# directivity

**System object: **phased.ReplicatedSubarray

**Namespace: **phased

Directivity of replicated subarray

## Syntax

`D = directivity(H,FREQ,ANGLE)`

D = directivity(H,FREQ,ANGLE,Name,Value)

## Description

`D = directivity(`

returns
the Directivity (dBi) of a replicated
array of antenna or microphone element, `H`

,`FREQ`

,`ANGLE`

)`H`

, at
frequencies specified by `FREQ`

and in angles of
direction specified by `ANGLE`

.

The integration used when computing array directivity has a minimum sampling grid of 0.1 degrees. If an array pattern has a beamwidth smaller than this, the directivity value will be inaccurate.

`D = directivity(`

returns
the directivity with additional options specified by one or more `H`

,`FREQ`

,`ANGLE`

,`Name,Value`

)`Name,Value`

pair
arguments.

## Input Arguments

`H`

— Replicated subarray

System object™

Replicated subarray, specified as a `phased.ReplicatedSubarray`

System object.

**Example: **`H = phased.ReplicatedSubarray;`

`FREQ`

— Frequency for computing directivity and patterns

positive scalar | 1-by-*L* real-valued row vector

Frequencies for computing directivity and patterns, specified
as a positive scalar or 1-by-*L* real-valued row
vector. Frequency units are in hertz.

For an antenna, microphone, or sonar hydrophone or projector element,

`FREQ`

must lie within the range of values specified by the`FrequencyRange`

or`FrequencyVector`

property of the element. Otherwise, the element produces no response and the directivity is returned as`–Inf`

. Most elements use the`FrequencyRange`

property except for`phased.CustomAntennaElement`

and`phased.CustomMicrophoneElement`

, which use the`FrequencyVector`

property.For an array of elements,

`FREQ`

must lie within the frequency range of the elements that make up the array. Otherwise, the array produces no response and the directivity is returned as`–Inf`

.

**Example: **`[1e8 2e6]`

**Data Types: **`double`

`ANGLE`

— Angles for computing directivity

1-by-*M* real-valued row vector | 2-by-*M* real-valued matrix

Angles for computing directivity, specified as a 1-by-*M* real-valued
row vector or a 2-by-*M* real-valued matrix, where *M* is
the number of angular directions. Angle units are in degrees. If `ANGLE`

is
a 2-by-*M* matrix, then each column specifies a direction
in azimuth and elevation, `[az;el]`

. The azimuth
angle must lie between –180° and 180°. The elevation
angle must lie between –90° and 90°.

If `ANGLE`

is a 1-by-*M* vector,
then each entry represents an azimuth angle, with the elevation angle
assumed to be zero.

The azimuth angle is the angle between the *x*-axis and the projection of the
direction vector onto the *xy* plane. This angle is positive when
measured from the *x*-axis toward the *y*-axis. The
elevation angle is the angle between the direction vector and *xy*
plane. This angle is positive when measured towards the *z*-axis. See
Azimuth and Elevation Angles.

**Example: **`[45 60; 0 10]`

**Data Types: **`double`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`PropagationSpeed`

— Signal propagation speed

speed of light (default) | positive scalar

Signal propagation speed, specified as the comma-separated pair
consisting of `'PropagationSpeed'`

and a positive
scalar in meters per second.

**Example: **`'PropagationSpeed',physconst('LightSpeed')`

**Data Types: **`double`

`Weights`

— Subarray weights

1 (default) | *N*-by-1 complex-valued column vector | *N*-by-*L* complex-valued
matrix

Subarray weights, specified as the comma-separated pair consisting
of `'Weights`

' and an *N*-by-1 complex-valued
column vector or *N*-by-*M* complex-valued
matrix. The dimension *N* is the number of subarrays
in the array. The dimension *L* is the number of
frequencies specified by the `FREQ`

argument.

`Weights` dimension | `FREQ` dimension | Purpose |
---|---|---|

N-by-1 complex-valued column vector | Scalar or 1-by-L row vector | Applies a set of weights for the single frequency or for all L frequencies. |

N-by-L complex-valued
matrix | 1-by-L row vector | Applies each of the L columns of `‘Weights’` for
the corresponding frequency in the `FREQ` argument. |

**Example: **`'Weights',ones(N,M)`

**Data Types: **`double`

`SteerAngle`

— Subarray steering angle

`[0;0]`

(default) | scalar | 2-element column vector

Subarray steering angle, specified as the comma-separated pair
consisting of `'SteerAngle'`

and a scalar or a 2-by-1
column vector.

If `'SteerAngle'`

is a 2-by-1 column vector,
it has the form `[azimuth; elevation]`

. The azimuth
angle must be between –180° and 180°, inclusive.
The elevation angle must be between –90° and 90°,
inclusive.

If `'SteerAngle'`

is a scalar, it specifies
the azimuth angle only. In this case, the elevation angle is assumed
to be 0.

This option applies only when the `'SubarraySteering'`

property
of the System object is set to `'Phase'`

or `'Time'`

.

**Example: **`'SteerAngle',[20;30]`

**Data Types: **`double`

`ElementWeights`

— Weights applied to elements within subarray

`1`

(default) | complex-valued *N*_{SE}-by-*N*
matrix

_{SE}

Subarray element weights, specified as complex-valued *N _{SE}*-by-

*N*matrix. Weights are applied to the individual elements within a subarray. All subarrays have the same dimensions and sizes.

*N*is the number of elements in each subarray and

_{SE}*N*is the number of subarrays. Each column of the matrix specifies the weights for the corresponding subarray.

#### Dependencies

To enable this name-value pair, set the `SubarraySteering`

property of the array to `'Custom'`

.

**Data Types: **`double`

**Complex Number Support: **Yes

## Output Arguments

`D`

— Directivity

*M*-by-*L* matrix

## Examples

### Directivity of Replicated Subarray

Compute the directivity of an array built up from ULA subarrays. Determine the directivity of the replicated subarray when the array is steered to towards 30 degrees azimuth.

Set the signal propagation speed to the speed of light. Set the signal frequency to 300 MHz.

```
c = physconst('LightSpeed');
fc = 3e8;
lambda = c/fc;
```

Create a 4-element ULA of isotropic antenna elements spaced 0.4-wavelength apart.

myArray = phased.ULA; myArray.NumElements = 4; myArray.ElementSpacing = 0.4*lambda;

Construct a 2-by-1 replicated subarray.

myRepArray = phased.ReplicatedSubarray; myRepArray.Subarray = myArray; myRepArray.Layout = 'Rectangular'; myRepArray.GridSize = [2 1]; myRepArray.GridSpacing = 'Auto'; myRepArray.SubarraySteering = 'Time';

Steer the array to 30 degrees azimuth and zero degrees elevation.

ang = [30;0]; mySV = phased.SteeringVector; mySV.SensorArray = myRepArray; mySV.PropagationSpeed = c;

Find the directivity at 30 degrees azimuth.

d = directivity(myRepArray,fc,ang,... 'PropagationSpeed',c,... 'Weights',step(mySV,fc,ang),... 'SteerAngle',ang)

d = 7.4776

## More About

### Directivity (dBi)

Directivity describes the directionality of the radiation pattern of a sensor element or array of sensor elements.

Higher directivity is desired when you want to transmit more radiation in a specific direction. Directivity is the ratio of the transmitted radiant intensity in a specified direction to the radiant intensity transmitted by an isotropic radiator with the same total transmitted power

$$D=4\pi \frac{{U}_{\text{rad}}\left(\theta ,\phi \right)}{{P}_{\text{total}}}$$

where
*U*_{rad}*(θ,φ)* is the radiant
intensity of a transmitter in the direction *(θ,φ)* and
*P*_{total} is the total power transmitted by an
isotropic radiator. For a receiving element or array, directivity measures the sensitivity
toward radiation arriving from a specific direction. The principle of reciprocity shows that
the directivity of an element or array used for reception equals the directivity of the same
element or array used for transmission. When converted to decibels, the directivity is
denoted as *dBi*. For information on directivity, read the notes on Element Directivity and Array Directivity.

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