Fast Sub-Hz Phase Noise Estimation Using a Simple Approximation in Terms of Allan Variance

For a quasi-periodic signal, the stability of the signal across time and frequency scales can be quantified with the Allan variance or the phase error power spectral density (PSD). Whether a closed-form approximation for phase noise in terms of Allan variance exists has remained a longstanding open question. By making an assumption about the phase error, we develop a simple approximation for phase noise in terms of Allan variance. Numerical tests show that the approximation is accurate for phase error spectra that decrease with frequency, as is the case for most clock signals at time scales for which error is not dominated by thermal noise. Experimental tests show that the approximation can be used to estimate phase noise at low-frequency offsets with a relatively short-time Allan variance measurement.