## 1 Introduction
However, these methods are developed to apply to single-direction localization problems where the target actively sends signals to the sensors. However, the localization problem to be solved in this study is where the sensors send signals to the target and receive the echo signals to estimate the position of the target. Some of these LS methods can be modified to apply to this problem, but the performance is not satisfactory because these methods are not specifically designed for this condition. The existing TOF estimation methods are mostly aimed at stationary targets whose ultrasonic echo signals are complete and undistorted waveforms, as shown in Figure 1. However, in practical applications, it is often impossible for the target object to be completely stationary, and the target object is often a moving object. These conditions often lead to the distortion of the envelope waveform, resulting in inaccurate TOF estimation, as shown in Figure 2. Furthermore, inaccurate TOF estimation results often lead to inaccurate object localization or unstable object tracking. This paper analyzes the most distorted ultrasonic echo waveform, and the proposed system integrates various methods to handle TOF estimation in most distortion cases, as shown in Figure 2. Furthermore, the remaining distortion cases that lead to inaccurate TOF estimation are processed by the stage of object localization. The object localization stage integrates several methods to provide a more stable and smoother result for the localization and tracking of stationary and moving objects. In this work, an ultrasonic array system consisting of one transmitter and five receivers for object localization and tracking is implemented. The transmitter transmits ultrasonic signals, and the five receivers receive the echo signals returned by the target. The localization system uses the proposed algorithm to analyze the raw data of the echo signal. First, the system detects whether there is an echo signal through a simple VAD algorithm and then starts to locate the target by the proposed algorithms. The demodulation algorithm with a low-pass filter is used to extract the envelope of the echo signal. In order to improve the accuracy and stability of TOF estimation, the GN model and the DE model are used to characterize the ultrasonic echo signal, and the NR algorithm is applied to fit the envelope curve to estimate the parameters of the models. Furthermore, the LM modification is applied to an NR algorithm to solve the ill-conditioned problem and improve the stability of the TOF estimation. Moreover, through the analysis of various distortion waveforms, a strategy is proposed to determine the optimal fitting area according to the ideal characteristics of the GN model and the DE model, which avoids the fitting of other distorted measurement data and reduces the deviation of TOF estimation. The next stage of the proposed system is to estimate the target position; thus a discrete EKF is designed to consider the interference of TOF estimation, which uses TOF to estimate the distance to locate the target position. Aiming at the problem of outliers in the measurement, the update process of EKF is improved, which effectively detects and eliminates outliers and improves the robustness of target tracking. In summary, the following algorithms are proposed in this study: one demodulation algorithm for envelope extraction, the NRLM with two models (GN and DE) for TOF estimation, and the EKF method with outlier rejection for object localization. These algorithms are used to build a system platform, and different algorithm combinations are realized in experiments to test the accuracy and stability of the proposed system. Figure 1: Ideal shape of an ultrasonic waveform. (Blue line: echo envelope; red line: raw ultrasonic signal.) Figure 2: Distorted shapes of envelope waveforms. stability of object localization and tracking. These methods are compared with each other to test which combination has the best performance.