Channel Polarization and Polar Codes; Capacity Achieving

Transcription

1 Chael Polarzato ad Polar Codes; Capacty chevg Peyma Hesam Tutoral of Iformato Theory Course Uversty of otre Dame December, 9, 009 bstract: ew proposed method for costructg codes that acheves the symmetrc capacty (the capacty of the chael wth the same probabltes for the puts), I ( W ), of ay Bary Dscrete Memoryless Chael W (B-DMC) wll be troduced ths tutoral. Ths method s based o chael polarzato. Chael polarzato s the fact that we ca sythesze chaels, { W : } goes to fty, the fracto of dces for whch fracto of dces for whch, out of depedet copes of DMC chaels W, such that f s aroud approaches s aroud 0 approaches I ( W ) ad the. We ca clearly see some sort of polarzato that ca help us to costruct such codes that mprove the performace terms of probablty of block error; we sed data through the chaels wth aroud wth rate ad wth rate 0 through the rest of chaels. The whole dea of chael polarzato ad polar codes ca be stated oe precse theorem: Gve ay B-DMC W wth I ( W ) > 0 ad ay target rate R < I ( W ), there exsts a sequece of polar codes { ; } block-legth =, rate R decodg bouded as (, ) C such that C has, the probablty of block error uder successve cacellato 4 Pe R O depedetly of the code rate. Ths performace s achevable ecoder ad decoder complexty of O( log ) for each. Idex Terms: Capacty achevg codes, chael polarzato, polar codes, successve cacellato decodg, ML, B-DMC, BEC. I. Itroducto: Sce 948, whe Shao proved hs osy chael codg theorem, a lot of efforts have bee doe to costruct a code that acheves the capacty of the chael troduced by Shao. I 990 s Low Party Desty Code (LDPC) was troduced (veted 960 s) ad t was show that ths codes have a better performace terms of probablty of block error tha all the

2 exstg codes. Recetly, rka proposed aother codes called Polar Codes that seems to outperform the exstg codes DMC cases []. However, t has several smlartes wth the class of Reed-Mullet (RM) codes code costructo ad the way of ecodg [5]. The ma dea of polar codes ad chael polarzato was frst troduced for the bary DMC (B-DMC) []. However, later ths dea was geeralzed for ay DMC [3]. I ths tutoral, we ll just go through the bary case that has a wde rage of applcatos commucato systems. II. We use otato otatos: j a for deotg a subvector (,..., j ) j a a ad a, o to deote a subvector wth odd j dces ad a, e to deote a subvector wth eve dces. deotes the modulo- addto ad deotes Kroecker product of two matrces wth dfferet dmesos. The Kroecker power s defed as ( ) for all 0 wth the tal value of [ ]. We deote a DMC wth put alphabet χ ad output alphabet ϒ ad trasto probablty of W y x, x χ, y ϒ as W : χ ϒ. Furthermore, we deote depedetly use of DMC W W χ ϒ wth trasto probablty of W ( y x ) W ( y x ) as : = =. Wth these otatos we ca ow calculate the symmetrc capacty of DMC as follow: ( ) W y x I ( W ) W ( y x) log y ϒ x χ W y + W y ( 0 ) ( ) () d we ca defe the Bhattacharyya parameter as: ( 0 ) ( ) Z W W y W y y ϒ () whch s a upper boud o the probablty of MP decso error whe we trasmt a sgle bt wth equal probablty to be 0 or. It ca be show that ths parameter s a covex fucto of the chael trasto probabltes. III. Chael polarzato The process of chael polarzato whch we sythesze chaels, { W : }, out of depedetly copes of DMC chaels W ( ths tutoral B-DMC) cossts of two depedet steps: - chael combg phase - chael splttg phase. s we sad before the polarzato refers to the fact that I ( W ) wll polarze to 0 ad as goes to fty [4].

3 ) Chael combg I ths phase we combe copes of DMC W a recursve maer to produce a vector chael W : χ ϒ where =, 0. I ths recursve procedure we talze W ( = 0) as oe copy of B-DMC. I the ext recurso we combe two B-DMCs as show Fg. to costruct the chael W : χ ϒ (,, ) ( ) ( ) = wth the trasto probablty of: W y y u u = W y u u W y u (3) W Fg.. Chael W [] For the ext levels, we ca costruct the chael the same recursve maer. I geeral we ca costruct the chael W : χ ϒ by combg two depedet copes of Fg.. W as show We ca see from the fgure that s = u u ad s = u for. The block R s a permutato operator kow as reverse shuffle operato whch coverts ts puts s to (,,...,,,,..., ) v = s s s s s s. We ca readly see that the there s a lear trasformato from 3 4 the puts ofw, u, to the puts of chaels W. So, there s a geerator matrx G so that x = u G. It ca be show that G = B F 0 for ay =, 0 where F ad B s a permutato matrx kow as bt-reversal ad ca be computed recursvely as: B = R I B (4) Where I s the -dmesoal detty matrx. Ths recurso wll be talzed wth B = I. 3

4 Fg.. Recursve costructo of uderlyg raw chael W s lear over GF W from combg two copes of W from [].. So, these operatos to sythesze the chael W, are remscet of lear codes. ctually, polar code has much commo wth Reed- Muller (RM) code whch s a class of lear codes [4]. ) Chael splttg: So far, we have sytheszed chael W from depedetly copes of B-DMC W. ow t s tme to splt the chael W to costruct chaelsw : χ ϒ χ,. We costruct these chaels by the followg trasto probabltes: W y u u W y u ( ) (, ) ( ) u + χ (5) To see how we use these chaels operatoally ad ga a tutve uderstadg of the chael { W }, cosder a gee-aded successve cacellato decoder whch the th decso 4

5 elemet estmates u after observg y ad the past chael puts u (suppled correctly by the gee regardless of ay decso errors at earler stages). If u s a-pror uform o the W s the effectve chael see by the th decso elemet ths scearo. 3) Chael polarzato: ow we ca state the polarzato effect B-DMC chaels the followg theorem: Theorem : For ay B-DMCW, the chaels { W } χ, polarze the sese that, for ay fxed δ ( o,), as goes to fty through powers of two, the fracto of dces {,... } whch ( δ,] goes to ad the fracto for whch 0, δ ) for goes to I ( W ) []. Hece, as, chaels polarze, ether completely osy or ose free ad we kow these chaels exactly at the trasmtter. So, we fx bad chaels (completely osy oes) ad trasmt ucoded bts over the good oes. Example: Bary Erasure Chael (BEC) wth erasure probablty ofε = 0.5. I ths example we wat to explore the polarzato effect a BEC ad so we wat to compute { I ( W )} I ( W ) { } for dfferet values of. Ufortuately, o effcet algorthm s kow for calculatg for a geeral B-DMC. Ths s oe of future works that ca be explored. However, a effcet recursve formula s kow for computg ( ) ( ) = ( ), =, Italzg wth,..., { I ( W )} for a BEC as follows: = ε. Usg these formulas ad plottg the symmetrc capacty versus 0 = = Fg.3 the polarzato effect ca be observed clearly. We kow that symmetrc capacty for a BEC s ε = 0.5 = 0.5. So, accordg to the theory for δ very small, almost half of dots the above plot should be ear ad the rest half should be ear 0 f we assume dagram. 0 = to be large eough whch s clearly observable from the plot (6) 5

6 Fg.3. Plot of vs. 0 =,..., = for a BEC wth 0.5 ε = from []. IV. Polar codg We beeft from the chael polarzato to costruct polar codes that acheves I ( W ) based o the dea that we oly utlze chaels ear. ) G -coset codes: W for whch Z W s ear 0 or equvaletly These codes are more geeral ad cota polar codes as well. The block legth s the form of =, 0 ad ecodg s as follow: s x = u G = u G u G c c (7) Where G s the geerator matrx of order, s ay arbtrary subset of {,..., } ad G a submatrx of bts) ad leave (,,, c ) G formed wth dces. We fx (formato se) ad u as a free varable. K u where K s the dmeso of the code ad. u c K χ s (froze G -coset codes wll be determed by a parameter vector 6

7 For example, let G4 = 0 0. The the ( 4,, {, 4 },(,0 )) code has the ecoder mappg: x = u G 4 = ( u, u 4 ) (,0 ) (8) 4 For a source block ( u, u ) = (,), the coded block s 4 x =,, 0,. By a partcular rule for choosg the set the ext two subsectos, the polar codes wll be specfed. ) successve cacellato (SC) decoder: Cosder a G -coset code wth parameter (,,, c ) K u. The put u wll be ecoded to x ad ths code wll be sed over the chaelw ad we receve the output y. The the decoder wll decde o y ad estmate u as the orgal puts. ˆ block error wll be occurred f u uˆ. Decso fuctos resemble ML decso fuctos but ot the same because these fuctos treat the froze bts as RV s but they are fxed. However, the loss of performace due to ths suboptmalty of decodg s eglgble ad the symmetrc capacty s stll achevable. However, ML decodg s a effcet decodg algorthm for short legth codes of polar codes but s more complex [6]. 3) Code performace: It ca be show that for ay B-DMC W ad ay choces of (, K, ) the probablty of block error for ths code uder SC decodg, e (,,, c ) e (,,, c ) ( ) P K u ca be bouded as follows: P K u Z W (9) Ths suggests that we should choose from all K -subsets of {,..., } such that the rght had sde of (9) becomes as small as possble. 4) Polar codes: From the two prevous parts we ca coclude that a set s chose as a K -subsets of {,..., } such that G -coset code s called the polar code f the ( j) ( ) Z W Z W c for all ad j. 7

8 Ths choce of leads us to polar codes but we ca choose the code s sestve to choce of u. 5) Codg theorems []: c Theorem: for ay B-DMC W wth I ( W ) > 0, ad ay fxed R I ( W ), such that of sets {,..., }, {,,...,,...}. u c R ad arbtrarly as the performace of <, there exsts a sequece 5 O( 4 ) Z W The results of theorem ad (9) lead us to corollary stated as a theory below: Theorem3: for ay gve B-DMC W ad fxed R I ( W ) codg uder successve cacellato decodg satsfes: for all <, block error probablty for polar 4 Pe (, R) = O (0) 6) Complexty: The recursve structure of chael polarzato costructo mposes a low complexty ecodg ad decodg algorthms. s we metoed before, the ecodg algorthm of the polar codes wll be determed maly by F whch ca be show that has a complexty of ( log ) O. There s a decodg algorthm for polar codes wth complexty of O ( ). There s aother effcet decodg algorthm. The complexty of ths algorthm s O( log ) state ad prove the followg theorem: Theorem4: For the class of successve cacellato decodg are both O( log ) depedet of the code rate.. I geeral we ca G -coset codes, the complexty of ecodg ad the complexty of as fuctos of code block-legth ad I comparso to ecodg complexty of well kow Low Desty Party Check (LDPC) codes, the polar code do better as the ecodg complexty of LDPC codes s O ( ) whch s much hgher tha the ecodg complexty of polar codes. Furthermore, several low complextes decodg algorthm has bee troduced for LDPC codes from whch the best oe has the complexty of O( log ) that s equal to the decodg complexty of polar codes. So, the geeral complexty of the system (both ecoder ad decoder) for polar codes s less tha LDPC codes (the best capacty achevg code so far) ad ths makes the polar codes more terestg. 8

9 So far, there are o low complexty code costructos for polar codes geeral. However, there are certa cases for whch some low complexty code costructo has bee foud. Oe of these case s BEC whch a low complexty code costructo of O( log ) has bee foud for []. V. Cocluso I ths tutoral a ew proposed method called chael polarzato was troduce whch ca be used to costruct code costructos that acheve the symmetrc capacty of a B-DMC. Varous aspects of ths method ad a clear procedure to how to costruct such codes usg ths method was expressed. The ecodg ad decodg process ad also performace evaluato of the polar code was stated ad through a buch of theores t became clear that these code ca acheves the symmetrc capacty of a B-DMC wth low complexty of O( log ) for both ecodg ad decodg algorthms. Furthermore, t was stated ths dea ca be geeralze to ay DMC chaels ad so these codes ca acheves the symmetrc capacty of ay DMC. However, there are lots of ope problems that ca be doe future to mprove the performace of these codes order to be applcable the real world dfferet stadards. Oe of these ope problems s troducg some more effcet decodg algorthm to mprove the performace of the code. Refereces [] E. rka, Chael polarzato: method for costructg capacty-achevg codes for symmetrc bary-put memoryless chaels, IEEE Tras. Iform. Theory, July 009. [] S.B. Korada, E. Sasoglu, ad R. Urbake, Polar Codes: Characterzato of Ex-poet, Bouds, ad Costructo, submtted to IEEE Tras. Iform. Theory, Jauary 009 ( [3] E. Sasoglu, E. Telatar, ad E. rka, Polarzato for rbtrary Dscrete Memoryless Chaels, submtted to IEEE Tras. Iform. Theory, ugust 009 ( [4] E. rka, Chael Combg ad Splttg for Cutoff Rate Improvemet, IEEE Tras. Iform. Theory, Vol.5, o., February 006. [5] E. rka, Chael polarzato: method for costructg capacty-achevg codes, Proceedg of Iteratoal Symposum o Iformato Theory (ISIT 08), Toroto, Caada, July 008. [6] E. rka, H. Km, G. Markara, U. Ozgur, E. Poyraz, Performace of Short Polar Codes Uder ML Decodg, Proceedg of ICT-MobleSummt Coferece, Satader, Spa, Jue