Toward resonator anharmonic sensors for precision crystal oscillators: a Gaussian model

This paper addresses the statistical properties of eigenfrequency modes (anharmonics, for instance) of BAW crystal resonators in oscillators excited by noise or intended modulation and considered to be sensors of environmental impact. The modulated noisy model of a closed loop oscillator is studied for its amplitude-frequency and phase-frequency modulation characteristics caused by anharmonic influence. Pursuing the aim, we consider in detail amplitude, phase, and the time derivatives of an anharmonic sensor signal and present corresponding probability distributions for the different signal-to-noise ratios (SNR). Both statistical effects caused by noise and intended modulation are considered. Experimental results are also given.