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In order to solve the problem that the original decoding algorithm of multi-band LDPC codes is high and is not conducive to hardware implementation, two simplified decoding algorithms for multi-band LDPC codes are studied: the reliability propagation based on fast Fourier transform Code algorithm (FFT-BP) and log-BP decoding algorithm based on logarithmic operations. The simulation results show that the FFT-BP decoding algorithm is more convenient and efficient.

Multicomponent coding is the key technology of multivariate wireless transmission, and its core lies in multivariate codec. Most of the current multi-code modulation using multiple LDPC code, multi-LDPC code was first proposed by Gallager, 1998 Davey and MacKay studied the finite field of multiple LDPC code. Compared with the binary LDPC code, the main advantages of multiple LDPC codes are reflected in the following aspects: 1) better error correction performance: multiple LDPC code will be a number of bits into a multi-symbol, which has the potential of eliminating LDPC code small ring of binary map and getting better error correction performance; 2) Strong anti-burst error capability: Because of Multiple LDPC codes making multiple burst bit errors can be combined into fewer multi-symbol errors, the error correction performance is better than binary LDPC codes; 3) Suitable for high-speed transmission: Based on high-or- der finite-domain multivariate LDPC codes are designed and are well suited for combining high-order modulation schemes with multi-antenna systems to provide higher data rates and spectral efficiency. It can be seen that multi-LDPC coding and modulation technology is more suitable for wireless channel transmission [

Field F is a set of elements that define two operations, plus “+” and multiplication “ ”and both of which satisfy the law of union, distribution law, and exchange law, both of these operations are closed [

If there is a domain F which has only a finite number of q elements, then the domain is called a finite field, or Galois Filed, denoted as GF (q). There is a finite field GF (q) only if q is a power of prime number, the extended domain GF (

The last element in the field is obtained by multiplying the previous element

Then

So the elements of the finite field GF (

The multi-band LDPC code defined on the finite field GF (q), which evaluates the value of the nonzero element

1) Row weight p, column weight q, p and q are very small with respect to m and n.

2) In H, any two rows and two columns of the corresponding position on the same non-zero elements no more than one.

The irregular polygon LDPC codes p and q are not fixed. The second row constraint ensures that the constructed parity check matrix has no four rings on its Tanner graph [

Compared with the random construction method, LDPC codes constructed by combinatorial and algebraic methods have little difference in performance loss and low hardware implementation. The algebraic construction method with the theoretical research value is composed of the LDPC code and the quasi-cyclic LDPC codes with quasi-cyclic algebra structure proposed by Tanner and Fossorier et al., however, the parameters of the algebraic structure of the regular LDPC code length, bit rate and other parameters of the selectivity is not strong, cannot be pre-set code length and bit rate, which can be based on these parity check matrix itself to determine the code length and bit rate and other parameters. On the basis of Tanne and Fossorier et al., Lin S. propose a method for constructing quasi-cyclic LDPC codes based on finite field GF (q) and affine mapping [

For multivariate QC-LDPC coding, since the generation matrix of multiple LDPC codes and binary LDPC is similar in construction, therefore, according to the construction of binary QC-LDPC, and each sub-matrix is independent, it is only necessary to obtain the generator in its generation matrix G, which can be encoded in serial or parallel way.

The information sequence is denoted as

At present, the practical implementation of multi-band LDPC code coding structure is no longer difficult, but the multi-band LDPC decoding of the highly complex problem has not been a very good solution. Davey and MacKay put forward the classic belief transmission (Belief Propagation, BP), the core idea is to pass the Tanner map to probability information [

Binary and multivariate LDPC codes can adopt BP decoding algorithm, but the two are slightly different in the process, in which the multiple BP decoding algorithm has message replacement and inverse permutation process. In the decoding process, we must first calculate the initial probability information, in practice, usually among the multi-band values are independent of each other, therefore, an element on the GF (q) (

Among them,

The decoding process of multiple LDPC codes can be summarized as follows:

1) Initialization: The variable node

2) Information replacement: Unlike binary LDPC code BP decoding, it is necessary to transpose the variable node information sequence. The rule is that the value of each variable in the information sequence is multiplied by

3) Check node update: Check the node information

where

4) Replacement: This process is the inverse of the check node transfer message and the parameter

5) Variable Node Update: After the transcoding of the check node information by the process (4), the new check sequence

In order for equation

cient

6) Posterior probability of the calculation, the decision:

The decision to use the information from the information and outside the information, rather than the variable node update, just update the external information.

Perform a hard decision expression:

Selecting the highest codeword as the decision codeword, and finally obtaining

In the traditional BP decode algorithm, the check node update process uses many addition and multiplication operations. In the hardware implementation, it will seriously consume the logic resources, which will affect the practical application to a large extent. (3)-(4) can be regarded as convolution in the BP decoding algorithm, and the algorithm is optimized for this performance. A

When GF (4), the corresponding butterfly chart as shown in

The initialization process in the

1) FFT transforms the information variables to obtain intermediate values

2) Make

3) Transform the results obtained by the above formula:

Due to the fast Fourier transform introduced in the

The FFT-BP decoding algorithm is based on the traditional BP decoding algorithm, and the fast Fourier transform algorithm is used to replace the convolution operation in the variable node updating process of the BP decoding algorithm, reducing the amount of computation and speeding up the decoding rate. At the same time, compared with the traditional BP decoding algorithm, because of the use of fewer multipliers, so the hardware is also more convenient to achieve, improve the hardware resource utilization. The simulation results are shown in

It can be seen from the simulation results that the multivariate BP decoding algorithm and the FFT-BP decoding algorithm are simplified by the fast Fourier transform performance and simplification of the operation due to the same information probabilistic initialization method. The performance of the decoding algorithm is the same, but the FFT-BP decoding algorithm is more convenient and efficient.

This paper is funded by the International Exchange Program of Harbin Engineering University for Innovation Oriented Talents Cultivation, International Science & Technology Cooperation Program of China (2014 DFR10240), National Nature Science Foundation of China (No. 61401115), National Natural

Science Foundation of China (No. 61301095), National Natural Science Foundation of China (No. 61671167), China Postdoctoral Science Foundation (2013- T60346), Harbin Science and Technology Research Projects (P083313026), Natural Science Foundation of Heilongjiang Province (P083014025).

Wei, X., Li, Z.G. and Dou, Z. (2017) Simulation Analysis of Multiple BP Decoding Algorithm and FFT-BP Decoding Algorithm. Int. J. Communications, Network and System Sciences, 10, 255-262. https://doi.org/10.4236/ijcns.2017.108B027