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

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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|Uncertainty, Variety and Information

When the concept and phenomenon of information are used to establish the meaning and existence of uncertainty, information becomes the primary category of knowing while uncertainty becomes the derived category of knowing which gives rise to the concept and phenomenon of risk as information-derived. In this case, the distribution of universal varieties and the path of knowing correspond to the distribution of the characteristic-signal dispositions where information is claimed as an essential part of matter just as energy, and that any definitional structure of information must fulfill this universal role. In this conceptual system, information is also a property of energy through matter. Similarly, when the concept and phenomenon of uncertainty are used to define the existence of information, then uncertainty becomes the primary category of knowing while information becomes the derived category of knowing which gives rise to the conceptual system, where all things are defined and known. In this alternative view, the distribution of universal varieties and the path of knowing correspond to the distribution of uncertainties as conceived and defined.

A problem arises as to whether uncertainty is ontological or epistemological? If uncertainty is ontological, then it must be shown that it is an organic property of matter and energy at the level of static states of elements. In the epistemic frame that is being projected, uncertainty is epistemological and arises through cognitive limitations associated in informing, knowing, deciding and choosing of varieties. The concepts of information and uncertainty are composed of quality and quantity that are reflected in linguistic measures as linguistic variables for inter-categorial and intra-categorial comparisons and ordering of varieties where quantitative and qualitative dispositions exist in a relationally inseparable continuum and unity. The cognitive activities of distinguishing and categorizing varieties for inter-categorial and intra-categorial comparison and ordering require information at the static level. The development of the required theory for the nature and behavior of these static properties of categories belongs to the subject matter of info-statics. The general structure of the theory of info-statics has been presented in a companion monograph on definitional foundation of information [R17.17].

## 数学代写|信息论作业代写information theory代考|Uncertainty and Categories of Measures

In the general knowledge production regarding concepts and measurements, one can conceive of three measurement problems of exactness, inexactness and the combination of the two and the instrumentation that may be imposed. The demand of the development of theories of measurement is over the epistemological space where cognitive agents operate over the space of cognitive actions with epistemic limitations. The exact, inexact and exact-inexact measures are numerical and non-numerical in structure to deal with the comparability problems of inter-categorial and intra-categorial ranking of varieties in understanding and utilization. Since every elemental variety exists in quality-quantity duality, and hence in an exact-inexact duality, it is analytically useful to speak of measurements in terms of degrees of exactness which place them in a fuzzy space of reasoning. The relational structure of quantity quality and exactness – inexactness dualities under the principles of relational continuum and unity allows an epistemic examination of information and its communication in terms of volume and meaning within the source-destination duality, where every source has a destination support and vice versa. It is useful to work with an analytical principle that the volume and meaning of message from the source agent are information exact and complete even though it may contain deceptive components at some level of intentionality as have been explained in a monograph entitled the theory of Philosophical Consciencism [R17.16].

The message contains encoded information defined by characteristic-signal disposition which constitutes the primary category at the source. The characteristic disposition presents the contents which is carried by the signal disposition. The destination decodes the message into an interpreted message which becomes the derived category at the destination. The concepts of primary and derived categories of informing, knowing and learning in the decision-choice systemicity and complexity must be clearly understood if one is to understand the theories of info-statics and info-dynamics and how these theories relate to categorial existence of matter and energy. A newly defined conceptual space arises when the primary and derived categories are related to the measurement problem over the epistemological space.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|CSYS5030

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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|Measurements and Their Relevance in Science

Any socio-natural phenomenon is composed of qualitative and quantitative dispositions in continuum and unity. The qualitative and qualitative dispositions present themselves in differential combinations in each socio-natural elements to create a distribution through the activities of their internal energies. The nature of the distribution of the deferential quality-quantity combinations imposes a second order distribution of socio-natural identities of varieties and categorial varieties in dividedness. The varieties and dividedness produce elemental diversity at their ontological setting with relational dependency, continuity and unity without ontological disagreements of their being. These ontological elements are what they are in the past and present, and will be what they would be in the future. Disagreements over the nature of their identities tend to arise over the epistemological space under the operations of cognitive agents.

Each identity of a variety raises a problem of some degree of either agreement or disagreement among cognitive agents due to conditions of deficiencies in individual acquaintances and assessments of the encounters that present differential awareness. The degree of agreement or disagreement of any identity may be seen and specified as a fuzzy set equipped with membership characteristic functions in duality with a relational continuum and unity, where an increasing degree of agreement leads to a decreasing degree of disagreement and vice versa. The solution to the measurement problem in all areas of knowledge production is to increase the zone of degrees of agreement and reduce the zone of disagreement. In this respect, measures of degrees of agreement and disagreement may be constructed on the basis of conditions associated with any signal disposition [R4.10, R17.17]. The reduction of the degree of disagreement is a reduction of subjectivity in qualitative disposition and an increase in quantitative disposition.

## 数学代写|信息论作业代写information theory代考|Information-Measurement Problem Defined

The measurement problem in knowledge production over the epistemological space may be seen in terms of characteristic sets that present a general distribution of identities in the process of acquaintance for informing, learning and knowing of varieties and categorial varieties. The universe is teeming with an infinite system of phenomena $\Phi$ with a generic element $\phi \in \Phi$ that generates an infinite set of universal complexities associated with elements of universal object space $\Omega$ with a generic element $\omega \in \Omega$. The universal phenomenon and object space have an infinitely corresponding universal characteristic set $\mathbb{X}$ with a generic element $x \in \mathbb{X}$. The universal characteristic set exist as a duality made up of a positive characteristic sub-set $\mathbb{X}^{\mathrm{P}}$ and a negative characteristic sub-set, $\mathbb{X}^{\mathrm{N}}$ such that $\mathbb{X}=\left(\mathbb{X}^{\mathrm{N}} \cup \mathbb{X}^{\mathrm{P}}\right)$ and that the intersection may or may not be empty, $\left(\mathbb{X}^{N} \cap \mathbb{X}^{\mathrm{P}}\right) \lesseqgtr \varnothing$ to allow for a relational continuum and unity. For each phenomenon $\phi_{i} \in \Phi$ and objective $\omega_{i} \in \Omega$ there is a corresponding characteristic sub-set $\left(\mathbb{X}{\phi{i}} \subset \mathbb{X}\right)$ for $\forall i \in \mathbb{I}^{\infty}$ which is an infinite index set for the elements except when indicated. The corresponding characteristic sub-set presents the identity of a variety $v_{i} \in \mathbb{V}$, which specifies the space of the actual and possible universal varieties. The universal characteristic set is a negative-positive duality of subsets of elements. Each $\mathbb{X}{\phi{i}}$ characteristic sub-set is composed of negative characteristic sub-set $\mathbb{X}{\phi{i}}^{N}$ and positive characteristic sub-sets $\mathbb{X}{\phi{i}}^{\mathrm{P}}$ such that $\mathbb{X}{\phi{i}}=\left(\mathbb{X}{\phi{i}}^{\mathrm{N}} \cup \mathbb{X}{\phi{i}}^{\mathrm{N}}\right)$ with the condition that $\mathbb{X}^{\mathrm{N}}=\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{N}}$, $\mathbb{X}^{\mathrm{P}}=\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{P}}$ and $\mathbb{X}=\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{N}}\right) \cup\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{P}}\right)$. By the principle of relational continuum and unity of duality and polarity, it is not the case that $\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{N}}\right) \cap\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X}{\phi{i}}^{\mathrm{P}}\right)=\varnothing$. It is analytically useful to keep in mind that every negative has a positive support and vice versa, the nature of which is constrained by the general existing conditions. The positive and the negative elements in the transformation dynamics may not only be inter-changeable but inter-transformable [R17.15, R17.16, R4.13] in the general conceptual framework of cost-benefit conditions of elements and time where every cost has a benefit support and vice versa. In other words, the negative-positive duality is not different from the cost benefit duality defined in the fuzzy space under the Asantrofi-anoma rationality where every decision-choice action comes with inseparable cost-benefit conditions in the decision-choice transformation process [R17.15].

## 数学代写|信息论作业代写information theory代考|Information-Measurement Problem Defined

$\mathbb{X}^{\mathrm{N}}=\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{N}}, \mathbb{X}^{\mathrm{P}}=\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{P}}$ 和 $\mathbb{X}=\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{N}}\right) \cup\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{P}}\right)$. 根据关系连续统和二元 对立统一原则，不是这样的 $\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{N}}\right) \cap\left(\bigcup_{\phi_{i} \in \Phi} \mathbb{X} \phi i^{\mathrm{P}}\right)=\varnothing$. 记住，每个负面都有一个正面的支持，反 之亦然，这在分析上是有用的，其性质受到一般现有条件的限制。在要素和时间的成本效益条件的一般概念框架 中，转换动态中的积极和消极因素不仅可以互换，而且可以相互转换[R17.15，R17.16，R4.13]，其中每个成本有收 益支持，反之亦然。换句话说，负正对偶与在 Asantrofi-anoma 理性下的模糊空间中定义的成本收益对偶没有什么 不同，其中每个决策选择动作在决策选择转换过程中都伴随看不可分割的成本收益条件 [R17.15]。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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 Framework of Reflections and Critique

The framework of reflections and critique of traditional theories on, and of information in this monograph is built on the notion of variety as the central epistemic tool in the concepts of identity, identity identification and identity distribution of elements and categorial elements. The framework demands that the concepts of variety, categorial identity and the distribution of categorial identities be philosophically defined and explicated for scientific usage. The framework for the critique further demands that the concept and phenomenon of information must be general with defined specificities that relate to the distribution of varieties and categorial varieties. The same framework must allow one to relate ontological identities to epistemological identities and how the relational structure of the two identities establishes the conditions of stock-flow dynamics of information and knowledge. The notions of ontological and epistemological identities must be clearly defined and distinguished in this framework of the concept and phenomenon of information. The concept and phenomenon of information must be independent of their representation, such as epistemic decoding of ontological encoding.

The available materials for epistemological analysis and decision are the signal dispositions which allow the establishment of distribution of epistemological varieties from which knowledge is claimed through their association to ontological characteristics and varieties. The set of problems of the concept and phenomenon of information must be separated from the set of problems of the ontological-epistemological transmissions and intra-epistemological communication of information. The first set of problems comes within the purview of the material theory of information to establish what there is in terms of content and meaning from the source. The second set of problems comes within the purview of the communication theory of information in terms of transmission of what there is from the source through a transmission mechanism to a destination. The general theory of information is composed of the material and communication theories about what there is and the transmission of what there is under the mechanism of source-destination interactive processes. What there is presents itself as a set of varieties and categorial varieties of ontological elements, the distribution of which presents elemental differences and similarities to establish varieties and identities for knowing and decisiding.

## 数学代写|信息论作业代写information theory代考|The Traditions on the Concept

There are few epistemic initiatives that must be discussed in an attempt to reflect on traditional and other views on the concept and phenomenon of information from the structure that have been provided in Fig. 1.1. They are qualitative and quantitative dispositions, materiality, and transmission over the ontological space, communications over the epistemological space and communications among elements in the ontological space and the elements of the epistemological space, and communications among elements of epistemological space and elements of ontological space. Here, a distinction is being made between the concepts of transmission and communication, where the concept of communication is contained in the concept of transmission. The understanding of the concept and phenomenon of information cannot be separated from their definitions and contents. The epistemic problems that are confronted are to identify the agency of the source and the acquaintances of the destination. The solution to the problems are important for examining the possible noises in the transmission mechanism and the available framework in constructing measures of the contents of information.

The transmission behavior over the ontological space is different from over the epistemological space. Over the epistemological space, the transmission behavior is defined in the context of noise system due to cognitive interactions. The behavior of such a noise system is not definable over the ontological space that holds identities of elements. Over the ontological space, the communications are multidimensional among ontological agents without noise from the source of ontological agents to the destination of ontological agents. The problem that arises in all transmission and communication is the nature of the codding from the source agent and the decoding by the destination agents. The decoding of the encoded message from the ontological source agent by the ontological destination agents may produce different contents through the medium of acquaintance and interpretations of the elements of signal disposition sent where different destination agents may have different interpretations of the same signal disposition. The differences are produced by the decoding mechanisms used by different ontological destination agents.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|INFM130

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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|Timelimited Signals

If $f(t)$ is also timelimited to $[-T / 2, T / 2]$, then the non-zero terms in (2.10) occur only bandlimited and timelimited signal is finite, and it is given by the Nyquist number ${ }^{3}$
$$N_{0}=\Omega T / \pi .$$

It follows that a waveform $f(t)$, viewed in a simple environment such as the real plane, can be replaced by a simpler entity such as a point, viewed in a complex environment such as an $N_{0}$-dimensional space. In the context of communication, this means that using signals bandlimited to $\Omega$ one can transmit only $N_{0}$ real numbers in time $T$, and these numbers can be used to completely identify any one signal of duration $T$.

This remarkable intuition, due to Shannon $(1948,1949)$, allows engineers to treat continuous signals as discrete ensembles corresponding to their coordinates in an $N_{0}$-dimensional space, and is the essential idea supporting today’s digital communication technology. As powerful as it is, the intuition is nevertheless imprecise, and it requires a great deal of mathematical argument to make it stand on solid ground.

## 数学代写|信息论作业代写information theory代考|Impossibility of Time–Frequency Limiting

The first problem that we encounter is that the only signal that is both bandlimited and timelimited is the trivial always-zero signal. A mathematical proof of the above statement is the following: consider the complex extension of $f(t)$ defined in (2.3) to the upper complex half-plane, by taking ${t=x+j y: y>0, x, y \in \mathbb{R}}$. If $F(\omega)$ is bandlimited to $\Omega$, then by the Paley-Wiener theorem stated in Appendix A.1, this extension is an entire function of exponential type $\Omega$. Such a function is holomorphic over the whole complex plane and therefore it equals its own Taylor series everywhere. Now, if $f(t)=0$ in an interval, then all of its derivatives would also be zero inside this interval and the Taylor expansion would require it to be zero everywhere. The above proof uses tools from analysis; the solution to Problem $2.10$ provides a more elementary derivation.
It follows that the perfect reconstruction formula (2.10) is only valid if one observes bandlimited signals over an infinite time, and this is not possible in practice. Signals of finite time support cannot be rigorously identified by $N_{0}$ numbers, and if one attempts to use this finite number of samples then a certain reconstruction error must occur. This interpolation error is due to the missing samples of the tails of the signals in the time domain. On the other hand, a timelimited signal cannot be bandlimited and its reconstruction from $N_{0}$ samples has an aliasing error due to the overlaps of the tails of the replicas in the spectral domain. The situation is depicted in Figure 2.2, and creates a problem in defining precisely the amount of information that is carried by the transmitted signals.

One possible way around these issues is to argue that signals can be, if not perfectly timelimited and bandlimited, at least approximately so. The aliasing and interpolation errors can then be neglected, if the approximation is sufficiently accurate. The problem is then to appropriately define the meaning of the term “approximately” and to relate this definition to the quality of the reconstruction. After Shannon’s breakthrough proposal, this objective kept communication engineers busy for a long time building rigorous foundations for their methods.

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

$$N_{0}=\Omega T / \pi .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|CSYS5030

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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|Information Content

The information a signal carries is its form, in the sense of Plato’s eidos. Signals with different shapes can have different meanings, and choosing one among them implies that a certain amount of information is transmitted via the selection process. Using mathematics, this concept can be made rigorous by geometrically representing a signal as a point that can be uniquely identified by its coordinates in a high-dimensional space. These coordinates allow us to distinguish among different signals, numerically representing the amount of information that can be communicated by selecting one of them.

A bandlimitation of the spectral support of the signal reflects into a limitation on how diverse the corresponding waveform can be, and in turn on how many coordinates are required to identify it. The number of coordinates that uniquely identify any bandlimited signal undergoes a phase transition around a critical value representing the effective dimension of the signal’s space. This means that a number of coordinates slightly below the critical value does not suffice to identify any signal, while a number slightly above it does suffice. The meaning of the word “slightly” here refers to the width of the transition around the critical point that can be characterized precisely, and the sharp cut-off occurring within this interval provides a fundamental limit on the amount of information that can be communicated by transmitting an arbitrary signal in the space.

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

The Kotelnikov-Shannon-Whittaker interpolation formula (cardinal series, for short) is widely used in engineering. It states that if $f(t)$ is bandlimited to $\Omega$, then
$$f(t)=\sum_{n=-\infty}^{\infty} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi)$$
with the conventional notation $\sin c t=(\sin t) / t$.
It follows that any bandlimited signal can be identified by specifying a discrete sequence of real numbers representing the sampled values of the signal spaced by $\pi / \Omega$. a sampled point and whose amplitude is adjusted to that of the corresponding sample. The interpolating functions have value zero at all sampled points except the one where they are centered.

In the context of linear filters, the interpolation can be viewed as applying an ideal low-pass filter of cut-off frequency $\Omega$, whose impulse response is $\operatorname{sinc}(\Omega t)$, and whose input is the ideal $\delta$-pulse train modulated by the samples $f(n \pi / \Omega)$, as shown in Figure 2.1. This interpretation makes the formula attractive and widely used in practical settings. Relaxing the condition on the ideal filter cut-off and on the ideal $\delta$-pulses leads to a reconstruction error that can be minimized using higher sampling rates.

A mathematical derivation of the formula can be easily obtained by replicating the spectrum $F(\omega)$ along the $\omega$ axis at every interval of size $2 \Omega$. This gives a periodic signal $F_{s}(\omega)$ of the fundamental period $2 \Omega$. By applying (2.6) and (2.8) with the substitutions $t \rightarrow \omega, T_{0} \rightarrow 2 \Omega$, and $\omega_{0} \rightarrow \pi / \Omega$, we obtain
$$F_{s}(\omega)=\sum_{n=-\infty}^{\infty} F(\omega-2 \Omega n)=\frac{\pi}{\Omega} \sum_{n=-\infty}^{\infty} \int(n \pi / \Omega) \exp [-j n(\pi / \Omega) \omega]$$

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

Kotelnikov-Shannon-Whittaker揷值公式 (简称基数级数) 在工程中被广泛使用。它指出，如果 $f(t)$ 带宽限制为 $\Omega$ ，然后
$$f(t)=\sum_{n=-\infty}^{\infty} f(n \pi / \Omega) \operatorname{sinc}(\Omega t-n \pi)$$

$$F_{s}(\omega)=\sum_{n=-\infty}^{\infty} F(\omega-2 \Omega n)=\frac{\pi}{\Omega} \sum_{n=-\infty}^{\infty} \int(n \pi / \Omega) \exp [-j n(\pi / \Omega) \omega]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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

1.1 There is no such thing as a “thin shell of small volume.” Consider the $N$-dimensional box $C^{N}$ (1) of side length 1 . Show that all the volume in high dimensions concentrates on the surface of the box; namely, for any $0<\epsilon<1$,
$$\lim {N \rightarrow \infty} \operatorname{Vol}\left(C^{N}(1)-C^{N}(1-\epsilon)\right)=1 .$$ $1.2$ Consider a shell that gets thinner as $N \rightarrow \infty$. Compute, for all $a>0$, $$\lim {N \rightarrow \infty} \operatorname{Vol}\left(C^{N}(1)-C^{N}(1-a / N)\right)$$
and notice that the choice of $a$ determines the fraction of volume that is trapped inside the shell.
$1.3$ The volume of the $N$-dimensional ball $B^{N}(r)$ of radius $r$ is
$$\operatorname{Vol}\left(B^{N}(r)\right)=\frac{2 \pi^{N / 2} r^{N}}{\Gamma(N / 2) N}$$
where $\Gamma(\cdot)$ is Euler’s Gamma function. Verify that for $r=1$ this has a maximum at $N=5$ and decreases to zero as $N \rightarrow \infty$.
1.4 Consider a box of side length two and a ball of radius one inscribed inside it. Show that the ball contains only a negligible fraction of the whole volume in high dimensions, namely
$$\lim _{N \rightarrow \infty} \frac{\operatorname{Vol}\left(C^{N}(2)\right)}{\operatorname{Vol}\left(B^{N}(1)\right)}=\infty$$
1.5 Consider a ball of radius one. Compute the side length of the largest box inscribed inside it. Show that in this case the box contains only a negligible fraction of the whole volume in high dimensions.

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

We consider a class of real functions that we call signals that are defined on a real line. Signals have important physical interpretations in terms of real, measured quantities. For example, $f(t)$ can represent the voltage measured at time $t$ across two terminals of an antenna. In what follows, we do not deal directly with units of measure, however these units should be made explicit whenever a given physical interpretation is considered.
The instantaneous power of a signal $f(t)$ is defined by $f^{2}(t)$, and its energy by
$$\int_{-\infty}^{\infty} f^{2}(t) d t<\infty$$
From (2.1) it follows that the space of signals for the communication engineer coincides with the $L^{2}(-\infty, \infty)$ space of the square-integrable functions for the mathematician.
An alternative representation of the signal $f(t)$ is given by the Fourier transform
$$F(\omega)=\int_{-\infty}^{\infty} f(t) \exp (-j \omega t) d t,$$
where $\omega$ is a real line called angular frequency and the engineering notation $j$ is used to denote the imaginary unit. $F(\omega)$ is called the spectrum of the signal. For real $f(t)$, we have that $|F(\omega)|=\left|F^{*}(-\omega)\right|$. and the absolute value of the spectrum is an even function of the frequency.

## 信息论代写

$1.1$ 不存在“小体积薄壳”。考虑 $N$ 维箱 $C^{N}$ (1) 边长为 1 。表明所有高维度的体积都集中在盒子的表面上；即，对 于任何 $0<\epsilon<1$ ， $$\lim N \rightarrow \infty \operatorname{Vol}\left(C^{N}(1)-C^{N}(1-\epsilon)\right)=1 .$$ $1.2$ 考虑一个变薄的壳 $N \rightarrow \infty$. 计算，为所有人 $a>0$ ，
$$\lim N \rightarrow \infty \operatorname{Vol}\left(C^{N}(1)-C^{N}(1-a / N)\right)$$

$1.3$ 的体积 $N$ 维球 $B^{N}(r)$ 半径 $r$ 是
$$\operatorname{Vol}\left(B^{N}(r)\right)=\frac{2 \pi^{N / 2} r^{N}}{\Gamma(N / 2) N}$$

$1.4$ 考虑一个边长为 2 的盒子和一个内接半径为 1 的球。证明球在高维中仅包含整个体积的可忽略不计的一部 分，即
$$\lim _{N \rightarrow \infty} \frac{\operatorname{Vol}\left(C^{N}(2)\right)}{\operatorname{Vol}\left(B^{N}(1)\right)}=\infty$$
$1.5$ 考虑一个半径为 1 的球。计算内接最大盒子的边长。证明在这种情况下，盒子只包含高维度整个体积的可忽 略不计的一部分。

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

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

$$F(\omega)=\int_{-\infty}^{\infty} f(t) \exp (-j \omega t) d t$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|INFM130

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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 Number of Degrees of Freedom

The physics of propagation dictate that any observed electromagnetic field is an essentially bandlimited function. This basic property allows us to define the size of the signals’ space in terms of the number of degrees of freedom. Consider a one-dimensional, real, scalar waveform $f$ of a single scalar variable $t$. We assume that $f$ is square-integrable, and
$$\int_{-\infty}^{\infty} f^{2}(t) d t \leq E$$
This ensures that the waveform can be expanded in a series of, possibly complex, orthonormal basis functions $\left{\psi_{n}\right}$,
$$f(t)=\sum_{n=1}^{\infty} a_{n} \psi_{n}(t),$$
where
$$a_{n}=\int_{-\infty}^{\infty} f(t) \psi_{n}^{*}(t) d t .$$
The equality in (1.2) is intended in the “energy” sense:
$$\lim {N \rightarrow \infty} \int{-\infty}^{\infty}\left[f(t)-f_{N}(t)\right]^{2} d t=0,$$
where
$$f_{N}(t)=\sum_{n=1}^{N} a_{n} \psi_{n}(t) .$$
In the language of mathematics, $f$ is in $L^{2}(-\infty, \infty)$, and it can be viewed as a point in an infinite-dimensional space of coordinates given by the coefficients $\left{a_{n}\right}$ in (1.3). By varying the values of these coefficients, we can create distinct waveforms and use them to communicate information. If the orthonormal set of basis functions $\left{\psi_{n}\right}$ is complete, then using (1.2) we can construct any element in the space of signals defined by (1.1). By associating a waveform in this space with a given message that the transmitter wishes to communicate, the correct selection of the same waveform at the receiver implies that a certain amount of information is transferred between the two. One may reasonably expect that only a finite number of coefficients is in practice needed to specify the waveform up to any given accuracy, while using a larger number does not significantly improve the resolution at the receiver. It turns out that the question of what the smallest $N$ is beyond which varying higher-order coefficients does not change the form of the waveform significantly has a remarkably precise answer.

## 数学代写|信息论作业代写information theory代考|Space–Time Fields

The electromagnetic field is in general a function of four scalar variables: three spatial and one temporal. It follows that in order to appreciate the total field’s informational content in terms of degrees of freedom, we need to extend the treatment above to higher dimensions.

Let us first consider the canonical case of a two-dimensional domain of cylindrical symmetry, in which an electromagnetic field is radiated by current sources located inside a circular domain of radius $r$, and oriented perpendicular to the domain. The sources can also be induced by multiple scattering inside the domain. In any case, the radiated field away from the sources is completely determined by the field on the cut-set boundary surrounding the sources and through which it propagates – see Figure 1.7. On this boundary, we can refer to a scalar field $f(\phi, t)$ that is a function of only two scalar variables: one angular and one temporal. The corresponding four representations, linked by Fourier transforms, are depicted in Figure $1.8$, where $\omega$ indicates the transformed coordinate of the time variable $t$ and $w$ indicates the wavenumber that is the transformed coordinate of the angular variable $\phi$.

Letting $\Omega$ be the angular frequency bandwidth and $W$ be the wavenumber bandwidth, we now wish to determine the total number of degrees of freedom of the space-time field $f(\phi, t)$. To visualize the phase transition, we fix the bandwidth $\Omega$ and the size of the angular observation interval $S=2 \pi$, and scale the time support where the signal is observed $T \rightarrow \infty$ and the wavenumber bandwidth $W \rightarrow \infty$. Using the results of the monodimensional case, we have that as $T \rightarrow \infty$ the number of time-frequency degrees of freedom is of the order of
$$N_{0}=\frac{\Omega T}{\pi} .$$

## 数学代写|信息论作业代写information theory代考|The Number of Degrees of Freedom

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

$$f(t)=\sum_{n=1}^{\infty} a_{n} \psi_{n}(t)$$

$$a_{n}=\int_{-\infty}^{\infty} f(t) \psi_{n}^{*}(t) d t$$
(1.2) 中的等式意在“能量”意义上:
$$\lim N \rightarrow \infty \int-\infty^{\infty}\left[f(t)-f_{N}(t)\right]^{2} d t=0$$

$$f_{N}(t)=\sum_{n=1}^{N} a_{n} \psi_{n}(t)$$

## 数学代写|信息论作业代写information theory代考|Space–Time Fields

$$N_{0}=\frac{\Omega T}{\pi}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|CSYS5030

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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 Dimensionality of the Space

To quantify the amount of information carried by electromagnetic waves of finite energy, we represent the possible messages that the transmitter may send by an ensemble of waveforms, and then attempt to quantify the amount of information transferred by the correct selection at the receiver of one element from this ensemble, according to Shannon’s theory. This selection implies that a certain amount of information has been transported between the two. To realize this program, we are faced with a first fundamental question:
How many distinct waveforms can we possibly select from?
The answer depends on the size of the signals’ space, or the number of degrees of freedom, available for communication and on the resolution at which each degree of freedom can be observed. The number of degrees of freedom is limited by the nature of the wave propagation process, while the resolution is limited by the uncertainty associated with the observation process.

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

Any radiated waveform has an essentially finite bandwidth. Due to the interaction with the propagation medium and with the measurement apparatus used to detect the signal, the high-frequency components are cut off, and any signal appears as the output of a linear filter. Typically this filter also shapes the frequency profile, distorting the transmitted waveform. A simple example of this phenomenon occurs in a scattering environment. Due to multiple scattering, multiple copies of the transmitted waveform, carrying different delays and attenuations while traveling on different paths, may overlap at the receiver, creating interference and distorting the original signal – see Figure 1.3. While in ideal free space the magnitude of the frequency response of the propagation environment would be flat across all frequencies, and the phase of the response would be proportional to the distance between transmitter and receiver, in the presence of a large amount of scattering the response is a highly varying signal, due to the interference over the multiple scattered paths.

An analogous effect occurs in the spatial domain. The signal observed along any spatial cut-set that separates transmitters and receivers is the image of all the elements radiating from one side of the cut to the other. These radiating waveforms may interfere along the cut, leading to spatial filtering – see Figure 1.4.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|信息论作业代写information theory代考|INFSCI2120

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

• 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|Work in London and Down

The Beagle was now on its way home, and Darwin impatiently counted the days and miles which separated him from his family and friends. To his sisters he wrote: “I feel inclined to write about nothing else but to tell you, over and over again, how I long to be quietly seated among you”; and in a letter to Henslow he exclaimed: “Oh the degree to which I long to be living quietly, without one single novel object near me! No one can imagine it until he has been whirled around the world, during five long years, in a Ten Gun Brig.”

Professor Sedgwick had told Darwin’s father that he believed that Charles would take his place among the leading scientific men of England. This encouraging news from home reached Darwin on Ascension Island. “After reading this letter”, Darwin wrote, “I clambered over the mountains with a bounding step and made the rocks resound under my geological hammer.”

On October 2, 1836, the Beagle docked at Falmouth, and Darwin, “giddy with joy and confusion”, took the first available coach to The Mount, his family’s home in Shrewsbury. After a joyful reunion with his family, he visited the Wedgwood estate at Maer, where his Uncle Jos and his pretty cousins were equally impatient to see him. To Henslow he wrote: “I am in the clouds, and neither know what to do or where to go… My chief puzzle is about the geological specimens – who will have the charity to help me in describing their mineralogical nature?”

Soon Darwin found a collaborator and close friend in none other than Sir Charles Lyell, the great geologist whose book had so inspired him. One of Lyell’s best characteristics was his warmth in encouraging promising young scientists. Darwin’s theory of the formation of coral barrier reefs and atolls had supplanted Lyell’s own theory, but far from being offended, Lyell welcomed Darwin’s ideas with enthusiasm. According to Lyell’s earlier theory, coral atolls are circular in shape because they are based on the circular rims of submerged volcanos. However, Darwin showed that any island gradually sinking beneath the surface of a tropical ocean can develop into an atoll. He showed that the reef-building organisms of the coral are poisoned by the stagnant water of the central lagoon, but they flourish on the perimeter, where new water is constantly brought in by the waves. Darwin was able to use the presence of coral atolls to map whole regions of the Pacific which are gradually sinking. He pointed out that in the subsiding regions there are no active volcanos, while in regions where the land is rising, there is much volcanic activity.

## 数学代写|信息论作业代写information theory代考|The Origin of Species

In 1837 Darwin had begun a notebook on Transmutation of Species. During the voyage of the Beagle he had been deeply impressed by the great fossil animals which he had discovered, so like existing South American species except for their gigantic size. Also, as the Beagle had sailed southward, he had noticed the way in which animals were replaced by closely allied species. On the Galapagos Islands, he had been struck by the South American character of the unique species found there, and by the way in which they differed slightly on each island.

It seemed to Darwin that these facts, as well as many other observations which he had made on the voyage, could only be explained by assuming that species gradually became modified. The subject haunted him, but he was unable to find the exact mechanism by which species changed. Therefore he resolved to follow the Baconian method, which his friend Sir Charles Lyell had used so successfully in geology. He hoped that by the wholesale collection of all facts related in any way to the variation of animals and plants under domestication and in nature, he might be able to throw some light on the subject. He soon saw that in agriculture, the key to success in breeding new varieties was selection; but how could selection be applied to organisms living in a state of nature?

## 数学代写|信息论作业代写information theory代考|Work in London and Down

1836 年 10 月 2 日，小猎犬号停靠在法尔茅斯，达尔文“喜出望外”，带着第一辆可用的教练前往他家在什鲁斯伯里的家——山。在与家人欢聚一堂后，他参观了位于 Maer 的 Wedgwood 庄园，他的乔斯叔叔和他漂亮的堂兄弟们同样迫不及待地想见他。他对亨斯洛写道：“我在云端，不知道该做什么，也不知道该去哪里……我的主要难题是关于地质标本——谁将有慈善事业帮助我描述它们的矿物学性质？”

## 数学代写|信息论作业代写information theory代考|The Origin of Species

1837 年，达尔文开始写一本关于物种嬗变的笔记本。在小猎犬号的航行中，他发现的巨大化石动物给他留下了深刻的印象，除了体型巨大外，与现存的南美物种一样。此外，随着小猎犬向南航行，他注意到动物被密切相关的物种所取代的方式。在加拉帕戈斯群岛，他被那里发现的独特物种的南美特征以及每个岛上它们的细微差别所震惊。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。