# Computation of Each Ray’s Contribution

Mathematical background and equations show how WinProp calculates the contribution to the prediction for each ray.

For the computation of the rays, not only the free space loss has to be considered but also the loss due to the reflections and (multiple) diffractions. This is either done using a physical deterministic model or using an empirical model.
Note: This only affects the determination of the reflection and diffraction coefficients. The prediction itself always remains a deterministic one, thus the same rays are taken into account.

The deterministic model uses Fresnel equations for the determination of the reflection and transmission loss and the GTD/UTD for the determination of the diffraction loss.1 This model has a slightly longer computation time and uses three physical material parameters (permittivity, permeability and conductivity). For details see Landron2.

The empirical model uses five empirical material parameters (minimum loss of incident ray, maximum loss of incident ray, loss of diffracted ray, reflection loss, transmission loss). For correction purposes or the adaptation to measurements, an offset to those material parameters can be specified.

The empirical model has the advantage that the needed material properties are easier to obtain than the physical parameters required for the deterministic model. Also, the parameters of the empirical model can more easily be calibrated with measurements. It is, therefore, easier to achieve high accuracy with the empirical model.

Both diffraction models are based on the angles shown in Figure 1

For the empirical diffraction model the loss ${L}_{B}$ of the diffracted rays is computed depending on the angles $\phi$ and $\phi \text{'}$ using the following equations:

(1) $\Psi =\Phi -\Phi \text{'}$
(2) ${a}_{b}\text{=}\left\{\begin{array}{cc}{a}_{{b}_{\mathrm{max}}}+\frac{{a}_{{b}_{\mathrm{min}}}-{a}_{{b}_{\mathrm{max}}}}{90°}\Phi \text{'}& \Phi \text{'}<90°\\ {a}_{{b}_{\mathrm{min}}}& \Phi \text{'}\ge 90°\end{array}\right\}$
(3)
(4) ${a}_{k}\text{=}{a}_{d}-{a}_{{k}_{aus}}$
(5) ${L}_{B}\text{=}{a}_{d}-{a}_{k}$

The angle dependencies are derived from the uniform diffraction theory (UTD) by the evaluation of measurements with different materials (brick, concrete) in an anechoic chamber and can be varied with the parameters ${a}_{{b}_{\mathrm{min}}}$ , ${a}_{{\text{b}}_{\text{max}}}$ and ${a}_{{k}_{aus}}$ within appropriate limits. With these three parameters, the model can be calibrated with measurements.

## Breakpoint

In free space, there is a reverse proportional relation between the square of the distance ${d}_{0}$ from the transmitter to the receiver and the power at the receiver ( ${A}_{0}$ is the propagation factor):

(6) ${A}_{\text{0}}=\frac{1}{{d}_{\text{0}}{}^{2}}$

To account for different propagation scenarios and to allow the user to manipulate the free space loss computation, the above conditions were considered by an extension of the equation:

(7)

With this approach a smooth transition of the free space loss is ensured. In general the breakpoint distance depends on the transmitter height, the height of the antenna at the mobile station and the frequency. The parameter BP is the breakpoint distance that is set to a default value according to the following formula:

(8) $\text{BP=}\frac{4\pi {h}_{TX}{h}_{RX}}{{\lambda }_{\text{0}}}$

For distances larger than the breakpoint distance, the angles ${\alpha }_{i}$ and ${\alpha }_{r}$ approach 90° and the reflection coefficient approaches -1. The two rays approach destructive interference. The received power then depends on distance as indicated in Equation 7, where the exponent P2 tends to be significantly larger than the free-space exponent.

The parameters ${p}_{1}$ (exponent before breakpoint) and ${p}_{2}$ (exponent after breakpoint) can also be set by the user. The default values are ${p}_{\text{1}}=2.0$ and ${p}_{\text{2}}=4.0$ .

1 Balanis, “Advanced Engineering Electromagnetics”, Wiley, New York, 1989.
2 O. Landron, M. J. Feuerstein, and T. S. Rappaport: “A Comparison of Theoretical and Empirical Reflection Co-efficients for Typical Exterior Wall Surfaces in a Mobile Radio Environment,” IEEE Transactions on Antennas and Propagation, vol. 44, pp. 341–351, Mar. 1996