Abstract
The collaborative efforts of multi-frequency receivers have proven to be a significant advantage in the challenging environments of global navigation satellite system carrier tracking. This paper delves into the exploration of multi-carrier hybrid tracking, employing both phase-locked loop and frequency-locked loop observations to maximize tracking capability. This hybrid approach conducts distributed phase tracking and centralized Doppler frequency tracking within decoupled local filters to accommodate the frequency dispersion and reduce dimensionality. Subsequently, global assimilation is performed to adjust filter weights for performance optimization. Theoretical analysis affirms that this hybrid design can achieve tracking performance close to optimality when compared with classic centralized estimation, significantly outperforming fully distributed approaches. Additionally, simulations of ionospheric scintillation validate the effectiveness of the proposed hybrid carrier tracking design.
1 INTRODUCTION
With the rapid advancement of high-precision navigation technology, global navigation satellite system (GNSS) receivers are inevitably confronted with significant challenges in various safety-critical or remote sensing applications (Kaplan & Hegarty, 2017). For instance, in demanding environments such as urban or equatorial areas, GNSS signals are susceptible to strong multipath interference or ionospheric scintillations, which result in deep amplitude fading and random phase changes. These effects profoundly degrade signal quality, leading to frequent cycle slips or loss of lock, ultimately compromising GNSS positioning accuracy and robustness (Vila-Valls et al., 2020). Among the various baseband signal processing techniques applied to GNSS receivers, carrier tracking stands out as the most critical function, primarily limiting tracking ability and warranting further enhancement in harsh environments (Vila-Valls, Closas, Monica et al., 2017).
Two closed-loop architectures, namely, the phase-locked loop (PLL) and frequency-locked loop (FLL), are commonly employed for GNSS carrier tracking. Extensive research has been conducted to enhance carrier tracking performance, and the evolution of carrier tracking can be primarily represented by three structures, as illustrated in Figure 1. Here, Δθi(k) denotes the PLL measurement, Δϖi(k) denotes the FLL measurement, and xi(k) denotes the carrier state on the Li band at the k-th epoch.
Multi-carrier observations and state estimations: (a) distributed structure, (b) centralized structure, and (c) hybrid structure
Initial investigations focused on robust PLL techniques on a single band, as depicted in Figure 1(a). The PLL-based tracking model has been discussed and optimized in both the state space and frequency domain (Curran, 2015; Jwo, 2001; Won et al., 2010; Yang, Morton, et al., 2017). Robust carrier tracking in challenging environments has been achieved by using adaptive PLL techniques (Harsha & Ratnam, 2016; Razavi et al., 2008; Roncagliolo & García, 2007; Roncagliolo et al., 2012; Vila-Valls et al., 2015; Won & Eissfeller, 2013).
To enhance tracking robustness, an FLL is typically introduced to assist the PLL, leveraging its wide pull-in range and insensitivity to phase rotations (Curran et al., 2012; Li et al., 2018). This architecture, known as FLL-assisted PLL (Ward, 1998), offers excellent noise suppression and dynamic flexibility under various challenging conditions (Jiang & Cui, 2014; Lu et al., 2014; Xu et al., 2015).
However, recent work found that according to the minimum mean square error (MMSE) optimization criterion, FLL measurements contribute minimally in FLL-assisted PLL design (Yang et al., 2022). The likely reason is that the FLL discriminator is equivalent to a differential PLL discriminator, leading to increased FLL measurement noise and time correlation. In response, a distributed design and optimization approach have been applied in FLL-assisted PLL, aiming to integrate FLL within the loop filter to enhance tracking performance (Yang et al., 2021). However, this design is only valid for single-band carrier tracking and can be classified as a self-contained distributed design approach that relies solely on individual observations in each signal channel or band.
Modern GNSSs typically provide multiple civil signals on different carrier bands, making frequency diversity a valuable feature for robust carrier tracking (Henkel et al., 2009). By leveraging this intrinsic frequency dependency, collective filters have been designed and optimized to mitigate ionospheric scintillation, multipath, and interference (Bolla et al., 2018; Fan et al., 2020; Siddakatte et al., 2017; Vila-Valls, Closas, & Curran, 2017). Inter-frequency assistance in multi-carrier tracking can significantly enhance tracking performance (Yang et al., 2019). Most multi-carrier tracking loop designs only utilize PLL measurements and can be categorized as a centralized approach, as shown in Figure 1(b). Theoretically, the centralized estimator achieves optimal estimation performance. However, existing centralized multi-carrier PLLs have two limitations: 1) increased tracking model complexity due to the inclusion of multi-carrier states and measurements and 2) underutilization of the advantages of FLL.
Driven by this insight, we aim to explore collaborative tracking designs for multi-carrier PLL and FLL. The representative tracking architecture is illustrated in Figure 1(c), where PLL and FLL measurements are combined to maximize the benefits of multi-carrier tracking. To address computational complexity and the time-correlated nature of FLL measurement models, multi-carrier states are decoupled and estimated separately within a hybrid tracking architecture. Here, centralized and distributed estimators are integrated to form a hybrid tracking solution within the decoupling filter design. This approach maintains centralized cooperative Doppler frequency tracking with multi-carrier FLL measurements to ensure robustness. Simultaneously, distributed phase tracking is conducted independently on each band with PLL measurements, with a consideration of dispersion effects.
Note that theoretically, only the centralized estimator achieves global optimization, whereas local estimators are prone to being trapped in local optimization. To prevent potential performance degradation in hybrid tracking, the decoupled filter will be designed and optimized with respect to global assimilation, and the computational complexity will be analyzed accordingly. Performance analysis and comparison will also be provided through ionospheric scintillation simulations, considering strong deep fading and dynamic variations. The remainder of this paper is organized as follows. Section 2 describes multi-carrier signal models. Section 3 presents the multi-carrier tracking loop design, where centralized, distributed, and hybrid approaches are all discussed. Section 4 provides a performance analysis, and Section 5 presents a simulation verification. Finally, Section 6 concludes this work.
2 SIGNAL MODELS
2.1 System Model
For multi-frequency GNSS signals, the carrier state xi(k) is typically used for each carrier representation:
where φi, ωi, and
where Ai is the system transition matrix and ni(k) represents the system noise. Here, ni(k) is assumed to be white Gaussian noise with a covariance matrix Qi. The expressions for Ai and Qi are as follows (Yang et al., 2019):
where T denotes the coherent integration time, fLi is the carrier frequency on the Li band, and c is the speed of light. qφ = h0/2, qω = 2π2h−2, and qa (m2/s6/Hz) are the power spectral densities of the oscillator noises on the phase and Doppler frequency and the line-of-sight (LOS) dynamic random-walk process noise on the Doppler rate, respectively. h0 and h−2 are the oscillator h-parameters with units of s2/Hz and 1/Hz, respectively. qa is a function of LOS jerk and can be set to different values to represent various receiver dynamics (Bar-Shalom et al., 2002; Yang et al., 2019).
In most cases, the Doppler frequency and Doppler rate on different signal bands are assumed to be linearly correlated. However, each carrier phase should follow an independent trend because of the dispersive propagation effects in realistic conditions. Hence, a centralized multi-carrier state can be constructed as follows:
where φ1, φ2, and φ5 denote the carrier phases on GPS L1, L2, and L5, respectively. ω0 and
Note that under some critical ionospheric conditions, the GNSS signal will experience severe fading and phase dispersion. Existing work has considered ionospheric effects and built auto-regressive phase models to approximate the physical phenomena of ionospheric scintillation within the state-space design framework (Vila-Valls, Closas, & Curran, 2017; Vila-Valls et al., 2020). Therefore, in this work, to accommodate phase dispersion effects, e.g., caused by ionospheric scintillation or clock-induced phase errors, we utilize only the connection concerning the Doppler effect, while maintaining independent phase variables in the state model. Certainly, further separate modeling and estimation of ionospheric phase dispersion in the state model can better identify and consistently mitigate ionospheric errors. However, because this work mainly focuses on the structural design of multi-carrier tracking, such a phase model can be considered and discussed in the future.
The centralized system model for x(k) can be written as follows:
Similarly, A is the centralized system transition matrix, and n(k) represents the centralized system noise with covariance matrix Q.
Because of the co-existence of independent and dependent carrier states in x(k), we can group the carrier states into the following form:
where xφ(k) denotes the independent phase state and xω(k) denotes the frequency-dependent Doppler state. This hybrid state can be written as follows:
Note that the carrier phase state xφ(k) can be further split into three one-dimensional (1D) scalars to reduce computational complexity, e.g., xφi(k) = φi(k).
Then, n(k) in Equation (5) can be written as follows:
According to Equation (3), A and Q can be reformulated as follows (Yang et al., 2019):
where Aφ is a identity matrix, Aφ = I3×3. Aφω and Aω have the following form:
Here, the vector η denotes the frequency dependency weights with the form of η = [η1 η2 η5]T.
Similarly, the expressions for Qφ, Qφω, and Qω can be derived based on Equation (3):
As a result, Equation (5) can be factorized as two subsystems:
Then, based on the diverse measurements, the state can be separately estimated and collaboratively utilized for multi-carrier tracking.
2.2 Measurement Model
Measurements for multi-carrier tracking include the discriminator outputs Δθi(k) (radians) and Δϖi(k) (2π Hz) from the PLL and FLL on each band. For the distributed measurement state, we have the following:
The distributed measurement model follows the linear dependency on the error state Δxi(k) with
where Hi represents the measurement matrix on the Li band:
and vi(k) = [vi(k)ui(k)]T denotes the measurement noise vector, including the measurement noises vi(k) and ui(k) from the PLL and FLL, respectively. Then, the autocorrelation covariance Ri(k) can be written as follows (Yang et al., 2022):
and the cross-correlation covariance Mi(k) can be written as follows (Yang et al., 2022):
with:
Here, C/N0,i denotes the carrier-to-noise ratio on the Li band.
Let z(k) be the centralized measurement state, written as follows:
Similarly, z(k) can be linearized as a function of the centralized error state Δx(k):
where H represents the centralized measurement matrix. v(k) denotes the centralized measurement noise vector as follows:
with an autocorrelation covariance v(k) and temporal cross-correlation covariance M(k).
Following the state factorizations in Equations (6) and (8), we have the following:
where:
Then, based on Equation (18), H in Equation (23) can be decomposed as follows:
where Hφ = I3×3 and Hω = η[1 T/2].
Hence, Equation (23) can finally be split as follows:
Similarly, we have the following:
After derivation and simplification, Equation (32) has the following form:
where
3 MULTI-CARRIER TRACKING LOOP DESIGN
3.1 Distributed Approach
Given the distributed system model in Equation (2) and the distributed measurement model in Equation (17), a single-carrier state estimate xi(k) can be obtained:
where Ki(k) denotes the filter gain. Let Pi(k + 1) be the state error covariance matrix
Because of the non-white measurement noise feature, here we use
Please note that this estimator defines a direct-state model but obtains estimates based on error measurements from the discriminator outputs. Strictly speaking, the direct-state filter represents the linear model of the entire tracking channel, including correlators, discriminators, loop filters, and numerical control oscillators. The states consist of the carrier phase, Doppler frequency, and angular acceleration, while the measurements include the carrier phase and Doppler frequency of the incoming GNSS signal. The matrix zi(k) in Equation (34) represents the innovation rather than the measurement. In contrast, the error-state filter replaces only the loop filter of the tracking loop, with states being the error of the carrier phase, Doppler frequency, and angular acceleration.
In our previous work (Yang et al., 2017), we demonstrated the equivalence of the direct-state and error-state designs from the perspective of the entire tracking procedure. As a result, the direct-state and error-state designs lead to a unified estimation form, as in Equation (34). For simplicity, we approximate this design as a direct-state estimator. Subsequently, the following centralized and hybrid approaches all adopt this approximation.
3.2 Centralized Approach
Given the system model in Equation (5) and the measurement model in Equation (23), the global state estimate x(k) can be obtained as follows:
where K(k) denotes the centralized filter gain. Let P(k + 1) be the centralized state error covariance matrix P(k + 1) = E [Δx(k + 1)ΔxT (k + 1)], derived as follows (Yang et al., 2021):
Let G(k) = E[Δx(k)vT(k)] represent the cross-correlation between the error state and the measurement noise; then, we have G(k) = –AK(k – 1) M(k) (see the work by Yang et al. (2021)). It is known that K(k) is designed to minimize the trace of P(k + 1); thus, we have the following:
3.3 Hybrid Approach
As shown in Equations (6)–(15), the multi-carrier state x(k) can be grouped as two sub-states. One is the phase state xφ(k) on each band, whereas the other is the reference Doppler frequency state xω(k) to be linearly correlated across signal bands. It can be seen that the estimations on xφ(k) and xω(k) should work in a diametrically opposite manner, where the phase estimate
where Kω(k) and Kφ(k) denote the local gains in the centralized Doppler frequency estimator and the distributed phase estimator, respectively. Such distributed and centralized state estimations form a mixed tracking structure, as shown in Figure 1. Therefore, the proposed design is termed a “hybrid approach” in this work.
The details on the phase and Doppler frequency estimator design will be discussed next. Note that
The estimator gain matrix K can be written as follows:
Please note that the gain mentioned above is derived under the assumption that the phase and Doppler frequency estimators can be fully decoupled, without accounting for cross-correlation. Therefore, the gain of the principal diagonal element is primarily estimated in the local filter, after which the remaining elements of the gain matrix can be optimized through global assimilation.
3.3.1 Centralized Doppler Frequency Estimator
From Equations (15) and (40), the Doppler frequency state error Δxω(k + 1) can be obtained as follows:
Let Pω(k + 1) be the Doppler frequency state error covariance matrix; then, we have the following:
Here, Gω(k) is the cross-correlation, as mentioned before, which can be derived as follows:
For the Doppler frequency state estimator design, the MMSE is adopted to optimize the estimation performance. Hence, Kω(k) can be written as follows:
3.3.2 Distributed Phase Estimator
From Equations (14) and (41), the phase state error Δxφ(k + 1) can be obtained as follows:
The corresponding covariance matrix Pφ(k + 1) can be derived as follows:
Here, Pφω(k) represents the cross-covariance matrix with the following expression:
where Gφω can be expressed as follows:
Similarly, the local phase estimator gain Kφ(k) is designed to minimize the phase tracking variance, e.g., the trace of the covariance matrix Pφ(k + 1) (Laue et al., 2020); therefore, we have the following:
3.3.3 Global Assimilation
The above local estimators only achieve local optimal performance. To realize global optimization, assimilation should be performed. A necessary and sufficient condition for global assimilation is that there exists a mapping matrix S that meets the following condition (Chung & Speyer, 1995):
From Equation (29), we have Sφ = [I3×3 | 03×2] and Sω = [02×3 | I2×2]. Then, given Pω in Equation (45), Pφ in Equation (49), and Pφω in Equation (50), the global covariance matrix P can be formulated as in Equation (42). By using the weighting matrices Wφ and Wω, the proportional contribution of each local estimate can be mapped into the global estimate. After derivation, Wφ and Wω can be expressed as follows:
Then, the globally assimilative gain matrix
By substituting Equations (54) and (55) into the above equation,
It is interesting to note that because of the cross-correlation between two local states, the assimilative global gain becomes a non-diagonal matrix. Therefore, to achieve near-optimal tracking performance, the non-diagonal elements should be included in the state estimator design. By substituting
and
In summary, we first define the system using Equations (14) and (15) and establish the measurement model with Equations (30) and (31). Next, we compute the phase and Doppler frequency estimates based on Equations (40) and (41). Subsequently, we determine the local Kalman gains for the carrier phase using a distributed approach and for the carrier Doppler frequency using a centralized approach based on the MMSE optimization criteria. Finally, global assimilation is performed to identify the non-diagonal elements of the Kalman gains.
By comparing the original design objective in Equations (40) and (41), it can be seen that the local estimators include the cross-correlation effects so as to alleviate the overly optimistic issue in the distributed approach. Because this is a decoupled approximation of the conventional centralized design, it can only achieve suboptimal performance. In particular, there does not exist a local estimator that can completely decouple the optimal state estimation if the system model is theoretically correlated. However, by comparing the above to the conventional implementation in Equation (37), it can be found that the tracking complexity can be reduced by using the lower-dimensional iteration. In addition, the tracking structure is more flexible and applicable for some remote sensing applications. For example, this design is beneficial for ionosphere activity monitoring because it can independently extract phase features due to environmental variations while maintaining robust Doppler frequency tracking.
4 IMPLEMENTATION AND PERFORMANCE ANALYSIS
4.1 Cascaded Implementation
The hybrid design involves two local estimators with a cascaded implementation process. The specific implementation procedure is summarized in Algorithm 1, where the functions zeros, dare, and inv create a codistributed matrix of zeros, solve discrete-time algebraic Riccati equations, and return the inverse of a symbolic matrix, respectively. Note that in the calculation of the local phase gain Kφ(k), the cross-correlation Pφω(k) does not initially need to be considered, because the local gain will be adjusted once again in the global assimilation with respect to Pφω(k). Basically, the calculation of the gain matrix is regarded as an optimization problem. The iterative convergence speed is related to the selection of the starting point as well as the gradient descent direction. It is reasonable to start with the local optimal candidates to ultimately approach the global optimal. To balance efficiency and performance, it is acceptable to sacrifice a certain level of accuracy to achieve sub-optimal performance with fewer iterations.
Hybrid tracking
For comparison, the implementation procedures of the distributed and centralized approaches will also be presented here. The procedures all initialize from the simplified calculations on P0 and K0 with the overlooked cross-correlation and then converge to the steady-state solutions via an iteration approach. The difference between these approaches lies in the gain iteration process and the estimation update. The distributed tracking is driven by each carrier measurement independently, the centralized tracking combines the multi-carrier measurements together, and the hybrid tracking formulates the carrier estimation problem into phase and Doppler frequency groups and executes the corresponding update procedures.
4.2 Computational Complexity
The computational complexity of the state estimation primarily comes from the gain matrix calculation, whereas the matrix calculus primarily depends on the matrix dimension. The precise computational complexity should encompass the consideration of additions, multiplications, and other relevant operations. However, it is crucial to note that the computational complexity of matrix multiplication far surpasses that of other operations. Therefore, the focus primarily lies on analyzing matrix multiplication. For matrix multiplication, if we have an invertible matrix
Computational Complexity for the Distributed, Centralized, and Hybrid Tracking Approaches
Distributed tracking
Centralized tracking
Using the complexity calculation in the distributed approach as an example, each iteration for the signals of each band necessitates the computation of Gi, K0, AX, QX, and P0, as outlined in Algorithm 2. Employing the rules of matrix multiplication, we determine the computational complexity for each as follows: O(Gi) = O(3 × 3 × 2) + O(3 × 2 × 2) = O(30), O(K0) = O(3 × 3 × 2) + O(2 × 3 × 3) + O(2 × 3 × 2) + O(2 × 3 × 2) + O(2 × 3 × 2) + O(2 × 2 × 2) + O(3 × 2 × 2) = O(92), O(AX) = O(3 × 3 × 2) + O(3 × 2 × 3) = O(36), O(QX) = O(3 × 3 × 2) + O(3 × 3 × 2) + O(3 × 2 × 2) + O(3 × 2 × 3) + O(3 × 3 × 2) + O(3 × 3 × 2) + O(3 × 2 × 3) + O(3 × 3 × 2) + O(3 × 3 × 2) + O(3 × 2 × 3) = O(174), and O(P0) = O(3 × 3 × 3) + O(3 × 3 × 3) = O(54). Thus, the total complexity per iteration is O(386). Consequently, each carrier incurs a computational complexity of O(386N) for N iterations. For Global Positioning System (GPS) triple-carrier tracking, the cumulative computational complexity becomes O(1158N). Similar calculations apply to the remaining entries in Table 1.
It is known that centralized tracking utilizes multi-carrier measurements simultaneously to estimate multi-carrier states; thus, its computational complexity is up to O(3646N) because of the higher matrix dimension. Because hybrid tracking combines distributed and centralized estimation together, the total computational complexity comes from both estimation operations. It can be seen that the Doppler frequency state takes the centralized estimation procedure, resulting in a higher complexity of O(319N) for N iterations. While the phase estimation is driven by an independent measurement, the complexity for the Pφ and Kφ calculations is O(192N). In contrast, for Gφω and Pφω, which we only calculate once, the complexity is approximately O(318). In this context, we omitted the calculation of Gφω and Pφω from the iteration process because we found that their exclusion has a negligible impact on the final gain values after the global assimilation.
Based on the testing, the matrix calculation converges within dozens of iterations, typically N<100. The hardware platform should be able to manage this computational load. Furthermore, this computational complexity analysis demonstrates that the hybrid approach boasts the lowest complexity among the three options. Splitting the phase state into three 1D vectors could further decrease the complexity, although it is worth noting that our current optimization efforts have already significantly reduced computational demands compared with conventional centralized tracking approaches. Unless this level of complexity is deemed unacceptable, the other two designs are considerably less viable for real-world applications.
4.3 Theoretical Performance Analysis
This section provides a theoretical performance analysis for comparison among the conventional centralized, distributed, and proposed hybrid designs for multi-carrier tracking, where L1, L2, and L5 signals are all set from 15 to 45 dB-Hz. In our previous work (Yang et al., 2019), we extensively discussed the centralized design and provided a theoretical analysis of tracking sensitivity. We demonstrated that tracking sensitivity can be substantially enhanced by utilizing multi-frequency architectures. Further information can be found in Yang et al. (2019). As a result, in this study, we will refrain from delving into the specifics of tracking sensitivity analysis, instead focusing on cross-comparisons and theoretical elucidation of the hybrid design.
In the analysis, the integration time T is set to 10 ms, and the four-quadrant arctangent (Atan2) discriminator is adopted to obtain the multi-carrier PLL and FLL measurements (Yang et al., 2022). The dynamic parameter qa is set to 1 (m2/s6)/Hz to cope with the typical dynamics (Yang, Morton, et al., 2017). A high-quality oscillator with phase noise parameters of h0 = 8.7 × 10−26 (s2/Hz) and h−2 = 3.6 × 10−26(1/Hz) is considered. Given these parameters, the corresponding gain matrices in Equations (39), (43), and (57) can be calculated. The corresponding error covariance matrices in Equations (35), (38), and (42) can be obtained as well. Note that in Yang et al. (2021 2019), we substantiated the validity of error covariance derivations through Monte Carlo simulation, where the standard deviation of the error between the estimated measurement and the true measurement is calculated for each C/N0 level. The statistical standard deviations align closely with the theoretical deviations in error covariance matrices. In this study, the error covariances in Equations (35), (38), and (42) will serve as the performance evaluation indicators for theoretical analysis.
To present a visualization of theoretical tracking errors from the three approaches, we take L5 at 15 dB-Hz as an example and plot a two-dimensional (2D) view of phase and Doppler frequency tracking errors in Figures 2 and 3, respectively. The figures show that as C/N0 increases, the phase and Doppler frequency errors both decrease. In Figure 2, the phase errors on L1 are slightly larger than those on L2 because of the larger frequency dependency ratio in the oscillator noise. In general, the phase tracking errors from the centralized and hybrid approaches are comparable. However, in the distributed approach, the L1 and L2 tracking errors depend solely on the strength of the L1 and L2 signals, respectively. When C/N0 is as low as 15 dB-Hz, the phase tracking errors exceed 15°. In the distributed approach, the maximum tracking error can reach up to 40°. However, to provide better visualization in the color map, we have set specific values for the color bar limits.
Phase tracking errors (degrees) with respect to various signal strengths
Doppler frequency tracking errors (Hz) and Doppler rate tracking errors (Hz/s) with respect to various signal strengths
The Doppler frequency observations exhibit a similar behavior, as depicted in Figure 3. The results indicate that the distributed estimator operates with an independent tracking error mode, devoid of inter-frequency aiding. The centralized approach achieves the best Doppler frequency tracking accuracy, as it is theoretically optimal. Meanwhile, the hybrid approach achieves only sub-optimal performance, even with the global assimilation. Please note that because of cross-correlation among multiple carriers, the centralized and hybrid approaches may both experience performance degradation when combining strong and weak signals. This phenomenon was previously reported in Yang et al. (2019). To address this issue, optimizing the inter-frequency aiding strategy can be beneficial. For further details, please refer to Yang et al. (2019).
The behavior of
Diagonal elements in the hybrid gain
To supplement the observations in Figures 2 and 3, the theoretical L1 tracking performances for the phase, Doppler frequency, and Doppler rate are shown in Figure 5, where the error covariance matrix P is used to represent the theoretical tracking error variance on different carrier states across different methods, with the square root given by the diagonal elements. This theoretical derivation has been verified in previous work (Yang, Ling, et al., 2017; Yang et al., 2022). Here, we use this theoretical representation to evaluate the tracking performance for comparison. Two cases are presented: L1 is varied from 15 to 45 dB-Hz while (a) L2 and L5 remain at 15 dB-Hz or (b) L2 stays at 15 dB-Hz and L5 remains at 45 dB-Hz. The conservative 3-sigma rules for both two-quadrant and four-quadrant arctangent discriminators are provided. Obviously, a greater signal strength corresponds to a higher tracking accuracy. Theoretically, the tracking accuracy of centralized tracking should be the highest, followed by hybrid tracking and distributed tracking.This trend can be confirmed by the phase tracking results in Figure 5. For the Doppler frequency tracking accuracy, the hybrid approach is slightly worse than the distributed approach, most likely because of cross-correlation effects.
L1 tracking errors from centralized, distributed, and hybrid approaches when L1 is varied from 15 to 45 dB-Hz while (a) L2 and L5 remain at 15 dB-Hz or (b) L2 stays at 15 dB-Hz and L5 remains at 45 dB-Hz
For a better comparison, we also took partial elements in P to visualize covariance ellipsoids (Johnson, 2022) for the hybrid and distributed approaches, where Algorithms 1 and 2 are carried out for the calculations, respectively. In the hybrid approach, we obtain P as in Equation (42) and extract the relevant elements of L1, e.g., P(1, 1), P(1, 4), P(1, 5), P(4, 1), P(4, 4), P(4, 5), P(5, 1), P(5, 4), and P(5, 5), to form the ellipsoid. For the distributed approach, we took P1 in Equation (35) to form the ellipsoid. In Figure 6, the X, Y, and Z axes correspond to the L1 phase, Doppler frequency, and Doppler rate, respectively. The green, red, and blue ellipses correspond to the ellipsoid projection in the X-Z, Y-Z, and X-Y planes, respectively. Because L1 and L2 both vary from 15 to 45 dB-Hz, here we only present the covariance examples when L1 and L2 are 45 dB-Hz for simplicity. The central ellipsoid represents the error accuracy of the estimator with three states involved. The results show that the hybrid ellipsoid is smaller than the distributed ellipsoid, which corresponds to a higher tracking accuracy.
L1 covariance ellipsoids for the hybrid and distributed approaches when L1, L2, and L5 are all at 45 dB-Hz (a) Hybrid Approach (b) Distributed Approach
After verifying the performance, it is desirable to know the relative weights of PLL or FLL measurements and how they were utilized in the phase and Doppler frequency estimations within the hybrid tracking. Figure 7 presents a 2D view of the contribution of zφ and zω to the phase or Doppler frequency estimations. We separately calculated the coefficients of zφ and zω in Equations (59) and (60). By applying normalization operations to the coefficients, the proportion of contribution of the two observations zφ and zω to the estimations for the phase state and Doppler frequency state can be obtained, respectively. It is evident that the phase estimation primarily depends on the PLL measurements zφ. Over 60% of the contributions stem from zφ, and as the signal strength decreases, this ratio increases to over 70%. This result indicates that as the signal strength decreases, the phase measurements encompass more randomly varying components, which are not adequately reflected in zω. Therefore, the phase estimation relies more heavily on zφ than on zω. For the Doppler frequency estimation, the dependency of FLL measurements zω gradually declines. The contribution of PLL and FLL measurements is approximate. The FLL weights are over 54%, slightly larger than the PLL weights. Moreover, the FLL weights decrease as C/N0 decreases. This tendency is opposite to the trend observed for phase estimation. Thus, zω can be neglected in the phase estimator in Equation (59), whereas zφ should be maintained in the Doppler frequency estimator in Equation (60) because of its large proportion from a global optimization perspective.
Contributions of the PLL or FLL measurements to phase and Doppler frequency estimations in hybrid tracking (a) Contribution of zφ on
The effectiveness of the above carrier tracking design hinges on an accurate characterization of system and measurement noises. This characterization ensures that the error covariance matrix accurately represents the tracking performance and that Kalman filter gains effectively suppress noise effects. However, accurately modeling the system is challenging in real applications. Consequently, the error covariance matrix can be misleading as an indicator of actual tracking performance and may lead to overly ideal adjustments in Kalman filter gains based on the MMSE criterion. To better reflect practical tracking states, the standard deviation of the innovation value may serve as a more reliable indicator of tracking performance than the error covariance matrix. Further optimization of the implementation can be achieved through the innovation covariance.
5 SIMULATION VERIFICATION
To evaluate the tracking performance in the hybrid approach and make a proper comparison to the distributed and centralized approaches, a multi-frequency ionospheric scintillation simulator based on a low-latitude physical model is applied (Jiao et al., 2017). This is a two-parameter simulator that primarily requires the amplitude index S4 and the decorrelation time τ0 as scintillation inputs (Xu et al., 2020). The simulator supports GPS and BeiDou multi-frequency signal generation (Huang et al., 2020). In this paper, 150-s triple-band GPS intermediate-frequency (IF) data with strong scintillation (S4 = 0.9 and τ0 = 0.75) after 30 s were generated for verification. In the simulation, there is no data modulation on the triple-band GPS signals. The nominal signal strengths on the triple bands are initially 42 dB-Hz. The IF was 0 MHz, with 4 quantization bits. The sampling rates were 5, 5, and 25 MHz for GPS L1, L2, and L5 signals, respectively. The tracking loops were configured with the same parameters as in Section 4.3. Four-quadrant arctangent discriminators were adopted to obtain both phase and frequency error measurements. The normalized signal intensity (Figure 8) and excess phase (Figure 9) caused by ionospheric scintillation in the simulator are discussed below.
Normalized signal intensity in the presence of ionospheric scintillation
Excess phase variations caused by ionospheric scintillation
Figure 8 shows that the signal amplitude attenuation caused by the ionosphere scintillation reaches almost 30 dB after the first 30 s, and the excess phase can reach up to 5 cycles at the maximum. Although the GPS L1, L2, and L5 signals are all disturbed by the ionospheric scintillation, the related signal fading and phase behaviors on the three bands are diverse. This random dispersion effect will cause tracking errors and degrade receiver performances. The three tracking approaches were tested, and their tracking results were compared to the geometric truth. The phase and Doppler frequency tracking errors are presented in Figure 12.
The C/N0 estimations and the measurements Δθi(k) from the PLL discriminator output are shown in Figure 10. The fluctuations observed in C/N0 estimations across the three bands closely correspond to the signal fading patterns in Figure 8. In the presence of strong ionospheric scintillation, the signal can weaken to nearly 20 dB-Hz. Because of the combined effects of ionospheric signal fading and random phase delays, phase errors across the three carriers show a notable increase. Instances of sudden signal drops or phase jumps, such as those at approximately 30 s and 100 s, can result in phase errors exceeding 60° due to ionospheric scintillation effects. These effects pose a great challenge for base processing in typical GNSS receivers. Because the carrier tracking performance of the proposed hybrid design, which combines the distributed and centralized methods, relies heavily on accurate modeling, we calculated standard deviation statistics based on the phase discriminator output and compared them with theoretical values estimated from the C/N0 results, as in Equation (21). The comparison results for L1, L2, and L5 are illustrated in Figure 11. It is evident that the statistical results align with theoretical expectations, demonstrating the effectiveness of C/N0 estimates and measurement modeling in reflecting ionospheric scintillation effects.
C/N0 estimations and the PLL discriminator output Δθi(k) for triple bands of the hybrid tracking approach
Theoretical and statistical standard derivation of the PLL discriminator output Δθi(k) for triple bands of the hybrid tracking approach
Obtained by subtracting the LOS phase and Doppler frequency truths from the tracking results, the phase tracking errors and Doppler frequency tracking errors of the three approaches are presented in Figure 12. It can be seen that the hybrid tracking errors are generally consistent with those of the centralized design, verifying that the hybrid approach achieves near-optimal performance comparable to that of the centralized design. In general, as shown in Figure 12(d)–(f), the Doppler frequency tracking accuracies of the hybrid and centralized approaches are better than that of the distributed approach, demonstrating that the inter-frequency aiding is effective for Doppler frequency error smoothing. Interestingly, for phase tracking, the hybrid and centralized approaches decrease phase tracking errors on L2 and L5, but introduce an additional 1–2 cycle errors for the L1 phase when compared to the distributed design. This result is primarily caused by the weak correlation effects among multi-carrier states, as discussed before. From Figure 9, we can see that the phase scintillation effects for L2 and L5 are more distinct than those for L1 after 90 s. Therefore, the L1 phase tracking performance in the hybrid and centralized approaches might be degraded by the L2 and L5 measurements when the filter gain is off-diagonal. This performance can be further optimized by incorporating the scintillation effects into the system models.
Tracking errors from the distributed, centralized, and hybrid approaches for (a–c) phase and (d–f) Doppler frequency
We also present the phase and frequency lock indicators of the three tracking approaches in Figure 13. Most values fluctuate above 0.8, with occasional drops below 0.6 during deep fading events. It is evident that all three approaches show similar levels of phase lock, whereas the distributed approach displays a less pronounced performance in frequency lock. This result aligns with the findings presented in Figure 12.
Phase and frequency lock indicators for the distributed, centralized, and hybrid approaches
6 CONCLUSIONS
This paper has proposed a hybrid design approach to leverage multi-carrier FLL measurements for collaboratively estimating Doppler frequency-related states while using PLL measurements to independently estimate individual phase states. This approach aims to enhance frequency tracking robustness while retaining the capability to track independent phase variations. In terms of estimation fusion, three fusion architectures were considered and compared: distributed, centralized, and hybrid. Theoretical performance analysis demonstrated that the hybrid approach achieves near-optimal performance similar to that of the centralized design but with reduced computational complexity. Simulation verification was conducted using triple-band signals with strong ionospheric scintillation, demonstrating that the proposed hybrid approach achieves improved Doppler frequency tracking accuracy over that of the distributed approach. Moreover, the phase tracking performance in the hybrid approach closely approximates that of the centralized approach. However, when dispersion effects are present, phase tracking performance may degrade for both the hybrid and centralized approaches because of weak cross-correlation among multi-carrier state models. This issue can be addressed by further splitting the phase states or incorporating dispersion effects into the system models, which will be the focus of our future research endeavors. In addition, the hybrid design can be further optimized by monitoring and evaluating the innovation covariance rather than relying solely on the typical error covariance.
HOW TO CITE THIS ARTICLE
Yang, R., Huang, J., & Zhan, X. (2024). Hybrid carrier tracking with decoupled local filters in multi-frequency GNSS receivers. NAVIGATION, 71(4). https://doi.org/10.33012/navi.672
CONFLICT OF INTEREST
The authors declare no potential conflicts of interest.
ACKNOWLEDGMENTS
This research was jointly supported by the National Key Research and Development Program of China (No. 2022YFB3904404) and the National Science Foundation of Shanghai (No. 22ZR1434500). We extend our sincere gratitude to Grammarly and ChatGPT 3.5 for helping us polish the language.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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