An Efficient Two-Fold Marginalized Bayesian Filter for Multipath Estimation in Satellite Navigation Receivers
© Bernhard Krach et al. 2010
Received: 27 March 2010
Accepted: 17 September 2010
Published: 21 September 2010
Multipath is today still one of the most critical problems in satellite navigation, in particular in urban environments, where the received navigation signals can be affected by blockage, shadowing, and multipath reception. Latest multipath mitigation algorithms are based on the concept of sequential Bayesian estimation and improve the receiver performance by exploiting the temporal constraints of the channel dynamics. In this paper, we specifically address the problem of estimating and adjusting the number of multipath replicas that is considered by the receiver algorithm. An efficient implementation via a two-fold marginalized Bayesian filter is presented, in which a particle filter, grid-based filters, and Kalman filters are suitably combined in order to mitigate the multipath channel by efficiently estimating its time-variant parameters in a track-before-detect fashion. Results based on an experimentally derived set of channel data corresponding to a typical urban propagation environment are used to confirm the benefit of our novel approach.
Within global navigation satellite systems (GNSS), such as the Global Positioning System (GPS) or the future European satellite navigation system Galileo, the user position is determined based upon the code division multiplex access (CDMA) navigation signals received from different satellites using the time-of-arrival method . A major error source for positioning comes from multipath, the reception of additional signal replicas due to reflections caused by the receiver environment. The reception of multipath introduces a bias into the time-delay estimate of the delay-lock loop (DLL) of a conventional navigation receiver, which finally leads to a bias in the receiver's position estimate. Multipath is today still one of the most critical problems in GNSS, as the error occurs as a result of the local environment and can not be corrected through the use of correction data, which is provided by reference receiver stations or networks.
The advances in the development of signal processing techniques for multipath mitigation have led to a continual improvement of performance. Basically, two major approaches can be distinguished. Firstly, the class of techniques that actually mitigate the effect of multipath by modifications of the antenna pattern (either by means of hardware design or with signal processing techniques) or by aligning the more or less traditional receiver components (e.g., the early/late correlator). Secondly, the class of multipath estimation techniques, which treat multipath (in particular the delay of the paths) as something to be estimated from the received signal so that its effects can be trivially removed at a later processing stage. Most of the conventional mitigation techniques in some way align the discriminator/timing error detector of the DLL to the signal received in the multipath environment. Well-known examples of this category are, amongst others, the Narrow Correlator , the Strobe Correlator , the Gated Correlator , or the Pulse Aperture Correlator .
For the estimation techniques, static and dynamic approaches can be distinguished, according to the underlying assumption of the channel dynamics. Examples for static multipath estimation are those belonging to the family of maximum likelihood (ML) estimators, where the probably best-known technique is the multipath estimating delay-lock loop (MEDLL) . In the ML approach, the signal parameters that maximize the probability of the received signal are determined. For this purpose, different maximization strategies exist, which basically characterize the different approaches. Most of these maximization algorithms are based on iterative techniques such as the Space-Alternating Generalized Expectation-Maximization algorithm (SAGE) [7, 8] and Newton-type methods. Newton-type methods have been considered with analytical  and numerical  expressions for the gradient and Hessian terms. Further ML algorithms have been reported in [11, 12].
During the last years, sequential estimation algorithms in the form of Bayesian filters [13–16] have gained some attention in the field of multipath mitigation. These algorithms exploit prior knowledge about the temporal channel statistics through the use of statistical channel models, which allows one to improve the multipath performance of the receiver. Bayesian filters for estimation of time-varying synchronization parameters in spread spectrum systems have already been suggested in the field of communications using the extended Kalman filter  as well as the sequential Monte Carlo approach [18, 19]. For navigation systems, an estimator based on sequential importance sampling (SIS) methods (particle filtering) was proposed in , which was shown to successfully mitigate multipath in a static channel scenario. An adaptation to dynamic multipath channels capable of coping with a time-variant number of multipath replicas was presented in . To reduce the complexity of these approaches, it was proposed in  to employ reduced complexity methods for the computation of the likelihood function, which previously have been considered for ML estimation . To improve the efficiency of the particle filter (PF) approach, a Rao-Blackwellized/marginalized filter was presented in , where the signal amplitudes are efficiently estimated via Kalman filters and where a novel proposal density for the particle filter based on a Gaussian approximation of the likelihood function was introduced. Furthermore,  includes a comprehensive analysis of the performance of various other Bayesian filters, and also the corresponding posterior Cramer-Rao bound (PCRB) is derived.
We believe that a key for successful application of the Bayesian approach in the future is to determine correctly the number of actually received replicas, which is unknown in practice. It is well known for the signal parameter estimation approaches that it is crucial to properly adjust the order of the employed signal model, since an improper number of degrees of freedom in the assumed model may lead to a heavy performance degradation. In previous work, however, often a known number of received replicas is assumed, and the problem of how to determine this number is not addressed . To tackle this problem, we introduce in this paper a further structuring of the Bayesian approach by means of a two-fold marginalized Bayesian filter (TFMBF). The filter operates in line with filters that were presented previously, but is capable of simultaneously estimating all possible system models in terms of the number of received multipath replicas along with their respective probabilities. We achieve this by introducing an intermediate step of marginalization, which estimates the number of impinging replicas and their parameters in a track-before-detect (TBD) fashion [25, 26].
The paper is organized as follows: first, the Bayesian approach is reviewed, and the underlying signal and dynamic models are introduced. After that, we address the implementation of our two-fold marginalized filter. Subsequently, we present results based upon a set of experimentally derived realistic dynamic channel data, which corresponds to a typical satellite-to-user propagation channel in urban environments . Finally, we conclude the paper by a discussion of our findings.
2. The Sequential Bayesian Approach
2.1. The Sequential Bayesian Framework
The goal is to determine the a posteriori probability density function (PDF) of every possible channel characterization given all channel observations: , in which represents the characterization of the hidden channel state. Once the a posteriori PDF is evaluated, either that channel configuration that maximizes it can be determined—the so called maximum a posteriori (MAP) estimate, or the expectation can be chosen—equivalent to the minimum mean square error (MMSE) estimate.
The noise affecting successive channel outputs is independent of the past noise values, so each channel observation depends only on the present channel state.
Future channel parameters, given the present state of the channel and all its past states, depend only on the present channel state and not on any past states.
2.2. Multipath Propagation and Receiver Requirements
High noise level: the power level of the received signal is usually more than 25?dB below the thermal noise, thus the algorithm has to be robust against false detection of line-of-sight (LOS) signals.
Low bandwidth: since in particular those echoes which arrive within a chip period of the CDMA signal cause the most heavy errors, the signal bandwidth is relatively low with respect to the desired timing resolution.
Multiple paths: multiple paths may impinge simultaneously at the receiver, so the algorithm must properly determine which of them is the actual LOS path.
LOS blockage: the algorithm shall ensure that tracking of the LOS path is maintained also during periods where it is heavily attenuated.
Close-in echoes: tracking of closely spaced signals with the proper multiple path signal model may lead to increased mean square errors compared to the case when tracking is based on a single-path model .
Track-before-detect: since it is difficult to declare distinct detections of multipath replicas due to the high noise and the low signal bandwidth, the algorithm will consider the echo detection in a probabilistic fashion. The track-before-detect approach  is suitable for this purpose, since for any echo both hypotheses (echo present/echo not present) are estimated simultaneously in a probabilistic sense.
These requirements are challenging, since weak signals need to be detected and tracked properly in a noisy environment, with echoes very close to the LOS signal.
3. Channel Model
3.1. Multipath Channel Signal Model
where refers to the variance of the elements within the noise vector . The purpose of the likelihood function is to quantify the conditional probability of the received signal conditioned on the unknown signal (specifically the channel parameters , , and ).
3.2. Markovian Channel Process Model
The channel is totally characterized by a LOS path (index ) and at most echoes.
Each path has complex amplitude and delay , where echoes are constrained to have delay , , to reflect that multipath replicas are physically constrained to arrive later at the receiver than the LOS path.
- (iii)The delay of each path follows the process(7)
with noise , , where is the same value for all indices .
- (iv)Each parameter that specifies the rate of the change of the path delay follows its own process(8)
with noise , , in which has the same value for all indices .
- (v)Each echo is either "on" or "off", as defined by the channel parameter , where , follows a simple two-state Markov process with a-symmetric crossover and same-state probabilities(9)
The LOS component is always present, and consequently for all .
- (vii)Newly appearing echoes ( with ) are initialized with(10)
with noise , and the characteristic constant (cf. ).
Blockage and shadowing of the LOS signal is considered through variations of the LOS amplitude .
- (ix)Following , the complex amplitudes depend on the previous amplitudes through(11)
with complex noise and the carrier frequency . Thus, the rate of change in the delay affects the evolution of the complex amplitude in a statistical manner in order to consider the physical relationships between phase, Doppler-frequency, and time delay adequately.
4. Estimator Implementation
4.1. Estimation of Amplitudes
The notation above indicates that dimension and values of the respective matrices and vectors correspond to the active paths as given by . We assume in our implementation that (21) is still a circular symmetric complex normal PDF , which is enforced by the approximation via (23).
The value of follows directly from (6).
4.2. Estimation of Path Activity
A proof for (32) can be found in .
4.3. Estimation of Path Delays
where each particle with index μ has a state and has a weight . Due to the marginalization, each particle carries in addition a grid-based filter, in which for each of the discrete states a Kalman filter is associated to the particle, resulting thus in Kalman filters per particle (see Figure 1).
The particle filter approach allows us to enforce the nonlinear constraint , in an easy way: when drawing new realizations of according to our proposal density via (7), we reinitialize and according to (10) in case . In our implementation of the particle filter, we apply resampling at every time instant. To tackle this potential bottleneck, advanced resampling strategies may be applied [33, 34].
5. Algorithm Assessment
5.2. Model Matching
Simulation and algorithm parameters.
Complexity of TFMBF depending on .
5.5. Filter Behavior
In this paper, we have introduced a novel two-fold marginalized Bayesian filter for multipath mitigation in satellite navigation receivers. Our approach allows us to exploit the constrained channel dynamics within a typical satellite-to-user propagation scenario in an urban environment. We have proposed an efficient implementation of the filter by applying the concept of marginalization, where we proposed to estimate impinging multipath replicas in a typical track-before-detect approach. Our approach is able to adapt to the channel dynamics and favors implicitly the most likely channel configuration for a given sequence of channel observations. This has been shown to be of particular benefit in case the LOS path is shadowed or blocked, since unlike other approaches, the presented filter does not synchronize on powerful replicas during such periods. We have shown that our approach requires a significantly reduced number of particles compared to previous work, which is achieved as a result of the implicit use of phase information. Our results for a real urban environment show that our approach is practically viable and confirm its benefits. They also provide insights on how many simultaneous multipath replicas a future Bayesian navigation receiver should consider. Our findings reveal that the LOS tracking performance of our Bayesian filter tends to saturate rapidly when increasing of the number of simultaneously detectable multipath replicas.
The authors would kindly thank the anonymous reviewers for their most valuable suggestions and comments.
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