The Parabolic variance (PVAR), a wavelet variance based on least-square fit

TL;DR

The paper introduces the Parabolic Variance (PVAR), a wavelet-based variance derived from least-square phase data fitting, combining advantages of AVAR and MVAR for improved long-term and short-term noise analysis.

Contribution

It presents the theoretical framework of PVAR, demonstrating its superior detection capabilities for various noise types compared to existing variances.

Findings

PVAR effectively detects weak noise processes at transition points.

PVAR outperforms MVAR in all tested cases.

PVAR is nearly as effective as AVAR for certain noise detections.

Abstract

This article introduces the Parabolic Variance (PVAR), a wavelet variance similar to the Allan variance, based on the Linear Regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the $\Omega$ frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and MVAR. PVAR is good for long-term analysis because the wavelet spans over $2 \tau$, the same of the AVAR wavelet; and good for short-term analysis because the response to white and flicker PM is $1/\tau^3$ and $1/\tau^2$, same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition - or corner - where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift.