Simulations of the Hadamard Variance: Probability Distributions and Confidence Intervals.

Area of Science:

• Physics
• Electrical Engineering
• Signal Processing

Background:

• Power-law noise is a significant factor in clock and oscillator performance.
• Allan variance is commonly used for noise characterization but can overestimate intrinsic noise in the presence of frequency drift.
• Hadamard variance offers an alternative metric insensitive to frequency drift.

Purpose of the Study:

• To extend simulation methods for power-law noise in oscillators.
• To predict the Hadamard variance for common types of power-law noise.
• To introduce a matrix-based approach for calculating Hadamard variances and their modified forms.

Main Methods:

• Simulating power-law noise by Fourier transforming modified white phase noise spectra.
• Extending simulation methods to predict Hadamard variance.
• Introducing symmetric real matrices whose traces equal Hadamard variances.

Main Results:

• The simulation method successfully predicts Hadamard variance for power-law noise.
• A novel matrix method is presented for calculating overlapping and nonoverlapping Hadamard and modified Hadamard variances.
• The eigenvalues of these matrices provide probability distributions for variance estimation.

Conclusions:

• The extended simulation method accurately predicts Hadamard variance, overcoming limitations of Allan variance in drifting oscillators.
• The matrix-based approach offers a new framework for analyzing oscillator noise and estimating measurement confidence.
• This work provides enhanced tools for understanding and simulating noise in precision timing devices.