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# 数学代写|信息论代写Information Theory代考|TELE9754

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## 数学代写|信息论代写Information Theory代考|The bit in communication

Originally, a bit was an abbreviation for “binary unit.” In a decimal number we use 10 digits, or 10 symbols: $0,1,2,3,4,5,6,7,8,9$. When we write a decimal number (base 10 ) such as 25 , we mean $25=5+20=5 \times 10^0+2 \times 10^1$, and similarly $256=6+50+200=6 \times 10^0+5 \times 10^1+2 \times 10^2$. In binary numbers we express a number with only two digits; 0 and 1 . In this case we use the powers of 2 , instead of powers of ten.

In computation and telecommunication, we use the binary language which essentially is a two-symbol or two-letter language. These could be “Yes” and “No, ” 0 ” and “1,” or a magnet “up” and “down,” etc. When we communicate a message in English we first translate each letter into a code, and transmit the encoded message to the receiving terminal, where it is decoded back into English letters.

In the process of transmission of information, we are usually interested in achieving a highest accuracy (or faithfulness) of transmission of a given information (in spite of noise), at a lowest cost.

One such code used in telegraph is the Morse code, Table 1.1, which uses a sequence of dots and dashes as code-word for representing the different letters. Clearly, the “length” of a dot, is shorter than that of a dash, let’s say a dot takes a unit of time to transmit, then the dash will take three units. Clearly, to transmit a given text we would like to have a shorter code-word for the more frequent letters, and longer code-word for the less frequent letters. This is roughly how the Morse code was constructed. As you can see from the table, this proportionality is not always true in the Morse code.

Another code known as ASCII (American Standard Code for Information Interchange) is used in electronic communication.

Each of the digit, 0 or 1 is referred to as a bit. Thus, when we send a sequence of bits such as: $1,0,1,1,0,1,0,1,1,0$, we say that we sent 10 bits. This is equivalent to saying that we sent ten symbols, which happen to be zeros and ones. Sometimes, it is said that we sent ten bits of information. This is true only when we use a binary digit as a “unit” of information. Another meaning of the “bit,” as a unit of information, follows.

## 数学代写|信息论代写Information Theory代考|The bit as a unit of information

We begin with the statement that the “bit” in IT is a measure of information, and it is not the same as the “bit” in “binary digit.”

The definition of the bit in IT arose from Shannon’s measure of information (SMI) when applied to the case of two outcomes, Shannon [2]. If an experiment, or a random variable (rv), has only two possible outcomes, say, 1 and 2 , with probabilities $p_1$ and $p_2$, the corresponding SMI is:
$$H\left(p_1, p_2\right)=-p_1 \log p_2-p_2 \log p_2$$
Since $p_1=1-p_2$ we have in fact, a one-parameter function. Setting $p_1=$ $p$, and $p_2=1-p$, we rewrite (1.1) as:

$$H(p)=-p \log p-(1-p) \log (1-p)$$
Here, $\log$ is the logarithm with base 2. The function $H(p)$ is shown in Fig. 1.1. Note that this function has a single maximum at $p_{\max }=\frac{1}{2}$, and the corresponding value of $H(p)$ is: $H\left(p_{\max }\right)=1$ (when base 2 is used).

The interpretation of $H\left(p_{\max }=\frac{1}{2}\right)=1$ as a unit of information follows from the interpretation of the SMI in the general case of $n$ outcomes as the amount of information associated with a probability distribution. In the case of the two outcomes the unit of information is called the bit. This is the amount of information one gets when asking a binary question about two outcomes having equal probabilities.

The emphasis on “equal probabilities” is important. It is very common to find statements, in popular science books referring to a “bit” as the amount of information one gets to a binary question. To see why this is not true consider the following examples of coins with different probability distributions.
Suppose we have four coins with distributions given in Table 1.2.
Now, suppose you play the following game ten times: You are given the distribution as in Table 1.2 and you have to guess the outcome of throwing the coin. You can ask either “is the outcome H?” or “is the outcome T?” And you can get either a Yes, or No answer.

To make the game more dramatic let us add that you get a dollar when you get an answer Yes, and get nothing when the answer is No. Clearly, your interest is to ask questions such that you will earn maximum dollars.

## 数学代写|信息论代写Information Theory代考|The bit as a unit of information

IT中比特的定义源于Shannon的信息度量(SMI)，当它应用于两个结果的情况下，Shannon[2]。如果一个实验或随机变量(rv)只有两种可能的结果，例如1和2，概率分别为$p_1$和$p_2$，则相应的SMI为:
$$H\left(p_1, p_2\right)=-p_1 \log p_2-p_2 \log p_2$$

## 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。