A stochastic analysis of an anharmonic sensor phase response

The probability density function (pdf) of a modulo 2π phase response slope of an intrinsic anharmonic sensor of a crystal oscillator is studied in detail. It is noted that without an external drive, the sensor is excited by the oscillator noise floor with a signal-to-noise ratio (SNR) of around unity. The slope pdf is provided both in the rigorous integral form and in the T-distribution-based approximation. It is shown that the slope mean value is estimated to be zero with SNR =0. It then gradually tends toward actual value as SNR rises so that with SNR >2 the bias of slope estimates is almost negligible. With 0≤ SNR <0.7, the slope variance stays at a maximum and then asymptotically diminishes toward zero as the SNR rises. The importance of these studies resides in a shown fact that, practically, having SNR <2 in anharmonic sensors may result in substantial bias and variance for phase response slope mod 2π estimates.