Time-Invariant Spatially Coupled Low-Density Parity-Check Codes with Small Constraint Length
Marco Baldi, Massimo Battaglioni, Franco Chiaraluce, Giovanni Cancellieri
aa r X i v : . [ c s . I T ] M a y Time-Invariant Spatially Coupled Low-DensityParity-Check Codes with Small Constraint Length
Marco Baldi, Massimo Battaglioni, Franco Chiaraluce and Giovanni Cancellieri
Dipartimento di Ingegneria dell’InformazioneUniversità Politecnica delle MarcheAncona, ItalyEmail: {m.baldi, f.chiaraluce, g.cancellieri}@univpm.it, [email protected]
Abstract —We consider a special family of spatially coupledlow-density parity-check (SC-LDPC) codes, that is, time-invariantlow-density parity-check convolutional (LDPCC) codes, which areknown in the literature for a long time. Codes of this kind areusually designed by starting from quasi-cyclic (QC) block codes,and applying suitable unwrapping procedures. We show that,by directly designing the LDPCC code syndrome former matrixwithout the constraints of the underlying QC block code, it ispossible to achieve smaller constraint lengths with respect to thebest solutions available in the literature. We also find theoreticallower bounds on the syndrome former constraint length for codeswith a specified minimum length of the local cycles in theirTanner graphs. For this purpose, we exploit a new approachbased on a numerical representation of the syndrome formermatrix, which generalizes over a technique we already used tostudy a special subclass of the codes here considered.
Index Terms —Constraint length, convolutional codes, LDPCcodes, local cycles length, spatially coupled codes, time-invariantcodes.
I. I
NTRODUCTION
Spatially coupled low-density parity-check (SC-LDPC)codes represent a cutting-edge innovation in the context ofmodern channel coding in general and of low-density parity-check (LDPC) coding in particular. In fact, classical LDPCblock codes [1] are known to approach the channel capacityunder belief propagation decoding [2]. SC-LDPC codes rep-resent a further step in this direction, since they are able tofurther reduce the gap to capacity [3] thanks to the thresholdsaturation phenomenon.A special class of SC-LDPC codes is that of low-densityparity-check convolutional (LDPCC) codes, which have beenshown to outperform their block counterparts [4]. These codesare usually designed by starting from quasi-cyclic low-densityparity-check (QC-LDPC) codes [5] and using a techniqueknown as unwrapping to produce a semi-infinite description ofthe convolutional code [4], [6], [7]. This approach has allowedto design LDPCC codes with very good performance [8], [9].However, despite some attempts to achieve small constraintlengths have been done [10], starting from QC-LDPC codesand then unwrapping them usually results in LDPCC codeswith large constraint lengths.In fact, shift register-based circuits like that in [6, Fig.4] can be used to perform encoding of an LDPCC code,while decoding can be performed through iterative messagepassing algorithms working on a window sliding over the received sequence [6]. Complexity of these encoding anddecoding techniques increases linearly with the syndromeformer constraint length of the code. Therefore, designingcodes with small constraint length is a valuable target fromthe complexity standpoint.In this paper we study the design of time-invariant LDPCCcodes without starting from QC-LDPC block codes. This isdone by directly designing the syndrome former matrix whichthen forms the semi-infinite parity-check matrix of the LDPCCcode. We follow an approach similar to that proposed in [11],where we introduced a special class of LDPCC codes namedprogressive differences convolutional low-density parity-check(PDC-LDPC) codes. The codes considered in [11] have rate a − a , with a being an integer > , and local cycles with length ≥ in their associated Tanner graph. Another solution todesign codes with the same parameters has been proposed in[12]. Here we generalize the approach proposed in [11] to thedesign of codes with rate a − ca , with a and c being two positiveintegers such that a > c , and minimum length g of the localcycles in their Tanner graphs. The numerical representation weadopt for the syndrome former matrix significantly facilitatessearching for short cycles in the code Tanner graph. Similarefficient searches have recently been performed for QC-LDPCblock codes [13]. This approach permits us to perform theo-retical and exhaustive analyses for g = 6 and g = 8 , as wellas Montecarlo assessments for larger values of g .The organization of the paper is as follows. In Section IIwe remind the definition of time-invariant SC-LDPC codesand their relevant parameters. In Section III we introduce anumerical description of the syndrome former matrix whichfacilitates the search of local cycles. In Section IV we providetheoretical bounds on the minimum constraint length whichis needed to avoid local cycles up to a given length. InSection V we provide a comparative assessment of the boundswith exhaustive searches as well as some results based onMontecarlo simulations. Section VI concludes the paper.II. T IME - INVARIANT SPATIALLY COUPLED LOW - DENSITYPARITY - CHECK CODES
The codes we consider are defined by semi-infinite parity-check matrices in the form (1), where each block H i , i =0 , , , . . . , m h , is a binary matrix with size c × a . The syn-drome former matrix H s = (cid:2) H T | H T | H T | . . . | H Tm h (cid:3) , where T enotes transposition, has a rows and L h columns. As evidentin (1), H is obtained by H Ts and its replicas, shifted one eachother by c positions. The time invariant LDPCC code definedby (1) has asymptotic code rate R = a − ca , syndrome formermemory order m h = (cid:6) L h c (cid:7) − and syndrome former constraintlength v s = ( m h + 1) a = (cid:6) L h c (cid:7) a . H = H . . . H H . . . H H H . . .... H H . . . H m h ... H . . . m h ... . . . m h . . . . . .... ... ... . . . . (1)An alternative representation of H s which is often used inthe literature exploits polynomials ∈ F [ x ] . In this case, thecode is described by a c × a matrix with polynomial entries,that is H ( x ) = h , ( x ) h , ( x ) . . . h ,a − ( x ) h , ( x ) h , ( x ) . . . h ,a − ( x ) ... ... . . . ... h c − , ( x ) h c − , ( x ) . . . h c − ,a − ( x ) , (2)where each h i,j ( x ) , i = 0 , , , . . . , c − , j = 0 , , , . . . , a − ,is a polynomial ∈ F [ x ] or a null term. The code representationbased on H s can be converted into that based on H ( x ) throughthe following simple procedure. First of all, starting from H s ,the multiset I containing the sets of indexes (beginning fromzero) of the symbols in each row of H s must be computed.Then, the j -th column of H ( x ) is obtained from the set I j ∈ I , j = 0 , , , . . . , a − , as follows:1) Initialize h i,j ( x ) = 0 , i = 0 , , , . . . , c − .2) ∀ l ∈ I j , compute l d = ⌊ l/c ⌋ , l m = l mod c and add x l d to h l m ,j ( x ) .This procedure is an inverse of the unwrapping techniquesproposed in [4], [6]. In fact, most previous works are devotedto the design of H ( x ) and then H s is obtained throughunwrapping. However, designing H ( x ) requires to first choosethe form of the polynomials h i,j ( x ) (null, monomials, bi-nomials, etc.) and then optimize their exponents. Such anapproach has also been followed in [14], where some low rateLDPCC codes with small constraint length have been found.The matrix H ( x ) is also used in [15] to find unavoidablecycles and design LDPCC codes free of short local loops.In this paper we aim at finding the codes with minimumconstraint length over all possible configurations. For thispurpose, working with H s is advantageous in that it allows toperform a single step optimization over all possible choices. Therefore, we focus on H s and we need the transformationfrom H s to H ( x ) described above to perform comparisonswith the design examples reported in previous works. As wewill see in Section V, our approach allows to find codes withshorter constraint length than those in [15].III. L OCAL CYCLES
Local cycles are closed loops starting from a node of theTanner graph associated to an LDPC code and returning to thesame node by passing only once through any edge. Since theTanner graph is derived from the code parity-check matrix,local cycles can be defined over such a matrix as well. Thisway, we are able to directly relate the constraint length of anSC-LDPC code to its local cycles length.Following an approach similar to that introduced in [11],we describe the matrix H s through a set of integer valuesrepresenting the differences between each pair of ones in eachrow of H s . These differences are denoted as δ i,j , where i isthe row of H s ( i = 0 , , , . . . , a − and j is the column of H s corresponding to the first of the two symbols forming thedifference ( j = 0 , , , . . . , L h − . The index of the secondsymbol forming the difference is easily found as j + δ i,j .For each difference we also compute the values of two levels which are relative to the value of the parameter c . The startinglevel is defined as l s = j mod c , while the ending level isdefined as l e = ( j + δ i,j ) mod c . Both levels obviously takevalues in { , , . . . , c − } .Based on this representation of the syndrome former ma-trix, it is easy to identify closed loops in the Tanner graphassociated to H . In fact, a local cycle occurs every time asum of the type δ i ,j ± δ i ,j ± . . . ± δ i l ,j l equals zero,and the length of the cycle is l , with l being an integer > . An example is reported in Fig. 1, where a cycle withlength corresponds to the relation δ , + δ , − δ , = 0 .Not all the possible sums or differences of δ i,j are valid togenerate local cycles. In fact, δ x,y can be added to δ i,j iff thestarting level of the former coincides with the ending levelof the latter. Instead, δ x,y can be subtracted to δ i,j iff theirending levels coincide. In addition, the first and the last levelsof the sum δ i ,j ± δ i ,j ± . . . ± δ i l ,j l must coincide. Letus denote as δ i,j ( l s )( l e ) the difference δ i,j with its associatedstarting and ending levels. For the example reported in Fig.1, we have δ , + δ , − δ , = 0 , which thereforecomplies with the above rules. Owing to the special structureof H , some further rules hold concerning the existence ofclosed loops. In fact, it must be taken into account that theshift of the replicas of H Ts within H is neither cyclic norquasi-cyclic. Therefore, a closed loop due to the differencesin a single row of H s can or cannot exist depending on thepositions of the symbols in that row. For example, a cyclewith length due to a single row of H s with weight w ≥ exists iff at least symbols are at the same level. Instead,a circulant matrix with row weight ≥ always yields length cycles. Moreover, in a sum of differences, the same δ i,j cannot appear with both signs in two adjacent terms. Basedon these considerations, for a given matrix H s a very efficient L h l = 0 l = 1 l = 2 l = 0 l = 1 l = 2 l = 0 l = 1 l = 2 a (cid:1) (cid:1) (cid:1) Fig. 1. Example of H with a local cycle of length . numerical procedure can be exploited to find all the localcycles with a given maximum length. Such a procedure hasbeen implemented in software, and has allowed to performexhaustive (when possible) or Montecarlo (otherwise) analysesof the syndrome former matrices with minimum constraintlength and free of local cycles up to a given size. Moreover,by studying the cases in which differences may or may not besummed or subtracted, it is possible to obtain lower bounds onthe minimum constraint length which is needed to avoid localcycles up to a given length, as described in the next section.IV. M INIMUM CONSTRAINT LENGTH
Let us consider some practical values of the minimum localcycles length g and aim at estimating the minimum syndromeformer constraint length which is needed to ensure that shortercycles do not exist. In the following we provide theoreticallower bounds of this type for g = 6 and g = 8 . A. Absence of cycles with length < g = 6
In order to meet the condition g = 6 , we must ensure thatlocal cycles with length do not exist. Such short cycles occurwhen, for some i, j, i ′ , j ′ , j = j ′ , δ i,j = δ i ′ ,j ′ and l s = l ′ s , (3) i.e. , in order to avoid length cycles there must not be anytwo equal differences starting from the same level. We observethat the two differences may even be in the same row of H s .Let us first consider a regular H s with row weight w = 2 .In this case, each row of H s only contains one difference δ i,j and each difference can be used up to c times withoutincurring length cycles (by using all the possible c levels asstarting levels). For a given L h , the differences starting from the first one of the c available levels can take up to L h − values. Similarly, the differences starting from the second levelcan take up to L h − values, and so on, until up to L h − c values for the differences starting from the last level. Sincethe differences corresponding to any two of the a rows of H s must be different in value and/or starting level, we have a ≤ c − X i =0 ( L h − i −
1) = cL h − (cid:18) c + 12 (cid:19) , (4)that is L h ≥ & a + (cid:0) c +12 (cid:1) c ' . Considering that it must be L h > c , we have L h ≥ max ( c + 1 , & a + (cid:0) c +12 (cid:1) c ') . (5)We can extend (4) to the case of a regular H s with rowweight w > by considering that, in such a case, each row of H s corresponds to (cid:0) w (cid:1) differences that must meet condition(3). Hence (4) becomes a (cid:18) w (cid:19) ≤ cL h − (cid:18) c + 12 (cid:19) , while (5) becomes L h ≥ max ( c + 1 , & a (cid:0) w (cid:1) + (cid:0) c +12 (cid:1) c ') . (6)When we have an irregular H s with row weights w i , i = 0 , , , . . . , a − , each row of H s corresponds to (cid:0) w i (cid:1) differences. Therefore (6) becomes L h ≥ max ( c + 1 , & P a − i =0 (cid:0) w i (cid:1) + (cid:0) c +12 (cid:1) c ') . (7) B. Absence of cycles with length < g = 8
The minimum length of local cycles is g = 8 whencondition (3) is met and length cycles of the type shownin Fig. 1 and described in Section III are avoided.Let us first consider the case with c = 1 and H s with rowweight w = 2 . Since summing two odd integers we alwaysget an even number, the following proposition easily follows. Proposition IV.1
For c = 1 and w = 2 , if all the δ i,j aredifferent and odd, then local cycles with length < g = 8 donot exist.From Proposition IV.1 it follows that, if we wish to min-imize L h , we can choose the values of δ i,j equal to { , , , . . . , a − } and the code will be free of cycles withlength < g = 8 .Another possible choice yielding absence of cycles withlength < g = 8 follows from the fact that, for a given oddinteger x , summing two values ∈ (cid:2) x +12 ; x (cid:3) always gives aresult > x . Therefore, the following proposition holds. roposition IV.2 For c = 1 and w = 2 , if the δ i,j values areequal to { a, a + 1 , a + 2 , . . . , a − } , then local cycles withlength < g = 8 do not exist.Based on these propositions, we can prove the followinglemma. Lemma IV.1
For c = 1 and w = 2 , local cycles with length < g = 8 can be avoided iff L h ≥ a. (8) Proof:
From Propositions IV.1 and IV.2 we have that themaximum value of a difference that is needed to avoid cycleswith length < g = 8 is a − . Therefore we have L h ≥ a − a . In order to prove the converse, let us considerthat, for a given even integer y , summing two values ∈ (cid:2) y (cid:3) always gives a result ∈ (cid:2) y + 1; y (cid:3) . In general, from the set [1; y ] we can select at most y values which may be summedpairwise resulting in other values in the same set. If we choosethe values of the differences from the set [1; 2 a − , we onlyhave a − values which may be summed pairwise resulting inother values in the same set. Therefore, we can only allocate a − differences without introducing length cycles, whichis not sufficient to cover all the a rows of H s .Equation (8) can be extended to the case c > byconsidering that, in such a case, each difference value canbe repeated up to c times (by exploiting all the c availablelevels as starting levels). Therefore, for w = 2 and c > wehave L h ≥ max (cid:26) c + 1 , ac (cid:27) . (9)Let us consider larger values of w , i.e. , w ≥ . For c = 1 ,each row of H s has one or more cycles with length , sinceat least symbols are at the same level (as described inSection III). Instead, for w ≥ and c > we can follow thesame approach used for the case with g = 6 , thus obtaining L h ≥ max ( c + 1 , a (cid:0) w (cid:1) c ) . (10)When the rows of H s are irregular with weights w i , i =0 , , , . . . , a − , as done for the case with g = 6 , we canconsider that each row of H s corresponds to (cid:0) w i (cid:1) differencesand hence (10) becomes L h ≥ max ( c + 1 , & P a − i =0 (cid:0) w i (cid:1) c ') . (11)V. E XAMPLES
In Figs. 2-4 we report the bounds on L h obtained asdescribed in Section IV as a function of a , for some valuesof w , g and c . We also compare these bounds with the resultsobtained through exhaustive searches over all the possiblechoices of H s , performed through efficient numerical tools.From Fig. 2 we observe that, for the cases with w = 2 and g = 6 , the matching between the theoretical bound and a L h Sim, c=1Bound, c=1Sim, c=2Bound, c=2Sim, c=3Bound, c=3Sim, c=4Bound, c=4
Fig. 2. Bounds on L h and values found through exhaustive searches as afunction of a , for w = 2 , g = 6 and some values of c . a L h Sim, c=1Bound, c=1Sim, c=2Bound, c=2Sim, c=3Bound, c=3Sim, c=4Bound, c=4
Fig. 3. Bounds on L h and values found through exhaustive searches as afunction of a , for w = 3 , g = 6 and some values of c . the values found through exhaustive searches is perfect forall the considered values of c . Indeed, in this situation, all thepractical cases are modeled by the bound, therefore it is alwayspossible to find a solution achieving the bound. Instead, whenwe have larger row weights of H s , the theoretical bound maynot be achievable in practical terms. This results from Fig. 3for w = 3 . However, we also observe that the deviations ofthe experimental values from the theoretical curves are rathersmall. The results of exhaustive searches are well matchedwith the theoretical bounds also for the case with w = 2 and g = 8 , as we observe from Fig. 4. In this case, we note thatthe gap to the bound increases for increasing values of c .The same efficient tools used to perform exhaustive searchescan also be exploited to perform Montecarlo experimentsaimed at finding codes with small constraint length andabsence of cycles with length up to some value g . This way, a L h Sim, c=1Bound, c=1Sim, c=2Bound, c=2Sim, c=3Bound, c=3Sim, c=4Bound, c=4
Fig. 4. Bounds on L h and values found through exhaustive searches as afunction of a , for w = 2 , g = 8 and some values of c . it has been possible to find improved results with respect toprevious solutions from the constraint length standpoint. Forexample, in [8] a code with a = 6 , c = 3 , w = 3 and g = 10 is provided with H ( x ) = x x x x x x x x x x , having m h = 85 and L h = 258 . Through a Montecarlo searchperformed with the tools described above, we have found acode with the same parameters and girth, having H ( x ) = x x x x x x x x x x x x , i.e. , m h = 38 and L h = 117 , thus resulting in a considerablereduction over the former. Similarly, in [15] a code with a = 5 , c = 3 , w = 3 and g = 12 is provided with H ( x ) = x x x x x x x x x x x x , having m h = 185 and L h = 558 , while we were able to finda code with the same parameters and girth, having H ( x ) = x x x x x x x x x x . This code has m h = 52 and L h = 159 , which also is aconsiderable improvement. Another example in [15] with thesame choice of the parameters achieves m h = 134 , which stillis considerably larger than the value we have found.Concerning performance of these codes, there is a trade-offwith their constraint length. However, codes with moderatelysmall constraint lengths may still achieve better performancethan their block counterparts. For example, we have verified through Montecarlo simulations of BPSK modulated transmis-sion over the AWGN channel that one of our LDPCC codeswith w = 3 , a = 9 , c = 3 , g = 8 and v s = 1143 exhibitsa gain of about . dB at BER = 10 − with respect to theWiMax standard LDPC block code with the same rate ( / )and length . VI. C ONCLUSION
We have studied the design of time-invariant SC-LDPCcodes with small constraint length and free of local cycles upto a given length. By directly designing the syndrome formermatrix, we have obtained codes with smaller constraint lengthwith respect to those designed by unwrapping QC-LDPC blockcodes. We have also provided theoretical lower bounds on theminimum constraint length which is needed to achieve codeswith a fixed minimum length of the local cycles, and shownthrough exhaustive searches that practical codes achieving or,at least, approaching these bounds can be found.R
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