phased.WidebandTwoRayChannel

Wideband two-ray propagation channel

Description

The phased.WidebandTwoRayChannel models a wideband two-ray propagation channel. A two-ray propagation channel is the simplest type of multipath channel. You can use a two-ray channel to simulate propagation of signals in a homogeneous, isotropic medium with a single reflecting boundary. This type of medium has two propagation paths: a line-of-sight (direct) propagation path from one point to another and a ray path reflected from the boundary.

You can use this System object™ for short-range radar and mobile communications applications where the signals propagate along straight paths and the earth is assumed to be flat. You can also use this object for sonar and microphone applications. For acoustic applications, you can choose nonpolarized fields and adjust the propagation speed to be the speed of sound in air or water. You can use phased.WidebandTwoRayChannel to model propagation from several points simultaneously.

Although the System object works for all frequencies, the attenuation models for atmospheric gases and rain are valid for electromagnetic signals in the frequency range 1–1000 GHz only. The attenuation model for fog and clouds is valid for 10–1000 GHz. Outside these frequency ranges, the System object uses the nearest valid value.

The phased.WidebandTwoRayChannel System object applies range-dependent time delays to the signals, as well as gains or losses, phase shifts, and boundary reflection loss. When either the source or destination is moving, the System object applies Doppler shifts to the signals.

Signals at the channel output can be kept separate or be combined. If you keep the signals separate, both signals arrive at the destination separately and are not combined. If you choose to combine the signals, the two signals from the source propagate separately but are coherently summed at the destination into a single quantity. Choose this option when the difference between the sensor or array gains in the directions of the two paths is insignificant.

In contrast to the phased.WidebandFreeSpace and phased.WidebandLOSChannel System objects, this System object does not support two-way propagation.

To compute the propagation delay for specified source and receiver points:

1. Define and set up your two-ray channel. See Construction.

2. Call the step method to compute the propagated signal using the properties of the phased.WidebandTwoRayChannel System object.

Note

Alternatively, instead of using the step method to perform the operation defined by the System object, you can call the object with arguments, as if it were a function. For example, y = step(obj,x) and y = obj(x) perform equivalent operations.

Construction

channel = phased.WidebandTwoRayChannel creates a two-ray propagation channel System object, channel.

channel = phased.WidebandTwoRayChannel(Name,Value) creates a System object, channel, with each specified property Name set to the specified Value. You can specify additional name and value pair arguments in any order as (Name1,Value1,...,NameN,ValueN).

Properties

PropagationSpeed — Signal propagation speed

physconst('LightSpeed') (default) | positive scalar

Signal propagation speed, specified as a positive scalar. Units are in meters per second. The default propagation speed is the value returned by physconst('LightSpeed'). See physconst for more information.

Example: 3e8

Data Types: double

OperatingFrequency — Operating frequency

300e6 (default) | positive scalar

Operating frequency, specified as a positive scalar. Units are in Hz.

Example: 1e9

Data Types: double

SpecifyAtmosphere — Enable atmospheric attenuation model

false (default) | true

Option to enable the atmospheric attenuation model, specified as a false or true. Set this property to true to add signal attenuation caused by atmospheric gases, rain, fog, or clouds. Set this property to false to ignore atmospheric effects in propagation.

Setting SpecifyAtmosphere to true, enables the Temperature, DryAirPressure, WaterVapourDensity, LiquidWaterDensity, and RainRate properties.

Data Types: logical

Temperature — Ambient temperature

15 (default) | real-valued scalar

Ambient temperature, specified as a real-valued scalar. Units are in degrees Celsius.

Example: 20.0

Dependencies

To enable this property, set SpecifyAtmosphere to true.

Data Types: double

DryAirPressure — Atmospheric dry air pressure

101.325e3 (default) | positive real-valued scalar

Atmospheric dry air pressure, specified as a positive real-valued scalar. Units are in pascals (Pa). The default value of this property corresponds to one standard atmosphere.

Example: 101.0e3

Dependencies

To enable this property, set SpecifyAtmosphere to true.

Data Types: double

WaterVapourDensity — Atmospheric water vapor density

7.5 (default) | positive real-valued scalar

Atmospheric water vapor density, specified as a positive real-valued scalar. Units are in g/m³.

Example: 7.4

Dependencies

To enable this property, set SpecifyAtmosphere to true.

Data Types: double

LiquidWaterDensity — Liquid water density

0.0 (default) | nonnegative real-valued scalar

Liquid water density of fog or clouds, specified as a nonnegative real-valued scalar. Units are in g/m³. Typical values for liquid water density are 0.05 for medium fog and 0.5 for thick fog.

Example: 0.1

Dependencies

To enable this property, set SpecifyAtmosphere to true.

Data Types: double

RainRate — Rainfall rate

0.0 (default) | nonnegative scalar

Rainfall rate, specified as a nonnegative scalar. Units are in mm/hr.

Example: 10.0

Dependencies

To enable this property, set SpecifyAtmosphere to true.

Data Types: double

SampleRate — Sample rate of signal

1e6 (default) | positive scalar

Sample rate of signal, specified as a positive scalar. Units are in Hz. The System object uses this quantity to calculate the propagation delay in units of samples.

Example: 1e6

Data Types: double

NumSubbands — Number of processing subbands

64 (default) | positive integer

Number of processing subbands, specified as a positive integer.

Example: 128

Data Types: double

EnablePolarization — Enable polarized fields

false (default) | true

Option to enable polarized fields, specified as false or true. Set this property to true to enable polarization. Set this property to false to ignore polarization.

Data Types: logical

GroundReflectionCoefficient — Ground reflection coefficient

-1 (default) | complex-valued scalar | complex-valued 1-by-N row vector

Ground reflection coefficient for the field at the reflection point, specified as a complex-valued scalar or a complex-valued 1-by-N row vector. Each coefficient has an absolute value less than or equal to one. The quantity N is the number of two-ray channels. Units are dimensionless. Use this property to model nonpolarized signals. To model polarized signals, use the GroundRelativePermittivity property.

Example: -0.5

Dependencies

To enable this property, set EnablePolarization to false.

Data Types: double
Complex Number Support: Yes

GroundRelativePermittivity — Ground relative permittivity

15 (default) | positive real-valued scalar | real-valued 1-by-Nrow vector of positive values

Relative permittivity of the ground at the reflection point, specified as a positive real-valued scalar or a 1-by-N real-valued row vector of positive values. The dimension N is the number of two-ray channels. Permittivity units are dimensionless. Relative permittivity is defined as the ratio of actual ground permittivity to the permittivity of free space. This property applies when you set the EnablePolarization property to true. Use this property to model polarized signals. To model nonpolarized signals, use the GroundReflectionCoefficient property.

Example: 5

Dependencies

To enable this property, set EnablePolarization to true.

Data Types: double

CombinedRaysOutput — Option to combine two rays at output

true (default) | false

Option to combine the two rays at channel output, specified as true or false. When this property is true, the object coherently adds the line-of-sight propagated signal and the reflected path signal when forming the output signal. Use this mode when you do not need to include the directional gain of an antenna or array in your simulation.

Data Types: logical

MaximumDistanceSource — Source of maximum one-way propagation distance

'Auto' (default) | 'Property'

Source of maximum one-way propagation distance, specified as 'Auto' or 'Property'. The maximum one-way propagation distance is used to allocate sufficient memory for signal delay computation. When you set this property to 'Auto', the System object automatically allocates memory. When you set this property to 'Property', you specify the maximum one-way propagation distance using the value of the MaximumDistance property.

Data Types: char

MaximumDistance — Maximum one-way propagation distance

10000 (default) | positive real-valued scalar

Maximum one-way propagation distance, specified as a positive real-valued scalar. Units are in meters. Any signal that propagates more than the maximum one-way distance is ignored. The maximum distance must be greater than or equal to the largest position-to-position distance.

Example: 5000

Dependencies

To enable this property, set the MaximumDistanceSource property to 'Property'.

Data Types: double

MaximumNumInputSamplesSource — Source of maximum number of samples

'Auto' (default) | 'Property'

The source of the maximum number of samples of the input signal, specified as 'Auto' or 'Property'. When you set this property to 'Auto', the propagation model automatically allocates enough memory to buffer the input signal. When you set this property to 'Property', you specify the maximum number of samples in the input signal using the MaximumNumInputSamples property. Any input signal longer than that value is truncated.

To use this object with variable-size signals in a MATLAB^® Function Block in Simulink^®, set the MaximumNumInputSamplesSource property to 'Property' and set a value for the MaximumNumInputSamples property.

Example: 'Property'

Dependencies

To enable this property, set MaximumDistanceSource to 'Property'.

Data Types: char

MaximumNumInputSamples — Maximum number of input signal samples

100 (default) | positive integer

Maximum number of input signal samples, specified as a positive integer. The input signal is the first argument of the step method, after the System object itself. The size of the input signal is the number of rows in the input matrix. Any input signal longer than this number is truncated. To process signals completely, ensure that this property value is greater than any maximum input signal length.

The waveform-generating System objects determine the maximum signal size:

• For any waveform, if the waveform OutputFormat property is set to 'Samples', the maximum signal length is the value specified in the NumSamples property.

• For pulse waveforms, if the OutputFormat is set to 'Pulses', the signal length is the product of the smallest pulse repetition frequency, the number of pulses, and the sample rate.

• For continuous waveforms, if the OutputFormat is set to 'Sweeps', the signal length is the product of the sweep time, the number of sweeps, and the sample rate.

Example: 2048

Dependencies

To enable this property, set MaximumNumInputSamplesSource to 'Property'.

Data Types: double

Methods

reset	Reset states of System object
step	Propagate wideband signal from point to point using two-ray channel model

Common to All System Objects
release	Allow System object property value changes

Examples

Scalar Wideband Signal Propagating in Two-Ray Channel

This example illustrates the two-ray propagation of a wideband signal, showing how the signals from the line-of-sight path and reflected path arrive at the receiver at different times.

Note: You can replace each call to the function with the equivalent step syntax. For example, replace myObject(x) with step(myObject,x).

Create and Plot Transmitted Waveform

Create a nonpolarized electromagnetic field consisting of two linear FM waveform pulses at a carrier frequency of 100 MHz. Assume the pulse width is 20 μs and the sampling rate is 10 MHz. The bandwidth of the pulse is 1 MHz. Assume a 50% duty cycle so that the pulse width is one-half the pulse repetition interval. Create a two-pulse wave train. Set the GroundReflectionCoefficient to –0.9 to model strong ground reflectivity. Propagate the field from a stationary source to a stationary receiver. The vertical separation of the source and receiver is approximately 10 km.

c = physconst('LightSpeed');
fs = 10e6;
pw = 20e-6;
pri = 2*pw;
PRF = 1/pri;
fc = 100e6;
lambda = c/fc;
bw = 1e6;
waveform = phased.LinearFMWaveform('SampleRate',fs,'PulseWidth',pw,...
    'PRF',PRF,'OutputFormat','Pulses','NumPulses',2,'SweepBandwidth',bw,...
    'SweepDirection','Down','Envelope','Rectangular','SweepInterval',...
    'Positive');
wav = waveform();
n = size(wav,1);
plot([0:(n-1)]/fs*1e6,real(wav),'b')
xlabel('Time (\mu s)')
ylabel('Waveform Magnitude')

Specify the Location of Source and Receiver

Place the source and receiver about 1 km apart horizontally and approximately 5 km apart vertically.

pos1 = [0;0;100];
pos2 = [1e3;0;5.0e3];
vel1 = [0;0;0];
vel2 = [0;0;0];

Create a Wideband Two-Ray Channel System Object

Create a two-ray propagation channel System object™ and propagate the signal along both the line-of-sight and reflected ray paths. The same signal is propagated along both paths.

channel = phased.WidebandTwoRayChannel('SampleRate',fs,...
    'GroundReflectionCoefficient',-0.9,'OperatingFrequency',fc,...
    'CombinedRaysOutput',false);
prop_signal = channel([wav,wav],pos1,pos2,vel1,vel2);

[rng2,angs] = rangeangle(pos2,pos1,'two-ray');

Calculate time delays in μs.

tm = rng2/c*1e6;
disp(tm)

   16.6815   17.3357

Display the calculated propagation paths azimuth and elevation angles in degrees.

disp(angs)

         0         0
   78.4654  -78.9063

Plot the Propagated Signals

1. Plot the real part of the signal propagated along the line-of-sight path.

2. Plot the real part of the signal propagated along the reflected path.

3. Plot the real part of the coherent sum of the two signals.

n = size(prop_signal,1);
delay = [0:(n-1)]/fs*1e6;
subplot(3,1,1)
plot(delay,real([prop_signal(:,1)]),'b')
grid
xlabel('Time (\mu sec)')
ylabel('Real Part')
title('Direct Path')

subplot(3,1,2)
plot(delay,real([prop_signal(:,2)]),'b')
grid
xlabel('Time (\mu sec)')
ylabel('Real Part')
title('Reflected Path')

subplot(3,1,3)
plot(delay,real([prop_signal(:,1) + prop_signal(:,2)]),'b')
grid
xlabel('Time (\mu sec)')
ylabel('Real Part')
title('Combined Paths')

The delay of the reflected path signal agrees with the predicted delay. The magnitude of the coherently combined signal is less than either of the propagated signals. This result indicates that the two signals contain some interference.

Compare Wideband Two-Ray Channel Propagation to Free Space

Calculate the result of propagating a wideband LFM signal in a two-ray environment from a radar 10 meters above the origin (0,0,10) to a target at (3000,2000,2000) meters. Assume that the radar and target are stationary and that the transmitting antenna is isotropic. Combine the signal from the two paths and compare the signal to a signal propagating in free space. The system operates at 300 MHz. Set the CombinedRaysOutput property to true to combine the direct path and reflected path signals when forming the output signal.

Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent step syntax. For example, replace myObject(x) with step(myObject,x).

Create a linear FM waveform.

fop = 300.0e6;
fs = 1.0e6;
waveform = phased.LinearFMWaveform();
x = waveform();

Specify the target position and velocity.

posTx = [0; 0; 10];
posTgt = [3000; 2000; 2000];
velTx = [0;0;0];
velTgt = [0;0;0];

Model the free space propagation.

fschannel = phased.WidebandFreeSpace('SampleRate',waveform.SampleRate);
y_fs = fschannel(x,posTx,posTgt,velTx,velTgt);

Model two-ray propagation from the position of the radar to the target.

tworaychannel = phased.WidebandTwoRayChannel('SampleRate',waveform.SampleRate,...
    'CombinedRaysOutput',true);
y_tworay = tworaychannel(x,posTx,posTgt,velTx,velTgt);
plot(abs([y_tworay y_fs]))
legend('Wideband two-ray (Position 1)','Wideband free space (Position 1)',...
    'Location','best')
xlabel('Samples')
ylabel('Signal Magnitude')
hold on

Move the radar by 10 meters horizontally to a second position.

posTx = posTx + [10;0;0];
y_fs = fschannel(x,posTx,posTgt,velTx,velTgt);
y_tworay = tworaychannel(x,posTx,posTgt,velTx,velTgt);
plot(abs([y_tworay y_fs]))
legend('Wideband two-ray (Position 1)','Wideband free space (Position 1)',...
    'Wideband two-ray (Position 2)','Wideband free space (Position 2)',...
    'Location','best')
hold off

The free-space propagation losses are the same for both the first and second positions of the radar. The two-ray losses are different due to the interference effect of the two-ray paths.

Wideband Polarized Field Propagation in Two-Ray Channel

Create a polarized electromagnetic field consisting of linear FM waveform pulses. Propagate the field from a stationary source with a crossed-dipole antenna element to a stationary receiver approximately 10 km away. The transmitting antenna is 100 m above the ground. The receiving antenna is 150 m above the ground. The receiving antenna is also a crossed-dipole. Plot the received signal.

Note: You can replace each call to the function with the equivalent step syntax. For example, replace myObject(x) with step(myObject,x).

Set Radar Waveform Parameters

Assume the pulse width is$$10\mu s$$ and the sampling rate is 10 MHz. The bandwidth of the pulse is 1 MHz. Assume a 50% duty cycle in which the pulse width is one-half the pulse repetition interval. Create a two-pulse wave train. Assume a carrier frequency of 100 MHz.

c = physconst('LightSpeed');
fs = 20e6;
pw = 10e-6;
pri = 2*pw;
PRF = 1/pri;
fc = 100e6;
bw = 1e6;
lambda = c/fc;

Set Up Required System Objects

Use a GroundRelativePermittivity of 10.

waveform = phased.LinearFMWaveform('SampleRate',fs,'PulseWidth',pw,...
    'PRF',PRF,'OutputFormat','Pulses','NumPulses',2,'SweepBandwidth',bw,...
    'SweepDirection','Down','Envelope','Rectangular','SweepInterval',...
    'Positive');
antenna = phased.CrossedDipoleAntennaElement(...
    'FrequencyRange',[50,200]*1e6);
radiator = phased.Radiator('Sensor',antenna,'OperatingFrequency',fc,...
    'Polarization','Combined');
channel = phased.WidebandTwoRayChannel('SampleRate',fs,...
    'OperatingFrequency',fc,'CombinedRaysOutput',false,...
    'EnablePolarization',true,'GroundRelativePermittivity',10);
collector = phased.Collector('Sensor',antenna,'OperatingFrequency',fc,...
    'Polarization','Combined');

Set Up Scene Geometry

Specify transmitter and receiver positions, velocities, and orientations. Place the source and receiver approximately 1000 m apart horizontally and approximately 50 m apart vertically.

posTx = [0;100;100];
posRx = [1000;0;150];
velTx = [0;0;0];
velRx = [0;0;0];
laxRx = rotz(180);
laxTx = rotx(1)*eye(3);

Create and Radiate Signals from Transmitter

Compute the transmission angles for the two rays traveling toward the receiver. These angles are defined with respect to the transmitter local coordinate system. The phased.Radiator System object(TM) uses these angles to apply separate antenna gains to the two signals.

[rng,angsTx] = rangeangle(posRx,posTx,laxTx,'two-ray');
wav = waveform();

Plot the transmitted waveform.

n = size(wav,1);
plot([0:(n-1)]/fs*1000000,real(wav))
xlabel('Time ({\mu}sec)')
ylabel('Waveform')

sig = radiator(wav,angsTx,laxTx);

Propagate the signals to the receiver via a two-ray channel.

prop_sig = channel(sig,posTx,posRx,velTx,velRx);

Receive Propagated Signal

Compute the reception angles for the two rays arriving at the receiver. These angles are defined with respect to the receiver local coordinate system. The phased.Collector System object(TM) uses these angles to apply separate antenna gains to the two signals.

[rng1,angsRx] = rangeangle(posTx,posRx,laxRx,'two-ray');
delays = rng1/c*1e6

delays = 1×2

    3.3564    3.4544

Collect and combine the received rays.

y = collector(prop_sig,angsRx,laxRx);

Plot the received waveform.

plot([0:(n-1)]/fs*1000000,real(y))
xlabel('Time ({\mu}sec)')
ylabel('Received Waveform')

Two-Ray Propagation of Wideband LFM Waveform with Atmospheric Losses

Propagate a wideband linear FM signal in a two-ray channel. The signal bandwidth is 15% of the carrier frequency. Assume there is signal loss caused by atmospheric gases and rain. The signal propagates from a transmitter located at (0,0,0) meters in the global coordinate system to a receiver at (10000,200,30) meters. Assume that the transmitter and the receiver are stationary and that they both have cosine antenna patterns. Plot the received signal. Set the dry air pressure to 102.0 Pa and the rain rate to 5 mm/hr.

Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent step syntax. For example, replace myObject(x) with step(myObject,x).

Set Radar Waveform Parameters

c = physconst('LightSpeed');
fs = 40e6;
pw = 10e-6;
pri = 2.5*pw;
PRF = 1/pri;
fc = 100e6;
bw = 15e6;
lambda = c/fc;

Set Up Radar Scenario

Create the required System objects.

waveform = phased.LinearFMWaveform('SampleRate',fs,'PulseWidth',pw,...
    'PRF',PRF,'OutputFormat','Pulses','NumPulses',2,'SweepBandwidth',bw,...
    'SweepDirection','Down','Envelope','Rectangular','SweepInterval',...
    'Positive');
antenna = phased.CosineAntennaElement;
radiator = phased.Radiator('Sensor',antenna);
collector = phased.Collector('Sensor',antenna);
channel = phased.WidebandTwoRayChannel('SampleRate',waveform.SampleRate,...
    'CombinedRaysOutput',false,'GroundReflectionCoefficient',0.95,...
    'SpecifyAtmosphere',true,'Temperature',20,...
    'DryAirPressure',102.5,'RainRate',5.0);

Set up the scene geometry. Specify transmitter and receiver positions and velocities. The transmitter and receiver are stationary.

posTx = [0;0;0];
posRx = [10000;200;30];
velTx = [0;0;0];
velRx = [0;0;0];

Specify the transmitting and receiving radar antenna orientations with respect to the global coordinates. The transmitting antenna points along the positive x-direction and the receiving antenna points close to the negative x-direction.

laxTx = eye(3);
laxRx = rotx(5)*rotz(170);

Compute the transmission angles which are the angles at which the two rays traveling toward the receiver leave the transmitter. The phased.Radiator System object™ uses these angles to apply separate antenna gains to the two signals. Because the antenna gains depend on path direction, you must transmit and receive the two rays separately.

[~,angTx] = rangeangle(posRx,posTx,laxTx,'two-ray');

Create and Radiate Signals from Transmitter

Radiate the signals along the transmission directions.

wavfrm = waveform();
wavtrans = radiator(wavfrm,angTx);

Propagate the signals to the receiver via a two-ray channel.

wavrcv = channel(wavtrans,posTx,posRx,velTx,velRx);

Collect Signal at Receiver

Compute the angle at which the two rays traveling from the transmitter arrive at the receiver. The phased.Collector System object™ uses these angles to apply separate antenna gains to the two signals.

[~,angRcv] = rangeangle(posTx,posRx,laxRx,'two-ray');

Collect and combine the two received rays.

yR = collector(wavrcv,angRcv);

Plot Received Signal

dt = 1/waveform.SampleRate;
n = size(yR,1);
plot([0:(n-1)]*dt*1e6,real(yR))
xlabel('Time ({\mu}sec)')
ylabel('Signal Magnitude')

More About

Two-Ray Propagation Paths

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.

$$\begin{array}{l}\overrightarrow{R}={\overrightarrow{x}}_s-{\overrightarrow{x}}_r\\ R_{los}=\left|\overrightarrow{R}\right|=\sqrt{{\left(z_r-z_s\right)}^2+L^2}\\ R_1=\frac{z_r}{z_r+z_z}\sqrt{{\left(z_r+z_s\right)}^2+L^2}\\ R_2=\frac{z_s}{z_s+z_r}\sqrt{{\left(z_r+z_s\right)}^2+L^2}\\ R_{rp}=R_1+R_2=\sqrt{{\left(z_r+z_s\right)}^2+L^2}\\ \tan {\theta }_{los}=\frac{\left(z_s-z_r\right)}{L}\\ \tan {\theta }_{rp}=-\frac{\left(z_s+z_r\right)}{L}\\ {{\theta }^{\prime }}_{los}=-{\theta }_{los}\\ {{\theta }^{\prime }}_{rp}={\theta }_{rp}\end{array}$$

Two-Ray Attenuation

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

• L_tworay is the two-ray geometric path attenuation

• L_G is the ground reflection attenuation

• L_g is the atmospheric path attenuation

• L_c is the fog and cloud path attenuation

• L_r is the rain path attenuation

Each component is in magnitude units, not in dB.

Ground Reflection and Propagation Loss

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

$$E_{los}=E_0\left(\frac{R_0}{R_{los}}\right)e^{i\omega \left(t-R_{los}/c\right)}$$

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

$$E_{rp}=L_G{E}_0\left(\frac{R_0}{R_{rp}}\right)e^{i\omega \left(t-R_{rp}/c\right)}$$

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.

$$\begin{array}{l}{G}_p=\frac{Z_1\cos {\theta }_1-Z_2\cos {\theta }_2}{Z_1\cos {\theta }_1+Z_2\cos {\theta }_2}=\frac{\cos {\theta }_1-\frac{Z_2}{Z_1}\cos {\theta }_2}{\cos {\theta }_1+\frac{Z_2}{Z_1}\cos {\theta }_2}\\ G_s=\frac{Z_2\cos {\theta }_1-Z_1\cos {\theta }_2}{Z_2\cos {\theta }_1+Z_1\cos {\theta }_2}=\frac{\cos {\theta }_2-\frac{Z_2}{Z_1}\cos {\theta }_1}{\cos {\theta }_2+\frac{Z_2}{Z_1}\cos {\theta }_1}\\ Z_1=\sqrt{\frac{{\mu }_1}{{\epsilon }_1}}\\ Z_2=\sqrt{\frac{{\mu }_2}{{\epsilon }_2}}\end{array}$$

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

$$\begin{array}{l}{G}_p=\frac{\sqrt{\rho }\cos {\theta }_1-\cos {\theta }_2}{\sqrt{\rho }\cos {\theta }_1+\cos {\theta }_2}\\ G_s=\frac{\sqrt{\rho }\cos {\theta }_2-\cos {\theta }_1}{\sqrt{\rho }\cos {\theta }_2+\cos {\theta }_1}\end{array}$$

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

$$L_{tworay}=\frac{{(4\pi R)}^2}{{\lambda }^2},$$

where λ is the signal wavelength.

Atmospheric Gas Attenuation Model

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

$$\gamma ={\gamma }_o(f)+{\gamma }_w(f)=0.1820f{N}^{″}(f).$$

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

$$N^{″}(f)={\displaystyle \sum _i{S}_i{F}_i+{N^{″}}_D^{}(f)}$$

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

$$S_i=a_1\times {10}^{-7}{\left(\frac{300}{T}\right)}^3\exp \left[a_2(1-\left(\frac{300}{T}\right)\right]P.$$

For atmospheric water vapor, each spectral line strength is

$$S_i=b_1\times {10}^{-1}{\left(\frac{300}{T}\right)}^{3.5}\exp \left[b_2(1-\left(\frac{300}{T}\right)\right]W.$$

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

$$W=\frac{\rho T}{216.7}.$$

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.

Fog and Cloud Attenuation Model

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

$${\gamma }_c=K_l\left(f\right)M,$$

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.

Rainfall Attenuation Model

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

$${\gamma }_R=k{R}^{\alpha },$$

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

$$r=\frac{1}{0.477d^{0.633}{R}_{0.01}^{0.073\alpha }{f}^{0.123}-10.579\left(1-\exp \left(-0.024d\right)\right)}$$

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.

Subband Frequency Processing

Subband processing decomposes a wideband signal into multiple subbands and applies narrowband processing to the signal in each subband. The signals for all subbands are summed to form the output signal.

When using wideband frequency System objects or blocks, you specify the number of subbands, N_B, in which to decompose the wideband signal. Subband center frequencies and widths are automatically computed from the total bandwidth and number of subbands. The total frequency band is centered on the carrier or operating frequency, f_c. The overall bandwidth is given by the sample rate, f_s. Frequency subband widths are Δf = f _s/N_B. The center frequencies of the subbands are

$$f_m=\{\begin{array}{c}{f}_c-\frac{f_s}{2}+\left(m-1\right)\Delta f\text{, }{N}_B\text{ even}\\ f_c-\frac{\left(N_B-1\right)f_s}{2N_B}+\left(m-1\right)\Delta f\text{, }{N}_B\text{ odd}\end{array},m=1,\dots ,N_B$$

Some System objects let you obtain the subband center frequencies as output when you run the object. The returned subband frequencies are ordered consistently with the ordering of the discrete Fourier transform. Frequencies above the carrier appear first, followed by frequencies below the carrier.

References

[1] Proakis, J. Digital Communications. New York: McGraw-Hill, 2001.

[2] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill.

[3] Saakian, A. Radio Wave Propagation Fundamentals. Norwood, MA: Artech House, 2011.

[4] Balanis, C. Advanced Engineering Electromagnetics. New York: Wiley & Sons, 1989.

[5] Rappaport, T. Wireless Communications: Principles and Practice, 2nd Ed New York: Prentice Hall, 2002.

[6] Radiocommunication Sector of the International Telecommunication Union. Recommendation ITU-R P.676-10: Attenuation by atmospheric gases. 2013.

[7] Radiocommunication Sector of the International Telecommunication Union. Recommendation ITU-R P.840-6: Attenuation due to clouds and fog. 2013.

[8] Radiocommunication Sector of the International Telecommunication Union. Recommendation ITU-R P.838-3: Specific attenuation model for rain for use in prediction methods. 2005.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).