数学代写|信息论作业代写information theory代考|STEM2004

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数学代写|信息论作业代写information theory代考|The Low-Energy Regime

To explore the low-energy regime, we consider bandlimited signals with $N_{0}$ degrees of freedom subject to the fixed energy constraint (1.22), and assume the addition of random Gaussian noise independently to each degree of freedom, subject to (1.68). In this case, the energy of the signal is bounded, while the total amount of noise is proportional to $N_{0}$, and we have
$$\frac{1}{N_{0}} \sum_{n=1}^{N_{0}} x_{n}^{2} \leq \frac{E}{N_{0}}$$
which tends to zero as $N_{0} \rightarrow \infty$. Substituting $E / N_{0}$ for $P$ into (1.73) and using a first-order Taylor expansion of the logarithmic function, we have
\begin{aligned} C &=\frac{1}{2} \log \left(1+\frac{E}{N_{0} \epsilon^{2}}\right) \ & \simeq \frac{E}{2 N_{0} \epsilon^{2}} \log e \text { bits per degree of freedom. } \end{aligned}
It follows that in a regime where the energy of the signal is negligible compared to the energy of the noise, the total amount of information carried by any one signal in the space and expressed in bits is proportional to the energy of the signal, and remains bounded even if the number of degrees of freedom tends to infinity. On the other hand, the capacity per degree of freedom in (1.77) vanishes as $N_{0} \rightarrow \infty$. This is due to the signal being spread over a large number of degrees of freedom, while a constant amount of noise is added to each degree of freedom.

数学代写|信息论作业代写information theory代考|The High-Energy Regime

In both the deterministic model of Kolmogorov and the stochastic model of Shannon, we can increase the amount of information associated with the waveforms in the signals’ space by increasing the signal-to-noise ratio. By (1.24), (1.26), (1.46), and (1.73), this increases entropy and capacity by a logarithmic factor. We now ask whether we can also spend energy to obtain a linear increase of the amount of information, keeping a fixed signal-to-noise ratio. A possible strategy seems to be to increase the number of degrees of freedom, since this increases entropy and capacity linearly, and by (1.15) and (1.18) it can be accomplished by increasing the frequency of radiation. It turns out, however, that high-frequency signals are also observed at a coarser resolution, so that increasing the frequency while keeping the signal-to-noise ratio constant requires a corresponding increase of the energy per degree of freedom of the radiated signal, and an ultimate limit to the amount of information is imposed by the laws of high-energy physics.

To view these effects in more detail, let us have a closer look at the quantities determining the number of degrees of freedom. By (1.15), in a two-dimensional setting the number of space-wavenumber degrees of freedom at every frequency $\omega$ depends on size of the cut-set boundary and on the frequency of radiation. For any arbitrary configuration of sources and scatterers, we can increase the number of space-wavenumber degrees of freedom by transmitting at higher and higher frequencies. This improves the spatial resolution of the received waveform on the cut-set boundary. Similarly, in a three-dimensional setting (1.18) shows that the number of spatial degrees of freedom at each frequency $\omega$ increases with the frequency of radiation.
When radiation occurs over a range of frequencies of support $2 \Omega$ centered around the origin, the total number of degrees of freedom is given by (1.16) and (1.19), in two and three dimensions respectively. These equations show that the number of degrees of freedom grows with the largest frequency $\Omega$ of the radiated signal.

Finally, when radiation occurs over a bandwidth $\Omega$ centered around a carrier frequency $\omega_{\mathrm{c}} \gg \Omega$, as depicted in Figure 1.25, a computation analogous to (1.16) gives the following total number of degrees of freedom in the two-dimensional setting:
$$N_{0}=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \int_{\omega_{1}}^{\omega_{2}} \omega d \omega$$ $=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \frac{\left(\omega_{2}^{2}-\omega_{1}^{2}\right)}{2}$
$=\frac{\Omega T}{\pi} \frac{2 \pi r \omega_{c}}{c \pi} .$

信息论代写

数学代写|信息论作业代写information theory代考|The Low-Energy Regime

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

数学代写|信息论作业代写information theory代考|The High-Energy Regime

\begin{aligned} &=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \frac{\left(\omega_{2}^{2}-\omega_{1}^{2}\right)}{2} \ &=\frac{\Omega T T}{\pi} \frac{2 \pi r \omega_{c}}{c \pi} . \end{aligned}
$$N_{0}=\frac{T}{\pi} \frac{2 \pi r}{c \pi} \int_{\omega_{1}}^{\omega_{2}} \omega d \omega$$

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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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