物理代写|傅立叶光学代写Fourier optics代考|ECE500

statistics-lab™ 为您的留学生涯保驾护航 在代写傅立叶光学Fourier optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅立叶光学Fourier optics代写方面经验极为丰富，各种代写傅立叶光学Fourier optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

物理代写|傅立叶光学代写Fourier optics代考|Sampling by averaging, distributions

We will now learn the important idea of the delta function that will be handy later in this book. Sampling a continuous function is a part of every measurement or digitization process. Suppose we have a signal $I(x)$ – say the intensity of ambient light along a line on a screen – that we wish to sample at discrete set of points $x_1, x_2, \ldots$. How do we go about it? Every detector we can possibly use to measure the intensity will have some finite width $2 L$ and a sample of $I(x)$ will be an average over this detector area. So we may write the intensity at point $x=0$, or $I(0)$ as:
$$I(0) \approx \frac{1}{2 L} \int_{-\infty}^{\infty} d x I(x) \operatorname{rect}\left(\frac{x}{2 L}\right) .$$
Now how do we improve this approximation so that we go to the ideal $I(0)$ ? Clearly we have to reduce the size $2 L$ over which the average is carried out. We may say that:
$$I(0)=\lim {2 L \rightarrow 0} \frac{1}{2 L} \int{-\infty}^{\infty} d x I(x) \operatorname{rect}\left(\frac{x}{2 L}\right) .$$
Notice that as the length $2 L \rightarrow 0$ the width of the function $\frac{1}{2 L} \operatorname{rect}\left(\frac{x}{2 L}\right)$ keeps reducing whereas its height keeps increasing such that the area under the curve is unity. This limiting process leads us to an impulse which is also commonly known by the name delta function. We may write:
$$\delta(x)=\lim _{2 L \rightarrow 0} \frac{1}{2 L} \operatorname{rect}\left(\frac{x}{2 L}\right) .$$
Although it is commonly referred to as the “delta function” and we will often call it that way, you will appreciate that it is not a function in the usual sense. When we say $f(x)=x^2$ we are associating a value for every input number $x$. The impulse or delta distribution is more of an idea that is the result of a limiting process. Anything that is equal to zero everywhere in the limit except at $x=0$, where it tends to infinity cannot be a function in the sense you may have learnt in your mathematics classes.

物理代写|傅立叶光学代写Fourier optics代考|Properties of delta function

1. Sampling property At points of continuity of a function $g(x)$ we have:
$$\int_{-\infty}^{\infty} d x g(x) \delta\left(x \quad x^{\prime}\right)=g\left(x^{\prime}\right) .$$
If at $x=x^{\prime}$ the function $g(x)$ has a finite jump discontinuity, the right hand side of the above equation is an average value of the two limits $g\left(x_{+}^{\prime}\right)$ and $g\left(x_{-}^{\prime}\right)$.
2. Derivatives of delta function All operations with delta function are to be associated with a test function under integral sign. We evaluate the integral below by parts.
\begin{aligned} \int_{-\infty}^{\infty} & d x g(x) \frac{d}{d x} \delta\left(x-x^{\prime}\right) \ &=\left.g(x) \delta\left(x-x^{\prime}\right)\right|{-\infty} ^{\infty}-\int{-\infty}^{\infty} d x \frac{d}{d x} g(x) \delta\left(x-x^{\prime}\right) \ &=-g^{\prime}\left(x^{\prime}\right) . \end{aligned}
This property applies to multiple order derivatives. So continuing along the lines of above equation we get:
$$\int_{-\infty}^{\infty} d x g(x) \frac{d^n}{d x^n} \delta\left(x-x^{\prime}\right)=(-1)^n g^{(n)}\left(x^{\prime}\right),$$
where $g^{(n)}(x)$ is the $\mathrm{n}$-th order derivative of $g(x)$.
3. Delta function with scaling First of all we note that the delta function is even: $\delta(-x)=\delta(x)$. This leads to:
\begin{aligned} \int_{-\infty}^{\infty} d x g(x) \delta(a x) &=\frac{1}{|a|} \int_{-\infty}^{\infty} d x g\left(\frac{x}{a}\right) \delta(x) \ &=\frac{1}{|a|} g(0) \end{aligned}
We may therefore write: $\delta(a x)=\frac{1}{|a|} \delta(x)$.

物理代写|傅立叶光学代写傅里叶光学代考|通过平均采样，分布

$$I(0) \approx \frac{1}{2 L} \int_{-\infty}^{\infty} d x I(x) \operatorname{rect}\left(\frac{x}{2 L}\right) .$$

$$I(0)=\lim {2 L \rightarrow 0} \frac{1}{2 L} \int{-\infty}^{\infty} d x I(x) \operatorname{rect}\left(\frac{x}{2 L}\right) .$$

$$\delta(x)=\lim _{2 L \rightarrow 0} \frac{1}{2 L} \operatorname{rect}\left(\frac{x}{2 L}\right) .$$

物理代写|傅立叶光学代写傅里叶光学代考|函数的性质

1. 采样性质在函数的连续性点上 $g(x)$ 我们有:
$$\int_{-\infty}^{\infty} d x g(x) \delta\left(x \quad x^{\prime}\right)=g\left(x^{\prime}\right) .$$
如果at $x=x^{\prime}$ 函数 $g(x)$ 有一个有限跳跃不连续，上面方程的右边是两个极限的平均值 $g\left(x_{+}^{\prime}\right)$ 和 $g\left(x_{-}^{\prime}\right)$
2. δ函数的导数所有关于δ函数的运算都与积分符号下的测试函数相关联。我们用分部计算下面的积分。
\begin{aligned} \int_{-\infty}^{\infty} & d x g(x) \frac{d}{d x} \delta\left(x-x^{\prime}\right) \ &=\left.g(x) \delta\left(x-x^{\prime}\right)\right|{-\infty} ^{\infty}-\int{-\infty}^{\infty} d x \frac{d}{d x} g(x) \delta\left(x-x^{\prime}\right) \ &=-g^{\prime}\left(x^{\prime}\right) . \end{aligned}
此属性适用于多个阶导数。所以继续沿着上面的等式，我们得到:
$$\int_{-\infty}^{\infty} d x g(x) \frac{d^n}{d x^n} \delta\left(x-x^{\prime}\right)=(-1)^n g^{(n)}\left(x^{\prime}\right),$$
where $g^{(n)}(x)$ 是 $\mathrm{n}$的-阶导数 $g(x)$.
3. 带缩放的函数首先我们注意到函数是偶的: $\delta(-x)=\delta(x)$。这将导致:
\begin{aligned} \int_{-\infty}^{\infty} d x g(x) \delta(a x) &=\frac{1}{|a|} \int_{-\infty}^{\infty} d x g\left(\frac{x}{a}\right) \delta(x) \ &=\frac{1}{|a|} g(0) \end{aligned}因此，我们可以这样写: $\delta(a x)=\frac{1}{|a|} \delta(x)$.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

物理代写|傅立叶光学代写Fourier optics代考|ECE502

statistics-lab™ 为您的留学生涯保驾护航 在代写傅立叶光学Fourier optics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅立叶光学Fourier optics代写方面经验极为丰富，各种代写傅立叶光学Fourier optics相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

物理代写|傅立叶光学代写Fourier optics代考|Fourier transform as a limiting case of Fourier series

We will now consider the limiting case of the Fourier series as the period $T$ goes to $\infty$. It is clear that the function $g(x)$ is no more periodic. We denote the discrete frequencies as:
$$f_{x n}=\frac{n}{T},$$
The difference between the consecutive discrete frequencies is given by $\Delta f_x=f_{x(n+1)}-f_{x n}=1 / T$. Further we define:
$$G\left(f_{x n}\right)=\int_{-\infty}^{\infty} d x g(x) \exp \left(-i 2 \pi f_{x n} x\right) .$$
We may write the Fourier series expansion as:
\begin{aligned} g(x) &=\sum_{n=-\infty}^{\infty} G_n \exp \left(i 2 \pi f_{x n} x\right) \ &=\sum_{n=-\infty}^{\infty}\left[\frac{1}{T} \int_{-T / 2}^{T / 2} d x^{\prime} g\left(x^{\prime}\right) \exp \left(-i 2 \pi f_{x n} x^{\prime}\right)\right] \exp \left(i 2 \pi f_{x n} x\right) . \end{aligned}
In the limit $T \rightarrow \infty$ we have:
\begin{aligned} g(x) & \rightarrow \sum_{n=-\infty}^{\infty} G\left(f_{x n}\right) \exp \left(i 2 \pi f_{x n} x\right) \Delta f_x \ & \rightarrow \int_{-\infty}^{\infty} d f_x G\left(f_x\right) \exp \left(i 2 \pi f_x x\right) \end{aligned}

物理代写|傅立叶光学代写Fourier optics代考|Fourier transform of the rectangle distribution

The Fourier transform of the rect function may be evaluated as:
\begin{aligned} G\left(f_x\right) &=\int_{-L}^L d x \exp \left(-i 2 \pi f_x x\right) \ &=\left[\frac{\exp \left(-i 2 \pi f_x x\right)}{-i 2 \pi f_x}\right]{-L}^L \ &=2 L \frac{\sin \left(2 \pi f_x L\right)}{\left(2 \pi f_x L\right)} \ &=2 L \operatorname{sinc}\left(2 L f_x\right) \end{aligned} Here the sinc-function is as defined in Eq. (2.21). Note that although $\operatorname{rect}(x)$ has a discontinuity, its transform is continuous. Further it is somewhat surprising to know that $\left|\operatorname{sinc}\left(f_x\right)\right|$ is not absolutely integrable to have a Fourier transform or Fourier inverse in the conventional sense required by the Dirichlet sufficiency conditions. To show this consider the intervals along $f_x$ axis where $\left|\sin \left(\pi f_x\right)\right| \geq$ $0.5$. These intervals are given by $f_x \in[n+1 / 6, n+5 / 6]$. We therefore have: $$\int{-\infty}^{\infty} d f_x\left|\operatorname{sinc}\left(f_x\right)\right|>2 \frac{1}{2 \pi} \sum_{n=0}^{\infty} \frac{2 / 3}{n+5 / 6} \rightarrow \infty$$
The series on the right hand side diverges and as a result the absolute integral of $\operatorname{sinc}\left(f_x\right)$ does not exist. Functions such as spikes, steps and even ever extending sines and cosines do not have a Fourier transform in traditional theory. Defining Fourier transforms for such functions is however a practical necessity when representing images as $2 \mathrm{D}$ matrices in digital form. In fact edges or spikes contain most important visual information in images. The traditional Fourier transform theory must therefore be extended to take these cases into account. We will study the important case of the Dirac delta function in this context. This class of functions (spikes, steps, etc) with no Fourier transform in conventional theory is known by the name generalized functions. We will not deal with theory of generalized functions in detail but study some specific cases of interest starting with the Dirac delta function.

物理代写|傅立叶光学代写傅立叶光学代考|傅立叶变换作为傅立叶级数的极限情况

$$f_{x n}=\frac{n}{T},$$

$$G\left(f_{x n}\right)=\int_{-\infty}^{\infty} d x g(x) \exp \left(-i 2 \pi f_{x n} x\right) .$$

\begin{aligned} g(x) &=\sum_{n=-\infty}^{\infty} G_n \exp \left(i 2 \pi f_{x n} x\right) \ &=\sum_{n=-\infty}^{\infty}\left[\frac{1}{T} \int_{-T / 2}^{T / 2} d x^{\prime} g\left(x^{\prime}\right) \exp \left(-i 2 \pi f_{x n} x^{\prime}\right)\right] \exp \left(i 2 \pi f_{x n} x\right) . \end{aligned}

\begin{aligned} g(x) & \rightarrow \sum_{n=-\infty}^{\infty} G\left(f_{x n}\right) \exp \left(i 2 \pi f_{x n} x\right) \Delta f_x \ & \rightarrow \int_{-\infty}^{\infty} d f_x G\left(f_x\right) \exp \left(i 2 \pi f_x x\right) \end{aligned}

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。