1. Introduction
During recent years, the use of Global Positioning Systems (GPSs) has remarkably increased, thus increasing the demand for accurate navigation and timing services. These requirements have led to the fact that a special role is assigned to research in the field of GPS receiver technology, facing new challenges such as the operation of GPS receivers in a high-noise environment, e.g., in urban, forest and indoor areas.
In conventional GPS receivers, digital processing is based on the channel structure. Each channel is designed to process data from a single GPS satellite, including acquisition, tracking, demodulation of navigation data, and solving the navigation solution.
In traditional GPS signal tracking approaches, there are three kinds of tracking loop: phase tracking loop (PLL), frequency lock loop (FLL), and delay lock loop (DLL). These loops estimate the following radio navigation parameters (RNPs): carrier phase, Doppler frequency, and code phase, respectively [
1,
2,
3,
4]. Since the above approaches use maximum-likelihood estimation, the accuracy of the RNP estimate is low [
2,
5]. In recent years, a different approach to tracking GPS signals based on Kalman filtering has been investigated. It is known that a linear Kalman filter (LKF) is the optimal solution for the Bayesian problem [
6]. In an LKF-based tracking system, the optimal tracking filter, i.e., the LKF, replaces the tracking loop filter. In this case, the outputs of the loop discriminators are used as measurements. According to the literature, the KF-based tracking system shows superior tracking performance. The authors of [
7] proposed an explicit tuning investigation and validation of a full KF-based tracking loop in GNSS receivers. In [
8], a multifrequency carrier tracking architecture based on the LKF was proposed to improve the robustness and accuracy of carrier tracking loop in frequency-selective fading environments. This architecture combines a traditional single frequency-independent tracking mode, which only uses measurements from a single carrier, a multifrequency joint tracking mode, which linearly combines measurements from multiple carriers, and a multifrequency optimal tracking mode, which incorporates signal strength in multicarrier combinations via a generalized LKF. In ref. [
9], an approach was introduced for designing a system of automated positioning of the engineer reconnaissance devices under high-dynamic-reception conditions with a weak GPS signal. Accordingly, a two-stage carrier tracking system was proposed, which involved both coarse and fine tracking. The first stage was implemented through the use of a conventional scheme, where the PLL and FLL were combined to provide a rough estimate of both the carrier phase and the Doppler frequencies of the received signal. The fine tracking was implemented using LKF, starting after successful bit synchronization. It was demonstrated that the proposed method improved the tracking performance and prevented the bit synchronization loss while simultaneously improving the Doppler frequency measurement accuracy under highly dynamic conditions.
However, the LKF approach assumes that the posteriori probability density at the outputs of discriminators is Gaussian. One disadvantage of this design is that the measurement noise at the discriminator output is not always white Gaussian. This follows from the fact that the use of nonlinear discriminators leads to loss of the white Gaussian property at low values of signal-to-noise ratio [
10]. In other words, in the LKF algorithm, the phase estimated accuracy is limited by the nonlinearity of the phase discriminator. This reduces the performance of the LKF when receiving weak signals. In this case, other approaches have been considered that use several different strategies for approaching the optimal solution. These approaches are suboptimal and include the extended Kalman filter (EKF) and unscented Kalman filter (UKF) [
11,
12,
13,
14,
15,
16,
17,
18,
19]. In this variant, a suboptimal nonlinear tracking filter (EKF or UKF) replaces both the discriminator and the tracking loop filter. In this case, the measurements are the quadrature components of the signals at the outputs of the nonlinear correlators.
The EKF uses the linearization of a nonlinear system by Taylor series expansion. An EKF-based tracking approach was proposed in [
11] to address the nonlinearity issue of the phase discriminator in LKF-based tracking system. In [
12], a practical implementation and performance assessment of an EKF-based signal tracking loop were presented. In [
13,
14], an EKF algorithm was proposed to track highly dynamic L1 and L2 GPS signals.
However, the same problem exists in the above algorithms, such as loss of optimality, accuracy reduction, and filter divergence. That is, there is a bias of estimates and errors in the calculation of covariance matrices [
6]. An approach [
15] based on the extended strong tracking filter (ESTF) was developed to improve the carrier tracking estimation instead of the traditional EKF approach. However, it was concluded that the class of strong tracking filters such as the EKF will suffer the kinds of problems induced by the subjective fading factor.
The UKF is one of the modifications of the sigma-point Kalman filter (SPKF) [
6]. That is, the UKF algorithm is based on a set of symmetrically distributed sample points (so-called sigma points), which are used to determine the mean and the covariance [
6]. This algorithm gives a more correct solution to nonlinear problems as compared with the EKF and forms suboptimal estimates of the state vector according to the criterion of the minimum mean-square error of the estimate. The authors of [
16,
17] proposed an SPKF algorithm in the problem of GPS signal parameter estimation in coherent and noncoherent tracking modes in spacecraft autonomous navigation equipment. The obtained results suggested that the SPKF algorithm could form RNP estimates with appropriate accuracy under the conditions of space consumer functioning. The results also demonstrated that the accuracy of the noncoherent tracking mode was worse than that of the coherent tracking system. In [
18], a new adaptive unscented Kalman filter (AUKF) was proposed to estimate the RNP of a GPS signal tracking system. The experimental results showed that the proposed AUKF-based method improved the GPS tracking margin by approximately 8 and 3 dB as compared to the conventional algorithm and LKF-based tracking, respectively. The authors of [
19] described an additive strong tracking filter using the UKF tracking algorithm for a GPS vector tracking loop. It should be noted that, from a practical point of view, the vector tracking approach requires large computational costs compared to the scalar tracking approach.
However, implementation of the UKF algorithm requires the determination of several coefficients, which are called scaling parameters (α, β, and κ) [
6]. It should be noted that not much research effort has been spent on determining if there is a global optimal setting of the UKF scaling parameters [
20]. It must also be noted that the choice of the scaling parameter (α) noticeably affects the result of the UKF algorithm. Traditionally, the choice of this parameter is left to the developer. If we follow the recommendations of the authors [
6], then α should be chosen as a value other than 0, but as small as possible (α ≤ 0.01). In this paper, the value of this scale parameter was empirically selected as 0.7. With this choice of parameter α, it was possible to improve the quality characteristics of the UKF-tracking based system. This improvement occurred due to a wider spread of sigma points, which led to better consideration of nonlinearities in observations. However, it took several attempts.
Thus, due to the tuning parameters, the rounding errors of numerical calculation for UKF may destroy the non-negative nature and asymmetry of the covariance matrix; thus, the convergence rate of the UKF approach is slow, and the system may also be unstable. Therefore, in order to avoid the influence of scaling parameters on the quality of the UKF algorithm, it is necessary to find other approaches to building a high-quality tracking system instead of UKF.
Recently, other approaches based on statistical processing theory have been proposed to deal with nonlinear systems. A typical example of these novel techniques is the particle filter (PF). PF is a sequential Monte Carlo method based on the Bayesian estimation principle. The PF uses a series of weighted random samples, which are also called particles, to approximate the posterior probability density, so that many kinds of statistical estimates such as the mean and variance can be computed, which makes it capable of dealing with any nonlinear models [
6]. In [
21,
22], PF-based carrier tracking approaches were proposed for GNSS receivers. However, the main drawback of the PF approach is the lack of computational efficiency [
6].
To overcome the problems of low UKF quality and high PF complicity, this paper proposes a new approach to the nonlinear tracking system, which is based on the Gauss–Hermite Kalman filter (GHKF). This approach, which belongs to the family of SPKF, ensures an additional justification of the UKF development and offers further options to improve the accuracy of the approximation. Furthermore, GHKF can provide estimation accuracy comparable to that of a much more computationally intensive simulation-based PF [
23]. As a direct quadrature formula, the proposed filter does not need the careful selection of tuning parameters like the UKF [
6].
The proposed tracking system was used to estimate the following RNPs of the received GPS signal: amplitude , carrier phase , Doppler frequency , and its change rate .
In the present paper, RNP estimates were compared to those of the EKF- and UKF-based tracking approaches. It should be noted that, in the available literature, there is no performance comparison between nonlinear KF-based tracking loop architectures operating under the same conditions.
The remainder of this paper is organized as follows: in
Section 2, the problem description is presented. Then,
Section 3 describes the problem solution. In
Section 4, the proposed GHKF-based tracking architecture is described in detail. In
Section 5, simulations are performed to evaluate the tracking performance and weak dynamic signal performance. Lastly, conclusions are presented in
Section 6.
2. Problem Description
The development of a tracking system based on the Bayesian theory of nonlinear filtering involves setting mathematical models of states and measurements.
Figure 1 shows a block diagram of the proposed tracking system model in one channel of the receiver.
This tracking system model can be described as a discrete linear state model with additive white Gaussian noise (AWGN) and a discrete nonlinear measurement model with AWGN [
6,
13].
where
is the state vector of the RNP of the GPS signal,
is the state transition matrix,
is a Gaussian distributed process noise with a spectral power density
and a covariance matrix
Q,
T is the coherent integration time,
is the measurement vector at the
moment,
hk is a nonlinear function of measurements, and
is a Gaussian-distributed measurement noise with a Gaussian probability density function
. The nonlinear function of measurements
h takes the following form [
2]:
where
Ak is the signal amplitude,
is the autocorrelation function of the C/A code,
is the C/A code chip length (
µs) [
2],
is the code phase error,
is the carrier phase error,
is the carrier Doppler shift, and
is the carrier Doppler shift rate. It should be noted that,
, in Equation (2), is assumed to be known. In this case, the estimate of
can be taken at the output of the traditional delay lock loop subsystem [
1]. The covariance matrix
Q of the process noise can be described as follows [
2,
3,
4]:
where
denotes the nominal carrier frequency in Hz (
MHz for GPS L1 signal), and
c denotes the velocity of light. A more detailed description of the process noise spectral power density vector
is presented in [
12,
13]. The measurement covariance matrix
R can be written as follows [
2,
3,
4]:
where
N0 is the power spectral density of the receiver white noise.
3. Problem Solution
In probability theory, Bayesian estimation is a general probabilistic approach to recursive estimation of an unknown probability density function (PDF) of the state using a mathematical model of the process and a set of receiving measurements. Many problems require an estimation of the state at each receiving measurement. In this case, a recursive filter is a convenient solution. Such a filter consists of mainly two stages: prediction and update. In modern GPS technologies, statistical signal processing methods based on Bayesian filtering theory are widely used. When constructing algorithms for a GPS signal tracking system, it is necessary to describe mathematical models of the system state and measurements. The mathematical model for discrete system state and measurements of a Gaussian process can be described as follows [
6]:
Here,
fk is a possibly nonlinear function of the state
,
hk is a possibly nonlinear function of measurements
zk,
is an independent and identically Gaussian distributed process noise sequence, and
is an independent and identically Gaussian distributed measurement noise sequence. According to the Bayes method, it is required to form a posteriori PDF
at the moment
k, if there is a sequence of all measurements up to the moment
k. In this case, it is assumed that the priori PDF
is known. Then, the PDF
can be obtained recursively in two stages: prediction and update. It is assumed that the required PDF
is known. Then, the prediction stage uses the system model in Equation (5) to obtain the priori pdf of the state at time
k using the Chapman–Kolmogorov equation [
6].
At the timestep
k, a new measurement is received, which is used to predict the posterior PDF and implement the update stage using Bayes’ rule.
Here, the normalizing constant
in Equation (7) has the following form:
It can be seen from Equation (8) that
depends on the observation likelihood function
determined by the observation model in Equation (5) and the known statistics of
. The recurrence relations in Equations (6) and (7) form the basis for the optimal Bayesian solution of the estimating problem. In practice, solving the Bayesian problem is possible in a number of cases using the Kalman filtering approach. As is known, Kalman obtained a recursive algorithm for generating linear estimates of the state by the criterion of the minimum mean square error for linear models. In this case, the optimal Bayesian estimate
of the state is the posteriori mean value of
xk [
6].
This algorithm is called the LKF and can be applied for estimating linear dynamical systems by forming Gaussian approximations to the joint distribution of the state
xk and measurement
zk. Therefore, the first and second moments (mean values and covariances) of posteriori estimates are sufficient. In nonlinear systems, other modifications of the traditional KF were considered to find Bayesian estimates of
xk in Equation (9). The most common algorithms for nonlinear filtering problems are EKF and UKF. The EKF extends the scope of KF to nonlinear optimal filtering problems by forming a Gaussian approximation to the nonlinear model using a Taylor series-based transformation. The drawback of this method is the filter divergence in some circumstances [
6]. The UKF approach is based on a set of symmetrically distributed sample points used to form the mean and the covariance. There is another approach to selecting the set of sample points for obtaining the mean and covariance. This approach is based on so-called Gaussian filters. In [
24], it was shown that the UKF can be considered as a special case of Gaussian filters, where the nonlinear filtering problem is solved using Gaussian-assumed density approximations.
In this paper, we present the GHKF as a Gaussian filtering framework to be used in a GPS carrier tracking system. Next, we conduct a detailed analysis of this filter [
6,
24,
25,
26].
In GHKF, the linear equations of KF can be adapted to the nonlinear state space model. Desired moments (mean and covariance) of the original distribution of
can be obtained exactly by calculating the following integrals [
6]:
These integrals can be evaluated with practically any analytical or numerical integration method. The Gauss–Hermite quadrature rule is a method for solving special integrals of the previous form in a Gaussian kernel with an infinite domain. More specifically, the Gauss–Hermite quadrature can be applied to integrals of the following form:
where
xi denotes the sample points (quadrature points),
wi denotes the associated weights to use for the approximation, and
p is the number of distinct quadrature points. The quadrature points
xi are chosen to be the zeros of the
-order Hermite polynomial. The Hermite polynomial of degree
p is defined as follows:
The first two Hermite polynomials are
Further polynomials can be found from the following recursion:
The weights
wi are given by
Therefore, one-dimensional Gauss–Hermite quadrature integration refers to the special case of Gaussian quadrature with a unit Gaussian weight function
, i.e., to approximations of the following form:
Now, consider a multidimensional random variable
x having a Gaussian density
, where
nx is the identity matrix of dimensions
. The one-dimensional integration method can be extended to multidimensional integrals of the following form:
where
is the
column of the quadrature matrix of dimensions
, which can be described as follows:
Additionally, the weight of the quadrature point vector
can be written as
Therefore, if
p points are required to calculate the one-dimensional integral, then
points are required for the
nx-dimensional case. This set of quadrature points are given by the Cartesian product of
nx vectors, each of which consists of
p points. These points are the roots
of the Hermite polynomial
. Thus, the quadrature matrix
can be written as
Now, consider a multidimensional random variable
x having a Gaussian density
. In this case, the Gaussian multidimensional integral can be written as
where the square root of the covariance matrix
can be obtained by Cholesky factorization.