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, s_r, you can compute the arrival angles of both paths, θ′_los and θ′_rp. 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, θ_los and θ_rp. In the global coordinates, the angle of reflection at the boundary is the same as the angles θ_rp and θ′_rp. The reflection angle is important to know when you use angle-dependent reflection-loss data. You can determine the reflection angle by using the 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_los which is equal to the geometric distance between source and receiver. The total path length for the reflected path is R_rp= R₁ + R₂. The quantity 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.

Attenuation or path loss in the two-ray channel is the product of five components, L = L_tworay L_G L_g L_c L_r, where

Each component is in magnitude units, not in dB.

Losses occurs when a signal is reflected from a boundary. You can obtain a simple model of ground reflection loss by representing the electromagnetic field as a scalar field. This approach also works for acoustic and sonar systems. Let E be a scalar free-space electromagnetic field having amplitude E₀ at a reference distance R₀ from a transmitter (for example, one meter). The propagating free-space field at distance R_los from the transmitter is

for the line-of-sight path. You can express the ground-reflected E-field as

where R_rp is the reflected path distance. The quantity L_G represents the loss due to reflection at the ground plane. To specify L_G, use the GroundReflectionCoefficient property. In general, L_G depends on the incidence angle of the field. If you have empirical information about the angular dependence of L_G, you can use rangeangle to compute the incidence angle of the reflected path. The total field at the destination is the sum of the line-of-sight and reflected-path fields.

For electromagnetic waves, a more complicated but more realistic model uses a vector representation of the polarized field. You can decompose the incident electric field into two components. One component, E_p, is parallel to the plane of incidence. The other component, E_s, is perpendicular to the plane of incidence. The ground reflection coefficients for these components differ and can be written in terms of the ground permittivity and incidence angle.

where Z is the impedance of the medium. Because the magnetic permeability of the ground is almost identical to that of air or free space, the ratio of impedances depends primarily on the ratio of electric permittivities

where the quantity ρ = ε₂/ε₁ is the ground relative permittivity set by the GroundRelativePermittivity property. The angle θ₁ is the incidence angle and the angle θ₂ is the refraction angle at the boundary. You can determine θ₂ using Snell’s law of refraction.

After reflection, the full field is reconstructed from the parallel and perpendicular components. The total ground plane attenuation, L_G, is a combination of G_s and G_p.

When the origin and destination are stationary relative to each other, you can write the output Y of step as Y(t) = F(t-τ)/L. The quantity τ is the signal delay and L is the free-space path loss. The delay τ is given by R/c. R is either the line-of-sight propagation path distance or the reflected path distance, and c is the propagation speed. The path loss

where λ is the signal wavelength.

This model calculates the attenuation of signals that propagate through atmospheric gases.

Electromagnetic signals attenuate when they propagate through the atmosphere. This effect is due primarily to the absorption resonance lines of oxygen and water vapor, with smaller contributions coming from nitrogen gas. The model also includes a continuous absorption spectrum below 10 GHz. The ITU model Recommendation ITU-R P.676-10: Attenuation by atmospheric gases is used. The model computes the specific attenuation (attenuation per kilometer) as a function of temperature, pressure, water vapor density, and signal frequency. The atmospheric gas model is valid for frequencies from 1–1000 GHz and applies to polarized and nonpolarized fields.

The formula for specific attenuation at each frequency is

The quantity N"() is the imaginary part of the complex atmospheric refractivity and consists of a spectral line component and a continuous component:

The spectral component consists of a sum of discrete spectrum terms composed of a localized frequency bandwidth function, F(f)_i, multiplied by a spectral line strength, S_i. For atmospheric oxygen, each spectral line strength is

For atmospheric water vapor, each spectral line strength is

P is the dry air pressure, W is the water vapor partial pressure, and T is the ambient temperature. Pressure units are in hectoPascals (hPa) and temperature is in degrees Kelvin. The water vapor partial pressure, W, is related to the water vapor density, ρ, by

The total atmospheric pressure is P + W.

For each oxygen line, S_i depends on two parameters, a₁ and a₂. Similarly, each water vapor line depends on two parameters, b₁ and b₂. The ITU documentation cited at the end of this section contains tabulations of these parameters as functions of frequency.

The localized frequency bandwidth functions F_i(f) are complicated functions of frequency described in the ITU references cited below. The functions depend on empirical model parameters that are also tabulated in the reference.

To compute the total attenuation for narrowband signals along a path, the function multiplies the specific attenuation by the path length, R. Then, the total attenuation is L_g= R(γ_o + γ_w).

You can apply the attenuation model to wideband signals. First, divide the wideband signal into frequency subbands, and apply attenuation to each subband. Then, sum all attenuated subband signals into the total attenuated signal.

This model calculates the attenuation of signals that propagate through fog or clouds.

Fog and cloud attenuation are the same atmospheric phenomenon. The ITU model, Recommendation ITU-R P.840-6: Attenuation due to clouds and fog is used. The model computes the specific attenuation (attenuation per kilometer), of a signal as a function of liquid water density, signal frequency, and temperature. The model applies to polarized and nonpolarized fields. The formula for specific attenuation at each frequency is

where M is the liquid water density in gm/m³. The quantity K_l(f) is the specific attenuation coefficient and depends on frequency. The cloud and fog attenuation model is valid for frequencies 10–1000 GHz. Units for the specific attenuation coefficient are (dB/km)/(g/m³).

To compute the total attenuation for narrowband signals along a path, the function multiplies the specific attenuation by the path length R. Total attenuation is L_c = Rγ_c.

You can apply the attenuation model to wideband signals. First, divide the wideband signal into frequency subbands, and apply narrowband attenuation to each subband. Then, sum all attenuated subband signals into the total attenuated signal.

This model calculates the attenuation of signals that propagate through regions of rainfall. Rain attenuation is a dominant fading mechanism and can vary from location-to-location and from year-to-year.

Electromagnetic signals are attenuated when propagating through a region of rainfall. Rainfall attenuation is computed according to the ITU rainfall model Recommendation ITU-R P.838-3: Specific attenuation model for rain for use in prediction methods. The model computes the specific attenuation (attenuation per kilometer) of a signal as a function of rainfall rate, signal frequency, polarization, and path elevation angle. The specific attenuation, ɣ_R, is modeled as a power law with respect to rain rate

where R is rain rate. Units are in mm/hr. The parameter k and exponent α depend on the frequency, the polarization state, and the elevation angle of the signal path. The specific attenuation model is valid for frequencies from 1–1000 GHz.

To compute the total attenuation for narrowband signals along a path, the function multiplies the specific attenuation by the an effective propagation distance, d_eff. Then, the total attenuation is L = d_effγ_R.

The effective distance is the geometric distance, d, multiplied by a scale factor

where f is the frequency. The article Recommendation ITU-R P.530-17 (12/2017): Propagation data and prediction methods required for the design of terrestrial line-of-sight systems presents a complete discussion for computing attenuation.

The rain rate, R, used in these computations is the long-term statistical rain rate, R_0.01. This is the rain rate that is exceeded 0.01% of the time. The calculation of the statistical rain rate is discussed in Recommendation ITU-R P.837-7 (06/2017): Characteristics of precipitation for propagation modelling. This article also explains how to compute the attenuation for other percentages from the 0.01% value.

You can apply the attenuation model to wideband signals. First, divide the wideband signal into frequency subbands and apply attenuation to each subband. Then, sum all attenuated subband signals into the total attenuated signal.