## 1 Introduction  
Indoor positioning is important for location-based service, internet of things and health care [1; 2]. Many wireless systems, such as Ultra-Wideband, Bluetooth and cellular communication systems, have been used for indoor positioning. As one of the most widely used wireless systems, the cellular communication system supports mobile phone positioning from 2G to 5G [3]. In recent years, with the development of the location service market, the cellular communication system based positioning can be used in many new fields such as vehicle positioning [4; 5], location-aware communications [6]and location-based applications [7]. The 3 rd Generation Partnership Project (3 GPP) also puts higher requirements on the positioning accuracy of the 5G system [8; 9]. The TDOA-only positioning technology based on cellular communication system is relatively mature. For a long time, the cellular communication system mainly realizes the user equipment (UE) positioning through the TDOA positioning method. Long Term Evolution (LTE) system provides a TDOA estimation technology called Observed Time Difference of Arrival (OTDOA) [10; 11]. In general, UE can only receive downlink signals from the nearest base station. Using OTDOA technology, base stations broadcast positioning reference signal (PRS) to UE. PRS has a unique mechanism named power boosting to make sure that the UE could receive PRSs from multiple sources so that multiple time-difference-of-arrival (TDOA) measurements can be obtained [12]. UE estimates the TDOA measurements by finding the peak of correlation of the received PRS with a shifted and conjugated version of PRS [13]. When the signal sources' locations are known, a nonlinear equation can be constructed using sources' locations and TDOA measurements to calculate UE's position. There are many existing localization algorithms for the solution of the TDOA measurement equations [14]. These algorithms can be divided into iterative methods and closed-form solutions [15]. The iterative methods usually use the Taylor-series to linearize the nonlinear equations and use Gauss-Newton method to iterate location estimation errors. Those methods can provide accurate position estimation when the initial position is appropriate, otherwise, it cannot guarantee convergence [16]. Closed-form solutions could avoid the convergence problem. The most used closed-form algorithm is a two-step Weighted Least squares (WLS) estimators developed by [PERSON] and [PERSON] [17]. This solution can reach the Cramer-Rao lower bound (CRLB) when TDOA error is small enough. However, as the error increases, its performance would degrade. [PERSON] also developed an improved version of the algorithm [18]. The improved algorithm has a better performance when the variance of the TDOA measurement errors is known. The constraint equation could improve the accuracy of the closed-form algorithm. An algorithm using earth constraint can improve accuracy [19], but this constraint is not suitable for indoor positioning. The above algorithms are often used for UE position estimation after obtaining TDOA measurements through OTDOA technology. There are still many problems in the exiting system that make the positioning accuracy not high enough. Since the outdoor base station cannot meet the indoor signal coverage requirements, cellular communication system uses the indoor distribution system for indoor communication services [20]. For the indoor distribution system, all TDOA-only positioning methods have the same shortcoming. In the indoor environment, deployment of communication indoor distribution system leads to a higher geometric dilution of precision (GDOP), which is not suitable for TDOA-only based positioning. The GDOP is defined as the ratio of the accuracy limitation of a position fix to the accuracy of measurements and the concept is widely used in positioning performance analysis [21].