Application of Nonbinary LDPC Codes for Communication over Fading Channels Using Higher Order Modulations

Transcription

1 Applcaton of Nonbnary LDPC Codes for Communcaton over Fadng Channels Usng Hgher Order Modulatons Rong-Hu Peng and Rong-Rong Chen Department of Electrcal and Computer Engneerng Unversty of Utah Ths work s supported n part by NSF under grant ECS

2 Outlne Motvaton Apply nonbnary LDPC codes over large Galos felds to fadng channels Low complexty nonbnary LDPC decodng Quas-cyclc constructon Smulaton results Concluson

3 Motvaton Bnary LDPC coded system has been studed extensvely. Optmal bnary code has been desgned to approach channel capacty. Nonbnary LDPC code desgn has been studed for AWGN and shows better performance than bnary codes [1][]. [1] A. Bennatan and D. Burshten Desgn and analyss of nonbnary LDPC codes for arbtrary dscrete-memoryless channels IEEE Trans. Inform. Theory vol. 5 pp Feb. 6. [] S. Ln S. Song L. Lan L. Zeng and Y. Y. Ta Constructons of nonbnary quas-cyclc ldpc codes: a fnte feld approach n Info.Theory and Applcaton Workshop (UCSD 6.

4 Motvaton Our contrbuton Apply large feld nonbnary LDPC codes to fadng channel Propose effcent nonbnary LDPC decodng algorthm. Construct nonbnary QC LDPC codes based on QPP[3] Provde comparson wth optmal bnary LDPC coded systems [3] Oscar. Y. Takeshta A New Constructon for LDPC Codes usng Permutaton Polynomals over Integer Rngs Submtted to IEEE Trans. Inform. Theory

5 Applcaton to fadng channels Channel model ρ X = HS + M V Assume each entry of channel matrx s ndependent follows Raylegh fadng and s known by recever

6 System block dagram Non-teratve system: the detecton s performed only once. Iteratve system: Soft messages are exchanged between detector and decoder teratvely. R.-H. Peng and R.-R. Chen Good LDPC Codes over GF(q for Multple-Antenna Transmsson" Presented on MILCOM 6

7 Non-teratve system Used for the systems wth small number of antennas log q = N sm m : the number of constellaton bts N denote the number of ndependent channel use Large GF(q s

8 Log-lkelhood rato vector Soft message n bnary system s LLR. p( b = ln p( b = 1 Soft message n nonbnary system s a vector-llrv denote the log-lkelhood rato of beng one element n GF(q. z = { z z1 zq 1} p( β = wherez = ln p( β = {1 q 1}

9 Symbol-wse MAP detecton Symbol-wse MAP detecton No pror nformaton feed back from LPDC decoder s requred: Detecton s only performance once Large complexty could be saved GF( transmtted constellaton symbols correspondng the collecton of denotes ( } { ( q N φ z s N l l l N l l l l s s = = = = β σ X X X H X Y H X Y

10 Nonbnary LDPC decodng from channel ( rk Repetton code r k (1 l k ( l k Varable node decoder (VND Vertcal step: In log doman: r r k ' k = r r d 1 v ( k n= 1 '( k d v l + n= ( n k 1 1 l '( n k

11 Nonbnary LDPC decodng r a1 r a (1 ( l k + Check node decoder (CND Sngle party check code Horzon step: l k = g n a n = g d c k d c 1 n= 1 r ( n a n Drect computaton has huge complexty!

12 Nonbnary LDPC decodng Horzon step can be consdered as a multple convoluton over GF(q Mult-dmensonal FFT can be appled R ( n The complexty s O(qlogq l = = DFT( r IDFT( d c 1 n= 1 ( n R ( n

13 Log doman mplementaton R (n may be negatve value can be represented by sgn/logarthmc number system (LNS sgn( u LNS( u = log( u In FFT lots of LNS addtons and subtractons requred LNS addton/subtracton requres one comparson two addtons and one table look-up.

14 Log doman mplementaton To avod LNS addton/subtracton we propose to convert data from LNS to plan lkelhood before the FFT and IFFT operatons and then convert them back afterwards. Only addtons subtractons and conversons between log to normal doman are requred. Complexty savng: Full log : qdc LNS addton/subtracton q Partal log : 4 dc Converson between log and normal doman p p = log q 75% computaton can be saved for GF(56 codes Accumulated errors could be reduced.

15 Quas-cyclc constructon Propose to use regular ( d c code for MIMO channels Modfy the QPP method to construct nonbnary QC codes wth flexble code length support pre-determned crculant sze allow lnear-tme encodng perform close to the PEG constructon

16 Quas-cyclc constructon Quas-cyclc structure H = A A A 11 1 m1 A A A 1 m A A A 1 n n m n A j s a crculant: each row s a rght cycle-shft of the row above t and the frst row s the rght cycle-shft of the last row The advantage of QC structure Allow lnear-tme encodng usng shft regster Allow partally parallel decodng Save memory

17 QC structure for GF(q j A β ary ' q = ' q j j j j j β δ β δ β δ δ A s a multpled crculant permutaton matrx GF( prmtve element of s a } randomly choosen from { s ' GF( prmtve element of s a ; GF( ' GF( 1 1 ' 1/( ( 1 q q q q q q j α α α α δ β

18 QC structure for GF(q Wth above structure the nonzero elements are chosen as randomly as possble wth equal probablty for each element For each crculant only the cyclc shft and the power of the frst nonzero element need to be saved Many exstng bnary QC constructon methods may be extended to nonbnary LDPC codes usng ths structure. Code sze : n( q' 1 m( q' 1 Crculant : ( q' 1 ( q' 1 Use QPP based method to construct nonbnary QC codes wth large grth

19 QPP based method Code constructon s based on edge nterleaver f(x f( = 3 f(1 = f( = 1 f(3 = 5 f(4 = 4 f(5 = Quadratc permutaton polynomal over nteger rngs (QPP f ( x = f x + 1 N : the number of f x (mod edges N

20 QPP based method To be Quas-cyclc need to search wth largest grth such that Gven (1 we have By groupng varable nodes check nodes obtan a QC code. f 1 f (1 (mod (mod ( ( ( N d f N d f x d f x f d x f v v v v + + β β β β c v d d f q k k j k H j H ( ' 1 ( ( β γ γ β = = + + = } { ' ( β β + + q v v v } { ' ( γ γ + + q c c c

21 QPP based method Code example: Regular ( 4 GF(56 code Code length: 3 Crculant: 15x15 f ( x = 17x + 3x Each node has local grth 14 Compare wth PEG constructon 68% varable nodes have local grth 14 9% have local grth 1 3% have local grth 1

22 1-1 BER reg. GF(56 Smulaton results 1 - BER optmzed rreg. GF( BLER reg. GF(56 BLER optmzed rreg. GF( Nonbnary codes outperform optmzed rrgular bnary codes by.6db n BER and by.35db n BLER Performance comparson of regular GF(56 LDPC code wth the optmzed rregular GF( (bnary LDPC code for a SISO channel wth 16QAM modulaton.

23 Smulaton results BER reg. GF(56 PEG BER reg. GF(56 QPP BER optmzed rreg. GF( BLER reg. GF(56 PEG BLER reg. GF(56 QPP BLER optmzed rreg. GF( Nonbnary codes outperform optmzed rrgular bnary codes by.16db n BER and by.db n BLER QPP codes have very close performance wth PEG codes Performance comparson of a regular GF(56 LDPC codes (both PEG and QPP constructons wth the optmzed rregular GF( (bnary LDPC code for a MIMO channel wth 4 transmt and receve antennas and QPSK modulaton.

24 Concluson Study the applcaton of nonbnary LDPC codes for MIMO system Propose an effcent decodng algorthm for nonbnary LDPC codes Construct nonbnary LDPC codes based on QPP methods that are flexble n code length and crculant sze Provde performance comparsons between regular nonbnary LDPC codes wth optmzed rregular bnary LDPC codes Demonstrate that nonbnary LDPC codes are good canddates for MIMO channels based on both performance and complexty

25 Thanks!