**System object: **phased.ULA

**Package: **phased

Directivity of uniform linear array

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

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

`D = directivity(`

computes
the Directivity (dBi) of a uniform linear
array (ULA) of antenna or microphone elements, `H`

,`FREQ`

,`ANGLE`

)`H`

,
at frequencies specified by `FREQ`

and in angles
of direction specified by `ANGLE`

.

`D = directivity(`

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

,`FREQ`

,`ANGLE`

,`Name,Value`

)`Name,Value`

pair
arguments.

`H`

— Uniform linear arraySystem object™

Uniform linear array specified as a `phased.ULA`

System
object.

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

`FREQ`

— Frequency for computing directivity and patternspositive scalar | 1-by-

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 directivity1-by-

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`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'PropagationSpeed'`

— Signal propagation speedspeed 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 weights1 (default) |

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` . |

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

`D`

— DirectivityCompute the directivities of two different uniform linear arrays (ULA). One array consists of isotropic antenna elements and the second array consists of cosine antenna elements. In addition, compute the directivity when the first array is steered in a specified direction. For each case, calculated the directivities for a set of seven different azimuth directions all at zero degrees elevation. Set the frequency to 800 MHz.

**Array of isotropic antenna elements**

First, create a 10-element ULA of isotropic antenna elements spaced 1/2-wavelength apart.

c = physconst('LightSpeed'); fc = 3e8; lambda = c/fc; ang = [-30,-20,-10,0,10,20,30; 0,0,0,0,0,0,0]; myAnt1 = phased.IsotropicAntennaElement; myArray1 = phased.ULA(10,lambda/2,'Element',myAnt1);

Compute the directivity

`d = directivity(myArray1,fc,ang,'PropagationSpeed',c)`

`d = `*7×1*
-6.9886
-6.2283
-6.5176
10.0011
-6.5176
-6.2283
-6.9886

**Array of cosine antenna elements**

Next, create a 10-element ULA of cosine antenna elements spaced 1/2-wavelength apart.

myAnt2 = phased.CosineAntennaElement('CosinePower',[1.8,1.8]); myArray2 = phased.ULA(10,lambda/2,'Element',myAnt2);

Compute the directivity

`d = directivity(myArray2,fc,ang,'PropagationSpeed',c)`

`d = `*7×1*
-1.9838
0.0529
0.4968
17.2548
0.4968
0.0529
-1.9838

The directivity of the cosine ULA is greater than the directivity of the isotropic ULA because of the larger directivity of the cosine antenna element.

**Steered array of isotropic antenna elements**

Finally, steer the isotropic antenna array to 30 degrees in azimuth and compute the directivity.

w = steervec(getElementPosition(myArray1)/lambda,[30;0]); d = directivity(myArray1,fc,ang,'PropagationSpeed',c,... 'Weights',w)

`d = `*7×1*
-297.5224
-13.9783
-9.5713
-6.9897
-4.5787
-2.0536
10.0000

The directivity is greatest in the steered direction.

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.

Computing directivity requires integrating the far-field transmitted radiant intensity over all directions in space to obtain the total transmitted power. There is a difference between how that integration is performed when Antenna Toolbox™ antennas are used in a phased array and when Phased Array System Toolbox antennas are used. When an array contains Antenna Toolbox antennas, the directivity computation is performed using a triangular mesh created from 500 regularly spaced points over a sphere. For Phased Array System Toolbox antennas, the integration uses a uniform rectangular mesh of points spaced 1° apart in azimuth and elevation over a sphere. There may be significant differences in computed directivity, especially for large arrays.

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