# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 1 Introduction  
### Motivation  
Radio-based positioning has rapidly grown into one of the most significant features in future wireless networks. As stated in the specifications of the upcoming 5 th generation of new radio networks in [1], positioning is considered part of basic network capability, and it offers a wide variety of performance requirements tailored to specific needs of numerous use cases and industry verticals. Path loss (PL) or received signal strength (RSS)-based positioning, studied earlier, for example, in [2; 3; 4; 5; 6; 7; 8], enables low-cost positioning capability. PL-based positioning is especially useful for use cases with limited power and computational resources. In addition, PL-based positioning can introduce additional support to various other high-precision positioning and tracking solutions. As shown, for example, in [8; 9] PL or RSS positioning can provide increased positioning accuracy, availability, stability or reliability when combined with other methods. Since PL and RSS are power-related measurements, they are typically continually measured and monitored in mobile networks over multiple base stations (BS) to support mobility management and other radio resource management functionalities. Thus, PL-based positioning can rely on regular reference signals of the underlying communications system without introducing any additional training overhead due to positioning capability. However, due to a challenging and highly dynamic propagation environment, PL-based positioning methods aretypically limited to positioning accuracy of tens or hundreds of meters in outdoor cellular networks [3; 5; 6].  
Conventionally PL-based positioning has two phases (e.g., [3; 4; 5]): (i) fitting a PL model to training data, if such training data are available, and (ii) determining link distance estimates based on the PL model and calculating the position estimate. Channel measurements can be used to measure the training data and then a PL model can be fitted to the data [3; 10; 11; 12]. The PL model describes the link distance dependency and the variation from the expected value, i.e., shadow fading (SF). Sometimes training data may be unavailable and the PL model can be taken from a standard channel model, e.g., [14]. In practice, the maximum measurable PL is limited by measurement noise. Therefore, in both the training data and in the positioning it may happen that the PL value cannot be determined. In that case, the PL is known to be more than the noise threshold level. When data above or below a certain range are immeasurable, meaning that all data above or below a certain range are counted, but not measured, the term is censored data [10; 11; 13; 15; 16; 17; 18]. The PL model can be fitted to censored data by using Tobit maximum likelihood estimation (MLE) [10; 11; 13]. Just as in the training phase, the noise threshold limits the maximum PL in the positioning phase. When PL measured from a certain BS is larger than the noise threshold, the true location is likely to be far from the BS, and the true distance is subject to the distance dependence of the PL and the threshold level. Therefore, measuring PL at the positioning phase is censoring data in the same fashion as in the case of the training data. Censored PL can be taken into account in the likelihood function in the positioning phase [3; 13; 19; 20; 21; 22; 23]. Likelihood function examples of a measured and censored PL are presented in Figure 1.