Conversion of 1/f fluctuations in crystal resonator within an inter resonance gap

This paper presents general relationships for transformation coefficients of BAW or SAW crystal resonator amplitude and phase fluctuations through the 1/f flicker noises of its motional and static equivalent parameters within the resonator inter resonance gap. Approximate functions of phase and amplitude power spectral densities are found based on Leeson's oscillator open loop model and are given with detailed consideration of Butler and Colpitts modes of operation with the assumption of full and zero inter noise correlation. It is also substantiated that a low-noise frequency region of crystal resonator operation exists in which the fluctuation influence of its motional inductance and capacity tend to zero in oscillators. Five examples are given as an illustration of a good agreement of the measured data with the prediction curves, giving a possibility of resonator power phase and amplitude spectral densities valuation at an arbitrary offset frequency from the carrier through the 1/f flicker noises of resonator parameters. Emphasis is laid in conclusion on the possible way of parameter spectral densities definition.     

Threshold-based identification of wireless SAW RFID-tags with pulse position encoding

In recent decades, there have been designed and developed passive surface acoustic wave radio frequency identification tags with time position encoding in groups of slots. For such tags identified when the received pulse-burst exceeds a threshold, we specify the generic identification (ID) error probability and optimize the threshold employing the Marcum Q-function of first order. As examples, we estimate the ID errors for the one-group 28-slot tag and Barker encoding. It is shown that, under the ideal conditions of equal SNRs (14.8 dB) in On-pulses and zeroth in Off-pulses, the ID error probability lesser 1% (one slip per 100 readings) can be guarantied with single reading for any reasonable number of slots. It becomes lesser 0.1% (1 slip per 1000 readings) with seven or lesser slots.     

Approximate estimates of limiting errors of passive wireless SAW sensing with DPM

This paper discusses approximate statistical estimates of limiting errors associated with single differential phase measurement of a time delay (phase difference) between two reflectors of the passive surface acoustic wave (SAW) sensor. The remote wireless measurement is provided at the ideal coherent receiver using the maximum likelihood function approach. Approximate estimates of the mean error, mean square error, estimate variance, and Cramér-Rao bound are derived along with the error probability to exceed a threshold in a wide range of signal-to-noise ratio (SNR) values. The von Mises/Tikhonov distribution is used as an approximation for the phase difference and differential phase diversity. Simulation of the random phase difference and limiting errors also is applied.     

Probability density of the phase of a random RF pulse in the presence of Gaussian noise

In this paper, we show that the probability density function (pdf) of the phase of a random nonstationary radio frequency (RF) impulse signal perturbed by Gaussian noise does not depend on the multiple time derivatives of the signal amplitude and phase at a given time instance and is the Bennett's pdf. Employing this revealing, we derive an alternative form of the conditional pdf of the phase representing it with the von Mises/Tikhonov pdf conditional on the envelope, signal-to-noise ratio (SNR), and signal phase. We expend this new form to the Fourier series and investigate for the locked and unlocked reference phases. The error probability for the phase to exceed a threshold is also analyzed.     

The modulational method of quartz crystal oscillator frequency stabilization

A new trend in self-contained (without the use of quantum discriminators) frequency stabilization of an oven controlled crystal oscillator (OCXO) and frequency standard is proposed and discussed. The method developing the trend is called a modulational method and is based on the use of the reference properties of crystal resonator natural bulk vibrations (double-frequency and multi-frequency oscillators are not used in this case). The concept is based on dynamic modulation characteristics of an oscillator, and basic relationships are found for their calculation. The construction principles of the frequency control systems are formulated substantiating mathematically the essence of the method. Basic ratios of modulating signals are determined, the solution of which shows only a slight influence of the modulation signal on the Allan variance and spectral density of an OCXO. The results of the method's practical use are considered. Their subject is the OCXO with the oven system adapted to the ambient temperature and crystal frequency standard with aging rate compensation.     

Efficient predictive estimator for holdover in GPS-based clock synchronization

This paper addresses an unbiased p-step predictive finite impulse response (FIR) filter of the local clock K-degree time interval error (TIE) polynomial model with applications to the global positioning system (GPS)-based clock synchronization. Generic coefficients are derived for a 2-parameter family of the polynomial filter gains. A generalization is provided for the p-step linear (ramp) gain allowing for close to optimal predictive filtering of the TIE. Basic holdover algorithms are discussed along with their most critical properties. Efficiency of the proposed filter in holdover is demonstrated by simulation and in real applications to GPS-based (sawtooth and sawtoothless) measurements of the TIE of a crystal clock.     

Optimal horizons for a one-parameter family of unbiased FIR filters

In this paper, we find the optimal horizons and sampling intervals, both in the sense of the minimum mean square error (MSE), for a one-parameter family of the discrete-time unbiased finite impulse response (FIR) filters. On a horizon of N_l points in the nearest past, the FIR and the model k-state are represented with the l-degree and m-degree polynomials, respectively. The noise-free state space model is observed in the presence of zero-mean noise of an arbitrary distribution and covariance. The approach is based on the following. The FIR filter produces an unbiased estimate if lm. In order to reduce the noise, N_l needs to be increased. The model fits the increased horizon with a higher degree polynomial, m>l. Minimization of the mean square error for m>l gives the optimal horizon and sampling interval. Justification is provided for the global positioning system (GPS)-based measurements of the first state of a local crystal clock provided in the presence of uniformly distributed sawtooth noise induced by the GPS timing receiver.     

A thinning algorithm for GPS-based unbiased FIR estimation of a clock TIE model

A thinning algorithm is proposed for real-time unbiased finite impulse response (FIR) estimation of the local clock time interval error (TIE) model (time error, fractional frequency offset, linear frequency drift rate, etc.) employing GPS-based sawtooth measurements. We show that the approach allows obtaining practically optimal estimates of the clock states, by large horizons (number of the points in the average). The algorithm is applied to the TIE measurements allowing for different time steps and averaging horizons for each of the clock states and compared to the three state Kalman filter. It is demonstrated that, in the presence of the sawtooth noise induced by the GPS receiver, the unbiased FIR estimates with thinning out fit the clock states better than the Kalman filter, in terms of the Allan deviation and precision time protocol deviation.     

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.