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

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## 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. ${ }^{1}$

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

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}

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}

$$C=\max {\left{\left.P{\mathcal{X}}\left(x_{i}\right)\right|_{1 \leq i \leq M}\right。 } I(\mathcal{X} ; \mathcal{R}) 。 一世F吨H和一世np在吨和n吨r这p是一世(X)一世ss米一个ll和r吨H一个n吨H和CH一个nn和lC一个p一个C一世吨是C I(\mathcal{X}) \stackrel{!}{<} C$$

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

C=1+e日志⁡2(e)+(1−e)日志⁡2(1−e)

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