数学代写|编码理论作业代写Coding Theory代考|AWGN Channel

如果你也在 怎样代写编码理论Coding Theory这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。


statistics-lab™ 为您的留学生涯保驾护航 在代写编码理论Coding 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。我们的专家在代写编码理论Coding Theory代写方面经验极为丰富,各种代写编码理论Coding Theory相关的作业也就用不着说。

我们提供的编码理论Coding Theory及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|编码理论作业代写Coding Theory代考|AWGN Channel

数学代写|编码理论作业代写Coding Theory代考|AWGN Channel

Up to now we have exclusively considered discrete-valued symbols. The concept of entropy can be transferred to continuous real-valued random variables by introducing the so-called differential entropy. It turns out that a channel with real-valued input and output symbols can again be characterised with the help of the mutual information $I(\mathcal{X} ; \mathcal{R})$ and its maximum, the channel capacity $C$. In Figure $1.5$ the so-called $A W G N$ channel is illustrated which is described by the additive white Gaussian noise term $\mathcal{Z}$.
With the help of the signal power
and the noise power
the channel capacity of the AWGN channel is given by
C=\frac{1}{2} \log _{2}\left(1+\frac{S}{N}\right)
The channel capacity exclusively depends on the signal-to-noise ratio $S / N$.
In order to compare the channel capacities of the binary symmetric channel and the AWGN channel, we assume a digital transmission scheme using binary phase shift keying (BPSK) and optimal reception with the help of a matched filter (Benedetto and Biglieri, 1999; Neubauer, 2007; Proakis, 2001). The signal-to-noise ratio of the real-valued output

$\mathcal{R}$ of the matched filter is then given by
\frac{S}{N}=\frac{E_{\mathrm{b}}}{N_{0} / 2}
with bit energy $E_{\mathrm{b}}$ and noise power spectral density $N_{0}$. If the output $\mathcal{R}$ of the matched filter is compared with the threshold 0 , we obtain the binary symmetric channel with bit error probability
\varepsilon=\frac{1}{2} \operatorname{erfc}\left(\sqrt{\frac{E_{\mathrm{b}}}{N_{0}}}\right)
Here, erfc(•) denotes the complementary error function. In Figure $1.6$ the channel capacities of the binary symmetric channel and the AWGN channel are compared as a function of $E_{\mathrm{b}} / N_{0}$. The signal-to-noise ratio $S / N$ or the ratio $E_{\mathrm{b}} / N_{0}$ must be higher for the binary symmetric channel compared with the AWGN channel in order to achieve the same channel capacity. This gain also translates to the coding gain achievable by soft-decision decoding as opposed to hard-decision decoding of channel codes, as we will see later (e.g. in Section 2.2.8).

Although information theory tells us that it is theoretically possible to find a channel code that for a given channel leads to as small an error probability as required, the design of good channel codes is generally difficult. Therefore, in the next chapters several classes of channel codes will be described. Here, we start with a simple example.

数学代写|编码理论作业代写Coding Theory代考|A Simple Channel Code

As an introductory example of a simple channel code we consider the transmission of the binary information sequence
00 1 0 110
over a binary symmetric channel with bit error probability $\varepsilon=0.25$ (Neubauer, 2006b). On average, every fourth binary symbol will be received incorrectly. In this example we assume that the binary sequence
00 0 0 0 110
is received at the output of the binary symmetric channel (see Figure 1.7).

In order to implement a simple error correction scheme we make use of the so-called binary triple repetition code. This simple channel code is used for the encoding of binary data. If the binary symbol 0 is to be transmitted, the encoder emits the code word 000 . Alternatively, the code word 111 is issued by the encoder when the binary symbol 1 is to be transmitted. The encoder of a triple repetition code is illustrated in Figure 1.8.

For the binary information sequence given above we obtain the binary code sequence
at the output of the encoder. If we again assume that on average every fourth binary symbol is incorrectly transmitted by the binary symmetric channel, we may obtain the received sequence
0 0 000 0 110 10 111 0 1 0 1110 10 .
This is illustrated in Figure 1.9.

数学代写|编码理论作业代写Coding Theory代考|Algebraic Coding Theory

In this chapter we will introduce the basic concepts of algebraic coding theory. To this end, the fundamental properties of block codes are first discussed. We will define important code parameters and describe how these codes can be used for the purpose of error detection and error correction. The optimal maximum likelihood decoding strategy will be derived and applied to the binary symmetric channel.

With these fundamentals at hand we will then introduce linear block codes. These channel codes can be generated with the help of so-called generator matrices owing to their special algebraic properties. Based on the closely related parity-check matrix and the syndrome, the decoding of linear block codes can be carried out. We will also introduce dual codes and several techniques for the construction of new block codes based on known ones, as well as bounds for the respective code parameters and the accompanying code characteristics. As examples of linear block codes we will treat the repetition code, paritycheck code, Hamming code, simplex code and Reed-Muller code.

Although code generation can be carried out efficiently for linear block codes, the decoding problem for general linear block codes is difficult to solve. By introducing further algebraic structures, cyclic codes can be derived as a subclass of linear block codes for which efficient algebraic decoding algorithms exist. Similar to general linear block codes, which are defined using the generator matrix or the parity-check matrix, cyclic codes are defined with the help of the so-called generator polynomial or parity-check polynomial. Based on linear feedback shift registers, the respective encoding and decoding architectures for cyclic codes can be efficiently implemented. As important examples of cyclic codes we will discuss $\mathrm{BCH}$ codes and Reed-Solomon codes. Furthermore, an algebraic decoding algorithm is presented that can be used for the decoding of BCH and Reed-Solomon codes.

In this chapter the classical algebraic coding theory is presented. In particular, we will follow work (Berlekamp, 1984; Bossert, 1999; Hamming, 1986; Jungnickel, 1995; Lin and Costello, 2004; Ling and Xing, 2004; MacWilliams and Sloane, 1998; McEliece, 2002; Neubauer, 2006b; van Lint, 1999) that contains further details about algebraic coding theory.

数学代写|编码理论作业代写Coding Theory代考|AWGN Channel


数学代写|编码理论作业代写Coding Theory代考|AWGN Channel

到目前为止,我们只考虑了离散值符号。通过引入所谓的微分熵,可以将熵的概念转移到连续实值随机变量上。事实证明,具有实值输入和输出符号的通道可以再次在互信息的帮助下进行表征一世(X;R)及其最大值,通道容量C. 如图1.5所谓的一种在Gñ通道被说明,它由加性高斯白噪声项描述从.


AWGN 信道的信道容量由下式给出

为了比较二进制对称信道和 AWGN 信道的信道容量,我们假设使用二进制相移键控 (BPSK) 的数字传输方案和借助匹配滤波器的最佳接收 (Benedetto and Biglieri, 1999; Neubauer, 2007 年;普罗基斯,2001 年)。实值输出的信噪比


有点能量和b和噪声功率谱密度ñ0. 如果输出R将匹配滤波器的值与阈值0进行比较,得到具有误码概率的二进制对称信道

这里,erfc(•) 表示互补误差函数。如图1.6将二进制对称信道和 AWGN 信道的信道容量作为以下函数进行比较和b/ñ0. 信噪比小号/ñ或比率和b/ñ0与 AWGN 信道相比,二进制对称信道必须更高,以实现相同的信道容量。这个增益也转化为软判决解码可实现的编码增益,而不是信道码的硬判决解码,我们将在后面看到(例如,在第 2.2.8 节中)。


数学代写|编码理论作业代写Coding Theory代考|A Simple Channel Code

00 1 0 110的传输

在二进制对称通道的输出端接收到00 0 0 0 110
(见图 1.7)。

为了实现一个简单的纠错方案,我们使用了所谓的二进制三重重复码。这个简单的通道代码用于二进制数据的编码。如果要传输二进制符号0,则编码器发出代码字000。或者,码字111由编码器在要传输二进制符号1时发出。三重重复码的编码器如图 1.8 所示。


0 0 000 0 110 10 111 0 1 0 1110 10。
如图 1.9 所示。

数学代写|编码理论作业代写Coding Theory代考|Algebraic Coding Theory


有了这些基础知识,我们将介绍线性分组码。由于其特殊的代数性质,这些信道码可以借助所谓的生成矩阵生成。基于密切相关的奇偶校验矩阵和校验子,可以进行线性分组码的译码。我们还将介绍双码和几种基于已知码构建新块码的技术,以及各个码参数的界限和伴随的码特征。作为线性块码的示例,我们将处理重复码、奇偶校验码、汉明码、单工码和 Reed-Muller 码。


本章介绍了经典的代数编码理论。我们将特别关注工作(Berlekamp,1984;Bossert,1999;Hamming,1986;Jungnickel,1995;Lin 和 Costello,2004;Ling 和 Xing,2004;MacWilliams 和 Sloane,1998;McEliece,2002;Neubauer,2006b; van Lint,1999),其中包含有关代数编码理论的更多细节。

数学代写|编码理论作业代写Coding Theory代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。







术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



您的电子邮箱地址不会被公开。 必填项已用*标注