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

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## 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
$$S=\mathrm{E}\left{\mathcal{X}^{2}\right}$$
and the noise power
$$N=\mathrm{E}\left{Z^{2}\right}$$
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
$$000000111000111111111000$$
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

S=\mathrm{E}\left{\mathcal{X}^{2}\right}S=\mathrm{E}\left{\mathcal{X}^{2}\right}

N=\mathrm{E}\left{Z^{2}\right}N=\mathrm{E}\left{Z^{2}\right}
AWGN 信道的信道容量由下式给出

C=12日志2⁡(1+小号ñ)

R匹配的过滤器由下式给出

e=12erfc⁡(和bñ0)

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

$$00 1 0 110的传输 这在和r一个b一世n一个r是s是米米和吨r一世CCH一个nn和l在一世吨Hb一世吨和rr这rpr这b一个b一世l一世吨是e=0.25(ñ和在b一个在和r,2006b).这n一个在和r一个G和,和在和r是F这在r吨Hb一世n一个r是s是米b这l在一世llb和r和C和一世在和d一世nC这rr和C吨l是.一世n吨H一世s和X一个米pl和在和一个ss在米和吨H一个吨吨H和b一世n一个r是s和q在和nC和 在二进制对称通道的输出端接收到00 0 0 0 110$$
（见图 1.7）。

000000111000111111111000

$$0 0 000 0 110 10 111 0 1 0 1110 10。$$

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