# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 1 Introduction  
Positioning approaches can be either direct or indirect. Global navigation satellite systems (GNSSs) rely on the same indirect estimation approach, which is a two-step procedure (2 SP): the signals received by the GNSS receiver from satellites enter an independent processing channel to obtain the corresponding synchronization parameters, and these parameters are used to estimate the position-velocity-time (PVT) state of the receiver through a multilateration procedure. The first step is to perform a two-dimensional search for the actual time delay and Doppler shift of each satellite separately by correlating the received signals with the known orthogonal direct-sequence spread spectrum emitted by each satellite. This is typically accomplished with two modules for acquisition and tracking. Subsequently, the second step involves solving a nonlinear least square (LS) minimization problem, which is commonly performed by iteratively linearizing the cost function around an initial estimate. While 2 SP-based receivers are widely utilized for their near-optimal performance in open-sky environments, their accuracy is usually limited in complex scenarios characterized by jamming, multiple paths, and channel fading. These effects pose a greater challenge for positioning accuracy and are not easily mitigated with conventional methods, such as employing differential measurements or assisted error modeling. Direct position estimation (DPE), conversely, uses a single-step procedure to estimate the PVT state from the received GNSS signal and has emerged as an attractive alternative to the 2 SP. It was first introduced for locating narrowband radio-frequency transmitters andhandling multiple radio signals [1]. This concept known as DPE for GNSS receivers was presented in [2] for the single-antenna case, and it was extended to the array antenna receiver case in [3]. The DPE exploits the fact that the signals emitted by satellites are all received from the same PVT state. Hence, all channels are processed simultaneously, enabling the sharing of information among them. In contrast, the 2 SP handles each channel separately, potentially leading to incompatible parameter combinations and erroneous estimates. Avoiding the intermediate estimation step has proven to be effective in mitigating some of the inherent limitations of the 2 SP and improving performance in challenging scenarios [4; 5]. Directly incorporating PVT estimation through a single-step procedure also facilitates the integration of prior information within a natural framework, as it involves the direct manipulation of the PVT state of the receiver. Hence, the DPE has been extended to incorporate prior information within the Bayesian paradigm [6; 7]. Several DPE-based GNSS receivers have been proposed [8; 9; 10], and various extended applications have been explored [11; 12; 13; 14; 15]. However, the potential benefits of the DPE do not come without costs. The optimal direct estimation processing of the DPE entails a significant computational burden, hindering its application. The nonlinear multi-dimensional optimization problem can be computationally intensive. To address this challenge, various approaches have been proposed. The most intuitive method is to employ the method of exhaustion, evaluating the function on a grid to identify the global minimum. A multi-resolution grid method involving a three-level search is proposed in [16] to reduce the number of computational points required. Additionally, Ref. [17] analyzes the features of a multi-resolution search algorithm and proposes an optimization-based 3D dichotomous search scheme to improve the overall efficiency of the multi-resolution grid method. Moreover, heuristic algorithms offer a promising approach for optimizing the relative position vector of the user with good resolution and without exhaustively searching the entire search space [18; 19].