# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 2 Noise-Limited Path Loss and Positioning  
In this section, we present some definitions and prior works related to PL-based positioning. These include the definitions of RSS and PL, the determination of a PL value,and a review of prior works on the two phases of PL-based positioning with noise-limited PL (or RSS). The PL and antenna models used in this study are then presented in Section 3. Path loss is the inverse of small-scale-averaged path gain between the base station (BS) and the mobile station (MS) calculated as the instantaneous local channel gain averaged over the small-scale fading. The positioning can be done based on RSS or PL. Provided that the RSS is small-scale averaged, the only relevant difference is that for PL the transmit power needs to be known. In both cases, the measured RSS or PL is noise limited, and it is therefore censored data. Noise-limited RSS is under the receiver noise threshold level and noise-limited PL is over the path loss noise threshold. In this paper, the notation is given for PL, but the methods and equations apply also for RSS positioning with minor changes. The instantaneous local channel gain is the difference between transmitted and received power, and importantly, it includes the small-scale fading, i.e., multipath fading. For PL, the small-scale-averaging can be performed by averaging over traveled distance, time, antenna elements, or frequency. If the multipath powers can be resolved, the PL can be calculated as the inverse of sum of the multipath powers. Measurement of RSS or the multipath powers is limited by noise which affects the PL values in case of poor or limited signal to noise ratio (SNR). Therefore, even when calculating a single PL value the data are truncated [24] or censored [25]. Reference [24] examines how to calculate PL from instantaneous RSS that is truncated by noise. In [25], the detectable multipaths are limited by the measurement noise, and therefore, the sum of the multipath powers is censored data. Naturally, a PL measurement can also be unaffected by noise when the SNR is good or simply censored by the noise power when the measurement is practically only noise. For simplicity, in this paper, we assume that the PL values are either available or censored by noise. Censored PL samples are typically ignored as outages and are not taken into account, which can lead to significant error in the PL model distribution. The censored PL samples can be taken into account when fitting the PL model to the training data using Tobit MLE [10; 11; 13]. For example, in [10] it is shown with measured path loss data that by ignoring the censored samples the slope of the PL model, i.e., path loss exponent, is drastically underestimated at 1.3 instead of 2.2. Thus, it is important to use the Tobit MLE when fitting the model to the training data. Note that ignoring the censored PL data in the training phase may lead to a similarly erroneous PL model as when getting the model from the literature without conducting the laborious training measurements. In the training data, there are two types of data samples--ones with measured PLs and those with PLs larger than the threshold. In the positioning phase, there are two types of data samples--ones with distance estimates and ones with distance estimates more than a threshold distance. The noise-limited measurements can be taken into account in the positioning phase [3; 19; 20; 21; 22; 23]. In [20; 22; 23], the likelihood of connecting or not connecting to a BS is taken into account in the case of time-of-arrival (TOA) positioning. In [20], it is noted that the audibility information resolves most of the ambiguity when a unique solution is not available with less than three signals measured in 2D. Reference [19] proposes a likelihood function for failing to detect a device in the case when the received power is below a threshold value. The noise level is taken into account in the likelihood function using proximity and quantized RSS positioning in [21]. Two-phased RSS-based positioning method is presented in [3].