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Updated: Dec 13, 2025

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A Fast Algorithm for Calculation of Thêo1.

Ben Lewis

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    Summary
    This summary is machine-generated.

    A new algorithm significantly speeds up the calculation of Thêo1 frequency stability statistics, making them as efficient as Allan variance. This advancement enables more reliable stability estimates for longer averaging times.

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    Area of Science:

    • Metrology
    • Signal Processing
    • Statistical Analysis

    Background:

    • Thêo1 is a frequency stability statistic offering higher confidence and longer averaging factors than Allan variance.
    • Traditional Thêo1 calculation is computationally intensive, limiting its use with large datasets.
    • Existing methods suffer from poor computational complexity compared to Allan variance.

    Purpose of the Study:

    • To develop a computationally efficient algorithm for calculating the "all- τ " version of Thêo1.
    • To reduce the computational complexity of Thêo1 to match that of the Allan variance.
    • To enable faster and more accessible frequency stability analysis.

    Main Methods:

    • Developed a recurrence relation to identify and remove repeated sums in Thêo1 calculations.
    • Reduced the computational complexity of the "all- τ " Thêo1 algorithm.
    • Investigated and addressed potential floating-point errors in large datasets.

    Main Results:

    • Achieved a computational complexity equal to the Allan variance for Thêo1.
    • Reduced computation time by orders of magnitude for numerous datasets.
    • Demonstrated that floating-point errors can be compensated for by increasing numerical precision.

    Conclusions:

    • The new algorithm provides a significant speedup for Thêo1 frequency stability calculations.
    • This advancement makes Thêo1 a more practical tool for analyzing frequency stability at longer averaging factors.
    • The algorithm's efficiency can be extended to related statistics like ThêoBr and ThêoH.