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

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  • 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

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
and the noise power
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
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.

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


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.


AWGN 信道的信道容量由下式给出

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


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

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)。


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

00 1 0 110的传输

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。


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


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。


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。

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