Method of steepest descent - Revision history

en>Hydraton31: /* The case of multiple non-degenerate saddle points */ -Corrected spelling.

2014-12-30T06:13:54Z

The case of multiple non-degenerate saddle points: -Corrected spelling.

← Older revision		Revision as of 08:13, 30 December 2014
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	~~The '''modified Allan deviation (MDEV)''',<ref name=NISTSP1065/> also known~~ as ~~mod σ<sub>''y''</sub>(τ),~~ is the ~~deviation variant~~ of the ~~modified Allan variance~~.

	~~==Background==~~
	The ~~[[Allan variance]]~~ has a ~~drawback~~ in that it is ~~unable~~ to ~~separate~~ the ~~white phase modulation (WPM) from~~ the ~~flicker phase modulation (FPM)~~. ~~Looking that their response~~ to ~~[[Allan variance#Power~~-~~law noise\|Power~~-~~law noise]]~~ it is ~~clearly seen~~ that ~~WPM~~ and ~~FPM~~ has ~~almost~~ the ~~same response to tau, but WPM~~ is ~~linearly sensitive~~ to the ~~system bandwidth~~ ''f''<~~sub~~>~~''H''~~<~~/sub~~> ~~whereas FPM is only weakly dependent~~ on it. ~~Thus~~, ~~by varying~~ the ~~system bandwidth~~ the ~~WPM and FPM noise forms may be separated~~. ~~However, it~~ is ~~impractical~~ to ~~alter~~ the ~~hardware of~~ the ~~measurement system~~. ~~By post-processing the sample-series~~ and ~~implementing~~ a software ~~bandwidth~~ a ~~modified Allan variance measure can be given capable~~ of ~~resolving~~ the ~~noise forms.~~

	~~==Definition==~~
	~~The modified Allan variance is defined for~~ <~~ref name=FCS35.470-1981~~>~~David W~~. ~~Allan and~~ James A. ~~Barnes: ''A modified "Allan variance" with increased oscillator characterization ability''~~, ~~35th Annual Frequency Control Symposium~~, ~~1981<~~/~~ref> using time error samples as~~

	:<~~math~~>~~\operatorname{mod}\sigma_y^2(n\tau_0) = \frac{1}{2\tau^2}\left\langle \left[ \frac{1}{n}\sum_{i=0}^{n-1}x_{i+2n}-2x_{i+n}+x_i\right]^2 \right\rangle~~<~~/math~~>

	or with [~~[Allan variance#Average fractional frequency\|average fractional frequency]] time series~~

	:~~<math>\operatorname{mod}\sigma_y^2(n\tau_0) = \frac{1}{2}\left\langle \left[ \frac{1}{n}\sum_{i=0}^{n~~-~~1}\bar{y}_{i+n}-\bar{y}_i\right~~]~~^2 \right\rangle</math>~~

	~~where ''n''~~ is the ~~integer number~~ of ~~samples averaged over~~.

	~~==Estimators==~~
	The ~~modified Allan variance estimator~~ for ~~time error time series is~~

	:<~~math~~>~~\operatorname{mod}\sigma_y^2(n\tau_0) = \frac{1}{2n^4\tau_0^2(N-3n+1)}\sum_{j=0}^{N-3n} \left\{ \sum_{i=j}^{j+n-1} x_{i+2n}-2x_{i+n}+x_i\right\}^2~~<~~/math~~>

	~~or with~~ [~~[Allan variance#Average fractional frequency\|average fractional frequency]] time series~~

	:~~<math>\operatorname{mod}\sigma_y^2(n\tau_0) = \frac{1}{2n^4(M~~-~~3n+2)}\sum_{j=0}^{M~~-~~3n+1} \left\{ \sum_{i=j}^{j+n~~-~~1} \left( \sum_{k=i}^{i+n~~-~~1}\bar{y}_{k+n}~~-~~\bar{y}_k\right) \right\}^2</math>~~

	~~==References==~~
	~~<references~~/>

	~~[[Category:Clocks]]~~
	~~[[Category:Signal processing metrics]~~]

140.112.54.158: /* Complex Morse LemmaLemma 3.3.2 on page 113 in {{harvtxt|Fedoryuk|1987}} */

2014-01-24T10:03:29Z

Complex Morse LemmaLemma 3.3.2 on page 113 in {{harvtxt|Fedoryuk|1987}}

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en>McKay: /* The other cases */ typo

2012-08-23T04:38:54Z

The other cases: typo

New page

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