A Fast Algorithm for Calculation of Thêo1

• Published in: IEEE transactions on ultrasonics, ferroelectrics, and frequency control Vol. 67; no. 10; pp. 2187 - 2190
• Main Author: Lewis, Ben
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Acoustics; Algorithms; Complexity; Complexity theory; Confidence; Datasets; Error compensation; Floating point arithmetic; Frequency control; Frequency stability; Heuristic algorithms; Mathematical analysis; noise; Numerical stability; software; stability analysis; Stability criteria; Thêo1; ThêoH; Variance
• ISSN: 0885-3010; 1525-8955; 1525-8955
• DOI: 10.1109/TUFFC.2020.2996313

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-<inline-formula> <tex-math notation="LaTeX">\tau </tex-math></inline-formula>" 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.