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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 `phased.TwoRayChannel`

and `phased.WidebandTwoRayChannel`

System objects model
propagation time delay, phase shift, Doppler shift, and loss effects
for both paths. For the reflected path, loss effects include reflection
loss at the boundary.

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

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

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

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