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# 物理代写|傅立叶光学代写Fourier optics代考|EE238 Linear Systems and Transforms

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## 物理代写|傅立叶光学代写Fourier optics代考|Linear Systems and Transforms

Diffraction as well as imaging can often be modeled as linear systems. First of all, a system is an input-output mapping. Thus, given an input, the system generates an output. For example, in a diffraction or imaging problem, the input and output are typically a wave at an input plane and the corresponding diffracted wave at a distance from the input plane.

Optical systems are quite analogous to communication systems. Both types of systems have a primary purpose of collecting and processing information. Speech signals processed by communication systems are 1-D whereas images are 2-D. Onedimensional signals are typically temporal whereas 2-D signals are typically spatial. For example, an optical system utilizing a laser beam has spatial coherence. Then, the signals can be characterized as 2-D or 3-D complex-valued field amplitudes. Spatial coherence is necessary in order to observe diffraction. Illumination such as ordinary daylight does not have spatial coherence. Then, the signals can be characterized as 2-D spatial, real-valued intensities.

Linear time-invariant and space-invariant communication and optical systems are usually analyzed by frequency analysis using the Fourier transform. Nonlinear optical elements such as the photographic film and nonlinear electronic components such as diodes have similar input-output characteristics.

In both types of systems, Fourier techniques can be used for system synthesis as well. An example is two-dimensional filtering. Theoretically optical matched filters, optical image processing techniques are analogous to matched filters and image processing techniques used in communications and signal processing.

## 物理代写|傅立叶光学代写Fourier optics代考|LINEAR SYSTEMS AND SHIFT INVARIANCE

Linearity allows the decomposition of a complex signal into elementary signals often called basis signals. In Fourier analysis, basis signals or functions are sinusoids.
In a linear system, a given input maps into a unique output. However, more than one input may map into the same output. Thus, the mapping may be one-to-one, or many-to-one.

A 2-D system is shown in Figure 2.1, where $u(x, y)$ is the input signal, and $g(x, y)$ is the output signal. Mathematically, the system can be written as
$$g(x, y)=O[u(x, y)]$$
in the continuous-space case. $O[\bullet]$ is an operator, mapping the input to the output. In the discrete-space case, the point $(x, y)$ is sampled as $[\Delta x \bullet m, \Delta y \bullet n]$, where $\Delta x$ and $\Delta y$ are the sampling intervals along the two directions. $[\Delta x \bullet m, \Delta y \bullet n]$ can be simply represented as $[m, n]$, and the system can be written as
$$g[m, n]=O[u[m, n]]$$
Below the continuous-space case is considered. The system is called linear if any linear combination of two inputs $u_1(x, y)$, and $u_2(x, y)$ generates the same combination of their respective outputs $g_1(x, y)$ and $g_2(x, y)$. This is called superposition principle and written as
$$O\left[a_1 u_1\left(t_1, t_2\right)+a_2 u_2(x, y)\right]=a_1 O\left[u_1(x, y)\right]+a_2 O\left[u_2(x, y)\right]$$

where $a_1$ and $a_2$ are scalars. Above $(x, y)$ is replaced by $[m, n]$ in the case of a linear discrete-space system.

## 物理代写|傅立叶光学代写傅里叶光学代考|线性系统和移位不变性

$$g(x, y)=O[u(x, y)]$$
。$O[\bullet]$是一个操作符，将输入映射到输出。在离散空间的情况下，点$(x, y)$被采样为$[\Delta x \bullet m, \Delta y \bullet n]$，其中$\Delta x$和$\Delta y$是沿两个方向的采样间隔。$[\Delta x \bullet m, \Delta y \bullet n]$可以简单地表示为$[m, n]$，系统可以写成
$$g[m, n]=O[u[m, n]]$$

$$O\left[a_1 u_1\left(t_1, t_2\right)+a_2 u_2(x, y)\right]=a_1 O\left[u_1(x, y)\right]+a_2 O\left[u_2(x, y)\right]$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。