Theoretical Upper and Lower Limits for Normalized Bandwidth of Digital Phase-Locked Loop in GNSS Receivers.

Determining the loop noise bandwidth and the coherent integration time is essential and important for the design of a reliable digital phase-locked loop (DPLL) in global navigation satellite system (GNSS) receivers. In general, designers set such parameters approximately by utilizing the well-known fact that the DPLL is stable if the normalized bandwidth, which is the product of the integration time and the noise bandwidth, is much less than one. However, actual limit points are not fixed at exactly one, and they vary with the loop filter order and implementation method. Furthermore, a lower limit on the normalized bandwidth may exist. This paper presents theoretical upper and lower limits for the normalized bandwidth of DPLL in GNSS receivers. The upper limit was obtained by examining the stability of DPLL with a special emphasis on the digital integration methods. The stability was investigated in terms of z-plane root loci with and without the consideration of the computational delay, which is a delay induced by the calculation of the discriminator and the loop filter. The lower limit was analyzed using the DPLL measurement error composed of the thermal noise, oscillator phase noise, and dynamic stress error. By utilizing the carrier-to-noise density ratio threshold which indicates the crossing point between the measurement error and the corresponding threshold, the lower limit of the normalized bandwidth is obtained.

Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver.

Direct Position Estimation (DPE) is a rather new Global Navigation Satellite System (GNSS) technique to estimate the user position, velocity and time (PVT) directly from correlation values of the received GNSS signal with receiver internal replica signals. If combined with Bayesian nonlinear filters-like particle filters-the method allows for coping with multi-modal probability distributions and avoids the linearization step to convert correlation values into pseudoranges. The measurement update equation (particle weight update) is derived from a standard GNSS signal model, but we show that it cannot be used directly in a receiver implementation. The numerical evaluation of the formulas needs to be carried out in a logarithmic scale including various normalizations. Furthermore, the residual user range errors (coming from orbit, satellite clock, multipath or ionospheric errors) need to be included from the very beginning in the stochastic signal model. With these modifications, sensible probability functions can be derived from the GNSS multi-correlator values. The occurrence of multipath yields a natural widening of the probability density function. The approach is demonstrated with simulated and real-world.