Optimization Design and Analysis of Systematic LT codes over AWGN Channel

Transcription

1 Optmzato Desg ad Aalyss of Systematc LT codes over AWGN Chael Shegka Xu, Dazhua Xu, Xaofe Zhag ad Haq Shao College of Electroc ad Iformato Egeerg Najg Uversty of Aeroautcs ad Astroautcs Emal: Abstract I ths paper, we study systematc Luby Trasform (SLT) codes over addtve whte Gaussa ose (AWGN) chael. We troduce the ecodg scheme of SLT codes ad gve the bpartte graph for teratve belef propagato (BP) decodg algorthm. Smlar to low-desty party-check codes, Gaussa approxmato (GA) s appled to yeld asymptotc performace of SLT codes. Recet work about SLT codes has bee focused o provdg better ecodg ad decodg algorthms ad desg of degree dstrbutos. I our work, we propose a ovel lear programmg method to optmze the degree dstrbuto. Smulato results show that the proposed dstrbutos ca provde better bterror-rato (BER) performace. Moreover, we aalyze the lower boud of SLT codes ad offer closed form expressos. Idex terms fouta codes; systematc LT codes; Gaussa approxmato; lear programmg; lower boud; AWGN chael

2 I. INTRODUCTION Fouta codes [, ] were frst troduced over bary erasure chaels (BEC). Due to packets loss, such as Iteret, automatc repeat request (AR) was adopted. However, request ad retrasmsso occur frequetly AR ad that leads to low effcecy. Ths problem does ot exst Luby Trasform (LT) codes. LT codes [3] are the frst class of fouta codes to be realzed. Gve some put symbols, fte output symbols are geerated ad the trasmtted to the recever tll all the put symbols are recovered. Thus by usg LT codes, we do ot ecessarly eed to kow the chael state formato (CSI). However, decodg complexty of LT codes creases rapdly as the legth of put symbols creases, whch s ot good for practcal applcatos. Raptor codes [4] fx the problem because the decodg complexty of Raptor codes s learly depedet o the legth of put symbols. The dea of fouta codes over BEC was exteded to other chaels, for stace, bary symmetrc chaels (BSC), addtve whte Gaussa ose (AWGN) chaels, ad fadg chaels. The performace of fouta codes o these chaels s also satsfyg [5-7]. Lowdesty party-check (LDPC) codes [8] were frst veted by Gallager 96. Belef propagato (BP) decodg algorthm was used to decode LDPC codes o osy chaels ad t s based o the bpartte graph of party-check matrx. As LT codes ad LDPC codes both have lowdesty geerator matrx, BP algorthm s stll sutable for LT codes [9]. Eve though LT codes ca deal wth dfferet chael stuatos, they stll suffer error floor o osy chaels [0]. Systematc codes are popular may practcal applcatos; however LT codes ad Raptor codes are ot desged systematcally. Takg BEC for a example, f packets are trasmtted through the chael ad there s few or eve o loss, systematc codes would performace way better tha osystematc codes cosderg the cost. Several works have bee doe o systematc

3 Luby Trasform (SLT) codes []. Yua [] proposed a famly of systematc rateless codes for BEC. I [3], Nguye provded a desged dstrbuto for systematc LT codes. Che [4] proposed aother desg of degree dstrbuto. Zhag [5] gave a ew soft decodg method to mprove performaces. I [6], Hayajeh provded a ew ecodg algorthm by shapg the left degree dstrbuto away from Posso dstrbuto. I [7], Asters troduced a ew famly of fouta codes that are systematc ad have sparse partes. Besdes, Che [8] studed systematc Raptor codes wth effcet ecodg method. I ths paper, we focus o SLT codes over AWGN chael ad aalyze the asymptotc performace usg Gaussa approxmato. More mportatly, we gve a ovel lear programmg method to optmze check ode degree dstrbuto. Our optmal dstrbutos ca provde better performace tha the desged dstrbuto [3]. Last but ot least, we derve some lower boud expressos of SLT codes. The remaders of the paper are orgazed as follows. Secto II wll troduce the ecodg ad decodg algorthms of SLT codes. I secto III we wll use Gaussa approxmato to aalyze the asymptotc performace ad a ovel optmzato of degree dstrbuto s proposed. Several lower boud expressos are derved for the frst tme Secto IV ad some smulato results are show Secto V. Secto VI wll coclude the paper. II. SYSTEM MODEL We cosder SLT codes trasmtted over AWGN chael. The ose s Gaussa 0,. Oe put symbol ca be just oe bt or a block of bts. Ether way, t has o mpact o the aalyss of SLT codes. So we use a bt to represet a put symbol ths paper. After SLT ecodg, we adopt bary phase shft keyg (BPSK) modulato. By c we deote the ecoded

4 SLT codes ad xcthe BPSK modulated symbols trasmtted to the recever. By addg chael ose, the receved symbols are y x. A. Geerato of SLT codes Let K deote the legth of put symbols ad,,, D be the degree dstrbuto o set,,, D so that deotes the probablty that degree s chose. Geerally we deote such dstrbuto by ts geerator polyomal,.e., x The geerato of LT codes takes several steps as follows [3]. Sample a degree wth probablty x ; D x where D s the maxmum degree. Sample dfferet put symbols uformly at radom from K total ad the XOR them; Repeat above two steps to get output LT ecoded symbols. LT codes are desged to be rateless but actual applcatos the output symbols are fte. Suppose we have M ecoded output symbols ad let M K deote extra percetage of put symbols that are eeded at the recever, whch s called overhead. I SLT codes, put symbols are trasmtted alog wth LT ecoded symbols. Let u u u u,, K deote K put symbols. By the geerato of LT codes, we ca get M ecoded symbols,.e., c LT LT c, c,, cm ug where G LT s the geerator matrx of sze K M ad thus SLT codes are, cslt u c LT. By addg a uty matrx of sze K K to G LT we ca get the geerator matrx of SLT codes G [ I G ]. Let N K M deote the legth of SLT codes. LT Smlar to LT codes, we defe overhead as N K M K. Afterwards, SLT codes are modulated as BPSK symbols ad trasmtted to the recever through AWGN chael.

5 B. BP deodg algorthm for SLT codes Smlar to LDPC codes, decodg SLT codes over AWGN chael s based o the bpartte graph. By H we deote the party-check matrx of SLT codes as H [ G I] of sze M N, T LT where T G LT s the trasposto of G LT. Each row ad colum H represets a check ode ad a varable ode, respectvely. Fg. gves a stace of bpartte graph based o party-check matrx of SLT codes, where varable odes ad check odes are o opposte sdes. As ca be see, varable odes ca be dvded to two dfferet types, amely, source odes ad check odes. The log-doma of BP decodg algorthm uses log-lkelhood ratos (LLRs), whch s Pr ck 0 y LLR ck l Pr c y k k k () where y k s the receved symbol of the ecoded symbol c k. By Z we deote the tal LLRs related to chael Z y () I the followg, we deote by R m, ad m, the message passg from -th check ode to m -th varable ode ad the message passg from m -th varable ode to -th check ode, respectvely. By S \ m we deote the set of all odes adjacet to ode except m. I roud 0 of BP decodg, we talze varable odes wth messages Z. I roud l, messages passg through bpartte graph [] wll be updated as R l m, l Z m K - K k, tah tah tah, ks \ m m K l - tah tah k,, ks \ m (3)

6 l m, l Zm R, m K mk, ksm \ Zm, m K (4) Sce we oly focus o source odes ad t s mpled from (3) ad (4) that messages betwee source odes ad check odes o the same sde of bpartte graph are rrelevat ad besdes messages of those check odes are uchageable, we may adjust (3) ad (4) to smple oes whch we are more terested,.e., messages exchagg betwee source odes ad check odes o opposte sdes of bpartte graph, as (5) ad (6). The correspodg bpartte graph s modfed ad show Fg.. From ths pot o, further dscussos wll all be based o the ew graph. R l - K m, tah tah tah k S m l k, (5) \ Z l l Z R (6) m, m mk, ksm \ Durg the decodg teratos the graph of Fg. we may focus o the degree dstrbuto wth respect to edges rather tha odes. We deote by the fracto of edges coected to source odes of degree. Ad deotes the probablty a source ode chose the bpartte graph s of degree. We deote by x ad x the geerator polyomal d s x ad ds x, respectvely, where the maxmum degree of source odes s d s. d j j deote edge degree dstrbuto of check ode, where j c j Let x x s the fracto of edges coected to check odes of degree j ad d c s the maxmum degree of check odes. Recall that x s the check ode degree dstrbuto. The we have

7 x x x x (7) (8) Fg.. Bpartte graph of SLT codes, wth varable odes ad check odes o each sde. Fg.. New bpartte graph of SLT codes, wth source odes ad check odes o each sde. where f x deotes the formal dervatve of f x wth respect to x. x ad x are obvously depedet of umber of check odes whle x ad x umber of check odes. However whe M s large eough, may deped o the x s cosdered as Posso dstrbuto. We deote by the average degrees of source odes thus x x e ad x s approxmately Posso dstrbuto as well.

8 III. ASYMPTOTIC ANALYSIS AND OPTIMIZATION OF SLT CODES I ths secto, we use Gaussa approxmato to study the asymptotc performace of SLT codes over AWGN chael. Based o BP teratve decodg algorthm, closed form expresso of BER s derved. By eforcg the desty evoluto formulae, we propose a ovel lear programmg method to optmze degree dstrbuto of SLT codes. A. Gaussa approxmato(ga) Wberg [9] observed that LLR message dstrbutos for AWGN chaels resemble Gaussas for LDPC codes. Due to the smlartes of SLT codes ad LDPC codes, we assume R ad (5) ad (6) ca be well approxmated by Gaussa destes over AWGN chael. Sce a Gaussa varable s totally determed by ts mea ad varace, we oly eed to keep a eye o meas ad varaces durg BP teratos. There s a mportat symmetrc codto f x f xe x whch s preserved uder desty evoluto for all messages, where f x s a LLR message desty [0]. For approxmate Gaussa desty wth mea ad varace, by eforcg ths symmetrc codto we ca get. I such case we oly eed meas of Gaussa varables durg teratos. Meawhle we trasmt all-zero codes over AWGN chael. thus Z s Gaussa,4 Takg expectatos of both sdes (5), we get l l R Z Etah E tah Etah j (9) where we have omtted the dces because they are..d. Gaussa varables. Product (5) has also bee smplfed for the same reaso gve the check ode has j adjacet source odes for j d. Recall that both c l R ad l are Gaussa varables,.e., l l, ad R R

9 l l,, respectvely. Apparetly, l d c l ad R j j R, j l d s l,, where l ad R, j l, are meas of message dstrbutos of check odes wth degree j ad source odes wth degree, respectvely. Sce we defe a fucto r R r 4R tah, R E tah e dr 4 x [] as (0) for all x 0, Thus, (9) ca be rewrtte as x R rx r 4x tah e dr, x 0 4 x (0), x 0 d j s l l R, j, () where x s the verse fucto of x ad because there s o exst of closed form expresso for x, we ormally use a approxmate expresso for all x 0 x x 0.08 e, 0<x<0 x 0 4 e, x0 x 7x () Smlarly, takg expectatos of (6) we get l (3) l, After l teratos, the mea of LLR messages for decso s l l ad the R R asymptotc performace of BP decodg after l teratos as all-zero codeword trasmtted s

10 l l ds e P e, 0 l l d t 4 s e P ds 4 l l / dt (4) where x s the tal probablty of a stadard ormal dstrbuto. B. Optmzato of degree dstrbuto The performace of SLT codes maly depeds o check ode degree dstrbuto x so ths part we wll fd a way to optmze the dstrbuto. Recall that all messages passg from source odes to check odes some roud l are..d. Gaussa wth mea l. Durg BP decodg, the error probablty wll decrease from terato to terato whe the codto (5) satsfes. l l (5) By ad we deote the average degrees of source odes ad check odes, respectvely. Posso dstrbuto x s specfed by ts mea. By usg the updatg rules we have acheved wth GA, (5) ca be expaded as [6] where dc l l j f j j (6) j l l f j

11 Note that (6) s a lear equalty wth ukow coeffcets of check ode degree dstrbuto for all l l 0. Actually t s ot ecessary for all postve to hold ad we just l l eed to assume (6) holds for a rage of, say 0 0,. Meawhle, to acheve a certa error probablty, we may wat to be as small as possble. As d c j j, our objectve s to mmze j d c j j j. I practce, we ca use a lear programmg (LP) to solve the problem. As the LP procedure (7), we fx d c,, 0 ad teger L advace. k 0,, L are L equdstat pots the terval0,. k 0 j m dc st.. f k 0,, L, 0, j j j k k k dc j dc 0 j, d j j j j c 0 (7) Note that there s o eed to fx because we ca treat j as a whole. By dog so, the secod costrat (7) should be deleted. Oce LP procedure s doe, (8) ca be used to acheve the check ode degree dstrbuto x x 0 0 z dz z dz (8) We clude our optmzato results for the value Table I. Fg. 6 gves a plot of error probablty versus overhead by usg those optmal dstrbutos. We also compare our dstrbuto wth that of [3] Fg. 7.

12 TABLE I. LP RESULTS FOR DIFFERENT MAXIMUM DEGREES Maxmum degree d c Check ode degree dstrbuto x x 0.736x 0.639x x 0.389x 0.573x 0.098x x x x 0.070x IV. LOWER BOUND ANALYSIS I ths secto we wll carry out some lower boud aalyses to vestgate the performace of SLT codes over AWGN chael. Lemma : For a source ode of degree, whe overhead goes to fty, LLR messages are Gaussa,4. Proof: As we metoed (5), mea of LLR messages of varable odes must crease uder teratos so a successful decodg progress ca be assured. Ad tah x s close to whe x s large eough, so we ca assume that LLR messages passg from source odes to check odes are perfect [0] the last terato of BP decodg. Uder such assumpto, (5) ca be smplfed as R Z tah tah l - (9) whch meas R shares the same dstrbuto wth Z. Recall that LLR messages of the chael are Gaussa,4, so R,4. Furthermore, LLR messages passg

13 from check odes to source odes (6) are sum of..d Gaussa varables. Specfcally, for source odes of degree, messages are Gaussa,4 after teratos. I prevous secto, we have gaed BER expresso by usg GA. Furthermore, the lower boud of SLT codes ca be acheved by eforcg Lemma to (4),.e., here we defe (0) d P s e d LB s () We demostrate GA asymptotc performace ad lower boud LB Fg. 5. Note that base- 0 logarthm of asymptotc performace Fg. 5 s approxmately lear wth respect to overhead whe s large eough. So the followg, we wll gve aother form of lower boud. Theorem : Whe BPSK modulated SLT codes are trasmtted through AWGN chael, the lower boud of BER performace ca be approxmately expressed as LB g g e () where g e. Proof: By takg base-0 logarthm of equalty (0) ad eforcg Posso dstrbuto x we get

14 Tal probablty of a stadard ormal dstrbuto ad actually (4) s the upper boud of lg Pe lg elg! (3) x ca be approxmately expressed [] as x x 3 x e e (4) 4 x. Note that there are two terms of approxmate x ad the followg we have to just keep oly oe of them to obta a lear relatoshp betwee base-0 logarthm of BER ad overhead. Let x ad x deote the frst ad the secod term, respectvely. Namely, x e x x e 4 ad they are both depcted wth respect to degrees Fg. 3. It s mpled that x tha x 3 x s closer to x for large degrees. Cosder the product of Posso dstrbuto ad -fucto (0) ad we demostrate t Fg. 4. Clearly, BER. So x s approxmately expressed as (5) ths paper x e x x s much more accurate whe calculatg (5)

15 ((d+)/ ) ((d+)/ ) ((d+)/ ) fucto d Fg. 3. Comparso betwee x ad x wth respect to degree d, where s set to..4 x 0-7. d *((d+)/ ) d * ((d+)/ ) d * ((d+)/ ) 0.8 * d Fg. 4. Comparso betwee product of Posso dstrbuto ad dfferet -fuctos, where s set to. By usg (5), (3) ca be expaded as

16 lg Pe lg elg! lg elg e! lg elg e e! e lg elg e e e lgelg e (6) Obvously, a ew approxmate lower boud ca be defed as where Ad we are doe wth the proof. LB 0 g g e lge lg e g e e (7) The logarthm of our ew lower boud LB s a lear fucto of, whch verfes the prevous observato. LB s depcted Fg. 5 as well. V. SIMULATION RESULTS I ths secto, we offer some smulato results of our work. Throughout the whole smulatos, the chael s AWGN wth, whch meas sgal-to-ose-rato (SNR) s 0dB uder BPSK modulato.

17 Frst dfferet lower bouds LB ad LB are plotted Fg. 5 as well as asymptotc performace of GA. 3 x s chose as the check ode degree dstrbuto. The result shows that these three curves are close to each other gradually as creases ad t verfes our lower boud aalyses. The we demostrate BER performace of our optmal check ode degree dstrbutos Fg. 6, where x ad 3 x are chose. I order to study the mpact of the legth of SLT codes, we set K 000, 000, 4000 for dfferet dstrbutos. Clearly, the legth s fty whe usg GA. As ca be see, for a certa dstrbuto, BER performace gets better ad better whe K creases to fty. Ad also, performace of GA s the deal extreme for fte legth. Moreover, SLT codes of large average check ode degree ca outperform those of small average degree, whch s clear Fg. 6. At last, we compare our optmal dstrbuto wth that of [3]. The result s show Fg. 7. To be far, crtero of comparso s the same average check ode degree. We choose 3 x wth 4, whch s almost the same as the dstrbuto [3] wth parameters c 0.3, 0.5. Smulatos wth K 000 ad GA dcate that our optmal dstrbuto ca offer better BER performace. Besdes, lower bouds of those two dstrbutos ted to be the same. Ths ca be explaed usg lower boud expresso (7) due to the same ad.

18 GA LB LB 0 - BER overhead Fg. 5. Low bouds of SLT codes over AWGN chael BER (x) wth K=000 (x) wth K=000 (x) wth K=4000 (x), GA 3 (x) wth K=000 3 (x) wth K=000 3 (x) wth K= (x), GA overhead Fg. 6. BER performace of SLT codes over AWGN chael. Optmal dstrbutos are used wth dfferet source legth.

19 BER Dstrbuto [3] wth K=000 3 (x) wth K=000 Dstrbuto [3], GA 3 (x), GA overhead Fg. 7. Optmal dstrbuto Gaussa approxmato. x are compared wth that of [3], wth K 000 as well as 3 VI. CONCLUSION I ths paper we fully demostrate the ecodg ad decodg algorthms of SLT codes frst. SLT codes are the systematc form of LT codes, whch cosst of put symbols ad ecoded LT symbols. Afterwards, bpartte graph for BP decodg s provded. We smplfy the bpartte graph by just keepg source odes ad check odes o opposte sdes ad gve the updatg rules of LLR messages. The we use GA to aalyze asymptotc error probablty of teratve decodg. A LP programmg s proposed to optmze check ode degree dstrbuto ad results are provded. Lower boud of SLT codes are also studed ad we propose two low boud expressos. Fally, we do some smulatos to valdate our work. Results of fte legth SLT codes verfy the asymptotc performace. Moreover, our optmal dstrbuto outperforms the dstrbuto [3]. ACKNOWLEDGMENT The work was supported by the Natoal Natural Scece Foudato of Cha (grat. No 647

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