# A Symbolic Encapsulation Point as Tool for 5G Wideband Channel Cross-Layer Modeling
## 4 Symbolic Encapsulation Roadmap  
Thanks to the concept of microsimulation semi-symbolic analysis [36], all variables and operations are encapsulated in one framework, which is accessed by simply calling its reference. The encapsulation process is the store of all relevant parameters, data, and behavior in one component--a symbolic encapsulation point. The complete procedure is shown in Figure 4. The figure represents a roadmap from inputs to symbolic and numerical outputs. Symbolic outputs represent expressions in closed-form, while numerical outputs are values for generating visual characteristics. The entrance of the roadmap is input signal \(\circleddelta\) that can be measured or simulated. In Section 3, the way to define the parameters of second-order statistics \(\circleddelta\) is shown. The next phase \(\circleddelta\) involves complex calculations in order to bring the parameters of the second-order statistics (LCR and AFD) into the form for fast computation [37; 38; 39; 40]. Microsimulation semi-symbolic elements \(\circleddelta\) are then applied to obtain closed-form expressions \(\circleddelta\) that provide encapsulated frames \(\circleddelta\). They can be further simplified \(\circleddelta\) if possible or manipulated \(\circleddelta\)At the same time, numerical values can be obtained 1 and use them to obtain the characteristics 1. If the input signal is simulated, in fact computer-generated, these parameters can be set via \(\circled{\textregistered}\). In this framework, it is possible to set the speed of the mobile users, the number of scatterers, fast Fourier transform (FFT) size, simulation length, sampling rate, frequency free space loss, base station distance, and radius of scatterers. In [37], the methodology of solving very complex scenarios in wireless communications is presented. The results of the error function of symbol error probability in terms of the number of the iteration are illustrated. Paper [38] presents, in detail, the methodology of the key parameters that simplify complex calculations. The paper [39] shows how it is possible to speed up a complex calculation where the relative errors do not exceed more than 8% between the original and the approximate value. In [36], the calculation speed was taken into account, which speeds up the calculation of outage probability by 955 times, and reduces the number of mathematical operations by almost 4 times, while for statistical parameters of the second-order, LCR is accelerated by 20 times and AFD by \(15\times 10^{3}\) times, for a relative error ranging between 0.5% and 2.5%.  
Figure 4: 5G tool roadmap.  
Figure 5 shows a full graphical environment for monitoring all parameters. Figure 5a shows the magnitudes of the complex envelope in time (s) and dB as well as the complex envelope normalized to RMS. Figure 5b shows modulo \(\pi\) of the complex envelope, absolute phase of the complex envelope, and phase histogram. Figure 5c provides the autocorrelation function and cumulative distribution function. Figure 5d shows a normalized RF spectrum and a normalized baseband spectrum. Figure 5e shows the LCR and AFD parameters. The visual environment provides the display of 3D features and in the next session, complex examples for the three scenarios are presented.  
Figure 5: Full visual environment of tool: (**a**) magnitude; (**b**) phase; (**c**) autocorrelation function (ACF) and cumulative distribution functions (CDF); (**d**) Doppler analysis; (**e**) level-crossing rate (LCR) and average fade duration (AFD).