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Main advantages of ARQ

Main disadvantages of ARQ

Principle block diagram of ARQ system

Characteristics of Hamming code:

H---Supervision matrix

Properties of H matrix

G---generating matrix

Properties of G matrix

The relationship between G and H

Correction and error patterns

Properties of linear block codes

QPSK is a 4-phase phase shift keying system that transmits 2 bits of information per symbol. If the signal phase is misjudged to an adjacent phase due to interference during the receiving end decision, a code error will occur. Now the system is changed to 8PSK, and each symbol of it can transmit 3 bits of information. However, each symbol is still allowed to transmit 2 bits of information, and the third bit is used for error correction code. For example, a convolutional code with a code rate of 2/3 is used. At this time, the demodulation and decoding at the receiving end are completed as one step. Unlike the traditional method, the baseband signal is first demodulated and then decoded for error correction.

Divide the signal constellation diagram into several subsets so that the distance between signal points in the subsets is larger than the original one. Each time it is divided, the distance between signal points in the new subset increases.

Demodulation algorithm of TCM signal

Free Euclidean distance Fed:

LDPC code is a linear block code, which belongs to the same category of composite codes as Turbo code. The performance of the two is similar, and the decoding delay of both is quite long, so they are more suitable for some communications that do not require very high real-time performance. . However, the decoding of LDPC codes is simpler and easier to implement than Turbo codes.

Regular LDPC code: H matrix has the same number of "1"s in each column Irregular LDPC code: The number of "1's in each column of the H matrix is ​​not necessarily the same. The irregular LDPC code is developed on the basis of the regular LDPC code. It improves the decoding performance and makes the bit error rate performance better than the Turbo code. .

The LDPC code is the same as the ordinary parity supervised code and can be determined by the supervision matrix H with n columns and m rows; n is the code length and m is the number of syndromes. However, its H matrix is ​​different from that of ordinary parity supervision codes: it is a sparse matrix, that is, the number of "1"s in the matrix is ​​very small, and the density is very low; suppose that the H matrix has j "1"s in each column and j "1"s in each row. With k "1"s, there should be j<<m, k<<n, and j3. The elements of any two rows of its H matrix cannot be "1" at the same position, that is, there is no rectangle with four corners composed of "1" in the H matrix. LDPC codes are usually represented by the above three parameters (n, j, k). During coding, after designing the H matrix, the generator matrix G can be derived from the H matrix. In this way, for a given information bit, it is not difficult to calculate the code group.

The decoding method of LDPC codes is also different from the decoding method of general parity supervised codes. The basic decoding algorithm is called the belief propagation algorithm, often referred to as the BP algorithm. This algorithm essentially seeks the maximum posterior probability, similar to the general maximum likelihood criterion decoding algorithm, but it requires multiple iterative operations to gradually approach the optimal decoding value. The specific encoding and decoding algorithm of LDPC code is very complicated and will not be discussed in depth here.

Since the complexity of block codes and convolutional codes increases exponentially with the increase of code group length or constraint degree, in order to improve the error correction capability, do not simply increase the length of the code, but combine two or more simple Codes are combined into composite codes.

The encoder of Turbo code adds an interleaver between two parallel or series component code encoders, so that it has a large code group length and can obtain close to ideal performance under low signal-to-noise ratio conditions.

The decoder of the Turbo code has two component code decoders. The decoding is iteratively decoded between the two component decoders. Therefore, the entire decoding process works like a turbine, so it is also vividly called Turbo. code

The main difference between RSCC encoders and convolutional code encoders: There is a feedback path from the output of the shift register to the input of the information bits. The original convolutional code encoder is like a FIR digital filter. After adding a feedback path, it becomes an IIR filter, or recursive filter.

N---coding constraint degree, indicating the number of code segments that constrain each other during the encoding process;

nN---encoding constraint length, indicating the number of code elements that are constrained to each other during the encoding process.

N or nN also reflects the complexity of the convolutional code encoder.

R=k/n

Encoder principle block diagram

Supervision matrix H

The general form of truncated supervision matrix H1 is:

Basic supervision matrix h

Generating matrix G

Truncated generating matrix G1

Basic generating matrix g

Classification

Large number logic decoding

Geometric representation of convolutional codes

Viterbi decoding algorithm

Cyclicity:

code polynomial

Cyclic code encoding

Decoding of cyclic codes

truncation purpose

truncation method

Truncated cyclic code performance

definition

——The problem of the relationship between the generator polynomial and the error correction capability is solved, and the generator polynomial of the code can be found under the given error correction capability requirements. ——With the generator polynomial, the basic problem of encoding is solved.

Primitive BCH code: Its generating polynomial g(x) contains a primitive polynomial with the highest degree m, and the code length is n= 2^m-1, (m≥3, a positive integer).

Non-primitive BCH code: its generator polynomial does not contain this primitive polynomial, and the code length n is a factor of (2^m-1), that is, the code length n must be divided by 2^m-1.

The relationship between the code length n, supervision bits, and the number of error corrections t: For a positive integer m (m 3) and a positive integer t < m / 2, there must be a code length n = 2^m – 1 with supervision bits is n – k  mt, which can correct all BCH codes with no more than t random errors. If the code length n = (2^m - 1) / i (i > 1, and all divisions can be made (2^m -1)), it is a non-original BCH code.

Hamming codes are codes that can correct a single random error. It can be proved that the Hamming code with cyclic properties is the original BCH code that can correct a single random error.

In engineering design, it is generally not necessary to use calculation methods to find the generating polynomial g(x). Because predecessors have already listed the found g(x) in a table, we can use the table lookup method to find the required generator polynomial.