# Methodology for Simulating 5G and GNSS High-Accuracy Positioning
## 3 Characterization of the Observables
### cmWave  
The exploitation of cellular networks for positioning purposes can be based on opportunistic or dedicated approaches. Opportunistic positioning uses the cellular transmissions in the downlink without any interaction with the network operator. The main impairments of this approach are the inter-cell interference, multipath, receiver noise and lack of knowledge of the network, such as the precise position and time synchronization of the BSs. Most of these impairments can be avoided with dedicated positioning services, where the network operator dedicates spectrum resources, protocols and network equipment to determine the mobile location with downlink or uplink transmissions. For instance, the LTE standard specifies the positioning reference signal (PRS) within low-interference subframes and LTE positioning protocol (LPP), in order to perform downlink observed time-difference of arrival (OTDoA) positioning. In addition, the cellular network can increase the amount of consecutive positioning subframes, in order to compute ranging measurements over an extended integration time to cope with the noise effects. Therefore, the main source of ranging error in cellular networks is due to the effect of multipath, especially with the relative narrow bandwidth allocated at cmWave bands. In this sense, our characterization of the cmWave observables is based only on the multipath-induced error. The ranging bias is here determined by the accuracy of a threshold-based time-delay estimator in a certain multipath channel. These estimators compute the cross-correlation function \(R_{\mathrm{yx}}\left(\tau\right)\) between the received signal and the transmitted positioning pilots, and then they provide a time-delay estimate for the first peak above a certain threshold. Due to their low complexity, they are widely adopted in the  
Figure 5: \(C/N_{0}\) and standard deviation of the GNSS observables as a function of the satellite elevation, by considering a GPS L1 C/A signal sampled at \(F_{S}=4\) MHz. literature. In addition, the performance of advanced multipath mitigation techniques is comparable to these threshold-based estimators for non-resolvable multipath and NLoS conditions. The time-delay estimation of the threshold-based approach is defined as  
\[\hat{\tau}=\min_{\tau_{\text{min}}\leq\tau<\tau_{\text{max}}}\left\{\tau\right\} \quad\text{s.t.}\quad\Lambda\left(\tau\right)\geq\Lambda_{\text{thr}}, \tag{4}\]  
where the estimation range is bounded by \(\tau_{\text{min}}\) and \(\tau_{\text{max}}\), and the likelihood function is \(\Lambda\left(\tau\right)=\left|R_{\text{yx}}\left(\tau\right)\right|^{2}\). Let us define the cross-correlation function in the absence of noise as  
\[R_{\text{yx}}\left(\tau\right)=R_{\text{xx}}\left(\tau\right)*h\left(\tau \right), \tag{5}\]  
where \(R_{\text{xx}}\left(\tau\right)\) is the auto-correlation function, and \(h\left(\tau\right)\) is the channel impulse response (CIR). Since the 5G physical layer is based on the orthogonal frequency division multiplexing (OFDM), the auto-correlation of the cmWave signal is  
\[R_{\text{xx}}\left(\tau\right)=\sum_{n\in\mathcal{N}}e^{-j\frac{2\pi n\tau}{N }}, \tag{6}\]  
where \(n\) is the pilot index within the set of subcarriers \(\mathcal{N}\) from a total of \(N\) subcarriers. The CIR is modeled as  
\[h\left(\tau\right)=\sum_{k=0}^{L-1}h_{k}\cdot\delta\left(\tau-\tau_{k}\right), \tag{7}\]  
where \(h_{k}\) is the channel coefficient and \(\tau_{k}\) is the tap delay for the \(k\)-th tap from a total of \(L\) multipath taps. The 5G cellular standard specifies in [16] geometry-based stochastic channels, where the delay distribution of the multipath rays mainly depends on the propagation scenario, the LoS conditions, and the carrier frequency. These channel models are an extension (to mmWave transmissions) of the WINNER II channel models in [28], where the main differences are based on the scaling of the channel model parameters as a function of the carrier frequency.