## 数学代写|信息论作业代写information theory代考|BUSN4002

2022年7月21日

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## 数学代写|信息论作业代写information theory代考|Capacity with Gaussian Noise

In Chapter 12, we provide a derivation of the capacity of the additive Gaussian noise channnel considering communication with waveforms of $N_{0}$ degress of freedom, sukject to the constraint
$$\left(\sum_{n=1}^{N_{0}} x_{n}^{2}\right)^{1 / 2} \leq \sqrt{P N_{0}}$$
Compared to (1.59), the more stringent constraint (1.70) is an empirical average rather than a statistical one, since from (1.70) we have
$$\frac{1}{N_{0}} \sum_{n=0}^{N_{0}} x_{n}^{2} \leq P$$

This ensures that all transmitted signals are inside a hypersphere of radius $\sqrt{P N_{0}}$, and requires that any realization of a random waveform must draw $N_{0}$ coefficients in a dependent fashion. Geometrically, since we are only allowed to pick signals at random inside the hypersphere of radius $\sqrt{P N_{0}}$, signals that are close to the boundary along one dimension must necessarily be far from it along the other dimensions. In contrast, in Section 1.4.6 we drew coefficients independently, subject to the weaker constraint (1.59). Fortunately, however, probabilistic concentration ensures that for any independent construction,
$$\lim {N{0} \rightarrow \infty} \frac{1}{N_{0}} \sum_{n=1}^{N_{0}} x_{n}^{2}=\mathbb{E}\left(\mathbf{X}^{2}\right) \leq P,$$
and the probability of constructing a random signal that violates the deterministic constraint (1.70) is arbitrarily small as $N_{0} \rightarrow \infty$. The geometric interpretation of (1.72) is that by drawing independent coefficients subject to the constraint (1.59), signals are contained with high probability inside the high-dimensional sphere of radius $\sqrt{P N_{0}}$, as $N_{0} \rightarrow \infty$.

## 数学代写|信息论作业代写information theory代考|Energy Limits

Electromagnetic waveforms can transport an amount of information that scales linearly with the number of degrees of freedom, representing the dimensionality of the space, and logarithmically with the ratio of the maximum energy of the signal to the energy of the noise. In the deterministic model of Kolmogorov, in which a bounded amount of noise of norm at most $\epsilon$ is added to the signal, we can communicate a number of bits proportional to the number of degrees of freedom. In the stochastic model of Shannon, in which random noise of standard deviation at most $\epsilon$ is added independently on each coordinate of the signal’s space and the signal’s energy is proportional to the number of coordinates, we can communicate a number of bits proportional to the number of degrees of freedom, with arbitrarily low probability of error.

We now explore two additional limiting regimes. In a low signal-to-noise ratio regime, where the energy of the signal vanishes compared to the energy of the noise, the capacity is directly proportional to the energy of the signal and inversely proportional to the number of degrees of freedom. On the other hand, if we keep a constant signal-to-noise ratio and increase the number of degrees of freedom, the capacity increases. However, this also requires energy expenditure, and it turns out that the number of degrees of freedom is ultimately limited by the laws of high-energy physics, and cannot grow beyond what is imposed by general relativity to keep the radiating system gravitationally stable.

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Capacity with Gaussian Noise

$$\left(\sum_{n=1}^{N_{0}} x_{n}^{2}\right)^{1 / 2} \leq \sqrt{P N_{0}}$$

$$\frac{1}{N_{0}} \sum_{n=0}^{N_{0}} x_{n}^{2} \leq P$$

$$\lim N 0 \rightarrow \infty \frac{1}{N_{0}} \sum_{n=1}^{N_{0}} x_{n}^{2}=\mathbb{E}\left(\mathbf{X}^{2}\right) \leq P$$

## 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。