### 数学代写|信息论作业代写information theory代考|ELEN90030

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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|The Physics of Information

This book describes the limits for the communication of information with waves. How many ideas can we communicate by writing on a sheet of paper? How well can we hear a concert? How many details can we distinguish in an image? How much data can we get from our internet connection? These are all questions related to the transport of information by waves. Our sensing ability to capture the differences between distinct waveforms dictates the limits of the amount of information that is delivered by a propagating wave. The problem of quantifying this precisely requires a mathematical description and a physical understanding of both the propagation and the communication processes.

We focus on the propagation of electromagnetic waves as described by Maxwell’s theory of electromagnetism, and on communication as described by Shannon’s theory of information. Although our treatment is mostly based on classical field theory, we also consider limiting regimes where the classical theory must give way to discrete quantum formulations. The old question of whether information is physics or mathematics resounds here. Information is certainly described mathematically, but we argue that it also has a definite physical structure. The central theme of this book is that Shannon’s information-theoretic limits are natural. They are revealed by observing physical quantities at certain asymptotic scales where finite dimensionality emerges and observational uncertainties are averaged out. These limits are also rigorous, and obey the mathematical rules that govern the model of reality on which the physical theories are based.

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

At the basis of the asymptotic arguments leading to information-theoretic limits is the notion of concentration.

Consider a space-time waveform $f(x, y, z, t)$ of finite energy, transmitted for $T$ seconds. As $T \rightarrow \infty$, we can define the effective frequency bandwidth of the waveform as the effective spectral support in the Fourier-transformed angular frequency domain see Figure 1.1. This definition is made possible by the mathematics at the basis of wave theory that predict spectral concentration. As the time domain support is stretched, the signal, when viewed in the frequency domain, can be more and more concentrated inside the bandwidth. Thanks to this phenomenon, electromagnetic signals can be considered, for large $T$, as occupying an essentially finite bandwidth. Signals of finite energy and finite bandwidth enjoy another important mathematical property.

They exhibit a limit on the amount of variation they can undergo in any given time interval and thus, when viewed at finite resolution, on the amount of information they can carry over time. The same limitation also applies to the spatial domain. As the region where the signal is observed is stretched by scaling all of its coordinates, spectral concentration occurs, and this allows the definition of the effective bandwidth in the wavenumber domain, that is the Fourier transform of the spatial domain. This limits the number of spatial configurations of the waveform, and thus, when viewed at finite resolution,the amount of information it can carry over space. This limitation is important in the context of network information theory, when multiple transmitters and receivers in a communication system are distributed in space. It is also important in the context of imaging systems, where it leads to spatial resolution limits of the constructed image.
When considering space and time asymptotics, another kind of concentration phenomenon also occurs. The precision level at which the signal can be observed probabilistically concentrates around its typical value. Every physical apparatus measuring a signal is affected by a measurement error: repeated measurements appear to fluctuate randomly by a small amount. This is a consequence of the quantized nature of the world observed at the microscopic scale. Over many repetitions, the uncertainty with which the signal is observed is typically contained within its standard deviation see Figure 1.2. This allows us to view the uncertainty of the observation as concentrated around its typical value and determines a resolution limit at which the signal can be observed. Combined with the constraints on the form of the signal due to spectral concentration mentioned above, it poses an ultimate limit on the amount of information that can be transported by waves in time and space.

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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。