A Comparison of ARA- and Protograph-Based LDPC Block and Convolutional Codes

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1 A Comparson of ARA- and Protograph-Based LDPC Block and Convolutonal Codes Danel J. Costello, Jr., Al Emre Pusane Department of Electrcal Engneerng Unversty of Notre Dame Notre Dame, IN 46556, U.S.A. {Costello.2, Chrstopher R. Jones, Darush Dvsalar Jet Propulson Laboratory Pasadena, CA U.S.A. {crjones, Abstract ARA- and protograph-based LDPC codes are capable of achevng error performance smlar to randomly constructed codes whle enjoyng several mplementaton advantages as a result of ther structure. LDPC convolutonal codes can be derved from these codes through an unwrappng process. In ths paper, we revew the unwrappng process as well as the ppelne decoder that allows contnuous decodng of LDPC convolutonal codes. Computer smulatons are then used to demonstrate that the unwrapped convolutonal codes acheve a convolutonal gan n error performance. We conjecture that ths s due to the concatenaton of many constrant lengths worth of receved symbols n the ppelne decodng process. The consequences of ths mproved performance are examned n terms of factors related to decoder mplementaton: processor sze, memory requrements, and decodng delay (latency). Fnally, gven dentcal protograph kernels, we compare derved block and convolutonal codes based on the above measures. I. INTRODUCTION The convolutonal counterparts of low-densty partycheck (LDPC) block codes, LDPC convolutonal codes, were frst proposed n [1]. Analogous to LDPC block codes, LDPC convolutonal codes are defned by sparse party-check matrces, whch allow them to be decoded usng teratve message-passng algorthms. The socalled ppelne decoder, that s typcally used to decode these codes, employs several small dentcal processors that perform the message-passng decodng teratons n parallel. In [2], the frst two authors presented a general comparson of LDPC block and convolutonal codes, nvestgatng several practcal encodng and decodng aspects of these codes. In ths paper, we extend that work by lookng specfcally at LDPC convolutonal codes derved from ARA- and Protograph-Based LDPC block codes. II. PRELIMINARIES We start wth a bref defnton of a rate R = b/c bnary LDPC convolutonal code C. (A more detaled descrpton can be found n [1].) Let u [0,t 1] = [u 0,u 1,...,u t 1 ], (1) where u = (u (1), u (2),..., u (b) ), 0 < t, t Z +, and u ( ) GF(2), be an nformaton sequence. The encoder maps ths sequence nto the code sequence v [0,t 1] = [v 0,v 1,...,v t 1 ], (2) where v = (v (1), v (2),..., v (c) ), 0 < t, t Z +, and v ( ) GF(2). A code sequence v [0, ] satsfes the equaton where, for all t t, 2 H T [t,t ] = 6 4 v [0, ] H T [0, ] = 0, (3) H T 0 (t) HT (t + ms) 0 ms 0 H T 0 (t + 1) HT ms (t + ms + 1) H T 0 (t ) H T ms (t + m s) s a transposed party check matrx, also called the syndrome former of the convolutonal code C. The submatrces H (t), = 0, 1,, m s, are bnary (c b) c submatrces gven by H (t) = h (1,1) (t) h (1,c) (t).. (t) h (c b,c) (t) h (c b,1) They satsfy the followng propertes: 1) H (t) = 0, < 0 and > m s, t. 2) There s a t such that H ms (t) 0.. (4) 3

2 Varable nodes Edge types Check nodes protograph Copy 3 tmes Permute the edges 2 Fg. 1. Copy and permute operaton for a protograph to generate larger graphs. 3) H 0 (t) 0 and has full rank t. We call m s the syndrome former memory and ν s = (m s + 1) c s the overall constrant length. These parameters determne the heght of the nonzero dagonal regon of H T [0, ]. The sparsty of the syndrome former s ensured by demandng that ts columns have very low Hammng weght,.e., w H (h ) << (m s +1) c, Z +, where h denotes the -th column of H T [0, ]. The code s sad to be regular f ts syndrome former H T [0, ] has exactly J ones n every row and K ones n every column, startng from the (m s (c b)+1)-th column. The other entres are zeros. We refer to a code wth these propertes as an (m s, J, K)-regular LDPC convolutonal code, and we note that n general the code s tme-varyng and has rate R = 1 J/K. An (m s, J, K)-regular tme-varyng LDPC convolutonal code s perodc wth perod T f H (t) s perodc,.e., H (t) = H (t + T),, t, and f H (t) = H,, t, the code s tme-nvarant. We wll descrbe a constructon technque, smlar to the one gven n [1], for dervng a perodcally tmevaryng party-check matrx H [0, ] from the partycheck matrx of an LDPC block code n Secton V. The LDPC convolutonal codes consdered n ths paper are derved from the well-known classes of ARA- and protograph-based LDPC block codes. III. ARA-BASED LDPC CODES A protograph [3] s a Tanner graph wth a relatvely small number of nodes. A protograph G = (V, C, E) conssts of a set of varable nodes V, a set of check nodes C, and a set of edges E. Each edge e E connects a varable node v e V to a check node c e C. Parallel edges are permtted, so the mappng e (v e, c e ) V C s not necessarly 1:1. As a smple example, we consder the protograph shown n Fg. 1. Ths graph conssts of 3 varable nodes and 2 check nodes, connected by 5 edges. In ths example we have 5 edge types,.e., each edge n the base protograph represents an edge type. For mult-edge LDPC codes, a group of edges (the number of edges n each group can be dfferent) represents an edge type. For unstructured rregular LDPC codes, there s only one edge type. Havng the base protograph, we can obtan a larger graph by a copy-and-permute operaton as shown n Fg. 1. Ths operaton conssts of frst makng N copes of the protograph and then permutng the endponts of each edge type among the N varable and N check nodes connected to the set of N edges coped from the same edge type n the protograph. The derved or lfted graph s the graph of a code N tmes as large as the code correspondng to the protograph, wth the same rate and the same dstrbuton of varable and check node degrees. As can be seen from the protograph representaton n the fgures, those varable nodes, say n trans nodes, that are connected to the channel (transmtted nodes) wll be shown as dark flled crcles. Those varable nodes that are not connected to the channel (punctured nodes or not transmtted nodes) wll be depcted by blank crcles. The check nodes wll be depcted by crcles wth a plus sgn nsde. The code rate for the protograph s R = nv nc n trans, provded that the party check matrx of the derved or lfted graph s full rank. As an example, consder the rate-1/2 systematc repeat-accumulate (RA) code wth repetton 3 shown n Fg. 2. n p u t n p u t Fg. 2. R e p e a t 3 p r o t o g r a p h o f R A C o d e T h r e s h o l d d B a c c u m u l a t o r Protograph for a rate 1/2 RA code. In [4] t was shown that the threshold can be further mproved by precodng the repetton code wth an accumulator. The desgn of the precoder n [4] was guded by an analyss of the extrnsc sgnal-to-nose rato (SNR) behavor of repetton codes and punctured accumulator codes usng densty evoluton. The use of a rate-1 accumulator as a precoder dramatcally mproves the extrnsc SNR behavor of a repetton 3 outer code n the hgh extrnsc SNR regon and hence mproves the teratve decodng threshold of the overall code. An RA code wth an accumulator precoder s called an Accumulate-Repeat-Accumulate (ARA) code [4]. An

3 example of a smple rate-1/2 ARA code and ts correspondng threshold s shown n Fg. 3. The ARA encoder n Fg. 3 uses a punctured accumulator as the precoder. n p u t p r e c o d e r Fg. 3. n p u t p r o t o g r a p h o f A R A c o d e T h r e s h o l d d B R e p e a t 3 a c c u m Protograph for a rate-1/2 ARA code. u l a t o r In an ARA code protograph the number of degree 2 varable nodes s equal to the number of nner checks (checks that are connected to these degree 2 varable nodes). If we decrease the number of degree 2 varable nodes wth respect to nner checks, then the ensemble asymptotc mnmum dstance of the code may grow wth n. For example, f we replace 50% of the degree 2 varable nodes wth degree 3 varable nodes, then the mnmum dstance grows wth n. We call such constructed codes ARJA [5] [6] codes because the nner accumulator now has a Jagged appearance. The protograph of a rate-1/2 AR4JA code and ts correspondng threshold s shown n Fg. 4. We call ths protograph AR 4 JA due to the repetton by 4 on the rght hand sde of the punctured degree-6 varable node. nput precoder (accumulator) Fg. 4. nput protograph Threshold 0.62 db repetton The rate-1/2 AR4JA protograph. Jagged accumulator IV. DECODING OF LDPC CONVOLUTIONAL CODES LDPC convolutonal codes can be teratvely decoded usng a message-passng algorthm. Although the correspondng Tanner graph has an nfnte number of nodes, the dstance between two varable nodes that are connected to the same check node s lmted by the syndrome former memory. Ths allows contnuous decodng wth a decoder that operates on a fnte wndow sldng along the receved sequence, smlar to a Vterb decoder wth fnte path memory [7], [8]. The decodng of two varable nodes that are at least (m s + 1) tme unts apart can be performed ndependently, snce the correspondng bts cannot partcpate n the same partycheck equaton. Ths allows the parallelzaton of the I C teratons by employng I C ndependent dentcal processors workng on dfferent regons of the Tanner graph smultaneously. A ppelne decoder that s based on ths dea was ntroduced by Jménez-Felström and Zgangrov n [1]. The operaton of ths decoder on the Tanner graph for a smple tme-nvarant rate R = 1/3 LDPC convolutonal code wth m s = 2 s shown n Fg. 5. Assumng c receved symbols enter the ppelne decoder per unt tme, t takes I C (m s + 1) tme unts for ths c-tuple to reach the output of the decoder, where a decson on these symbols s made. Thus, once a decodng delay of I C (m s + 1) tme unts has elapsed, the decoder produces a contnuous output stream,.e., at each tme unt, c newly receved symbols enter the decoder and c decoded bts leave t. The I C processors perform the I C decodng teratons n parallel. At each tme unt, the -th processor begns by actvatng the frst column of (c b) check nodes n ts operatng regon and then proceeds to actvate the last column of c varable nodes that are about to leave the operatng regon. A check node actvaton s the step where the check node collects all the ncomng messages from ts neghborng varable nodes, calculates the outgong messages, and sends the new messages to each neghborng varable node. Correspondngly, durng a varable node actvaton, all the ncomng messages to a varable node from the neghborng check nodes are collected and the new outgong messages are calculated. Then the new messages are sent to each neghborng check node. Ths message passng schedule corresponds to a parallel message passng schedule of a block code decoder where all varable nodes are updated at once and then all check nodes are actvated. We wll dscuss a bass for comparson between LDPC block and convolutonal decodng n Secton VI. V. DERIVATION OF CONVOLUTIONAL CODES FROM BLOCK CODES In ths secton, we present a graphcal unwrappng procedure based on the bnary party-check matrx H of an arbtrary LDPC block code. In order to derve a perodcally tme-varyng LDPC convolutonal code of rate R = b/c from an LDPC block code, we cut the bnary party-check matrx H of the

4 Proc.1 Proc.2 Proc.(I c 1) Proc.I c r (0) t m s v (0) t I(ms+1)+1 r (1) t v (1) t I(ms+1)+1 r (2) t v (2) t I(ms+1)+1 decodng wndow (sze = I C (m s + 1) ) varable node check node Fg. 5. The Tanner graph of an R=1/3 LDPC convolutonal code and an llustraton of ppelne decodng. underlyng block code along the dagonal n steps of sze (c b) c. Next, we remove the upper-dagonal porton and paste t to the bottom of the lower-dagonal porton. The resultng dagonal shaped matrx s then repeated ndefntely, gvng us the party-check matrx H conv of a tme-varyng convolutonal code wth perod T = m s + 1. We refer to ths cuttng and pastng operaton as the unwrappng procedure. In general, f we start from an (n, J, K)-regular LDPC block code of length n and rate R 1 J/K, we obtan an (m s, J, K)-regular LDPC convolutonal code wth rate R = b/c = 1 J/K and syndrome former memory m s = (n/c) 1,.e., overall constrant length ν s = n. Ths procedure s llustrated n Fg. 6, where we derve a rate R = 1/3, (6, 2, 3)- regular LDPC convolutonal code from a rate R = 8/21, (21, 2, 3)-regular LDPC block code, where n ths case the H matrx of the block code contans one redundant row. The unwrappng step sze of (c b) c produces a rate R = b/c convolutonal code. Smlarly, a step sze of (c b)k ck, where 0 < ck n and k Z +, produces a rate bk/bc code. The specal case of ck = n, for example, corresponds to repeatng the orgnal block code ndefntely, and s therefore of no practcal sgnfcance. On the other hand, ths specal case helps to show the connecton between the block and convolutonal codes and llustrates that by decreasng the value of k we arrve at a more convolutonal structure. Although the above example uses regular LDPC block codes as the startng pont of the unwrappng procedure, there s, n general, no constrant on the row and column weghts of the party-check matrces, and the derved tme-varyng convolutonal code has the exact same node degree dstrbuton as that of the underlyng block code. In other words, the unwrappng procedure preserves the degree dstrbuton durng the cuttng and pastng (and also the repeatng) steps, so the same approach can be employed to derve rregular LDPC convolutonal codes from rregular LDPC block codes. VI. DECODING COMPARISONS In ths secton, we compare several aspects of decodng LDPC convolutonal and block codes. A. Computatonal Complexty Let C check (C var ) denote the number of computatons requred for a degree-k (J) check (varable) node update. Regardless of the code structure, C check and C var depend only on the values J and K. For a rate R = b/c, (m s, J, K)-regular LDPC convolutonal code and a ppelne decoder wth I C teratons/processors, at every tme nstant each processor actvates c b check nodes and c varable nodes. The computatonal complexty per decoded bt s therefore gven by C conv bt = ((c b) C check + c C var ) I C /c (5) = ((1 R) C check + C var ) I C, whch s ndependent of the constrant length ν s.

5 (a) (b) Fg. 6. Dervng a tme-varyng LDPC convolutonal code from an LDPC block code: (a) H matrx for the block code, (b) H conv matrx for the convolutonal code after unwrappng. Smlarly, the decodng complexty for a rate R 1 J/K, (n, J, K)-regular LDPC block code wth I B teratons s gven by C block bt = (n J K C check + n C var ) I B /n (6) = ( J K C check + C var ) I B = ((1 R) C check + C var ) I B, whch s agan ndependent of the code length n. Thus the dfference n computatonal complexty between block and convolutonal codes (on a per decoded bt bass) s gven by the rato I B /I C. B. Internal Observaton Memory The sldng wndow decoder mplementaton of an LDPC convolutonal code requres the storage of I C ν s symbols. However, snce decodng can be carred out by ppelnng I C dentcal ndependent parallel processors, each operatng on only ν s symbols, the smaller sze of the ndvdual processors may be useful n smplfyng hardware desgn and reducng routng congeston. For an LDPC block code of length n, the processor must be capable of storng all n symbols. C. Internal Edge Memory For the ppelne decoder, we need a storage element for each edge n the correspondng Tanner graph. Thus a total of I C (J) ν s storage elements are requred for I C teratons of decodng. Smlarly, we need n (J) storage elements for the decodng of an LDPC block code of length n. D. Decodng Delay (Latency) and External Observaton Memory Let T s denote the tme between the arrval of successve symbols,.e., the symbol rate s 1/T s. Then the maxmum tme from the arrval of a symbol untl t s decoded n the ppelne decoder s conv o = ((c 1) + (m s + 1) c I C ) T s. (7) The frst term (c 1) n (7) represents the tme between the arrval of the frst and last of the c encoded symbols output by a rate R = b/c convolutonal encoder n each encodng nterval. The domnant second term (m s + 1) c I C s the tme each symbol spends n the decodng wndow. Snce c symbols are loaded nto the decoder smultaneously, the ppelne decoder requres a buffer to hold the frst (c 1) symbols. Wth LDPC block codes, data s typcally transmtted n a sequence of blocks. Dependng on the data rate and the processor speed, several scenaros are possble. We consder the best case for block codes,.e., each block s decoded by the tme the last bt of the next block arrves. Ths results n an nput-output delay of = 2n T s. Note that the block decoder needs a buffer to hold the n 1 symbols that arrve before the next block s complete. Ths scenaro s overly optmstc snce some blocks wll consume sgnfcantly more than the average number of teratons to decode, and therefore more than n symbols should be accommodated n the decoder s nput buffer. Fogal, Dolnar, and Andrews [9] have shown an addtonal block of storage reduces the lkelhood of buffer overflow below typcal frame error rates at operatng SNR s of nterest. Therefore, we block o choose block o = 3n T s as an upper bound on block code decodng latency.

6 VII. A BASIS FOR COMPARISON As noted n the prevous secton, there are many ways to compare possble decoder realzatons for LDPC block and convolutonal codes. However, no one approach tells the whole story. Ths s due n part to the very dfferent structures of block and convolutonal codes. In fact, even before the LDPC codng era, t has always been a controversal topc to determne how best to compare block and convolutonal codes. Notons such as trells complexty, mnmum dstance bounds, and error exponents have all been employed to ths end. Smlarly, t s very hard to determne an deal bass of comparson between LDPC block and convolutonal codes. In the fnal analyss, any comparson must be a functon of the partcular applcaton, e.g., whle a large storage requrement or processor sze mght not be problems for moble communcatons, a large latency may be undesrable, especally for real-tme voce transmsson. In ths paper, we choose a comparson method based on decoded bt error rate (BER) performance. For each of the code constructons, we target a specfc channel SNR at a fxed dstance from the teratve decodng threshold of the employed code famly. We then compare LDPC block and convolutonal codes that acheve the same BER and/or frame error rate (FER) performance at ths SNR based on the crtera presented n the prevous secton. VIII. COMPARISON OF BLOCK AND CONVOLUTIONAL CODES BUILT FROM THE AR4JA PROTOGRAPH Our comparson begns wth the AR4JA protograph of Secton III. Before constructng ether convolutonal or block LDPC codes, we expand the graph of Fg. 4 by a factor of 4 to remove double edges. Ths workng base graph s gven the name AR4JAx4. The constructon of two unque 1 LDPC convolutonal codes s descrbed n Fgs. 7(a)(b). In both cases, a block code s frst derved va expanson by a factor of 128, Fg. 7(a), and expanson by a factor of 125, Fg. 7(b). The number of rows and columns n the transposed party-check matrx of each of these block codes s ndcated n the fgure. Note that permutatons assocated wth the aforementoned expansons (x4, x125, x128) were performed usng the well-known progressve edge growth technque [10]. The performance of these two block codes s denoted by the k = 1000 and k = 1024 curves n Fg These convolutonal codes are unque, but are constructed to acheve relatvely comparable convolutonal LDPC codes m +1=512 copes s m +1=125 copes s T H blk= T H blk= m s+1=512 copes m s elements Perodc repetton of m s+1 elements from top row Elements n orgnal block matrx (2560x1536) Elements n syndrome former matrx H T m s+1=125 copes (a) m s elements Perodc repetton of m s+1 elements from top row Elements n orgnal block matrx (2500x1500) Elements n syndrome former matrx H T (b) Fg. 7. Dervng an LDPC convolutonal code from an LDPC block code: (a) AR4JAx4x128, k=1024, n=2048 (+ 512 punctured nodes) protograph unwrappng. (b) AR4JAx4x125 protograph unwrappng. These two block codes (wth k roughly 1000) are unwrapped to form LDPC convolutonal codes, agan followng the parameters denoted n Fg. 7. For nstance, the AR4JAx4x128 (Fg. 7(a)) block code s broken nto sub-blocks of sze c = 5 and c b = 3, whle the AR4JAx4x125 (Fg. 7(b)) block code s broken nto subblocks of sze c = 20 and c b = 12. Note that the latter dmensons adhere to those nherent n the AR4JAx4 base graph. An addtonal factor s that the AR4JA code has all degree-6 nodes punctured (see Fg. 4). Therefore, n terms of transmtted nodes, Fg. 7(a) has c trans = 4 and rate R = (c (c b))/c trans = 1/2. Smlarly, Fg. 7(b) has c trans = 16 and rate R = (c (c b))/c trans = 1/2. Much of the nterest assocated wth LDPC convolutonal codes stems from the ncrease n codng gan that can be obtaned through smple perodc repetton, by a factor of I C, of graphs lke those shown

7 AR4JAx4x125 Conv AR4JAx4x128 Conv AR4JAx4x2048 Blk Complexty Per Decoded Bt [2-nput Ops] 12I C=1202I C=1202I B=436 Observaton Memory [Depth] ν si C= ν si C= n=40960 Edge Storage Memory [Depth] ν sj avgi C= ν sj avgi C= nj avg= Latency [T s] ν si C= ν si C= n= TABLE I COMPLEXITY MEASURES FOR CONVOLUTIONAL AND BLOCK LDPC CODES BUILT FROM IDENTICAL PROTOGRAPHS WITH COMPARABLE BER VS. SNR PERFORMANCE. n Fgs. 7(a)(b) (I C = 2 n the fgure). For nstance, whle the performance of the block codes underlyng the convolutonal codes n Fgs. 7(a)(b) s gven by the crcle and square marked lnes n Fg. 8, perodc repetton by a factor of I C = 100 of the unwrapped versons of these block codes yelds the performance shown by the damond and pentagram marked lnes n the same fgure. At a BER=10 6, nearly a full decbel of performance s ganed wthout the expense of any addtonal rate whatsoever. A man motvaton of ths paper s to examne whether or not the cost of the performance ganed by convolutonal constrant length repetton s more or less than the cost of the gan assocated wth further expanson of the underlyng block code. Note that the performance of any code constructed va repetton of the AR4JA protograph can do no better than the protograph s threshold, whch s E b /N o = 0.62 db. Based on the smulaton results shown n Fg. 8, we see that the performance of AR4JAx4x2048 (the AR4JAx4 base graph expanded va crculant permutaton by a factor of 2048, whch has k = and n trans = (n = 40960)) concdes nearly exactly wth the performance of the I C = 100 unwrapped convolutonal codes at E b /N o = 1.0 db and BER = In addton, the average number of teratons, I B = 36.3, performed by the block decoder s gven for ths SNR and others across an operatng regon of nterest n Fg. 9. We can apply the complexty analyss outlned for regular codes n Secton VI to AR4JA based codes wth a few addtonal assumptons. Frst, let the parameters K and J (the constrant and varable node degrees) be denoted by K avg = 5 and J avg = 3. Second, assume that a degree K avg constrant node requres C check = 2K avg 2-nput operatons to perform an update; smlarly a degree J avg varable node requres C var = 2J avg 2-nput operatons (see, for nstance, [11]). Gven these assumptons, we refer the reader to Table I. Wthout basng results to one approach over the other, the results n the table assume that all varable nodes n BER E b /N o (db) Conv AR4JAx4x125 (c = 20 b = 8 m s = 124) I C = 100 Conv AR4JAx4x128 (c = 5 b = 2 m s = 511) I C = 100 Blk AR4JA (k = 1000) Imax = 200 Blk AR4JA (k = 1024) Imax = 200 Blk AR4JA (k = 16384) Imax = 200 Fg. 8. Rate-1/2 convolutonal and block LDPC codes based on the AR4JA protograph. the code are transmtted and must be buffered n the decodng process. For essentally equal performance the block code requres 6 tmes less nternal observaton and edge storage memory and ncurs at worst (due to the upper bound of 3n on block code latency) 2 tmes less decodng latency. Table I also shows that the block code requres 2.57 tmes fewer computatons per decoded bt than the convolutonal code, where we note that the number of computatons performed s proportonal to the number of teratons. (Because of the dentcal graph connectvty of the LDPC block codes and the LDPC convolutonal codes derved from them, they perform the same number of computatons f they use the same number of teratons.) However, for the smulaton results shown n Fg. 8, the LDPC convolutonal codes employed a

8 Avg. Iteratons Blk AR4JA (k = 16384) Avg. Iter processors may allow for hgher clock frequences and provde hgher throughputs than s possble wth a sngle block decoder. We note that ppelned LDPC block decoders can also be desgned to use ndvdual processors, wth each processor performng only a fracton of the total number of teratons. In ths case, however, the block code would lose the storage memory and decodng latency advantages noted above compared to ther convolutonal counterparts E b /N o (db) Fg. 9. Average decodng teratons vs. SNR for Rate-1/2 AR4JA wth k = constant number of I C = 100 teratons, whle the LDPC block codes made use of a syndrome-based stoppng rule to decrease ther average number of teratons to I B = 36.3, wth the maxmum number of teratons fxed at I B = 200. Ths leads to the dfference n the number of requred computatons noted above. As a result of the extended graph of LDPC convolutonal codes, stoppng rules are not as obvous as for LDPC block codes. One such stoppng rule has been proposed, however, n [12], where the average number of teratons can be reduced wthout affectng the error performance. Usng ths rule, the ndependent processors n the ppelne decoder can sometmes be put nto a sleep mode n order to save computatons. Adjustng the above computatonal comparson to take advantage of ths ppelne decodng stoppng rule s currently beng nvestgated. We note that the sze of a ppelne decodng processor s roughly 1/I C tmes that any of the convolutonal decoder complexty measures n Table I. Convolutonal LDPC codes utlzng a ppelne decoder therefore lend themselves to a fne granularty that allows a trade-off between performance and complexty. The smulaton results of Fg. 8 ndcate that t s possble to acheve low error rate performance by keepng the processor sze ν s relatvely small whle ncreasng the number of processors I C. (In Fg. 8, for example, the convolutonal code processor s only 1/16 the sze of the block code processor.) The resultng cascade of relatvely small IX. CONCLUSION An advantage of the convolutonal structure s that performance can easly be ganed by smply addng more constrant length multples (teratons I C ) at the decoder wthout changng the encodng structure at all. Also, the natural ppelne structure of the convolutonal decoder facltates the realzaton of low error rates and potentally hgh throughputs by desgnng a processor of relatvely modest sze and replcatng t n the ppelne. However, f one has a partcular target error rate n mnd, desgnng a dedcated block code acheves the desred result wth less latency and reduced memory requrements. We are currently ntatng a practcal hardware-based complexty comparson of the LDPC codes presented n ths paper. Ths ncludes realzatons of the LDPC block decodng and ppelned LDPC convolutonal decodng archtectures on feld programmable gate arrays (FPGAs) and applcaton specfc ntegrated crcuts (ASICs). These realzatons wll provde further nsght nto the comparsons presented n ths paper, ncludng determnng possble practcal mplementaton bottlenecks n terms of memory and logc element usage. ACKNOWLEDGMENT Ths work was supported n part by NSF Grants CCR , CCF , NASA Grant NNG05GH73G, and at the Jet Propulson Laboratory / Calforna Insttute of Technology under a contract wth NASA. REFERENCES [1] A. Jménez-Feltström and K. Sh. Zgangrov, Tme-varyng perodc convolutonal codes wth low-densty party-check matrx, IEEE Trans. Inform. Theory, vol. IT-45, pp , Sept [2] D. J. Costello, Jr., A. E. Pusane, S. Bates, and K. Sh. Zgangrov, A comparson between LDPC block and convolutonal codes, n Proc. Informaton Theory and Applcatons Workshop, (San Dego, CA, USA), February [3] J. Thorpe, Low densty party check (LDPC) codes constructed from protographs, JPL IPN Progress Report, pp , August 2003.

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