Least-square fit, Ω counters, and quadratic variance

This work is motivated by the wish to have the most precise measurement of a frequency ν and of the variance σ_y² of its fractional fluctuations in a given time τ, out of high-end general-purpose instruments. Thanks to the progress of digital electronics, new time-interval analyzers have been made available in the last few years. Such instruments measure the time stamp of the input events at high sampling speed (MS/s), and with high resolution (10-100 ps). We propose the linear regression as a means to estimate the frequency from time stamps of the input signal. The frequency counter based on the linear regression is called Ω counter. The linear regression is interpreted as a finite impulse response filter which takes the frequency samples as the input, and delivers the estimated frequency at the output. We derive the transfer function of such filter, which turns out to be parabolic shaped. As compared to the H and Λ counters, the Ω counter features better rejection of the background noise. We define the quadratic variance (QVAR), a wavelet variance similar to the Allan variance, and we derive its statistical properties. The QVAR is superior to the AVAR and MVAR in the rejection of the background noise.