We propose a new class of parallel data convolutional
codes (PDCCs) in this paper. The PDCC encoders inputs
are composed of an original block of data and its interleaved version.
A novel single self-iterative soft-in/soft-out a posteriori probability
(APP) decoder structure is proposed for the decoding of
the PDCCs. Simulation results are presented to compare the performance
of PDCCs

Pietrobon, Steven S.

Xiang, Wei

English

University of Southern Queensland ePrints

A New Class of Parallel Data Convolutional CodesWei Xiang and Steven S. PietrobonAbstract— We propose a new class of parallel data convolu­tional codes (PDCCs) in this paper. The PDCC encoders inputsare composed of an original block of data and its interleaved ver­sion. A novel single self­iterative soft­in/soft­out a posteriori prob­ability (APP) decoder structure is proposed for the decoding ofthe PDCCs. Simulation results are presented to compare the per­formance of PDCCs to that of parallel concatenated convolutionalcodes (PCCCs).Index Terms—PDCCs, Self­iterative, APP.I. INTRODUCTIONThe original turbo codes proposed by Berrou et al. [1] arebinary turbo codes in that those codes accept only single binaryinputs. The so­called non­binary turbo codes are based on aparallel concatenation of RSC component codes withm inputs(m � 2) [2]. The advantages of non­binary turbo codes includebetter convergence in iterative decoding, large minimum dis­tances, less sensitivity to puncturing patterns and suboptimumdecoding algorithms and reduced latency [2]. Double­binaryturbo codes [3] (m = 2) usually possess better error­correctingcapabilities than binary turbo codes for equivalent implementa­tion complexity and coding rate. This observation led to the useof circular recursive systematic convolutional (CRSC) codes byBerrou et. al. [4]. CRSC codes have the advantage of a grace­ful degradation to increasing coding rate, and is less suscepti­ble to puncturing and suboptimal decoding algorithms [5]. Asa consequence, a CRSC code was chosen for the DVB­RCSstandard for return channel via satellite [6] as an alternative toconcatenated Reed­Solomon (RS) and non­systematic convolu­tional codes due to their outstanding performance.Inspired by a paper submitted recently to Electronic Let­ters [7], we propose a new class of parallel data convolutionalcodes (PDCCs) in this paper. The PDCC encoder inputs arecomposed of an original block of data and its interleaved ver­sion. A novel single self­iterative soft­in/soft­out a posterioriprobability (APP) decoder structure is proposed for the decod­ing of the PDCCs.The remainder of this paper is organised as follows. Sec­tion II­A briefly reviews the circular recursive systematic con­volutional code adopted in the DVB­RCS standard. A new classof parallel convolutional codes is proposed in Section II­B. Sec­tion III discusses the MAP decoding and self­iterative decodingof PDCCs. Section IV is dedicated to simulation results. Theconcluding remarks are presented in Section V.II. PARALLEL DATA CONVOLUTIONAL CODESA. Circular Recursive Systematic Convolutional CodesFig. 1 depicts the CRSC code adopted in the DVB­RCS stan­dard [6]. The data sequence to be encoded consists of a blockof N bits grouped into M couples (N = 2M ). The incomingdata is first demultiplexed and fed into A and B of a CRSCencoder (the first bit to A, second bit to B and so on). Aand B are two systematic bits, whereas Y and W are two par­ity bits. Since a CRSC encoder has two inputs and four out­puts, it can provide an ample set of seven coding rates, i.e.,R = 1/3, 2/5, 1/2, 2/3, 3/4, 4/5 and 6/7. These rates areachievable through puncturing the parity bits.We show the derivation of the parity check equations and itscanonical form for the CRSC code adopted by the DVB­RCSstandard as shown in Fig. 1.+ S1 + S2+++ S3ABW Y Redundancy partSystematic partFig. 1. CRSC component code structure.Through some mathematic derivation, it is not difficult toshow that the parity check equations of the parity output bitsY andW are expressed by (1) and (2) as follows:(1 +D2 +D3)A+ (1 +D +D2 +D3)B+ (1 +D +D3)Y = 0 (1)(1 +D3)A+ (1 +D2)B + (1 +D +D3)W = 0. (2)Furthermore, the canonical form [8] of the CRSC depicted inFig. 1 can be derived, which is illustrated in Fig. 2. Refer to [9]for detailed derivation of the canonical form shown in Fig. 2.B. Parallel Data Convolutional Codeswe propose a new class of parallel data convolutional codes.Fig. 3 depicts a PDCC encoder in its canonical form whichadopts the CRSC code described in Section II­A as the con­stituent convolution code. It is assumed that S� is the MSB1(most significant bit) and S� = 4S� + 2S� + S3.1 2 �As depicted in Fig. 3, the block of data sequence to be en­coded A and its interleaved version A� constitute two inputsinto the encoder. The fact that a PDCC encoder has two parallelA+ + S �1++ S �2++ S �3 +XB X�YWFig. 2. Canonical systematic convolutional code of coding rate 1/2.A+ + S �1++ S �2++ S �3 +π A�XX�YWFig. 3. PDCC encoder.data inputs is the reason that we name it parallel data convolu­tional codes. X and X � are two systematic outputs, whereas YandW are two parity bits. The parity check relationships of YandW resembling (1) and (2) are given by(1 +D2 +D3)A+ (1 +D +D2 +D3)A�+ (1 +D +D3)Y = 0 (3)(1 +D3)A+ (1 +D2)A� + (1 +D +D3)W = 0. (4)The data stream A and its interleaved version A� are fed intothe decoder at the same time. However, A� is decorrelated rela­tive toA due to the presence of the interleaver. For a reasonablygood interleaver, like the S­interleaver used in our simulations,this should not adversely affect the performance of the code.The systematic bitX � is not transmitted asX � is the interleavedversion ofX . Thus, the PDCC encoder shown in Fig. 3 can typ­ically provide a code rate of 1/2 by transmitting the systematicbit X and the parity bit Y , and a code rate of 1/3 by transmit­ting the systematic bit X and the parity bits Y and W . It canalso provide other coding rates through puncturing the paritybits Y andW if needed.It is noted that the idea of self­concatenation in [10] is dif­ferent from that of PDCCs. For the idea of self­concatenation,data X and its interleaved version X � are joined together andencoded as a single data stream. In other words, the end stateof X is the starting state of X �. However, this is quite differ­ent from the idea of PDCCs where X and X � are fed into thedecoder at the same time.III. SELF­ITERATIVE DECODING OF PDCCSA. MAP Decoding of PDCCsThe key difference between the MAP algorithm for PDCCsand the MAP algorithm presented in [11] is that the PDCC en­coder has two input bits and four output bits, including twosystematic bits A,A� and two parity bits Y,W . The MAP al­gorithm described in [11], however, is applicable to the softdecoding of rate 1/2 systematic convolution codes which haveone input bit and two output bits, including one systematic bitand one parity bit. Fig. 4 illustrates the trellis diagram of thePDCC presented in Fig. 3. It is an 8­state trellis. The numbersshown on the left of each state represent a format of system­atic bits/parity bits, i.e., AA�/YW . The numbers from top tobottom correspond to the state transitions diverging from eachstate from top to bottom.AA�/YW k−1 k00/0011/0001/1110/1111/1000/1010/0101/0111/0100/0110/1001/1000/1111/1101/0010/0001/1010/1000/0111/0110/0001/0011/1100/1110/1101/1111/0000/0001/0110/0100/1011/10S0S1S2S3S4S5S6S7S0S1S2S3S4S5S6S7Fig. 4. PDCC’s 8­state trellis diagram.Assume that the outputs of the PDCC encoder depicted inFig. 3 at time index k are the systematic bit Ak, and the par­ity bits Yk and Wk. These outputs are BPSK modulated andtransmitted through an AWGN channel. At the receiver end,the received symbols at time index k are defined as1RAk = (1− 2Ak) + n (5)k2RYk = (1− 2Yk) + n (6)k3RWk = (1− 2Wk) + nk, (7)1 3with nk, n2 and nk being three independent normally dis­ktributed Gaussian random variables with variance σ2. Ak� , theinterleaved version of the received symbol Ak, is obtained byinterleaving Ak at the receiver end.As shown in Fig. 4, each branch in the PDCC trellis diagramis associated with two input bits and four output bits. Therefore,the branch metric γi,m, which denotes the branch exiting fromk� �� �� �� �� ������Sk = m with Ak = i, can be expressed asjγi,mξi ξ� 1 � �2= k k exp RAk − (1− 2Ak) dRAkk 23√2πσ −2σ21 1 � �2exp RA� − (1− 2A�k k·√2πσ −2σ2 k) dRA�1 1 � �2exp RYk − (1− 2Yk) dRYk·√2πσ −2σ21 1 � �2exp RWk − (1− 2Wk) dRWk·√2πσ −2σ2j= χk ξi k k +RYkYkk ξ� exp − Lc RAkAk +RA� A�k+RWkWk , (8)jwhere χk is a constant, ξi = Pr(Ak = i), ξ� = Pr(A� = j),k k kLc = 2/σ2, and dRAk , dRA� , dRYk and dRWk are differentialskof RAk , RA� , RYk and RWk .kThe forward state metric αm at time k and state m can bekshown as3= γi,b(i,m)αb(i,m),αm (9)k k−1 k−1i=0where b(i,m) denotes the backward state whose next state ismgiven input i at the previous time. Likewise, the backward statemetric βm at time k and statem can be expressed ask3= γi,m f(i,m),βm (10)k k βk+1i=0where f(i,m) denotes the forward state given current input iand statem.The likelihood ratio λk associated with each decoded bit Akis defined asPr(Ak = 1|RA, RY , RW )λk = Pr(Ak = 0|RA, RY , RW )3��αm γi,mf(i,m)k k βk+1m i=2=1��αm γi,mf(i,m), (11)k k βk+1m i=0where i = 0, 1, 2, 3 corresponds to A,A� inputs of 00, 01, 10and 11. Similarly, the log­likelihood ratio λ� of the interleavedkbit A� can be written askPr(A� = 1|RA, RY , RW )kλ� =k Pr(A� = 0|RA, RY , RW )k� �αm γi,mf(i,m)k k βk+1m i=1,3= � �αm γi,mf(i,m). (12)k k βk+1m i=0,2Decisions on decoded bits �Ak are then made by the PDCCMAPdecoder by comparing λk to a threshold equal to one.B. Self­iterative DecodingThe novelty of decoding the PDCCs lies in self­iterative de­coding. The self­iterative PDCC decoder operates like a normalMAP decoder except it feeds the extrinsic outputs after inter­leaving or deinterleaving back as a priori inputs. Fig. 5 showsa schematic of a self­iterative PDCC decoder.LcRALcRYLcRWπ π π−1++LoutAA�YWZeAZeA�MAPLcRA�DecoderZaAZaA�Fig. 5. Self­iterative PDCC decoder.The inputs to the decoder are the soft outputs of a noisychannel LcRA, LcRY and LcRW , respectively. The decoderreconstructs LcRA by interleaving LcRA. The idea of the self­iterative decoding comes from the fact that R� is the inter­Aleaved version of RA, so that the extrinsic information of RAcan be fed back as the a priori information for R� after inter­Aleaving and the extrinsic information of R� can be fed back asAthe a priori information for RA after deinterleaving.We denote the a priori information ofRA andRA� by Za andAZaA� , while the extrinsic information ofRA andR� are denotedAby Ze and ZeA� , respectively. The self­iterative MAP decoderAcomputes the APP of the information bit A. The LLR output ofthe decoder can be expressed asLout = LcRA + LcRA� + Za + Ze + Za A� . (13)A� + ZeA AThe self­iterative PDCC decoder proceeds as follows. At thefirst decoding iteration, Za and Za are initialised to zero. ForA A�the subsequent iterations, Ze is interleaved and fed back as theAa priori information for A�, i.e., Za = π(ZeA) where π(·) de­A�notes an interleaving mapping. Likewise, Ze is deinterleavedA�and fed back as the a priori information for A, i.e., Za =Aπ−1(ZeA�) where π−1(·) denotes a deinterleaving mapping. Atthe final iteration, the decoder delivers the log­likelihood out­put Lout. The self­iterative decoding process can be clearlyseen from the two feed back connections between Ze and ZaA A� ,and Ze and ZaA in Fig. 5.A�IV. SIMULATION RESULTSIn this section, we compare the performance of the proposedPDCC to a parallel concatenated CRSC coding system used inthe DVB­RCS standard and present simulation results. The sys­tem functional block diagram is graphically shown in Fig. 6.The horizontal and vertical constituent codes used in Fig. 6are the same as the CRSC presented in Fig. 1. The incom­ing information data sequence consists of a block of N bitsπ ENC−ENC |ABYA�B�WFig. 6. Parallel concatenated CRSC encoder.grouped into M couples (N = 2M ). The horizontal con­stituent encoder is fed with the information bits in the natu­ral order of the data, whereas the vertical constituent encoderis fed with the same information bits in an interleaved orderof the data. A� and B� bits are not transmitted since they canbe reconstructed by interleaving the A and B bits at the re­ceiver side. The parallel concatenated CRSC encoder can pro­vide seven code rates as defined in the DVB­RCS standard, i.e.,R = 1/3, 2/5, 1/2, 2/3, 3/4, 4/5, 6/7 [6]. The decoderstructure is similar to that of turbo codes [1] except that theMAP decoders for a rate 1/2 systematic convolutional code arereplaced by the MAP decoders for the CRSC described in Sec­tion II­A.Simulations were conducted to compare the performance forthe proposed PDCC system shown in Fig. 3 to the parallel con­catenated CRSC system shown in Fig. 6. The simulation con­figurations are as follows. An S­type interleaver [12] is adoptedwith S equal to 47. Randomly generated data of length 8K(8192) bits is used for both systems. The channel coding ratefor both systems is 1/2. Simulation results are presented inFig. 7, where the PCCC curve refers to the performance of theparallel concatenated CRSC system.As depicted in the figure, the performance of the PDCC isvery close to that of the parallel concatenated CRSC at lowEb/N0 up to 0.6 dB. However, PDCC performs at least 0.2 dBworse than the parallel concatenated CRSC at low BERs. Weconjecture the relatively poor performance of PDCCs at lowBER may be due to the self­terminating phenomena with oneinput bit. Fig. 8 illustrates the trellis terminating properties ofthe PCCCs (turbo codes) and PDCCs. For the PCCCs, an errorbit could cause the trellis path to divert from the two all­zeropaths as shown in Fig.8–(a). The same bit is interleaved andfed into the second encoder. That bit would not cause the di­verted trellis path to re­emerge earlier. On the other hand, forthe PDCCs, an error causes a diversion from the all­zero trellispath. The same bit is interleaved and then fed into the samePDCC encoder. That bit could cause an earlier trellis remergeas shown in Fig.8–(b). The decoding complexity of PDCCs issimilar to that of the PCCCs. This is because the PDCC de­coder consists of one constituent MAP decoder which decodesone bit in each iteration, whereas the PCCC decoder consists of0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210−710−610−510−410−310−210−1100Eb/N0 (dB)Bit Error Rate (BER)PDCC: 8 iterationsPDCC: 16 iterationsPDCC: 8 iterations + trellis terminationPDCC: 16 iterations + trellis terminationPCCC: 8 iterationsFig. 7. Simulation results.two constituent MAP decoders which decode two bits in eachiteration.( a )( b )Fig. 8. Self­terminating property of (a) PCCC; (b) PDCC.A way of overcoming the self­terminating property may beby designing the interleaver so as to avoid self­terminations.For example, if Ak = 1, with zeros elsewhere except atA� = 1, then we would have a self­terminating sequencek+3going through states S0 � S3 � S4 � S0. Thus, we could de­sign the trellis such that if the interleaver maps position i to j,then (i− j) mod 3 is not equal to zero. This will remove manyself­terminating sequences and hopefully lower the error floor.Another possibility is designing the convolutional code so thatthese period 3 terminations do not occur. It may be possible toincrease the self­termination length to 2ν − 1 = 7, which is themaximum that can be expected with a primitive divisor polyno­mial. These longer lengths should be easier to design out of theinterleaver.Nevertheless, the idea of self­iterative decoding could be use­ful in some cases, e.g., space­time coding. We could combinethe two trellises of a space­time code into a super trellis. Subse­quently, self­iterative decoding could be applied to such a trelliswith non modulo­2 operation.V. CONCLUSIONSIn this paper, a new class of parallel data convolutional codesis presented. The PDCC encoder takes two parallel data inputs,with one being the original data and the other being the inter­leaved data. The PDCC decoder has an innovative self­iterativedecoding structure. Unlike a turbo decoder where the extrinsicoutput of one MAP decoder is passed on to the other as the apriori input, the PDCC decoder operates like a normal MAP de­coder but feeds the extrinsic outputs back as its own interleaveda priori inputs.The performance of PDCCs was compared to that of a paral­lel concatenated CRSC, equivalent to the one used in the DVB­RCS standard. The two schemes performed close to each otherat low SNRs, however the parallel concatenated CRSC outper­formed PDCCs by at least 0.2 dB at low BERs. Also, the PDCChas a higher error floor than the PCCC. We conjecture this isdue to the self­terminating property of PDCCs with single biterrors. Although the performance of PDCCs is not encourag­ing, the idea of self­iterative decoding is worth exploring andcan be applied to some codes like space­time codes.REFERENCES[1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error­correcting coding and decoding: turbo­codes,” in Proc. IEEE Int. Conf.Commun. (ICC’93), Geneva, Switzerland, May 1993, pp. 1064–1070.[2] C. Berror, M. Je´ze´quel, C. Douillard, and S. Keroue´dan, “The advantagesof non­binary turbo codes,” in Proc. IEEE Information Theory Workshop(ITW’01), Cairns, Australia, Sept. 2001, pp. 61–63.[3] C. Berrou and M. Jezequel, “Non­binary convolutional codes for turbocoding,” Electron. Lett., vol. 35, no. 1, pp. 39–40, Jan. 1999.[4] C. Berrou, C. Douillard, and M. Jezequel, “Multiple parallel concate­nation of circular recursive convolutional (CRSC) codes,” Annals ofTelecommunications, vol. 54, no. 3­4, pp. 166–172, Mar.­Apr. 1999.[5] M. R. Soleymani, Y. Gao, and U. Vilaipornsawai, Turbo Coding for Satel­lite and Wireless Communications. Norwell, MA: Kluwer AcademicPublishers, 2002.[6] ETSI EN 301 790, Digital Video Broadcasting (DVB); Interaction chan­nel for satellite distribution systems. ETSI reference EN 301 790, v.1.3.1,Mar. 2003.[7] Y.­H. Li and D. Li, “Two­dimensional log­MAP decoding algorithm andits applications,” Submitted to Electron. Lett., Mar. 2003.[8] G. D. Forney, “Convolutional codes I: Algebraic structure,” IEEE Trans.Inform. Theory, vol. IT­16, pp. 720–738, Nov. 1970.[9] W. Xiang, “Joint source­channel coding for image transmission and re­lated topics,” Ph.D. dissertation, University of South Australia, Adelaide,Australia, 2003.[10] C. Berrou, “Some clinical aspects of turbo codes,” in Proc. Int. Symp. onTurbo Codes & Related Topics, Brest, France, Sept. 1997, pp. 26–31.[11] S. S. Pietrobon, “Implementation and performance of a Turbo/MAP de­coder,” Int. J. Satellite Commun., vol. 16, pp. 23–46, Jan.­Feb. 1998.[12] D. 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Convolutional codes I: Algebraic structure,”

Implementation and performance of a Turbo/MAP de coder,”

Joint sourcechannel coding for image transmission and re lated topics,”

Multiple parallel concate nation of circular recursive convolutional (CRSC) codes,”

Multiple turbo codes for deepspace com munications,”

Near Shannon limit error correcting coding and decoding: turbocodes,” in

Nonbinary convolutional codes for turbo coding,”

Some clinical aspects of turbo codes,” in

The advantages of nonbinary turbo codes,” in

Turbo Coding for Satel lite and Wireless Communications.

Twodimensional logMAP decoding algorithm and its applications,” Submitted to Electron.

A new class of parallel data convolutional codes

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A new class of parallel data convolutional codes

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