# Influence of Noise-Limited Censored Path Loss on Model Fitting and Path Loss-Based Positioning +
## 3 Path Loss and Antenna Model  
A PL model describes the probability distribution as a function of the link distance, and possibly, also as a function of frequency, BS height, etc. The model has two parts--a model for the expected value and a model for SF, i.e., the variation from the expected value. The expected value is typically modeled as a simple linear function of the logarithm of the link distance and the SF is modeled as a zero-mean log-normal distribution. If needed, the SF model may include the auto-correlation function and correlation distance. Usually, the standard deviation of the SF, \(\sigma\), is assumed to be a constant, e.g., [14], but it can be also a function of the link distance [11]. In channel models, such as [14; 30], there are two models, one for the line-of-sight (LOS) scenario and one for the non-line-of-sight (NLOS) scenario, depending on whether the line-of-sight is clear of obstacles. In positioning, typically there is no distinction between LOS and NLOS, and the same model is used for both--e.g., [2; 3; 19; 13; 21; 22; 23; 9]. Path loss-based positioning always has two PL distributions that may be more or less different. These are the true PL distribution and the PL model fitted to training data and used in the positioning to estimate the link distances based on the measured PL to the BSs. The purpose of the training phase is to attain accurate information on the PL distribution. In practice, there are several possible reasons why the model used in the positioning phase may either underestimate or overestimate the link distances as compared to the true PL distribution. In this paper, we examine one such reason, namely, the effect of ignoring the noise-limited censored PL data. A simple log-distance PL model is assumed as follows:  
\[PL(d)=\overline{PL}(d)+S, \tag{1}\]  
\[\overline{PL}(d)=10\cdot\alpha\cdot\log_{10}(d/d_{0})+\beta, \tag{2}\]  
where \(S\) is the zero-mean shadow fading with variance \(\sigma^{2}\), \(\overline{PL}(d)\) is the link distance-dependent expected path loss, and \(d_{0}=1\) m is a reference distance. Equation (2) has two free parameters: path loss exponent \(\alpha\) and floating intercept point \(\beta\), which can be interpreted as the mean PL at \(d_{0}\). Parameters \(\alpha\), \(\beta\), and \(\sigma\) can be attained by fitting the model to measurement data or, e.g., taken from a channel model. Additionally, other PL models exist, such as the close-in (CI) reference model, where \(\beta\) is fixed to free-space path loss at the reference distance [31; 32], and the dual slope model in which the path loss exponent changes after a break point [11; 14]. In this work, we use the single-slope log-distance PL model (1) and (2) when fitting the model to training data and in the positioning phase. It should be noted that the methods presented in this paper are not limited only to the single-slope model. Models such as [14] also define SF correlation properties. SF inter-site cross-correlation is small for widely spaced sites and large for closely spaced BSs [33; 34]. Therefore, we use zero correlation between BS sites and the same SF for different beams of the same BS location. In this work, we do not use any tracking algorithm, and therefore SF auto-correlation function and correlation distance are not defined. In the case of a directive antenna, we approximate  
\[PL(d,\theta)=PL(d)-G(\theta), \tag{3}\]  
\[PL(d,\theta)=10\cdot\alpha\cdot\log_{10}(d/d_{0})+\beta-G(\theta)+S, \tag{4}\]where \(G(\theta)\) is the antenna gain pattern. A simple BS antenna pattern from [14; 35] is used:  
\[A(\theta)=-\min(12(\theta/\theta_{\rm 3 dB})^{2},20\;{\rm dB}), \tag{5}\]  
where \(\min(\cdot)\) denotes the minimum function, \(\theta\) is the offset angle from boresight, and \(\theta_{\rm 3 dB}\) is the antenna half-power beam-width. The relative sidelobe level is fixed at a constant 20 dB below the maximum gain.