M-sample variance
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A M-sample variance is the Allan variance of M samples, T time between measures and [math]\displaystyle{ \tau }[/math] observation time .
References
2016
- (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/Allan_variance#/M-sample_variance Retrieved 2016-07-30
- 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
- [math]\displaystyle{ \sigma_y^2(\tau). \, }[/math]
- The Allan deviation (ADEV) is the square root of Allan variance. It is also known as sigma-tau, and is expressed mathematically as
- [math]\displaystyle{ \sigma_y(\tau).\, }[/math]
- The M-sample variance is a measure of frequency stability using M samples, time T between measures and observation time [math]\displaystyle{ \tau }[/math]. M-sample variance is expressed as
- [math]\displaystyle{ \sigma_y^2(M, T, \tau).\, }[/math]
- (...)The [math]\displaystyle{ M }[/math]-sample variance is defined (here in a modernized notation form) as
- [math]\displaystyle{ \sigma_y^2(M, T, \tau) = \frac{1}{M-1}\left\{\sum_{i=0}^{M-1}\left[\frac{x(iT+\tau )-x(iT)}{\tau}\right]^2 - \frac{1}{M}\left[\sum_{i=0}^{M-1}\frac{x(iT+\tau)-x(iT)}{\tau}\right]^2\right\} }[/math]
- where [math]\displaystyle{ x(t) }[/math] is the phase angle (in radians) measured at time [math]\displaystyle{ t }[/math], or with average fractional frequency time series
- [math]\displaystyle{ \sigma_y^2(M, T, \tau) = \frac{1}{M-1}\left\{\sum_{i=0}^{M-1}\bar{y}_i^2 - \frac{1}{M}\left[\sum_{i=0}^{M-1}\bar{y}_i\right]^2\right\} }[/math]
- where [math]\displaystyle{ M }[/math] is the number of frequency samples used in variance, [math]\displaystyle{ T }[/math] is the time between each frequency sample and [math]\displaystyle{ \tau }[/math] is the time-length of each frequency estimate.
- An important aspect is that [math]\displaystyle{ M }[/math]-sample variance model can include dead-time by letting the time [math]\displaystyle{ T }[/math] be different from that of[math]\displaystyle{ \tau }[/math].