A fast algorithm for calculation of Thêo1

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Thêo1 is a frequency stability statistic that is similar to the Allan variance but can provide stability estimates at longer averaging factors and with higher confidence. However, the calculation of Thêo1 is significantly slower than that of the Allan variance, particularly for large data sets, due to a worse computational complexity. A faster algorithm for calculating the "all- τ " version of Thêo1 is developed by identifying certain repeated sums and removing them with a recurrence relation. The new algorithm has a reduced computational complexity, which is equal to that of the Allan variance. Computation time is reduced by orders of magnitude for many data sets. The new, faster algorithm does introduce an error due to accumulated floating-point errors in very large data sets. The error can be compensated for by increasing the numerical precision used at critical steps. The new algorithm can also be used to increase the speed of ThêoBr and ThêoH that are more sophisticated statistics derived from Thêo1.

Original languageEnglish
Pages (from-to)2187-2190
Number of pages4
JournalIEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control
Issue number10
Early online date21 May 2020
Publication statusPublished - 1 Oct 2020


  • frequency stability
  • software
  • stability analysis
  • Theo1


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