Belief Propagation Decoding of Polar Codes on Permuted Factor Graphs
Ahmed Elkelesh, Moustafa Ebada, Sebastian Cammerer, Stephan ten Brink
BBelief Propagation Decoding of Polar Codes onPermuted Factor Graphs
Ahmed Elkelesh, Moustafa Ebada, Sebastian Cammerer and Stephan ten Brink
Institute of Telecommunications, Pfaffenwaldring 47, University of Stuttgart, 70569 Stuttgart, Germany { elkelesh,ebada,cammerer,tenbrink } @inue.uni-stuttgart.de Abstract —We show that the performance of iterative beliefpropagation (BP) decoding of polar codes can be enhanced bydecoding over different carefully chosen factor graph realiza-tions. With a genie-aided stopping condition, it can achieve thesuccessive cancellation list (SCL) decoding performance whichhas already been shown to achieve the maximum likelihood(ML) bound provided that the list size is sufficiently large.The proposed decoder is based on different realizations of thepolar code factor graph with randomly permuted stages duringdecoding. Additionally, a different way of visualizing the polarcode factor graph is presented, facilitating the analysis of theunderlying factor graph and the comparison of different graphpermutations. In our proposed decoder, a high rate CyclicRedundancy Check (CRC) code is concatenated with a polar codeand used as an iteration stopping criterion (i.e., genie) to evenoutperform the SCL decoder of the plain polar code (without theCRC-aid). Although our permuted factor graph-based decoderdoes not outperform the SCL-CRC decoder, it achieves, to thebest of our knowledge, the best performance of all iterative polardecoders presented thus far.
I. I
NTRODUCTION
Recently, polar codes [1] have been considered as channelcodes for the upcoming as part of the control channel [2] and thus,decoding of polar codes has more and more become a practicalimplementation challenge. As successive cancellation (SC)decoding [1] for finite length polar codes is sub-optimal,successive cancellation list (SCL) decoding [3] is applied,at the cost of increased complexity due to the list decodingnature. It was shown in [3] that this decoder can approach themaximum likelihood (ML) bound for a sufficiently large listsize. Furthermore, an additional high rate Cyclic RedundancyCheck (CRC) code can be easily used under SCL decodingin order to enhance the code itself via increasing its mini-mum distance. This combination renders polar codes into apowerful coding scheme. However, SCL suffers from both,high-complexity and an inherently serial decoding nature.In this work, we focus on iterative decoding of polar codesbased on the message passing algorithm over the encodinggraph of polar codes [4]. In contrast to SCL decoding, thebelief propagation (BP) algorithm can be easily parallelized[5]. Additionally, this approach inherently enables soft-in/soft-out decoding and allows joint iterative detection and decodingloops. Although the BP algorithm can outperform SC decod-ing, it is yet not competitive to SCL decoding and, thus, not
This work has been supported by DFG, Germany, under grant BR 3205/5-1. attractive for many applications. In [6], [7], it has been shownthat the finite length polar codes under BP decoding can beenhanced when the semi-polarized channels are additionallyprotected by check nodes, or an augmented (shorter) polarcode [8]. Unfortunately, these approaches are still of an infe-rior performance when compared to SCL decoding. Besides,they require an adjusted code structure and are thus notcompatible to the expected standardized polar code (i.e., polarcode concatenated with CRC code). The proposed algorithmin this paper enhances the decoder without any modificationsneeded at the encoder. The proposed algorithm works wellwith any polar code concatenated with a CRC code.For a polar code of length N , the stages of the encodinggraph can be permuted [9] leading to ( log N ) ! different graphswith the same encoding behavior. As a result, an almostinfinite amount of different decoder permuted realizations existeven for moderate length polar codes (e.g., 10! > × permutations for an N = a r X i v : . [ c s . I T ] A p r I. P
OLAR CODES AND ITERATIVE DECODING
In this section, we briefly mention fundamental concepts ofpolar codes, besides introducing the notations used throughoutthis work. We also briefly discuss the BP decoder of polarcodes and review some of the recent work pursued on it.
A. Polar Codes
Polar codes introduced in [1] are constructed based ona 2 × G = (cid:20) (cid:21) is themost commonly used). After recursively acquiring the n th − kronecker power of the kernel G , N synthesized bit channelsare obtained, where N = n . The term “polarized channels”indicates that one portion of the synthesized channels is purelynoisy, and thus, would be frozen (i.e., cannot be used foruseful data transmission and would be set to known values,e.g., 0 in this work), and another portion would be purelynoiseless which will be the information bit channels, i.e., usedfor data transmission. However, the phenomenon of channelpolarization requires sufficiently large N , where the channelsconverge to either purely noisy or purely noiseless channels.The selection of good and bad bit channels out of the N synthesized channels is called code construction , where theinformation set A is the set of indices denoting the informa-tion bit channels. Several code construction algorithms exist,with different bit channel “quality” criteria, e.g., [14][15][16].Throughout this paper, we use the polar code constructionbased on Arıkan’s Bhattacharyya bounds of bit channels [1],however, any other polar code construction algorithm could beused straightforwardly.A polar code of length N with k information bits and coderate R = kN is denoted by P ( N , k ) . Encoding requires the com-putation of the N × N generator matrix G N by computing thekronecker product G ⊗ n . A vector u of length N is constructedcontaining k information bits placed in the A indices and zerosin the remaining indices ¯ A . The N coded bits x are calculatedas follows x = u · G N .For the family of polar codes, there exist two main decodingschemes: SC decoding (and its variants, e.g., SCL) and BPdecoding. Although polar codes were theoretically proven toachieve the symmetric channel capacity of a Binary InputDiscrete Memoryless Channel (BI-DMC) under SC decodingassuming an infinite length code, finite length polar codesshow a degraded performance because of the incompletechannel polarization phenomenon [6]. An alternative iterativedecoding algorithm based on the idea of message passing overthe encoding graph was introduced in [4], and was shown tooutperform the SC decoding for finite length polar codes. B. Belief Propagation Decoding
The flooding BP decoding of polar codes is a messagepassing algorithm in which the information bits are retrievedthrough iterations conducted on the factor graph correspondingto the polar code generator matrix G N . As depicted in Fig. 1a,the polar code factor graph consists of n = log N stages. In the following, all messages are assumed to be log-likelihoodratios (LLR) and are defined as LLR ( x ) = log P ( x = | y ) P ( x = | y ) .
1) Types of LLR messages:
Two types of messages areinvolved, left-to-right messages ( R -messages) and right-to-leftmessages ( L -messages). The R -messages at stage 1 representthe a priori information available to the decoder and, thus,are either zero or infinity for non-frozen and frozen bits,respectively. The L -messages at stage n + L ch . All other messages are initialized withzero (i.e., no initial information).
2) Types of LLR updates:
Each single iteration is composedof one left-to-right message propagation, updating the LLRvalues of the R -messages and one right-to-left message prop-agation, updating the LLR values of the L -messages.
3) Factor graph and processing element (PE):
The polarfactor graph (see Fig. 1) consists of N · log ( N ) PEs. The L -and R -messages are updated in each PE (shown in [17, Fig.2]) as follows: R out , = f ( R in , , L in , + R in , ) R out , = f ( R in , , L in , ) + R in , L out , = f ( L in , , L in , + R in , ) L out , = f ( R in , , L in , ) + L in , where f ( L , L ) = L (cid:1) L is commonly referred to as boxplus operator, which can be expressed as f ( x , y ) = x (cid:1) y = log 1 + e x + y e x + e y .
4) Decoding termination and stopping conditions:
Tra-ditionally, the conventional BP decoder terminates when itreaches a predefined maximum number of BP iterations N it , max . The LLRs of the estimated message ˆu and the esti-mated transmitted codeword ˆx are computed according to L ( ˆ u i ) = L , i + R , i L ( ˆ x i ) = L n + , i + R n + , i However, early stopping conditions introduced in [18] are usedto speed up the decoding process by terminating the decodingprocess if a certain stopping condition is met. One of theconditions proposed is G -matrix based, where ˆ u is said tobe a valid estimate of u if ˆ x = ˆ u · G is fulfilled. Throughoutthis work, this stopping condition is called “practical stoppingcondition” . Note that this multiplication is of high complexityand, thus, can be avoided by encoding over the polar encodingcircuit which has a complexity of O ( N · log N ) . One canobviously infer that if the BP decoder is terminated using thecondition ˆu = u , this would act like a lower bound on the BPdecoder performance (continue iterating till reaching the cor-rect transmitted information bits, i.e., perfect knowledge-baseddecoding). Throughout this work, this stopping condition iscalled “perfect knowledge-based stopping condition” , since itis more of a bound rather than representing a real decoding c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v u x (a) Conventional factor graph v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v v c v c v c v u x (b) Stage-shuffled factor graph v , u v , u c c v v c c v , u v , u c c v v c c v , u v , u c c v , u v , u c c v v c c v v c c v , x v , x v v c c v , x v , x c c v v c c v , u v , u v , u v , u c c v v c c v v c c v , x v , x v v c c v , x v , x c c v v c c v v c c v , u v , u c c v v c c v , u v , u v v c c v , x v , x c c v v v , x v , x v v c c v , x v , x c c v v v , x v , x (c) Unfolded factor graph labeled according to (a) and (b) Fig. 1: Conventional and permuted factor graph of P ( , k ) .behavior. A further stopping (and detection) condition wasintroduced in [19], where a high rate CRC code is used asan outer code on the information bit vector to overcome thesituation of undetected errors (where the estimate ˆx is a validcodeword but ˆx = x ). Thus, stop the BP iterations when theCRC on the information bits is satisfied.III. D ECODING ON PERMUTED FACTOR GRAPHS
As exemplary shown in Fig. 1 for two permutations, thereare ( log N ) ! different permutations of the polar code factorgraph based on the generator matrix G . The BP decoding algo-rithm can be performed over any of such permutations [9]. Theleftmost stage (i.e., stage 1) of all of the different permutationsof the polar factor graph represents the vector u containingthe frozen known bits and the non-frozen information bits.The rightmost stage (i.e., stage n +
1) of all of the differentpermutations of the polar factor graph represents the codeword x or its corresponding LLRs. Polar decoding on permuted factor graphs was referredto as “BP decoding on an overcomplete representation of afactor graph” in [9], or simply “multi-trellis” BP decoding. Itwas used in [17] to overcome error floors due to inadequateLLR-clipping values. This decoding algorithm (Algorithm 1)works as follows. BP decoding iterations are performed overa random permutation of the polar code factor graph shownin Fig. 1a until a certain early stopping condition (Algorithm2) [18] is fulfilled, or until a predefined maximum number ofBP iterations per trellis N it , max is reached. If decoding on onefactor graph permutation fails (i.e., stopping condition is neversatisfied and a maximum number of iterations is reached), theinformation from the channel L ch and the a priori informationto the decoder (i.e., frozen and non-frozen bit positions) arepassed on to a new factor graph permutation (e.g., Fig. 1b).When successively reaching a predefined maximum number offactor graph permutations q max , decoding terminates. This canbe viewed as a multi-stage decoding process in which a newstage is invoked if the previous stage(s) failed to converge. Algorithm 1
Multi-trellis BP decoder
Input:L ch , . LLR channel output A , . information set q max , . max. no. of factor graphs N it , max , . max. no. of iterations per factor graph u , . transmitted information bits stopID . . stopping criterion Output:ˆu . . estimated information bits N ← length ( L ch ) n ← log N iq ← for iq ≤ q max do ( L , R ) ← initializeLandR ( L ch , A ) schedule ← permute ( { , ..., n } ) iI ← for iI ≤ N it , max do ( L , R ) ← OneBPiteration ( N , n , L , R , schedule ) if checkStopCondition( L , R , u , stopID ) then ˆu = LLR2bit ( L ( , : ) + R ( , : )) return ˆu end if iI ← iI + end for iq ← iq + end for ˆu = LLR2bit ( L ( , : ) + R ( , : )) . no stop. cond. satisfied return ˆu The soft values in the intermediate stages of a specificfactor graph are cleared in subsequent decoding iterationson a permuted factor graph realization. This is because theinformation that L - and R -messages at certain stages hold isifferent for each factor graph realization. This can be inferredfrom Fig. 1c. Some soft messages are shared between differentrealizations of the factor graph, and thus, these messagescan be re-used (i.e, not dismissed). Some soft messages ina certain factor graph represents certain messages which arenot explicitly seen in another factor graph (i.e., should beeither dismissed or somehow translated according to the newfactor graph permutation). For simplicity we clear all internalmessages after each factor graph permutation in this work. Algorithm 2 checkStopCondition
Input:L , .
L-matrix of BP factor graph R , . R-matrix of BP factor graph u , . transmitted information bits stopID . . stopping criterion Output: isSatis f ied . . indicate if stop. condition is satisfied ˆu ← LLR2bit ( L ( , : ) + R ( , : )) ˆx ← LLR2bit ( L ( n + , : ) + R ( n + , : )) switch stopID do case . practical stopping criterion if ˆu · G = ˆx then return true end if case . perfect knowledge-based stop. condition if ˆu = u then return true end if case . CRC-aided stop. condition if ˆu satisfiesCRCcheck then return true end if end switch return falseFor all simulations in this work, 10 codewords per SNRpoint were simulated to get “stable” B(L)ER curves. The error-rate performance of this multi-trellis-based BP decoder whileusing the previously mentioned ( practical stopping condition )is shown in Fig. 2a and 2b, respectively. The multi-trellisBP has a better error-rate performance when compared tothe conventional BP decoder. It can be also depicted fromFig. 2a and 2b that the use of more different permutations ofthe factor graph enhances the error-rate performance, up to acertain point, when enhancement is no longer possible withincreasing the number of factor graphs used. In Fig. 2a and2b, it is also shown that the gain in performance is due tothe use of different permutations of factor graphs not a resultof the increased number of BP iterations. One reason for thisperformance improvement is due to the different structure ofloops from one factor graph permutation to another, as seenfrom Fig. 1c. As shown, two loops are highlighted in theunfolded factor graph, whereas the nodes belonging to oneloop in a graph permutation are spread among different loops 2 2 . . . . . − − − − increasing q max E b / N [dB] B E R BP 200 iterations BP 10 iterationsSCL L =
32 Multi-trellis (a) BER . . . . . − − − − increasing q max E b / N [dB] B LE R (b) BLER Fig. 2: BER and BLER comparison between BP, SCL withlist size L =
32 and Multi-trellis BP for a P ( , ) -code. Multi-trellis BP: the maximum number of iterations pertrellis is N it , max = q max = { , , } , all with the practical stopping criterion.in another graph permutation. Eventually, this means that onefactor graph might be better for a specific noise realizationthan another. Besides, in the high SNR region (i.e., errorfloor region), the main cause of errors are the stopping sets.Therefore, shuffling the used factor graphs effectively leads toshuffling the variable nodes of the decoding graph, and, thus,resulting in totally different stopping sets [17].IV. E XTENSIONS AND C OMPLEXITY
A. Perfect knowledge-based termination
During our simulations, we noticed that in many cases inwhich the decoder fails, the decoder really converged to the setof correct information bits at a certain point but the stoppingcondition was not satisfied (i.e., ˆx and ˆu did not satisfy thestopping condition due to oscillating errors on both sides). Thisled to the conclusion that the performance of the multi-trellisBP decoder can be enhanced with a better stopping criterion,e.g., a genie to decide when to stop the iterations.To test the validity of this lemma, the following experimentwas conducted. The BP decoding stops when the estimatedinformation bits in ˆu is equal to the transmitted informationbits in u , or when a maximum number of iterations pertrellis N it , max over the maximum number of trellises q max isreached (second stopping condition in Algorithm 2). It is 2 . . . . . − − increasing q max E b / N [dB] B E R BP 200 iterations BP 10 iterationsSCL L =
32 Multi-trellis (a) BER . . . . . − − − − increasing q max E b / N [dB] B LE R (b) BLER Fig. 3: BER and BLER comparison between BP, SCL L = P ( , ) -code. Multi-trellisBP: N it , max = q max = { , , , , , } , all it-erative decoders use the perfect knowledge-based stoppingcriterion (i.e., the results provide a lower bound).worth mentioning that this is not a practical decoder sinceit assumes perfect knowledge of the transmitted informationbits. However, it provides a bound on the achievable decod-ing performance. The error-rate performance of this decoder(shown in Fig. 3a and 3b) approaches, and even outperforms,the SCL decoder performance (due to the perfect knowledge-based stopping criterion), indicating that one major cause ofdecoding failure in the multi-trellis BP decoder is the lack ofproper stopping criterion. One can also see that the enhancederror-rate performance is due to the multiple different factorgraph permutations used and not due to the increased numberof BP iterations.This means that for a specific noise realization, if theSCL decoder (which achieves the ML bound) can decodesuccessfully, then there exists a polar factor graph such thatthe BP decoder can also decode successfully while using thisspecific factor graph with a carefully chosen stopping criterion. B. CRC-aided termination
Inspired by [3], one type of a practical genie that can beused is a high rate CRC code with r redundancy bits. TheCRC code can be considered as an outer code applied overthe information bits, while the polar code is the inner code.The multi-trellis BP decoder iterations terminate if the CRC 2 2 . . . . . − − − increasing q max E b / N [dB] B LE R BP 200 iterations BP+CRC-32 10 iterationsSCL L =
32 SCL+CRC-32 L = Fig. 4: BLER comparison between BP, BP+CRC, SCL L =
32, SCL+CRC L =
32 and Multi-trellis BP+CRC for a P ( , ) -code. The CRC used is r =
32 bits long. Multi-trellis BP: N it , max = q max = { , , , , , } ,with CRC-aided stopping criterion.is satisfied, or when a maximum number of iterations pertrellis N it , max over the maximum number of trellises q max isused (third stopping condition in Algorithm 2). Although weintroduce a rate loss penalty (cid:0) rN (cid:1) , this CRC-based stoppingcriterion is less complex than the previously mentioned prac-tical stopping condition, which requires an additional polarre-encoding step. Fig. 4 shows that this CRC-aided decodingwill help in approaching and even outperforming, in the highSNR region, the error-rate performance of the SCL decoderof a conventional polar code. Additionally, we provide resultsfor BP decoding with the CRC-based stopping criterion for N it , max = iterations. This shows that the gain observed bythe multi-trellis decoder is neither a result of the increasednumber of iterations nor of the CRC stopping condition.Thus the error-rate performance gain is due to the differentpermutations of the factor graphs used. It is worth mentioningthat the concatenation of a polar code with a high rate CRCcode has been proposed for the uplink control channel ofthe upcoming 5th generation mobile standard [2]. Thus, theproposed multi-trellis BP decoder is compatible with the(expected) standardized polar codes.We want to emphasize that the performance of this CRC-aided decoder is not as good as the CRC-aided version ofthe SCL decoder (i.e., SCL-CRC). The CRC codes are well-suited for the task of picking the correct codeword from thelist in the SCL decoder but in the multi-trellis BP decoder it isjust used as a stopping guideline, thus, it is not adding muchgain in the sense of information transfer (outer-inner iterativedecoding strategy). C. Complexity
The main objective of this work is to propose this newvariant of the BP decoder for polar codes and present its error-rate performance when compared to the conventional BP andSCL decoders. However, in its naive implementation, the price . . . . . E b / N [dB] A v e r a g e nu m b e r o f it e r a ti on s BP 200 iterations: practical stopping criterionBP 10 iterations: practical stopping criterionMulti-trellis: N it , max = q max =
10, practical stop.Multi-trellis: N it , max = q max = N it , max = q max = Fig. 5: Average number of iterations performed per iterativedecoding algorithm; polar code P ( , ) .to pay is a rather high decoding complexity which we brieflyevaluate next.As can be seen from Fig. 5, the average number of per-formed decoding iterations does not drastically increase undermulti-trellis decoding in the SNR range of interest, as thedecoder typically operates in the low error-rate region. Obvi-ously, the average number of iterations depends on the BLERof the plain BP decoder as each block failure causes additionaliterations over the permuted trellises. Thus, decoding overmultiple trellises is only required in some rare cases.In the CRC-aided multi-trellis decoder, the complexity inthe low SNR region is high, because the decoder will stopiterating when the CRC is satisfied, which is very unlikelydue to bad channel conditions (i.e., the CRC-aided stoppingcondition is stricter than the practical stopping condition).Various ideas can be implemented in order to reduce thecomplexity of the proposed decoder. The concept of a par-titioned successive cancellation list (PSCL) decoder [20] canbe used in this decoder where we treat a polar code of length N as two polar codes of length N (i.e., two partitions), andthe proposed decoder is applied over each partition. This willreduce the complexity indeed, but on the expense of error-rateperformance.Furthermore, wasting useless iterations on non-convergingfactor graphs can be later avoided by using an LLR-basedmetric which quantifies the convergence behavior of a spe-cific factor graph permutation. This will certainly reduce thecomplexity (and maximum latency) by a significant factor viaskipping some (non-useful) permutations only after perform-ing a few BP iterations over them.V. C ONCLUSION
In this paper, we presented a new variant of the BP decoderfor polar codes based on different permutations of the polarcode factor graph. The proposed decoder, with a perfectknowledge-based termination criterion, approaches the error- rate performance of the state-of-the-art SCL decoder, suggest-ing that the current early BP decoding stopping criteria arenot yet optimum and can be further optimized. A CRC-aidedversion of this decoder is proposed, which can outperform, inthe high SNR region, the state-of-the-art SCL decoder of aplain polar code. Obviously, a CRC is a well-suited “genie”in the context of SCL decoding, and thus the performance ofthe CRC-aided SCL decoder is still better than any iterativedecoding-based scheme for polar codes. Yet, to the best ofour knowledge, the permuted factor graph-based BP decoderpresented in this paper is the best iterative decoder for polarcodes known thus far. R
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