# comm.MIMOChannel

Filter input signal through MIMO multipath fading channel

## Description

The `comm.MIMOChannel`

System object™ filters an input signal through a multiple-input/multiple-output (MIMO)
multipath fading channel. This object models Rayleigh and Rician fading and employs the
Kronecker model for modeling the spatial correlation between the links. For processing
details, see the Algorithms
section.

To filter an input signal through a MIMO multipath fading channel:

Create the

`comm.MIMOChannel`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Description

creates a MIMO frequency-selective or frequency-flat fading channel System object. `mimochannel`

= comm.MIMOChannel

sets properties using one or more name-value arguments. For example,
`mimochannel`

= comm.MIMOChannel(`Name`

,`Value`

)`'SampleRate',2`

sets the input signal sample rate to
2.

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`SampleRate`

— Input signal sample rate

`1`

(default) | positive scalar

Input signal sample rate in hertz, specified as a positive scalar.

**Data Types: **`double`

`PathDelays`

— Discrete path delay

`0`

(default) | scalar | row vector

Discrete path delay in seconds, specified as a scalar or row vector.

When you set

`PathDelays`

to a scalar, the MIMO channel is frequency flat.When you set

`PathDelays`

to a vector, the MIMO channel is frequency selective.

The `PathDelays`

and `AveragePathGains`

properties must be the same length.

**Data Types: **`double`

`AveragePathGains`

— Average path gains

`0`

(default) | scalar | row vector

Average path gains in decibels, specified as a scalar or row vector. The
`AveragePathGains`

and `PathDelays`

properties must be the same length.

**Data Types: **`double`

`NormalizePathGains`

— Normalize path gains

`true`

or `1`

(default) | `false`

or `0`

Normalize path gains, specified as one of these logical values:

`1`

(`true`

) — The fading processes are normalized so that the total power of the path gains, averaged over time, is 0 dB.`0`

(`false`

) — The total power of the path gains is not normalized.

The `AveragePathGains`

property specifies the average powers of
the path gains.

**Data Types: **`logical`

`FadingDistribution`

— Fading distribution

`'Rayleigh'`

(default) | `'Rician'`

Fading distribution to use for the channel, specified as
`'Rayleigh'`

or `'Rician'`

.

**Data Types: **`char`

| `string`

`KFactor`

— K-factor of Rician fading channel

`3`

(default) | positive scalar | 1-by-*N*_{P} vector of nonnegative
values

K-factor of a Rician fading channel, specified as a positive scalar or a
1-by-*N*_{P} vector of nonnegative
values. *N*_{P} is the number of
discrete path delays specified by the `PathDelays`

property.

When you set

`KFactor`

to a scalar, the first discrete path is a Rician fading process with a Rician K-factor of`KFactor`

. Any remaining discrete paths are independent Rayleigh fading processes.When you set

`KFactor`

to a vector, the discrete path corresponding to a positive element of the`KFactor`

vector is a Rician fading process with a Rician K-factor specified by that element. The discrete path corresponding to any zero-valued elements of the`KFactor`

vector are Rayleigh fading processes. At least one element must be nonzero.

#### Dependencies

To enable this property, set the `FadingDistribution`

property to
`'Rician'`

.

**Data Types: **`double`

`DirectPathDopplerShift`

— Doppler shifts for line-of-sight components

`0`

(default) | scalar | row vector

Doppler shifts for the line-of-sight components of the multipath Rician
fading channel, specified as a scalar or row vector. Units are in hertz.
This property must be the same size as the `KFactor`

property.

When you set

`DirectPathDopplerShift`

to a scalar, the value represents the line-of-sight component Doppler shift of the first discrete path. This path exhibits a Rician fading process.When you set

`DirectPathDopplerShift`

to a row vector, the discrete path corresponding to a positive element of the`KFactor`

vector is a Rician fading process. The corresponding element of`DirectPathDopplerShift`

specifies the line-of-sight component for the Doppler shift of that discrete path.

#### Dependencies

To enable this property, set the `FadingDistribution`

property to
`'Rician'`

.

**Data Types: **`double`

`DirectPathInitialPhase`

— Initial phases for line-of-sight components

`0`

(default) | scalar | row vector

Initial phases for the line-of-sight components of the multipath Rician
fading channel, specified as a scalar or row vector. Units are in radians.
This property must be the same size as the `KFactor`

property value.

When you set

`DirectPathInitialPhase`

to a scalar, the value represents the line-of-sight component initial phase of the first discrete path. This path exhibits a Rician fading process.When you set

`DirectPathInitialPhase`

to a row vector, the discrete path corresponding to a positive element of the`KFactor`

vector is a Rician fading process. The corresponding element of`DirectPathInitialPhase`

specifies the line-of-sight component for the initial phase of that discrete path.

#### Dependencies

To enable this property, set the `FadingDistribution`

property to
`'Rician'`

.

**Data Types: **`double`

`MaximumDopplerShift`

— Maximum Doppler shift for all channel paths

`0.001`

(default) | nonnegative scalar

Maximum Doppler shift for all channel paths, specified as a nonnegative scalar. Units are in hertz.

The maximum Doppler shift limit applies to each channel path. When you set
this property to `0`

, the channel remains static for the
entire input. You can use the `reset`

object function to generate a new channel realization. The `MaximumDopplerShift`

property value must be
smaller than `SampleRate`

/10/*f*_{c}
for each path, where *f*_{c} is the
cutoff frequency factor of the path. For most Doppler spectrum types, the
value of *f*_{c} is 1. For Gaussian and
bi-Gaussian Doppler spectrum types,
*f*_{c} is dependent on the Doppler
spectrum structure fields. For more details about how
*f*_{c} is defined, see the Cutoff
Frequency Factor section.

**Data Types: **`double`

`DopplerSpectrum`

— Doppler spectrum shape for all channel paths

`doppler('Jakes')`

(default) | Doppler spectrum structure | 1-by-*N*_{P} cell array of Doppler
spectrum structures

Doppler spectrum shape for all channel paths, specified as a Doppler
spectrum structure or a 1-by-*N*_{P}
cell array of Doppler spectrum structures. These Doppler spectrum structures
must be outputs of the form returned from the `doppler`

function.
*N*_{P} is the number of discrete
path delays specified by the `PathDelays`

property. The `MaximumDopplerShift`

property defines the maximum Doppler
shift value that the `DopplerSpectrum`

property permits
when you specify the Doppler spectrum.

When you set

`DopplerSpectrum`

to a single Doppler spectrum structure, all paths have the same specified Doppler spectrum.When you set

`DopplerSpectrum`

to a cell array of Doppler spectrum structures, each path has the Doppler spectrum specified by the corresponding structure in the cell array.

Specify options for the spectrum type by using the
`specType`

input to the `doppler`

function. If you set
the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

, you must set `DopplerSpectrum`

to `doppler('Jakes')`

.

#### Dependencies

To enable this property, set the `MaximumDopplerShift`

property to a positive
scalar.

**Data Types: **`struct`

| `cell`

`SpatialCorrelationSpecification`

— Spatial correlation specification

`'Separate Tx Rx'`

(default) | `'None'`

| `'Combined'`

Spatial correlation specification, specified as ```
'Separate Tx
Rx'
```

, `'None'`

, or
`'Combined'`

.

Choose

`'Spatial Tx Rx'`

to separately specify the transmit and receive spatial correlation matrices from which the number of transmit antennas (*N*_{T}) and number of receive antennas (*N*_{R}) are derived.Choose

`'None'`

to specify the number of transmit and receive antennas.Choose

`'Combined'`

to specify a single correlation matrix for the whole channel from which the product of*N*_{T}and*N*_{R}is derived.

**Data Types: **`char`

| `string`

`NumTransmitAntennas`

— Number of transmit antennas

`2`

(default) | positive integer

Number of transmit antennas, specified as a positive integer.

#### Dependencies

To enable this property, set the `SpatialCorrelationSpecification`

property to
`'None'`

or `'Combined'`

.

**Data Types: **`double`

`NumReceiveAntennas`

— Number of receive antennas

`2`

(default) | positive integer

Number of receive antennas, specified as a positive integer.

#### Dependencies

To enable this property, set the `SpatialCorrelationSpecification`

property to
`'None'`

or `'Combined'`

.

**Data Types: **`double`

`TransmitCorrelationMatrix`

— Spatial correlation of transmitter

`[1 0; 0 1]`

(default) | *N*_{T}-by-*N*_{T}
matrix | *N*_{T}-by-*N*_{T}-by-*N*_{P}
array

Spatial correlation of the transmitter, specified as an
*N*_{T}-by-*N*_{T}
matrix or
*N*_{T}-by-*N*_{T}-by-*N*_{P}
array. *N*_{T} is the number of
transmit antennas. *N*_{P} is the
number of discrete path delays specified by the `PathDelays`

property.

If you set

`PathDelays`

to a scalar, the channel is frequency flat and`TransmitCorrelationMatrix`

must be an*N*_{T}-by-*N*_{T}Hermitian matrix. The magnitude of any off-diagonal element must be no larger than the geometric mean of the two corresponding diagonal elements.If you set

`PathDelays`

to a vector, the channel is frequency selective and you can specify`TransmitCorrelationMatrix`

as one of these options:An

*N*_{T}-by-*N*_{T}matrix. In this case, each path has the same transmit spatial correlation matrix.An

*N*_{T}-by-*N*_{T}-by-*N*_{P}array. In this case, each path has its own specified transmit spatial correlation matrix.

#### Dependencies

To enable this property, set the `SpatialCorrelationSpecification`

property to
`'Separate Tx Rx'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`ReceiveCorrelationMatrix`

— Spatial correlation of receiver

`[1 0; 0 1]`

(default) | *N*_{R}-by-*N*_{R}
matrix | *N*_{R}-by-*N*_{R}-by-*N*_{P}
array

Spatial correlation of the receiver, specified as an
*N*_{R}-by-*N*_{R}
matrix or
*N*_{R}-by-*N*_{R}-by-*N*_{P}
array. *N*_{R} is the number of receive
antennas. *N*_{P} is the number of
discrete path delays specified by the `PathDelays`

property.

If you set

`PathDelays`

to a scalar, the channel is frequency flat, and`ReceiveCorrelationMatrix`

must be an*N*_{R}-by-*N*_{R}Hermitian matrix. The magnitude of any off-diagonal element must be no larger than the geometric mean of the two corresponding diagonal elements.If you set

`PathDelays`

to a vector, the channel is frequency selective and you can specify`ReceiveCorrelationMatrix`

as one of these options:An

*N*_{R}-by-*N*_{R}matrix. In this case, each path has the same receive spatial correlation matrix.An

*N*_{R}-by-*N*_{R}-by-*N*_{P}array. In this case, each path has its own specified receive spatial correlation matrix.

#### Dependencies

To enable this property, set the `SpatialCorrelationSpecification`

property to
`'Separate Tx Rx'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`SpatialCorrelationMatrix`

— Combined spatial correlation matrix

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

(default) | *N*_{TR}-by-*N*_{TR}
matrix | *N*_{TR}-by-*N*_{TR}-by-*N*_{P}
array

Combined spatial correlation matrix, specified as an
*N*_{TR}-by-*N*_{TR}
matrix or
*N*_{TR}-by-*N*_{TR}-by-*N*_{P}
array. *N*_{TR} =
(*N*_{T} ✕
*N*_{R}), and
*N*_{P} is the number of discrete
delay paths (the length of the `PathDelays`

property).

If

`PathDelays`

is a scalar, the channel is frequency flat, and`SpatialCorrelationMatrix`

must be an*N*_{TR}-by-*N*_{TR}Hermitian matrix. The magnitude of any off-diagonal element must be no larger than the geometric mean of the two corresponding diagonal elements.If you set

`PathDelays`

to a vector, the channel is frequency selective and you can specify`SpatialCorrelationMatrix`

as one of these options:An

*N*_{TR}-by-*N*_{TR}matrix. In this case, each path has the same combined spatial correlation matrix.An

*N*_{TR}-by-*N*_{TR}-by-*N*_{P}array. In this case, each path has its own specified combined spatial correlation matrix.

#### Dependencies

To enable this property, set the `SpatialCorrelationSpecification`

property to
`'Combined'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

`AntennaSelection`

— Antenna selection scheme

`'Off'`

(default) | `'Tx'`

| `'Rx'`

| `'Tx and Rx'`

Antenna selection scheme, specified as `'Off'`

,
`'Tx'`

, `'Rx'`

, or ```
'Tx and
Rx'
```

.

`Tx`

represents transmit antennas, and
`Rx`

represents receive antennas. When you configure
any antenna selection other than the default setting, the object requires
one or more inputs to specify which antennas are selected for signal
transmission. For more information, see Antenna
Selection.

**Data Types: **`char`

| `string`

`NormalizeChannelOutputs`

— Normalize channel outputs

`true`

or `1`

(default) | `false`

or `0`

Normalize channel outputs, specified as one of these logical values:

`1`

(`true`

) — The channel outputs are normalized by the number of receive antennas.`0`

(`false`

) — The channel outputs are not normalized.

**Data Types: **`logical`

`ChannelFiltering`

— Channel filtering

`true`

or `1`

(default) | `false`

or `0`

Channel filtering, specified as one of these logical values:

`1`

(`true`

) — The channel accepts an input signal and produces a filtered output signal.`0`

(`false`

) — The object does not accept an input signal, produces no filtered output signal, and outputs only channel path gains. You must specify the duration of the fading process by using the`NumSamples`

property.

**Data Types: **`logical`

`PathGainsOutputPort`

— Output channel path gains

`false`

or `0`

(default) | `true`

or `1`

Output channel path gains, specified as a logical `0`

(`false`

) or `1`

(`true`

). Set this property to `true`

to output the channel path gains of the underlying fading process.

#### Dependencies

To enable this property, set the `ChannelFiltering`

property to
`true`

.

**Data Types: **`logical`

`NumSamples`

— Number of samples

`100`

(default) | nonnegative integer

Number of samples used for the duration of the fading process, specified as a nonnegative integer.

**Tunable: **Yes

#### Dependencies

To enable this property, set the `ChannelFiltering`

property to
`false`

.

**Data Types: **`double`

`OutputDataType`

— Path gain output data type

`'double'`

(default) | `'single'`

Path gain output data type, specified as `'double'`

or
`'single'`

.

#### Dependencies

To enable this property, set the `ChannelFiltering`

property to
`false`

.

**Data Types: **`char`

| `string`

`FadingTechnique`

— Channel model fading technique

`'Filtered Gaussian noise'`

(default) | `'Sum of sinusoids'`

Channel model fading technique, specified as ```
'Filtered Gaussian
noise'
```

or `'Sum of sinusoids'`

.

**Data Types: **`char`

| `string`

`NumSinusoids`

— Number of sinusoids

`48`

(default) | positive integer

Number of sinusoids used to model the fading process, specified as a positive integer.

#### Dependencies

To enable this property, set the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

.

**Data Types: **`double`

`InitialTimeSource`

— Source to control start time of fading process

`'Property'`

(default) | `'Input port'`

Source to control the start time of the fading process, specified as
`'Property'`

or `'Input port'`

.

When you set

`InitialTimeSource`

to`'Property'`

, set the initial time offset by using the`InitialTime`

property.When you set

`InitialTimeSource`

to`'Input port'`

, specify the start time of the fading process by using the`inittime`

input argument. The input value can change in consecutive calls to the object.

#### Dependencies

To enable this property, set the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

.

**Data Types: **`char`

| `string`

`InitialTime`

— Initial time offset

`0`

(default) | nonnegative scalar

Initial time offset for the fading model in seconds, specified as a nonnegative scalar.

`InitialTime`

must be greater than the end time of the
last frame. When `mod`

(`InitialTime`

/`SampleRate`

) is nonzero, the object rounds the initial time offset up
to the nearest sample position.

#### Dependencies

To enable this property, set the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

and the `InitialTimeSource`

property to
`'Property'`

.

**Data Types: **`double`

`RandomStream`

— Source of random number stream

`'Global stream'`

(default) | `'mt19937ar with seed'`

Source of the random number stream, specified as ```
'Global
stream'
```

or `'mt19937ar with seed'`

.

When you specify

`'Global stream'`

, the object uses the current global random number stream for random number generation. In this case, the`reset`

object function resets only the filters.When you specify

`'mt19937ar with seed'`

, the object uses the mt19937ar algorithm for random number generation. In this case, the`reset`

object function resets the filters and reinitializes the random number stream to the value of the`Seed`

property.

**Data Types: **`char`

| `string`

`Seed`

— Initial seed of mt19937ar random number stream

`73`

(default) | nonnegative integer

Initial seed of the mt19937ar random number stream, specified as a
nonnegative integer. When you call the `reset`

object
function, it reinitializes the mt19937ar random number stream to the
`Seed`

value.

#### Dependencies

To enable this property, set the `RandomStream`

property to ```
'mt19937ar with
seed'
```

.

**Data Types: **`double`

`Visualization`

— Channel visualization

`'Off'`

(default) | `'Impulse response'`

| `'Frequency response'`

| `'Impulse and frequency responses'`

| `'Doppler spectrum'`

Channel visualization, specified as `'Off'`

,
`'Impulse response'`

, ```
'Frequency
response'
```

, ```
'Impulse and frequency
responses'
```

, or `'Doppler spectrum'`

. When you
set the channel visualization to a value other than
`'Off'`

, the selected channel characteristics, such as
impulse response or Doppler spectrum, display in a separate window. For more
information, see Channel Visualization.

#### Dependencies

To enable this property, set the `FadingTechnique`

property to ```
'Filtered Gaussian
noise'
```

.

**Data Types: **`char`

| `string`

`AntennaPairsToDisplay`

— Transmit-receive antenna pair to display

`[1 1]`

(default) | two-element row vector

Transmit-receive antenna pair to display, specified as a two element row vector. The first element corresponds to the desired transmit antenna, and the second element corresponds to the desired receive antenna. Only a single pair can be displayed.

#### Dependencies

To enable this property, set the `Visualization`

property to ```
'Impulse
response'
```

, `'Frequency response'`

,
`'Doppler spectrum'`

, or ```
'Impulse and
frequency responses'
```

.

**Data Types: **`double`

`PathsForDopplerDisplay`

— Path for which Doppler spectrum is displayed

`1`

(default) | positive integer in the range [1,
*N*_{P}]

Path for which the Doppler spectrum is displayed, specified as an integer
in the range [1, *N*_{P}].
*N*_{P} is the number of discrete
path delays specified by the `PathDelays`

property. Use this property to select the discrete path used in constructing
a Doppler spectrum plot.

#### Dependencies

To enable this property, set the `Visualization`

property to ```
'Doppler
spectrum'
```

.

**Data Types: **`double`

`SamplesToDisplay`

— Percentage of samples to display

`'25%'`

(default) | `'10%'`

| `'50%'`

| `'100%'`

Percentage of samples to display, specified as `'25%'`

,
`'10%'`

, `'50%'`

, or
`'100%'`

. Increasing the percentage improves display
accuracy at the expense of simulation speed.

#### Dependencies

To enable this property, set the `Visualization`

property to ```
'Impulse
response'
```

, `'Frequency response'`

, or
`'Impulse and frequency responses'`

.

**Data Types: **`char`

| `string`

## Usage

### Syntax

### Description

filters the input signal `y`

= mimochannel(`x`

)`x`

through the MIMO fading
channel and returns the result in `y`

.

To enable this syntax, set the `ChannelFiltering`

property to `true`

.

filters the input signal through the MIMO fading channel by using the
transmit antennas specified by `y`

= mimochannel(`x`

,`seltx`

)`seltx`

.

To enable this syntax set the `AntennaSelection`

property to `'Tx'`

.

For example, this code shows how to select the first and third transmit antenna index as active.

mimochannel = comm.MIMOChannel('AntennaSelection','Tx'); seltx = [1 0 1]; ... y = mimochannel(x,seltx);

filters the input signal through the MIMO fading channel by using the
receive antennas selected by `y`

= mimochannel(`x`

,`selrx`

)`selrx`

.

To enable this syntax set the `AntennaSelection`

property to `'Rx'`

.

For example, this code shows how to select the second receive antenna index as active.

mimochannel = comm.MIMOChannel('AntennaSelection','Rx'); selrx = [0 1]; ... y = mimochannel(x,selrx);

filters the input signal through the MIMO fading channel by using the
transmit and receive antennas selected by `y`

= mimochannel(`x`

,`seltx`

,`selrx`

)`seltx`

and
`selrx`

.

To enable this syntax set the `AntennaSelection`

property to ```
'Tx and
Rx'
```

.

For example, this code shows how to select the first and second transmit antenna and the second receive antenna as active.

mimochannel = comm.MIMOChannel( ... 'AntennaSelection','Tx and Rx'); seltx = [1 1]; selrx = [0 1]; ... y = mimochannel(x,selrx);

specifies a start time for the fading process in addition to an input
argument combination from any of the previous syntaxes.`y`

= mimochannel(___,`inittime`

)

To enable this syntax, also set the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

and the `InitialTimeSource`

property to ```
'Input
port'
```

.

returns the channel path gains of the underlying fading process. In this
case, the channel requires no input signal and acts as a source of path
gains.`pathgains`

= mimochannel()

To enable this syntax, set the `ChannelFiltering`

property to
`false`

.

returns the channel path gains of the underlying fading process by using the
transmit antennas specified by `pathgains`

= mimochannel(`seltx`

)`seltx`

.

To enable this syntax set `ChannelFiltering`

property to `false`

and
the `AntennaSelection`

property to `'Tx'`

.

returns the channel path gains of the underlying fading process by using the
transmit antennas specified by `pathgains`

= mimochannel(`selrx`

)`selrx`

.

To enable this syntax set the `ChannelFiltering`

property to `false`

and
the `AntennaSelection`

property to `'Rx'`

.

returns the channel path gains of the underlying fading process by using the
transmit and receive antennas selected by `pathgains`

= mimochannel(`seltx`

,`selrx`

)`seltx`

and
`selrx`

.

To enable this syntax set the `ChannelFiltering`

property to `false`

and
the `AntennaSelection`

property to ```
'Tx and
Rx'
```

.

specifies a start time for the fading process in addition to an input
argument combination from any of the previous syntaxes.`pathgains`

= mimochannel(___,`inittime`

)

To enable this syntax, also set the `FadingTechnique`

property to ```
'Sum of
sinusoids'
```

and the `InitialTimeSource`

property to ```
'Input
port'
```

.

### Input Arguments

`x`

— Input signal

scalar | vector | matrix

Input signal, specified as a scalar, an
*N*_{S}-element column vector,
or an
*N*_{S}-by-*N*_{T}
or
*N*_{S}-by-*N*_{ST}
matrix.

*N*_{S}is the number of samples.*N*_{T}is the number of transmit antennas and is determined by the`TransmitCorrelationMatrix`

or`NumTransmitAntennas`

property values.*N*_{ST}is the number of selected transmit antennas and is determined by the number of elements that are set to`1`

in the vector provided by the`seltx`

input.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

`seltx`

— Select active transmit antennas

1-by-*N*_{T} binary-valued
vector

Select active transmit antennas, specified as a
1-by-*N*_{T} binary-valued
vector. *N*_{T} is the number of
transmit antennas. Elements set to `1`

identify
selected antenna indices, and the elements set to `0`

identify nonselected antenna indices.

**Data Types: **`single`

| `double`

`selrx`

— Select active receive antennas

1-by-*N*_{R} binary-valued
vector

Select active receive antennas, specified as a
1-by-*N*_{R} binary-valued
vector. *N*_{R} is the number of
receive antennas. Elements set to `1`

identify selected
antenna indices, and the elements set to `0`

identify
nonselected antenna indices.

**Data Types: **`single`

| `double`

`inittime`

— Initial time offset

`0`

(default) | nonnegative scalar

Initial time offset for the fading model in seconds, specified as a nonnegative scalar.

When `mod`

(`inittime`

/`SampleRate`

) is nonzero, the initial time offset is rounded up to
the nearest sample position.

**Data Types: **`single`

| `double`

### Output Arguments

`y`

— Output signal

matrix

Output signal, returned as an
*N*_{S}-by-*N*_{R}
or
*N*_{S}-by-*N*_{SR} matrix.

*N*_{S}is the number of samples.*N*_{R}is the number of receive antennas and is determined by the`ReceiveCorrelationMatrix`

or`NumReceiveAntennas`

property values.*N*_{SR}is the number of selected receive antennas and is determined by the number of elements that are set to`1`

in the vector provided by the`selrx`

input.

`pathgains`

— Output path gains

*N*_{S}-by-*N*_{P}-by-*N*_{T}-by-*N*_{R}
array

Output path gains, returned as an
*N*_{S}-by-*N*_{P}-by-*N*_{T}-by-*N*_{R}
array with `NaN`

values for the unselected
transmit-receive antenna pairs. `pathgains`

contains
complex values.

*N*_{S}is the number of samples.*N*_{P}is the number of discrete path delays specified by the`PathDelays`

property.*N*_{T}is the number of transmit antennas.*N*_{R}is the number of receive antennas.

When you set the `ChannelFiltering`

property to `false`

,
the data type of this output has the same precision as the input signal
`x`

. When you set the `ChannelFiltering`

property to `true`

,
the data type of this output is specified by the `OutputDataType`

property.

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

**Note**

## Examples

### Pass QPSK Data Through 4-by-2 MIMO Channel

Create a 4-by-2 MIMO channel by using the MIMO channel System object. Modulate and spatially encode data, and then pass the data through the channel.

Generate QPSK-modulated data.

data = randi([0 3],1000,1); modData = pskmod(data,4,pi/4);

Create an orthogonal space-time block encoder System object to encode the modulated data into four spatially separated streams. Then, encode the data.

ostbc = comm.OSTBCEncoder( ... 'NumTransmitAntennas',4, ... 'SymbolRate',1/2); txSig = ostbc(modData);

Create a MIMO channel System object, using name-value pairs to set the properties. The channel consists of two paths, each with a maximum Doppler shift of 5 Hz. Set the `SpatialCorrelationSpecification`

property to `'None'`

, which requires that you specify the number of transmit and receive antennas. Specify four transmit antennas and two receive antennas.

mimochannel = comm.MIMOChannel( ... 'SampleRate',1000, ... 'PathDelays',[0 2e-3], ... 'AveragePathGains',[0 -5], ... 'MaximumDopplerShift',5, ... 'SpatialCorrelationSpecification','None', ... 'NumTransmitAntennas',4, ... 'NumReceiveAntennas',2);

Pass the modulated and encoded signal through the MIMO channel.

rxSig = mimochannel(txSig);

Create a time vector, `t`

, to use for plotting the power of the received signal.

ts = 1/mimochannel.SampleRate; t = (0:ts:(size(txSig,1)-1)*ts)';

Calculate and plot the power of the signal received by antenna 1.

pwrdB = 20*log10(abs(rxSig(:,1))); plot(t,pwrdB) title('Channel Response Power (dBW)') xlabel('Time (s)') ylabel('Power (dBW)')

### Examine Spatial Correlation Characteristics of 2-by-2 Rayleigh Fading Channel

Generate path gains for a 2-by-2 Rayleigh fading channel and examine the spatial correlation characteristics of the channel realization. Use the `release`

object function to unlock the object to set the `AntennaSelection`

property to `'Tx and Rx'`

and then confirm the unselected transmit-receive antenna pairs.

Create a 2-by-2 MIMO channel System object with two discrete paths and channel filtering disabled. Each path has different transmit and receive correlation matrices, specified by the `TransmitCorrelationMatrix`

and `ReceiveCorrelationMatrix`

properties.

mimoChan = comm.MIMOChannel( ... 'SampleRate',1000, ... 'PathDelays',[0 1e-3], ... 'AveragePathGains',[3 5], ... 'NormalizePathGains',false, ... 'MaximumDopplerShift',5, ... 'TransmitCorrelationMatrix',cat(3,eye(2),[1 0.1;0.1 1]), ... 'ReceiveCorrelationMatrix',cat(3,[1 0.2;0.2 1],eye(2)), ... 'RandomStream','mt19937ar with seed', ... 'Seed',33, ... 'ChannelFiltering',false);

Generate channel response path gains using the MIMO channel object.

pathGains = mimoChan();

The transmit spatial correlation for the first discrete path at the first receive antenna is specified as an identity matrix in the `TransmitCorrelationMatrix`

property. Confirm that the channel output `pathGains`

exhibits the same statistical characteristics by using the `corrcoef`

function to display the transmit spatial correlation for the first discrete path and the first receive antenna.

corrcoef(squeeze(pathGains(:,1,:,1)))

`ans = `*2×2 complex*
1.0000 + 0.0000i -0.3391 + 0.4285i
-0.3391 - 0.4285i 1.0000 + 0.0000i

The transmit spatial correlation for the second discrete path at the second receive antenna is specified as `[1 0.1;0.1 1]`

in the `TransmitCorrelationMatrix`

property. Confirm that the channel output `pathGains`

exhibits the same statistical characteristics by using the `corrcoef`

function to display the transmit spatial correlation for the second discrete path and the second receive antenna.

corrcoef(squeeze(pathGains(:,2,:,2)))

`ans = `*2×2 complex*
1.0000 + 0.0000i -0.8989 - 0.2663i
-0.8989 + 0.2663i 1.0000 + 0.0000i

The receive spatial correlation for the first discrete path at the second transmit antenna is specified as `[1 0.2;0.2 1]`

in the `ReceiveCorrelationMatrix`

property. Confirm that the channel output `pathGains`

exhibits the same statistical characteristics by using the `corrcoef`

function to display the receive spatial correlation for the first discrete path and the second transmit antenna.

corrcoef(squeeze(pathGains(:,1,2,:)))

`ans = `*2×2 complex*
1.0000 + 0.0000i 0.9170 + 0.3141i
0.9170 - 0.3141i 1.0000 + 0.0000i

The receive spatial correlation for the second discrete path at the first transmit antenna is specified as an identity matrix in the `ReceiveCorrelationMatrix`

property. Confirm that the channel output `pathGains`

exhibits the same statistical characteristics by using the `corrcoef`

function to display the receive spatial correlation for the second discrete path and the first transmit antenna.

corrcoef(squeeze(pathGains(:,2,1,:)))

`ans = `*2×2 complex*
1.0000 + 0.0000i 0.9227 - 0.3435i
0.9227 + 0.3435i 1.0000 + 0.0000i

### Display Impulse and Frequency Responses of Frequency-Selective Channel

Create a frequency-selective MIMO channel, and then display its impulse and frequency responses.

Set the sample rate to 10 MHz. Specify path delays and gains using the extended vehicular A (EVA) channel parameters. Set the maximum Doppler shift to 70 Hz.

fs = 10e6; % Hz pathDelays = [0 30 150 310 370 710 1090 1730 2510]*1e-9; % Seconds avgPathGains = [0 -1.5 -1.4 -3.6 -0.6 -9.1 -7 -12 -16.9]; % dB fD = 70; % Hz

Create a 2-by-2 MIMO channel System object, specifying the previously defined parameters and setting channel visualization to plot the impulse and frequency responses. By default, the plot displays the antenna pair corresponding to first transmit and receive antennas.

mimoChan = comm.MIMOChannel('SampleRate',fs, ... 'PathDelays',pathDelays, ... 'AveragePathGains',avgPathGains, ... 'MaximumDopplerShift',fD, ... 'Visualization','Impulse and frequency responses');

Generate random binary data, and then pass it through the MIMO channel. The impulse response plot enables you to easily identify the individual paths and their corresponding filter coefficients. The frequency response plot shows the frequency-selective nature of the EVA channel.

x = randi([0 1],1000,2); y = mimoChan(x);

To view the antenna pair corresponding to the second transmit and first receive antennas, release the MIMO channel System object, and then set its `AntennaPairsToDisplay`

property to [2 1]. Because the `AntennaPairsToDisplay`

property is nontunable, to change its value, you must release the System object.

release(mimoChan) mimoChan.AntennaPairsToDisplay = [2 1]; y = mimoChan(x);

### Display Doppler for 2-by-2 MIMO Channel

Create and visualize the Doppler spectra of a MIMO channel that has two paths.

Construct a cell array of Doppler structures to be used in creating the channel. Set the Doppler spectrum of the first path to be bell shaped and the second path to be flat.

dp{1} = doppler('Bell'); dp{2} = doppler('Flat');

Create a 2-by-2 MIMO channel System object, specifying two paths and a maximum Doppler shift of 100 Hz, disabling the channel filtering, and enabling the visualization of the Doppler spectrum for the first Doppler path.

mimoChan = comm.MIMOChannel('SampleRate',1000, ... 'PathDelays',[0 0.002], ... 'AveragePathGains',[0 -3], ... 'MaximumDopplerShift',100, ... 'DopplerSpectrum',dp, ... 'ChannelFiltering',false, ... 'NumSamples',10000, ... 'Visualization','Doppler spectrum', ... 'PathsForDopplerDisplay',1);

Use the MIMO channel to generate the Doppler spectrum of the first path. Because the Doppler spectrum plot does not update until its buffer is filled, call the MIMO channel object multiple times to help improve the accuracy of the estimate. Observe that the spectrum has a bell shape and that its minimum and maximum frequencies fall within the limits set by the `MaximumDopplerShift`

property.

for k = 1:25 mimoChan(); end

Release the MIMO channel object, and set its `PathsForDopplerDisplay`

property to display the second path. Because the `PathsForDopplerDisplay`

property is nontunable, to change its value, you must release the System object. Call the object multiple times to display the Doppler spectrum of the second path. The results show that the spectrum is flat.

release(mimoChan) mimoChan.PathsForDopplerDisplay = 2; for k = 1:25 y = mimoChan(); end

### Model MIMO Channel Using Sum-of-Sinusoids Technique

Show that the channel state is maintained for discontinuous transmissions by using MIMO channel System objects configured to use the sum-of-sinusoids fading technique. Observe discontinuous channel response segments overlaid on a continuous channel response.

Set the channel properties.

fs = 1000; % Sample rate (Hz) pathDelays = [0 2.5e-3]; % Path delays (s) pathPower = [0 -6]; % Path power (dB) fD = 5; % Maximum Doppler shift (Hz) ns = 1000; % Number of samples nsdel = 100; % Number of samples for delayed paths

Define a continuous time span and three discontinuous time segments over which to plot and view the channel response. View a 1000-sample continuous channel response starting at time 0 and three 100-sample channel responses starting at times 0.1, 0.4, and 0.7 seconds, respectively.

to0 = 0.0; to1 = 0.1; to2 = 0.4; to3 = 0.7; t0 = (to0:ns-1)/fs; % Transmission 0 t1 = to1+(0:nsdel-1)/fs; % Transmission 1 t2 = to2+(0:nsdel-1)/fs; % Transmission 2 t3 = to3+(0:nsdel-1)/fs; % Transmission 3

Create a flat-fading 2-by-2 MIMO channel System object, disabling channel filtering and specifying a 1000 Hz sampling rate, the sum-of-sinusoids fading technique, and the number of samples to view. Specify a seed value so that results can be repeated. Use the default `InitialTime`

property setting so that the fading channel is simulated from time 0.

mimoChan1 = comm.MIMOChannel('SampleRate',fs, ... 'MaximumDopplerShift',fD, ... 'RandomStream','mt19937ar with seed', ... 'Seed',17, ... 'FadingTechnique','Sum of sinusoids', ... 'ChannelFiltering',false, ... 'NumSamples',ns);

Create a clone of the MIMO channel System object. Change the number of samples for the delayed paths and the source for the initial time so that you can specify the fading channel offset time as an input argument when calling the System object.

```
mimoChan2 = clone(mimoChan1);
mimoChan2.InitialTimeSource = 'Input port';
mimoChan2.NumSamples = nsdel;
```

Save the path gain output for the continuous channel response by using the `mimoChan1`

object and for the discontinuous delayed channel responses by using the `mimoChan2`

object with initial time offsets provided as input arguments.

pg0 = mimoChan1(); pg1 = mimoChan2(to1); pg2 = mimoChan2(to2); pg3 = mimoChan2(to3);

Compare the number of samples processed by the two channels by using the `info`

method. The results show that `mimoChan1`

processed 1000 samples and that `mimoChan2`

processed only 300 samples.

G = info(mimoChan1); H = info(mimoChan2); [G.NumSamplesProcessed H.NumSamplesProcessed]

`ans = `*1×2*
1000 300

Convert the path gains into decibels for the path corresponding to the first transmit and first receive antenna.

pathGain0 = 20*log10(abs(pg0(:,1,1,1))); pathGain1 = 20*log10(abs(pg1(:,1,1,1))); pathGain2 = 20*log10(abs(pg2(:,1,1,1))); pathGain3 = 20*log10(abs(pg3(:,1,1,1)));

Plot the path gains for the continuous and discontinuous cases. The results show that the gains for the three segments match the gain for the continuous case. The alignment of the two shows that the sum-of-sinusoids technique is ideally suited to the simulation of packetized data because the channel characteristics are maintained even when data is not transmitted.

plot(t0,pathGain0,'r--') hold on plot(t1,pathGain1,'b') plot(t2,pathGain2,'b') plot(t3,pathGain3,'b') grid title('Continuous and Discontinuous Channel Response') xlabel('Time (sec)') ylabel('Path Gain (dB)') legend('Continuous','Discontinuous','location','nw')

### Calculate Execution Time Advantage Using Sum-of-Sinusoids Technique

Demonstrate the advantage of using the sum-of-sinusoids fading technique when simulating a channel with burst data.

Set the simulation parameters such that the sampling rate is 100 kHz, the total simulation time is 100 seconds, and the duty cycle for the burst data is 25%.

fs = 1e5; % Hz tsim = 100; % seconds dutyCycle = 0.25;

Create a flat-fading 2-by-2 MIMO channel System object, specifying the sample rate and using the default filtered Gaussian noise technique.

`fgn = comm.MIMOChannel('SampleRate',fs);`

Create a similar MIMO channel System object, specifying the same sample rate as the previous MIMO channel object but using the sum-of-sinusoids technique. Additionally specify 48 sinusoids and for the fading process start times to be given as an input argument.

sos = comm.MIMOChannel('SampleRate',fs, ... 'FadingTechnique','Sum of sinusoids', ... 'NumSinusoids',48, ... 'InitialTimeSource','Input port');

Run a continuous sequence of random bits through the filtered Gaussian noise MIMO channel object. Use the `tic`

and `toc`

stopwatch timer functions to measure the execution time of the System object call.

tic y = fgn(randi([0 1],fs*tsim,2)); tFGN = toc;

To transmit a data burst each second, pass random bits through the sum-of-sinusoids MIMO channel object by calling it inside of a `for`

loop. Use the `tic`

and `toc`

stopwatch timer functions to measure the execution time.

tic for k = 1:tsim z = sos(randi([0 1],fs*dutyCycle,2),0.5+(k-1)); end tSOS = toc;

Compare the ratio of the sum-of-sinusoids execution time to the filtered Gaussian noise execution time. The ratio is less than one, which indicates that the sum-of-sinusoids technique is faster than the filtered Gaussian noise technique.

tSOS/tFGN

ans = 0.3331

### Reciprocal Downlink and Uplink Transmissions in MIMO Channel

Using one MIMO channel System object™ and two identically configured channel filter System objects, switch a link-level simulation between 3-by-2 downlink and reciprocal 2-by-3 uplink signal transmissions.

Define system parameters.

modOrder = 256; % Modulation order Nant1 = 3; % Number of 'transmit' antennas Nant2 = 2; % Number of 'receive' antennas Rs = 1e6; % Sample rate pd = [0 1.5 2.3]*1e-6; % Path delays frmLen = 1e3; % Frame length

Create a MIMO channel System object™, configuring it for path gain generation by disabling channel filtering.

chan = comm.MIMOChannel( ... 'SampleRate',Rs, ... 'PathDelays',pd, ... 'AveragePathGains',[1.5 1.2 0.2], ... 'MaximumDopplerShift',300, ... 'SpatialCorrelationSpecification','none', ... 'NumTransmitAntennas',Nant1, ... 'NumReceiveAntennas',Nant2, ... 'ChannelFiltering',false, ... 'NumSamples',frmLen);

Create identical channel filter System objects for both transmission directions: one channel filter for the `Nant1`

-by-`Nant2`

downlink channel (3 transmit antennas to 2 receive antennas) and a reciprocal channel filter for the `Nant2`

-by-`Nant1`

uplink channel (2 transmit antennas to 3 receive antennas).

chanFiltDownlink = comm.ChannelFilter( ... 'SampleRate',Rs, ... 'PathDelays',pd); chanFiltUplink = clone(chanFiltDownlink);

**Downlink Transmission**

Generate random path gains for one frame of the downlink 3-by-2 channel. Pass randomly generated 256-QAM signals through the 3-by-2 downlink channel.

pgDownlink = chan(); x = qammod(randi([0 modOrder-1],frmLen,Nant1),modOrder); yDL = chanFiltDownlink(x,pgDownlink);

**Uplink Transmission**

Switch the link direction. Run the channel object to generate another frame of path gains, permuting its 3rd (Tx) and 4th (Rx) dimensions for the reciprocal uplink 2-by-3 channel. Pass randomly generated 256-QAM signals through the 2-by-3 reciprocal uplink channel.

pgUplink = permute(chan(),[1 2 4 3]); x = qammod(randi([0 modOrder-1],frmLen,Nant2),modOrder); yUL = chanFiltUplink(x,pgUplink);

**Downlink and Uplink Array Dimensions**

Show the sizes of the downlink and uplink path gain arrays returned by the MIMI channel object as an ${\mathit{N}}_{\mathrm{S}}$-by-${\mathit{N}}_{\mathrm{P}}$-by-${\mathit{N}}_{\mathrm{T}}$-by-${\mathit{N}}_{\mathrm{R}}$ array.

${\mathit{N}}_{\mathrm{S}}$ is the number of samples.

${\mathit{N}}_{\mathrm{P}}$ is the number of path delays.

${\mathit{N}}_{\mathrm{T}}$ is the number of transmit antennas.

`Nant1`

for downlink and`Nant2`

for uplink.${\mathit{N}}_{\mathrm{R}}$ is the number of receive antennas.

`Nant2`

for downlink and`Nant1`

for uplink.

size(pgDownlink)

`ans = `*1×4*
1000 3 3 2

size(pgUplink)

`ans = `*1×4*
1000 3 2 3

Show the size of the channel output matrices returned by the MIMI channel object as an ${\mathit{N}}_{\mathrm{S}}$-by-${\mathit{N}}_{\mathrm{R}}$ matrix. ${\mathit{N}}_{\mathrm{S}}$ is the number of samples. ${\mathit{N}}_{\mathrm{R}}$ is the number of receive antennas.

size(yDL)

`ans = `*1×2*
1000 2

size(yUL)

`ans = `*1×2*
1000 3

## Algorithms

The fading processing per link is described in Methodology for Simulating
Multipath Fading Channels and assumes the same parameters for all
(*N*_{T} ×
*N*_{R}) links of the MIMO channel. Each link comprises all
multipaths for that link.

### The Kronecker Model

The Kronecker model assumes that the spatial correlations at the transmit and receive sides are separable. Equivalently, the direction of departure (DoD) and directions of arrival (DoA) spectra are assumed to be separable. The full correlation matrix is:

$${R}_{H}=E\left[{R}_{t}\otimes {R}_{r}\right]$$

The ⊗ symbol represents the Kronecker product.

*R*_{t}is the correlation matrix at the transmit side, $${R}_{t}=E\left[{H}^{H}H\right]$$, and is of size*N*_{T}-by-*N*_{T}.*R*_{r}is the correlation matrix at the receive side, $${R}_{r}=E\left[H{H}^{H}\right]$$, and is of size*N*_{R}-by-*N*_{R}.

You can obtain a realization of the MIMO channel matrix as:

$$H={R}_{r}^{\frac{1}{2}}A{R}_{t}^{\frac{1}{2}}$$

*A* is an
*N*_{R}-by-*N*_{T}
matrix of independent identically distributed complex Gaussian variables with zero mean and
unit variance.

### Cutoff Frequency Factor

The cutoff frequency factor,
*f*_{c}, is dependent on the type of Doppler spectrum.

For any Doppler spectrum type other than Gaussian and bi-Gaussian,

*f*_{c}equals 1.For a

`doppler`

`('Gaussian')`

spectrum type,*f*_{c}equals`NormalizedStandardDeviation`

$$\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\sqrt{2\mathrm{log}2}$$.For a

`doppler`

`('BiGaussian')`

spectrum type:If the

`PowerGains`

`(1)`

and`NormalizedCenterFrequencies`

`(2)`

field values are both`0`

, then*f*_{c}equals`NormalizedStandardDeviation`

`(1)`

$$\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\sqrt{2\mathrm{log}2}$$.If the

`PowerGains`

`(2)`

and`NormalizedCenterFrequencies`

`(1)`

field values are both`0`

, then*f*_{c}equals`NormalizedStandardDeviation`

`(2)`

$$\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\sqrt{2\mathrm{log}2}$$.If the

`NormalizedCenterFrequencies`

field value is`[0,0]`

and the`NormalizedStandardDeviation`

field has two identical elements, then*f*_{c}equals`NormalizedStandardDeviation`

`(1)`

$$\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\sqrt{2\mathrm{log}2}$$.In all other cases,

*f*_{c}equals 1.

### Antenna Selection

When the object is in antenna-selection mode, it uses these algorithms to process an input signal.

All random path gains are always generated and keep evolving for each link, whether or not a given link is selected. The path gain values output for the nonselected links are populated with

`NaN`

.The spatial correlation applies to only the selected transmit and receive antennas, and the correlation coefficients are the corresponding entries in the transmit, receive, or combined correlation matrices. That is, the spatial correlation matrix for the selected transmit or receive antennas is a submatrix of the transmit, receive, or combined spatial correlation matrix property value.

For signal paths that are associated with nonactive antennas, a signal with zero power is transmitted to the channel filter.

Channel output normalization happens over the number of selected receive antennas.

## References

[1] Oestges, Claude, and Bruno Clerckx. *MIMO Wireless Communications: From Real-World Propagation to Space-Time Code Design*. 1st ed. Boston, MA: Elsevier, 2007.

[2] Correia, Luis M., and European Cooperation in the Field of Scientific and Technical Research (Organization), eds. *Mobile Broadband Multimedia Networks: Techniques, Models and Tools for 4G*. 1st ed. Amsterdam ; Boston: Elsevier/Academic Press, 2006.

[3] Kermoal, J.P., L. Schumacher, K.I. Pedersen, P.E. Mogensen, and F. Frederiksen. “A Stochastic MIMO Radio Channel Model with Experimental Validation.” *IEEE Journal on Selected Areas in Communications* 20, no. 6 (August 2002): 1211–26. https://doi.org/10.1109/JSAC.2002.801223.

[4] Jeruchim, Michel C., Philip Balaban, and K. Sam Shanmugan. *Simulation of Communication Systems*. Second edition. Boston, MA: Springer US, 2000.

[5] Patzold, M., Cheng-Xiang Wang, and B. Hogstad. “Two New Sum-of-Sinusoids-Based Methods for the Efficient Generation of Multiple Uncorrelated Rayleigh Fading Waveforms.” *IEEE Transactions on Wireless Communications* 8, no. 6 (June 2009): 3122–31. https://doi.org/10.1109/TWC.2009.080769.

## 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 R2012a**

### R2022b: Updates to channel visualization display

The channel visualization feature now presents:

Configuration settings in the bottom toolbar on the plot window.

Plots side-by-side in one window when you select the

`Impulse and frequency response`

channel visualization option.

## See Also

### Objects

`comm.AWGNChannel`

|`comm.RayleighChannel`

|`comm.RicianChannel`

|`comm.RayTracingChannel`

|`comm.ChannelFilter`

### Blocks

### Topics

## Open Example

You have a modified version of this example. Do you want to open this example with your edits?

## MATLAB Command

You clicked a link that corresponds to this MATLAB command:

Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.

Select a Web Site

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

You can also select a web site from the following list:

## How to Get Best Site Performance

Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.

### Americas

- América Latina (Español)
- Canada (English)
- United States (English)

### Europe

- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)

- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)