# A Symbolic Encapsulation Point as Tool for 5G Wideband Channel Cross-Layer Modeling
## 3 Methods Applied
### Level Crossing Rate and Average Fade Duration  
Before going onward, it is desirable to illustrate subject metrics, as they show how often a complex envelope crosses the defined threshold and how long, on average, the signal remains below that level (Figure 2b).For comparison with our deterministic approach, let us briefly recall some basic analytics. Rate that complex envelope \(r\) is crossing level \(L\), it is derived from its definition:  
\[LCR_{L}=\int\limits_{0}^{\infty}\dot{r}p_{rr}(L,\ \dot{r})\ \dot{d}r,\ r\geq 0 \tag{11}\]  
where \(p_{rr}\big{(}L,\dot{r}\big{)}\) denotes joint probability density function of stochastic variable \(r(t)\) particularized at level \(L\) and its derivative \(\dot{r}\ (t)=\frac{dr}{dt}\) at the same time instant. By definition, AFD is the expected value of the length of the time intervals in which the stochastic process is below a given signal level \(L\) and is calculated by  
\[AFD=\frac{CDF(r<L)}{LCR_{L}} \tag{12}\]  
and \(CDF(r<L)\) is cumulative distributive function probability that stochastic process (complex envelope \(r\)) is less or equal to \(L\). These metrics, although often hard to express analytically, numerically are not intensive to calculate and measure. The urban environment, on the other hand, is often statistically described with Rice or Rayleigh distribution, where it is shown that  
\[LCR_{L}=\sqrt{\frac{\beta}{2\pi}}\frac{L}{\sigma_{0}^{2}}e^{-\frac{l^{2}}{2 \sigma_{0}^{2}}}=\sqrt{\frac{\beta}{2\pi}}R_{r}(L),\ L\geq 0, \tag{13}\]  
Showing that practically, LCR depends on autocorrelation function \(R_{r}(L)\), where \(\sigma_{0}^{2}\) denotes mean power and \(\beta\) is short notation of negative curvature for the autocorrelation function at the origin,  
\[\beta=-\left.\frac{d^{2}}{d^{2}\tau}R(\tau)\right|_{\tau=0} \tag{14}\]  
At the same time, the autocorrelation function is found to be  
\[R(\Delta x)=J_{0}\bigg{(}\frac{2\pi}{\lambda}\Delta x\bigg{)} \tag{15}\]with \(J_{0}\) noted the well-known Bessel function of first kind and zero order. Note that distance and time are connected here through \(\Delta x=V\Delta t=\mathrm{V}\tau\). Analogously, for AFDin the Rayleigh channel, we can express  
\[AFD_{Rayleigh}=\sqrt{\frac{2\pi}{\beta}\frac{v_{0}^{2}}{L}}\bigg{(}e^{-\frac{L ^{2}}{2v_{0}^{2}}}-1\bigg{)},\ L\geq 0, \tag{16}\]  
Although it is often hard to analytically express for different statistical distributions, LCR and AFD are not intensive to calculate and measure. For given \(\mathbf{r}\), the first thing is to configure a deterministic LCR vector. This is done by taking integer values of signal levels from lowest to highest \(\mathbf{r}\). Then, for the single integer threshold level L, count the number of crossings (samples) across vector \(\mathbf{r}\). Using the same levels, we calculate AFD vector across \(\mathbf{r}\) by counting the number of samples under the threshold and dividing with number of fading intervals. In order to evaluate wideband values, here is proposed an extension over the arbitrary frequency band where, for single frequency, LCR and AFD are calculated. Steps related to the appropriate matrices and vectors calculation are given in Table 1.  
Relating the Table 1 parameters to the 5G architecture, let us observe the physical resource block (PRB), representing the basic scheduling unit, comprised of 12 subcarriers over one orthogonal frequency-division multiplexing (OFDM) symbol, spaced apart by \(\Delta f=2^{\mu}\times 15\) KHz, where \(\mu\ \in\ \{0,\ 1,\ 2,\ 3,\ 4\}\), denotes the applied 5G transmission scheme or, so called, numerology. Therefore, for our \(\mathbf{\mathrm{f}}_{\mathrm{axis}}\) parameter, let us use frequency range starting from 3500 MHz with the step aligned according to \(\Delta f\). Another view is from the PRB granularity level, where we could set the resolution to \(12\cdot\Delta f\). In the time scale, the basic scheduling unit in 5G is indicated by symbol length indicator value (SLIV).