The six-rays model is applied in an urban or indoor environment where a
radio signal
Radio waves are a type of electromagnetic radiation with the longest wavelengths in the electromagnetic spectrum, typically with frequencies of 300 gigahertz (GHz) and below. At 300 GHz, the corresponding wavelength is 1 mm (shor ...
transmitted will encounter some objects that produce reflected,
refracted
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
or scattered copies of the transmitted signal. These are called multipath signal components, they are attenuated, delayed and shifted from the original signal (LOS) due to a finite number of reflectors with known location and
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
properties, LOS and multipath signal are summed at the receiver.
This model approach the propagation of
electromagnetic waves
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) lig ...
by representing
wavefront
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
as simple particles. Thus reflection, refraction and scattering effects are approximated using simple
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
equation instead Maxwell's wave equations.
The simplest model is two-rays which predicts signal variation resulting from a
ground reflection interfering with the loss path. This model is applicable in isolated areas with some reflectors, such as rural roads or hallway.
The above two-rays approach can easily be extended to add as many rays as required. We may add rays bouncing off each side of a street in an urban corridor, leading to a six-rays model. The deduction of the six-rays model is presented below.
Mathematical deduction
Antennas of heights equal located in the center of the street
For the analysis of
antennas
In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies a ...
with equal heights then
, determining that for the following two rays that are reflected once in the wall, the point in which they collide is equal to said height
. Also for each ray that is reflected in the wall, there is another ray that is reflected in the ground in a number equal to the reflections in the wall plus one, in these rays there are diagonal distances for each reflection and the sum of these distances is denominated
.
Being located in the center of the street the distance between the antennas
and
, the buildings and the width of the streets are equal in both sides so that
, defining thus a single distance
.
The mathematical model of propagation of six rays is based on the model of two rays, to find the equations of each ray involved. The distance
that separates the two antennas, is equal to the first direct ray
or line of sight (LOS), that is:
For the ray reflected under
applies the
theorem of Pythagoras, in the
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
that forms between the reflection of
as the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
and the direct ray obtaining:
For
the Pythagorean theorem is reapplied, knowing that one of the hinges is double the distances between the
transmitter
In electronics and telecommunications, a radio transmitter or just transmitter is an electronic device which produces radio waves with an antenna (radio), antenna. The transmitter itself generates a radio frequency alternating current, which i ...
and the building due to the reflection of
and the diagonal distance to the wall:
For
the second ray is multiplied twice but it is taken into account that the distance is half of the third ray to form the equivalent triangle considering that
is the half of the distance of
and these must be the half of the line of sight distance
:
For
y
the deduction and the distances are equals, therefore:
Antennas of heights equal located in any point of the street
As the direct ray LOS does not vary and has not angular variation between the rays, the distance of the first two rays
and
of model does not vary and deduced according to the
mathematic model for two rays. For the other four rays it applies the next mathematical process:
is obtained through a
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
of the top view for the model and it applies the Pythagorean Theorem triangles, taking into account the distance between the wall and the antennas
,
,
,
are different:
For likeness of triangles in the top view for model is determined the equation
:
For
and
the deduction and the distances are equal then:
Antennas of heights different located in the center of the street
For antennas of different heights with rays that rebound in the wall, it is noted that the wall is the half point, where the two transmitted rays they fall on such wall. This wall has half the height between the height of the
and
, it means smaller than the transmitter and higher than the receiver and this high is where the two rays impact in the point, then rebound to the receiver. The ray reflected leaves two reflections, one that it has the same high of the wall and the other the receiver, and the ray of the line of sight maintains the same direction between the
and the
. The diagonal distance ''d´'' that separates the two antennas divides in two distances through of the wall, one is called
and the other
''.''
Antennas of heights different located in any point of the street
For the mathematical model of six-ray propagation for antennas of different heights located at any point in the street,
, there is a direct distance
that separates the two antennas, the first ray is formed by applying The Pythagorean theorem from the difference of heights of the antennas with respect to the line of sight:
The second ray or reflected ray is calculated as the first ray but the heights of the antennas are added to form the right triangle.
For deducing the third ray it is calculated the angle between the direct distance
and the distance of line of sight
Now deducing the height that subtraction of the wall with respect the height of the receiver called
by similarity the triangles:
By similarity of triangles it can deduce the distance where the ray hits the wall until the
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
of the receiver called ''a'' achieved:
By similarity of the triangles can be deduced the equation of the fourth ray:
For
y
the deduction and the distances are equal, therefore:
Free-space path loss on the model
Consider a transmitted signal in the free space a receptor located a distance ''d'' of the transmitter. One may add rays bouncing off each side of a street in an urban corridor, leading to a six-rays model, with rays
,
and
each one having a direct and a ground bouncing ray.
An important assumption must be made to simplify the model:
is small compared to the symbol length of the useful information, that is
. For the rays rebound outside the earth and on each side of the street, this assumption is fairly safe, but in general will have remembered that these assumptions mean the dispersion of delays (diffusion of the values
) is smaller than symbols speed of transmission.
Free-space path loss of six rays model is defined as:
is the wavelength.
Is the time difference between the two paths.
Is the
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
of ground reflection.
Gain of the transmitter.
Receiver gain.
See also
*
Two-ray ground-reflection model The two-rays ground-reflection model is a multipath radio propagation model which predicts the path losses between a transmitting antenna and a receiving antenna when they are in line of sight (LOS). Generally, the two antenna each have different ...
*
Ten-rays model
*
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, ...
References
{{Radio frequency propagation models
Antennas
Radio frequency propagation model