## Two-Ray Multipath Propagation

A two-ray propagation channel is the next step up in complexity from a
free-space channel and is the simplest case of a multipath propagation environment. The
free-space channel models a straight-line *line-of-sight * path from
point 1 to point 2. In a two-ray channel, the medium is specified as a homogeneous,
isotropic medium with a reflecting planar boundary. The boundary is always set at *z
= 0*. There are at most two rays propagating from point 1 to point 2. The first
ray path propagates along the same line-of-sight path as in the free-space channel (see the `phased.FreeSpace`

System object™). The line-of-sight path is often called the *direct
path*. The second ray reflects off the boundary before propagating to point 2.
According to the Law of Reflection , the angle of reflection equals the angle of incidence.
In short-range simulations such as cellular communications systems and automotive radars,
you can assume that the reflecting surface, the ground or ocean surface, is flat.

The figure illustrates two propagation paths. From the source
position, *s _{s}*, and the receiver
position,

*s*, you can compute the arrival angles of both paths,

_{r}*θ′*and

_{los}*θ′*. The arrival angles are the elevation and azimuth angles of the arriving radiation with respect to a local coordinate system. In this case, the local coordinate system coincides with the global coordinate system. You can also compute the transmitting angles,

_{rp}*θ*and

_{los}*θ*. In the global coordinates, the angle of reflection at the boundary is the same as the angles

_{rp}*θ*and

_{rp}*θ′*. The reflection angle is important to know when you use angle-dependent reflection-loss data. You can determine the reflection angle by using the

_{rp}`rangeangle`

function and
setting the reference axes to the global coordinate system. The total
path length for the line-of-sight path is shown in the figure by *R*which is equal to the geometric distance between source and receiver. The total path length for the reflected path is

_{los}*R*. The quantity

_{rp}= R_{1}+ R_{2}*L*is the ground range between source and receiver.

You can easily derive exact formulas for path lengths and angles in terms of the ground range and object heights in the global coordinate system.

$$\begin{array}{l}\overrightarrow{R}={\overrightarrow{x}}_{s}-{\overrightarrow{x}}_{r}\\ {R}_{los}=\left|\overrightarrow{R}\right|=\sqrt{{\left({z}_{r}-{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{1}=\frac{{z}_{r}}{{z}_{r}+{z}_{z}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{2}=\frac{{z}_{s}}{{z}_{s}+{z}_{r}}\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ {R}_{rp}={R}_{1}+{R}_{2}=\sqrt{{\left({z}_{r}+{z}_{s}\right)}^{2}+{L}^{2}}\\ \mathrm{tan}{\theta}_{los}=\frac{\left({z}_{s}-{z}_{r}\right)}{L}\\ \mathrm{tan}{\theta}_{rp}=-\frac{\left({z}_{s}+{z}_{r}\right)}{L}\\ {{\theta}^{\prime}}_{los}=-{\theta}_{los}\\ {{\theta}^{\prime}}_{rp}={\theta}_{rp}\end{array}$$