Editing Allan variance (section)

==Measurement issues==
When making measurements to calculate Allan variance or Allan deviation, a number of issues may cause the measurements to degenerate. Covered here are 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 theorem]]. As can be seen in the power-law noise formulas, the white and flicker noise modulations both depends on the upper corner frequency <math>f_H</math> (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, whereas for noise types with greater slope the upper frequency limit becomes of less importance, assuming that the measurement system bandwidth is wide relative the <math>\tau</math> as given by

:<math>\tau \gg \frac{1}{2\pi f_H}.</math>

When this assumption is not met, the effective bandwidth <math>f_H</math> needs to be notated alongside the measurement. The interested should consult NBS TN394.<ref name=NBSTN394/>

If, however, one adjust the bandwidth of the estimator by using integer multiples of the sample time <math>n\tau_0</math>, 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<ref name=ITUTG813>ITU-T Rec. G.813: [http://www.itu.int/rec/T-REC-G.813/recommendation.asp?lang=en&parent=T-REC-G.813-200303-I ''Timing characteristics of SDH equipment slave clock (SEC)''], ITU-T Rec. G.813 (03/2003).</ref> for the TDEV measurement.

It can be recommended that the first <math>\tau_0</math> 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 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.

===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 minimal 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 time]], 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 <math>\tau</math> 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 create 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 ''B''<sub>1</sub>, ''B''<sub>2</sub> and ''B''<sub>3</sub>. 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 interval|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|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 be plotted along with the data, such that the reader of the plot knows of the statistical uncertainty of the values.
* The length of the sample sequence (i.e. the number of samples ''N'') must be kept as high as possible to ensure that confidence interval is small over the ''τ'' range of interest.
* 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.
* The ''τ'' range as swept by the ''τ''<sub>0</sub> multiplier ''n'' is limited in the upper end relative ''N'', such that the reader of the plot may not be confused by highly unstable estimator values.

===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<ref name=Snyder1981/> allowed 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,<ref name="Rubiola2005">{{Cite journal|url=http://www.femto-st.fr/~rubiola/pdf-articles/journal/2005rsi-hi-res-freq-counters.pdf |doi=10.1063/1.1898203 |title=On the measurement of frequency and of its sample variance with high-resolution counters |year=2005 |last1=Rubiola |first1=Enrico |journal=Review of Scientific Instruments |volume=76 |issue=5 |pages=054703–054703–6 |arxiv=physics/0411227 |bibcode=2005RScI...76e4703R |s2cid=119062268 |url-status=dead |archive-url=https://web.archive.org/web/20110720220221/http://www.femto-st.fr/~rubiola/pdf-articles/journal/2005rsi-hi-res-freq-counters.pdf |archive-date=20 July 2011 }}</ref><ref name=Rubiola2005ifcs>Rubiola, Enrico: [http://www.femto-st.fr/~rubiola/pdf-articles/conference/2005-ifcs-counters.pdf ''On the measurement of frequency and of its sample variance with high-resolution counters''] {{webarchive |url=https://web.archive.org/web/20110720220233/http://www.femto-st.fr/~rubiola/pdf-articles/conference/2005-ifcs-counters.pdf |date=20 July 2011 }}, Proc. Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval Systems and Applications Meeting pp. 46–49, Vancouver, Canada, 29–31 August 2005.</ref><ref name=Rubiola2008cntpres>Rubiola, Enrico: [http://www.femto-st.fr/~rubiola/pdf-slides/2008T-femto-counters.pdf ''High-resolution frequency counters (extended version, 53 slides)''] {{webarchive |url=https://web.archive.org/web/20110720220251/http://www.femto-st.fr/~rubiola/pdf-slides/2008T-femto-counters.pdf |date=20 July 2011 }}, seminar given at the FEMTO-ST Institute, at the Université Henri Poincaré, and at the Jet Propulsion Laboratory, NASA-Caltech.</ref> 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.