The Parabolic Variance (PVAR): A Wavelet Variance Based on the Least-Square Fit

• Published in: IEEE transactions on ultrasonics, ferroelectrics, and frequency control ferroelectrics, and frequency control, 2016-04, Vol.63 (4), p.611-623
• Main Authors: Vernotte, Francois, Lenczner, Michel, Bourgeois, Pierre-Yves, Rubiola, Enrico
• Format: Article
• Language: English
• Subjects: Acoustics; Astrophysics; Convolution; Frequency control; Frequency estimation; Kernel; Linear regression; Physics; Radiation detectors
• ISSN: 0885-3010; 1525-8955
• DOI: 10.1109/TUFFC.2015.2499325

This paper introduces the parabolic variance (PVAR), a wavelet variance similar to the Allan variance (AVAR), based on the linear regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the Ω frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and modified AVAR (MVAR). PVAR is good for long-term analysis because the wavelet spans over 2τ, the same as the AVAR wavelet, and good for short-term analysis because the response to white and flicker PM is 1/τ 3 and 1/τ 2 , the 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.