# Performance Evaluation of Adaptive Tracking Techniques with Direct-State Kalman Filter +
## 1 Introduction  
Global navigation satellite system (GNSS) receivers synchronize with GNSS signals to decode the navigation message, measure the pseudo-range and pseudo-range rate, and calculate a position, velocity, and time (PVT) solution [1, 2]. The synchronization consists of two stages: Acquisition and tracking. Acquisition performs a coarse estimate of the synchronization parameters, whereas the tracking stage provides an improved estimate. The latter stage uses the scalar tracking loop (STL) to refine the synchronization of the incoming GNSS signals [3, 4]. The STL replicates a synchronization parameter for every loop iteration. The synchronization lock is achieved when the difference between the true parameter and its replica (i.e., the estimation error) tends to zero [3]. The carrier phase\(\theta\), the carrier Doppler \(f_{\rm d}\), and the code phase \(\tau\) are the GNSS signal parameters in which the GNSS receiver must synchronize. Therefore, a tracking channel comprises three STLs: phase locked loop (PLL), frequency locked loop (FLL), and delay locked loop (DLL). A correlator, a discriminator, a loop filter, and a numerically controlled oscillator (NCO) compose the STL [5; 6]. The STL's configuration parameters are the type of discriminator, the loop bandwidth \(B\), the integration time \(\tau_{\rm int}\), the order \(p\), and the correlator spacing. These parameters determine the robustness against noise and signal dynamics. The well-known trade-off between noise filtering capabilities and signal dynamics resistance is the main problem of standard STLs with fixed configurations [7]. For instance, a high-order STL with wide loop bandwidth and short integration time is adequate to track rapidly changing parameters. In contrast, a low-order STL with narrow loop bandwidth and long integration time is preferable to track noisy parameters. Time-varying scenarios are characterized by different realizations of signal dynamics, noise, and fading effects. These changing effects challenge the synchronization capability of the tracking stage [1; 7]. Since traditional tracking lacks resilience due to its fixed configuration, there has been significant research towards robust tracking solutions to solve this problem [8; 9]. However, there is still ample investigation to find the optimal technique in terms of performance and complexity [7]. The Kalman filter (KF) is an optimal infinite impulse response (IIR) estimator under the assumption of linear Gaussian error statistics [10; 11; 12]. Good knowledge of the process noise covariance \(\mathbf{Q}\) and measurement noise covariance \(\mathbf{R}\) allows the KF to optimally adapt its coefficients to achieve the minimum mean square error (MMSE) [13]. There are several KF implementation methods in STLs [14] grouped into error-state Kalman-filter (ESKF) and direct-state Kalman filter (DSKF) [15]. The former replaces the loop filter of the STL with a KF whereas the latter considers the whole STL part of the KF. In the ESKF, the measurement is the discriminator's output, and the predicted measurement drives the NCO. A detailed study of this architecture has been done in previous research [16; 17; 18; 19]. The complexity of the ESKF is a limiting factor but it can be reduced taking advantage of the Kalman gains' convergence in the steady-state [19]. The DSKF is more straightforward to implement than the ESKF since it considers the whole STL part of the KF. In this case, the measurement is the synchronization parameter, the innovation is the discriminator's output, and the loop filter and the NCO are the core of the KF. This method benefits from the simplicity of relating the coefficients of the standard STL with the Kalman gains of the DSKF [20]. The MMSE is only achieved if a priori knowledge of \(\mathbf{Q}\) and \(\mathbf{R}\) is available or if these are accurately estimated [13]. If this is not the case, the KF tends to be a sub-optimal solution [21].