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数学代写|信息论代写Information Theory代考|INFM130 Principles of information transmission and information reception in the presence of noise

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数学代写|信息论代写Information Theory代考|Principles of information transmission and information reception in the presence of noise

Consider some communication channel. We denote its input variable (at a selected time moment), which we call a transmitted character or letter, as $x$. It can assume discrete values from some set $X$. It is convenient to suppose that probabilities $P(x)$ are also given. Then $x$ will serve as a given random variable.

We consider a noisy channel. This means that for a fixed value of $x$ a variable on a channel output (at a fixed time moment) is random, i.e. it is described by conditional probabilities $P(y \mid x)$. Random variable $y$ can be called a received character or letter.
It is assumed that a process of the letter $x$ transmission and the letter $y$ reception can occur many times with the same probabilities $P(x), P(y \mid x)$ (although a generalization to the case of varied probabilities is possible, see Theorem 7.2). Let $n$ letters constitute a block or word, for instance,
$$\xi=\left(x_{1}, \ldots, x_{n}\right), \quad \eta=\left(y_{1}, \ldots, y_{n}\right) \quad\left(\xi \in X^{n}, \eta \in Y^{n}\right) .$$
If we desire to transmit some messages through a channel, we have to associate these messages with input words with the help of some code. Then a recipient on the receiving end will read the received word and try to restore the transmitted message using the code he knows.

Since there is noise in the channel, it is possible that the recipient makes a mistake and receives a different message (not the one having been transmitted). The code should be selected in such a way that the probability of a similar mistake is as small as possible. The next questions are of fundamental and practical interest: what can we achieve if we choose good codes, which codes are good and how to choose them?
The case of simple disturbances (noise) is especially clear (see Section 6.1). In this case a transmitted message apparently needs to be connected with this or that region-subset $E_{k}$ of values $x$ and thereby region $G_{k}$ of values $y$. For simple noise when transmitting letter $x$ from region $E_{k}$, the received letter $y$ definitely belongs to region $G_{k}$. Therefore, if a message is confronted with regions $E_{k}$ and $G_{k}$ (or, equivalently, with their index $k$ ), then a message reception will be errorless despite of the presence of noise in the communication channel. In addition, of course, the number of transmitted messages does not have to exceed the number $L$ of regions $E_{k}(k=1, \ldots, L)$, which is equal to the number of regions $G_{k}$. Evidently, every letter is capable to faultlessly transmit $\ln L$ units of information and every $n$-character word $-n \ln L$ units. An attempt to send more information through the channel will inevitably lead to emergence of errors.

数学代写|信息论代写Information Theory代考|Random code and the mean probability of error

For a fixed decoding rule, such as the optimal rule described in the previous section, the probability of error (i.e. the probability that a recipient selects a wrong word $\xi_{k}$ different from a word actually transmitted) depends on a chosen code. In order to decrease the frequency of decoding errors caused by noise, it is desirable to select code words that are ‘dissimilar’, lying one from another, in some sense, as ‘far’ as possible. Because we cannot simultaneously increase the ‘distance’ between the code points $\xi_{1}, \ldots, \xi_{M}$ without decreasing their number $M$, it is desirable to arrange code points in the space $X^{n}$ of values $\xi$ ‘as uniformly as possible’. The desired ‘uniformity’ is achieved due to the Laws of Large Numbers for large $M$ (and $n$ ) if we select the code points randomly and independently of each other.

The Shannon’s random code is constructed as follows. Code point $\xi_{1}$ is obtained as a result of sampling random variable $\xi$ with probabilities $P(\xi)$. The second point (also the third one and the others) is sampled independently of other ones and by the same method. Consequently, the second point is an independent random variable with probabilities $P\left(\xi_{2}\right)$. In aggregate, all code points $\xi_{1}, \ldots, \xi_{M}$ are described by the probability distribution $P\left(\xi_{1}\right) \cdots P\left(\xi_{M}\right)$.

For every fixed code $\left(\xi_{1}, \ldots, \xi_{M}\right)$ obtained by the specified method and a fixed message $k$ there is some probability of decoding error. We denote that probability as $P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right)$. According to (7.1.9) it is equal to

$$P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right)=1-\sum_{\eta \in G_{k}} P\left(\eta \mid \xi_{k}\right) .$$
According to the definition of region $G_{k}$ given in Section 7.1, the summation in (7.2.1) has to be carried out over a region where all
$$P\left(\eta \mid \xi_{l}\right)<P\left(\eta \mid \xi_{k}\right), \quad l=1, \ldots, k-1, k+1, \ldots, M$$
and thereby
$$P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{k}\right)=1-\sum_{\forall P\left(\eta \mid \xi_{l}\right)<P\left(\eta \mid \xi_{k}\right)} P\left(\eta \mid \xi_{k}\right) .$$
Instead of calculating the error probabilities
$$P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right) \quad \text { and } \quad P_{\mathrm{er}}\left(\mid \xi_{1}, \ldots, \xi_{M}\right)=\frac{1}{M} \sum_{k=1}^{M}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right)$$
it is much easier to determine the probability
$$P_{\mathrm{er}}(\mid k)=\sum_{\xi_{1}, \ldots, \xi_{M}} P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right) P\left(\xi_{1}\right) \ldots P\left(\xi_{M}\right),$$
averaged by different random codes. This convenience is an important advantage of random codes. As is evident due to a symmetry standpoint, probability (7.2.3) is independent of the index of the transmitted message. The additional averaging by $k$ involved in (7.1.11) is not required in the given case:
$$P_{\mathrm{er}}=P_{\mathrm{er}}(\mid k) .$$

数学代写|信息论代写INFORMATION THEORY代 考|PRINCIPLES OF INFORMATION TRANSMISSION AND INFORMATION RECEPTION IN THE PRESENCE OF NOISE

althoughageneralizationtothecaseofvariedprobabilitiesispossible, seeTheorem7.2. 让n字母构成一个块或单词，例如，
$$\xi=\left(x_{1}, \ldots, x_{n}\right), \quad \eta=\left(y_{1}, \ldots, y_{n}\right) \quad\left(\xi \in X^{n}, \eta \in Y^{n}\right) .$$

$$P_{\text {er }}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right)=1-\sum_{\eta \in G_{k}} P\left(\eta \mid \xi_{k}\right) .$$

$$P\left(\eta \mid \xi_{l}\right)<P\left(\eta \mid \xi_{k}\right), \quad l=1, \ldots, k-1, k+1, \ldots, M$$

$$P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{k}\right)=1-\sum_{\forall P\left(\eta \mid \xi_{i}\right)<P\left(\eta \mid \xi_{k}\right)} P\left(\eta \mid \xi_{k}\right) .$$

$$P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right) \quad \text { and } \quad P_{\mathrm{er}}\left(\mid \xi_{1}, \ldots, \xi_{M}\right)=\frac{1}{M} \sum_{k=1}^{M}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right)$$

$$P_{\mathrm{er}}(\mid k)=\sum_{\xi_{1}, \ldots, \xi_{M}} P_{\mathrm{er}}\left(\mid k, \xi_{1}, \ldots, \xi_{M}\right) P\left(\xi_{1}\right) \ldots P\left(\xi_{M}\right)$$

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