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• 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

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Convolution

Convolution is a special operation between two functions involving integration to obtain a third function. ${ }^{2}$ Convolution has various applications in communication and therefore has to be understood thoroughly before we proceed.

To be specific, if the two time-dependant functions are $f(t)$ and $g(t)$, then the convolution ‘product’ is represented by the mathematical formula
$$f(t) * g(t)=h(t)=\int_{-\infty}^{\infty} f(t-\tau) g(\tau) d \tau$$
where the * represents convolution. Notice that integration is over the dummy variable $\tau$, while $t$ is treated as a constant. The value of $t$ lies in the interval, $-\infty<$ $t<\infty$. From the above remarks, it is obvious that the result of the convolution is a function of time. From Eq. (2.33), we observe that if the integration is performed numerically, then an integration has to be performed for every value of $t$.

If we observe Eq. (2.33), then we see that the function $f(t-\tau)$ is involved in the integration of the variable $\tau$. Let us determine what the function $f(t-\tau)$ looks like.
Figure $2.16$ shows three plots. The first plot is of $f(\tau)$ shown as a solid curve. For convenience, the function is considered to be zero when $\tau \leq 0$. The second plot with the dashed lines is the function $f(-\tau)$ which is the mirror image of $f(\tau)$ or the function $f(\tau)$ folded about the $y$-axis. The third trace, $f(t-\tau)$, is the dot-dash line which is the function $f(-\tau)$ shifted to the right by $t$. In Fig. $2.16$ where the three functions are shown, the line $\mathrm{AB}$ cuts the three functions at different points, $\tau_{0}, \tau_{1}$ and $\tau_{2}$ such that each of the functions has the value of $f\left(\tau_{0}\right)$. Therefore, in general $$f(\tau)=f(-\tau)=f(t-\tau)$$
In particular, the value $\tau=0$ of $f(\tau)$ maps to 0 in $f(-\tau)$ and $t$ in $f(t-\tau)$.

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|The Sinc Function

The envelope of the sinc function is proportional to $\pm 1 / \omega$ as $\omega$ increases in the positive direction and decreases in the negative direction. The minus sign occurs since the sine function oscillates between $\pm 1$. The sinc function is an even function since it consists of an odd function divided by another odd function.

Similarly, if we plot the function $\operatorname{sinc}(\omega \tau)$, then the zero crossings exist when the numerator is zero,
$$\sin (\omega \tau)=0$$
or when
$$\omega_{z, m} \tau=m \pi \quad \text { for } m=\pm 1, \pm 2, \ldots$$
which excludes $\omega=0$ for the reason that both the numerator and denominator both tend to zero. Or
$$\omega_{z, m}=\frac{m \pi}{\tau} \quad m=\pm 1, \pm 2, \ldots$$
and the sinc(.) graph expands and contracts as per the value of $\tau:$ if $\tau>1$ it contracts; if $\tau<1$ it expands, as given in the section on ‘Transformation of the Independent Variable.’ Generally, we will be interested in the case when $\tau<1$, when we realise from Eq. (2.30) (given above) the zeros are widely spaced.

# 电子系统工程代考

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|Convolution

$$f(t) * g(t)=h(t)=\int_{-\infty}^{\infty} f(t-\tau) g(\tau) d \tau$$

$$f(\tau)=f(-\tau)=f(t-\tau)$$

## 电子工程代写|电子系统工程代写Digital Systems Engineering代考|The Sinc Function

$\operatorname{sinc}$ 函数的包络与 $\pm 1 / \omega$ 作为 $\omega$ 正方向增加，负方向减少。出现负号是因为正弦函数在 $\pm 1$. sinc 函数是偶函数，因为它由一个奇函数除以另一个奇函数组成。

$$\sin (\omega \tau)=0$$

$$\omega_{z, m} \tau=m \pi \quad \text { for } m=\pm 1, \pm 2, \ldots$$

$$\omega_{z, m}=\frac{m \pi}{\tau} \quad m=\pm 1, \pm 2, \ldots$$
$\operatorname{sinc}(.)$ 图根据 $\tau$ :如果 $\tau>1$ 它收缩；如果 $\tau<1$ 它会扩展，如“自变量变换“部分所述。通常，我们会对以下情况感兴趣 $\tau<1$ ，当我们从方程式中实现时。(2.30)（如 上所示) 零点间距很大。

## 有限元方法代写

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## 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。

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