uv2azelpat

Convert radiation pattern from u/v form to azimuth/elevation form

Syntax

pat_azel = uv2azelpat(pat_uv,u,v)
pat_azel = uv2azelpat(pat_uv,u,v,az,el)
[pat_azel,az_pat,el_pat] = uv2azelpat(___)

Description

example

pat_azel = uv2azelpat(pat_uv,u,v) expresses the antenna radiation pattern pat_azel in azimuth/elevation angle coordinates instead of u/v space coordinates. pat_uv samples the pattern at u angles in u and v angles in v. The pat_azel matrix uses a default grid that covers azimuth values from –90 to 90 degrees and elevation values from –90 to 90 degrees. In this grid, pat_azel is uniformly sampled with a step size of 1 for azimuth and elevation. The function interpolates to estimate the response of the antenna at a given direction.

example

pat_azel = uv2azelpat(pat_uv,u,v,az,el) uses vectors az and el to specify the grid at which to sample pat_azel. To avoid interpolation errors, az should cover the range [–90, 90] and el should cover the range [–90, 90].

example

[pat_azel,az_pat,el_pat] = uv2azelpat(___) returns vectors containing the azimuth and elevation angles at which pat_azel samples the pattern, using any of the input arguments in the previous syntaxes.

Examples

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Convert a radiation pattern to azimuth/elevation form with the angles spaced 1° apart.

Define the pattern in terms of u and v. Because u and v values outside the unit circle are not physical, set the pattern values in this region to zero.

u = -1:0.01:1;
v = -1:0.01:1;
[u_grid,v_grid] = meshgrid(u,v);
pat_uv = sqrt(1 - u_grid.^2 - v_grid.^2);
pat_uv(hypot(u_grid,v_grid) >= 1) = 0;

Convert the pattern to azimuth/elevation space.

pat_azel = uv2azelpat(pat_uv,u,v);

Convert a radiation pattern to azimuth/elevation form with the angles spaced 1° apart.

Define the pattern in terms of u and v. Because u and v values outside the unit circle are not physical, set the pattern values in this region to zero.

u = -1:0.01:1;
v = -1:0.01:1;
[u_grid,v_grid] = meshgrid(u,v);
pat_uv = sqrt(1 - u_grid.^2 - v_grid.^2);
pat_uv(hypot(u_grid,v_grid) >= 1) = 0;

Convert the pattern to azimuth/elevation space. Store the azimuth and elevation angles for plotting.

[pat_azel,az,el] = uv2azelpat(pat_uv,u,v);

Plot the pattern.

H = surf(az,el,pat_azel);
H.LineStyle = 'none';
xlabel('Azimuth (degrees)')
ylabel('Elevation (degrees)')
zlabel('Pattern')

Convert a radiation pattern to azimuth/elevation form, with the angles spaced 5° apart.

Define the pattern in terms of u and v. Because u and v values outside the unit circle are not physical, set the pattern values in this region to zero.

u = -1:0.01:1;
v = -1:0.01:1;
[u_grid,v_grid] = meshgrid(u,v);
pat_uv = sqrt(1 - u_grid.^2 - v_grid.^2);
pat_uv(hypot(u_grid,v_grid) >= 1) = 0;

Define the set of azimuth and elevation angles at which to sample the pattern. Then convert the pattern.

az = -90:5:90;
el = -90:5:90;
pat_azel = uv2azelpat(pat_uv,u,v,az,el);

Plot the pattern.

H = surf(az,el,pat_azel);
H.LineStyle = 'none';
xlabel('Azimuth (degrees)')
ylabel('Elevation (degrees)')
zlabel('Pattern')

Input Arguments

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Antenna radiation pattern in u/v form, specified as a Q-by-P matrix. pat_uv samples the 3-D magnitude pattern in decibels in terms of u and v coordinates. P is the length of the u vector and Q is the length of the v vector.

Data Types: double

u coordinates at which pat_uv samples the pattern, specified as a vector of length P. Each coordinate is between –1 and 1.

Data Types: double

v coordinates at which pat_uv samples the pattern, specified as a vector of length Q. Each coordinate is between –1 and 1.

Data Types: double

Azimuth angles at which pat_azel samples the pattern, specified as a vector of length L. Each azimuth angle is in degrees, between –90 and 90. Such azimuth angles are in the hemisphere for which u and v are defined.

Data Types: double

Elevation angles at which pat_azel samples the pattern, specified as a vector of length M. Each elevation angle is in degrees, between –90 and 90.

Data Types: double

Output Arguments

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Antenna radiation pattern in azimuth/elevation form, returned as an M-by-L matrix. pat_azel samples the 3-D magnitude pattern in decibels, in terms of azimuth and elevation angles. L is the length of the az vector, and M is the length of the el vector.

Azimuth angles at which pat_azel samples the pattern, returned as a vector of length L. Angles are expressed in degrees.

Elevation angles at which pat_azel samples the pattern, returned as a vector of length M. Angles are expressed in degrees.

More About

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U/V Space

The u and v coordinates are the direction cosines of a vector with respect to the y-axis and z-axis, respectively.

The u/v coordinates for the hemisphere x ≥ 0 are derived from the phi and theta angles, as follows:

u=sinθcosϕv=sinθsinϕ

In these expressions, φ and θ are the phi and theta angles, respectively.

In terms of azimuth and elevation, the u and v coordinates are

u=coselsinazv=sinel

The values of u and v satisfy the inequalities

1u11v1u2+v21

Conversely, the phi and theta angles can be written in terms of u and v using

tanϕ=u/vsinθ=u2+v2

The azimuth and elevation angles can also be written in terms of u and v

sinel=vtanaz=u1u2v2

Phi Angle, Theta Angle

The φ angle is the angle from the positive y-axis toward the positive z-axis, to the vector’s orthogonal projection onto the yz plane. The φ angle is between 0 and 360 degrees. The θ angle is the angle from the x-axis toward the yz plane, to the vector itself. The θ angle is between 0 and 180 degrees.

The figure illustrates φ and θ for a vector that appears as a green solid line. The coordinate system is relative to the center of a uniform linear array, whose elements appear as blue circles.

The coordinate transformations between φ/θ and az/el are described by the following equations

sin(el)=sinϕsinθtan(az)=cosϕtanθcosθ=cos(el)cos(az)tanϕ=tan(el)/sin(az)

Azimuth Angle, Elevation Angle

The azimuth angle of a vector is the angle between the x-axis and the orthogonal projection of the vector onto the xy plane. The angle is positive in going from the x axis toward the y axis. Azimuth angles lie between –180 and 180 degrees. 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. These definitions assume the boresight direction is the positive x-axis.

Note

The elevation angle is sometimes defined in the literature as the angle a vector makes with the positive z-axis. The MATLAB® and Phased Array System Toolbox™ products do not use this definition.

This figure illustrates the azimuth angle and elevation angle for a vector that appears as a green solid line. The coordinate system is relative to the center of a uniform linear array, whose elements appear as blue circles.

Extended Capabilities

Introduced in R2012a