Resumen de Efficient time-variant synchronization in spread-spectrum navigation receivers
Ingmar Groh
Navigation receivers in systems like GNSS (global navigation satellite system) estimate their position and their velocity from the time of arrival (TOA) of several incoming signals. In this estimation, the multipath phenomenon is one of the main error sources. This phenomenon consists in the propagation along different paths of a transmitted wave, and produces a superposition of signal replicas at the receiver input. The main effect of multipath is a bias in the delay, phase, amplitude and Doppler shift estimates that translate into biases in the position and velocity estimates.
To mitigate multipath, navigation receivers resort to closed-loop or to open-loop approaches. The closed-loop approach is an attempt to keep the conventional operation of a navigation receiver, which is based on the delay-locked loop (DLL), but giving a multipath mitigation capability. And the open-loop approach is a direct approach in which the multipath parameters are directly estimated.
Among the closed-loop approaches there are fundamentally two kinds. The first mitigates biased delay estimates by a simple modification of the loop discriminator to enhance its mitigation capability, like in the narrow correlator. And the second assumes that there is some delay and complex amplitude estimates available and performs an interference cancelation, where the loop discriminator remains unchanged. On the contrary, all open-loop multipath mitigation approaches dispense with any loop estimates for the channel delays and the channel phasors. Thus, open-loop multipath mitigation approaches must resort to multidimensional high resolution estimators of the channel parameters.
Maximum likelihood (ML) estimators, or Bayesian estimators, such as the extended Kalman filter (EKF) or the particle filter (PF), are candidates for the multidimensional high resolution estimators.
However, the main drawback of ML and Bayesian estimators is their high computational complexity. The high computational complexity precludes a simple hardware implementation in mass-market receivers such as mobile handsets; for applications such as multipath mitigation, low-complexity algorithms are required. The closed-loop and the open-loop multipath mitigation algorithms have to cope with the complexity of the estimation and with the time-variant nature of the multipath channel. The closed-loop approach accounts for the variations fundamentally. And the open-loop approach handles the complex estimation of several signal replicas. Obviously, a combination of closed-loop and open-loop approaches needs to account for the complexity constraint and the time-variant channel properties at the same time. Thus, especially for time-variant channels, ML estimators are prohibitive, since the high complexity is even required for each time instant. Additionally, the independent ML multipath channel delay and channel phasor estimators in each time slot prevent an additional noise averaging over several time instants.
A suitable combination of open-loop and closed-loop approaches works as follows. Within one time slot, where the channel delays are approximately constant and where only the channel phasors alternate significantly, we use the open-loop approach to get low complexity high resolution multipath delay estimators. These low complexity high resolution multipath delay estimators enable the two closed-loop approaches that keep the conventional operation of a navigation receiver. In the successive time slot, the open-loop approach computes again a low complexity high resolution multipath delay estimator as input for the closed-loop approach. To minimize the complexity of multidimensional high resolution time-variant estimators, we present as an open-loop approach novel low complexity high resolution channel estimators. We modify two subspace based algorithms, the multiple signal classification (MUSIC) and the estimation of signal parameters via rotational invariant techniques (ESPRIT) for application to the task of TOA estimation. Both of our novel methods reuse existing receiver sub-systems, such as sliding correlator outputs corresponding to several chip-delayed matched filters (MFs) that have already been computed at the acquisition stage. We focus on the unitary ESPRIT method since it achieves a decomposed estimation in the delay, Doppler, and angle of arrival (AOA) domains and yields automatically paired delay, Doppler, and AOA estimates. Our simulation results show that unitary ESPRIT approximates the Cramer-Rao bound (CRB) quite closely, even though the computational complexity is far smaller in comparison to ML estimators. The decomposed estimation directions enable a much faster multidimensional acquisition compared to the two- or three-dimensional maximization of MUSIC spectra. Because of the time-variant channel delays that are typical for real environments for positioning applications, however, repeated computation of the corresponding signal subspaces is required. Thus, we provide both algorithms for the open-loop approach with a suitable projector tracking algorithm to account for the variation of the mobile channel over time and space.
The projector tracking algorithm allows the already small computational complexity of the two subspace methods to be reduced even further. The ESPRIT projector tracking variant performs an additional noise averaging in the time direction. We thereby obtain a considerably enhanced estimation accuracy at low signal to noise ratios (SNRs) with a reduced computational complexity.
Our combination of high resolution channel estimators, such as MUSIC and ESPRIT, and projector tracking yields besides the open-loop multipath mitigation approach also a closed-loop multipath mitigation approach. The high resolution time-variant multipath channel estimators enable the application of interference minimization or interference cancelation to eliminate the timevariant loop delay bias and the time-variant phasor bias. For the closed-loop multipath mitigation approach, our novel time-variant multipath mitigation techniques are based on the acquisition stage and hence permit the tracking complexity to be reduced. Our novel approach based on high resolution time-variant projector tracking needs one DLL only contrary to conventional closed-loop multipath mitigation approaches, where each time-variant channel path is tracked with a separate DLL. To evaluate the performance of the closed-loop multipath mitigation approach, we need to compute the jitter and the mean time to lose lock (MTLL) of the closed-loop multipath mitigation approaches. Standard methods for DLLs fail to provide analytical jitter and MTLL results; these figures are usually obtained from simulations. At high SNRs, however, the simulation time grows extremely rapidly. To address this high simulation complexity, we design a novel algorithm based on the Ornstein-Uhlenbeck (OU) stochastic differential equation (SDE) and the corresponding OU random processes. The main merit of our algorithm, besides its simplicity, is the joint analytical jitter and MTLL computation. We investigate the jitter and MTLL of closed-loop multipath mitigation approaches. Also, we demonstrate the relation between closed-loop multipath mitigation approaches with jitter and MTLL figures and show the influence of the multipath propagation on the jitter and MTLL figures. We reveal how the closed-loop multipath mitigation approaches that involve interference minimization or interference cancelation remove the multipath bias component of the jitter and increase or decrease theMTLL. Furthermore, our novel closed-loop multipath mitigation algorithms also yield valuable approaches for the jitter and MTLL of related multipath mitigation algorithms, such as the EKF or the PF.
