**2. Case study 1: two-ray model**

Initially, the accuracy of the RT and SCM tools are verified using a simple scenario involving the channel capacity of the classical Two-Ray channel model over a flat Perfect Electrically Conducting (PEC) surface since an analytical solution is already available. Wireless InSite by Remcom is used for the RT approach while MIMObit is used to calculate the channel characteristics for a set of Tx and Rx antennas inserted in a half free-space environment above an infinite PEC surface.

To test the various settings and parameters for the proper operation of both software tools, the first case focused on evaluating the SISO channel capacity in a Two-Ray model and compare it to the theoretical calculations. The Two-Ray model consists of a single large flat layer of PEC. A single half-wavelength dipole transmitting antenna operating at 2.45 GHz and an identical receiving antenna are placed 30 m apart at a height of 2 m. The model consists of two rays, a Line-of-Sight (LOS) component and a Non-Line-of-Sight (NLOS) component that is reflected over the ground plane. **Figure 1** shows the two rays between the Tx antenna and the Rx antenna.

To evaluate the received power at the receiver, Eq. (1) is used [17].

$$P\_r = 4P\_t \left[\frac{\lambda}{4\pi d}\right]^2 \mathbf{G}\_t \mathbf{G}\_r \cos^2\left(\frac{\Delta \mathcal{O}}{2}\right) \tag{1}$$

where **Pr** is the received power in W, **Pt** is the transmitted power in W, **λ** is the wavelength, **d** is the distance between the Tx and Rx, **Gt** and **Gr** are the gain of the

**Figure 1.** *Two-ray model.*

Tx and Rx antennas, respectively, and the phase difference, **Δ***ϕ*, can be calculated using Eq. (2).

$$\Delta\phi = \frac{2\pi\left(\sqrt{\left(h\_t + h\_r\right)^2 + d^2} - \sqrt{\left(h\_t - h\_r\right)^2 + d^2}\right)}{\lambda} \tag{2}$$

The Shannon capacity formula is used to find the SISO channel capacity [18] as shown in Eq. (3).

$$\mathbf{C} = \mathbf{B} \, \log\_2(\mathbf{1} + \text{SNR}) \tag{3}$$

where **C** is the channel capacity in bits/s, **B** is the channel bandwidth in Hz and **SNR** is the Signal to Noise Ratio. The thermal noise is assumed to be equal to �100.99 dBm which is the standard used in [15, 16]. The geometry of the model simulated using Wireless InSite is shown in **Figure 2**. An equal total noise and interference of �100.99 dBm is used for both software tools. The channel capacity obtained from the three approaches are displayed in **Table 1**.

**Table 1** shows a close agreement between the SISO channel capacity results obtained from the RT tool, SCM tool and analytical calculations. A difference of

**Figure 2.**

*Two-ray model in wireless insite.*


**Table 1.** *Two-ray SISO model.*

*Stochastic versus Ray Tracing Wireless Channel Modeling for 5G and V2X Applications… DOI: http://dx.doi.org/10.5772/intechopen.101625*

0.09% is noted between the RT tool and theoretical channel capacity while 0.48% difference is observed between the SCM capacity and the theoretical one. This provides impetus to perform precise channel modeling comparisons when moving to the more realistic indoor and outdoor environments.

To provide further validation over the reliability of the software tools, the next comparison uses the same Two-Ray model, however, this time using a 2x2 MIMO scenario. Two-vertically polarized, half-wavelength dipole antennas with a separation of 2λ between the elements, operating at 2.45 GHz are used as the Tx and Rx antennas, centered at the same locations as the SISO two-ray model, separated by 30 m horizontally at a height of 2.5 m and 1.5 m, respectively. A single LOS ray and a single NLOS ray propagate from each Tx antenna element to each Rx antenna element, resulting in a total of eight rays in the model. The MIMO channel capacity can be expressed as shown in Eq. (4) [19]:

$$\mathbf{C} = \mathbf{B} \cdot \log\_2 \left| I + \frac{\text{SNR}}{N\_t} \mathbf{H} \mathbf{H}^H \right| \tag{4}$$

where **I** is the identity matrix, **Nt** is the number of transmitting antenna elements and **H** is the channel matrix between the Tx and the Rx.

The achieved MIMO channel capacity is 24.92 bps/Hz in RT and 23.01 bps/Hz in SCM with a difference of 7.98% which further validates the operation of these channel modeling tools.
