstable estimator values.
It is recommended that estimators providing better degrees of freedom values be used in replacement of the Allan variance estimators or as complementing them where they outperform the Allan variance estimators. Among those the Total variance and Theo variance estimators should be considered.
Dominant noise type
A large number of conversion constants, bias corrections and confidence intervals depends on the dominant noise type. For proper interpretation shall the dominant noise type for the particular τ of interest be identified through noise identification. Failing to identify the dominant noise type will produce biased values. Some of these biases may be of several order of magnitude, so it may be of large significance.Linear drift
Systematic effects on the signal is only partly cancelled. Phase and frequency offset is cancelled, but linear drift or other high degree forms of polynomial phase curves will not be cancelled and thus form a measurement limitation. Curve fitting and removal of systematic offset could be employed. Often removal of linear drift can be sufficient. Use of linear drift estimators such as the Hadamard variance could also be employed. A linear drift removal could be employed using a moment based estimator.Measurement instrument estimator bias
Traditional instruments provided only the measurement of single events or event pairs. The introduction of the improved statistical tool of overlapping measurements by J.J. Snyder allowed for much improved resolution in frequency readouts, breaking the traditional digits/time-base balance. While such methods is useful for their intended purpose, using such smoothed measurements for Allan variance calculations would give a false impression of high resolution, but for longer τ the effect is gradually removed and the lower τ region of the measurement has biased values. This bias is providing lower values than it should, so it is an overoptimistic (assuming that low numbers is what one wishes) bias reducing the usability of the measurement rather than improving it. Such smart algorithms can usually be disabled or otherwise circumvented by using time-stamp mode which is much preferred if available.Practical measurements
While several approaches to measurement of Allan variance can be devised, a simple example may illustrate how measurements can be performed.Measurement
All measurements of Allan variance will in effect be the comparison of two different clocks. Lets consider a reference clock and a device under test (DUT), and both having a common nominal frequency of 10 MHz. A time-interval counter is being used to measure the time between the rising edge of the reference (channel A) and the rising edge of the device under test.In order to provide evenly spaced measurements will the reference clock be divided down to form the measurement rate, triggering the time-interval counter (ARM input). This rate can be 1 Hz (using the 1 PPS
Pulse per second
A pulse per second is an electrical signal that very accurately repeats once per second . PPS signals are output by various types of precision clocks, including atomic clocks, radio clocks and some models of GPS receivers...
output of a reference clock) but other rates like 10 Hz and 100 Hz can also be used. The speed of which the time-interval counter can complete the measurement, output the result and prepare itself for the next arm will limit the trigger frequency.
A computer is then useful to record the series of time-differences being observed.
Post-processing
The recorded time-series require post-processing to unwrap the wrapped phase, such that a continuous phase error is being provided. If necessary should also logging and measurement mistakes be fixed. Drift estimation and drift removal should be performed, the drift mechanism needs to be identified and understood for the sources. Drift limitations in measurements can be severe, so letting the oscillators become stabilized by long enough time being powered on is necessary.The Allan variance can then be calculated using the estimators given, and for practical purposes the overlapping estimator should be used due to its superior use of data over the non-overlapping estimator. Other estimators such as Total or Theo variance estimators could also be used if bias corrections is applied such that they provide Allan variance compatible results.
To form the classical plots, the Allan deviation (square root of Allan variance) is plotted in log-log format against the observation interval tau.
Equipment and software
The time-interval counter is typically an off the shelf counter commercially available. Limiting factors involve single-shot resolution, trigger jitter, speed of measurements and stability of reference clock. The computer collection and post-processing can be done using existing commercial or public domain software. Highly advanced solutions exists which will provide measurement and computation in one box.Research history
The field of frequency stability has been studied for a long time, however it was found during the 1960s that there was a lack of coherent definitions. The NASA-IEEE Symposium on Short-Term Stability in 1964 was followed with the IEEE Proceedings publishing a special issue on Frequency Stability in its February 1966 issue.The NASA-IEEE Symposium on Short-Term Stability in November 1964 brings together many fields and uses of short and long term stability with papers from many different contributors. The articles and panel discussions is interesting in that they concur on the existence of the frequency flicker noise and the wish for achieving a common definition for short and long term stability (even if the conference name only reflect the short-term stability intention).
The IEEE proceedings on Frequency Stability 1966 included a number of important papers including those of David Allan, James A. Barnes, L. S. Cutler and C. L. Searle and D. B. Leeson. These papers helped shape the field.
The classical M-sample variance of frequency was analysed by David Allan in along with a initial bias function. This paper tackles the issues of dead-time between measurements and analyses the case of M frequency samples (called N in the paper) and variance estimators. It provides the now standard α to µ mapping. It clearly builds on James Barnes work as detailed in his article in the same issue. The initial bias functions introduced assumes no dead-time, but the formulas presented includes dead-time calculations. The bias function assumes the use of the 2-sample variance as a base-case, since any other variants of M may be chosen and values may be transferred via the 2-sample variance to any other variance for of arbitrary M. Thus, the 2-sample variance was only implicitly used and not clearly stated as the preference even if the tools where provided. It however laid the foundation for using the 2-sample variance as the base case of comparison among other variants of the M-sample variance. The 2-sample variance case is a special case of the M-sample variance which produces an average of the frequency derivative.
The work on bias functions was significantly extended by James Barnes in in which the modern B1 and B2 bias functions was introduced. Curiously enough it refers to the M-sample variance as "Allan variance" while referencing to. With these modern bias functions full conversion among M-sample variance measures of variating M, T and τ values could used, by conversion through the 2-sample variance.
James Barnes and David Allan further extended the bias functions with the B3 function in to handle the concatenated samples estimator bias. This was necessary to handle the new use of concatenated sample observations with dead time in between.
The IEEE Technical Committee on Frequency and Time within the IEEE Group on Instrumentation & Measurements provided a summary of the field in 1970 published as NBS Technical Notice 394. This paper could be considered first in a line of more educational and practical papers aiding the fellow engineers in grasping the field. In this paper the 2-sample variance with T = τ is being the recommended measurement and it is referred to as Allan variance (now without the quotes). The choice of such parametrisation allows good handling of some noise forms and to get comparable measurements, it is essentially the least common denominator with the aid of the bias functions B1 and B2.
An improved method for using sample statistics for frequency counters in frequency estimation or variance estimation was proposed by J.J. Snyder. The trick to get more effective degrees of freedom out of the available dataset was to use overlapping observation periods. This provides a square-root n improvement. It was included into the overlapping Allan variance estimator introduced in. The variable τ software processing was also included in. This development improved the classical Allan variance estimators as well as providing a direct inspiration going into the work on modified Allan variance
Modified Allan variance
The modified Allan variance , also known as mod σy2, is a variable bandwidth modified variant of Allan variance, a measurement of frequency stability in clocks, oscillators and amplifiers...
.
The confidence interval and degrees of freedom analysis, along with the established estimators was presented in.
Educational and practical resources
The field of time and frequency and its use of Allan variance, Allan deviation and friends is a field involving many aspects, for which both understanding of concepts and practical measurements and post-processing requires care and understanding. Thus, there is a realm of educational material stretching some 40 years available. Since these reflect the developments in the research of their time, they focus on teaching different aspect over time, in which case a survey of available resources may be a suitable way of finding the right resource.The first meaningful summary is the NBS Technical Note 394 "Characterization of Frequency Stability". This is the product of the Technical Committee on Frequency and Time of the IEEE Group on Instrumentation & Measurement. It gives the first overview of the field, stating the problems, defining the basic supporting definitions and getting into Allan variance, the bias functions B1 and B2, the conversion of time-domain measures. This is useful as it is among the first references to tabulate the Allan variance for the five basic noise types.
A classical reference is the NBS Monograph 140 from 1974, which in chapter 8 has "Statistics of Time and Frequency Data Analysis". This is the extended variant of NBS Technical Note 394 and adds essentially in measurement techniques and practical processing of values.
An important addition will be the Properties of signal sources and measurement methods. It covers the effective use of data, confidence intervals, effective degree of freedom as well as introducing the overlapping Allan variance estimator. It is a highly recommended reading for those topics.
The IEEE standard 1139 Standard definitions of Physical Quantities for Fundamental Frequency and Time Metrology is beyond that of a standard a comprehensive reference and educational resource.
A modern book aimed towards telecommunication is Stefano Bregni "Synchronisation of Digital Telecommunication Networks". This summarises not only the field but also much of his research in the field up to that point. It aims to include both classical measures as well as telecommunication specific measures such as MTIE. It is a handy companion when looking at telecommunication standard related measurements.
The NIST Special Publication 1065 "Handbook of Frequency Stability Analysis" of W.J. Riley is a recommended reading for anyone wanting to pursue the field. It is rich of references and also covers a wide range of measures, biases and related functions that a modern analyst should have available. Further it describes the overall processing needed for a modern tool.
Uses
Allan variance is used as a measure of frequency stability in a variety of precision oscillators, such as crystal oscillatorCrystal oscillator
A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a very precise frequency...
s, atomic clock
Atomic clock
An atomic clock is a clock that uses an electronic transition frequency in the microwave, optical, or ultraviolet region of the electromagnetic spectrum of atoms as a frequency standard for its timekeeping element...
s and frequency-stabilized laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
s over a period of a second or more. Short term stability (under a second) is typically expressed as phase noise
Phase noise
Phase noise is the frequency domain representation of rapid, short-term, random fluctuations in the phase of a waveform, caused by time domain instabilities...
. The Allan variance is also used to characterize the bias stability of gyroscopes, including fiber optic gyroscopes and MEMS
Microelectromechanical systems
Microelectromechanical systems is the technology of very small mechanical devices driven by electricity; it merges at the nano-scale into nanoelectromechanical systems and nanotechnology...
gyroscopes.
See also
- VarianceVarianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
- Semivariance
- VariogramVariogramIn spatial statistics the theoretical variogram 2\gamma is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z...
- MetrologyMetrologyMetrology is the science of measurement. Metrology includes all theoretical and practical aspects of measurement. The word comes from Greek μέτρον , "measure" + "λόγος" , amongst others meaning "speech, oration, discourse, quote, study, calculation, reason"...
- Network time protocolNetwork Time ProtocolThe Network Time Protocol is a protocol and software implementation for synchronizing the clocks of computer systems over packet-switched, variable-latency data networks. Originally designed by David L...
- Precision Time ProtocolPrecision Time ProtocolThe Precision Time Protocol is a protocol used to synchronize clocks throughout a computer network. On a local area network it achieves clock accuracy in the sub-microsecond range, making it suitable for measurement and control systems....
- SynchronizationSynchronizationSynchronization is timekeeping which requires the coordination of events to operate a system in unison. The familiar conductor of an orchestra serves to keep the orchestra in time....