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# 数学代写|信息论代写Information Theory代考|CPSC530

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## 数学代写|信息论代写Information Theory代考|ARITHMETIC CODING

We could alleviate this loss by using blocks of input symbols – however, the complexity of this approach increases exponentially with block length. We now describe a method of encoding without this inefficiency. In arithmetic coding, instead of using a sequence of bits to represent a symbol, we represent it by a subinterval of the unit interval.

The code for a sequence of symbols is an interval whose length decreases as we add more symbols to the sequence. This property allows us to have a coding scheme that is incremental (the code for an extension to a sequence can be calculated simply from the code for the original sequence) and for which the codeword lengths are not restricted to be integral. The motivation for arithmetic coding is based on Shannon-Fano-Elias coding (Section 5.9) and the following lemma:

Lemma 13.3.1 Let $Y$ be a random variable with continuous probability distribution function $F(y)$. Let $U=F(Y)$ (i.e., $U$ is a function of $Y$ defined by its distribution function). Then $U$ is uniformly distributed on $[0,1]$.
Proof: Since $F(y) \in[0,1]$, the range of $U$ is $[0,1]$. Also, for $u \in[0,1]$,
\begin{aligned} F_U(u) & =\operatorname{Pr}(U \leq u) \ & =\operatorname{Pr}(F(Y) \leq u) \ & =\operatorname{Pr}\left(Y \leq F^{-1}(u)\right) \ & =F\left(F^{-1}(u)\right) \ & =u, \end{aligned}
which proves that $U$ has a uniform distribution in $[0,1]$.

## 数学代写|信息论代写Information Theory代考|LEMPEL-ZIV CODING

In Section 13.3 we discussed the basic ideas of arithmetic coding and mentioned some results on worst-case redundancy for coding a sequence from an unknown distribution. We now discuss a popular class of techniques for source coding that are universally optimal (their asymptotic compression rate approaches the entropy rate of the source for any stationary ergodic source) and simple to implement. This class of algorithms is termed Lempel-Ziv, named after the authors of two seminal papers $[603,604]$ that describe the two basic algorithms that underlie this class. The algorithms could also be described as adaptive dictionary compression algorithms.

The notion of using dictionaries for compression dates back to the invention of the telegraph. At the time, companies were charged by the number of letters used, and many large companies produced codebooks for the frequently used phrases and used the codewords for their telegraphic communication. Another example is the notion of greetings telegrams that are popular in India-there is a set of standard greetings such as “25:Merry Christmas” and “26:May Heaven’s choicest blessings be showered on the newly married couple.” A person wishing to send a greeting only needs to specify the number, which is used to generate the actual greeting at the destination.

The idea of adaptive dictionary-based schemes was not explored until Ziv and Lempel wrote their papers in 1977 and 1978. The two papers describe two distinct versions of the algorithm. We refer to these versions as LZ77 or sliding window Lempel-Ziv and LZ78 or tree-structured Lempel-Ziv. (They are sometimes called LZ1 and LZ2, respectively.)
We first describe the basic algorithms in the two cases and describe some simple variations. We later prove their optimality, and end with some practical issues. The key idea of the Lempel-Ziv algorithm is to parse the string into phrases and to replace phrases by pointers to where the same string has occurred in the past. The differences between the algorithms is based on differences in the set of possible match locations (and match lengths) the algorithm allows.

## 数学代写|信息论代写Information Theory代考|ARITHMETIC CODING

\begin{aligned} F_U(u) & =\operatorname{Pr}(U \leq u) \ & =\operatorname{Pr}(F(Y) \leq u) \ & =\operatorname{Pr}\left(Y \leq F^{-1}(u)\right) \ & =F\left(F^{-1}(u)\right) \ & =u, \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。