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# 电子工程代写|通讯系统代写Communication System代考|EE343 Geometrical optics

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## 电子工程代写|通讯系统代写Communication System代考|Geometrical optics

The geometrical optics approximation describes wave propagation in the limit of high frequencies. In a homogeneous isotropic medium, the waves propagate in straight lines called rays; in an inhomogeneous isotropic medium, the rays are curved. Geometrical optics describes the ray trajectories in the limit where changes in the properties of the medium are slowly varying in space. ${ }^{4,5}$ Starting with the steady-state field equations, Equation $2.8$, the plane wave solution, Equation $2.12$ is obtained for a homogeneous medium. As a trial solution for an inhomogeneous equation, assume a generalized plane wave solution: ${ }^4$

$$\underline{E}(\underline{r}, \omega)=\underline{E}_G(\underline{r}) e^{-j k_0 \mathbf{S}(\underline{r})}$$
where $\underline{E}_G$, is a vector function of position and $\mathbf{S}$ is a scalar function of position. The surface $L=\operatorname{Re}[\mathbf{S}]$ equal a constant is a constant phase wavefront. Note $L$ has the dimension of length because $k_0 L$ is the value of phase in radians on the wavefront. By substituting the trial solution into Equation $2.8$, the following results are obtained:
\begin{aligned} &\nabla \times\left(\underline{H}_G e^{-j k_0 \mathbf{s}}\right)=\left[\nabla \times \underline{H}_G+j k_0 \underline{H}_G \times \nabla \mathbf{S}\right] e^{-j k_0 \mathbf{s}}=(\sigma+j \omega \varepsilon) \underline{E}_G e^{-j k_0 \mathbf{s}} \ &\nabla \times\left(\underline{E}_G e^{-j k_0 \mathbf{s}}\right)=\left[\nabla \times \underline{E}_G+j k_0 \underline{E}_G \times \nabla \mathbf{S}\right] e^{-j k_0 \mathbf{s}}=-j \omega \mu \underline{H}_G e^{-j k_0 \mathbf{s}} \ &\nabla \cdot\left(\varepsilon \underline{E}_G e^{-j k_0 \mathbf{s}}\right)=\left[\varepsilon \nabla \cdot \underline{E}_G+\underline{E}_G \cdot \nabla \varepsilon-j k_0 \varepsilon \underline{E}_G \cdot \nabla \mathbf{S}\right] e^{-j k_0 \mathbf{s}}=0 \ &\nabla \cdot\left(\mu \underline{H}_G e^{-j k_0 \mathbf{s}}\right)=\left[\mu \nabla \cdot \underline{H}_G+\underline{H}_G \cdot \nabla \mu-j k_0 \mu \underline{H}_G \cdot \nabla \mathbf{S}\right] e^{-j k_0 \mathbf{s}}=0 \end{aligned}
which may be simplified to:
\begin{aligned} &\nabla \mathbf{S} \times \underline{H}_G+\frac{(\sigma+j \omega \varepsilon)}{j k_0} \underline{E}_G=\frac{1}{j k_0} \nabla \times \underline{H}_G \ &\nabla \mathbf{S} \times \underline{E}_G-\frac{j \omega \mu \underline{H_G}}{j k_0}=\frac{1}{j k_0} \nabla \times \underline{E}_G \ &\nabla \mathbf{S} \cdot \underline{E}_G=\frac{1}{j k_0}\left(\nabla \cdot \underline{E}_G+\underline{E}_G \cdot \frac{\nabla \varepsilon}{\varepsilon}\right) \ &\nabla \mathbf{S} \cdot \underline{H}_G=\frac{1}{j k_0}\left(\nabla \cdot \underline{H}_G+\underline{H}_G \cdot \frac{\nabla \mu}{\mu}\right) \end{aligned}
Note that this equation is exact; no approximation has been made.

## 电子工程代写|通讯系统代写Communication System代考|Ray tracing

The Earth’s atmosphere is often modeled as spherically symmetric with the index of refraction varying with height (see Section 1.4.2.1). The index of refraction varies with time and is generally known only at a number of discrete heights. For computational convenience, the atmosphere is usually broken up into concentric spherical shells between the heights, $h_i$, where $n^{\prime}\left(h_i\right)$ is known. If $n^{\prime}(h)$ is assumed to vary linearly with height between the known values (a constant $n^{\prime}$ gradient layer), the integration of Equation $2.63$ in the layer requires the evaluation of an elliptic integral. For the range of index of refraction values expected in the atmosphere, additional approximations will be required to evaluate the integral. If instead, the modified index of refraction $m(h)$ is assumed to vary linearly with height between the known values, Equation $2.63$ can be readily integrated.

For the layer between $m_{\mathrm{i}}$ and $m_{i+1}$, the change in $\theta, \Delta \theta_{\mathrm{i}}=\theta_{i+1}-\theta_i$ is given by:

\begin{aligned} \Delta \theta_i &=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{m^2-K^2}}=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{(a r+b)^2-K^2}} \ &=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{(a r)^2+2 a b r+\left(b^2-K^2\right)}} \ &=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{u r^2+v r+w}} ; w=b^2-K^2, v=2 a b, u=a^2 \ &=-\frac{K}{\sqrt{w}} \ln \left[\frac{\sqrt{\left(m^2-K^2\right)}+\sqrt{w}}{r}+\frac{v}{2 \sqrt{w}}\right]{h_i+A} ; w>0 \ &\left.=\frac{K}{\sqrt{-w}} \sin \right]{h_i+A}^{h_{i+1}+A}\left[\frac{v r+2 w}{r \sqrt{v^2-4}-4 u}\right]{h_i+A}^{h{i+1}+A} ; w<0 \ &=-\frac{2 K}{v}\left[\frac{\sqrt{\left(m^2-K^2\right)}}{r}\right]{i+1}^{h_i+A} \ &=0 \end{aligned} where $A$ is the radius of the Earth. The integral for $\Delta L{S i}$ is:
$$\Delta L_{S i}=\int_{h_i+A}^{h_{i+1}+A} \frac{m d r}{\sqrt{m^2-K^2}}=\frac{1}{a}\left[\sqrt{m^2-K^2}\right]{h_i+A}^{h{i+1}+A}$$
where $a, b, u, v$, and $w$ are defined in Equation 2.66. The integral for $\Delta L_{P_i}$ is:
\begin{aligned} \Delta L_{P i} &=\int_{h_i+A}^{h_{i+1}+A} \frac{m^2 d r}{r \sqrt{m^2-K^2}} \approx \Delta L_{S i}\left(1+\frac{1}{2}\left(N_i^{\prime}+N_{i+1}^{\prime}\right) \times 10^{-6}\right) \ &=|a| \Delta L_{S i}+\frac{b^2}{K} \Delta \theta_i+b \ln \left[2 u r+v+2|a| \sqrt{m^2-K^2}\right]{h_i+A}^{h{i+1}+A} \end{aligned}

## 电子工程代写通讯系统代写Communication System代考|Geometrical optics

$$\underline{E}(\underline{r}, \omega)=\underline{E}G(\underline{r}) e^{-j k_0 \mathbf{S}(r)}$$ 在哪里 $\underline{E}{G^{\prime}}$ 是位置的向量函数，并且 $\mathbf{S}$ 是位置的标量函数。表面 $L=\operatorname{Re}[\mathbf{S}]$ 等于一个常数是一个恒定的相位诐前。笔迉 $L$ 有长度 的维度，因为 $k_0 L$ 是以弧度为单位的波前相位值。通过将试验解决方宴代入方程 $2.8$ ，得到以下结果:
$$\nabla \times\left(\underline{H}G e^{-j k k_0}\right)=\left[\nabla \times \underline{H}_G+j k_0 \underline{H}_G \times \nabla \mathbf{S}\right] e^{-j k \boldsymbol{s}}=(\sigma+j \omega \varepsilon) \underline{E}_G e^{-j k_0} \quad \nabla \times\left(\underline{E}_G e^{-j k{0 \boldsymbol{s}}}\right)=\left[\nabla \times \underline{E}G+j k_0 \underline{E}_G \times \nabla \mathbf{S}\right] e^{-j k 0 \mathbf{s}}=-j \omega \mu \underline{H}_G e^{-j k 0 \boldsymbol{s}}$$ 可以简化为: $$\nabla \mathbf{S} \times \underline{H}_G+\frac{(\sigma+j \omega \varepsilon)}{j k_0} \underline{E}_G=\frac{1}{j k_0} \nabla \times \underline{H}_G \quad \nabla \mathbf{S} \times \underline{E}_G-\frac{j \omega \mu \underline{H}_G}{j k_0}=\frac{1}{j k_0} \nabla \times \underline{E}_G \nabla \mathbf{S} \cdot \underline{E}_G=\frac{1}{j k_0}\left(\nabla \cdot \underline{E}_G+\underline{E}_G \cdot \frac{\nabla \varepsilon}{\varepsilon}\right) \quad \nabla \mathbf{S} \cdot \underline{H}_G=\frac{1}{j k_0}(，$$ 请注意，这个等式是精确的; 没有进行近似。

## 电子工程代写|通讯系统代写Communication System代考|Ray tracing

$$\Delta \theta_i=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{m^2-K^2}}=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{(a r+b)^2-K^2}} \quad=\int_{h_{i+A}}^{h_{i+1}+A} \frac{K d r}{r \sqrt{(a r)^2+2 a b r+\left(b^2-K^2\right)}}=\int_{h_i+A}^{h_{i+1}+A} \frac{K d r}{r \sqrt{u r^2+v r+w}} ; w=b^2-$$

$$\Delta L_{S i}=\int_{h_i+A}^{h_{i+1}+A} \frac{m d r}{\sqrt{m^2-K^2}}=\frac{1}{a}\left[\sqrt{m^2-K^2}\right] h_i+A^{h i+1+A}$$

$$\Delta L_{P i}=\int_{h_{i+}+A}^{h_{i+1}+A} \frac{m^2 d r}{r \sqrt{m^2-K^2}} \approx \Delta L_{S i}\left(1+\frac{1}{2}\left(N_i^{\prime}+N_{i+1}^{\prime}\right) \times 10^{-6}\right) \quad=|a| \Delta L_{S i}+\frac{b^2}{K} \Delta \theta_i+b \ln \left[2 u r+v+2|a| \sqrt{m^2-K^2}\right] h_i+A^{h i+1+A}$$

## MATLAB代写

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