Combined Effects of Frequency Quantization and Additive Input Noise in a First-order Digital PLL

AbstractA recent work by Gardner [Gardner, F.M., Frequency granularity in digital phase-locked loops, IEEE Trans. Commun., 44 (1996), 749758] on the subject of digital phase-locked loops (DPLLs) investigated, via simulation, the characteristics of the phase-jitter caused by frequency quantization in the numerically-controlled oscillator. Further works by Feely, Teplinsky et al [Feely, O., Rogers, A., and Teplinsky, A., Phase-jitter dynamics of digital phaselocked loops, IEEE Trans. Circuits and Systems, Part I: Fundamental Theory and Applications, 46 (1999), 545–558], [Feely, O., and Teplinsky, A., Phase-jitter dynamics of digital phase-locked loops: Part II, IEEE Trans. Circuits and Systems, Part I: Fundamental Theory and Applications., 47 (2000), 458–473] used the theory of nonlinear dynamics to provide a complete analytical explanation of this phase-jitter.This paper examines in detail the case where the input signal is embedded in additive noise, a scenario earlier investigated by Gardner where no satisfactory method of characterising the phase-jitter was found. Here, further numerical results for the 1-D DPLL are presented and it is shown analytically how the DPLL noise-jitter dynamics may be approximated by a noisy circle rotation map for reasonable levels of additive noise. The noise in this case is unique and highly nonlinear in nature and thus not amenable to traditional analysis. By considering the the probability flow over time, a time-dependent difference-delay equation is derived for the probability density function (PDF) of the phase-jitter. It is shown that this PDF reaches a steady-state and that this state is described by a non-local equation. The solutions of this equation are investigated, both numerically and analytically, and used to explain the interaction between the additive and quantization noise that was previously not understood

Laser Ranging Interferometry for Future Gravity Missions : Instrument Design, Link Acquisition and Data Calibration

The presented study aims to improve the design solution adopted for the Laser Ranging Instrument of the GRACE Follow-On mission in terms of instrument layout, algorithms for the laser link acquisition and techniques for mitigating the range measurement noise. The first part of this work describes viable layout solutions of a heterodyne interferometer employed for intra-satellite range metrology and the major noise contributions which degrade the overall accuracy of the instrument. Together with the optical layout of the instrument, novel design concepts of the instrumenta s subsystems are also analyzed and tested. Precisely, a phasemeter designed to autonomously acquire and track a heterodyne signal with low signal-to-noise ratio in a frequency band that spans from 1MHz to 25MHz is presented. Particular attention is also dedicated to the mathematical modeling of the steering mirror dynamics and to the enhancement of its pointing performance by means of feedforward control. In the second part of this work, solutions for autonomously acquiring a laser signal buried in noise are analyzed and put in relation with the boundary constraints of the acquisition problem. The acquisition algorithms presented and the robustness of their design is verified mainly using numerical simulations. Experimental tests have also been performed for validating the simulation hypothesis and verifying their compliancy to a realistic mission scenario. The last part of this work describes a calibration algorithm which has been developed for minimizing, during data post-processing, the noise due to the tilt-to-piston coupling which represents one of the highest contributors to the overall measurement noise