## 电气工程代写|通讯系统作业代写communication system代考|LDPC Code

Low density parity check (LDPC) code is one of the forward error correction (FEC) schemes; it is more effective by their better performance, decoding complexity, and to provide good quality of THz communication band, and LDPC remains a good coding scheme technique. The schema block of LDPC coding based TS-OOK modulation is given in Fig. 9. First rediscovered by Robert Gallager [24], many researches have shown novel LDPC codes which are well generalized providing the practical advantages over turbo codes.

LDPC codes can be used in long distance wireless communication. LDPC code is a linear error-correcting code who has a parity check matrix $\mathrm{H}$, and contains rows and the columns with less to 1’s compared to 0 ‘s.

LDPC codes are classified into two types of codes; the first is regular LDPC, and the second is irregular LDPC codes. Gallager codes is an original regular binary of LDPC codes. However, the matrix $H_{n \times(n-k)}$ is composed by number $w_{\mathrm{c}}$ of ones in any column and the number $w_{\mathrm{r}}$ of ones in any row, which $n$ is the number of bits in the code-word, and $k$ is bits information symbol [25]. If both conditions are satisfied (i.e., $(n-k) w_{\mathrm{r}}=n w_{\mathrm{c}}$ thus $\left.w_{\mathrm{c}}<w_{\mathrm{r}}\right)$ [21], the parity check matrix is said to be of low density.

We call an LDPC code $\left(n, w_{\mathrm{c}}, w_{\mathrm{r}}\right.$ ) of length $n$, and in general the matrix $H$ has a very high dimension, which a very low density of element ‘ 1 ‘.

Consider the parity check $H$ matrix, for example, $\left(n=20, w_{\mathrm{c}}=3, w_{\mathrm{r}}=4\right)$ first code proposed with Gallager [26]:

The code rate of the linear regular LDPC $\operatorname{code} C\left(n, w_c, w_r\right)$ is formulated by:
$$\mathcal{R}=\frac{w_{\mathrm{r}}-w_{\mathrm{c}}}{w_{\mathrm{r}}}=\frac{k}{n}$$
For LDPC coding, a generating matrix $G$ is derived from the parity check matrix $H$ to eliminate Gaussian in modulo-2 arithmetic, such that $H$ and $G$ are formed differently. LDPC encoding is giving by matrix multiplication as
$$\bar{C}=[b \mid m]=m \cdot \vec{G}$$
where $b$ is the parity vector corresponding of $m$ message vector.
The matrix generate LDPC codes is giving by:
$$G=\left[P_{k(n-k)} \mid I_k\right]=\left[H_2 H_1^{-1} \mid I_k\right]$$

## 电气工程代写|通讯系统作业代写communication system代考|Turbo Code

Turbo codes are the groundbreaking codes introduced in [28, 29]. Turbo encoders use the best known systematic convolutional codes. As shown in Fig. 12, the implementation of turbo codes by the parallel concatenation of two recursive systematic convolutional (RSC) encoders. We will see in the following, the turbo encoding and the iterative turbo decoding. The RSCs encoders are a short constraint length to avoid excessive decoding complexity, and it is a code rate $1 / 3$ encoder.

A row column interleave, one finds the data are written by row and read by column, in this way the random interleave provides in the data bits [30]. For a turbo encoder, the generating sequence is the feedback output. Figure 13 shows the recursive systematic convolutional (RSC) encoder [31] for the intended system. The generative sequence used for the encoding algorithm which is defined in the relationship that given by:
$$G(D)=\left[1, \frac{1+D^2}{1+D+D^2}\right]$$
where 1 is the systematic output.
Generation of the moderate weight turbo codes is realized by the combination of the low from RSC1 and the high-weight code from RSC2. Finally, one will have the transmission of the original input sequence $x$ next to the two parity bit ranges are transmitted over the channel, where it is dropped to noise interference and attenuation. However, the bit error rates (BER) are different because the BER can be changed, as the input-output of the encoder. At $\frac{E_b}{N_0}$ SNR less, the BER of an RSC code should be minimized. The demodulator can express its trust in the value of each bit by using the corresponding log likelihood ratio (LLR). Each LLR gives the logarithm of the probability ratio of the corresponding bit having the values ‘ 0 ‘ and ‘ 1 ‘, which receives the information iteratively, despite the uncertainty of the wireless channel.
The schematic diagram for Turbo decoder represent in Fig. 14. The output decoder produce a soft estimation of systematic bit expressed as LLRs.
$$L_i(\hat{x}(n))=\left(\frac{P\left(x(n)=1 \mid x^{\prime}, p_1^{\prime}, L_a(x)\right.}{P\left(x(n)=0 \mid x^{\prime}, p_1^{\prime}, L_a(x)\right.}\right) ; \quad n=1,2, \ldots, N$$
where $p_1^{\prime}$ is the noise of the parity check bits, and the external bits information received by the first decoder is:
$$L_{e 1}(x)=L_1(x)-L_a(x)-L_c x^{\prime}$$
The term $L_c x^{\prime}$ is the information provided by the noisy observation. The extrinsic information $L_{e 1}(x)$ where $x$ is interleaved before applying as input to the Bahl, Cocke, Jelinek, and Raviv algorithm (BCJR) in the second decoder [32].

# 通讯系统代考

## 电气工程代写|通讯系统作业代写communication system代考|LDPC Code

LDPC码可用于长距离无线通信。LDPC码是具有奇偶校验矩阵的线性纠错码 $\mathrm{H}$, 并且包含与 0 相比小于 1 的 行和列。

LDPC码分为两类码; 第一个是规则的LDPC码，第二个是不规则的LDPC码。Gallager 码是 LDPC 码的原始 正则二进制。然而，矩阵 $H_{n \times(n-k)}$ 由数字组成 $w_{\mathrm{c}}$ 任何列中的个数和数量 $w_{\mathrm{r}}$ 任何一行中的那些，其中 $n$ 是码 字中的位数，并且 $k$ 是位信息符号[25]。如果两个条件都满足（即 $(n-k) w_{\mathrm{r}}=n w_{\mathrm{c}}$ 因此 $\left.w_{\mathrm{c}}<w_{\mathrm{r}}\right)[21]$ ， 奇偶校验矩阵被认为是低密度的。

$$\mathcal{R}=\frac{w_{\mathrm{r}}-w_{\mathrm{c}}}{w_{\mathrm{r}}}=\frac{k}{n}$$

$$\bar{C}=[b \mid m]=m \cdot \vec{G}$$

$$G=\left[P_{k(n-k)} \mid I_k\right]=\left[H_2 H_1^{-1} \mid I_k\right]$$

## 电气工程代写|通讯系统作业代写communication system代考|Turbo Code

Turbo 码是 [28, 29] 中引入的开创性码。Turbo 编码器使用最著名的系统卷积码。如图 12 所示, turbo 码 通过两个递归系统卷积 (RSC) 编码器的并行级联实现。我们将在下面看到turbo编码和迭代turbo解码。 RSCs 编码器是一个较短的约束长度，以避免过度的解码筫杂性，它是一个码率 $1 / 3$ 编码器。

$$G(D)=\left[1, \frac{1+D^2}{1+D+D^2}\right]$$

$$L_i(\hat{x}(n))=\left(\frac{P\left(x(n)=1 \mid x^{\prime}, p_1^{\prime}, L_a(x)\right.}{P\left(x(n)=0 \mid x^{\prime}, p_1^{\prime}, L_a(x)\right.}\right) ; \quad n=1,2, \ldots, N$$

$$L_{e 1}(x)=L_1(x)-L_a(x)-L_c x^{\prime}$$

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