# directivity

**System object: **phased.UCA

**Package: **phased

Directivity of uniform circular array

## Syntax

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

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

## Description

`D = directivity(`

returns
the Directivity (dBi) of a uniform circular
array (UCA) of antenna or microphone elements, `sArray`

,`FREQ`

,`ANGLE`

)`sArray`

,
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 `sArray`

,`FREQ`

,`ANGLE`

,`Name,Value`

)`Name,Value`

pair
arguments.

## Input Arguments

`sArray`

— Uniform circular array

System object™

Uniform circular array, specified as a `phased.UCA`

System object.

**Example: **`sArray= phased.UCA;`

`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`

— Array weights

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

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

' and an *N*-by-1 complex-valued
column vector or *N*-by-*L* complex-valued
matrix. Array weights are applied to the elements of the array to
produce array steering, tapering, or both. The dimension *N* is
the number of elements in the array. The dimension *L* is
the number of frequencies specified by `FREQ`

.

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 `FREQ` . |

**Note**

Use complex weights to steer the array response toward different
directions. You can create weights using the `phased.SteeringVector`

System object or
you can compute your own weights. In general, you apply Hermitian
conjugation before using weights in any Phased Array System Toolbox™ function
or System object such as `phased.Radiator`

or `phased.Collector`

. However, for the `directivity`

, `pattern`

, `patternAzimuth`

,
and `patternElevation`

methods of any array System object use
the steering vector without conjugation.

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

**Data Types: **`double`

**Complex Number Support: **Yes

## Output Arguments

`D`

— Directivity

*M*-by-*L* matrix

## Examples

### Directivity of UCA

Compute the directivity of two uniform circular arrays (UCA) at zero degrees azimuth and elevation. The first array consists of isotropic antenna elements. The second array consists of cosine antenna elements. In addition, compute the directivity of the cosine element array steered to a 45 degrees elevation.

**Array of isotropic antenna elements**

First, create a 10-element UCA with a radius of one-half meter consisting of isotropic antenna elements. Set the signal frequency to 300 MHz.

c = physconst('LightSpeed'); fc = 300e6; sIso = phased.IsotropicAntennaElement; sArray = phased.UCA('Element',sIso,'NumElements',10,'Radius',0.5); ang = [0;0]; d = directivity(sArray,fc,ang,'PropagationSpeed',c)

d = -1.1423

**Array of cosine antenna elements**

Next, create a 10-element UCA of cosine antenna elements also with a 0.5 meter radius.

sCos = phased.CosineAntennaElement('CosinePower',[3,3]); sArray1 = phased.UCA('Element',sCos,'NumElements',10,'Radius',0.5); ang = [0;0]; d = directivity(sArray1,fc,ang,'PropagationSpeed',c)

d = 3.2550

The directivity is increased due to the added directivity of the cosine antenna elements.

**Steered array of cosine antenna elements**

Finally, steer the cosine antenna array toward 45 degrees elevation, and then examine the directivity at 45 degrees.

ang = [0;45]; lambda = c/fc; w = steervec(getElementPosition(sArray1)/lambda,ang); d = directivity(sArray1,fc,ang,'PropagationSpeed',c,... 'Weights',w)

d = -3.1410

The directivity is decreased because of the combined reduction of directivity of the elements and the array.

## 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.

## Version History

**Introduced in R2015a**

## See Also

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