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The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. 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 variance 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 proven useful outside the scope of timekeeping and are a set of improved statistical tools to use whenever the noise processes are not unconditionally stable, thus a derivative exists. The general ''M''-sample variance remains important since it allows dead time in measurements and bias functions allows conversion into Allan variance values. Nevertheless, for most applications the special case of 2-sample, or "Allan variance" with is of greatest interest. ==Background== When investigating the stability of crystal oscillators and atomic clocks 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 said to be divergent. Early efforts in analysing the stability included both theoretical analysis and practical measurements. An important side-consequence of having these types of noise 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 and make meaningful specifications to require from suppliers. Essentially all forms of scientific and commercial uses were then 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 allow all types of noise to be distinguished, it provided a means to meaningfully separate many 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 oscillators was analyzed by D. B. Leeson〔 and the result is now referred to as Leeson's equation. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Allan variance」の詳細全文を読む スポンサード リンク
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