A Novel QC-LDPC Code with Flexible Construction and Low Error Floor

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1 A Novel QC-LDPC Code wit Flexile Construction and Low Error Floor Hanxin WANG,2, Saoping CHEN,2,CuitaoZHU,2 and Kaiyou SU Department of Electronics and Information Engineering, Sout-Central University for Nationalities, Wuan, Cina 2 Huei key Laoratory of Intelligent Wireless Communications, Wuan, Cina wangx8888@63.com, spcen@scuec.edu.cn, zucuitao@63.com, sukaiyou@gmail.com Astract-- Slide rectangular window structure for QC-LDPC codes (SRW-QC-LDPC) wit flexile code lengts and code rates is proposed, wic aim to eliminate te cycles of lengt 4 witout computer searc. Te parity-ceck matrix would ave different extension factors and structures y using te slide rectangular window in te ase matrix, te degree distriution is optimized y te optimal diagonal metod. Because te dual-diagonal structure wit many variale nodes of degree-2 may lead to ig error floor, SRW-QC-LDPC codes wit quasi tri-diagonal structure are also proposed y canging te location of te tird diagonal to partly eliminate variale nodes of degree-2 for lower error floor. Simulation results sow tat SRW-QC-LDPC codes wit quasi tri-diagonal structure can not only flexily expand te code lengts and code rates ut also reduce te encoding complexity and improve te performance compared to quasi dual-diagonal structure in IEEE802.6e QC-LDPC codes. Te novel QC-LDPC codes are availale and suitale for te adaptive transmission systems and ardware implementation. Keywords QC-LDPC codes; Diagonal Structure; Degree Distriution; Encoding Complexity; Error Floor I. INTRODUCTION QC-LDPC codes are quasi cyclic low density parity ceck codes wit simple structure and efficient encoding algoritm, tey are caracterized y te parity-ceck matrix H wic consists of small square locks wic are te zero matrix or circulant permutation matrices. QC-LDPC codes ave een widely used in te fields of moile communication, suc as IEEE802.6e, DVB-T2 and IEEE802.n, etc. In te construction of LDPC code s matrix H, it sould e considered to avoid te existence of cycles of lengt 4. Cycle refers to te numer of edges wic form a closed loop in te corresponding Tanner grap of matrix H, teminimumlengt of cycles is called girt[] [2]. By optimizing matrix H, te larger girt can e realized to improve error correction capaility of QC-LDPC codes. BIBD algoritm is proposed to construct te girt 0 for matrix H in [3]. Te girt of matrix H is increased to 8 y constructing te moter matrix in [4]. Altoug te large girt is elpful to otain etter performance, only a few LDPC codes meet te requirements, tus te practical application of LDPC codes is limited. No cycles of lengt 4 is essential to optimize te numer of cyclic sifting and degree distriution of LDPC codes to improve teir performance. Te typical scemes include IEEE802.6e QC-LDPC codes wit girt 6, te maximized ACE and minimized criterion algoritm, ut te complexity of tese algoritms is iger. In addition, te aritmetic progression metod can also rapidly construct te matrix H witout cycles of lengt 4, ut te performance is not good enoug [5] [6] [7] [8]. Wen te ase matrix is larger, te lengt of te permutation matrix is also very long, te range of code lengt is limited so as to not e suitale for adaptive transmission system wit te varied code lengts and code rates [9]. QC-LDPC codes usually adopt lower triangular structure of quasi dual-diagonal, coding efficiency and performance is not good [0]. In order to acieve more efficient encoding, several improved structures ave een put forward, suc as tri-diagonal structure, new lower triangle structure and ackward iteration structure [] [2] [3] [4]. However, tri-diagonal structure is not flexile, te new lower triangle structure and ackward iteration structure contain many variale nodes of degree-2 similar to te quasi dual-diagonal structure, wic will lead to te iger error floor. Te aove improved QC-LDPC codes are mostly restricted in te code lengts and code rates flexiility. Tis paper firstly presents a novel slide rectangular window (SRW) sceme ased on te aritmetic progression to construct matrix H of QC-LDPC codes and uses te optimal diagonal metod to optimize te degree distriution of matrix H. Secondly, a kind of quasi tri-diagonal structure is proposed, it can partly eliminate te variale nodes of degree-2 y adjusting te location of te tird diagonal to acieve lower error floor. Te experimental results sow tat te proposed SRW-QC-LDPC codes wit quasi tri-diagonal structure can not only flexily expand te code lengts and code rates ut also reduce te encoding complexity and improve te performance compared to quasi dual-diagonal structure in IEEE802.6e QC-LDPC codes. Te novel QC-LDPC codes are availale and suitale for te adaptive transmission systems and ardware implementation. Te rest of tis paper is organized as follows. In section 2, we present te SRW sceme to construct QC-LDPC codes and use te optimal diagonal metod to optimize te degree distriution. In section 3, te proposed quasi tri-diagonal structure and fast encoding algoritm is introduced. Te simulation results for SRW-QC-LDPC codes wit te quasi ISBN Feruary 69, 204 ICACT204

2 tri-diagonal structure are sown in section 4 and te conclusions are drawn in te last section. II. QC-LDPC CODES WITH SLIDE RECTANGULAR WINDOW STRUCTURE A. Sliding rectangular window construction for matrix H TeasematrixH for parity-ceck matrix H of LDPC codes are composed of a group of zero matrix and circulant permutation matrices, it is sown as follows: S S2 S n = S S S () n H Sm S m S 2 mn were Sij( i m, j n)(-, 0, N) is te circulant permutation matrices in it row and jt column of te matrix H, - denotes zero matrix, 0 denotes identity matrix, te positive integer N denotes rigt cyclic sifting matrix from identity matrix. If te expansion factor z denotes dimension of permutation matrix, te matrix H will e extended to otain matrix H wit te size of (z*m)*( z*n). In te construction of te matrix H of QC-LDPC codes, it is important to avoid te existence of te cycles of lengt 4. In order to realize no cycles of lengt 4, z satisfies a certain value and S ij must satisfy: i j =, i m S = (2) ij Si( j) 2 j n, i m Te ase matrix H of QC-LDPC codes ased on te quasi dual-diagonal structure can e expressed as follows: H = [ H H ] (3) 2 H 2 k (4) H = r r 2 r k m 2 m mk H 2 () H = 2 ( r ) (5) 0 0 ( m ) 0 were H and H2 is te information its and te parity its portion of matrix H, respectively. H wit te size of m*k was sparse matrix witout cycles of lengt 4. H2 wit te size of mm is a fixed structure matrix, 0 and - denotes te identity matrix and zero matrix, respectively, ()=(m) is te prime numer less tan z, (r)=0. According to (2), matrix H witout cycles of lengt 4 is constructed as follows: f + ( i ) rt j = = (6) ij i + ( i ) ( ct + ( j )) < j k Te corresponding z sould satisfy: z ( m ) rt + ( k ) ( ct + m ) (7) were i m, f is elements value of te first row and first column in te matrix; ct and rt is te common difference of te first row and first column, respectively; m and k is te row and column numer of te matrix H, respectively. Due to te code rate R=k/(k+m), te code rates and code lengts can flexily cange y adjusting m and k value. For example, m=k=6, R=/2, te minimum code lengt is 6*62=372, step lengt is 2; If R want to /3, one way is to put k=3 wile m maintain, it is equivalent to delete tree columns in te matrix H, te corresponding minimum code lengt ecomes 9*6=44, step lengt 9; Anoter way is to put m=4 and k=2, it is equivalent to delete two rows and four columns of te matrix H, te minimum code lengt turn into 6*7=42, step lengt into 6. Oviously, te code rates and code lengts transformation is very flexile witout matrix reconstruction ut simple processing of row and column. In addition, te different matrix H and z can e constructed y modifying te value of f, ct and rt. For instance, wile f=, ct=0 and rt=, a matrix wit te size of 6*6 is constructed in te solid rectangular window and wen f, ct and rt value is 3, and 2, te matrix will ecome te dased rectangular window as sown in Fig.. It seems to slide rectangular window to te rigt and low position in matrix, tus named as Slide Rectangular Window (SRW) metod. Te specific steps of matrix H construction wit SRW metod are as follows: Step : On te asis of (6), determine a rectangular window matrix, te size is m*k. Step 2: According to (7), determine te corresponding extension factor z and slide rectangular window s position in te matrix. Step 3: Regard slide rectangular window matrix as te matrix H, and comine it wit matrix H 2 wose size is m *m, ten matrix H is finised. To construct matrix H wit SRW, te corresponding extension factor z is smaller, it can gain flexile and diverse code lengts and code rates of QC-LDPC codes, it is more suitale for adaptive transmission system H = Figure. Slide rectangular window B. Optimal diagonal metod for Matrix H In SRW metod, regards directly slide rectangular window matrix as H and ten constructs H, te degree of rows and columns is te same. It is equivalent to a regular LDPC code wose performance is poor []. We adopt te optimal diagonal metod to modify te matrix H and make it ave te different degree distriution to improve performance of ISBN Feruary 69, 204 ICACT204

3 LDPC codes. Specific means is to use - instead of certain diagonal elements in te matrix H wit SRW, i.e. using zero matrix instead of te permutation matrix, it will realize te optimal degree distriution. E.g. te solid rectangular window matrix in Fig. is modified to get te matrix H as follows: H 2 = Te optimized matrix H comined te matrix H2 to otain matrix H as follows:. H = After optimal diagonal metod for matrix H wit SRW, te performance of te constructed QC-LDPC codes is more improved. Meanwile, te code lengts and code rates are flexile and various, it is adaptive in wireless moile communication system. III. QC-LDPC CODES WITH QUASI TRI-DIAGONAL STRUCTURE A. Quasi tri-diagonal construction for matrix H2 QC-LDPC codes mostly use te quasi dual-diagonal structure for matrix H2. In order to improve coding efficiency and performance, we present a quasi tri-diagonal structure for matrix H 2 in tis section. Te existing optimal structures for matrix H2 include tri-diagonal structure, new lower tri-triangle structure and reverse iterative structure. In te tri-diagonal structure, tree diagonal elements are not -, one of tem is zero, te oter two are natural numers and teir values need to meet certain condition, so te code lengts and code rates are restricted. In new lower tri-triangle structure, tree elements in te first column is not - and two elements in te rest column is not -. In reverse iterative structure, te elements of two diagonal are 0, te rest are. Te variale nodes of degree-2 accounts for te vast majority and tese low degree variale nodes will lead to te ig error floor. To solve te disadvantages of tese existing structures, we propose a quasi tri-diagonal construction for matrix H 2 as follows: (0) 0 H 2 = r() 0 rl () 0 0 Were, te element of matrix 2 is only 0 or -, and only tree diagonal elements are 0, r(i)0, r() is te start position of te tird diagonal, r (l) is te end position. In order to H (8) (9) satisfy te no cycles of lengt 4, te row values of r () sould not e less tan 3. From (0), it can e seen tat te quasi tri-diagonal structure as te advantages of flexily adjusting te tird diagonal position compared wit tri-diagonal structure. And te quasi tri-diagonal structure can also partly eliminate te variale nodes of te degree-2 compared to te new lower tri-triangle and reverse iterative structure. Wit te increase of 2 H matrix dimension, te selected tird diagonal will e various, te degree distriution is also different, so we can properly select te tird diagonal position to optimize te degree distriution so as to improve performance of te QC-LDPC codes. Using SRW metod in section II, te matrix H is constructed at m=9 and k=9. According to (0), te matrix H 2 is constructed. Two matrices are comined to te matrix H as sown in figure 2. It can e seen tat te tird diagonals ave 6 kinds of selection wic uses olique line to lael te tird diagonal position. Wen one of te laeled tird diagonals is selected, ten te elements - on tis diagonal is replaced y 0. H = Figure 2.MatrixH wit SRW and quasi tri-diagonal structure B. Encoding algoritm for quasi tri-diagonal structure Te specific steps of fast encoding algoritm for te quasi tri-diagonal structure are as follows: Step : Information its s and parity ceck its p is segmented, eac segment lengt is z, te codeword c is expressed as follows: [ ] 2 k 2 c= s p = s s s p p p () m were k=k/z, m=m/z. Step 2: According to te parity-ceck matrix equation T Hc i = 0, p can e calculated y: k j= (, ) p = Z j i s (2) j k pi + Z( i, j) isj, i= 2,, cr j= pi = (3) k pi c p (, ),,, r + + i + Z r j sj i= cr m i j= were cr is te row value of r() in matrix H 2 and Z ( i, j) is te permutation matrices of it row and jt column in te matrix H. Step 3: Comine p = p p2 p m from step 2 and information its s to end up wit codeword c. In te quasi dual-diagonal structure, p is calculated y: ISBN Feruary 69, 204 ICACT204

4 m k ( ) (, ) p = Z + Z + Z i Z i j i s (4) r m j i= j= Oviously, te calculation amount of p in (2) is smaller tan tat in (4). Tat is to say, te computational complexity of te quasi tri-diagonal structure coding algoritm is lower tan te quasi dual-diagonal. Ta. sows te comparing results for p computation of quasi tri-diagonal and quasi dual-diagonal structure. Here, te code rate is /2, code lengt of n, extension factor is z. Tale. Comparison for p computation of two structures Diagonal structure Addition operation Multiplication operation quasi-dual (z-)*n/(4z) n 2 /4 quasi-triple (z-)*n/2 nz/2 IV. SIMULATION RESULTS AND ANALYSIS A. Simulation results of SRW-QC-LDPC codes Te simulation experiments are under te fast iterative coding, BPSK modulation, LLR BP decoding, 20 iterative numer and AWGN cannel. In Ta.2, m =6andk takes different values, ten te code rates cange from /7 to 5/8, and te code lengts also vary from 42 to 86 wit te different steps. It is sown tat te code lengts and code rates for SRW-QC-LDPC codes are flexile, it can meet te need of adaptive transmission system. Tale 2. Parameters of SRW-QC-LDPC m k Code Rate Code lengt (t=0,,2,...) 6 /7 42+7t 6 2 /4 88+8t 6 3 /3 44+9t 6 4 2/5 20+0t 6 5 5/ 286+t 6 6 / t 6 7 7/ t 6 8 4/ t 6 9 3/ t 6 0 5/8 86+6t Fig.3 gives te performance comparison of asic ceck matrix wit te different degrees efore and after te optimal diagonal metod. We can find tat te performance is a little improved wen te degree is less tan 6, ut te optimal diagonal metod as ovious advantages wen te degree is greater tan 6 and SRW-QC-LDPC codes performance is greater improved no optimization(m=4,n=8,l=04) optimization (m=4,n=8,l=04) 0-4 no optimization(m=6,n=2,l=372) optimization (m=6,n=2,l=372) no optimization(m=8,n=6,l=92) optimization (m=8,n=6,l=92) Figure 3. wit te optimal diagonal metod In IEEE802.6e QC-LDPC codes, tere are many code lengts and code rates, so we coose 4 code lengts 576, 960, 440 and 2304 to do simulations. Te experimental results for SRW-QC-LDPC and 802.6e-QC-LDPC codes are sown in Fig SRW-LDPC(576) 802.6e-LDPC(576) 0-4 SRW-LDPC(960) 802.6e-LDPC(960) SRW-LDPC(440) e-LDPC(440) SRW-LDPC(2304) 802.6e-LDPC(2304) SNR(dB) Figure 4. for SRW-QC-LDPC and 802.6e-QC-LDPC It can e seen from Fig.4 tat SRW-QC-LDPC codes are sligtly losses on te performance compared to 802.6e QC-LDPC codes, ut te signal-to-noise ratio (SNR) is more tan 2dB, te of SRW-QC-LDPC codes wit te code lengt 2304 can reac 0-5, it can satisfy te demand of system. Ta.3 sows te parameters of SRW-QC-LDPC codes compared wit 802.6e standard and literature [2]. It can e seen tat te code rates of 802.6e-QC-LDPC codes ave only four of /2, 2/3, 3/4 and 5/6, te code lengts ave a total of 9 from te minimum 576 to maximum 2304 wic te step lengt is 96. In literature [2], te code rates are expanded to a total of 0 from /4 to 3/4, te minimum code lengt is 336, te step lengt is on a total of 6 from 4 to 9. Te code rates of SRW-QC-LDPC codes can expand witout limits, te minimum code lengt reaces to 42, te minimum step lengt is 4. It oviously indicates tat SRW-QC-LDPC codes can flexily adjust te code rates and te code lengts to e more availale and suitale for adaptive transmission system. Tale 3. Parameters of QC-LDPC wit different matrices H IEEE802.6e Reference [2] SRW Code rates /2,2/3,3/4,5/6 /4,...,2/3,...,3/4 unlimited Code lengts ,... 42,... Step size 96(t+),(t=0,,......) 4,5,6,7,8,9 4,... B. Simulation results and analysis of SRW-QC-LDPC codes wit difference quasi tri-diagonal structure Te parity ceck matrix H can e otained from te ase matrix H in Fig.2. By selecting te different tird diagonal position of matrix H, te performance is simulated for SRW-QC-LDPC codes wit difference quasi tri-diagonal. Te simulation results are sown in Fig.5, were frame numer is 300, te code lengt 52, te code rate /2, extension factor z=64. It can e oserved from Fig.5 tat te performance in te and diagonal is significantly etter tan te oters, and te diagonal is worst. It implies tat te tird diagonal of matrix H selects position is te est in AWGN cannel. ISBN Feruary 69, 204 ICACT204

5 Due to te different degree distriutions in te different cannels, SRW-QC-LDPC codes wit different quasi tri-diagonal ave more advantages ecause of teir flexile and optimal degree distriutions. 0 - (8, 6) and (9, 8) Figure 5. for SRW-QC-LDPC wit different quasi tri-diagonal Fig.6 gives te performance for QC-LDPC codes under te same matrix H and te different matrix H2 wic is te quasi dual-diagonal and quasi tri-diagonal structure, respectively. Here, te code lengts take 576, 960 and 52, te frame numer is 000, 500 and 300, te code rate is /2, te matrix H of quasi tri-diagonal structure is taken te size of te (6, 2), (8, 6) and (9, 8) dual (576) 0-4 triple (576) dual (960) 0-5 triple (960) dual (52) triple (52) Figure 6. comparison for QC-LDPC wit quasi dual-diagonal and quasi tri-diagonal structure It is ovious in Fig.6 tat te performance of quasi tri-diagonal structure as more and more ovious advantage wit te increase of te code lengts. E.g. Wen reaces 0-4, te SNR of quasi dual-diagonal structure needs 2.4dB, ut te SNR of quasi tri-diagonal only needs 2.05 db, te coding gain is nearly 0.35dB. Tis is ecause wit te increase of te code lengt, te tird diagonal coice for te quasi tri-diagonal structure gets more, te degree distriution is more flexile variation, terefore te cance to near a good degree distriution is greater. Fig.7 gives te performance comparison etween te quasi tri-diagonal SRW-QC-LDPC codes and 802.6e QC-LDPC codes under te different code lengts. Here, te code lengts take 576, 960 and 52, te frame numer is 000, 500 and 300 respectively, te code rate is /2, te matrix H of quasi tri-diagonal structure is taken te size of te (6, 2), e (576) 0-4 SRW-Triple (576) 802.6e (960) 0-5 SRW-Triple (960) 802.6e (52) SRW-Triple(52) Figure 7. for 802.6e QC-LDPC and quasi tri-diagonal SRW-QC-LDPC It is clear in Fig.7 tat te performance of quasi tri-diagonal SRW-QC-LDPC codes is poorer wen te code lengt is 576. But te performance of quasi tri-diagonal SRW-QC-LDPC codes is near to IEEE802.6e LDPC codes wen te code lengt is 960 and 52. In addition, te performance of quasi tri-diagonal SRW-QC-LDPC codes is even etter tan te IEEE802.6e LDPC codes at te iger SNR and longer code lengts. V. CONCLUSIONS Tis paper proposed a novel SRW-QC-LDPC code wit quasi tri-diagonal structure wic can flexily cange te code rates and code lengts wit lower complexity, it is more availale and suitale for adaptive transmission system. Te simulation results sow tat te presented structure can not only improve te performance of QC-LDPC codes ut also reduce te encoding complexity compared wit dual-diagonal structure. Wen te code lengt is te longer, te performance of SRW-QC-LDPC codes wit quasi tri-diagonal construction is near to IEEE802.6e LDPC codes and is even etter tan te 802.6e standard at te iger SNR. ACKNOWLEDGMENT Tis work was sponsored y te National Natural Science Foundation of Cina wit Grant No and te Natural Science Foundation of Huei Province wit Grant No.203CFB448, it was also supported y te Key Project of te Fundamental Researc Funds for te Central Universities, Sout-Central University for Nationalities wit Grant No.CZZ300. REFERENCES []Fossorier M P. Quasi-Cyclic Low Density Parity Ceck Codes From Circulant Permutation Matrices[J]. IEEE Trans. Info. Teory, 2004, 50(8): [2] Tanner R M. A recursive approac to low complexity codes [J]. IEEE Trans. Info. Teory, 98, 27(5): [3] Zang Fan, Mao Xueong, Zou Wuyang.Girt-0 LDPC codes ased on 3-D cyclic lattices [J]. IEEE Trans. on Veicular Tecnology, 2008, 57(2): [4] Kim S, No J and Cung H. On te girt of Tanner (3,5) Quasi-Cyclic LDPC codes [J]. IEEE Trans. Info. Teory, 2006, 52(4): ISBN Feruary 69, 204 ICACT204

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