The two-rays ground-reflection model is a
multipath radio propagation model
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
which predicts the
path loss
Path loss, or path attenuation, is the reduction in power density (attenuation) of an electromagnetic wave as it propagates through space. Path loss is a major component in the analysis and design of the link budget of a telecommunication system. ...
es between a transmitting antenna and a receiving antenna when they are in
line of sight (LOS). Generally, the two
antenna
Antenna ( antennas or antennae) may refer to:
Science and engineering
* Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves
* Antennae Galaxies, the name of two collid ...
each have different height. The received signal having two components, the LOS component and the reflection component formed predominantly by a single ground reflected wave.
Mathematical derivation
From the figure the received line of sight component may be written as
:
and the ground reflected component may be written as
:
where
is the transmitted signal,
is the length of the direct line-of-sight (LOS) ray,
is the length of the ground-reflected ray,
is the combined antenna gain along the LOS path,
is the combined antenna gain along the ground-reflected path,
is the wavelength of the transmission (
, where
is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and
is the transmission frequency),
is ground reflection coefficient and
is the delay spread of the model which equals
. The ground reflection coefficient is
:
where
or
depending if the signal is horizontal or vertical polarized, respectively.
is computed as follows.
:
The constant
is the relative permittivity of the ground (or generally speaking, the material where the signal is being reflected),
is the angle between the ground and the reflected ray as shown in the figure above.
From the geometry of the figure, yields:
:
and
:
,
Therefore, the path-length difference between them is
:
and the phase difference between the waves is
:
The power of the signal received is
:
where
denotes average (over time) value.
Approximation
If the signal is narrow band relative to the inverse delay spread
, so that
, the power equation may be simplified to
:
where
is the transmitted power.
When distance between the antennas
is very large relative to the height of the antenna we may expand
,
:
using the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
of
:
:
and taking the first two terms only,
:
The phase difference can then be approximated as
:
When
is large,
,
:
and hence
:
Expanding
using
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
:
and retaining only the first two terms
:
it follows that
:
so that
:
which is accurate in the far field region, i.e. when
(angles are measured here in radians, not degrees) or, equivalently,
:
and where the combined antenna gain is the product of the transmit and receive antenna gains,
. This formula was first obtained by B.A. Vvedenskij.
Note that the power decreases with as the inverse fourth power of the distance in the far field, which is explained by the destructive combination of the direct and reflected paths, which are roughly of the same in magnitude and are 180 degrees different in phase.
is called "effective isotropic radiated power" (EIRP), which is the transmit power required to produce the same received power if the transmit antenna were isotropic.
In logarithmic units
In logarithmic units :
Path loss :
Power vs. distance characteristics
When the distance
between antennas is less than the transmitting antenna height, two waves are added constructively to yield bigger power. As distance increases, these waves add up constructively and destructively, giving regions of up-fade and down-fade. As the distance increases beyond the critical distance
or first Fresnel zone, the power drops proportionally to an inverse of fourth power of
. An approximation to critical distance may be obtained by setting Δφ to π as the critical distance to a local maximum.
An extension to large antenna heights
The above approximations are valid provided that
, which may be not the case in many scenarios, e.g. when antenna heights are not much smaller compared to the distance, or when the ground cannot be modelled as an ideal plane . In this case, one cannot use
and more refined analysis is required, see e.g.
Propagation modeling for high-altitude platforms,
UAVs
An unmanned aerial vehicle (UAV), commonly known as a drone, is an aircraft without any human pilot, crew, or passengers on board. UAVs are a component of an unmanned aircraft system (UAS), which includes adding a ground-based controlle ...
,
drones, etc.
The above large antenna height extension can be used for modeling a ground-to-the-air propagation channel as in the case of an airborne communication node, e.g. an UAV , drone, high-altitude platform. When the airborne node altitude is medium to high, the relationship
does not hold anymore, the clearance angle is not small and, consequently,
does not hold either. This has a profound impact on the propagation path loss and typical fading depth and the fading margin required for the reliable communication (low outage probability).
As a case of log distance path loss model
The standard expression of
Log distance path loss model in
Bis
:
where
is the large-scale (log-normal) fading,
is a reference distance at which the path loss is
,
is the path loss exponent; typically
.
This model is particularly well-suited for measurements, whereby
and
are determined experimentally;
is selected for convenience of measurements and to have clear line-of-sight. This model is also a leading candidate for 5G and 6G systems
and is also used for indoor communications, see e.g.
[; see als]
/ref> and references therein.
The path loss Bof the 2-ray model is formally a special case with :
:
where , , and
:,
which is valid the far field, = the critical distance.
As a case of multi-slope model
The 2-ray ground reflected model may be thought as a case of multi-slope model with break point at critical distance with slope 20 dB/decade before critical distance and slope of 40 dB/decade after the critical distance. Using the free-space and two-ray model above, the propagation path loss can be expressed as
where and are the free-space and 2-ray path losses; is a minimum path loss (at smallest distance), usually in practice; dB or so. Note that and also follow from the law of energy conservation (since the Rx power cannot exceed the Tx power) so that both and break down when is small enough. This should be kept in mind when using these approximations at small distances (ignoring this limitation sometimes produces absurd results).
See also
*Radio propagation model
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another in vacuum, or into various parts of the atmosphere.
As a form of electromagnetic radiation, like light waves, radio waves are affected ...
*Free-space path loss
In telecommunication, the free-space path loss (FSPL) (also known as Free Space Loss, FSL) is the attenuation of radio energy between the feedpoints of two antennas that results from the combination of the receiving antenna's capture area plus the ...
*Friis transmission equation
The Friis transmission formula is used in telecommunications engineering, equating the power at the terminals of a receive antenna as the product of power density of the incident wave and the effective aperture of the receiving antenna under i ...
*ITU-R P.525
ITU-R P.525 is the International Telecommunication Union radiocommunications standard for the calculation of free-space attenuation.
See also
*Free space loss
In telecommunication, the free-space path loss (FSPL) (also known as Free Space Loss, ...
*Link budget
A link budget is an accounting of all of the power gains and losses that a communication signal experiences in a telecommunication system; from a transmitter, through a communication medium such as radio waves, cable, waveguide, or optical fiber ...
*Ray tracing (physics)
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, ...
*Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ' ...
*Specular reflection
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
*Six-rays model
The six-rays model is applied in an urban or indoor environment where a radio signal transmitted will encounter some objects that produce reflected, refracted or scattered copies of the transmitted signal. These are called multipath signal compone ...
*Ten-rays model
The ten-rays model is a mathematical model applied to the transmissions of radio signal in an urban area,
to generate a model of ten rays typically four rays more are added to the six rays model, these are (R3 and R4 bouncing on both sides of ...
References
Further reading
* S. Salous, Radio Propagation Measurement and Channel Modelling, Wiley, 2013.
* J.S. Seybold, Introduction to RF propagation, Wiley, 2005.
* K. Siwiak, Radiowave Propagation and Antennas for Personal Communications, Artech House, 1998.
* M.P. Doluhanov, Radiowave Propagation, Moscow: Sviaz, 1972.
* V.V. Nikolskij, T.I. Nikolskaja, Electrodynamics and Radiowave Propagation, Moscow: Nauka, 1989.
* 3GPP TR 38.901, Study on Channel Model for Frequencies from 0.5 to 100 GHz (Release 16), Sophia Antipolis, France, 201
* Recommendation ITU-R P.1238-8: Propagation data and prediction methods for the planning of indoor radiocommunication systems and radio local area networks in the frequency range 300 MHz to 100 GHz
* S. Loyka, ELG4179: Wireless Communication Fundamentals, Lecture Notes (Lec. 2-4), University of Ottawa, Canada, 202
{{Radio frequency propagation models
Radio frequency propagation model