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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

## 数学代写|信息论作业代写information theory代考|Problems with the Folk Theorem

Let us have a closer look at the representation error of the cardinal series. We show that using the cardinal series, the reconstruction error decays slowly and makes it impossible to identify the desired sharp error cut-off required to define the effective dimensionality of the signals’ space. This raises the question of whether there is a better, more concentrated basis than the one used in the cardinal series to optimally represent any bandlimited signal.

In the following discussion we fix the bandwidth $\Omega$ and consider a signal of unbounded time support. The treatment by fixing the time duration and considering unbounded bandwidth is completely analogous. We consider signals in the set $\mathscr{E}\left(\epsilon_{T}\right)$ of bandlimited signals subject to the energy constraint
$$\int_{-\infty}^{\infty} f^{2}(t) d t \leq 1$$
and with at most a fraction of $\epsilon_{T}^{2}$ energy outside the interval $[-T / 2, T / 2]$, namely
$$1-\frac{\int_{-T / 2}^{T / 2} f^{2}(t) d t}{\int_{-\infty}^{\infty} f^{2}(t) d t} \leq \epsilon_{T}^{2}$$
As $T \rightarrow \infty$ we can also let $\epsilon_{T} \rightarrow 0$, as the signal resembles more and more a timelimited one. Consider the truncation error,
\begin{aligned} e_{N}(t) &=f(t)-\sum_{n=-N}^{N} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi) \ &=\sum_{|n|>N} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi) \end{aligned}

## 数学代写|信息论作业代写information theory代考|Slepian’s Concentration Problem

There are problems with the statement of the folk theorem and the way the approximation error decays. In the early days of communication theory these were overlooked for some time. The dimension of the space of signals the communication engineer worked with was set to approximately $N_{0}$, for large $\Omega T$. The precise meaning of “approximately” and “large” was cautiously swept under the carpet.

In the 1960s, David Slepian and his colleagues Henry Landau and Henry Pollak at Bell Laboratories, moved by an effort to have rigorous engineering models that are practically relevant, finally asked the right question, came up with a precise answer, and provided a theory that has been refined over the span of roughly two decades. In doing so, the discrete geometrical approach of representing continuous signals as points in a finite-dimensional space, put forth by Shannon, was placed on solid mathematical ground.

Slepian’s concentration problem can be stated as follows. For a given $T>0$, define the concentration of a signal $f(t)$ as
$$\alpha^{2}(T)=\frac{\int_{-T / 2}^{T / 2} f^{2}(t) d t}{\int_{-\infty}^{\infty} f^{2}(t) d t},$$
namely the fraction of the signal’s energy that lies in a given time interval of width $T$ centered at the origin. If $f(t)$ is timelimited to $T$, then the concentration has its largest value, one. Similarly, for a given $\Omega>0$, define the concentration of the spectrum of $f(t)$ as
$$\beta^{2}(\Omega)=\frac{\int_{-\Omega}^{\Omega}|F(\omega)|^{2} d \omega}{\int_{-\infty}^{\infty}|F(\omega)|^{2} d \omega},$$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Problems with the Folk Theorem

$$\int_{-\infty}^{\infty} f^{2}(t) d t \leq 1$$

$$1-\frac{\int_{-T / 2}^{T / 2} f^{2}(t) d t}{\int_{-\infty}^{\infty} f^{2}(t) d t} \leq \epsilon_{T}^{2}$$

$$e_{N}(t)=f(t)-\sum_{n=-N}^{N} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi) \quad=\sum_{|n|>N} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi)$$

## 数学代写|信息论作业代写information theory代考|Slepian’s Concentration Problem

Slepian 的浓度问题可以表述如下。对于给定的 $T>0$, 定义信号的浓度 $f(t)$ 作为
$$\alpha^{2}(T)=\frac{\int_{-T / 2}^{T / 2} f^{2}(t) d t}{\int_{-\infty}^{\infty} f^{2}(t) d t},$$

$$\beta^{2}(\Omega)=\frac{\int_{-\Omega}^{\Omega}|F(\omega)|^{2} d \omega}{\int_{-\infty}^{\infty}|F(\omega)|^{2} d \omega},$$

## 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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