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Protograph-Based LDPC Code Design for Bit-Metric Decoding Fabian Steiner, Georg Bo¨cherer Gianluigi Liva Institute for Communications Engineering Institute of Communication and Navigation of the Technische Universita¨t Mu¨nchen, Germany Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR) Email: {fabian.steiner, georg.boecherer}@tum.de 82234 Wessling, Germany Email: [email protected] Abstract—A protograph-based low-density parity-check structure and the mapping of the coded bits to the BICM bit- 5 (LDPC) code design technique for bandwidth-efficient coded levels. However, we propose a protograph-based design [12] 1 modulation is presented. The approach jointly optimizes the and use protograph EXIT (PEXIT) analysis [13] to determine 0 LDPC code node degrees and the mapping of the coded bits the ensemble iterative convergence threshold by accounting 2 to the bit-interleaved coded modulation (BICM) bit-channels. for the different bit-channels associated with the protograph n For BICM with uniform input and for BICM with probabilistic variable nodes. The protograph ensemble is optimized with shaping, binary-input symmetric-output surrogate channels a respect to the threshold by differential evolution. A design of J are constructed and used for code design. The constructed protographsforcodedmodulationwasfirstintroducedin[14], codes perform as good as multi-edge type codes of Zhang and 2 Kschischang (2013). For 64-ASK with probabilistic shaping, a whichreliesonavariabledegreematchedmapping(VDMM). 2 blocklength 64800 code is constructed that operates within 0.69 More specifically, in [14] each bit-level is associated to a dBof 1log (1+SNR)ataspectralefficiencyof4.2bits/channel specific protograph variable node following the waterfilling ] 2 2 T use and a frame error rate of 10−3. approach (i.e., assigning the most protected coded bits to I the bit-levels with highest bit-channel capacities). Recently, s. I. INTRODUCTION a protograph-based coded modulation scheme was introduced c in [15] by performing a one-to-one mapping between the [ Bit-interleaved coded modulation (BICM) combines high constellation symbols and the codeword symbols of a non- order modulation with binary error correcting codes [1]. This binary protograph LDPC code. This requires the constellation 1 makes BICM attractive for practical application and BICM v order to match the field order on which the LDPC code is is widely used in standards, e.g., in DVB-T2/S2/C2. At a 5 constructed. To the best of our knowledge, none of the above- BICM receiver, bit-metric decoding (BMD) is used [2, Sec. 9 mentionedapproachesleadstoajointbinaryLDPCprotograph II].AchievableratesforBMDwereinvestigatedforuniformly 5 ensemble and bit-mapping optimization. distributed inputs in [2] and for non-uniformly distributed 5 0 bits in [3]. These results were generalized to non-uniformly For the PEXIT analysis, we represent each bit-channel . distributed input symbols in [4] and [5]. by a surrogate. Our surrogates reflect both the BICM bit- 1 channelsandtheinputdistribution.Weoptimizecodesbothfor 0 When designing low-density parity-check (LDPC) codes uniformly distributed inputs and for the probabilistic shaping 5 for BICM, the key challenge is to take into account the 1 unequalerrorprotectionoftheLDPCcodedbitsandtheBICM strategy proposed in [4]. Our optimized codes show a finite v: bit-channels that are different for different bit-levels. A first length performance that is as good as the one presented by Zhang and Kschischang [9]. Moreover, our design approach i approach is to take an existing LDPC code and to optimize X can be applied to very large constellations. For 64-ASK with the mapping of the coded bits to the BICM bit-levels. This probabilistic shaping, our blocklength 64800 code operates r was done, e.g., in [6]–[8]. A more fundamental approach is a within 0.69dB of 1log (1 + SNR) at a spectral efficiency to directly incorporate the different bit-channels in the code 2 2 of 4.2 bits/channel use and a frame error rate of 10−3. design. This is done in [9], where the authors use multi- edge type (MET) codes [10] to parameterize the different bit- This paper is organized as follows. In Sec. II, we review channels. They then extend the extrinsic information transfer bit-metric decoding. We present our code design approach in charts (EXIT) [11] to multiple dimensions to design codes Sec. III. In Sec. IV and Sec. V, we discuss the performance for quadrature amplitude modulation with 16 signal points of our codes for uniform and shaped inputs, respectively. (16-QAM). As 16-QAM can be constructed as the Cartesian product of two four point amplitude-shift keying (4-ASK) constellations, two different bit-channels are apparent. For II. BIT-METRICDECODING constellations with more than two different bit-channels, the authorsof[9]observelongruntimesoftheirmultidimensional Consider the discrete time additive white Gaussian noise EXITapproach.Therefore,theysuggestahigh-orderextension (AWGN) channel based on nesting, i.e., starting from m=2, they successively Y =∆X+Z (1) extend their codes from m to m+1 bit-levels by optimizing in each step only the additional bit-level. where the noise term Z is zero mean, unit variance Gaussian, In this work, we follow [9] and jointly optimize the code and where the input X is distributed on the normalized 2m- 000 001 011 010 110 111 101 100 C1 C2 (cid:18) (cid:19) 2 1 1 Figure1.8-ASKconstellationwithBRGC[16]labeling. A= 0 2 1 B1 pL1|B1 L1 V1 V2 V3 B2 pL2|B2 L2 C1 C2 C1(cid:48) C2(cid:48) . . . Bm pLm|Bm Lm Figure2.mparallelbit-channelsthataredifferentfordifferentbit-levels. V V V V(cid:48) V(cid:48) V(cid:48) 1 2 3 1 2 3 ASK constellation Figure3.Above,thetannergraphofbasematrixAwithe=2andf =3is displayed.Below,anexampleliftingwithQ=2instancesoftheprotograph X ={±1,±2,...,±(2m−1)}. (2) isshown. The constellation spacing ∆ controls the average power III. LDPCCODEDESIGN E[|∆X|2], where E[·] denotes expectation. The signal-to-noise ratio (SNR) is SNR = E[|∆X|2]/1. Each signal point x ∈ X LDPC code ensembles are usually characterized by the is labeled by m bits B=(B ,B ,...,B ). The B represent degree profiles of the variable and check nodes of the Tan- 1 2 m i the bit-levels. Throughout this work, we label by the binary ner graph representation of the sparse binary parity-check reflectedGraycode(BRGC)[16],e.g.,seeFig.1.Letp be matrix H ∈ F(n−k)×n [17, Section 3.3]. For instance, Y|B 2 the memoryless channel with random input B and output Y. λ(x)=(cid:80)dv λ xd−1andρ(x)=(cid:80)dc ρ xd−1aretheedge- d=1 d d=1 d The bit-metric of bit-level i is the L-value perspective variable and check node degree polynomials with maximumdegreed andd ,respectively.However,thedegree P (0|y) v c L =log Bi|Y (3) profiles do not characterize the mapping of variable nodes to i PBi|Y(1|y) different bit-channels. In the following, we use protographs to incorporate the bit-mapping in our threshold analysis. where P can be calculated from the joint distribution Bi|Y A. Protograph-Based LDPC Codes (cid:88) p (b ,y)= p (y|a)P (a). (4) BiY i Y|B B Starting from a small bipartite graph represented via its a∈{0,1}m:ai=bi basematrix A = [b ] of size e×f, one applies a copy-and- lk A bit-metric decoder uses the L-values L ,L ,...,L to permuteoperation(alsoknownaslifting)tocreateQinstances 1 2 m estimate the transmitted data. For the decoder, the channel ofthesmallgraphandthenpermutestheedgessothatthelocal appears as m parallel bit-channels, see Fig. 2. Bit-metric edgeconnectivityremainsthesame.TheQreplicasofvariable decoding can achieve the rate [5, Theorem 1] nodeVk,k ∈{1,...,f}mustbeconnectedonlytoreplicasof theneighborsofV whilemaintainingtheoriginaldegreesfor k (cid:88)m that specific edge. The resulting bipartite graph representing RBMD =H(B)− H(Bi|Li) (5) the final parity-check matrix H possesses n=Q·f variable i=1 nodesandn−k =Q·echecknodes.Paralleledgesareallowed, but must be resolved during the copy-and-permute procedure. where H(·) denotes entropy. Anexampleprotographwiththecorrespondingbasematrixand Remark 1. If the bits B ,B ,...,B are independent, then an example lifting for Q=2 are shown in Figure 3. 1 2 m (5) can be written as [2], [3] B. Surrogate Channels m (cid:88) R = I(B ;L ) (6) Optimizing LDPC codes directly for the m bit-channels in BMD i i Fig. 2 is difficult, as they are usually not output-symmetric i=1 and do not possess uniform input in the case of probabilistic where I(·;·) denotes mutual information. shaping.Weinsteadoptimizeforasurrogatechannel.Theopti- Remark 2. If the bits B ,B ,...,B are independent and mizedcodeworkswellontheoriginalchannelifthesurrogate 1 2 m uniformly distributed, then the L-values (3) can be written as channel preserves some characteristics of the original channel and if the code is universal in the sense that its performance p (y|0) depends mainly on these characteristics. We propose to use Li =logpY|Bi(y|1). (7) the rate backoff as the key characteristic, i.e., if R∗ is the Y|Bi achievable rate of the original channel and R is the actual transmission rate of the code, we assume that the decoding We use L-values as defined in (3) and the achievable rate error probability depends on the rate backoff as defined in (5), since we will use non-uniformly distributed R∗−R. (11) inputs in Sec. V.   (cid:118) IEVk→Cl =J(cid:117)(cid:117)(cid:117)(cid:116)(cid:88)e bl(cid:48)k·J−1(cid:16)IECl(cid:48)→Vk(cid:17)2+(blk−1)·J−1(cid:16)IECl→Vk(cid:17)2+σc2hk (8) l(cid:48)=1 l(cid:48)(cid:54)=l   (cid:118) (cid:117)(cid:117)(cid:88)f (cid:16) (cid:17)2 (cid:16) (cid:17)2 IECl→Vk =1−J(cid:117)(cid:117)(cid:116)k(cid:48)=1blk(cid:48)J−1 1−IEVk(cid:48)→Cl +(blk−1)·J−1 1−IEVk→Cl  (9) k(cid:48)(cid:54)=k (cid:118)  (cid:117)(cid:117)(cid:88)f (cid:16) (cid:17)2 IAPPk =J(cid:116) bl(cid:48)k·J−1 IECl(cid:48)→Vk +σc2hk. (10) l(cid:48)=1 We design codes for a surrogate channel with the same rate PEXIT analysis requires tracking the extrinsic mutual infor- backoff. Let R˜ be the transmission rate and R˜∗ the achievable mation values I per (V ,C ) pair to accommodate for the E k l rate of the surrogate channel; we require edge connections imposed by the protograph basematrix and the (possibly different) reliability of the coded bits V . The R˜∗−R˜ =! R∗−R. (12) PEXIT algorithm is summarized as follows: k Supposeourcoderateiscsothat(1−c)mbitsperchanneluse 1) Initialize each σ ,k ∈ {1,...,f} with a corre- consist of redundancy bits on average. Thus, the transmission chk sponding σ ,i ∈ {1,...,m} (17) originating from rate is Bi thebiAWGNCsurrogates.Foralltuples(V ,C ),k ∈ k l H(B)−(1−c)m. (13) {1,...,f},l ∈ {1,...,e}, set the a priori mutual information values I =0. Forinstance,foruniformlydistributedinputs,thetransmission 2) Perform variable toEcChle→cVkk node update (8) for all rate is R=cm. By (5) and (13), the rate backoff is tuples (V ,C ). k l m 3) Perform check to variable node update (9) for all (cid:88) R∗−R=(1−c)m− H(Bi|Li). (14) tuples (Vk,Cl). i=1 4) Calculate IAPPk,k ∈{1,...,f} (10). Note that the term (1−c)m does not depend on the channel. 5) Stop if all IAPPk =1, otherwise go to 2). Thus,weoperatethesurrogatechannelatthesameratebackoff if IV. CODEDESIGNFORUNIFORMINPUTS m m (cid:88)H(B |L )=! (cid:88)H(cid:16)B˜ |Y˜ (cid:17). (15) A. Optimization Procedure for Protograph Ensembles i i i i i=1 i=1 Weimposerestrictionsonthedimensionofthebasematrix Since we want to account for the different bit-channels, we A∈{0,1,...,S}e×f.Theparameterseandf determinehow restrict our surrogate channel further by requiring equality for many different variable degrees per bit-channel can occur and the individual conditional entropies in (15) for each bit-level. are chosen to represent the desired code rate c = f−e. The f We use biAWGN channels with uniform input as surrogate maximum number of parallel edges S is crucial for both the channel, i.e., we have performance of the code and the success of the optimization procedure. The product e·S describes the maximum variable Y˜ =x +Z (16) i B˜i i node degree in the final code and thereby determines the where the B˜ are binary and uniformly distributed, x = 1, size of the optimization search space. The optimization of the i 0 basematrices is performed by differential evolution (DE), see x =−1, and where Z is zero mean Gaussian with variance 1 i σ2 where [18]. Bi (cid:16) (cid:17) σ2 : H(B |L )=! H B˜ |Y˜ . (17) B. Numerical Results Bi i i i i Inordertocompareourcodedesignapproachtothesetting C. PEXIT Analysis of Zhang and Kschischang [9], we design codes of rates 1/2 The analysis of the asymptotic decoding threshold can be and 3/4 for 4-ASK constellations with uniform inputs. The done by PEXIT analysis [13]. For EXIT chart analysis, the J optimized protographs are depicted in Table 4. We discuss the function is widely used. The value J(σ ) hereby represents rate 1/2 code in more detail. The coded bits are transmitted ch the capacity of the biAWGNC with variance σ2, where σ2 = overtwodifferentbit-channels.Foreachbit-channel,weallow ch 4 : 3(possiblydifferent)variabledegreessothatalways3variable σ2 nodes in the protograph are assigned to the same surrogate J(σch)=1− (cid:90)∞ (cid:112)2π1σ2 e−((cid:96)−2σσc2hc2h/2)2 log2(cid:0)1+e−(cid:96)(cid:1)d(cid:96). σchc2ha6nn=el:4σ/cσ2hB213.=Onσcc2eh2th=e oσpc2thim3 iz=ed4b/aσsB2e2m,aσtrc2ihc4es=havσec2hb5ee=n ch found, we construct quasi-cyclic parity-check matrices with −∞ (18) blocklengths n = 16200. We simulate the constructed codes Constellation(coderate) Basematrix PEXIT Decoding GaptoRBMD(5) Threshold   2 1 1 2 1 4 4-ASK(1/2) A= 1 1 1 2 2 5  5.57dB 0.28dB 1 0 0 1 0 6 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) B2 B1 (cid:18) (cid:19) 1 1 1 1 6 6 1 1 4-ASK(3/4) A= 9.57dB 0.26dB 1 1 2 2 6 6 2 2 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) B2 B1 (cid:18) (cid:19) 2 2 2 1 2 2 6 2 2 0 6 6 64-ASK(5/6) A= 25.52dB 0.29dB 1 1 1 2 1 1 6 1 0 2 6 6 (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125) B2 B3 B4 B5 B6 B1 Figure4.Optimizedprotographsanddesignparameters. 100 100 10−1 10−1 10−2 10−2 R R E E R/F 10−3 R/F 10−3 E E B B 10−4 10−4 METcode[9]BER METcode[9]BER METcode[9]FER METcode[9]FER OptimizedProtographBER OptimizedProtographBER 10−5 10−5 OptimizedProtographFER OptimizedProtographFER 16-ASKuniformBMD 16-ASKuniformBMD 10−6 10−6 5.2 5.4 5.6 5.8 6 6.2 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 SNR[dB] SNR[dB] Figure 5. 16-ASK: Performance of a rate 1/2 optimized protograph code Figure 6. 16-ASK: Performance of a rate 3/4 optimized protograph code comparedtotherate1/2METcodein[9]. comparedtotherate3/4METcodein[9]. using 100 decoding iterations. The bit error rates (BER) and • bit-levels (B ···B ) and bit-level B are indepen- 2 m 1 frame error rates (FER) in Fig. 5 and 6 show that the finite dent. bit-levels B ,...,B are correlated. 2 m length performance of our codes is equal to or slightly better thanthecodesin[9].ThedecodingthresholdsgiveninTable4 This suggests to mimic the capacity-achieving distribution are obtained by PEXIT analysis for the surrogate channels. in the following way: first, generate bit-levels B ···B ac- 2 m cording to P , e.g., by using a distribution matcher B2···Bm (see Fig. 7). Then encode by a systematic encoder of rate V. CODEDESIGNFORSHAPEDINPUTS (m−1)/m that copies the bits B ···B to its output and 2 m We briefly outline how our design procedure can be used leaves their distribution un-changed. The encoder appends to design LDPC codes for the probabilistic shaping scheme checkbitsB thatareapproximatelyuniformlydistributedbe- 1 proposed in [4]. causeeachcheckbitisamodulotwosumofmanyinformation bits [19, Sec. 7.1]. The signal point x selected by B1B2···Bm A. Probabilistic Shaping the bit-mapper then has approximately the capacity-achieving distribution. Fig. 7 illustrates the shaping scheme. The key observation is that the capacity-achieving distribution of ASK constellations in Gaussian noise is symmetric around the origin. B. Surrogate Channels Consequently, the capacity-achieving distribution induces a distribution P on the BRGC labeling with the fol- As discussed before, the channel appears to the receiver lowing properBti1eBs2:···Bm as a set of m parallel binary input channels, see Fig. 2. The non-uniformdistributionP influenceseachchannellaw B2···Bm • bit-level B is uniformly distributed (bit-level B p via (4). Consequently, it also influences the conditional decides on1the sign of the transmitted constellation1 enLitr|Boipies H(B |L ) in our surrogate channels (17) that we use i i point, see Fig. 1). for code optimization. Ukd Matcher (B2···Bm)nc LDPCEncoder (B2···Bm)ncBn1c Bit-Mapper Xnc Fii=gu1re,27,..P.r.o,bnacbtihliastticarsehdaipsitnrigbuatsedpraocpcoosreddinignt[o4]P.BIn2d··e·Bpmen.dTenhteusnyisftoermmaltyicdLisDtrPibCuteendcdoadtearbaiptspeUnkdds=ncUch1eUc2k·b·it·sUBkdn1car=emB1a1tcBh1ed2·to··nBc1sntcr.inTghse(bBi2tBm3ap·p·e·rBmma)pis, thisbitstreamtosignalpointsaccordingtotheBRGClabeling.Theoverallrateiskd/nc[bits/channeluse]andtherateoftheLDPCcodeis(m−1)nc/(mnc)= (thme −est1im)/amte.oAft(Bth2e·r·e·cBeimve)rn,cathbrito-umgehtriacddeemcoadtcehrecr.alFcourlattheseamnatecshtiemraatnedofth(eBd2e·m·a·tBchmer),ncwBen1cu.seAthdeataimepstliemmaetnetaUˆtikodn=httUˆp:1/U/ˆw2w·w··.lUnˆtk.dei.itsumob.dtaei/nfieldeabdymipna/ssstainffg/ gu92vox/matcher.zip. 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Bo¨cherer, “Capacity-achieving probabilistic shaping for noisy and proach for bandwidth-efficient coded modulation that is suit- noiseless channels,” Ph.D. dissertation, RWTH Aachen University, able both for uniform and shaped inputs. The different bit- 2012. [20] F. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for channels are replaced by biAWGN surrogates so that PEXIT Gaussianchannels,”IEEETrans.Inf.Theory,vol.39,no.3,pp.913– and differential evolution give ensembles with good decoding 929,May1993. thresholds. The performance of the new codes for uniform inputs are as good as the best codes in literature. For shaped inputs,thenewcodesoperatecloseto 1log (1+SNR).Future 2 2 research should investigate the influence of the surrogates on the code performance by employing a full-fledged density evolution for protographs. Furthermore, precoded protographs shouldbeconsideredtoimprovethethresholdwithoutincreas- ing the variable node degrees.