Dual Kalman Filtering based GNSS Phase Tracking for Scintillation Mitigation
Friederike Fohlmeister, German Aerospace Center (DLR), Germany; Felix Antreich, Federal University of Ceara (UFC), Brazil; Josef A. Nossek, Federal University of Ceara (UFC), Brazil/Technical University Munich (TUM)
Global Navigation Satellite Systems (GNSS) are used in a wide variety of applications to provide a globally and permanently available position solution. However, in case of ionospheric scintillations, the signal of one or more satellites can be disturbed at the same time. Ionospheric scintillations are rapid fluctuations in the parameters of an electromagnetic wave, caused by ionospheric irregularities, i.e. an increase or depletion in the total electron content (TEC) of the ionosphere. These irregularities lead to a diffraction or refraction of the electro-magnetic wave traveling through the ionosphere, and consequently to rapid fluctuations in the amplitude and phase of the received signal. The contribution of ionospheric scintillations to the overall signal amplitude and phase are called scintillation amplitude and scintillation phase, respectively. On the one hand, the decreased signal amplitude, increased phase noise, and sudden phase drifts can degrade the accuracy of the pseudorange and carrier phase estimate, as well as leading to cycle slips or induce loss of lock. This can directly degrade the position solution or make it unavailable if multiple satellites are affected at the same time. On the other hand, the ionospheric effects on the signal can be used to gain information on the structure of the ionosphere from the measured signal perturbations. The GNSS receiver, in this case, acts as an ionospheric monitor that uses probes, i.e. signal amplitude and phase measurements, from different satellites at different points of the ionosphere. For ionospheric monitoring the contribution of the ionosphere to the signal amplitude and phase, i.e. scintillation amplitude and scintillation phase, have to be separated from the contribution of the dynamics of the line of sight (LOS) path. Scintillation phase and amplitude than allow for a characterization of the ionosphere during the measurement. Ionospheric monitoring is not only interesting from a scientific point of view but can be used to forecast and broadcast ionospheric threats to the GNSS system, such that the user is aware of a possible degradation of the position solution.
The GNSS signal is commonly tracked by a closed-loop phase locked-loop (PLL) and delay locked-loop (DLL). If the signal phase or amplitude shows deep fades due to ionospheric scintillations, the PLL may not be able to follow the signal phase and therefore go into a non-linear state or loses lock. In this case, the tracked signal phase and amplitude are not purely representative of the LOS and scintillation effect but also includes non-linear receiver effects. This affects the receiver's positioning as well as the amplitude and phase measurement and therefore ionospheric monitoring capabilities.
Different methods have been investigated to overcome this problem. In order to maintain lock under ionospheric scintillations with a classical PLL/ DLL tracking loop structure the PLL noise bandwidth can be adopted. Alternatively frequency locked loop (FLL) can be used to assist the PLL to maintain signal lock when the signal amplitude is small and a pure PLL structure would lose lock . Usually, not all visible satellites are affected by ionospheric scintillations. In this case vector tracking PLLs  can offer means to assist the tracking of signals affected by scintillations.
A totally different approach is applied in so-called open loop or maximum likelihood (ML) structures. In this case, the phase error estimate is not fed-back into the tracking loop, i.e. the loop is not “closed’. Instead the signal is processed in batches. In each batch the phase is estimated with the ML principle , i.e. searching for the signal parameters which maximize the probability density function for a given batch of received signal samples. The drawback of this approach is its increased computational complexity as the a priori information from the preceding batch is not fully used.
In order to come up with a computationally more effective tracking approach than ML methods with increased mathematical flexibility in comparison to the PLL the Kalman Filter can be used. In  a three-state Kalman filtering with constant gain was proposed for phase tracking. In  an extended Kalman filter for joint tracking of code and carrier for weak signals, such as in the case of amplitude scintillations has been proposed. However, all of the mentioned approaches in general estimate the overall phase and amplitude. This estimate is based on the assumption that the signal amplitude and phase only follow the known LOS dynamics despite the scintillation amplitude and phase follow a process with a different characteristic. In the case of severe scintillation the overall amplitude and phase is therefore not tracked reliably. This can lead to cycle slips and loss of lock and the scintillation amplitude and phase cannot be fully extracted from the overall measurement. For ionospheric monitoring however, especially the scintillation amplitude and phase are of interest. To overcome this problem the dynamics of the scintillation amplitude and phase has to be respected in the tracking loop
Motivated by the low-pass characteristic of the power spectral density (PSD) of scintillation amplitude and phase  proposes to use an autoregressive (AR) process to model scintillation amplitude and phase. Scintillation phase and amplitude are incorporated into the Kalman filter as additional state-space variables which follow an AR model. Not only can the scintillation phase and amplitude, in this case, be directly used for monitoring of the ionosphere, also the LOS tracking becomes more robust in case of severe ionospheric scintillations. However, this concept requires not only a model of the signal's LOS dynamics, which are well known for static receivers but also the parameters of the AR model of the scintillation phase and amplitude. As the model order and the paremeters are time-varying, these parameters have to be estimated together with the signal parameters. In  the AR parameters and the model order are estimated parallel to the phase error from the scintillation phase estimates with the Yule-Walker method.
The Yule-Walker method assumes that the scintillation amplitude and phase process is measured without measurement noise. In the case of high LOS dynamics or additional distortion of the signal amplitude and phase, this cannot be ensured. In the paper, we therefore propose to use a second Kalman filter to estimate the process parameters of the AR model for scintillation amplitude and phase from the possibly noisy scintillation amplitude and phase process. In this case, the scintillation phase and amplitude are estimated by the first Kalman filter. They are than used as measurements in the second Kalman filter to estimate the parameters of the respective AR process. This concept is referred as dual Kalman filter . We implement the first Kalman filter as an extended Kalman filter which takes the correlation of the GNSS received signal with the local carrier signal replica as a measurement. This non-linear measurement of the phase and frequency error is processed with an extended Kalman filter which uses the AR model as a process model for the scintillation phase and amplitude. The second filter uses the scintillation phase and amplitude as measurement inputs to fit the underlying signal dynamics to an AR model. The LOS and scintillation phase error estimates are finally fed back into the carrier signal replica generator by a linear quadratic Gaussian (LQG) control approach. This general control approach allows optimizing the tracking algorithm with respect to robustness and/or tracking accuracy.
The algorithm is tested with computer simulations. Using the Cornell Scintillation Model (CSM)  we produce a time series which is representative for the statistics of amplitude and phase scintillation. The performance of the approach is compared to the case of a priori known AR model parameters and an estimation of the AR model parameters with the Yule-Walker equations. The results show that the proposed dual Kalman filter approach can achieve the performance of a Kalman filter with a priori known AR model parameters under scintillation conditions and outperforms the Yule-Walker approach in the case of high dynamics.
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