标签： TELE9754

数学代写|信息论代写information theory代考|TELE9754

Posted on 2023年8月22日2023年8月22日 by statistics-lab

如果你也在怎样代写信息论information 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。信息论information theory回答了通信理论中的两个基本问题:什么是最终的数据压缩(答案:熵$H$)，什么是通信的最终传输速率(答案:信道容量$C$)。由于这个原因，一些人认为信息论是通信理论的一个子集。我们认为它远不止于此。

信息论information theory在统计物理学(热力学)、计算机科学(柯尔莫哥洛夫复杂性或算法复杂性)、统计推断(奥卡姆剃刀:“最简单的解释是最好的”)以及概率和统计学(最优假设检验和估计的误差指数)方面都做出了根本性的贡献。

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。

数学代写|信息论代写information theory代考|$p_i$ is not a measure of information, and $-\log p_i$ is not measured in bits

In numerous textbooks on IT, as well as in popular science books one can find a description of $-\log p_i$ as a measure of information associated with the event $i$, hence, the SMI $=-\sum p_i \log p_i$ is interpreted as an average information. This erroneous misinterpretation of SMI is discussed further in Ben-Naim [1]. Here, we focus only on the single term $-\log p_i$, which is sometimes referred to as “self-information,” or the amount of information you get when you know that the event $i$ occurs. Some even assign to this the term a value in units of bits.
Here is how “self-information” is introduced in Wikipedia:
Definition: Claude Shannon’s definition of self-information was chosen to meet several axioms:

If two independent events are measured separately, the total amount of information is the sum of the self-information of the individual events…given an event $\mathrm{x}$ with probability $\mathrm{P}$, the information content is defined as follows:
$$
I_X(x)=-\log \left(P_X(x)\right)
$$
This whole quotation is not only untrue; it is misleading as well. First of all, Shannon never defined self-information, (neither in the original article, Shannon [2], nor in Shannon and Weaver [4], and, of course, this was never chosen to meet “several axioms.”

Shannon searched for a measure of information based on the whole distribution and not for a single event. His conditions (as in Shannon [2]: “it is reasonable to require of it the following properties”), were entirely different from the conditions or requirements stated in abovementioned quotation.

If an event with a probability 1 occurs, it is not surprising, it is very much expected, but it is not true that it yields no information. When I hear that an event $x$ with probability $100 \%$ occurred, I obtained the information that ” $x$ occurred”.

If an event with lower probability occurred, I am more surprised. This it is true. But it is not true that I obtained more information!
Suppose that we have four dice with different probability distributions, say
$$
\begin{aligned}
& \text { die } \mathrm{A}: p_1=1, p_2=p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{B}: p_1=0.9, p_2=0.1, p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{C}: p_1=0.8, p_2=0.2, p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{D}: p_1=0.7, p_2=0.3, p_3=p_4=p_5=p_6=0
\end{aligned}
$$

数学代写|信息论代写information theory代考|SMI is not a probability

In the beginning of this section we claimed that probability in general, may not be interpreted as SMI. It is true that in a special case when all $p_i=p_0=\frac{1}{n}$, then $-\log p_0$ may be interpreted as SMI. However, in general $-\log p_i$ is not SMI. From this particular example, one cannot conclude that SMI is, in general, probability.
The association of SMI with probability is probably due to Brillouin [6]. On page 120 of his book “Science and Information Theory,” we find:
The probability has a natural tendency to increase, and so does entropy. The exact relation is given by the famous Boltzmann-Planck formula:
$$
S=k \ln P
$$
It is difficult to overestimate the amount of misinformation that is packed in these two sentences. Probability has no natural tendency to increase! Probability does not behave as entropy! There is no exact relationship between entropy and probability! The quoted formula is not the Boltzmann-Planck formula.

The correct Boltzmann-Planck relationship for the entropy is $S=k \ln W$, where $W$ is the total number of accessible microstates in the system. This relationship is a special case SMI for the case when all the events have equal probabilities. As we showed above, in general, probability is not SMI (except when $p_i=p_0=\frac{1}{n}$ ).

Here, we claim that entropy (being a special case of SMI) is never related to probability by an equation $S=k \ln P$.

The simplest reason for my claim is that probability is a positive number between 0 to 1 . Therefore, $\ln P$ varies between minus infinity to 0 . Entropy, as well as SMI is always a positive number greater or equal to 0. More on this in Ben-Naim [7].

信息论代写

数学代写|信息论代写information theory代考|$p_i$ is not a measure of information, and $-\log p_i$ is not measured in bits

在许多关于IT的教科书以及通俗科学书籍中，人们可以找到$-\log p_i$作为与事件$i$相关的信息度量的描述，因此，SMI $=-\sum p_i \log p_i$被解释为平均信息。Ben-Naim[1]进一步讨论了这种对重度精神分裂症的错误误解。在这里，我们只关注单个术语$-\log p_i$，它有时被称为“自我信息”，或者当您知道事件$i$发生时所获得的信息量。有些人甚至给这个术语赋以比特为单位的值。
维基百科是这样介绍“自我信息”的:
定义:选择克劳德·香农对自我信息的定义来满足几个公理:

如果分别测量两个独立事件，则信息总量为单个事件的自信息之和，给定一个概率为$\mathrm{P}$的事件$\mathrm{x}$，则信息内容定义如下:
$$
I_X(x)=-\log \left(P_X(x)\right)
$$
这整句话不仅不真实;这也具有误导性。首先，香农从未定义过自我信息，(无论是在原文香农[2]中，还是在香农和韦弗[4]中)，当然，这也从未被选择来满足“几个公理”。

香农寻找的是一种基于整个分布而不是单一事件的信息度量。他的条件(如香农[2]:“合理地要求它具有下列性质”)与上述引文中所述的条件或要求完全不同。

如果一个概率为1的事件发生了，这并不奇怪，它是非常值得期待的，但它不产生任何信息并不是真的。当我听到发生了一个概率为$100 \%$的事件$x$时，我得到的信息是“$x$发生了”。

如果发生概率较低的事件，我会更惊讶。这是真的。但我获得更多的信息是不真实的!
假设我们有四个概率分布不同的骰子
$$
\begin{aligned}
& \text { die } \mathrm{A}: p_1=1, p_2=p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{B}: p_1=0.9, p_2=0.1, p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{C}: p_1=0.8, p_2=0.2, p_3=p_4=p_5=p_6=0 \
& \text { die } \mathrm{D}: p_1=0.7, p_2=0.3, p_3=p_4=p_5=p_6=0
\end{aligned}
$$

数学代写|信息论代写information theory代考|SMI is not a probability

在本节的开始，我们声称概率一般不能被解释为SMI。的确，在特殊情况下，当所有$p_i=p_0=\frac{1}{n}$，那么$-\log p_0$都可以被解释为SMI。然而，一般来说$-\log p_i$不是重度精神分裂症。从这个特殊的例子中，我们不能得出SMI通常是概率性的结论。
SMI与概率的关联可能是由于布里渊[6]。在他的书《科学与信息论》的第120页，我们发现:
概率有增加的自然趋势，熵也是如此。精确的关系由著名的玻尔兹曼-普朗克公式给出:
$$
S=k \ln P
$$
很难高估这两句话中包含的错误信息的数量。概率没有自然增加的趋势!概率不代表熵!熵和概率之间没有确切的关系!引用的公式不是玻尔兹曼-普朗克公式。

正确的玻尔兹曼-普朗克熵的关系是$S=k \ln W$，其中$W$是系统中可达到的微观状态的总数。当所有事件的概率相等时，这种关系是一种特殊情况SMI。如上所示，一般来说，概率不是SMI(除非$p_i=p_0=\frac{1}{n}$)。

在这里，我们声称熵(作为SMI的一个特例)永远不会通过一个方程$S=k \ln P$与概率相关。

我的说法最简单的原因是概率是0到1之间的正数。因此，$\ln P$在负无穷到0之间变化。熵和SMI总是一个大于或等于0的正数。Ben-Naim[7]对此有更详细的介绍。

数学代写|信息论作业代写information 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。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

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）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

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。

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随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

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。

回归分析代写

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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。

编码理论是研究编码的属性和它们各自对特定应用的适用性。编码被用于数据压缩、密码学、错误检测和纠正、数据传输和数据存储。各种科学学科，如信息论、电气工程、数学、语言学和计算机科学，都对编码进行了研究，目的是设计高效和可靠的数据传输方法。这通常涉及消除冗余和纠正或检测传输数据中的错误。

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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|Convolutional Codes

数学代写|编码理论作业代写Coding Theory代考|Encoding of Convolutional Codes

The easiest way to introduce convolutional codes is by means of a specific example. Consider the convolutional encoder depicted in Figure 3.1. Information bits are shifted into a register of length $m=2$, i.e. a register with two binary memory elements. The output sequence results from multiplexing the two sequences denoted by $\mathbf{b}^{(1)}$ and $\mathbf{b}^{(2)}$. Each output bit is generated by modulo 2 addition of the current input bit and some symbols of the register contents. For instance, the information sequence $\mathbf{u}=(1,1,0,1,0,0, \ldots)$ will be encoded to $\mathbf{b}^{(1)}=(1,0,0,0,1,1, \ldots)$ and $\mathbf{b}^{(2)}=(1,1,1,0,0,1, \ldots)$. The generated code sequence after multiplexing of $\mathbf{b}^{(1)}$ and $\mathbf{b}^{(2)}$ is $\mathbf{b}=(1,1,0,1,0,1,0,0,1,0,1,1, \ldots)$.
A convolutional encoder is a linear sequential circuit and therefore a Linear TimeInvariant (LTI) system. It is well known that an LTI system is completely characterized

by its impulse response. Let us therefore investigate the two impulse responses of this particular encoder. The information sequence $\mathbf{u}=1,0,0,0, \ldots$ results in the output $\mathbf{b}^{(1)}=$ $(1,1,1,0, \ldots)$ and $\mathbf{b}^{(2)}=(1,0,1,0, \ldots)$, i.e. we obtain the generator impulse responses $\mathbf{g}^{(1)}=(1,1,1,0, \ldots)$ and $\mathbf{g}^{(2)}=(1,0,1,0, \ldots)$ respectively. These generator impulse responses are helpful for calculating the output sequences for an arbitrary input sequence
$$
\begin{aligned}
&b_{i}^{(1)}=\sum_{l=0}^{m} u_{i-l} g_{l}^{(1)} \leftrightarrow \mathbf{b}^{(1)}=\mathbf{u} * \mathbf{g}^{(1)} \
&b_{i}^{(2)}=\sum_{l=0}^{m} u_{i-l} g_{l}^{(2)} \leftrightarrow \mathbf{b}^{(2)}=\mathbf{u} * \mathbf{g}^{(2)}
\end{aligned}
$$
The generating equations for $\mathbf{b}^{(1)}$ and $\mathbf{b}^{(2)}$ can be regarded as convolutions of the input sequence with the generator impulse responses $\mathrm{g}^{(1)}$ and $\mathrm{g}^{(2)}$. The code $\mathrm{B}$ generated by this encoder is the set of all output sequences $\mathbf{b}$ that can be produced by convolution of arbitrary input sequence $\mathbf{u}$ with the generator impulse responses. This explains the name convolutional codes.

The general encoder of a rate $R=k / n$ convolutional code is depicted in Figure 3.2. Each input corresponds to a shift register, i.e. each information sequence is shifted into its own register. In contrast to block codes, the $i$ th code block $\mathbf{b}{i}=b{i}^{1}, b_{i}^{2}, \ldots, b_{i}^{n}$ of a convolutional code sequence $\mathbf{b}=\mathbf{b}{0}, \mathbf{b}{1}, \ldots$ is a linear function of several information blocks $\mathbf{u}{j}=u{j}^{1}, u_{j}^{2}, \ldots, u_{j}^{k}$ with $j \in{i-m, \ldots, i}$ and not only of $\mathbf{b}_{i}$. The integer $m$

denotes the memory of the encoder. The $k$ encoder registers do not necessarily have the same length. We denote the number of memory elements of the $l$ th register by $v_{l}$. The $n$ output sequences may depend on any of the $k$ registers. Thus, we require $k \times n$ impulse responses to characterise the encoding. If the shift registers are feedforward registers, i.e. they have no feedback, than the corresponding generator impulse responses $\mathbf{g}{l}^{(j)}$ are limited to length $v{l}+1$. For this reason, $v_{l}$ is often called the constraint length of the lth input sequence. The memory $m$ of the encoder is
$$
m=\max {l} v{l} .
$$

数学代写|编码理论作业代写Coding Theory代考|Generator Matrix in the Time Domain

Similarly to linear block codes, the encoding procedure can be described as a multiplication of the information sequence with a generator matrix $\mathbf{b}=\mathbf{u G}$. However, the information sequence $\mathbf{u}$ and the code sequence $\mathbf{b}$ are semi-infinite. Therefore, the generator matrix of a convolutional code also has a semi-infinite structure. It is constructed from $k \times n$ submatrices $\mathbf{G}{i}$ according to Figure $3.5$, where the elements of the submatrices are the coefficients from the generator impulse responses. For instance, for the encoder in Figure $3.1$ we obtain $$ \begin{aligned} &\mathbf{G}{0}=\left(g_{0}^{(1)}, g_{0}^{(2)}\right)=(11) \
&\mathbf{G}{1}=\left(g{1}^{(1)}, g_{1}^{(2)}\right)=\left(\begin{array}{ll}
1 & 0)
\end{array}\right. \
&\mathbf{G}{2}=\left(g{2}^{(1)}, g_{2}^{(2)}\right)=\left(\begin{array}{ll}
1 & 1
\end{array}\right)
\end{aligned}
$$

Finally, the generator matrix is
$$
\mathbf{G}=\left(\begin{array}{ccccccc}
\mathbf{G}{0} & \mathbf{G}{1} & \ldots & \mathbf{G}{m} & \mathbf{0} & \mathbf{0} & \ldots \ \mathbf{0} & \mathbf{G}{0} & \mathbf{G}{1} & \ldots & \mathbf{G}{m} & \mathbf{0} & \ldots \
\mathbf{0} & \mathbf{0} & \mathbf{G}{0} & \mathbf{G}{1} & \ldots & \mathbf{G}_{m} & \ldots \
\mathbf{0} & \mathbf{0} & \mathbf{0} & \ddots & \ddots & & \ddots
\end{array}\right)=\left(\begin{array}{cccccc}
11 & 10 & 11 & 00 & 00 & \ldots \
00 & 11 & 10 & 11 & 00 & \ldots \
00 & 00 & 11 & 10 & 11 & \ldots \
00 & 00 & 00 & \ddots & & \ddots
\end{array}\right) .
$$
With this generator matrix we can express the encoding of an information sequence, for instance $\mathbf{u}=(1,1,0,1,0,0, \ldots)$, by a matrix multiplication
$$
\begin{aligned}
\mathbf{b} &=\mathbf{u G}=(1,1,0,1,0,0, \ldots)\left(\begin{array}{cccccc}
11 & 10 & 11 & 00 & 00 & \ldots \
00 & 11 & 10 & 11 & 00 & \ldots \
00 & 00 & 11 & 10 & 11 & \ldots \
00 & 00 & 00 & \ddots & & \ddots
\end{array}\right) \
&=(11,01,01,00,10,11,00, \ldots .
\end{aligned}
$$

数学代写|编码理论作业代写Coding Theory代考|State Diagram of a Convolutional Encoder

Up to now we have considered two methods to describe the encoding of a convolutional code, i.e. encoding with a linear sequential circuit, a method that is probably most suitable for hardware implementations, and a formal description with the generator matrix. Now we consider a graphical representation, the so-called state diagram. The state diagram will be helpful later on when we consider distance properties of convolutional codes and decoding algorithms.

The state diagram of a convolutional encoder describes the operation of the encoder. From this graphical representation we observe that the encoder is a finite-state machine. For the construction of the state diagram we consider the contents of the encoder registers as encoder state $\sigma$. The set $\mathrm{S}$ of encoder states is called the encoder state space.

Each memory element contains only 1 bit of information. Therefore, the number of encoder states is $2^{\nu}$. We will use the symbol $\sigma_{i}$ to denote the encoder state at time $i$. The state diagram is a graph that has $2^{v}$ nodes representing the encoder states. An example of the state diagram is given in Figure 3.6. The branches in the state diagram represent possible state transitions, e.g. if the encoder in Figure $3.1$ has the register contents $\sigma_{i}=(00)$ and the input bit $u_{i}$ at time $i$ is a 1 , then the state of the encoder changes from $\sigma_{i}=(00)$ to $\sigma_{i+1}=(10)$. Along with this transition, the two output bits $\mathbf{b}_{i}=(11)$ are generated. Similarly, the information sequence $\mathbf{u}=(1,1,0,1,0,0, \ldots)$ corresponds to the state sequence $\sigma=(00,10,11,01,10,01,00, \ldots)$, subject to the encoder starting in the all-zero state. The code sequence is again $\mathbf{b}=(11,01,01,00,10,11,00, \ldots)$. In general, the output bits only depend on the current input and the encoder state. Therefore, we label each transition with the $k$ input bits and the $n$ output bits (input/output).

编码理论代写

数学代写|编码理论作业代写Coding Theory代考|Encoding of Convolutional Codes

引入卷积码最简单的方法是通过一个具体的例子。考虑图 3.1 中描述的卷积编码器。信息位被移入一个长度为的寄存器米=2，即具有两个二进制存储元素的寄存器。输出序列来自多路复用两个序列，表示为b(1)和b(2). 每个输出位由当前输入位和寄存器内容的一些符号的模 2 加法生成。例如，信息序列在=(1,1,0,1,0,0,…)将被编码为b(1)=(1,0,0,0,1,1,…)和b(2)=(1,1,1,0,0,1,…). 多路复用后生成的代码序列b(1)和b(2)是b=(1,1,0,1,0,1,0,0,1,0,1,1,…).
卷积编码器是一个线性时序电路，因此是一个线性时不变 (LTI) 系统。众所周知，LTI 系统具有完全的特征

通过它的脉冲响应。因此，让我们研究这个特定编码器的两个脉冲响应。信息序列在=1,0,0,0,…结果输出b(1)= (1,1,1,0,…)和b(2)=(1,0,1,0,…)，即我们获得发生器脉冲响应G(1)=(1,1,1,0,…)和G(2)=(1,0,1,0,…)分别。这些发生器脉冲响应有助于计算任意输入序列的输出序列

b一世(1)=∑l=0米在一世−lGl(1)↔b(1)=在∗G(1) b一世(2)=∑l=0米在一世−lGl(2)↔b(2)=在∗G(2)
生成方程为b(1)和b(2)可以看作是输入序列与生成器脉冲响应的卷积G(1)和G(2). 编码乙该编码器生成的是所有输出序列的集合b可以通过任意输入序列的卷积产生在与发生器脉冲响应。这解释了卷积码的名称。

速率的通用编码器R=ķ/n卷积码如图 3.2 所示。每个输入对应一个移位寄存器，即每个信息序列都被移位到它自己的寄存器中。与块代码相比，一世第一个代码块b一世=b一世1,b一世2,…,b一世n卷积码序列的b=b0,b1,…是几个信息块的线性函数在j=在j1,在j2,…,在jķ和j∈一世−米,…,一世而不仅仅是b一世. 整数米

表示编码器的内存。这ķ编码器寄存器不一定具有相同的长度。我们表示存储单元的数量l注册由在l. 这n输出序列可能取决于任何ķ寄存器。因此，我们要求ķ×n脉冲响应来表征编码。如果移位寄存器是前馈寄存器，即它们没有反馈，则对应的发生器脉冲响应Gl(j)限于长度在l+1. 为此原因，在l通常称为第 l 个输入序列的约束长度。记忆米编码器是

米=最大限度l在l.

数学代写|编码理论作业代写Coding Theory代考|Generator Matrix in the Time Domain

与线性块码类似，编码过程可以描述为信息序列与生成矩阵的乘积b=在G. 但是，信息序列在和代码序列b是半无限的。因此，卷积码的生成矩阵也具有半无限的结构。它是由ķ×n子矩阵G一世根据图3.5，其中子矩阵的元素是来自发生器脉冲响应的系数。例如，对于图中的编码器3.1我们获得

G0=(G0(1),G0(2))=(11) G1=(G1(1),G1(2))=(10) G2=(G2(1),G2(2))=(11)

最后，生成矩阵为

G=(G0G1…G米00… 0G0G1…G米0… 00G0G1…G米… 000⋱⋱⋱)=(1110110000… 0011101100… 0000111011… 000000⋱⋱).
使用这个生成矩阵，我们可以表达信息序列的编码，例如在=(1,1,0,1,0,0,…), 通过矩阵乘法

b=在G=(1,1,0,1,0,0,…)(1110110000… 0011101100… 0000111011… 000000⋱⋱) =(11,01,01,00,10,11,00,….

数学代写|编码理论作业代写Coding Theory代考|State Diagram of a Convolutional Encoder

到目前为止，我们已经考虑了两种描述卷积码编码的方法，即使用线性时序电路进行编码，一种可能最适合硬件实现的方法，以及使用生成矩阵的形式描述。现在我们考虑一个图形表示，即所谓的状态图。稍后当我们考虑卷积码和解码算法的距离属性时，状态图将很有帮助。

卷积编码器的状态图描述了编码器的操作。从这个图形表示中，我们观察到编码器是一个有限状态机。对于状态图的构建，我们将编码器寄存器的内容视为编码器状态σ. 套装小号编码器状态称为编码器状态空间。

每个存储元素仅包含 1 位信息。因此，编码器状态数为2ν. 我们将使用符号σ一世表示时间的编码器状态一世. 状态图是一个具有2在表示编码器状态的节点。图 3.6 给出了一个状态图的例子。状态图中的分支表示可能的状态转换，例如，如果图3.1有寄存器内容σ一世=(00)和输入位在一世有时一世为 1 ，则编码器的状态从σ一世=(00)至σ一世+1=(10). 随着这种转变，两个输出位b一世=(11)被生成。同样，信息序列在=(1,1,0,1,0,0,…)对应状态序列σ=(00,10,11,01,10,01,00,…)，以编码器从全零状态开始。代码序列又是b=(11,01,01,00,10,11,00,…). 通常，输出位仅取决于当前输入和编码器状态。因此，我们用ķ输入位和n输出位（输入/输出）。

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

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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

数学代写|编码理论作业代写Coding Theory代考|Definition of Cyclic Codes

A cyclic code is characterised as a linear block code $\mathrm{B}(n, k, d)$ with the additional property that for each code word
$$
\mathbf{b}=\left(b_{0}, b_{1}, \ldots, b_{n-2}, b_{n-1}\right)
$$

all cyclically shifted words
$$
\begin{aligned}
&\left(b_{n-1}, b_{0}, \ldots, b_{n-3}, b_{n-2}\right) \
&\left(b_{n-2}, b_{n-1}, \ldots, b_{n-4}, b_{n-3}\right) \
&\vdots \
&\left(b_{2}, b_{3}, \ldots, b_{0}, b_{1}\right) \
&\left(b_{1}, b_{2}, \ldots, b_{n-1}, b_{0}\right)
\end{aligned}
$$
are also valid code words of $\mathrm{B}(n, k, d)$ (Lin and Costello, 2004; Ling and Xing, 2004). This property can be formulated concisely if a code word $\mathbf{b} \in \mathbb{F}{q}^{n}$ is represented as a polynomial $$ b(z)=b{0}+b_{1} z+\cdots+b_{n-2} z^{n-2}+b_{n-1} z^{n-1}
$$
over the finite field $\mathbb{F}{q} \cdot{ }^{16}$ A cyclic shift $$ \left(b{0}, b_{1}, \ldots, b_{n-2}, b_{n-1}\right) \mapsto\left(b_{n-1}, b_{0}, b_{1}, \ldots, b_{n-2}\right)
$$
of the code polynomial $b(z) \in \mathrm{F}{q}[z]$ can then be expressed as $$ b{0}+b_{1} z+\cdots+b_{n-2} z^{n-2}+b_{n-1} z^{n-1} \mapsto b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1} .
$$
Because of
$$
b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1}=z b(z)-b_{n-1}\left(z^{n}-1\right)
$$
and by observing that a code polynomial $b(z)$ is of maximal degree $n-1$, we represent the cyclically shifted code polynomial modulo $z^{n}-1$, i.e.
$$
b_{n-1}+b_{0} z+b_{1} z^{2}+\cdots+b_{n-2} z^{n-1} \equiv z b(z) \bmod z^{n}-1 .
$$
Cyclic codes $\mathrm{B}(n, k, d)$ therefore fulfil the following algebraic property
$$
b(z) \in \mathrm{B}(n, k, d) \quad \Leftrightarrow \quad z b(z) \bmod z^{n}-1 \in \mathbb{B}(n, k, d) .
$$
For that reason – if not otherwise stated – we consider polynomials in the factorial ring $F_{q}[z] /\left(z^{n}-1\right)$. Figure $2.38$ summarises the definition of cyclic codes.

Similarly to general linear block codes, which can be defined by the generator matrix G or the corresponding parity-check matrix $\mathbf{H}$, cyclic codes can be characterised by the generator polynomial $g(z)$ and the parity-check polynomial $h(z)$, as we will show in the following (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004).

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

A linear block code $\mathbb{B}(n, k, d)$ is defined by the $k \times n$ generator matrix
$$
\mathbf{G}=\left(\begin{array}{c}
\mathbf{g}{0} \ \mathbf{g}{1} \
\vdots \
\mathbf{g}{k-1} \end{array}\right)=\left(\begin{array}{cccc} g{0,0} & g_{0,1} & \cdots & g_{0, n-1} \
g_{1,0} & g_{1,1} & \cdots & g_{1, n-1} \
\vdots & \vdots & \ddots & \vdots \
g_{k-1,0} & g_{k-1,1} & \cdots & g_{k-1, n-1}
\end{array}\right)
$$

with $k$ linearly independent basis vectors $\mathbf{g}{0}, \mathbf{g}{1}, \ldots, \mathbf{g}{k-1}$ which themselves are valid code vectors of the linear block code $\mathrm{B}(n, k, d)$. Owing to the algebraic properties of a cyclic code there exists a unique polynomial $$ g(z)=g{0}+g_{1} z+\cdots+g_{n-k-1} z^{n-k-1}+g_{n-k} z^{n-k}
$$
of minimal degree $\operatorname{deg}(g(z))=n-k$ with $g_{n-k}=1$ such that the corresponding generator matrix can be written as
$$
\mathbf{G}=\left(\begin{array}{ccccccccccc}
g_{0} & g_{1} & \cdots & g_{n-k} & 0 & \cdots & 0 & 0 & \cdots & 0 & 0 \
0 & g_{0} & \cdots & g_{n-k-1} & g_{n-k} & \cdots & 0 & 0 & \cdots & 0 & 0 \
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \vdots \
0 & 0 & \cdots & 0 & 0 & \cdots & g_{0} & g_{1} & \cdots & g_{n-k} & 0 \
0 & 0 & \cdots & 0 & 0 & \cdots & 0 & g_{0} & \cdots & g_{n-k-1} & g_{n-k}
\end{array}\right)
$$
This polynomial $g(z)$ is called the generator polynomial of the cyclic code $\mathbb{B}(n, k, d)$ (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004). The rows of the generator matrix $\mathbf{G}$ are obtained from the generator polynomial $g(z)$ and all cyclic shifts $z g(z), z^{2} g(z), \ldots, z^{k-1} g(z)$ which correspond to valid code words of the cyclic code. Formally, we can write the generator matrix as
$$
\mathbf{G}=\left(\begin{array}{c}
g(z) \
z g(z) \
\vdots \
z^{k-2} g(z) \
z^{k-1} g(z)
\end{array}\right)
$$

数学代写|编码理论作业代写Coding Theory代考|Cyclic Redundancy Check

With the help of the generator polynomial $g(z)$ of a cyclic code $\mathrm{B}(n, k, d)$, the so-called cyclic redundancy check (CRC) can be defined for the detection of errors (Lin and Costello, 2004). Besides the detection of $e_{\text {det }}=d-1$ errors by a cyclic code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$, cyclic error bursts can also be detected. With a generator polynomial $g(z)$ of degree $\operatorname{deg}(g(z))=n-k$, all cyclic error bursts of length
$$
\ell_{\text {burst }} \leq n-k
$$
can be detected (Jungnickel, 1995). This can be seen by considering the error model $r(z)=$ $b(z)+e(z)$ with the received polynomial $r(z)$, the code polynomial $b(z)$ and the error polynomial $e(z)$ (see also Figure 2.46). Errors can be detected as long as the parity-check equation
$$
g(z) \mid r(z) \Leftrightarrow r(z) \in \mathrm{B}(n, k, d)
$$
of the cyclic code $\mathbb{B}(n, k, d)$ is fulfilled. Since $g(z) \mid b(z)$, all errors for which the error polynomial $e(z)$ is not divisible by the generator polynomial $g(z)$ can be detected. As long as the degree $\operatorname{deg}(e(z))$ is smaller than $\operatorname{deg}(g(z))=n-k$, the error polynomial $e(z)$ cannot be divided by the generator polynomial. This is also true if cyclically shifted variants $z^{i} e(z)$ of such an error polynomial are considered. Since for an error burst of length $\ell_{\text {burst }}$ the degree of the error polynomial is equal to $\ell_{\text {burst }}-1$, the error detection is possible if
$$
\operatorname{deg}(e(z))=\ell_{\text {burst }}-1<n-k=\operatorname{deg}(g(z)) \text {. }
$$

编码理论代写

数学代写|编码理论作业代写Coding Theory代考|Definition of Cyclic Codes

循环码的特征是线性块码乙(n,ķ,d)具有每个代码字的附加属性

b=(b0,b1,…,bn−2,bn−1)

所有循环移位的词

(bn−1,b0,…,bn−3,bn−2) (bn−2,bn−1,…,bn−4,bn−3) ⋮ (b2,b3,…,b0,b1) (b1,b2,…,bn−1,b0)
也是有效的代码字乙(n,ķ,d)（Lin 和 Costello，2004 年；Ling 和 Xing，2004 年）。如果一个代码字可以简明地表述这个性质b∈Fqn表示为多项式

b(和)=b0+b1和+⋯+bn−2和n−2+bn−1和n−1
在有限域上Fq⋅16循环移位

(b0,b1,…,bn−2,bn−1)↦(bn−1,b0,b1,…,bn−2)
码多项式的b(和)∈Fq[和]那么可以表示为

b0+b1和+⋯+bn−2和n−2+bn−1和n−1↦bn−1+b0和+b1和2+⋯+bn−2和n−1.
因为

bn−1+b0和+b1和2+⋯+bn−2和n−1=和b(和)−bn−1(和n−1)
并通过观察代码多项式b(和)是最大程度的n−1，我们表示循环移位码多项式模和n−1， IE

bn−1+b0和+b1和2+⋯+bn−2和n−1≡和b(和)反对和n−1.
循环码乙(n,ķ,d)因此满足以下代数性质

b(和)∈乙(n,ķ,d)⇔和b(和)反对和n−1∈乙(n,ķ,d).
出于这个原因——如果没有另外说明——我们考虑阶乘环中的多项式Fq[和]/(和n−1). 数字2.38总结了循环码的定义。

类似于一般的线性块码，可以由生成矩阵 G 或对应的奇偶校验矩阵来定义H, 循环码可以用生成多项式来表征G(和)和奇偶校验多项式H(和)，正如我们将在下文中展示的那样（Berlekamp，1984；Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004）。

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

线性块码乙(n,ķ,d)由定义ķ×n生成矩阵

G=(G0 G1 ⋮ Gķ−1)=(G0,0G0,1⋯G0,n−1 G1,0G1,1⋯G1,n−1 ⋮⋮⋱⋮ Gķ−1,0Gķ−1,1⋯Gķ−1,n−1)

和ķ线性独立的基向量G0,G1,…,Gķ−1它们本身是线性块代码的有效代码向量乙(n,ķ,d). 由于循环码的代数性质，存在唯一多项式

G(和)=G0+G1和+⋯+Gn−ķ−1和n−ķ−1+Gn−ķ和n−ķ
最低程度的你⁡(G(和))=n−ķ和Gn−ķ=1使得对应的生成矩阵可以写成

G=(G0G1⋯Gn−ķ0⋯00⋯00 0G0⋯Gn−ķ−1Gn−ķ⋯00⋯00 ⋮⋮⋱⋮⋮⋱⋮⋮⋱⋮⋮ 00⋯00⋯G0G1⋯Gn−ķ0 00⋯00⋯0G0⋯Gn−ķ−1Gn−ķ)
这个多项式G(和)称为循环码的生成多项式乙(n,ķ,d)（Berlekamp，1984；Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004）。生成矩阵的行G从生成多项式获得G(和)和所有循环移位和G(和),和2G(和),…,和ķ−1G(和)对应于循环码的有效码字。形式上，我们可以将生成矩阵写为

G=(G(和) 和G(和) ⋮ 和ķ−2G(和) 和ķ−1G(和))

数学代写|编码理论作业代写Coding Theory代考|Cyclic Redundancy Check

在生成多项式的帮助下G(和)循环码乙(n,ķ,d)，可以定义所谓的循环冗余校验 (CRC) 来检测错误 (Lin and Costello, 2004)。除了检测和这 =d−1循环码错误乙(n,ķ,d)具有最小汉明距离d，也可以检测到循环错误突发。使用生成多项式G(和)学位你⁡(G(和))=n−ķ, 所有长度的循环错误突发

ℓ爆裂 ≤n−ķ
可以检测到（Jungnickel，1995）。这可以通过考虑误差模型来看出r(和)= b(和)+和(和)与收到的多项式r(和), 码多项式b(和)和误差多项式和(和)（另请参见图 2.46）。只要奇偶校验等式就可以检测到错误

G(和)∣r(和)⇔r(和)∈乙(n,ķ,d)
循环码乙(n,ķ,d)被履行。自从G(和)∣b(和), 误差多项式的所有误差和(和)不能被生成多项式整除G(和)可以检测到。只要学历你⁡(和(和))小于你⁡(G(和))=n−ķ, 误差多项式和(和)不能被生成多项式整除。如果循环移位变体也是如此和一世和(和)考虑这样的误差多项式。因为对于长度的错误突发ℓ爆裂误差多项式的次数等于ℓ爆裂 −1，错误检测是可能的，如果

你⁡(和(和))=ℓ爆裂 −1<n−ķ=你⁡(G(和)).

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

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|编码理论作业代写Coding Theory代考|Bounds for Linear Block Codes

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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。

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|Bounds for Linear Block Codes

The code rate $R=k / n$ and the minimum Hamming distance $d$ are important parameters of a linear block code $\mathrm{B}(n, k, d)$. It is therefore useful to know whether a linear block code theoretically exists for a given combination of $R$ and $d$. In particular, for a given minimum Hamming distance $d$ we will consider a linear block code to be better than another linear block code with the same minimum Hamming distance $d$ if it has a higher code rate $R$. There exist several theoretical bounds which we will briefly discuss in the following (Berlekamp, 1984; Bossert, 1999; Ling and Xing, 2004; van Lint, 1999) (see also Figure 2.22). Block codes that fulfil a theoretical bound with equality are called optimal codes.
Singleton Bound
The simplest bound is the so-called Singleton bound. For a linear block code $\mathrm{B}(n, k, d)$ it is given by
$$
k \leq n-d+1 .
$$
A linear block code that fulfils the Singleton bound with equality according to $k=n-d+$ 1 is called MDS (maximum distance separable). Important representatives of MDS codes are Reed-Solomon codes which are used, for example, in the channel coding scheme within the audio compact disc (see also Section 2.3.7).
Sphere Packing Bound
The so-called sphere packing bound or Hamming bound can be derived for a linear $e_{\text {cor }}=$ $\lfloor(d-1) / 2\rfloor$-error correcting $q$-nary block code $\mathrm{B}(n, k, d)$ by considering the correction balls within the code space $\mathbb{F}{q}^{n}$. Each correction ball encompasses a total of $$ \sum{i=0}^{e_{\mathrm{cor}}}\left(\begin{array}{l}
n \
i
\end{array}\right)(q-1)^{i}
$$

vectors in the code space $F_{q}^{n}$. Since there is exactly one correction ball for each code word, we have $M=q^{k}$ correction balls. The total number of vectors within any correction ball of radius $e_{\text {cor }}$ is then given by
$$
q^{k} \sum_{i=0}^{e_{\text {cor }}}\left(\begin{array}{l}
n \
i
\end{array}\right)(q-1)^{i}
$$
Because this number must be smaller than or equal to the maximum number $\left|\mathbb{F}{q}^{n}\right|=q^{n}$ of vectors in the finite vector space $\mathbb{F}{q}^{n}$, the sphere packing bound
$$
q^{k} \sum_{i=0}^{e_{\operatorname{cor}}}\left(\begin{array}{l}
n \
i
\end{array}\right)(q-1)^{i} \leq q^{n}
$$

数学代写|编码理论作业代写Coding Theory代考|Plotkin’s (u|v) Code Construction

In contrast to the aforementioned code constructions, the (u|v) code construction originally proposed by Plotkin takes two linear block codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$ with code word lengths $n_{1}$ and $n_{2}$, information word lengths $k_{1}$ and $k_{2}$ and minimum Hamming distances $d_{1}$ and $d_{2}$. For simplification, we fill the code words of the block code with smaller code word length by an appropriate number of zeros. This step is called zero padding. Let $\mathbf{u}$ and $\mathbf{v}$ be two arbitrary code words thus obtained from the linear block codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$ respectively. ${ }^{12}$ Then we have
$$
\mathbf{u}= \begin{cases}\left(\mathbf{b}{1} \mid 0, \ldots, 0\right), & n{1}<n_{2} \ \mathbf{b}{1}, & n{1} \geq n_{2}\end{cases}
$$
and
$$
\mathbf{v}= \begin{cases}\mathbf{b}{2}, & n{1}<n_{2} \ \left(\mathbf{b}{2} \mid 0, \ldots, 0\right), & n{1} \geq n_{2}\end{cases}
$$
with the arbitrarily chosen code words $\mathbf{b}{1} \in \mathbb{B}\left(n{1}, k_{1}, d_{1}\right)$ and $\mathbf{b}{2} \in \mathbb{B}\left(n{2}, k_{2}, d_{2}\right)$. We now identify the code words $\mathbf{u}$ and $\mathbf{v}$ with the original codes $\mathrm{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathrm{B}\left(n_{2}, k_{2}, d_{2}\right)$, i.e. we write $\mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)$. The code resulting from Plotkin’s (u|v) code construction is then given by
$$
\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf{u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}
$$
This code construction creates all vectors of the form $(\mathbf{u} \mid \mathbf{u}+\mathbf{v})$ by concatenating all possible vectors $\mathbf{u}$ and $\mathbf{u}+\mathbf{v}$ with $\mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right)$ and $\mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)$. The resulting code is characterised by the following code parameters
$$
\begin{aligned}
&n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \
&k^{\prime}=k_{1}+k_{2}, \
&d^{\prime}=\min \left{2 d_{1}, d_{2}\right}
\end{aligned}
$$

数学代写|编码理论作业代写Coding Theory代考|Examples of Linear Block Codes

In this section we will present some important linear block codes. In particular, we will consider the already introduced repetition codes, parity-check codes and Hamming codes. Furthermore, simplex codes and Reed-Muller codes and their relationships are discussed (Berlekamp, 1984; Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004).

We have already introduced the binary triple repetition code in Section $1.3$ as a simple introductory example of linear block codes. In this particular code the binary information symbol 0 or 1 is transmitted with the binary code word 000 or 111 respectively. In general, a repetition code over the alphabet $\mathrm{F}{q}$ assigns to the $q$-nary information symbol $u{0}$ the $n$-dimensional code word $\mathbf{b}=\left(u_{0}, u_{0}, \ldots, u_{0}\right)$. Trivially, this block code is linear. The minimum Hamming distance is
$$
d=n .
$$
Therefore, the repetition code in Figure $2.27$ is a linear block code $\mathrm{B}(n, 1, n)$ that is able to detect $e_{\text {det }}=d-1=n-1$ errors or to correct $e_{\text {cor }}=\lfloor(d-1) / 2\rfloor=\lfloor(n-1) / 2\rfloor$ errors. The code rate is
$$
R=\frac{1}{n} .
$$
For the purpose of error correction, the minimum distance decoding can be implemented by a majority decoding scheme. The decoder emits the information symbol $\hat{u}_{0}$ which appears most often in the received word.
The weight distribution of a repetition code is simply given by
$$
W(x)=1+(q-1) x^{n}
$$
Although this repetition code is characterised by a low code rate $R$, it is used in some communication standards owing to its simplicity. For example, in the short-range wireless communication system Bluetooth ${ }^{\mathrm{TM}}$ a triple repetition code is used as part of the coding scheme of the packet header of a transmitted baseband packet (Bluetooth, 2004).

编码理论代写

数学代写|编码理论作业代写Coding Theory代考|Bounds for Linear Block Codes

码率R=ķ/n和最小汉明距离d是线性分组码的重要参数乙(n,ķ,d). 因此，对于给定的组合，了解理论上是否存在线性分组码是很有用的。R和d. 特别是，对于给定的最小汉明距离d我们将认为一个线性分组码优于另一个具有相同最小汉明距离的线性分组码d如果它有更高的码率R. 存在几个理论界限，我们将在下面简要讨论（Berlekamp，1984；Bossert，1999；Ling 和 Xing，2004；van Lint，1999）（另请参见图 2.22）。满足理论上相等界限的分组码称为最优码。
单例界
最简单的界就是所谓的单例界。对于线性块码乙(n,ķ,d)它是由

ķ≤n−d+1.
一个线性块代码，它满足具有相等性的 Singleton 约束，根据ķ=n−d+1称为MDS（最大可分离距离）。MDS 码的重要代表是 Reed-Solomon 码，例如，在音频光盘中的通道编码方案中使用（另见第 2.3.7 节）。
Sphere Packing Bound
所谓的 sphere Packing bound 或 Hamming bound 可以推导出一个线性的和心电图 = ⌊(d−1)/2⌋-纠错q-nary 块代码乙(n,ķ,d)通过考虑代码空间内的修正球Fqn. 每个修正球总共包含

∑一世=0和C这r(n 一世)(q−1)一世

代码空间中的向量Fqn. 由于每个码字都有一个修正球，我们有米=qķ修正球。任何半径修正球内的向量总数和心电图然后由

qķ∑一世=0和心电图 (n 一世)(q−1)一世
因为这个数必须小于等于最大数|Fqn|=qn有限向量空间中的向量Fqn, 球体包装界

qķ∑一世=0和心电图(n 一世)(q−1)一世≤qn

数学代写|编码理论作业代写Coding Theory代考|Plotkin’s (u|v) Code Construction

与上述代码结构相比，Plotkin 最初提出的 (u|v) 代码结构采用两个线性块代码乙(n1,ķ1,d1)和乙(n2,ķ2,d2)带码字长n1和n2, 信息字长ķ1和ķ2和最小汉明距离d1和d2. 为了简化，我们用适当数量的零填充具有较小码字长度的块码的码字。此步骤称为零填充。让在和在是从线性块码中获得的两个任意码字乙(n1,ķ1,d1)和乙(n2,ķ2,d2)分别。12然后我们有

在={(b1∣0,…,0),n1<n2 b1,n1≥n2
和

在={b2,n1<n2 (b2∣0,…,0),n1≥n2
用任意选择的代码字b1∈乙(n1,ķ1,d1)和b2∈乙(n2,ķ2,d2). 我们现在识别代码字在和在与原始代码乙(n1,ķ1,d1)和乙(n2,ķ2,d2)，即我们写在∈乙(n1,ķ1,d1)和在∈乙(n2,ķ2,d2). 然后由 Plotkin 的 (u|v) 代码构造产生的代码由下式给出

\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf {u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}\mathbb{B}^{\prime}\left(n^{\prime}, k^{\prime}, d^{\prime}\right)=\left{(\mathbf{u} \mid \mathbf {u}+\mathbf{v}): \mathbf{u} \in \mathbb{B}\left(n_{1}, k_{1}, d_{1}\right) \wedge \mathbf{v} \in \mathbb{B}\left(n_{2}, k_{2}, d_{2}\right)\right}
此代码构造创建形式的所有向量(在∣在+在)通过连接所有可能的向量在和在+在和在∈乙(n1,ķ1,d1)和在∈乙(n2,ķ2,d2). 生成的代码具有以下代码参数的特征

\begin{对齐} &n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \ &k^{\prime}=k_{1}+k_{2}, \ &d ^{\prime}=\min \left{2 d_{1}, d_{2}\right} \end{aligned}\begin{对齐} &n^{\prime}=2 \max \left{n_{1}, n_{2}\right} \ &k^{\prime}=k_{1}+k_{2}, \ &d ^{\prime}=\min \left{2 d_{1}, d_{2}\right} \end{aligned}

数学代写|编码理论作业代写Coding Theory代考|Examples of Linear Block Codes

在本节中，我们将介绍一些重要的线性分组码。特别是，我们将考虑已经介绍的重复码、奇偶校验码和汉明码。此外，还讨论了单纯形码和 Reed-Muller 码及其关系（Berlekamp，1984；Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004）。

我们已经在章节中介绍了二进制三重重复代码1.3作为线性分组码的简单介绍性示例。在此特定代码中，二进制信息符号 0 或 1 分别与二进制代码字 000 或 111 一起传输。一般来说，字母表上的重复代码Fq分配给q-nary 信息符号在0这n- 维码字b=(在0,在0,…,在0). 很简单，这个块代码是线性的。最小汉明距离为

d=n.
因此，图中的重复码2.27是一个线性块码乙(n,1,n)能够检测到和这 =d−1=n−1错误或纠正和心电图 =⌊(d−1)/2⌋=⌊(n−1)/2⌋错误。码率是

R=1n.
为了纠错，最小距离解码可以通过多数解码方案来实现。解码器发出信息符号在^0最常出现在接收到的单词中。
重复代码的权重分布简单地由下式给出

在(X)=1+(q−1)Xn
虽然这种重复码的特点是码率低R，由于其简单性，它被用于一些通信标准中。例如，在短距离无线通信系统蓝牙吨米三重重复码被用作传输的基带数据包的数据包头的编码方案的一部分（蓝牙，2004）。

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

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

数学代写|编码理论作业代写Coding Theory代考|Error Detection and Error Correction

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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。

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|Error Detection and Error Correction

Based on the minimum distance decoding rule and the code space concept, we can assess the error detection and error correction capabilities of a given channel code. To this end, let $\mathbf{b}$ and $\mathbf{b}^{\prime}$ be two code words of an $(n, k)$ block code $\mathrm{B}(n, k, d)$. The distance of these code words shall be equal to the minimum Hamming distance, i.e. $\operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=d$. We are able to detect errors as long as the erroneously received word $\mathbf{r}$ is not equal to a code word different from the transmitted code word. This error detection capability is guaranteed as long as the number of errors is smaller than the minimum Hamming distance $d$, because another code word (e.g. $\mathbf{b}^{\prime}$ ) can be reached from a given code word (e.g. b) merely by changing at least $d$ components. For an ( $n, k)$ block code $\mathrm{B}(n, k, d)$ with minimum Hamming distance $d$, the number of detectable errors is therefore given by (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004; van Lint, 1999)
$$
e_{\text {det }}=d-1 .
$$
For the analysis of the error correction capabilities of the $(n, k)$ block code $\mathbb{B}(n, k, d)$ we define for each code word $\mathbf{b}$ the corresponding correction ball of radius $\varrho$ as the subset of all words that are closer to the code word $\mathbf{b}$ than to any other code word $\mathbf{b}^{\prime}$ of the block code $\mathrm{B}(n, k, d)$ (see Figure 2.10). As we have seen in the last section, for minimum distance decoding, all received words within a particular correction ball are decoded into the respective code word $\mathbf{b}$. According to the radius $\varrho$ of the correction balls, besides the code word $\mathbf{b}$, all words that differ in $1,2, \ldots, \varrho$ components from $\mathbf{b}$ are elements of the corresponding correction ball. We can uniquely decode all elements of a correction ball into the corresponding code word $\mathbf{b}$ as long as the correction balls do not intersect. This condition is true if $\varrho<\frac{d}{2}$ holds. Therefore, the number of correctable errors of a block code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$ is given by (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004; van Lint, 1999) ${ }^{5}$
$$
e_{\mathrm{cor}}=\left\lfloor\frac{d-1}{2}\right\rfloor .
$$

For the binary symmetric channel the number of errors $w$ within the $n$-dimensional transmitted code word is binomially distributed according to $\operatorname{Pr}{w$ errors $}=\left(\begin{array}{c}n \ w\end{array}\right) \varepsilon^{w}(1-$ remaining detection error probability is bounded by
$$
p_{\mathrm{det}} \leq \sum_{w=e_{\mathrm{det}}+1}^{n}\left(\begin{array}{l}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}=1-\sum_{w=0}^{e_{\text {det }}}\left(\begin{array}{l}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}
$$ for a binary symmetric channel can be similarly bounded by
$$
p_{\mathrm{err}} \leq \sum_{w=e_{\mathrm{cor}}+1}^{n}\left(\begin{array}{l}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}=1-\sum_{w=0}^{e_{\mathrm{cor}}}\left(\begin{array}{l}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}
$$

数学代写|编码理论作业代写Coding Theory代考|Definition of Linear Block Codes

The $(n, k)$ block code $\mathbb{B}(n, k, d)$ with minimum Hamming distance $d$ over the finite field $\mathrm{F}{q}$ is called linear, if $\mathrm{B}(n, k, d)$ is a subspace of the vector space $\mathrm{F}{q}^{n}$ of dimension $k$ (Lin and Costello, 2004; Ling and Xing, 2004). The number of code words is then given by
$$
M=q^{k}
$$
according to the code rate
$$
R=\frac{k}{n} .
$$
Because of the linearity property, an arbitrary superposition of code words again leads to a valid code word of the linear block code $\mathrm{B}(n, k, d)$, i.e.
$$
\alpha_{2} \mathbf{b}{1}+\alpha{2} \mathbf{b}{2}+\cdots+\alpha{l} \mathbf{b}{l} \in \mathbb{B}(n, k, d) $$ with $\alpha{1}, \alpha_{2}, \ldots, \alpha_{l} \in \mathbb{F}{q}$ and $\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{l} \in \mathrm{B}(n, k, d)$. Owing to the linearity, the $n$ dimensional zero row vector $\mathbf{0}=(0,0, \ldots, 0)$ consisting of $n$ zeros is always a valid code word. It can be shown that the minimum Hamming distance of a linear block code $\mathrm{B}(n, k, d)$ is equal to the minimum weight of all non-zero code words, i.e.
$$
d=\min {\mathbf{b} \neq \mathbf{b}^{\prime}} \operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=\min {\mathbf{b} \neq \mathbf{0}} \mathbf{w t}(\mathbf{b}) .
$$
These properties are summarised in Figure 2.11. As a simple example of a linear block code, the binary parity-check code is described in Figure $2.12$ (Bossert, 1999).

For each linear block code an equivalent code can be found by rearranging the code word symbols. ${ }^{7}$ This equivalent code is characterised by the same code parameters as the original code, i.e. the equivalent code has the same dimension $k$ and the same minimum Hamming distance $d$.

数学代写|编码理论作业代写Coding Theory代考|Parity-Check Matrix

With the help of the generator matrix $\mathbf{G}=\left(\mathbf{I}{k} \mid \mathbf{A}{k, n-k}\right)$, the following $(n-k) \times n$ matrix $-$ the so-called parity-check matrix – can be defined (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004)
$$
\mathbf{H}=\left(\mathbf{B}{n-k, k} \mid \mathbf{I}{n-k}\right)
$$
with the $(n-k) \times(n-k)$ identity matrix $\mathbf{I}{n-k}$. The $(n-k) \times k$ matrix $\mathbf{B}{n-k, k}$ is given by
$$
\mathbf{B}{n-k, k}=-\mathbf{A}{k, n-k^{*}}^{\mathrm{T}}
$$
For the matrices $\mathbf{G}$ and $\mathbf{H}$ the following property can be derived
$$
\mathbf{H} \mathbf{G}^{\mathrm{T}}=\mathbf{B}{n-k, k}+\mathbf{A}{k, n-k}^{\mathrm{T}}=\mathbf{0}{n-k, k} $$ with the $(n-k) \times k$ zero matrix $\mathbf{0}{n-k, k}$. The generator matrix $\mathbf{G}$ and the parity-check matrix $\mathbf{H}$ are orthogonal, i.e. all row vectors of $\mathbf{G}$ are orthogonal to all row vectors of $\mathbf{H}$.
Using the $n$-dimensional basis vectors $\mathbf{g}{0}, \mathbf{g}{1}, \ldots, \mathbf{g}{k-1}$ and the transpose of the generator matrix $\mathbf{G}^{\mathrm{T}}=\left(\mathrm{g}{0}^{\mathrm{T}}, \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathrm{g}{k-1}^{\mathrm{T}}\right)$, we obtain
$$
\mathbf{H} \mathbf{G}^{\mathrm{T}}=\mathbf{H}\left(\mathrm{g}{0}^{\mathrm{T}}, \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathbf{g}{k-1}^{\mathrm{T}}\right)=\left(\mathbf{H g}{0}^{\mathrm{T}}, \mathbf{H} \mathbf{g}{1}^{\mathrm{T}}, \ldots, \mathbf{H} \mathbf{g}{k-1}^{\mathrm{T}}\right)=(\mathbf{0}, \mathbf{0}, \ldots, \mathbf{0})
$$
with the $(n-k)$-dimensional all-zero column vector $0=(0,0, \ldots, 0)^{\mathrm{T}}$. This is equivalent to $\mathbf{H} \mathbf{g}{i}^{\mathrm{T}}=\mathbf{0}$ for $0 \leq i \leq k-1$. Since each code vector $\mathbf{b} \in \mathbb{B}(n, k, d)$ can be written as $$ \mathbf{b}=\mathbf{u} \mathbf{G}=u{0} \mathbf{g}{0}+u{1} \mathbf{g}{1}+\cdots+u{k-1} \mathbf{g}_{k-1}
$$

编码理论代写

数学代写|编码理论作业代写Coding Theory代考|Error Detection and Error Correction

基于最小距离解码规则和码空间概念，我们可以评估给定信道码的检错和纠错能力。为此，让b和b′是一个的两个代码字(n,ķ)块代码乙(n,ķ,d). 这些码字的距离应等于最小汉明距离，即距离⁡(b,b′)=d. 只要错误接收到的字，我们就能够检测到错误r不等于与发送码字不同的码字。只要错误的数量小于最小汉明距离，就可以保证这种错误检测能力d，因为另一个代码字（例如b′) 可以从给定的代码字（例如 b）仅通过更改至少d组件。为（n,ķ)块代码乙(n,ķ,d)具有最小汉明距离d，因此可检测错误的数量由下式给出（Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004；van Lint，1999）

和这 =d−1.
用于分析的纠错能力(n,ķ)块代码乙(n,ķ,d)我们为每个码字定义b相应的半径修正球ϱ作为更接近码字的所有字的子集b比任何其他代码字b′块代码乙(n,ķ,d)（见图 2.10）。正如我们在上一节中看到的，对于最小距离解码，特定校正球内的所有接收字都被解码为相应的代码字b. 根据半径ϱ校正球，除了代码字b, 所有不同的词1,2,…,ϱ组件来自b是相应修正球的元素。我们可以将一个修正球的所有元素唯一地解码成对应的码字b只要修正球不相交。这个条件为真，如果ϱ<d2持有。因此，块码的可纠正错误数乙(n,ķ,d)具有最小汉明距离d由（Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004；van Lint，1999）给出5

和C这r=⌊d−12⌋.

对于二进制对称通道，错误数在内n-维传输码字按照二项式分布公关⁡在$和rr这rs$=(n 在)e在(1−剩余检测错误概率为

pd和吨≤∑在=和d和吨+1n(n 在)e在(1−e)n−在=1−∑在=0和这 (n 在)e在(1−e)n−在对于二进制对称通道，可以类似地由

p和rr≤∑在=和C这r+1n(n 在)e在(1−e)n−在=1−∑在=0和C这r(n 在)e在(1−e)n−在

数学代写|编码理论作业代写Coding Theory代考|Definition of Linear Block Codes

这(n,ķ)块代码乙(n,ķ,d)具有最小汉明距离d在有限域上Fq称为线性，如果乙(n,ķ,d)是向量空间的子空间Fqn维度的ķ（Lin 和 Costello，2004 年；Ling 和 Xing，2004 年）。代码字的数量由下式给出

米=qķ
根据码率

R=ķn.
由于线性特性，码字的任意叠加再次导致线性块码的有效码字乙(n,ķ,d)， IE

一种2b1+一种2b2+⋯+一种lbl∈乙(n,ķ,d)和一种1,一种2,…,一种l∈Fq和b1,b2,…,bl∈乙(n,ķ,d). 由于线性度，n维零行向量0=(0,0,…,0)包含由…组成nzeros 始终是有效的代码字。可以证明线性分组码的最小汉明距离乙(n,ķ,d)等于所有非零码字的最小权重，即

d=分钟b≠b′距离⁡(b,b′)=分钟b≠0在吨(b).
这些属性总结在图 2.11 中。作为线性块码的一个简单示例，二进制奇偶校验码在图 1 中描述2.12（博塞特，1999）。

对于每个线性块代码，可以通过重新排列代码字符号来找到等效代码。7该等价码的特点是与原码具有相同的码参数，即等价码具有相同的维数ķ和相同的最小汉明距离d.

数学代写|编码理论作业代写Coding Theory代考|Parity-Check Matrix

在生成矩阵的帮助下G=(一世ķ∣一种ķ,n−ķ)，以下(n−ķ)×n矩阵−所谓的奇偶校验矩阵——可以定义（Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004）

H=(乙n−ķ,ķ∣一世n−ķ)
与(n−ķ)×(n−ķ)单位矩阵一世n−ķ. 这(n−ķ)×ķ矩阵乙n−ķ,ķ是（谁）给的

乙n−ķ,ķ=−一种ķ,n−ķ∗吨
对于矩阵G和H可以推导出以下性质

HG吨=乙n−ķ,ķ+一种ķ,n−ķ吨=0n−ķ,ķ与(n−ķ)×ķ零矩阵0n−ķ,ķ. 生成矩阵G和奇偶校验矩阵H是正交的，即所有行向量G正交于所有行向量H.
使用n维基向量G0,G1,…,Gķ−1以及生成矩阵的转置G吨=(G0吨,G1吨,…,Gķ−1吨)，我们获得

HG吨=H(G0吨,G1吨,…,Gķ−1吨)=(HG0吨,HG1吨,…,HGķ−1吨)=(0,0,…,0)
与(n−ķ)维全零列向量0=(0,0,…,0)吨. 这相当于HG一世吨=0为了0≤一世≤ķ−1. 由于每个代码向量b∈乙(n,ķ,d)可以写成

b=在G=在0G0+在1G1+⋯+在ķ−1Gķ−1

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

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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。

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|Code Parameters

Channel codes are characterised by so-called code parameters. The most important code parameters of a general $(n, k)$ block code that are introduced in the following are the code rate and the minimum Hamming distance (Bossert, 1999; Lin and Costello, 2004; Ling and Xing, 2004). With the help of these code parameters, the efficiency of the encoding process and the error detection and error correction capabilities can be evaluated for a given $(n, k)$ block code.

Code Rate
Under the assumption that each information symbol $u_{i}$ of the $(n, k)$ block code can assume $q$ values, the number of possible information words and code words is given by ${ }^{2}$
$$
M=q^{k} .
$$
Since the code word length $n$ is larger than the information word length $k$, the rate at which information is transmitted across the channel is reduced by the so-called code rate
$$
R=\frac{\log {q}(M)}{n}=\frac{k}{n} . $$ For the simple binary triple repetition code with $k=1$ and $n=3$, the code rate is $R=$ $\frac{k}{n}=\frac{1}{3} \approx 0,3333$. Weight and Hamming Distance Each code word $\mathbf{b}=\left(b{0}, b_{1}, \ldots, b_{n-1}\right)$ can be assigned the weight wt(b) which is defined as the number of non-zero components $b_{i} \neq 0$ (Bossert, 1999), i.e. ${ }^{3}$
$$
\operatorname{wt}(\mathbf{b})=\left|\left{i: b_{i} \neq 0,0 \leq i<n\right}\right| .
$$
Accordingly, the distance between two code words $\mathbf{b}=\left(b_{0}, b_{1}, \ldots, b_{n-1}\right)$ and $\mathbf{b}^{\prime}=\left(b_{0}^{\prime}\right.$, $\left.b_{1}^{\prime}, \ldots, b_{n-1}^{\prime}\right)$ is given by the so-called Hamming distance (Bossert, 1999)
$$
\operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=\left|\left{i: b_{i} \neq b_{i}^{\prime}, 0 \leq i<n\right}\right| .
$$
The Hamming distance dist $\left(\mathbf{b}, \mathbf{b}^{\prime}\right)$ provides the number of different components of $\mathbf{b}$ and $\mathbf{b}^{\prime}$ and thus measures how close the code words $\mathbf{b}$ and $\mathbf{b}^{\prime}$ are to each other. For a code $\mathbb{B}$ consisting of $M$ code words $\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{M}$, the minimum Hamming distance is given by $$ d=\min {\mathbf{b} \mathbf{b} \neq \mathbf{b}^{\prime}} \operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right) .
$$
We will denote the $(n, k)$ block code $\mathrm{B}=\left{\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{M}\right}$ with $M=q^{k} q$-nary code words of length $n$ and minimum Hamming distance $d$ by $\mathrm{B}(n, k, d)$. The minimum weight of the block code $\mathrm{B}$ is defined as $\min {\mathbf{b} \neq \mathbf{0}}$ wt(b). The code parameters of $\mathbb{B}(n, k, d)$ are summarised in Figure 2.4.

数学代写|编码理论作业代写Coding Theory代考|Maximum Likelihood Decoding

Channel codes are used in order to decrease the probability of incorrectly received code words or symbols. In this section we will derive a widely used decoding strategy. To this end, we will consider a decoding strategy to be optimal if the corresponding word error probability
$$
p_{\mathrm{err}}=\operatorname{Pr}{\hat{\mathbf{u}} \neq \mathbf{u}}=\operatorname{Pr}{\hat{\mathbf{b}} \neq \mathbf{b}}
$$
is minimal (Bossert, 1999). The word error probability has to be distinguished from the symbol error probability
$$
p_{\mathrm{sym}}=\frac{1}{k} \sum_{i=0}^{k-1} \operatorname{Pr}\left{\hat{u}{i} \neq u{i}\right}
$$

which denotes the probability of an incorrectly decoded information symbol $u_{i}$. In general, the symbol error probability is harder to derive analytically than the word error probability. However, it can be bounded by the following inequality (Bossert, 1999)
$$
\frac{1}{k} p_{\mathrm{err}} \leq p_{\mathrm{sym}} \leq p_{\mathrm{err}} .
$$
In the following, a $q$-nary channel code $\mathrm{B} \in \mathrm{F}{q}^{n}$ with $M$ code words $\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{M}$ in the code space $\mathrm{F}{q}^{n}$ is considered. Let $\mathbf{b}{j}$ be the transmitted code word. Owing to the noisy channel, the received word $\mathbf{r}$ may differ from the transmitted code word $\mathbf{b}{j}$. The task of the decoder in Figure $2.6$ is to decode the transmitted code word based on the sole knowledge of $\mathbf{r}$ with minimal word error probability $p{\text {err }}$.

This decoding step can be written according to the decoding rule $\mathbf{r} \mapsto \hat{\mathbf{b}}=\hat{\mathbf{b}}(\mathbf{r})$. For hard-decision decoding the received word $\mathbf{r}$ is an element of the discrete code space $\mathbb{F}{q}^{n}$. To each code word $\mathbf{b}{j}$ we assign a corresponding subspace $\mathrm{D}{j}$ of the code space $\mathbb{F}{q}^{n}$, the so-called decision region. These non-overlapping decision regions create the whole code space $\mathbb{F}{q}^{n}$, i.e. $\bigcup{j=1}^{M} \mathbb{D}{j}=\mathbb{F}{q}^{n}$ and $\mathbb{D}{i} \cap \mathbb{D}{j}=\emptyset$ for $i \neq j$ as illustrated in Figure 2.7. If the received word $\mathbf{r}$ lies within the decision region $\mathbb{D}{i}$, the decoder decides in favour of the code word $\mathbf{b}{i}$. That is, the decoding of the code word $\mathbf{b}{i}$ according to the decision rule $\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}$ is equivalent to the event $\mathbf{r} \in \mathbb{D}{i}$. By properly choosing the decision regions $\mathbb{D}{i}$, the decoder can be designed. For an optimal decoder the decision regions are chosen such that the word error probability $p_{\mathrm{err}}$ is minimal.

The probability of the event that the code word $\mathbf{b}=\mathbf{b}{j}$ is transmitted and the code word $\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}$ is decoded is given by
$$
\operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} .
$$
We obtain the word error probability $p_{\text {err }}$ by averaging over all possible events for which the transmitted code word $\mathbf{b}=\mathbf{b}{j}$ is decoded into a different code word $\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}$ with

$i \neq j$. This leads to (Neubauer, 2006b)
$$
\begin{aligned}
p_{\mathrm{err}} &=\operatorname{Pr}{\hat{\mathbf{b}}(\mathbf{r}) \neq \mathbf{b}} \
&=\sum_{i=1}^{M} \sum_{j \neq i} \operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \
&=\sum_{i=1}^{M} \sum_{j \neq i} \operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \
&=\sum_{i=1}^{M} \sum_{j \neq i} \sum_{\mathbf{r} \in \mathrm{D}{i}} \operatorname{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right}
\end{aligned}
$$
With the help of Bayes’ rule $\operatorname{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\mathbf{b}=\mathbf{b}{j} \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}}$ and by changing the order of summation, we obtain
$$
\begin{aligned}
p_{e r r} &=\sum_{i=1}^{M} \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \operatorname{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \ &=\sum{i=1}^{M} \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \operatorname{Pr}\left{\mathbf{b}=\mathbf{b}_{j} \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}}
\end{aligned}
$$

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

In Section $1.2 .3$ we defined the binary symmetric channel as a memoryless channel with the conditional probabilities
$$
\operatorname{Pr}\left{r_{i} \mid b_{i}\right}=\left{\begin{array}{cc}
1-\varepsilon, & r_{i}=b_{i} \
\varepsilon, & r_{i} \neq b_{i}
\end{array}\right.
$$
with channel bit error probability $\varepsilon$. Since the binary symmetric channel is assumed to be memoryless, the conditional probability $\operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}$ can be calculated for code word $\mathbf{b}=$

$\left(b_{0}, b_{1}, \ldots, b_{n-1}\right)$ and received word $\mathbf{r}=\left(r_{0}, r_{1}, \ldots, r_{n-1}\right)$ according to
$$
\operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}=\prod_{i=0}^{n-1} \operatorname{Pr}\left{r_{i} \mid b_{i}\right}
$$
If the words $\mathbf{r}$ and $\mathbf{b}$ differ in $\operatorname{dist}(\mathbf{r}, \mathbf{b})$ symbols, this yields
$$
\operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}=(1-\varepsilon)^{n-\operatorname{dist}(\mathbf{r}, \mathbf{b})} \varepsilon^{\operatorname{dist}(\mathbf{r}, \mathbf{b})}=(1-\varepsilon)^{n}\left(\frac{\varepsilon}{1-\varepsilon}\right)^{\operatorname{dist}(\mathbf{r}, \mathbf{b})}
$$
Taking into account $0 \leq \varepsilon<\frac{1}{2}$ and therefore $\frac{\varepsilon}{1-\varepsilon}<1$, the MLD rule is given by
$$
\hat{\mathbf{b}}(\mathbf{r})=\underset{\mathbf{b} \in \mathrm{B}}{\operatorname{argmax}} \operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}=\underset{\mathbf{b} \in \mathrm{B}}{\operatorname{argmax}}(1-\varepsilon)^{n}\left(\frac{\varepsilon}{1-\varepsilon}\right)^{\operatorname{dist}(\mathbf{r}, \mathbf{b})}=\underset{\mathbf{b} \in \mathrm{B}}{\operatorname{argmin}} \operatorname{dist}(\mathbf{r}, \mathbf{b}),
$$
i.e. for the binary symmetric channel the optimal maximum likelihood decoder (Bossert, 1999)
$$
\hat{\mathbf{b}}(\mathbf{r})=\underset{\mathbf{b} \in \mathbb{B}}{\operatorname{argmin}} \operatorname{dist}(\mathbf{r}, \mathbf{b})
$$
emits that particular code word which differs in the smallest number of components from the received word $\mathbf{r}$, i.e. which has the smallest Hamming distance to the received word $\mathbf{r}$ (see Figure 2.9). This decoding rule is called minimum distance decoding. This minimum distance decoding rule is also optimal for a $q$-nary symmetric channel (Neubauer, 2006b). We now turn to the error probabilities for the binary symmetric channel during transmission before decoding. The probability of $w$ errors at $w$ given positions within the $n$-dimensional binary received word $\mathbf{r}$ is given by $\varepsilon^{w}(1-\varepsilon)^{n-w}$. Since there are $\left(\begin{array}{l}n \ w\end{array}\right)$ different possibilities

of choosing $w$ out of $n$ positions, the probability of $w$ errors at arbitrary positions within an $n$-dimensional binary received word follows the binomial distribution
$$
\operatorname{Pr}{w \text { errors }}=\left(\begin{array}{c}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}
$$
with mean $n \varepsilon$. Because of the condition $\varepsilon<\frac{1}{2}$, the probability $\operatorname{Pr}{w$ errors $}$ decreases with increasing number of errors $w$, i.e. few errors are more likely than many errors.

The probability of error-free transmission is $\operatorname{Pr}{0$ errors $}=(1-\varepsilon)^{n}$, whereas the probability of a disturbed transmission with $\mathbf{r} \neq \mathbf{b}$ is given by
$$
\operatorname{Pr}{\mathbf{r} \neq \mathbf{b}}=\sum_{w=1}^{n}\left(\begin{array}{l}
n \
w
\end{array}\right) \varepsilon^{w}(1-\varepsilon)^{n-w}=1-(1-\varepsilon)^{n} .
$$

编码理论代写

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

信道代码的特征在于所谓的代码参数。一般最重要的代码参数(n,ķ)下面介绍的块码是码率和最小汉明距离（Bossert，1999；Lin 和 Costello，2004；Ling 和 Xing，2004）。借助这些代码参数，可以评估给定编码过程的效率以及错误检测和纠错能力(n,ķ)块代码。

码率
假设每个信息符号在一世的(n,ķ)块代码可以假设q值，可能的信息字和代码字的数量由下式给出2

米=qķ.
由于码字长n大于信息字长ķ，通过信道传输信息的速率被所谓的码率降低

R=日志⁡q(米)n=ķn.对于简单的二进制三重重复代码ķ=1和n=3，码率为R= ķn=13≈0,3333. 权重和汉明距离每个码字b=(b0,b1,…,bn−1)可以分配权重 wt(b)，定义为非零分量的数量b一世≠0（博塞特，1999），即3

\operatorname{wt}(\mathbf{b})=\left|\left{i: b_{i} \neq 0,0 \leq i<n\right}\right| .\operatorname{wt}(\mathbf{b})=\left|\left{i: b_{i} \neq 0,0 \leq i<n\right}\right| .
因此，两个码字之间的距离b=(b0,b1,…,bn−1)和b′=(b0′, b1′,…,bn−1′)由所谓的汉明距离给出 (Bossert, 1999)

\operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=\left|\left{i: b_{i} \neq b_{i}^{\prime }, 0 \leq i<n\right}\right| .\operatorname{dist}\left(\mathbf{b}, \mathbf{b}^{\prime}\right)=\left|\left{i: b_{i} \neq b_{i}^{\prime }, 0 \leq i<n\right}\right| .
汉明距离 dist(b,b′)提供不同组件的数量b和b′从而测量代码字的接近程度b和b′是彼此的。对于一个代码乙包含由…组成米暗语b1,b2,…,b米，最小汉明距离由下式给出

d=分钟bb≠b′距离⁡(b,b′).
我们将表示(n,ķ)块代码\mathrm{B}=\left{\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{M}\right}\mathrm{B}=\left{\mathbf{b}{1}, \mathbf{b}{2}, \ldots, \mathbf{b}{M}\right}和米=qķq-nary码字长度n和最小汉明距离d经过乙(n,ķ,d). 块码的最小权重乙定义为分钟b≠0重量（b）。代码参数乙(n,ķ,d)总结在图 2.4 中。

数学代写|编码理论作业代写Coding Theory代考|Maximum Likelihood Decoding

使用信道代码是为了降低错误接收代码字或符号的概率。在本节中，我们将推导出一种广泛使用的解码策略。为此，我们将考虑一个解码策略是最优的，如果相应的单词错误概率

p和rr=公关⁡在^≠在=公关⁡b^≠b
是最小的（Bossert，1999）。字错误概率必须与符号错误概率区分开来

p_{\mathrm{sym}}=\frac{1}{k} \sum_{i=0}^{k-1} \operatorname{Pr}\left{\hat{u}{i} \neq u{我}\右}p_{\mathrm{sym}}=\frac{1}{k} \sum_{i=0}^{k-1} \operatorname{Pr}\left{\hat{u}{i} \neq u{我}\右}

表示错误解码的信息符号的概率在一世. 一般来说，符号错误概率比字错误概率更难解析推导。然而，它可以受到以下不等式的限制（Bossert，1999）

1ķp和rr≤ps是米≤p和rr.
在下文中，一个q-nary 频道代码乙∈Fqn和米暗语b1,b2,…,b米在代码空间Fqn被认为。让bj是传输的码字。由于信道嘈杂，接收到的词r可能与传输的代码字不同bj. 图中解码器的任务2.6是根据唯一的知识对传输的码字进行解码r具有最小的单词错误概率p呃 .

这个解码步骤可以根据解码规则来写r↦b^=b^(r). 对于接收到的字进行硬决策解码r是离散码空间的一个元素Fqn. 对每个码字bj我们分配一个相应的子空间Dj代码空间Fqn，即所谓的决策区域。这些不重叠的决策区域创建了整个代码空间Fqn， IE⋃j=1米Dj=Fqn和D一世∩Dj=∅为了一世≠j如图 2.7 所示。如果收到的话r位于决策区域内D一世，解码器决定支持码字b一世. 也就是码字的解码b一世根据决策规则b^(r)=b一世相当于事件r∈D一世. 通过正确选择决策区域D一世，可以设计解码器。对于最佳解码器，选择决策区域使得单词错误概率p和rr是最小的。

码字发生事件的概率b=bj被传输和代码字b^(r)=b一世被解码由下式给出

\operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}\right) \wedge\left(\mathbf{b}= \mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\ mathbf{b}=\mathbf{b}{j}\right)\right} 。\operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b}{i}\right) \wedge\left(\mathbf{b}= \mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\ mathbf{b}=\mathbf{b}{j}\right)\right} 。
我们得到单词错误概率p呃通过对传输代码字的所有可能事件进行平均b=bj被解码成不同的码字b^(r)=b一世和

一世≠j. 这导致（纽鲍尔，2006b）

\begin{aligned} p_{\mathrm{err}} &=\operatorname{Pr}{\hat{\mathbf{b}}(\mathbf{r}) \neq \mathbf{b}} \ &=\sum_ {i=1}^{M} \sum_{j \neq i} \operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b} {i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \ &=\sum_{i=1}^{M} \sum_{j \ neq i} \operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j }\right)\right} \ &=\sum_{i=1}^{M} \sum_{j \neq i} \sum_{\mathbf{r} \in \mathrm{D}{i}} \operatorname {Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \end{对齐}\begin{aligned} p_{\mathrm{err}} &=\operatorname{Pr}{\hat{\mathbf{b}}(\mathbf{r}) \neq \mathbf{b}} \ &=\sum_ {i=1}^{M} \sum_{j \neq i} \operatorname{Pr}\left{\left(\hat{\mathbf{b}}(\mathbf{r})=\mathbf{b} {i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \ &=\sum_{i=1}^{M} \sum_{j \ neq i} \operatorname{Pr}\left{\left(\mathbf{r} \in \mathbb{D}{i}\right) \wedge\left(\mathbf{b}=\mathbf{b}{j }\right)\right} \ &=\sum_{i=1}^{M} \sum_{j \neq i} \sum_{\mathbf{r} \in \mathrm{D}{i}} \operatorname {Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \end{对齐}
借助贝叶斯法则\operatorname{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\mathbf{ b}=\mathbf{b}{j} \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}}\operatorname{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right}=\operatorname{Pr}\left{\mathbf{ b}=\mathbf{b}{j} \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}}并且通过改变求和的顺序，我们得到

\begin{对齐} p_{e r r} &=\sum_{i=1}^{M} \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \运算符名{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \ &=\sum{i=1}^{M } \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \operatorname{Pr}\left{\mathbf{b}=\mathbf{b}_{j } \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}} \end{aligned}\begin{对齐} p_{e r r} &=\sum_{i=1}^{M} \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \运算符名{Pr}\left{\mathbf{r} \wedge\left(\mathbf{b}=\mathbf{b}{j}\right)\right} \ &=\sum{i=1}^{M } \sum_{\mathbf{r} \in \mathrm{D}{i}} \sum{j \neq i} \operatorname{Pr}\left{\mathbf{b}=\mathbf{b}_{j } \mid \mathbf{r}\right} \operatorname{Pr}{\mathbf{r}} \end{aligned}

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

在部分1.2.3我们将二进制对称通道定义为无记忆通道，条件概率为
$$
\operatorname{Pr}\left{r_{i} \mid b_{i}\right}=\left{

1−e,r一世=b一世 e,r一世≠b一世\正确的。
$$
与信道误码概率e. 由于假设二进制对称信道是无记忆的，因此条件概率公关⁡r∣b可以计算码字b=

(b0,b1,…,bn−1)并收到消息r=(r0,r1,…,rn−1)根据

\operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}=\prod_{i=0}^{n-1} \operatorname{Pr}\left{r_{i} \mid b_{i }\正确的}\operatorname{Pr}{\mathbf{r} \mid \mathbf{b}}=\prod_{i=0}^{n-1} \operatorname{Pr}\left{r_{i} \mid b_{i }\正确的}
如果的话r和b不同之处距离⁡(r,b)符号，这会产生

公关⁡r∣b=(1−e)n−距离⁡(r,b)e距离⁡(r,b)=(1−e)n(e1−e)距离⁡(r,b)
考虑在内0≤e<12因此e1−e<1, MLD 规则由下式给出

b^(r)=最大参数b∈乙公关⁡r∣b=最大参数b∈乙(1−e)n(e1−e)距离⁡(r,b)=精氨酸b∈乙距离⁡(r,b),
即对于二进制对称信道，最佳最大似然解码器（Bossert，1999）

b^(r)=精氨酸b∈乙距离⁡(r,b)
发出特定的代码字，该代码字与接收到的字的分量数量最少r，即与接收到的词的汉明距离最小r（见图 2.9）。这种解码规则称为最小距离解码。这个最小距离解码规则对于q-nary 对称通道（Neubauer，2006b）。我们现在转向在解码之前传输期间二进制对称信道的错误概率。的概率在错误在在在给定的位置n维二进制接收字r是（谁）给的e在(1−e)n−在. 既然有(n 在)不同的可能性

的选择在在……之外n位置，概率在内任意位置的误差n维二进制接收字服从二项分布

公关⁡在错误 =(n 在)e在(1−e)n−在
平均ne. 因为条件e<12, 概率公关⁡在$和rr这rs$随着错误数量的增加而减少在，即少数错误比很多错误更可能发生。

无差错传输的概率为公关⁡0$和rr这rs$=(1−e)n，而干扰传输的概率与r≠b是（谁）给的

公关⁡r≠b=∑在=1n(n 在)e在(1−e)n−在=1−(1−e)n.

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

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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。

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

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
$$
S=\mathrm{E}\left{\mathcal{X}^{2}\right}
$$
and the noise power
$$
N=\mathrm{E}\left{Z^{2}\right}
$$
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
$$
000000111000111111111000
$$
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.

编码理论代写

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.
借助信号电源

S=\mathrm{E}\left{\mathcal{X}^{2}\right}S=\mathrm{E}\left{\mathcal{X}^{2}\right}
和噪声功率

N=\mathrm{E}\left{Z^{2}\right}N=\mathrm{E}\left{Z^{2}\right}
AWGN 信道的信道容量由下式给出

C=12日志2⁡(1+小号ñ)
信道容量完全取决于信噪比小号/ñ.
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

R匹配的过滤器由下式给出

小号ñ=和bñ0/2
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

e=12erfc⁡(和bñ0)
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）。

尽管信息论告诉我们，理论上可以找到对于给定信道导致错误概率尽可能小的信道码，但通常很难设计出好的信道码。因此，在接下来的章节中将描述几类信道代码。在这里，我们从一个简单的例子开始。

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

作为简单信道代码的介绍性示例，我们考虑二进制信息序列
$$
00 1 0 110的传输

这在和r一个b一世n一个r是s是米米和吨r一世CCH一个nn和l在一世吨Hb一世吨和rr这rpr这b一个b一世l一世吨是$e=0.25$(ñ和在b一个在和r,2006b).这n一个在和r一个G和,和在和r是F这在r吨Hb一世n一个r是s是米b这l在一世llb和r和C和一世在和d一世nC这rr和C吨l是.一世n吨H一世s和X一个米pl和在和一个ss在米和吨H一个吨吨H和b一世n一个r是s和q在和nC和
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。

对于上面给出的二进制信息序列，我们得到二进制码序列

000000111000111111111000
在编码器的输出端。如果我们再次假设平均每四个二进制符号被二进制对称信道错误地传输，我们可以获得接收序列
$$
0 0 000 0 110 10 111 0 1 0 1110 10。
$$
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

在本章中，我们将介绍代数编码理论的基本概念。为此，首先讨论分组码的基本特性。我们将定义重要的代码参数并描述如何使用这些代码进行错误检测和纠错。将推导出最优最大似然解码策略并将其应用于二进制对称信道。

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。

虽然对于线性分组码可以有效地进行代码生成，但一般线性分组码的解码问题难以解决。通过引入进一步的代数结构，循环码可以作为线性块码的子类推导出来，其中存在有效的代数解码算法。类似于使用生成矩阵或奇偶校验矩阵定义的一般线性块码，循环码是在所谓的生成多项式或奇偶校验多项式的帮助下定义的。基于线性反馈移位寄存器，可以有效地实现循环码各自的编解码架构。作为循环码的重要例子，我们将讨论乙CH码和里德-所罗门码。此外，还提出了一种代数解码算法，可用于BCH和Reed-Solomon码的解码。

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。

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

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
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数学代写|编码理论作业代写Coding Theory代考| Communication Systems

Posted on 2022年5月28日2022年5月28日 by statistics-lab

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Statistical Inference 统计推断
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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|Communication Systems

The reliable transmission of information over noisy channels is one of the basic requirements of digital information and communication systems. Here, transmission is understood both as transmission in space, e.g. over mobile radio channels, and as transmission in time by storing information in appropriate storage media. Because of this requirement, modern communication systems rely heavily on powerful channel coding methodologies. For practical applications these coding schemes do not only need to have good coding characteristics with respect to the capability of detecting or correcting errors introduced on the channel. They also have to be efficiently implementable, e.g. in digital hardware within integrated circuits. Practical applications of channel codes include space and satellite communications, data transmission, digital audio and video broadcasting and mobile communications, as well as storage systems such as computer memories or the compact disc (Costello et al., 1998).

In this introductory chapter we will give a brief introduction into the field of channel coding. To this end, we will describe the information theory fundamentals of channel coding. Simple channel models will be presented that will be used throughout the text. Furthermore, we will present the binary triple repetition code as an illustrative example of a simple channel code.

According to Figure $1.1$ the modulator generates the signal that is used to transmit the sequence of symbols $\mathbf{b}$ across the channel (Benedetto and Biglieri, 1999; Neubauer, 2007; Proakis, 2001). Due to the noisy nature of the channel, the transmitted signal is disturbed. The noisy received signal is demodulated by the demodulator in the receiver, leading to the sequence of received symbols $\mathbf{r}$. Since the received symbol sequence $\mathbf{r}$ usually differs from the transmitted symbol sequence $\mathbf{b}$, a channel code is used such that the receiver is able to detect or even correct errors (Bossert, 1999; Lin and Costello, 2004; Neubauer, 2006b). To this end, the channel encoder introduces redundancy into the information sequence $\mathbf{u}$. This redundancy can be exploited by the channel decoder for error detection or error correction by estimating the transmitted symbol sequence $\hat{\mathbf{u}}$.

In his fundamental work, Shannon showed that it is theoretically possible to realise an information transmission system with as small an error probability as required (Shannon, 1948). The prerequisite for this is that the information rate of the information source be smaller than the so-called channel capacity. In order to reduce the information rate, source coding schemes are used which are implemented by the source encoder in the transmitter and the source decoder in the receiver (McEliece, 2002; Neubauer, 2006a).

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

With the help of the entropy concept we can model a channel according to Berger’s channel diagram shown in Figure $1.3$ (Neubauer, 2006a). Here, $\mathcal{X}$ refers to the input symbol and $\mathcal{R}$ denotes the output symbol or received symbol. We now assume that $M$ input symbol values $x_{1}, x_{2}, \ldots, x_{M}$ and $N$ output symbol values $r_{1}, r_{2}, \ldots, r_{N}$ are possible. With the help of the conditional probabilities
$$
P_{\mathcal{X} \mid \mathcal{R}}\left(x_{i} \mid r_{j}\right)=\operatorname{Pr}\left{\mathcal{X}=x_{i} \mid \mathcal{R}=r_{j}\right}
$$
and
$$
P_{\mathcal{R} \mid \mathcal{X}}\left(r_{j} \mid x_{i}\right)=\operatorname{Pr}\left{\mathcal{R}=r_{j} \mid \mathcal{X}=x_{i}\right}
$$
the conditional entropies are given by
$$
I(\mathcal{X} \mid \mathcal{R})=-\sum_{i=1}^{M} \sum_{j=1}^{N} P_{\mathcal{X}, \mathcal{R}}\left(x_{i}, r_{j}\right) \cdot \log {2}\left(P{\mathcal{X} \mid \mathcal{R}}\left(x_{i} \mid r_{j}\right)\right)
$$
and
$$
I(\mathcal{R} \mid \mathcal{X})=-\sum_{i=1}^{M} \sum_{j=1}^{N} P_{\mathcal{X}, \mathcal{R}}\left(x_{i}, r_{j}\right) \cdot \log {2}\left(P{\mathcal{R} \mid \mathcal{X}}\left(r_{j} \mid x_{i}\right)\right) .
$$
With these conditional probabilities the mutual information
$$
I(\mathcal{X} ; \mathcal{R})=I(\mathcal{X})-I(\mathcal{X} \mid \mathcal{R})=I(\mathcal{R})-I(\mathcal{R} \mid \mathcal{X})
$$
can be derived which measures the amount of information that is transmitted across the channel from the input to the output for a given information source.

The so-called channel capacity $C$ is obtained by maximising the mutual information $I(\mathcal{X} ; \mathcal{R})$ with respect to the statistical properties of the input $\mathcal{X}$, i.e. by appropriately choosing the probabilities $\left{P_{\mathcal{X}}\left(x_{i}\right)\right}_{1 \leq i \leq M}$. This leads to
$$
C=\max {\left{\left.P{\mathcal{X}}\left(x_{i}\right)\right|_{1 \leq i \leq M}\right.} I(\mathcal{X} ; \mathcal{R}) .
$$
If the input entropy $I(\mathcal{X})$ is smaller than the channel capacity $C$
$$
I(\mathcal{X}) \stackrel{!}{<} C
$$
then information can be transmitted across the noisy channel with arbitrarily small error probability. Thus, the channel capacity $C$ in fact quantifies the information transmission capacity of the channel.

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

As an important example of a memoryless channel we turn to the binary symmetric channel or BSC. Figure $1.4$ shows the channel diagram of the binary symmetric channel with bit error probability $\varepsilon$. This channel transmits the binary symbol $\mathcal{X}=0$ or $\mathcal{X}=1$ correctly with probability $1-\varepsilon$, whereas the incorrect binary symbol $\mathcal{R}=1$ or $\mathcal{R}=0$ is emitted with probability $\varepsilon$.

By maximising the mutual information $I(\mathcal{X} ; \mathcal{R})$, the channel capacity of a binary symmetric channel is obtained according to
$$
C=1+\varepsilon \log {2}(\varepsilon)+(1-\varepsilon) \log {2}(1-\varepsilon)
$$
This channel capacity is equal to 1 if $\varepsilon=0$ or $\varepsilon=1$; for $\varepsilon=\frac{1}{2}$ the channel capacity is 0 . In contrast to the binary symmetric channel, which has discrete input and output symbols taken from binary alphabets, the so-called AWGN channel is defined on the basis of continuous real-valued random variables. ${ }^{1}$

编码理论代写

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

在噪声信道上可靠地传输信息是数字信息和通信系统的基本要求之一。这里，传输被理解为空间传输，例如通过移动无线电信道，以及通过将信息存储在适当的存储介质中的时间传输。由于这一要求，现代通信系统严重依赖强大的信道编码方法。对于实际应用，这些编码方案不仅需要在检测或纠正信道上引入的错误的能力方面具有良好的编码特性。它们还必须是可有效实现的，例如在集成电路内的数字硬件中。信道码的实际应用包括空间和卫星通信、数据传输、

在这个介绍性章节中，我们将简要介绍信道编码领域。为此，我们将描述信道编码的信息论基础。将介绍将在整个文本中使用的简单通道模型。此外，我们将呈现二进制三重重复代码作为简单信道代码的说明性示例。

根据图1.1调制器生成用于传输符号序列的信号b跨越海峡（Benedetto 和 Biglieri，1999；Neubauer，2007；Proakis，2001）。由于信道的噪声特性，传输的信号受到干扰。接收到的噪声信号由接收器中的解调器解调，导致接收符号序列r. 由于接收到的符号序列r通常不同于传输的符号序列b，使用信道代码，以便接收器能够检测甚至纠正错误（Bossert，1999；Lin 和 Costello，2004；Neubauer，2006b）。为此，信道编码器在信息序列中引入了冗余在. 信道解码器可以通过估计传输的符号序列来利用这种冗余进行错误检测或纠错在^.

在他的基础工作中，Shannon 表明理论上可以实现一个错误概率尽可能小的信息传输系统（Shannon，1948 年）。其前提是信息源的信息速率小于所谓的信道容量。为了降低信息速率，使用了源编码方案，这些方案由发射器中的源编码器和接收器中的源解码器实现（McEliece，2002；Neubauer，2006a）。

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

借助熵概念，我们可以根据 Berger 的通道图对通道进行建模，如图1.3（纽鲍尔，2006a）。这里，X指输入符号和R表示输出符号或接收符号。我们现在假设米输入符号值X1,X2,…,X米和ñ输出符号值r1,r2,…,rñ是可能的。在条件概率的帮助下

P_{\mathcal{X} \mid \mathcal{R}}\left(x_{i} \mid r_{j}\right)=\operatorname{Pr}\left{\mathcal{X}=x_{i} \mid \mathcal{R}=r_{j}\right}P_{\mathcal{X} \mid \mathcal{R}}\left(x_{i} \mid r_{j}\right)=\operatorname{Pr}\left{\mathcal{X}=x_{i} \mid \mathcal{R}=r_{j}\right}
和

P_{\mathcal{R} \mid \mathcal{X}}\left(r_{j} \mid x_{i}\right)=\operatorname{Pr}\left{\mathcal{R}=r_{j} \mid \mathcal{X}=x_{i}\right}P_{\mathcal{R} \mid \mathcal{X}}\left(r_{j} \mid x_{i}\right)=\operatorname{Pr}\left{\mathcal{R}=r_{j} \mid \mathcal{X}=x_{i}\right}
条件熵由下式给出

一世(X∣R)=−∑一世=1米∑j=1ñ磷X,R(X一世,rj)⋅日志⁡2(磷X∣R(X一世∣rj))
和

一世(R∣X)=−∑一世=1米∑j=1ñ磷X,R(X一世,rj)⋅日志⁡2(磷R∣X(rj∣X一世)).
有了这些条件概率，互信息

一世(X;R)=一世(X)−一世(X∣R)=一世(R)−一世(R∣X)
对于给定的信息源，可以推导出它测量通过通道从输入到输出传输的信息量。

所谓通道容量C通过最大化互信息获得一世(X;R)关于输入的统计属性X，即通过适当地选择概率\left{P_{\mathcal{X}}\left(x_{i}\right)\right}_{1 \leq i \leq M}\left{P_{\mathcal{X}}\left(x_{i}\right)\right}_{1 \leq i \leq M}. 这导致
$$
C=\max {\left{\left.P{\mathcal{X}}\left(x_{i}\right)\right|_{1 \leq i \leq M}\right。 } I(\mathcal{X} ; \mathcal{R}) 。

一世F吨H和一世np在吨和n吨r这p是$一世(X)$一世ss米一个ll和r吨H一个n吨H和CH一个nn和lC一个p一个C一世吨是$C$
I(\mathcal{X}) \stackrel{!}{<} C
$$
那么信息可以以任意小的错误概率通过噪声信道传输。因此，信道容量C实际上量化了信道的信息传输能力。

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

作为无记忆信道的一个重要例子，我们转向二进制对称信道或 BSC。数字1.4显示了具有误码概率的二进制对称信道的信道图e. 该通道传输二进制符号X=0或者X=1正确概率1−e, 而不正确的二进制符号R=1或者R=0以概率发射e.

通过最大化互信息一世(X;R), 二元对称信道的信道容量由下式得到

C=1+e日志⁡2(e)+(1−e)日志⁡2(1−e)
如果这个信道容量等于 1e=0或者e=1; 为了e=12信道容量为 0 。与具有取自二进制字母表的离散输入和输出符号的二进制对称信道相比，所谓的 AWGN 信道是在连续实值随机变量的基础上定义的。1

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