Construction of Irregular QC-LDPC Codes in Near-Earth Communications

Transcription

1 Journal of Communications Vol. 9, No. 7, July 24 Construction of Irregular QC-LDPC Codes in Near-Earth Communications Hui Zhao, Xiaoxiao Bao, Liang Qin, Ruyan Wang, and Hong Zhang Chongqing University of Posts and elecommunication, School of Communication and Information Engineering, Chongqing 465, China Abstract High-seed, high-reliability requirements and strict ower limitation of near-earth communications yield that the channel coding must have high coding gain and lower error floor. In this aer, an irregular quasi-cyclic low-density aritycheck (QC-LDPC) code for secific use in near-earth communications is roosed. In order to eliminate the imact of short-cycles and further imrove the erformance of codes, a base matrix with girth at least 8 is emloyed. hen dualdiagonal structure is adoted in arity check matrix to reduce encoding comlexity. Besides, circulant ermutation matrices and zero matrices with different sizes are used to fill the nonzero ositions and zero ositions of the base matrix so as to remove the comlexity of the selection of cyclic shift coefficients. he adotion of circulant ermutation matrices with different sizes ensures that the arity check matrix is full rank, which makes the code rate controllable. Finally, simulation results are given to demonstrate the erformance of the constructed codes by comaring with several existing codes. Index erms Low-density arity-check codes, arity check matrix, circulant ermutation matrix, near-earth communications I. INRODUCION Due to high-seed, high-reliability requirements and strict ower limitation, excellent channel coding becomes one of the key technologies to ensure the reliability and validity of near-earth communications []. Low-density arity check (LDPC) codes [2] have received more and more attention in near-earth communications for their outstanding erformance, such as faster iterative convergence, larger throughut, lower error floor, etc. Among LDPC codes, the quasi-cyclic LDPC (QC-LDPC) codes are the most romising class of structured LDPC codes due to their ease of imlementation and excellent erformance over noisy channels when decoded with message-assing algorithms [3]. In general, the overall erformance of a LDPC code is described by two different regions, the waterfall region and the error-floor region [4], [5]. he erformance in the Manuscrit received Aril 8, 24; revised July 28, 24. his work was suorted by the National Natural Science Foundation of China (Grants , 625and 55392), and the National Natural Science Foundation of CQ (Grants Nos. CSC22jjA448, CSC 2BA24, and CSC 2jjA76). Corresonding author zhaohui@cqut.edu.cn doi:.272/jcm waterfall region is heavily affected both by the girth (he length of the smallest cycle in the arity check matrix) and by the variable and check nodes degree rofiles for irregular LDPC codes. Iteratively decoded LDPC codes demonstrate an abrut change in their error rate curves. An error floor in the erformance curve means that the decoding failure rate does not continue to decrease raidly as the signal to noise ratio (SNR) increases [6]. Eliminating or lowering error floors is articularly imortant for near-earth alications that have extreme reliability demands. he erformance in the error floor region is also affected by the girth. For examle, in erasure channels every stoing set contains cycles [7], so that increasing the girth turns out to increasing the smallest stoing set size [8] of the LDPC codes. Short cycles in the anner grah, which lead to inefficient decoding and limit the convergence rate of the sumroduct algorithm, affect the erformance of the LDPC codes. herefore, design of LDPC codes with large girth is of great interest [4]. here have been several aroaches, including direct construction aroaches [9]-[2] and cycle removal based aroaches [3], to remove short cycles in anner grah and construct LDPC codes with large girth. However, these aroaches are either not sufficiently flexible in the code arameter selection due to the constraint on girth [4], or unable to reserve the quasi-cyclic structure thereby increasing the encoding comlexity [5]. In order to reduce the comlexity of arity check matrix construction and to achieve an efficient encoding, we use the arity check matrix with systematic form, and design the check art and the information art indeendently. he check art of arity check matrix adots the dual-diagonal structure to achieve iterative encoding. he information art uses the quasi-cyclic structure whose nonzero ositions relaced by circulant ermutation matrices (CPMs) with different sizes and zero ositions relaced by corresonding-size zero matrices to reduce the comlexity of arity check matrix construction. he class of LDPC code derived from our construction ossesses large girth and much more flexibility in terms of code length and rate. What s more, the roosed codes also can achieve very good erformance while retaining the racticality of quasicyclic structured. 24 Engineering and echnology Publishing 54

2 Journal of Communications Vol. 9, No. 7, July 24 he organization of the rest of this aer is as follows. First, we resent the construction method of the check art and the information art of arity check matrix in Section II. hen, we show the erformance comarison between the roosed code and several existing codes over binary-inut Additive White Gaussian Noise (AWGN) channel in Section III. he aer is summarized in the last section. and then exand H into desired matrix Hk using CPMs or zero matrices with different sizes. ) Construction of sub-matrix B: Design a matrix B with dimension v v2, where B(,)=B(2,)=B(3,)= =B(v,)=, and other elements are. hen B is a v v2 matrix of the following form: II. CONSRUCION MEHOD In this section, we introduce the basic construction of the roosed irregular QC-LDPC codes. Suose the arity check matrix of QC-LDPC code is systematic form: H H H k () where H is an M M matrix corresonding to the check art of arity check matrix; Hk is an M (N-M) matrix that indicates the information art. he arity check matrix of irregular QC-LDPC code can be obtained through constructing H and Hk indeendently, where H is a dual-diagonal square matrix and Hk kees the quasi-cyclic form. M M (3) By cyclically shifting the elements of B one lace to right, we can obtain v2- new matrices, denoted by B, B2, B3,, B( v2 ). Combining B and the v2- new (4) B B B B2 B(v2 ) 2) Construction of sub-matrix B2: Design a matrix B2 with dimension v v2, where B2(,)=B2(2,2)=B2(3,3)= =B2(v,v)=, and other elements are. hen B2 is a v v2 matrix of the following form: B2 (2) v v 2 (5) Let B2 have the following form: B2 B2 Clearly, H is a non-singular matrix. It is well known that degree-2 variable nodes (VNs) make it ossible to achieve sarse and otimal irregular roerties and simlify the encoding rocess [6]. However, many works show that the VNs with low degree are bad for decoding erformance [8]. hus, in order to avoid the bad influence of degree-2 VNs, we make the number of VNs with degree-2 satisfy the following constraint. he maximum number of VNs with degree-2 should satisfy Nv,max (2) N K M, where M is the B2 B2 B2 (6) he total number of B2 is v. By cyclically shifting the elements of B2 v laces to right, we can get v- new matrices, denoted by B22, B23,,B2v. Combining B2 and the v- new matrices into a column, then we can get a v3 v2 matrix B2 of the following form: B2 B2 B22 B2v B23 (7) 3) Construction of sub-matrix B3: Design a matrix B3 also with dimension v v2, where B3(, )=B3(2, +v)=b3(3, +2v)= =B3(v, +v-v2)= and other elements are. hen we can obtain the following matrix: number of check nodes (CNs), N is the number of VNs and K is the number of information bits. B. Construction of Hk. In this subsection, we use the CPMs with different sizes to construct a class of irregular QC-LDPC codes. Comared with the direct construction methods [9]-[2], this method has a relatively simle structure, and there is no need to search the cyclic shift coefficients of base matrix. B3 v v C. Construction of Base Matrix We design three sub-matrices B, B2 and B3, combine them into a row and transose to obtain matrix H [7], 24 Engineering and echnology Publishing v v 2 matrices into a column, then we can obtain the following v3 v2 matrix B: A. Construction of H Dual-diagonal structure of H has the following form: H B v v v v 2 (8) Cyclically shifting the elements of B3 one lace to right, then we can get v- new matrices, denoted by B3, 542

3 Journal of Communications Vol. 9, No. 7, July 24 B3 B3( v ) B32 (9) By arranging v B3v into a column, we form the following v3 v2 matrix: B3 B3v B3v B3v B3v Coy Coy () B2 B3 3v2 v3 () B3 B2 Ga Ga2 hen we can get H as follows: H B Ga B3v B3 Ga2 B32,,B3(v-). Arranging B3 and the v- new matrices results in a column of v matrices: B Fig.. Princile of cyclic-coy. D. Exansion of the Base Matrix Following the above construction, we can construct three v3 v2 arrays B, B2 and B3. Relace the nonzero elements of the three sub-matrices with CPMs of different suitable sizes, and then relace the zero elements of the three sub-matrices with zero matrices of the corresonding sizes, we can obtain the desired matrix H k. ) Size selection of CPMs: Let P be the CPM defined by: P he null sace of H gives an irregular QC-LDPC code of length v3 3 whose check matrix with column and row weight at least 3 and v, resectively. he corresonding anner grah has girth at least 8. aking a sub-matrix as Hk of the arity check matrix, we can construct a class of irregular QC-LDPC codes. Based on the above analysis, the conditions that the sizes of the CPMs and the base matrix need to meet are as follows: a) he sizes of CPMs need to satisfy the condition: max{, 2, 3 } 2 min{, 2, 3} If max{, 2, 3 } 2 min{, 2, 3}, then Ga is larger than the row size of B. Similarly, Ga2 is larger than the row size of B2. he cyclic coy oeration will not be imlemented. b) he number of columns for a common quasi-cyclic matrix should be smaller than the least common multile (LCM) of any two of its CPMs sizes. hat is, the number of columns should satisfy the condition: (2) Since the three sub-matrices have the same dimension, using different-size CPMs and zero matrices added will lead them have different dimensions. So in order to make H kee the desired matrix form and avoid short cycles, the sizes of CPMs are needed to be selected carefully. Now, suose we use CPMs with size, 2 and 3 to relace the nonzero elements of B, B2 and B3, and exloit zero matrices with size, 2 and 3 to relace the zero elements of them. For simlicity and without loss of generality, we assume 2 3, then the sizes of the obtained sub-matrices are (v3 ) (v2 ), (v3 2) (v2 2) and (v3 3) (v2 3), resectively. Clearly, the size of row ga between B and B3 is v3(3-), and the row size difference between B2 and B3 is v3(3-2). Since H B B2 n min LCM{ i, j } (4) i, j 3{ i, j } Next we use an examle to illustrate the above roosition b. Let 2 and 3 be the sizes of CPMs. hen the LCM of 2 and 3 is 6. When the number of columns of the matrix is greater than 6, there will form cycles of length four. he four bold in matrix A form a 4-cycle. B3 3v2 v3, then we can negligent the column size difference among B, B2 and B3. he size of row ga originated from different size CPMs and zero matrices can be resolved through cyclic coy. he rincile of cyclic-coy is shown in Fig., where Ga indicates the row size difference between B and B3, and Ga2 indicates the row size difference between B2 and B3. If we coy a Ga from the head of B and lace it at the end of B, then it can make sure that B and B3 are with the same row size. In a similar manner, we can maintain B2 and B3 having the same row size. hen we will get the desired nearly quasi-cyclic form matrix H. 24 Engineering and echnology Publishing (3) (5) c), 2 and 3 are rime. Minimum distance is an imortant arameter of LDPC code, which determines the LDPC codes detecting and correcting random errors caacity. Gallager has demonstrated that when the degrees of the variable node are less than 3, the minimum distances of LDPC codes grow linearly with block length (for a fixed rate and fixed degree of variable node), and 543

4 Journal of Communications Vol. 9, No. 7, July 24 when the variable node degree is greater than 3, the minimum distance of LDPC codes increase linearly along with block length [2]. So we often choose, 2 and 3 are rime and also relatively rime to make sure that the code have large minimum distance, and further make it ossess owerful error detection and correction caability. 2) Girth of the roosed irregular QC-LDPC codes It is well-known that girth is one of the designing criteria to otimize the erformance of message-assing decoding algorithms for LDPC codes. he girths of QC- LDPC codes have some unique features, as show in Fig. 2. H Fig. 2 Cycles in arity check matrix of QC-LDPC code. A cycle is a ath through nodes in the anner grah of a code. It alternates between check and variable nodes, and starts and ends at the same node. In terms of the code s arity check matrix, each check node in the anner grah corresonds to a row in its arity check matrix, and each variable node corresonds to a column. However, the systematic form QC-LDPC codes also have some unique structures. Next, we will take advantage of the relationshi between anner grah and the arity check matrix to show the roosed code of girth 8. he roosed H matrix has irregular structure and the anner grah [7] of the code c is similar to those of (irregular) reeat-accumulate (RA) codes [9]. RA codes are tyically systematic. What s more, RA codes rovide two imortant advantages. First, they allow flexibility in the choice of the reetition rate for each information bit, so that high-rate codes may be designed. Second, their irregularity allows oeration closer to the caacity limit. In the anner grah of the code c, we call the nodes corresonding to the columns of H arity bit nodes, the nodes corresonding to the columns of H k information bit nodes, and the nodes corresonding to the rows of H check nodes [2], [2]. We also call information and arity bit nodes variable nodes. We now use a general examle to illustrate the relationshi between anner grah and the arity check matrix owning systematic form. Suose the arity check matrix of the basic construction is given by: k (6) As shown in Fig. 3, the columns of H and H k are numbered from to 5 from the left. he rows of H and H k are numbered from to 5 from the to. he bold lines in anner grah reresent a 4-cycle, and they corresond to the four bold in the arity check matrix Infor. nodes Fig. 3. anner grah of H in (6) Check nodes Parity. nodes Since there is no 4-cycle and 6-cycle in the base matrix, the girth of the QC-LDPC code designed by our base matrix is at least 8 [7]. Making the girth of the arity check matrix as large as ossible can imrove the erformance of the code. III. SIMULAIONS In this section, we resent some simulation results to demonstrate the erformance of the constructed code by comaring with several existing codes. Simulations are erformed on AWGN channel with binary hase-shift keying modulation. Let v=8, =, 2 =3 and 3 =7. Following the above construction, we can construct a array H. Relacing the nonzero elements of B, B 2 and B 3 with CPMs of sizes, 2 and 3 resectively, and then relacing the zero elements of B, B 2 and B 3 by zero matrices with corresonding sizes resectively, we can get a class of irregular QC-LDPC codes. o validate the advantage of different-size CPMs, the bit error rate (BER) erformance comarison between the codes with different-size CPMs and same-size CPMs are resented in Fig. 4. In Fig. 4, the bi-diagonal regular code reresents the code with same size CPMs and also 24 Engineering and echnology Publishing 544

5 Journal of Communications Vol. 9, No. 7, July 24 has systematic form, the regular code reresents the code with same-size CPMs and has non systematic form, the bi-diagonal irregular code means the code with different-size CPMs and has systematic form, and the irregular code means the code with different-size CPMs and has non systematic form. he different-size CPMs with sizes, 2 and 3, the same-size CPMs with size =7. Choosing a sub-matrix H qc with dimension , the null sace of H qc gives an irregular (5248, 2624) QC-LDPC code with rate.5. If we let H qc be H k, then H [ H H qc], thus we can construct a class of systematic QC-LDPC codes. As shown in Fig.4, for the systematic code, when the SNR is 3dB, the BER erformance of our roosed bi-diagonal irregular code is -5, while the code whose arity check matrix has the same size CPMs is -4. he two differ by an order of magnitude. he code whose arity check matrix is nonsystematic form has similar erformance. where bi-diagonal code indicates the code whose arity check matrix is systematic form and irregular code reresents the code whose arity check matrix is not systematic form. It can be seen that our roosed bidiagonal code outerforms the CCSDS standard (876, 756) QC-LDPC code. Secially, it can achieve.5db SNR imrovement at BER of -5. In addition, the designing of our base matrix without using another algorithm to select the cyclic shift coefficients simlifies the construction rocess of the arity check matrix. BER BER regular code irregular code bi-diagonal regular code our roosed bi-diagonal irregular code SNR(dB) Fig. 4. BER erformance of the codes based on same base matrix with various CPMs. BER (876, 756) LDPC code irregular LDPC code our roosed bi-diagonal code SNR(dB) Fig. 5. BER erformance of the roosed code and CCSDS standard (876, 756) QC-LDPC code. Choosing a sub-matrix H qc2 with size , the null sace of H qc2 gives an irregular (874, 766) QC- LDPC code with rate.87. he BER erformance comarison between the roosed bi-diagonal code and consultative committee for sace data systems (CCSDS) standard (876, 756) QC-LDPC code deicts Fig. 5, -5 quasi-cyclic code-[] algebric code-[9] irregular code -6 our roosed bi-diagonal LDPC code SNR(dB) Fig. 6. BER erformance of the roosed code and the code resented in [9] and []. o further illustrate the erformance of the code in this aer, the BER erformance of the code constructed by the null sace of H qc and the codes constructed by the methods given in [9] and [] is shown in Fig.6, where the roosed bi-diagonals LDPC code indicates the code whose arity check matrix is systematic form and irregular code reresents the code whose arity check matrix is not systematic form. he structure of arity check matrix in [9] is also systematic form. However, the information art of arity check matrix H k filled with the same dimension CPM, and need related algorithms to determine the cyclic shift coefficients of CPMs. Comared with the construction method in [9], using different sizes CPM filled the nonzero elements of H k, can ensure it is a full rank matrix. hat is, there is no redundant row in H k. When there are redundant rows in the arity check matrix, the actual code rate is greater than the theoretical design code rate, which make the code rate is not controlled. he construction of arity check matrix in [] is also emloying systematic form. And the notable feature is reflected on the structure of the base matrix, which is constructed by masking method. When the highly structured arrays are densely acked, the density of such an array can be reduced by relacing a set of CPMs by zero matrices. his relacement of CPMs by zero matrices is referred to as masking. Masking an array of CPMs can result in an array of CPMs and zero matrices whose anner grah have fewer edges and hence have fewer short cycles and ossibly larger girth. However, how to construct the masking matrix with large girth and minimum distance is a very challenging roblem. As seen in Fig. 6, comared with the code in [9], 24 Engineering and echnology Publishing 545

6 Journal of Communications Vol. 9, No. 7, July 24 the roosed bi-diagonal code can achieve.5db SNR erformance imrovement when BER is -5. Besides, when the SNR is 3.5dB, the BER erformance order of the roosed bi-diagonal code is -6, while the code in [] is -3, the ga between them is three orders of magnitude. What s more, the convergence of the code constructed in [] is slow. IV. CONCLUSION In summary, we have resented a new method to construct irregular QC-LDPC code based on CPMs of different sizes. he main advantages of the roosed QC- LDPC codes are with large girth and irregular structure. What s more, our arity check matrix is quasi-cyclic and comosed by zero matrix and CPMs of different sizes. Encoding rocess can be achieved in a recursive way with few calculations. Simulation results show that the roosed QC-LDPC codes outerform the CCSDS standard code and several existing codes with same size CPMs under AWGN channel. REFERENCES [] Exerimental Secifications, Low Density Parity Check codes for Use in Near-Earth and Dee Sace Alications, CCSDS 3.- O-2, 27. [2] R. G. Gallager, Low-density arity-check codes, IRE rans. Information heory., vol. I-8,. 2-28, Jan [3] L. Zhang, Q. Huang, S. Lin, K. Abdel-Ghaffar, and I. F. Blake, Quasi-cyclic LDPC codes: An algebraic construction, rank analysis, and codes on Latin squares, IEEE rans. Commun., vol. 58, , Nov. 2. [4]. Richardson, Error floors of LDPC codes, in Proc. 23 Allerton Conf. Commun., Control and Comuting, he University, 23, [5] G. Liva, W. E. Ryan, and M. Chiani, Quasi-cyclic generalized LDPC codes with low error floors, IEEE rans. Commun., vol. 56, , Jan 28. [6] Y. Wang, S. C. Draer, and J. S. Yedidia, Hierarchical and highgirth QC-LDPC codes, IEEE rans. Inf. heory, vol. 59, , July 23. [7]. ian, C. R. Jones, J. D. Villasenor, and R. D. Wesel, Selective avoidance of cycles in irregular LDPC code construction, IEEE rans. Commun., vol. 52, , Aug. 24. [8] J. Wang, L. Dolecek, and R. D. Wesel, he cycle consistency matrix aroach to absorbing sets in searable circulant-based LDPC codes, IEEE rans. Inf. heory, vol. 59, , Aril 23. [9] J. Chen, R. M. anner, J. Zhang, and M. P. Fossorier, Construction of irregular LDPC codes by quasi-cyclic extension, IEEE rans. Inf. heory, vol. 53, , Aril 27. [] S. J. Johnson and S. R. Weller, A family of irregular LDPC codes with low encoding comlexity, IEEE Commun. Lett., vol. 7,. 79-8, Feb. 23. [] L. Wang, X. Zhang, F. Yu, Y. Fu, and Y. Wang, QC-LDPC codes with girth eight based on indeendent row-column maing sequence, IEEE Commun. Lett., vol. 7, , Nov. 23. [2] S. Kim, J. No, H. Chung, and D. Shin, On the girth of tanner (3, 5) quasi-cyclic LDPC codes, IEEE rans. Inf. heory, vol. 52, , Aril 26. [3] P. Yang, X. Bao, and H. Zhao, Construction of quasi-cyclic LDPC codes with large girth based on circulant ermutation matirx, in Proc. 22nd Wireless and Otical Communication Conference (WOCC), Chong Qing, 23, [4] K. Liu, Z. Fei, J. Kuang, and X. Li, A novel algorithm for removing cycles in quasi-cyclic LDPC codes, in Proc. 2th International Symosium on Personal, Indoor and Mobile Radio Communications, Bei Jing, 29, [5] J. Zhang and G. Zhang, Deterministic girth-eight QC-LDPC codes with large column weight, IEEE Commun. Lett., vol. 8, , Aril 24. [6] M. Yang, W. E. Ryan, and Y. Li, Design of efficiently encodable moderate-length high-rate irregular LDPC codes, IEEE rans. Commun., vol. 52, , Aril 24. [7] J. Fan and Y. Xiao, A design of LDPC codes with large girth based on the sub-matrix shifting, in Proc. IE International Conference, Wireless, Mobile and Multimedia Networks, Hang Zhou, 26,. -4. [8] R. M. anner, A recursive aroach to low comlexity codes, IEEE rans. Inf. heory, vol. 27, , Se 98. [9] H. Jin and R. J. McEliece, Coding theorems for turbo code ensembles, IEEE rans. Inf. heory, vol. 48, , Jun 22. [2] Y. Zhang and W. E. Ryan, Structured IRA codes: erformance analysis and construction, IEEE rans. Commun., vol. 55, , May 27. [2] J. Wang, J. Lei, S. Ni, X. ang, and G. Ou, QC-IRA-d codes based on circulant ermutation matrices, IEEE Commun. Lett., vol. 5, , Nov. 2. Hui Zhao received the Ph.D. degree in Astronautics from Harbin Institute of echnology. She now is a rofessor and mainly engaged in the research direction for signal and information rocessing, digital communication, theory and technology of otical network. Xiaoxiao Bao is a master and mainly engaged in the research of channel code, esecially LDPC codes. In addition, she is articiating in roject study of the key technology research on dee sace laser reliable communication. Liang Qin is a master and mainly engaged in the research of LDPC codes in dee sace communications. In addition, he is articiating in roject study of the key technology research on dee sace laser reliable communications. 24 Engineering and echnology Publishing 546

7 Journal of Communications Vol. 9, No. 7, July 24 Ruyan Wang is a rofessor and mainly engaged in the research direction for wireless communication, sace otical communication, theory and technology of otical network, otical information rocessing, and communication network reliability fault management. Hong Zhang received the M.S. degree in Chongqing University of Posts and elecommunications in 23. He is currently a researcher in key laboratory of otical fiber communication. His research includes wireless communications, survivability technology and fault management of otical networks and next generation assive otical network. 24 Engineering and echnology Publishing 547