# A Symbolic Encapsulation Point as Tool for 5G Wideband Channel Cross-Layer Modeling
## 3 Methods Applied  
### Propagation Deterministic Mechanism  
Real-world multipath/multiscatter propagation scenarios for three typical 5G use cases are depicted as in Figure 1: high-speed train (HST), vehicle-to-vehicle (V2V) in an urban environment, and crossroad (V2 Vcrossroad) scenarios. First, case (a) comprises transmission from a fixed base station transmitter to a fast moving receiver in the presence of static point scatterers in the surrounding area. In case (b), either the transmitter or receiver is moving with static scatterers between, in the way that they are out of direct line of sight. Under (c), besides the transmitter/receiver, a larger number of scatterers are moving on their way, also. This way, in order to compare, we gradually increased the portion of channel non-stationarity. Under the effect of multipath propagation, the signal at the receiver is formed by a combination of multiple signal components, attenuated and delayed. The basic assumption to build our deterministic model was planar wave propagation of the transmitted and modulated (RF) baseband signal \(x_{CE}(t)\), which, in standard notation, is given by expression  
\[x_{RF}(t)=Re\Big{[}x_{CE}(t)e^{j2\pi f_{0}t}\Big{]} \tag{1}\]  
and the signal received through multipath scattering surrounding and free-space propagation is  
\[y_{RF}(t)=Re\Bigg{\{}\sum_{i=1}^{N(t)}c_{i}(t)e^{j\phi_{n}(t)}x_{CE}(t-\tau_{i })e^{j2\pi f_{0}t-\tau_{i}}\Bigg{\}}=Re\Big{[}r(t)e^{j2\pi f_{0}t}\Big{]} \tag{2}\]  
where \(N(t)\)is the time-varying number of scatterers, \(c_{i}(t)e^{j\phi_{n}(t)}\) magnitude and phase contribution from scattereri, \(\tau_{i}\) propagation delay of each multipath component, and \(f_{0}\)carrier frequency. Hence, the received complex (baseband) envelope is  
\[r(t)=\sum_{i=1}^{N(t)}c_{i}(t)e^{j\varphi_{n}(t)}e^{-j2\pi f_{0}\tau_{i}}x_{CE }(t-\tau_{i}) \tag{3}\]  
For planar wave propagation, at the receiver side, following that \(\tau_{i}=\frac{d_{i}}{c}\), with \(c\) speed of light, \(d_{i}\) is \(i^{\text{th}}\) component path length and \(\lambda_{0}=\frac{c}{f_{0}}\) carrier wavelength, we obtain  
\[r(t)=\sum_{i=1}^{N(t)}c_{i}(t)e^{j\varphi_{i}(t)}e^{-j\frac{2\pi}{\lambda_{0} }d_{i}}x_{CE}\Big{(}t-\frac{d_{i}}{c}\Big{)} \tag{4}\]  
Mobile user equipment is moving with some constant velocity \(V\), causing the existence of Doppler frequency at each beam \(i\) that arrives at the receiver with the angle \(\alpha_{i}\). At this instance, we focus on fast variations of a signal complex envelope at the receiver (fast fading) and when no direct line-of-sightis available. For analysis of fast fading separately from slow variations (slow fading due to distance change), one must define a small localarea, where observation takes route section \(\Delta\)of typically 10-40 wavelengths (\(\lambda\)) of transmitted signal, taken from our recent work [32]; see Figure 2a. Samples of \(r(t)\) are taken at some fractions of wavelength \(F\),at each \(\Delta x=\lambda_{0}/F\) point of mobile movement, equivalently each \(\lambda_{0}/FV\) second. The contribution with Doppler shift from \(i^{\text{th}}\) ray, in summary the received signal, depends on the angle of arrival \(f_{Di}=f_{max}cos\alpha_{i}\), where \(f_{max}=V/\lambda_{0}\) presents maximum Doppler shift. We observe changes either in amplitude or phase. Two factors in (4) comprise phase changes:  
* \(\varphi_{i}(t)\), introduced by the scatterer reflection coefficient and here, omitted due to complexity;
* \(\Delta\phi_{i}\), originating from phase rotation along the electrical distance of a wave. \[\Delta\phi_{i}=\frac{2\pi}{\lambda_{0}}d_{i} \tag{5}\]  
It should be noted that realization without term \(\varphi_{i}(t)\) shows comparable results with empirical measured models. Now, we observe amplitude changes of \(r(t)\):  
* \(c_{i}(t)\), which represent ith scatterer contribution to overall magnitude of received signal;
* \(x_{CE}\big{(}t-\frac{d_{i}}{c}\big{)}\), term that follows changes of transmitted signal.