directivity

Compute array directivity

Syntax

D = directivity(array,FREQ,ANGLE)

directivity(___,Name=Value)

Description

D = directivity(array,FREQ,ANGLE) returns the Directivity (dBi) of an array of antenna or microphone elements, array, at frequencies specified by FREQ and in angles of direction specified by ANGLE.

Note

The directivity object functions for arrays and elements differ. The PropagationSpeed Name-Value pair argument does not appear in the element version but does appear in the array version.

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.

directivity(___,Name=Value) plots the array pattern with additional options specified by one or more Name=Value pair arguments.

Examples

Directivity of Uniform Linear Array

Compute 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, calculate the directivities for a set of seven different azimuth directions all at zero degrees elevation. Set the frequency to 300 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 = 300e6;
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

   -6.9897
   -6.2294
   -6.5187
   10.0000
   -6.5187
   -6.2294
   -6.9897

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
   -2.0537
   10.0000

The directivity is greatest in the steered direction.

Compare Directivity to Pattern Function

Show that the pattern object function gives the same result as the directivity function. Create a 5x7-element URA operating at 1 GHz. Assume the elements are spaced one-half wavelength apart.

First, set up the URA.

elem = phased.CosineAntennaElement( ...
    FrequencyRange=[50e6,1000e6]);
fc = 500e6;
c = physconst("LightSpeed");
lam = c/fc;
array = phased.URA(Element=elem,Size=[5,7], ...
    ElementSpacing=0.5*lam);

Use the pattern object function to display the array directivity for azimuth angles from $-{45}^{\circ }$to ${45}^{\circ }$, all at ${45}^{\circ }$ elevation.

[pat,az_ang,el_ang] = pattern(array,fc, ...
    [-45:45],45, CoordinateSystem="polar", ...
    Type="directivity",PropagationSpeed=c);
plot(az_ang,pat)
hold on

Using the directivity object function, display the directivity at several discrete angles.

ang = [-30,-20,-10,0,10,20,30; ...
    45,45,45,45,45,45,45];
d = directivity(array,fc,ang, ...
    PropagationSpeed=c);
plot(ang(1,:),d,'*')
legend("pattern function","directivity function", ...
    "location","southwest")
hold off

The two computations coincide.

Input Arguments

array — Phased array

Phased Array System Toolbox™ System object™

Phased array, specified as a Phased Array System Toolbox System object.

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 Hz.

• 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.

Example: CoordinateSystem="polar",Type="directivity"

PropagationSpeed — Signal propagation speed

physconst("LightSpeed") (default) | positive scalar

Signal propagation speed, specified as the equal sign 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 equal sign 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

Directivity, returned as an M-by-L matrix. Each row corresponds to one of the M angles specified by ANGLE. Each column corresponds to one of the L frequency values specified in FREQ. Directivity units are in dBi where dBi is defined as the gain of an element relative to an isotropic radiator.

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.

Azimuth and Elevation Angles

Define the azimuth and elevation conventions used in the toolbox.

The azimuth angle of a vector is the angle between the x-axis and its orthogonal projection onto the xy-plane. The angle is positive when going from the x-axis toward the y-axis. Azimuth angles lie between –180° and 180° degrees, inclusive. The elevation angle is the angle between the vector and its orthogonal projection onto the xy-plane. The angle is positive when going toward the positive z-axis from the xy-plane. Elevation angles lie between –90° and 90° degrees, inclusive.

Numerical Uncertainty

Grating lobes due to array element spacing greater than ½ wavelength may introduce numerical errors in the computation of array directivity.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2021a