球形坐标-MATLAB和SIMULINK -MATHWORKS日本金宝app - 金宝app,下载188bet金宝搏,金宝搏官方网站

Spherical Coordinates

金宝app支持球形坐标

Spherical coordinates describe a vector or point in space with a distance and two angles. The distance,R，是通常的欧几里得规范。关于两个角度的规范有多个约定。他们包括：

方位角
phi和theta角度
u和vcoordinates

分阶段阵列系统工具箱™ software natively supports the azimuth/elevation representation. The software also provides functions for converting between the azimuth/elevation representation and the other representations. Seephi和theta角度和U和V坐标。

方位角

在分阶段数组系统工具箱软件中，球形坐标的主要约定如下：

Use the azimuth angle,az，和高程角，el, to define the location of a point on the unit sphere.
指定所有角度。
列出序列中的坐标（az,el,R）。

这方位角of a vector is the angle between thex-axis and the orthogonal projection of the vector onto thexy飞机。这angle is positive in going from thex朝向轴y轴。方位角位于–180至180度之间。这高程角is the angle between the vector and its orthogonal projection onto thexy-飞机。当朝向正时，角度是阳性的z-axis from thexy飞机。默认情况下，元素或数组的孔隙方向与正对齐x-轴。钻孔方向是元素或数组的主叶的方向。

Note

这高程角is sometimes defined in the literature as the angle a vector makes with the positivez-轴。matlab^®分阶段数组系统工具箱产品不使用此定义。下载188bet金宝搏

该图说明了显示为绿色实线的矢量的方位角和高程角。

phi和theta角度

作为方位角和高程角的替代方案，您可以使用表示的角度φ和θ表达单位球上的点的位置。转换φ/θ表示与相应方位角/高程表示的表示形式，使用坐标转换函数，phitheta2azel和azel2phitheta。

PHI角（φ) is the angle from the positivey-axis to the vector’s orthogonal projection onto theYz飞机。角度是正向positivez-轴。这phi angle is between 0 and 360 degrees. The theta angle (θ) is the angle from thex- 向量本身。角度是正向Yz飞机。theta角度在0到180度之间。

该图说明了作为绿色实线的矢量的phi和theta。

φ/θ和AZ/ELare described by the following equations

$\begin{array}{l} \sin e l = \sin ϕ \sin θ \\ 棕褐色 a z = \cos ϕ 棕褐色 θ \\ \cos θ = \cos e l \cos a z \\ 棕褐色 ϕ = 棕褐色 e l / \sin a z \end{array}$

U和V坐标

在雷达应用中，通常使用由表示的坐标来参数化半球x≥0u和v。

转换φ/θrepresentation to and from the correspondingu/v表示，使用坐标转换功能phitheta2uv和uv2phitheta。
将方位角/高程表示转换为相应的u/v表示，使用坐标转换功能azel2uv和uv2azel。

您可以定义u和v就φ和θ而言：

$\begin{array}{l} u = \sin θ \cos ϕ \\ v = \sin θ \sin ϕ \end{array}$

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

To convert azimuth and elevation tou和vuse the transformation

$\begin{array}{l} u = \cos e l \sin a z \\ v = \sin e l \end{array}$

which is valid only in the rangeABS（AZ）≤= 90。

这values ofu和vsatisfy the inequalities

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

Conversely, the phi and theta angles can be written in terms ofu和vusing

$\begin{array}{l} 棕褐色 ϕ = v / u \\ \sin θ = \sqrt{u^{2} + v^{2}} \end{array}$

方位角和高程角度也可以用u和v:

$\begin{array}{l} \sin e l = v \\ 棕褐色 a z = \frac{u}{\sqrt{1 - u^{2} - v^{2}}} \end{array}$

Conversion between Rectangular and Spherical Coordinates

这following equations define the relationships between rectangular coordinates and the (az,el,R）在分阶段数组系统工具箱软件中使用的表示。

To convert rectangular coordinates to (az,el,R）：

$\begin{array}{l} R = \sqrt{x^{2} + y^{2} + z^{2}} \\ a z = {棕褐色}^{- 1} (y / x) \\ e l = {棕褐色}^{- 1} (z / \sqrt{x^{2} + y^{2}}) \end{array}$

To convert (az,el,R）到矩形坐标：

$\begin{array}{l} x = R \cos (e l) \cos (a z) \\ y = R \cos (e l) \sin (a z) \\ z = R \sin (e l) \end{array}$

指定目标的位置相对于分阶段阵列，通常是指其距离和方向。距阵列的距离对应于R在球形坐标中。方向对应于方位角和高程角。

Tip

To convert between rectangular coordinates and (az,el,R), use the MATLAB functionsCART2SPH和sph2cart。这se functions specify angles in radians. To convert between degrees and radians, usedeg2rad和rad2deg。

宽边角度

Broadside anglesare useful when describing the response of a uniform linear array (ULA). The array response depends directly on the broadside angle and not on the azimuth and elevation angles. Start with a ULA and draw a plane orthogonal to the ULA axis as shown in blue in the figure. The broadside angle, β, is the angle between the plane and the signal direction. To compute the broadside angle, construct a line from any point on the signal path to the plane, orthogonal to the plane. The angle between these two lines is the broadside angle and lies in the interval [–90°,90°]. The broadside angle is positive when measured toward the positive direction of the array axis. Zero degrees indicates a signal path orthogonal to the array axis. ±90° indicates paths along the array axis. All signal paths having the same broadside angle form a cone around the ULA axis.

这conversion from azimuth angle,az和高程角，el侧向角,β，是

$β = \sin^{- 1} (\sin (a z) \cos (e l))$

这个等式表明

For an elevation angle of zero, the broadside angle equals the azimuth angle.
Elevation angles equally above and below thexy平面导致相同的宽角。

You can convert from broadside angle to azimuth angle but you must specify the elevation angle

$a z = \sin^{- 1} (\frac{\sin β}{\cos (e l)})$

Because the signals paths for a given broadside angle,β, form a cone around the array axis, you cannot specify the elevation angle arbitrarily. The elevation angle and broadside angle must satisfy

$| e l | + | β | \leq 90$

这following figure depicts a ULA with elements spacedd沿着y-轴。这ULA is irradiated by a plane wave emitted from a point source in the far field. For convenience, the elevation angle is zero degrees. In this case, the signal direction lies in thexy-飞机。这n, the broadside angle reduces to the azimuth angle.

由于到达角度，阵列元素不会同时被平面波照亮。入射波在数组元素之间传播的额外距离为d sinβwheredis the distance between array elements. The constant time delay,τ，数组元素之间

$τ = \frac{d \sin β}{c},$

wherecis the speed of the wave.

对于±90°的宽边角，信号入射在平行于阵列轴的阵列上，传感器之间的时间延迟等于±d/c。对于零的宽边角，平面波同时照明ULA的所有元素，并且元素之间的时间延迟为零。

分阶段阵列系统工具箱软件提供功能az2broadside和Broadside2AZ用于在方位角和宽边角度之间转换。

转换Between Broadside Angles and Azimuth and Elevation

打开实时脚本

这following examples show how to use theaz2broadside和Broadside2AZ功能。

一个目标位于45°的方位角，相对于ULA的高度为60°。确定相应的宽角角度。

bsang = az2broadside(45,60)

bsang = 20.7048

Calculate the azimuth for an incident signal arriving at a broadside angle of 45° and an elevation of 20°.

az = broadside2az(45,20)

AZ = 48.8063