Response and Uncertainty of the Parabolic Variance PVAR to Non-Integer Exponents of the Power Law

TL;DR

This paper extends the application of Parabolic Variance (PVAR) to non-integer power law exponents, providing analytical response expressions and uncertainty estimates, thereby enhancing noise process detection in oscillator analysis.

Contribution

It introduces analytical formulas for PVAR's response to non-integer exponents and an approximate expression for its statistical uncertainty, broadening its applicability.

Findings

Derived the response of PVAR to non-integer power law exponents.

Provided an approximate expression for PVAR's statistical uncertainty.

Enhanced the detection capabilities of PVAR for various noise processes.

Abstract

Oscillator fluctuations are described as the phase or frequency noise spectrum, or in terms of a wavelet variance as a function of the measurement time. The spectrum is generally approximated by the `power law,' i.e., a Laurent polynomial with integer exponents of the frequency. This article extends the domain of application of PVAR, a wavelet variance which uses the linear regression on phase data to estimate the frequency, and called `parabolic' because such regression is equivalent to a parabolic-shaped weight function applied to frequency fluctuations. In turn, PVAR is relevant in that it improves on the widely-used Modified Allan variance (MVAR) enabling the detection of the same noise processes at the same confidence level in a shorter measurement time. More specifically, we provide (i) the analytical expression of the response of the PVAR to the frequency-noise spectrum in the general case of non-integer exponents of the frequency, and (ii) a useful approximate expression of the statistical uncertainty.