Advanced channel coding for space mission telecommand links
Marco Baldi, Marco Bianchi, Franco Chiaraluce, Roberto Garello, Ignacio Aguilar Sanchez, Stefano Cioni
aa r X i v : . [ c s . I T ] O c t Advanced channel coding for space missiontelecommand links
M. Baldi, M. Bianchi, F. Chiaraluce, R. Garello, I. Aguilar Sanchez, S. Cioni
Abstract —We investigate and compare different options forupdating the error correcting code currently used in spacemission telecommand links. Taking as a reference the solutionsrecently emerged as the most promising ones, based on Low-Density Parity-Check codes, we explore the behavior of alter-native schemes, based on parallel concatenated turbo codes andsoft-decision decoded BCH codes. Our analysis shows that thesefurther options can offer similar or even better performance.
I. I
NTRODUCTION T He only error correcting code currently included in theCCSDS [1] recommendation and the ECSS [2] standardfor Telecommand (TC) synchronization and channel codingis the expurgated BCH(63, 56) code, with hard-decision de-coding. In order to improve such an “obsolete” scheme, alot of work has been recently done to propose solutions thattake into account the most recent progress and comply withincreasing demand for more and more sophisticated uplinkcoding capabilities [3]. Among the most attractive options, aprominent role is played by short Low-Density Parity-Check(LDPC) codes, proposed in binary [4] and non-binary [5] form.In this paper, we enlarge the grid of possible candidatesto replace the code of the current standard. In particular,we consider parallel turbo codes (already included in theCCSDS Telemetry (TM) recommendation [6]) and extendedBCH codes (eBCH) that however, contrary to the standard,use soft-decision decoding in place of hard-decision decoding.As a meaningful example, we focus on the case of codeswith length and rate / . The performance of the newoptions are evaluated, also in comparison with the asymptoticlimits achievable, i.e., Shannon’s sphere packing lower bound(SPLB). Assuming non-binary LDPC codes as a valuablepractical reference, we show that: i) new designed parallelturbo codes have performance close to non-binary LDPC codesdown to codeword error rate (CER) in the order of − ; ii)the eBCH(128, 64) code decoded through a maximum likeli-hood (ML) algorithm has nearly optimal performance, betterthan non-binary LDPC codes. We then focus on sub-optimalalgorithms that allow to approach the ML performance, at areasonable computational cost. Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected] work was supported in part by the ESA Contract No 4000106268: Ad-vanced Coding Schemes for Direct Sequence Spread Spectrum TelecommandLinks. M. Baldi, M. Bianchi and F. Chiaraluce are with Universit`a Politecnicadelle Marche, Dipartimento di Ingegneria dell’Informazione, Ancona, Italy.R. Garello is with Politecnico di Torino, Dipartimento di Elettronica eTelecomunicazioni, Torino, Italy. I. Aguilar Sanchez and S. Cioni are withESA-ESTEC, TEC-ETC, Noordwijk, The Netherlands.
The organization of the paper is as follows. In Section IIwe remind the current standard and the possibility to improveperformance by resorting to soft-decision decoding. In SectionIII, we present LDPC codes, that are at the basis of recentproposals for updating the standard. In Section IV we designparallel turbo codes that are competitive against LDPC codes.In Section V we discuss the possibility to apply sub-optimalsoft-decision decoding algorithms to eBCH codes, achievingperformance very close to that of ML decoding. Finally, someconclusions are drawn in Section VI, where we also highlightsome open issues.II. C
URRENT STANDARD
Let us refer to the CCSDS recommendation [1]: it specifiesthe functions performed in the “Synchronization and ChannelCoding sublayer” in TC ground-to-space (or space-to-space)communication links. In short, the sublayer takes transferframes produced by the upper sublayer (“Data Link Protocolsublayer”), elaborates them and outputs Communications LinkTransmission Units (CLTUs) that are passed to the lower layer(“Physical layer”) where they are mapped into the transmittedwaveform by adopting a proper modulation format. Within theSynchronization and Channel Coding sublayer, three functionsare realized: randomization (optional for CCSDS, mandatoryfor ECSS), error control coding and synchronization.The current CCSDS recommendation and ECSS standarduse a BCH(63, 56) code for error protection against noise andinterference. At the receiver side, hard decision is taken onthe received symbols. The performance of the hard-decisiondecoded BCH(63, 56) code, evaluated on the additive whiteGaussian noise channel, are quite unsatisfactory. Because of itsvery limited error correction capability, it requires very largesignal-to-noise ratios. For this code, however, it is possibleto perform an effective ML soft-decision decoding based onits trellis representation. More precisely, for each linear code C ( n, k ) it is possible to apply, for example, the techniquedescribed in [7] to build a time-variant trellis representationwith a maximum number of states equal to x , where x =min { k, n − k } . For the BCH(63, 56) the maximum numberof states is equal to = 128 , and then it is possible to applya soft-decision decoding by using the Viterbi algorithm or theBCJR algorithm [7] that, for states, still have reasonablecomplexity.In Fig. 1 we have reported the simulated CER of theBCH(63, 56) code when using BCJR, in comparison withthe performance achieved through hard-decision decoding. Thelatter can be also determined analytically. As a further bench-mark, we have also plotted the so-called “truncated union -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] BCJR ML, simulation Hard, analytical Hard, simulation TUB d*=8 TUB d*=4
Fig. 1. Performance of the BCH(63, 56) code with hard- and soft-decisiondecoding. bound” (TUB), which is given by the following expression:
CER
T UB = d ∗ X i =1 A i erfc r i kn E b N , (1)where A i is the weight- i codeword multiplicity and E b /N isthe signal-to-noise ratio per bit. The TUB represents an ap-proximation of the complete union bound (for which d ∗ = n )and, as shown in the figure, provides an excellent approx-imation of the ML decoding performance in the region ofsignificant CER ( ≤ − ). Although the whole code distanceprofile is known for the considered code, the figure showsthat d ∗ = 8 (or even d ∗ = 4 ) is enough to obtain an excellentapproximation. Looking at the figure, we can conclude that, atlow error rates, the BCJR ML soft-decision decoding providesa gain of more than 2 dB with respect to the hard-decisiondecoding curve. Though appreciable, such gain is not enoughfor the expectations of the updated standard. Thus, othersolutions must be explored, for example of the type discussedin the next sections. III. LDPC CODES
An obvious way to improve the error rate performance, withrespect to the current standard, consists in using more powerfulcodes, with lower rate [8]. A first significant proposal, inthis sense, has been advanced by the National Aeronauticsand Space Administration (NASA) and is described in [4].It is based on the adoption of three systematic short binaryLDPC codes, with rate / and length n = 128 , and ,respectively. These codes are designed using protographs withcirculant matrices. We have verified that different constructions[9]–[12] can be used as well, providing similar performance,with no significant impact on the encoding/decoding com-plexity. As an example, we have adopted the structure namedMultiple Serially-Concatenated Multiple-Parity-Check (M-SC-MPC) code [12]. These codes are a class of structured LDPCcodes obtained from the serial concatenation of very simplecomponent codes, named MPC codes, which results in LDPCcodes with good performance and very good flexibility in thedesign. The very simple structure of the component codes -8 -7 -6 -5 -4 -3 -2 -1 NASA code BER NASA code CER M-SC-MPC code BER M-SC-MPC code CER E b /N [dB] Fig. 2. Performance of different binary (128, 64) LDPC codes. also facilitates the encoder implementation. The bit error rate(BER) and CER performance of the ( n = 128 , k = 64) code designed this way are reported in Fig. 2 and comparedwith the performance of the NASA code. Both codes havebeen decoded by using the sum-product algorithm with log-likelihood ratios (LLR-SPA) (actually, for its code, NASAuses an optimized min* decoding algorithm that has, basically,the same performance) with a maximum number of iterations I max = 100 .Despite the structural differences, Fig. 2 confirms that theperformances of the two codes are very similar. To decideabout their goodness, however, an absolute reference is re-quired. Indeed, a valuable benchmark can be provided by theSPLB. Giving a lower bound on the CER performance of acoding scheme with a given codeword length, it is useful toestimate a code “optimality”, i.e., how far the performance ofthe considered code is from the best theoretical one, and howmuch gain is available for other coding schemes, if able tooutperform it. Among the various approaches available, themost suitable one is the so-called SP59, as introduced byShannon in 1959 [13]. It must be said that a modified versionof this bound is also available (called SP67 [14]), that is able totake into account the constraint put by the signal constellation(2-PSK in the present analysis). This further bound has beeneven improved more recently [15] but such improvements aresignificant only for high code rates or long codeword lengths,and these conditions are not satisfied by the codes here ofinterest. Thus, in the present study, we consider SP59 as themost significant SPLB.The SP59 is plotted in Fig. 3, for the considered caseof n = 128 and k = 64 , and there compared with theperformance of the NASA code, for different values of I max .From the figure, we observe that the performance of the NASALDPC code is good but not excellent. In fact, its distancefrom the SPLB is larger than dB. This result suggests thatfurther improvement is potentially achievable. An attractivesolution, in such a perspective, consists in using non-binaryLDPC codes. These codes have been analyzed in [5] and[16]. We refer to the implementation in [5], and in Fig. 4we report the performance of a non-binary LDPC code with -7 -6 -5 -4 -3 -2 -1 E b /N [dB] SP59 I max = 100 I max = 16 I max = 10 I max = 7 I max = 4
Fig. 3. Sphere packing lower bound against performance of the NASA (128,64) LDPC code. -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] SP59 Non-binary I max = 4 Non-binary I max = 100 Binary I max = 4 Binary I max = 100 Non-binary TUB
Fig. 4. Performance of the non-binary (128, 64) LDPC code in comparisonwith the binary code, the SPLB and the truncated union bound. n = 128 and k = 64 , constructed on the Galois FieldGF(256). Decoding is realized by using iterative algorithmsbased on fast Hadamard transforms. The TUB has been alsoplotted, as a further reference, in the figure, since the codewordmultiplicity of the non-binary LDPC code is known [17] (withreference to (1), d ∗ = 14 is enough for a good representation).However, we note that, being below the SPLB, it is notparticularly significant in the explored region. From Fig. 4,the improvement achievable by using the non-binary code isevident and the distance from the SPLB becomes very small(in the order of dB in the region of low CER). Hence,these results look excellent. In the next sections, however, wewill show that they can be approached and, in principle, evenoutperformed by using different solutions.IV. P ARALLEL TURBO CODES
Parallel turbo codes (PTCs) are one of the coding options ofthe CCSDS recommendation for TM links [6]. The CCSDSturbo encoder is based on the parallel concatenation of twoequal 16-state systematic convolutional encoders with polyno-mial description (1 , (1 + D + D + D ) / (1 + D + D )) .The interleavers are based on an algorithmic rule proposed by Berrou and described in [6]. The CCSDS turbo encoderhas four possible information frame lengths: , , and bits. The nominal code rate can be / , / , / ,and / , but higher rates are obtainable by puncturing.Maintaining unchanged the encoder structure, we haveconsidered frame lengths shorter than those in the TM rec-ommendation and fixed the nominal code rate to / , in sucha way as to comply with the NASA’s choices discussed inthe previous section. Because of the shorter length, we cannotuse the interleavers in [6] and we must design new smallerinterleavers. Among a large number of different options, wehave focused attention on: completely random, spread [18],Quadratic Permutation Polynomial (QPP) [19] and DitheredRelative Prime (DRP) [20] interleavers.Moreover, since the constituent CCSDS convolutional codeshave states, four extra-tail bits are needed for termination;then, the turbo codeword length is n = 2( k + 4) . For thespecific case of k = 64 , this implies to have n = 136 and anactual code rate . . In order to achieve the same code rateof the other schemes (that is necessary for fair comparison),we have implemented a suitable puncturing strategy. Moreprecisely, the algorithm [21] has been applied to identifythe codewords with smallest weight. Then, we have selectedthe positions that, with higher probability, do not correspondto bits equal to . Finally we have looked for puncturingpatterns insisting on these positions. The design criterionwas the maximization of the punctured turbo code minimumdistance and the minimization of its multiplicity. When nogood puncturing patterns were found by this method, we haveperformed a joint search for both interleaver permutation andpuncturing pattern looking for the best punctured turbo codes.In doing this we have adopted periodic puncturing patterns,according to the rules described in [22].As a result of this optimization process, the best interleaverwe have found, among the considered classes, is a DRPinterleaver. Using it, the (128, 64) PTC is characterized byminimum distance d min = 10 and weight- d min codewordmultiplicity A min = 5 . In Fig. 5 we compare the performanceof the (128, 64) PTC equipped with such an interleaver withthat of the (128, 64) binary LDPC code and the (128, 64)non-binary LDPC code discussed in the previous section. TheSP59 is also plotted for the sake of reference. From the figurewe see that the performance of the turbo code is very closeto that of the non-binary LDPC codes if the requested errorrates are not too low. As an example, at CER ≈ − theloss is about . dB, and becomes about . dB at CER ≈ − . The loss becomes greater for lower and lower CER,because of the higher error floor, due to the smaller minimumdistance. For the sake of completeness, it must be said thatwe have developed a similar comparison for the longer codes(that is, with k = 128 and k = 256 ). The corresponding CERcurves, not reported here because of lack of space, show thatthe loss for these codes is smaller. Taking this into account,PTCs seem a valid alternative to non-binary LDPC codesfor TC applications, at not too low CER values. Besides theerror rate performance, the choice of the former or the lattersolution may depend on complexity issues, whose evaluationis in progress and will be presented in a next paper. -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] Non-binary LDPC Binary LDPC PTC SP59
Fig. 5. Performance of the (128, 64) PTC against binary and non-binaryLDPC codes; the SPLB is reported as a reference. V. E BCH
CODES
Within the family of BCH codes, we have considered theeBCH(128, 64) code. The TUB for this code can be easilydetermined, since its codeword multiplicity is completelyknown [23]. In particular, it is possible to verify that d ∗ = 50 is enough to have a good description of the complete unionbound and to obtain a good estimate of an optimal ML soft-decision decoding algorithm, in the region of low error rates.The TUB of the eBCH is shown in Fig. 6 and there comparedwith the TUB and the simulated performance of the non-binary(128, 64) LDPC code described in Section III, as well as withthe SP59. From the figure, we see that the gap between theeBCH TUB and the SPLB is very small (for example, ≈ . dB for CER = 10 − ). Moreover, the eBCH TUB achieves again of about . dB with respect to the simulated non-binaryLDPC code with the same length and rate. In explicit terms,this means that if one is able to apply ML decoding to theeBCH(128, 64) code, this can provide performance better thanthat of all the other solutions discussed so far. Contrary to theBCH(63, 56) considered in Section II, soft-decision decodingof the eBCH(128, 64) code based on its trellis representation isunfeasible. In fact, its trellis has a maximum complexity of states. As a consequence, sub-optimal soft-decision decodingalgorithms must be applied. Many sub-optimal algorithmshave been presented in the literature. Most of them are basedon ordered statistics decoding. As a further option, one cantake advantage of proper LDPC-like code representations[24]. Among the huge amount of variants available, we havefocused attention on some solutions that, in our opinion,are particularly promising, as they permit to conciliate thedesire for good performance with the need to maintain limitedcomplexity. More precisely, we have considered: • The Box and Match Algorithm (BMA) [25]. • The Most Reliable Basis (MRB) algorithm [26].Details of these methods can be found in the quoted referencesand are here omitted, for the sake of brevity. Numericalexamples are given in Fig. 7. The BMA curve has been taken -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 E b /N [dB] eBCH TUB, d*=50 Non-binary LDPC, simulated Non-binary LDPC, TUB d*=14 SP59 Fig. 6. TUB of the eBCH(128, 64) code, compared with the TUB and thesimulated performance of the non-binary (128, 64) LDPC code and the SPLB. -7 -6 -5 -4 -3 -2 -1 E b /N [dB] eBCH BMA(4,20) eBCH TUB SP59 Non-binary LDPC Binary LDPC MRB(3) Binary LDPC LLR-SPA eBCH hard eBCH MRB(4) eBCH MRB(3) eBCH MRB(2) Fig. 7. Performance of different sub-optimal algorithms for soft-decisiondecoding of the eBCH(128, 64), in comparison with that of LDPC codes,TUB and SP59. from [25], while all the others have been simulated. For thesake of comparison, the figure also reports: • The hard-decision decoding performance of the eBCHcode. • The TUB of the eBCH code for d ∗ = 50 . • The SP59 for a (128, 64) code. • The performance of the non-binary (128, 64) LDPC code,taken from Section III. • The performance of the NASA binary (128, 64) LDPCcode, taken from Section II, and the result of its sub-optimal decoding by using the MRB algorithm.It should be noted that this comparison extends a preliminaryanalysis, of the same type, previously presented in [8].A number of interesting conclusions can be drawn fromthe figure. First of all, we observe that the performance ofthe BMA(4, 20) algorithm (the meaning of the parameters isexplained in [25]) is very close to that of the optimal ML soft-decision decoder: its gap from the SP59 is smaller than . B. Thus, its performance is excellent. The performance of theMRB(4) (where 4 is the order of the algorithm, see [26] for de-tails) is practically coincident with that of the BMA(4, 20). Onthe other hand, tolerating a slight performance degradation, thecomplexity can be reduced by using the MRB(3): the penalty islimited and the algorithm provides better performance than thenon-binary LDPC code down to CER = 2 · − . In general,the complexity of the sub-optimal algorithms depends on anumber of design parameters [26] that need to be optimized.However, it is not difficult to find a set of parameters that allowefficient decoding of the eBCH(128, 64) code. Also relevantin the figure, we observe that the MRB(3) algorithm appliedto the binary LDPC code yields a significant improvementwith respect to the LLR-SPA, and the achieved performance ispractically coincident with that of the eBCH code. Taking intoaccount that the complexity of the MRB algorithm is almostindependent of the code structure, being only a function of thecode parameters, this result confirms the convenience of theeBCH solution over the LDPC one.VI. C ONCLUSION AND OPEN ISSUES
This paper shows that valid alternatives to the solutionsbased on binary and non-binary LDPC codes can be found forupdating the current TC recommendation. Parallel turbo codesand eBCH codes can provide similar or even better features.More precisely, the turbo code can show a penalty with respectto the non-binary LDPC code but, as a counterpart, it exploitsa scheme that is already included in the CCSDS Recommenda-tions and, most of all, it does not suffer some problems relatedto the possible adoption of the eBCH code. The latter exhibitsthe best error rate performance. However, extending the sub-optimal decoding algorithms used for the eBCH(128, 64) codeto longer codes, while maintaining acceptable complexity, maybe difficult. Additionally, the sub-optimal algorithms generallydefine “complete” decoders. As well known, this may bea penalty for the undetected frame error rate (UFER) that,in TC applications, is at least as important as the CER (orthe frame error rate, FER). This problem does not exist if aCyclic Redundancy Check (CRC) code is used for detectingframe integrity, which makes UFER negligible. If the CRCcode is not used, the UFER performance can be improvedby making the decoder slightly incomplete but this has,obviously, an impact on the CER (and the FER) performance.Another important issue concerns CLTU termination that inthe current standard (whose hard-decision decoder is certainlyincomplete) is realized by introducing (at the transmitter) andsearching for (at the receiver) an uncorrectable pattern. Sincesuch a strategy cannot be applied with complete decoders,different approaches shall be identified for delimiting theCLTU and exploiting the error correction capabilities of theeBCH code. A
CKNOWLEDGMENT
The authors wish to thank Marco Chiani and Enrico Paolinifor having made available the software to simulate non-binaryLDPC codes. R
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