# azel2uvpat

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

## Description

example

pat_uv = azel2uvpat(pat_azel,az,el) expresses the antenna radiation pattern pat_azel in u/v space coordinates instead of azimuth/elevation angle coordinates. pat_azel samples the pattern at azimuth angles in az and elevation angles in el. The pat_uv matrix uses a default grid that covers u values from –1 to 1 and v values from –1 to 1. In this grid, pat_uv is uniformly sampled with a step size of 0.01 for u and v. The function interpolates to estimate the response of the antenna at a given direction. Values in pat_uv are NaN for u and v values outside the unit circle because u and v are undefined outside the unit circle.

example

pat_uv = azel2uvpat(pat_azel,az,el,u,v) uses vectors u and v to specify the grid at which to sample pat_uv. To avoid interpolation errors, u should cover the range [–1, 1] and v should cover the range [–1, 1].

example

[pat_uv,u_pat,v_pat] = azel2uvpat(___) returns vectors containing the u and v coordinates at which pat_uv samples the pattern, using any of the input arguments in the previous syntaxes.

## Examples

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Convert a radiation pattern to u-v space, with the u and v coordinates spaced by 0.01.

Define the pattern in terms of azimuth and elevation.

az = -90:90;
el = -90:90;
pat_azel = mag2db(repmat(cosd(el)',1,numel(az)));

Convert the pattern to u-v space.

pat_uv = azel2uvpat(pat_azel,az,el);

Plot the result of converting a radiation pattern to $u/v$ space with the $u$ and $v$ coordinates spaced by 0.01.

The radiation pattern is the cosine of the elevation angle.

az = -90:90;
el = -90:90;
pat_azel = repmat(cosd(el)',1,numel(az));

Convert the pattern to $u/v$ space. Use the $u$ and $v$ coordinates for plotting.

[pat_uv,u,v] = azel2uvpat(pat_azel,az,el);

Plot the result.

H = surf(u,v,mag2db(pat_uv));
H.LineStyle = 'none';
xlabel('u');
ylabel('v');
zlabel('Pattern');

Convert a radiation pattern to $u/v$ form, with the $u$ and $v$ coordinates spaced by 0.05.

The radiation pattern is cosine of the elevation angle.

az = -90:90;
el = -90:90;
pat_azel = repmat(cosd(el)',1,numel(az));

Define the set of $u$ and $v$ coordinates at which to sample the pattern. Then, convert the pattern.

u = -1:0.05:1;
v = -1:0.05:1;
pat_uv = azel2uvpat(pat_azel,az,el,u,v);

Plot the result.

H = surf(u,v,mag2db(pat_uv));
H.LineStyle = 'none';
xlabel('u');
ylabel('v');
zlabel('Pattern');

## Input Arguments

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Antenna radiation pattern in azimuth/elevation form, specified as a Q-by-P matrix. pat_azel samples the 3-D magnitude pattern in decibels, in terms of azimuth and elevation angles. P is the length of the az vector, and Q is the length of the el vector.

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

## Output Arguments

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Antenna radiation pattern in u/v form, returned as an M-by-L matrix. pat_uv samples the 3-D magnitude pattern in decibels, in terms of u and v coordinates. L is the length of the u vector, and M is the length of the v vector. Values in pat_uv are NaN for u and v values outside the unit circle because u and v are undefined outside the unit circle.

u coordinates at which pat_uv samples the pattern, returned as a vector of length L.

v coordinates at which pat_uv samples the pattern, returned as a vector of length M.

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### 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. By default, the boresight direction of an element or array is aligned with the positive x-axis. The boresight direction is the direction of the main lobe of an element or array.

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 shown as a green solid line.

### 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 by:

$\begin{array}{l}u=\mathrm{sin}\theta \mathrm{cos}\varphi \\ v=\mathrm{sin}\theta \mathrm{sin}\varphi \end{array}$

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

To convert azimuth and elevation to u and v use the transformation

$\begin{array}{l}u=\mathrm{cos}el\mathrm{sin}az\\ v=\mathrm{sin}el\end{array}$

which is valid only in the range abs(az)≤=90.

The values of u and v satisfy the inequalities

$\begin{array}{l}-1\le u\le 1\\ -1\le v\le 1\\ {u}^{2}+{v}^{2}\le 1\end{array}$

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

$\begin{array}{l}\mathrm{tan}\varphi =v/u\\ \mathrm{sin}\theta =\sqrt{{u}^{2}+{v}^{2}}\end{array}$

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

$\begin{array}{l}\mathrm{sin}el=v\\ \mathrm{tan}az=\frac{u}{\sqrt{1-{u}^{2}-{v}^{2}}}\end{array}$

### Phi Angle, Theta Angle

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

The figure illustrates phi and theta for a vector that appears as a green solid line.

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

$\begin{array}{l}\mathrm{sin}el=\mathrm{sin}\varphi \mathrm{sin}\theta \\ \mathrm{tan}az=\mathrm{cos}\varphi \mathrm{tan}\theta \\ \mathrm{cos}\theta =\mathrm{cos}el\mathrm{cos}az\\ \mathrm{tan}\varphi =\mathrm{tan}el/\mathrm{sin}az\end{array}$

## Version History

Introduced in R2012a