# phased.ScatteringMIMOChannel

Scattering MIMO channel

## Description

The `phased.ScatteringMIMOChannel`

System object™ models a multipath propagation channel in which radiated signals from a
transmitting array are reflected from multiple scatterers back toward a receiving array.
In this channel, propagation paths are line of sight from point to point. The object
models range-dependent time delay, gain, Doppler shift, phase change, and atmospheric
loss due to gases, rain, fog, and clouds.

The attenuation models for atmospheric gases and rain are valid for electromagnetic signals in the frequency range from 1 through 1000 GHz. The attenuation model for fog and clouds is valid from 10 through 1000 GHz. Outside of these frequency ranges, the object uses the nearest valid value.

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

Define and set up your scattering MIMO channel using the Construction procedure. You can set the System object properties during construction or leave them at their default values.

Call the

`step`

method to compute the propagated signals using the properties of the`phased.ScatteringMIMOChannel`

System object. You can change tunable properties before or after any call to the`step`

method.

**Note**

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.ScatteringMIMOChannel`

creates a scattering MIMO
propagation channel System object, `channel`

.

`channel = phased.ScatteringMIMOChannel(`

creates a System object, `Name`

,`Value`

)`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

`TransmitArray`

— Transmitting array

`phased.ULA`

(default) | Phased Array System Toolbox™ antenna array System object

Transmitting array, specified as a Phased Array System Toolbox antenna array System object. The default value for this property is a `phased.ULA`

array with its
default property values.

**Example: **`phased.URA`

`ReceiveArray`

— Receiving array

`phased.ULA`

(default) | Phased Array System Toolbox antenna array System object

Receiving array, specified as a Phased Array System Toolbox antenna array System object. The default value for this property is a `phased.ULA`

array with its
default property values.

**Example: **`phased.URA`

`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`

`CarrierFrequency`

— Signal carrier frequency

`300e6`

(default) | positive real-valued scalar

Signal carrier frequency, specified as a positive real-valued scalar. Units are in Hz.

**Example: **`100e6`

**Data Types: **`double`

`Polarization`

— Polarization configuration

`'None'`

(default) | `'Combined'`

| `'Dual'`

Polarization configuration, specified as `'None'`

,
`'Combined'`

, or `'Dual'`

. When you set this
property to `'None'`

, the output field is considered a scalar field.
When you set this property to `'Combined'`

, the radiated fields are
polarized and are interpreted as a single signal in the sensor's inherent polarization.
When you set this property to `'Dual'`

, the *H* and
*V* polarization components of the radiated field are independent
signals.

**Example: **`'Dual'`

**Data Types: **`char`

`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/m^{3}.

**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/m^{3}.
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`

`SimulateDirectPath`

— Enable propagation along direct path

`false`

(default) | `true`

Option to enable signal propagation along the direct path, specified as
`false`

or `true`

. The direct path is
a line-of-sight path from the transmitting array to the receiving array with
no scattering.

**Data Types: **`logical`

`ChannelResponseOutputPort`

— Enable output of channel response

`false`

(default) | `true`

Option to enable output of channel response, specified as
`false`

or `true`

. Set this property
to `true`

to output the channel response and time delay by
using the `chmatrix`

and `tau`

output
arguments of the `step`

method.

**Data Types: **`logical`

`MaximumDelaySource`

— Source of maximum delay

`'Auto'`

(default) | `'Property'`

Source of the maximum delay value, specified as `'Auto'`

or `'Property'`

.
When you set this property to `'Auto'`

, the channel
automatically allocates enough memory to simulate the propagation
delay. When you set this property to `'Property'`

,
you can specify the maximum delay by using the `MaximumDelay`

property.
Signals arriving after the maximum delay are ignored.

`MaximumDelay`

— Maximum signal delay

`10e-6`

(default) | positive scalar

Maximum signal delay, specified as a positive scalar. Delays greater than this value are ignored. Units are in seconds.

#### Dependencies

To enable this property, set the
`MaximumDelaySource`

property to
`'Property'`

.

**Data Types: **`double`

`TransmitArrayMotionSource`

— Source of transmitting array motion parameters

`'Property'`

(default) | `'Input port'`

Source of the transmitting array motion parameters, specified as
`'Property'`

or `'Input port'`

.

When you set this property to

`'Property'`

, the transmitting array is stationary. Then, you can specify the location and orientation of the array using the`TransmitArrayPosition`

and`TransmitArrayOrientationAxes`

properties.When you set this property to

`'Input port'`

, specify the transmitting array location, velocity, and orientation by using the`txpos`

,`txvel`

, and`txaxes`

input arguments of the`step`

method.

**Data Types: **`char`

`TransmitArrayPosition`

— Position of transmitting array phase center

`[0;0;0]`

(default) | real-valued three-element vector

Position of the transmitting array phase center, specified as a
real-valued three-element vector in Cartesian form,
`[x;y;z]`

, with respect to the global coordinate
system. Units are in meters.

**Example: **`[1000;-200;55]`

#### Dependencies

To enable this property, set the
`TransmitArrayMotionSource`

property to
`'Property'`

.

**Data Types: **`double`

`TransmitArrayOrientationAxes`

— Orientation of transmitting array

`eye(3,3)`

(default) | real-valued 3-by-3 orthonormal matrix

Orientation of transmitting array, specified as a real-valued 3-by-3
orthonormal matrix. The matrix specifies the three axes,
*(x,y,z)*, that define the local coordinate system of
the array with respect to the global coordinate system. Matrix columns
correspond to the axes of the local array coordinate system.

**Example: **`rotz(45)`

#### Dependencies

To enable this property, set the
`TransmitArrayMotionSource`

property to
`'Property'`

.

**Data Types: **`double`

`ReceiveArrayMotionSource`

— Source of receiving array motion parameters

`'Property'`

(default) | `'Input port'`

Source of the receiving array motion parameters, specified as
`'Property'`

or `'Input port'`

.

When you set this property to

`'Property'`

, the receiving array is stationary. Then, you can specify the location and orientation of the array by using the`ReceiveArrayPosition`

and`ReceiveArrayOrientationAxes`

properties.When you set this property to

`'Input port'`

, you can specify the receiving array location, velocity, and orientation by using the`rxpos`

,`rxvel`

, and`rxaxes`

input arguments of the`step`

method.

**Data Types: **`char`

`ReceiveArrayPosition`

— Position of receiving array

`[0;0;0]`

(default) | real-valued three-element vector

Position of the receiving array phase center, specified as a real-valued
three-element vector in Cartesian form,`[x;y;z]`

, with
respect to the global coordinate system. Units are in meters.

**Example: **`[1000;-200;55]`

#### Dependencies

To enable this property, set the
`ReceiveArrayMotionSource`

property to
`'Property'`

.

**Data Types: **`double`

`ReceiveArrayOrientationAxes`

— Orientation of receiving array

`eye(3,3)`

(default) | real-valued 3-by-3 orthonormal matrix

Orientation of receiving array, specified as a real-valued 3-by-3
orthonormal matrix. The matrix specifies the three axes,
*(x,y,z)*, that define the local coordinate system of
the array with respect to the global coordinate system. Matrix columns
correspond to the axes of the local array coordinate system.

**Example: **`roty(60)`

#### Dependencies

To enable this property, set the
`ReceiveArrayMotionSource`

property to
`'Property'`

.

**Data Types: **`double`

`ScattererSpecificationSource`

— Source of scatterer parameters

`'Auto'`

(default) | `'Property'`

| `'Input port'`

Source of scatterer parameters, specified as `'Auto'`

,
`'Property'`

, `'Input port'`

.

When you set this property to

`'Auto'`

, all scatterer positions and coefficients are randomly generated. Scatterer velocities are zero. The generated positions are contained within the region defined by the`ScattererPositionBoundary`

. To set the number of scatterers, use the`NumScatterers`

property.When you set this property to

`'Property'`

, you can set the scatterer positions by using the`ScattererPosition`

property and the scattering coefficients by using the`ScattererCoefficient`

property. All scatterer velocities are zero.When you set this property to

`'Input port'`

, you can specify the scatterer positions, velocities, and scattering coefficients using the`scatpos`

,`scatvel`

, and`scatcoef`

input arguments of the`step`

method.

**Example: **`'Input port'`

**Data Types: **`char`

`NumScatterers`

— Number of scatterers

`1`

(default) | nonnegative integer

Number of scatterers, specified as a nonnegative integer.

**Example: **`9`

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Auto'`

.

**Data Types: **`double`

`ScattererPositionBoundary`

— Boundary of scatterer positions

`[0,1000]`

(default) | 1-by-2 real-valued vector | 3-by-2 real-valued matrix

Boundary of the scatterer positions, specified as a 1-by-2 real-valued row
vector or a 3-by-2 real-valued matrix. The vector specifies the minimum and
maximum, `[minbdry maxbdry]`

, for all three dimensions. The
matrix specifies boundaries in all three dimensions in the form
```
[x_minbdry x_maxbdry;y_minbdry y_maxbdry; z_minbdry
z_maxbdry]
```

.

**Example: **`[-1000 500;-100 100;-200 0]`

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Auto'`

.

**Data Types: **`double`

`ScattererPosition`

— Positions of scatterers

`[0;0;0]`

(default) | real-valued 3-by-*K* matrix

Positions of the scatterers, specified as real-valued
3-by-*K* matrix. *K* is the number of
scatterers. Each column represents a different scatterer and has the
Cartesian form `[x;y;z]`

with respect to the global
coordinate system. Units are in meters.

**Example: **`[1050 -100;-300 55;0 -75]`

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Property'`

.

**Data Types: **`double`

`ScattererCoefficient`

— Scattering coefficients

`1`

(default) | complex-valued 1-by-*K* vector

Scattering coefficients, specified as a complex-valued
1-by-*K* vector. *K* is the number of
scatterers. Units are dimensionless.

**Example: **`2+1i`

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Property'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`ScatteringMatrix`

— Scattering matrices

`[1 0;0 1]`

| complex–valued 2-by-2-by-*N*_{s}
array

Scattering matrices of the scatterers, specified as a complex–valued
2-by-2-by-*N*_{s} array where
*N*_{s} is the number of
scatterers. Each page of this array represents the scattering matrix of a
scatterer. Each scattering matrix has the form ```
[s_hh s_hv;s_vh
s_vv]
```

. For example, the component `s_hv`

specifies the complex scattering response when the input signal is
vertically polarized and the reflected signal is horizontally polarized. The
other components are defined similarly. Units are in square meters.

#### Dependencies

To enable this property, set the
`ScatteringMatrixSource`

property to
`'Property'`

and the
`Polarization`

property to
`'Combined'`

or `'Dual'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`ScattererOrientationAxes`

— Orientation of scatterers

`[1 0 0;0 1 0;0 0 1]`

(default) | real–valued 3-by-3-by-*N*_{s}
array

Orientation of the scatterers, specified as a real–valued
3-by-3-by-*N*_{s} array where
*N*_{s} is the number of
scatterers. Each page of this array is an orthonormal matrix. Matrix columns
represent the axis of the local coordinates
(*x*,*y*,*z*) of the
scatterer with respect to the global coordinate system.

**Example: **`roty(45)`

#### Dependencies

To enable this property, set the
`ScatteringMatrixSource`

property to
`'Property'`

and the
`Polarization`

property to
`'Combined'`

or `'Dual'`

.

**Data Types: **`double`

`SeedSource`

— Source of random number generator seed

`'Auto'`

(default) | `'Property'`

Source of random number generator seed, specified as
`'Auto'`

or `'Property'`

.

When you set this property to

`'Auto'`

, random numbers are generated using the default MATLAB^{®}random number generator.When you set this property to

`'Property'`

, the object uses a private random number generator with the seed specified by the value of the`Seed`

property.

To use this object with Parallel Computing Toolbox™ software, set this property to
`'Auto'`

.

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Auto'`

.

`Seed`

— Random number generator seed

`0`

(default) | nonnegative integer

Random number generator seed, specified as a nonnegative integer less than
2^{32}.

**Example: **`5005`

#### Dependencies

To enable this property, set the
`ScattererSpecificationSource`

property to
`'Auto'`

and the `SeedSource`

property to `'Property'`

.

**Data Types: **`double`

## Methods

reset | Reset state of the System object |

step | Propagate signals in scattering MIMO channel |

Common to All System Objects | |
---|---|

`release` | Allow System object property value changes |

## Examples

### Propagate Signals in MIMO Channel

Create a 30 GHz MIMO channel with random scatterers. The scenario contains a stationary 21-element transmitting ULA array and a stationary 15-element receiving ULA array. The transmitting antennas have cosine responses and the receiving antennas are isotropic. Element spacing for both arrays is less than one-half wavelength. The channel has 50 randomly generated static scatterers within a specified bounding box. The transmit array is located at [0;20;50] meters and the receive array is located at [200;10;10] meters. Compute the propagated signal through this channel. The sample rate for the signal is 10 MHz.

fc = 30e9; c = physconst('LightSpeed'); lambda = c/fc; fs = 10e6; txarray = phased.ULA('Element',phased.CosineAntennaElement,... 'NumElements',21,'ElementSpacing',0.45*lambda); rxarray = phased.ULA('Element',phased.IsotropicAntennaElement,... 'NumElements',15,'ElementSpacing',0.45*lambda); channel = phased.ScatteringMIMOChannel('TransmitArray',txarray,... 'ReceiveArray',rxarray,'PropagationSpeed',c,'CarrierFrequency',fc,... 'SampleRate',fs,'TransmitArrayPosition',[0;20;50],... 'ReceiveArrayPosition',[200;10;10],'NumScatterers',50,... 'ScattererPositionBoundary',[10 180; -30 30; -30 30]);

Create a random data signal of ones and zeros for each transmitter.

x = randi(2,[100 21]) - 1;

Compute the received signals after propagating through the channel.

y = channel(x);

### Propagate Signals in MIMO Channel from Moving Transmitter

Create a MIMO channel containing 3 fixed scatterers. The scenario contains a 21-element transmitting ULA array operating at 72 GHz, and a 15-element receiving ULA array. The transmitting elements have cosine response shapes and the receiving antennas are isotropic. Only the transmitting antenna is moving. Element spacing for both arrays is less than one-half wavelength. The transmitting array starts at (0,20,50) meters and moves towards the receiver at 2 m/s. The receiving array is located at (200,10,10) meters. Compute the propagated signal through this channel. The sample rate for the signal is 10 MHz.

fc = 72e9; c = physconst('LightSpeed'); lambda = c/fc; fs = 10e6; txplatform = phased.Platform('MotionModel','Velocity','InitialPosition', ... [0;20;50],'Velocity',[2;0;0]); txarray = phased.ULA('Element',phased.CosineAntennaElement, ... 'NumElements',21,'ElementSpacing',0.45*lambda); rxarray = phased.ULA('Element',phased.IsotropicAntennaElement, ... 'NumElements',15,'ElementSpacing',0.45*lambda); channel = phased.ScatteringMIMOChannel('TransmitArray',txarray, ... 'ReceiveArray',rxarray,'PropagationSpeed',c,'CarrierFrequency',fc,... 'SampleRate',fs,'TransmitArrayMotionSource','Input port', ... 'ReceiveArrayMotionSource','Property','ReceiveArrayPosition',[200;10;10],... 'ReceiveArrayOrientationAxes',rotz(180),... 'ScattererSpecificationSource','Property','ScattererPosition', ... [75 100 120; -10 20 12; 5 -5 8],'ScattererCoefficient',[1i,2+3i,-1+1i]);

Move the platforms for two time steps at one second intervals. For each time instance:

Create a random data signal of ones and zeros for each transmitter element.

Move the transmitter and receiver. The orientations are fixed.

Propagate the signals from transmitters to scatterers to receiver.

for k =1:2 x = randi(2,[100 21]) - 1; [txpos,txvel] = txplatform(1); txaxes = eye(3); y = channel(x,txpos,txvel,txaxes); end

### Propagate Signals Through MIMO Channel to Moving Receiver

Create a MIMO channel containing 3 fixed scatterers. The scenario contains a 21-element transmitting ULA array and a 15-element receiving ULA array. Both arrays operate at 72 GHz. The transmitting elements have cosine response shapes and the receiving antennas are isotropic. Only the receiving antenna is moving. Element spacing for both arrays is less than one-half wavelength. The transmitting array is located at (0,20,50) meters. The receiving array starts at (200,10,10) meters and moves toward the transmitter at 2 m/s. Compute the propagated signal through this channel. The sample rate for the signal is 10 MHz.

fc = 72e9; c = physconst('LightSpeed'); lambda = c/fc; fs = 10e6; rxplatform = phased.Platform('MotionModel','Velocity','InitialPosition',... [200;10;10],'Velocity',[-2;0;0]); txarray = phased.ULA('Element',phased.CosineAntennaElement, ... 'NumElements',21,'ElementSpacing',0.45*lambda); rxarray = phased.ULA('Element',phased.IsotropicAntennaElement, ... 'NumElements',15,'ElementSpacing',0.45*lambda); channel = phased.ScatteringMIMOChannel('TransmitArray',txarray, ... 'ReceiveArray',rxarray,'PropagationSpeed',c,'CarrierFrequency',fc, ... 'SampleRate',fs,'TransmitArrayMotionSource','Property',... 'TransmitArrayPosition',[0;20;50],'TransmitArrayOrientationAxes',eye(3,3), ... 'ReceiveArrayMotionSource','Input port','ScattererSpecificationSource', ... 'Property','ScattererPosition',[75 100 120; -10 20 12; 5 -5 8], ... 'ScattererCoefficient',[1i,2+3i,-1+1i],'SpecifyAtmosphere',false);

Move the platforms for two time steps at one-second intervals. For each time instance:

Create a random data signal of ones and zeros for each transmitter element.

Move the transmitter and receiver. Fix the array orientations.

Propagate the signals from transmitters to scatterers to receiver.

for k =1:2 x = randi(2,[100 21]) - 1; [rxpos,rxvel] = rxplatform(1); rxaxes = rotz(45); y = channel(x,rxpos,rxvel,rxaxes); end

### Propagate Polarized Signals in MIMO Channel

Create a MIMO channel at 30 GHz with a 16-element transmit array and a 64-element receive array. Assume the elements are short-dipole antennas and the arrays are uniform linear arrays. The transmit array is located at [0;0;50] meters.

The receive array has an initial position at [200;0;0] meters and is moving at a speed of [10;0;0] meters/second. There are 200 static scatterers randomly located on the *xy* plane within a square centered at [200;0;0] and with a side length of 100 meters.

Use the channel to compute the propagated polarized signal. Assume the sample rate for the signal is 10 MHz and the frame length is 1000 samples. Collect 5 frames of received signal.

fc = 30e9; c = 3e8; lambda = c/fc; fs = 10e6; txarray = phased.ULA('Element',phased.ShortDipoleAntennaElement,... 'NumElements',16,'ElementSpacing',lambda/2); rxarray = phased.ULA('Element',phased.ShortDipoleAntennaElement,... 'NumElements',64,'ElementSpacing',lambda/2); Ns = 200; scatpos = [100*rand(1,Ns) + 150; 100*rand(1,Ns) + 150; zeros(1,Ns)]; temp = randn(1,Ns) + 1i*randn(1,Ns); scatcoef = repmat(eye(2),1,1,Ns).*permute(temp,[1 3 2]); scatax = repmat(eye(3),1,1,Ns); Nframesamp = 1000; Tframe = Nframesamp/fs; rxmobile = phased.Platform('InitialPosition',[200;0;0],... 'Velocity',[10;0;0],'OrientationAxesOutputPort',true); chan = phased.ScatteringMIMOChannel(... 'TransmitArray',txarray,... 'ReceiveArray',rxarray,... 'PropagationSpeed',c,... 'CarrierFrequency',fc,... 'SampleRate',fs,... 'Polarization','Dual',... 'TransmitArrayPosition',[0;0;50],... 'ReceiveArrayMotionSource','Input port',... 'ScattererSpecificationSource','Property',... 'ScattererPosition',scatpos,... 'ScatteringMatrix',scatcoef,... 'ScattererOrientationAxes',scatax); xh = randi(2,[Nframesamp 16])-1; xv = randi(2,[Nframesamp 16])-1; for m = 1:5 [rxpos,rxvel,rxax] = rxmobile(Tframe); [yh,yv] = chan(xh,xv,rxpos,rxvel,rxax); end

## More About

### Attenuation and Loss Factors

Attenuation or path loss in the scattering MIMO channel consists of four
components. *L =
L _{fsp}L_{g}L_{c}L_{r}*, where:

*L*is the free-space path attenuation._{fsp}*L*is the atmospheric path attenuation._{g}*L*is the fog and cloud path attenuation._{c}*L*is the rain path attenuation._{r}

Each component is in magnitude units, not in dB.

### Free-Space Time Delay and Loss

When the origin and destination are stationary relative to each
other, you can write the output signal of a free-space channel as *Y(t)
= x(t-τ)/L _{fsp}*.
The quantity

*τ*is the signal delay and

*L*is the free-space path loss. The delay

_{fsp}*τ*is given by

*R/c*, where

*R*is the propagation distance and

*c*is the propagation speed. The free-space path loss is given by

$${L}_{fsp}=\frac{{(4\pi R)}^{2}}{{\lambda}^{2}},$$

where λ is the signal wavelength.

This formula assumes that the target is in the far field of
the transmitting element or array. In the near field, the free-space
path loss formula is not valid and can result in a loss smaller than
one, equivalent to a signal gain. Therefore, the loss is set to unity
for range values, *R ≤ λ/4π*.

When the origin and destination have relative motion, the processing
also introduces a Doppler frequency shift. The frequency shift is *v/λ* for
one-way propagation and *2v/λ* for two-way
propagation. The quantity *v* is the relative speed
of the destination with respect to the origin.

For more details on free space channel propagation, see [8]

### 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

$$\gamma ={\gamma}_{o}(f)+{\gamma}_{w}(f)=0.1820f{N}^{\u2033}(f).$$

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

$${N}^{\u2033}(f)={\displaystyle \sum _{i}{S}_{i}{F}_{i}+{{N}^{\u2033}}_{D}^{}(f)}$$

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

$${S}_{i}={a}_{1}\times {10}^{-7}{\left(\frac{300}{T}\right)}^{3}\mathrm{exp}\left[{a}_{2}(1-\left(\frac{300}{T}\right)\right]P.$$

For atmospheric water vapor, each spectral line strength is

$${S}_{i}={b}_{1}\times {10}^{-1}{\left(\frac{300}{T}\right)}^{3.5}\mathrm{exp}\left[{b}_{2}(1-\left(\frac{300}{T}\right)\right]W.$$

*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,

*a*and

_{1}*a*. Similarly, each water vapor line depends on two parameters,

_{2}*b*and

_{1}*b*. The ITU documentation cited at the end of this section contains tabulations of these parameters as functions of frequency.

_{2}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.

For a complete description of this model, see [4].

### 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/m^{3}. 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/m

^{3}).

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.

For a complete description of this model, see [5]

### 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*_{eff}*γ*_{R}.

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

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

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.

## References

[1] Heath, R. Jr. et al. “An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems”, arXiv.org:1512.03007 [cs.IT], 2015.

[2] Tse, D. and P. Viswanath, *Fundamentals of Wireless
Communications*, Cambridge: Cambridge University Press,
2005.

[3] Paulraj, A. *Introduction to Space-Time Wireless
Communications*, Cambridge: Cambridge University Press,
2003.

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

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

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

[7] Seybold, J. *Introduction to RF Propagation*. New
York: Wiley & Sons, 2005.

[8] Skolnik, M. *Introduction to Radar
Systems*, 3rd Ed. New York: McGraw-Hill, 2001.

## 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).

## Version History

**Introduced in R2017a**

## See Also

### Functions

`rangeangle`

|`fogpl`

|`gaspl`

|`rainpl`

|`fspl`

|`diagbfweights`

|`scatteringchanmtx`

|`waterfill`

### Objects

`phased.FreeSpace`

|`phased.RadarTarget`

|`phased.BackscatterRadarTarget`

|`twoRayChannel`

(Radar Toolbox) |`phased.LOSChannel`

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