A new method for constructing parity check matrix of QC-LDPC codes based on shortening RS codes for wireless sensor networks
Introduction
Wireless sensor networks (WSNs) are the core of the Internet of things. A large number of sensor nodes in WSNs continuously pass the collected data to the data center to form a big data [1]. Transmission of big data consumes a large amount of network energy. How to reliably transmit big data while extending the network life is a key problem to be solved by WSNs for the future big data application. To solve this problem, error control coding (ECC) is a very good solution for enhancing the stability of wireless sensor nodes and network transmission as well as reducing the power consumption of nodes and improving the energy efficiency of WSNs [2].
Low density parity check (LDPC) code is a linear block code defined on a sparse parity check matrix [3], [4], [5], and has been increasingly adopted as the primary standard of the channel coding scheme of the actual system because of its excellent performance near the Shannon limit. Construction of the structure of its check matrix is the key for the design of LDPC codes. A well constructed parity check matrix of LDPC codes can take into account the excellent error-correcting performance of LDPC codes and the low hardware implementation complexity of coding and decoding algorithms.
Construction methods of the parity check matrix of LDPC codes can be divided into random construction method and structured method. The random constructed check matrix lacks a regular structure, which complicates the LDPC coding process and requires a large memory space to store the parity check matrix. This hinders the application of the random construction method. The structure of parity check matrix constructed by structured method has a definite structure and plays an important role in engineering applications. Especially, quasi-cyclic (QC) LDPC codes [6], [7], [8], [9] are an important research branch of LDPC codes. Its check matrix has characteristics of quasi-cyclic, which means that the encoding of LDPC codes can be achieved by the shift register, thus greatly reduces the implementation complexity. Therefore, QC-LDPC codes have a good application prospect, which has been adopted in the standard of IEEE802.16e, DVB-S2 and so on [10], [11]. As for the IEEE 802.11a/b/g WLAN application, when the LDPC code is used in the downlink, the battery life of the mobile terminal can be extended up to four times. It can be seen that the application of the QC-LDPC code used as the ECC for the node of WSNs is a good solution for reducing the energy consumption of the nodes and improving the network life of WSNs.
However, the construction of the parity check matrix of QC-LDPC codes requires that the base matrix is first constructed, and the nonzero elements of the base matrix are then shifted and expanded [12], [13], [14]. Only when the base matrix and the shift value both have the characteristics of the structured, the constructed parity check matrix of QC-LDPC codes has the characteristics of fully structured, so as to facilitates the practical application of LDPC codes. In [12], [13], [14], a method to construct the parity check matrix of QC-LDPC codes based on the Progressive Edge-Growth (PEG) algorithm was proposed. This method requires the long generation time and large storage space which is not convenient for the hardware design and implementation, because the base matrix is constructed by the PEG algorithm which belongs to the random construction method. Furthermore, reference [15] proposed an efficient algorithm based on the Chinese remainder theorem (named PEG-CRT algorithm). It was proved that the LDPC codes constructed by PEG-CRT algorithm have similar BER performance compared with the LDPC codes constructed by PEG algorithm, but PEG-CRT algorithm has lower complexities.
Papers [16], [17], [18], [19] proposed a method to construct LDPC codes by shortening RS codes based on two information symbols, which can solve the problem of the minimum code distance as large as possible. The shortening RS codes are subsets of cyclic codes, which can be generated by the cyclic encoder. Therefore, this method has the advantage of simple implementation [20]. However, although the cycle of length 4 is eliminated in the parity check matrix constructed by this method, the girth of the check matrix cannot be guaranteed to be large. If we can combine the advantage of LDPC codes construction of shortening RS codes and the advantages of parity check matrix construction of QC-LDPC codes, it is possible to construct a practical LDPC code with good performance and simple implementation.
The specific contributions of this paper are illustrated as follows. This paper proposes a new method to construct the parity check matrix of QC-LDPC codes based on the shortening RS codes. This method to construct the parity check matrix of QC-LDPC codes has the characteristic of fully structured and convenient for hardware implementation. Furthermore, the girth of the parity check matrix of LDPC codes constructed by this method is at least 8. Thus, the constructed QC-LDPC code still has the largest minimum code distance, which can ensure good error correction performance of LDPC codes. These characteristics guarantee that QC-LDPC codes constructed by the proposed method can reduce the power consumption of WSNs effectively in the case of guaranteeing transmission reliability.
The rest of this paper is organized as follows. The structural characteristics of check matrix of QC-LDPC codes are given in Section 2. The construction of base matrix based on shortening RS codes is studied in Section 3. In Section 4, how to construct the parity check matrix of QC-LDPC codes based on shortening RS codes is proposed. The effectiveness of the proposed approaches compared with the existing algorithm is examined in Section 5 through computer simulation. Finally, the paper is concluded in Section 6.
Section snippets
Structural characteristics of check matrix of QC-LDPC codes
The parity check matrix H of QC-LDPC codes consists of cyclic matrices called a cyclic permutation matrix [4], which can be represented by the Eq. (1)where each element I (Pij)(1 ≤ i ≤ m, 1 ≤ j ≤ n) is a cyclic permutation matrix obtained by rightward shifting each row of the unit matrix I with b × b dimension, and Pij ∈ {0, 1, 2, ⋯ , b − 1, ∞}, especially, I (Pij) is a unit matrix when Pij = 0, while I (Pij) is an all-zero square matrix when Pij
Shortening RS codes with two information symbols
Let α be a primitive element on Galois field GF(q), where q be an arbitrary power of a prime number, and {0, α0, α1, α2, ..., αq−2} be all elements on GF(q). If a positive integer ρ satisfies 2 ≤ ρ ≤ q − 1, the code length is q-1, the number of information bits is q + 1 − ρ, and the number of parity bits is ρ-2. The generator polynomial of the (q − 1, q-ρ + 1, ρ − 1) RS code whose minimum code distance is ρ − 1can be written as [24]where gi ∈ GF(q),gi ≠ 0,gρ−2 = 1. The
Construction of parity check matrix of QC-LDPC codes based on shortening RS codes
For constructing parity check matrix of QC-LDPC codes based on shortening RS codes, the base matrix A free of cycles of length 4 is first constructed by shortening RS codes. Then, the base matrix is optimized and expanded by the cyclic permutation matrix I(Pij) to obtain the parity check matrix H of QC-LDPC codes. Here, the optimal expansion means that the shift values of the cyclic shift matrix I(Pij) is selected to satisfy the condition of Eq. (4), so that the resulting of the parity check
Example of the proposed method and simulation experiment
In this subsection, in order to evaluate the effectiveness of the proposed method for constructing parity check matrix of QC-LDPC codes based on shortening RS codes, we give an example of the construction of parity check matrix for regular LDPC codes using the proposed method. Unless otherwise indicated, the LDPC decoder invokes the sum-product algorithm to execute the decoding, and the iteration number is set to 20, and BPSK is selected as the modulation scheme.
Conclusions
This paper presents a new method to construct QC-LDPC codes based on shortening RS codes for WSNs. This method takes advantage of constructing the base matrix of the parity check matrix of QC-LDPC based on shortening RS codes, and it also absorbs the characteristics of the parity check matrix construction of the QC-LDPC codes. The constructed parity check matrix not only has fully structured features, but also has a good minimum code distance. The proposed method can make different parity check
Acknowledgments
This work is supported in part by the Natural Science Foundation of Jiangsu Province (Grant No.BK20160294), the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No.14KJB510010), Changzhou Sci&Tech Program (Grant No.CJ20140058), and National Natural Science Foundation of China (Grant No. 61601208).
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These authors contributed equally to this work.