# Data-Driven Analysis of Outdoor-to-Indoor Propagation for 5G Mid-Band Operational Networks
## 2 Background, Related Work, and Contributions
### Channel Impulse Response, Power Delay Profile, and Propagation Parameters  
The Channel Impulse Response (CIR) mathematically formalizes the time and power dispersion experienced by a signal when propagating over a wireless channel. It is usually represented as the sum of all the paths a signal can undertake from a transmitter to a receiver. Paths are usually denoted as the multipath component (MPC) of the CIR, and reach the receiver at different time instants and with different power levels. The CIR of such a time-varying wireless channel, \(h(t)\), can thus be written as follows [32]:  
\[h(t)=\sum\limits_{n=0}^{N-1}a_{n}e^{i\theta_{n}}\delta(t-\tau_{n}), \tag{1}\]  
where \(a_{n}\), \(\tau_{n}\), and \(\theta_{n}\) are the amplitude, delay, and phase of the \(n\)th path, and \(N\) is the total number of paths.  
A well-known model for CIRs in indoor environments is the Saleh-Valenzuela (S-V) model [32]. In this model, the MPC is divided into clusters, where the power associated to each cluster and the power of the paths in each cluster decays exponentially. The interarrival times of the clusters, and of the paths within the clusters, also follow exponential distributions. Following [32], CIRs of indoor environments can be thus written as follows [32]:  
\[h(t)=\sum\limits_{l=0}^{L-1}\sum\limits_{n=0}^{N_{l}-1}a_{nl}e^{i\theta_{nl}} \delta(t-T_{l}-\tau_{n,l}), \tag{2}\]  
where \(a_{n,l}\), \(\tau_{n,l}\), and \(\theta_{nl}\) are the amplitude, delay, and phase of the \(n\)th path in the \(l\)th cluster, \(T_{l}\) is the arrival time of the \(l\)th cluster of paths, \(L\) is the total number of clusters, and \(N_{l}\) is the number of paths in the \(l\)th cluster.  
The propagation properties of a wireless channel are usually evaluated on the so-called Power Delay Profile (PDP). The PDP is directly derived from the CIR, and shows the intensity of the received power over the different channel paths, as a function of path delays. From Equation (1), the PDP can be represented as follows:  
\[\psi(t)=\sum\limits_{n=0}^{N-1}|a_{n}|^{2}\delta(t-\tau_{n})=\sum\limits_{n=0 }^{N-1}g_{n}\delta(t-\tau_{n}), \tag{3}\]  
where \(|a_{n}|^{2}=g_{n}\) is the power associated to the \(n\)th path with delay equal to \(\tau_{n}\).  
Different propagation parameters can be evaluated on PDPs and are used to quantify time-related and power-related channel propagation characteristics.  
As for time features, the _mean excess delay_\(\tau\) is the first moment of the PDP and measures the power-weighted average delay of all paths, that is:  
\[\tau=\frac{\sum\limits_{n=0}^{N-1}g_{n}\tau_{n}}{\sum\limits_{n=0}^{N-1}g_{n}}. \tag{4}\]  
The _RMS delay spread_\(T_{\tau}\) is defined as the square root of the second central moment of the PDP, and provides a measure of the multipath richness of a communication channel. It can be calculated as follows [33]:  
\[T_{\tau}=\sqrt{\frac{\sum\limits_{n=0}^{N-1}g_{n}(\tau_{n}-\tau)^{2}}{\sum \limits_{n=0}^{N-1}g_{n}}}. \tag{5}\]  
As for power features, the _path loss_ quantifies signal power reduction on the transmitter-to-receiver path. It increases with the distance between transmitter and receiver and is evaluated, in dB, as follows:  
\[PL_{dB}=10\text{log}_{10}\frac{P_{t}}{P_{t}}, \tag{6}\]where \(P_{t}\) is the transmitted signal power and \(P_{r}\) is the sum of the power received on the \(N\) paths in the PDP, i.e., \(P_{r}=\sum\limits_{n=0}^{N-1}g_{n}P_{t}\) (in linear units).  
The _Rician K-factor_ is commonly used to indicate the quality of a wireless channel [34]; it is defined as the ratio between the received power of the strongest path and the sum of powers of all other paths. Mathematically, it can be expressed as follows [35]:  
\[K_{R}=10\text{log}_{10}\frac{g_{max}}{\sum_{n=0}^{N-1}(g_{n}-g_{max})}, \tag{7}\]  
where \(g_{max}\) is the highest gain across paths.