# A Carrier Estimation Method Based on MLE and KF for Weak GNSS Signals
## 1 Introduction  
Global positioning systems (GPS) have been widely used in various fields. Whenever and wherever four or more satellites are visible, we can use GPS receivers to achieve positioning and navigation. The receiver can only capture and track the received signal from satellites which have an intensity above a certain power level. The signal intensity is usually referred to as signal-to-noise ratio (SNR) or carrier-to-noise spectral density ratio (\(C/N_{0}\)). However, it is not always possible to ensure high \(C/N_{0}\) in some harsh circumstances such as in urban, forest and indoor areas. Since the carrier tracking loop is always the weakest link in a stand-alone GPS receiver, its threshold characterizes the unaided GPS receiver performance [1]. For weak signal applications, the two important factors that affect the performance of the receiver are the signal strength and dynamics of the receiver. The most commonly used loop, the phase-locked loop (PLL), has low sensitivity [1]. Increasing the integration time is a common solution to improve the sensitivity of receivers, such as coherent integration, non-coherent integration and differential integration [2, 3]. However, increasing the integration time will reduce the dynamic performance of the phase-locked loop (PLL). The frequency-locked loop (FLL) is another common loop which has higher sensitivity and better dynamic performance. However, its accuracy is low. In urban and indoorcircumstances, the satellite signal is severely attenuated. Received signal intensity is typically 10-30 dB lower than actual received power level in open environments, which brings a serious challenge to the conventional loops [4]. The maximum likelihood estimation (MLE) approach is known to yield optimal performance for GNSS signal tracking. Some loops based on MLE have been researched and applied in some specific signal environments [5; 6; 7; 8; 9; 10; 11]. [PERSON] et al. [5] proposed an approximate maximum likelihood method to track Doppler frequency and code delay for a high-dynamic receiver. The multipath signal parameters are estimated by MLE in [6; 7]. Iterative and non-iterative MLE for Doppler frequency and code delay are discussed in [8; 9]. Joint maximum likelihood estimation for position is proposed in [10] and its Cramer-Rao bound (CRB) is derived in [11]. These methods can achieve high performance in some specific environments. However, all of these methods process the sampling data (intermediate frequency, IF, raw samples) directly and perform complex processing or iteration operations. This is because the sampling data has a high data rate from 5 MHz up to 30 MHz. That implies a large computation burden and means more hardware or software resources are needed, which is undesirable. This paper proposes a low-complexity MLE carrier estimation method. In contrast to the existing methods, this method processes the coherent integration results instead of sampling data. It therefore has a small computation burden because the processed data rate is reduced from 5.714 MHz (the sampling rate) to 1000 Hz. The main contributions of this paper include two key points which are included in Sections 3 and 4, respectively. In Section 3, the MLE discriminator is presented under the assumption that the pseudo random code (PRN) is stripped off. Its cost function with respect to the carrier phase, frequency and amplitude is given in Section 3.1. An efficient parameter estimation method by Levenberg-Marquardt (LM) algorithm is described in Section 3.2. Then, the performance of an MLE discriminator including CRB, dynamic performance and computation cost is analyzed in Sections 3.3-3.5, respectively. In Section 4, an adaptive Kalman filter (KF) is designed to improve the performance of the MLE discriminator. The basic equations of KF are given in Section 4.1 including the observation equation, state transition equation and their covariance matrix. It uses the amplitude estimate to adjust the observation noise matrix adaptively, so it is an adaptive KF.