Probability density of the differential phase difference in applications to passive wireless surface acoustic wave sensing

The probability density function (pdf) is discussed of the differential phase difference (DPD) in the radio frequency (RF) pulse-burst perturbed by Gaussian noise at the coherent receiver. Statistical properties of the DPD are of importance for error estimation in coherent systems such as remote passive wireless surface acoustic wave (SAW) sensing with multiple differential phase measurement. The rigorous probability density of the DPD is derived and its particular functions, all having no closed forms, are given for different signal-to-noise ratios (SNRs) in the RF pulses. Employing the von Mises/Tikhonov distribution, an efficient approximation is proposed via the modified Bessel functions of the first kind and zeroth order. Engineering features and small errors of the approximation are demonstrated. Applications are given for the phase difference drift rate and error probability for the drift rate to exceed a threshold.