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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|Converse to the Uncertainty Principle

The uncertainty principle provides an upper bound on the amount of simultaneous concentration in time and frequency of any signal in the space. One can also ask about the converse, namely what the most concentrated signals are that satisfy (2.22), (2.36), or (2.42). For example, we may wish to determine signals of given frequency concentration that achieve the largest time concentration, or vice versa.

When concentration is measured in terms of variance, or entropy, then the Gaussian achieves the largest simultaneous concentration in time and frequency see Problems $2.8$ and 2.9. On the other hand, when concentration is measured in terms of the fraction of the signal’s energy over given measurable sets in time and frequency, as in (2.40) and (2.41), then the most concentrated signals are somewhat more difficult to determine. These signals, however, have the additional remarkable property of providing an optimal orthogonal basis representation for any bandlimited signal. The error associated with this representation drops sharply to zero when slightly more than a critical number $N_{0}$ of basis functions are used for the approximation, and $N_{0}$ can be identified with the effective dimensionality of the space of bandlimited signals.
It turns out that these highly concentrated basis functions are the solutions of an integral equation defined on the sets of concentration. They can be obtained explicitly in the case of intervals, and the critical number of functions needed to represent any signal up to arbitrary accuracy can be determined in an asymptotic order sense.

It took communication engineers a great deal of effort to rigorously derive the above results. For a long time, the standard “hand waving” argument to determine the asymptotic dimensionality of the space of bandlimited signals using an orthogonal basis representation relied on the somewhat simpler, but sub-optimal, cardinal series sampling representation (2.10). The idea was to approximate bandlimited signals using a finite number $N_{0}=\Omega T / \pi$ of terms of the cardinal series, corresponding to sampled signal values collected inside a time interval of size $T$. Then, noticing that as $T \rightarrow \infty$ a vanishing portion $\epsilon_{T}^{2} \rightarrow 0$ of the signal’s energy is neglected and a better and better approximation is achieved, one may consider $N_{0}$ as being the asymptotic dimensionality of the space. In this way, any real bandlimited signal of unbounded time support can be approximated by a finite number of samples collected inside a finite interval, and thus appears as a point in a high-dimensional space. Since Shannon’s first outline of this argument in 1948, it has been the undisputed “folk theorem” of communication engineering.

## 数学代写|信息论作业代写information theory代考|The Folk Theorem

The communication engineer wants to work with signals that are somewhat concentrated in both the time and the frequency domains. These are the kind of signals that seem to be the most natural, because physical devices are characterized by a finite frequency response, and because signals are really observed for a finite time. Signals of this kind can be represented via the cardinal series by a discrete set of $N_{0}$ sampled points, and open the possibility of using a geometric approach to the design of communication systems.

On the other hand, the same mathematics that is at the basis of the fundamental physical indeterminacy laws of quantum mechanics seems to prohibit this. Signals that appear highly concentrated in frequency must be widely dispersed in time, and vice versa. For some time engineers ignored the issue. The dilemma was hand-waved as being only an apparent one, more of interest to the mathematician than the practitioner. They made the observation that as $N_{0}=\Omega T / \pi \rightarrow \infty$, the number of samples grows and both the sampled values of the original signal and the reconstruction error of the cardinal series become negligible. So – they argued – if the error cut-off is sufficiently sharp, then $N_{0}$ can still be considered the asymptotic dimension of the signals’ space. This led to the formulation of the following folk theorem that engineers have used with great success to design sophisticated, real communication systems.

# 信息论代写

## 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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