Band Splitting Permutations for Spatially Coupled LDPC Codes Enhancing Burst Erasure Immunity

0.16 MB·English

by  Hiroki Mori

Band Splitting Permutations for Spatially Coupled
LDPC Codes Enhancing Burst Erasure Immunity
Hiroki Mori and Tadashi Wadayama
Department Computer Science and Engineering,
Nagoya Institute of Technology, Nagoya, Japan
Email: [email protected], [email protected]
5 Abstract—It is well known that spatially coupled (SC) codes a domino toppling. Since a burst erasure interferes the propa-
1 with erasure-BP decoding have powerful error correcting capa- gation of a wave of such reliable estimations, it causes severe
0 bility over memoryless erasure channels. However, the decoding degradation on decoding performance. In order to overcome
2 performance of SC-codes significantly degrades when they are thisdifficulties,theyproposedanewclassofmultidimensional
n usedoverbursterasurechannels.Inthispaper,weproposeband SC-codes that shows higher immunity against burst erasures.
splitting permutations (BSP) suitable for (l,r,L) SC-codes. The
a
BSP splitsa diagonal band in a base matrix into multiplebands It is known that the burst erasure correcting capability of
J
in order to enhance the span of the stopping sets in the base LDPCcodesdependsonacolumnorderofparitycheckmatri-
9 matrix. As theoretical performance guarantees, lower and upper ces of LDPC codes [12]. This is because the minimum length
1 boundsonthemaximal burstcorrectablelengthofthepermuted
of stopping sets determining the burst correcting capability
(l,r,L) SC-codes are presented. Those bounds indicate that the dependsonthecolumnorderofaparitycheckmatrix.Inorder
] maximal correctable burst ratio of the permuted SC-codes is
T to enhance the burst erasure correctability, several heuristic
givenbyλmax ≃1/k wherek=r/l.Thisimpliestheasymptotic
I algorithms to improve the column order have been prensend
. optimality of the permuted SC-codes in terms of burst erasure
s correction. by Wadayama [9], Paolini and Chiani [15], Hosoya et al. [8].
c Of course, the column order of a parity check matrix does
[
not affect the decoding performance over memoryless erasure
1 I. INTRODUCTION channels.
v
Low-Density Parity-Check (LDPC) codes that are linear In this paper, we will propose a class of column permu-
4
codes defined by extremelysparse parity check matrices were tations that is called band splitting permutations suitable for
9
3 developedbyGallagerin1963[1].ThecombinationofLDPC (l,r,L) SC-codes. A band splitting permutation is applied to
4 codes and belief propagation provides remarkable error cor- the base matrix of (l,r,L) SC-codes having a single diagonal
0 rectingperformancewithreasonabletimecomplexity.Inrecent band and it results in a column-permuted base matrix with
. days, it is easy to find practical applications of LDPC codes severaldiagonalbands.Byliftingupthepermutedbasematrix,
1
inwireless/wiredcommunicationsystemsandstoragesystems. wecanobtainaparitycheckmatrixofapermuted(l,r,L)SC-
0
5 Not only a practical importance but also recent theoretical codes. Itwill be provedthatan appropriatebandsplitting per-
1 advancementproducesrenewedinterestsinthisfield.Kudekar mutation producespermuted (l,r,L) SC-codes that have near
: et. al [2] proposed a new class of LDPC codes, that is called optimal minimum length of stopping sets. The permuted SC-
v
spatially coupled codes (SC-codes) and they provided theo- codes constructed in such a way have burst erasure correcting
i
X reticalargumentsonthresholdsaturationofSC-codes[2].The superior to those of conventional SC-codes. Upper and lower
originofSC-codesisLDPC-convolutionalcodesthatdateback bounds on the minimum length of stopping sets to be proved
r
a to the workdue to Felstrom andZigangirov[3]. Lentmaieret. in this paper can provide theoretical performance guarantees
al[6]showedanensembleofanLDPC-convolutionalcodecan for burst erasure correcting capability of permuted SC-codes.
have a higher threshold than that of a componentLDPC code
The outline of this paper is as follows. Section 2 provides
ensemble. From these works on SC-codes, it is unveiled that
notion and fundamental definitions required throughout this
well-designed SC-codes have capacity achieving performance
paper. Section 3 presents several theorems regarding stopping
over symmetric memoryless channels.
sets in a base matrix. The band splitting permutations will
A burst erasure means a consecutive erased symbols. In be defined and analyzed in Section 4. Results on computer
manypracticalsituations,wecanobserveoccurrencesofburst experiments will be shown in Section 5.
erasuresduetoslowfadinginmobilewirelesscommunication,
buffer overflow at a congested router in a packet based II. PRELIMINARIES
network, and media flaw in a magnetic recording system.
A. (l,r,L) SC-codes
A strong erasure correcting code should have high erasure
correctingcapabilitynotonlyformemorylessrandomerasures In this subsection, the definition (l,r,L) SC-codes pro-
but also for burst erasures. Ohashi et. al [7] pointed out that posed by Kudekar et al. [2] is reviewed. The (l,r,L) SC-
SC-codes are not immune to burst erasures compared with codes belong to the class of protograph LDPC codes and
conventional LDPC codes such as regular LDPC codes. In its parity check matrix can be obtained by lifting up the
a typical decoding process of SC-codes, reliabilities of bit base matrix B(l,r,L). The base matrix B(l,r,L) is a binary
estimation gradually improves from both side into inside as (L+l−1)×kLmatrix(k =r/l)anditsstructureisillustrated
The subscript in H represents the column indices of
{i1,...,iu}
H corresponding to the column vectors in the sub-matrix.
Definition 1 (Stopping sets [13]) Let H be a parity check
matrix and S = {i ,i ,...,i } ⊆ [1,n] be an index set. The
1 2 u
notation [a,b] denotes the set of consecutive integers from a
to b. If the sub-matrix H has no rows with weight one, the
S
index set S is said to be a stopping set.
Fig.1. Basematrix B(l,r,L)that defines (l,r,L)SC-codes (gray area is It is well known that stopping sets are closely related
filledwithsymbolone). to correctability of erasure patterns if we exploit erasure-BP.
Assume that a transmitted word is a codeword of the code
defined by H and that some symbol erasures happen over the
inFig.1.Theparameterslandr representsthecolumnweight
channel. Let E = {e ,e ,...,e } ⊆ [1,n] be the indices
andmaximalrowweightofB(l,r,L),respectively.Weassume 1 2 w
corresponding to the symbol erasures. This erasure pattern
that the ratio k = r/l is integer throughout the paper. The
cannotbecorrectedwitherasure-BPifthereexistsanon-empty
parameter L denotes the number of sections.
stopping set S satisfying S ⊆ E. This fact indicates that the
A parity check matrix of an (l,r,L) SC-code can be set of stopping set in H determine Wmax(H) [14]
obtained by lifting up the base matrix B(l,r,L). A lift-
up process is summarized as follows: For each element one Assume that H = (h ,h ,...,h ) ∈ Fm×n is given and
in B(l,r,L), we can replace it with any binary M × M anindexsetS ={i ,i ,.1..,2i }⊆[1n,n]isg2ivenaswell.The
1 2 u
permutationmatrix.ThezerosinB(l,r,L)shouldbereplaced
length of S, that is denoted by Len(S), is defined by
with a binary M ×M zero matrix. Let a parity check matrix
obtained by the above process be H. The binary linear code Len(S)=1+ max|a−b|. (3)
defined by H is called an (l,r,L) SC-code. The size of the a,b∈S
permutation matrices, M, is said to be the lift up factor. The Let us denote the set of non-empty stopping sets of H by
number of rows of H is M(L + l − 1) and the number Q(H). The span of H, Span(H), is defined by
of columns is MkL. The design rate of (l,r,L) SC-codes,
Span(H)= min Len(S). (4)
R(l,r,L), is thus given by
S⊂Q(H)
1 l−1 It is clear thata burst erasureof lengthshorter than Span(H)
R(l,r,L)=1− − . (1)
k kL cannot cover any non-empty stopping set in H. This means
that we have
B. Maximal correctable burst length
W (H)=Span(H)−1. (5)
max
YangandRyan[12]introducedameasureforbursterasure
correctingcapabilityofLDPCcodesthatiscalledthemaximal Note that the quantity Wmax(H) can be evaluated efficiently
correctable burst length. Let H be a parity check matrix byusingerasure-BP[12].Fromthedefinition,we cansee that
that defines an LDPC code. The maximal correctable burst Span(H)stronglydependsontheorderofthecolumnvectors
length of this code is denoted by W (H). The meaning of in H. It has been shown that an appropriate rearrangementof
max
W (H) is the following. A burst erasure is a sequence of column order can increase the span of LDPC codes [8], [9].
max
consecutive erasures occurred on an erasure channel. In this
paper, we assume that only single burst erasure occurs in a
2)Irreducible stopping sets: We provide the definition of
code block. If the length of a single burst erasure is less than
irreducible stopping sets that will be required in the next
or equal to W (H), it can be perfectly corrected by belief
max
section.
propagation(BP) decoding for erasure channels. On the other
hand,thereexistsasinglebursterasureoflengthWmax(H)+1 Definition 2 (Irreducible stopping sets) Let S ⊆ [1,n] be
that cannot be corrected with erasure-BP. Namely, Wmax(H) a non-empty stopping set of H. If removing any subset of
representsthemaximumguaranteedcorrectablelengthforany elements fromS yields an index set that is nota stoppingset,
singlebursterasure.Asarelatedmeasureforbursterasurecor- then S is said to be irreducible stopping set.
recting capability, we here introduce the maximal correctable
burst ratio defined by λ = W (H)/n, where n is the
max max
code length. This quantity is useful for studying asymptotic Fromtheabovedefinitionofirreduciblestoppingsets, itis
behavior of the burst correcting capability. straightforward to see that the inequality Len(S′) ≤ Len(S)
holds for a pair of nested stopping sets where S is a stopping
C. Stopping sets and maximal correctable burst length set and S′ ⊆S is an irreducible stopping set in S. From this
inequality, we have
1)Stooping sets: Let H =(h ,h ,...,h )∈Fm×n be a
1 2 n 2
paritycheckmatrix.Thevectorhi isthei-thcolumnvectorof Span(H)=minS′⊂Q′(H)Len(S′), (6)
H. A sub-matrix of H consists of a subset of column vectors
where Q′(H) is the set of irreducible stopping set of H. This
in H; namely a sub-matrix of H has the form:
means that we only need to focus on the set of irreducible
H =(h ,h ,...,h )∈Fm×u. (2) stopping sets when we discuss the span of H.
{i1,...,iu} i1 i2 iu 2
III. IRREDUCIBLE STOPING SETSIN BASE MATRIX β).Letusfocusonthefirstnonzeroelementofthefirstcolumn
of the sub-matrix B(l,r,L) . Due to the assumption
In this section, we will prepareseveraltheoremsregarding that b 6= b and the defin(jit1i,o..n.,juo)f B(l,r,L), it is evident
the maximal correctable burst length that are required for the α β
that the row corresponding to the first nonzero element has
argument in Section IV.
Hamming weight 1. This means that S cannot be a stopping
set in this case.
A. Maximal correctable burst length of base matrices
By using this sufficient condition, we can immediately
Sridharan et. al [16] studied the maximalcorrectableburst show thatanystoppingsetof B(l,r,L)containstwo different
length of protograph LDPC codes. They showed a tight re- indices which belong to the same block. In other words, any
lationship between Wmax(B) and Wmax(H) where H is a stopping set must contain (α,β) satisfying bα = bβ(α 6= β).
parity check matrix obtained by lifting up a base matrix B. If a stopping set without such a pair exists, it contradicts the
The next theorem states this relationship. sufficient condition shown above.
Theorem 1 (Maximal correctable burst length ([16])) It is clear that a pair of indices (α,β)(α,β ∈[1,kL],α6=
Assume that a base matrix B ∈ Fm2 ×n is given. Let H be a β)isanirreduciblestoppingsetifbothindicesαandβ belong
parity check matrix obtained by lifting up B. The following to the same block. The last job is to show that there are no
inequalities hold: irreduciblestoppingsetswithsizelargerthan2.SupposethatS
is anirreduciblestoppingsetwith size largerthan2.Fromthe
(W (B)−1)M <W (H)<(W (B)+1)M. (7)
max max max aboveargument,S mustcontainatleastapairoftwoelements
thatbelongto thesameblock.Sincesucha pairconstitutesan
Theorem 1 indicates that the maximal correctable burst
irreduciblestoppingset,itcontradictstheassumptionthatS is
length of a protograph LDPC code is nearly determined by
anirreduciblestoppingset.Thiscompletesthecharacterization
Wmax(B).Thismeansthatanappropriatecolumnpermutation of the set of irreducible stopping sets of B(l,r,L). (cid:3)
for a base matrix B might be able to improve the maximal
correctable burst length of a resulting photograph code. Of
C. Burst erasure correcting capability of (l,r,L) SC-codes
course, (l,r,L) SC-codes belong to the class of protograph
LDPC codes. It is reasonableto devise an appropriatecolumn An immediate application of Theorem 2 is to analyze the
permutationforB(l,r,L),whichwillbediscussedinthenext burst erasure correcting capability of (l,r,L) SC-codes. The
section. size of irreducible stopping set is two and the minimal length
ofthestoppingsetisthustwo;wehaveSpan(B(l,r,L))=2.
B. Irreducible stopping sets in B(l,r,L) This givesWmax(B(l,r,L))=1 and we can utilize Theorem
1 to obtain lower and upper bounds on maximal correctable
The maximal correctable burst length of the base matrix burst length of (l,r,L) SC-codes:
B(l,r,L)is determinedby the set of irreduciblestoppingsets
in B(l,r,L). In this subsection, we will show a structural 0<Wmax(H)<2M, (10)
property on the set of irreducible stopping sets in B(l,r,L).
whereH representsaparitycheckmatrixof(l,r,L)SC-codes.
Let us denote the base matrix of the (l,r,L) SC-codes as By dividing both sides in (10) by the code length kLM, we
have inequalities for the maximal correctable burst ratio:
B(l,r,L)=(b ,b ,...,b )∈Fm×kL.
1 2 kL 2 2
0<λ < . (11)
A block Ti(i∈[1,L]) that is a subset of indices is defined by max kL
Ti ={(i−1)k+1,(i−1)k+2,...,(i−1)k+k}. (8) Itisclearthatλmax convergestozerowhenLgoestoinfinity.
This inequality presents that the conventional (l,r,L) SC-
From the structure of B(l,r,L) (i.e., Fig. 1), it is easy to codes have poor burst erasure correcting capability in the
see that bα = bβ holds if and only if α,β ∈ Ti. The next asymptotic regime when L → ∞. This result justifies the
theoremcharacterizesthestructureofirreduciblestoppingsets
observation made by Ohashi et. al [7].
in B(l,r,L).
Theorem 2 (Irreducible stopping sets of base matrix) IV. BAND SPLITTING PERMUTATIONS
The set of irreducible stopping sets in the base matrix
Inthissection,wewillproposebandsplittingpermutations
B(l,r,L) is given by
(BSP) for the base matrix B(l,r,L). The BSP is designed to
Q′(B(l,r,L))={{α,β}|α,β ∈T ,i∈[1,L]}. (9) improve the span of B(l,r,L).
i
The theorem states that an irreducible stopping set consists of A. Definition
two column indices belonging to the same block.
WhenaBSPσ isappliedtoabasematrixB(l,r,L),we
k,L
(Proof) Suppose an ordered index set S = (j ,...,j ) ⊆ have permuted base matrix with multiple bands as shown in
1 u
[1,n] is given where j1 < j2 < ··· < ju. The sub- Fig. 2. The formaldefinition of BSP σk,L is givenas follows:
matrix corresponding to S is written as B(l,r,L) = AccordingtoCauchy’stwo-linenotationonapermutation,the
(j1,...,ju)
(bj1,bj2,...,bju). permutation σk,L is described as
We will first show a sufficient condition that S is not a 1 2 ... kL
σ = . (12)
stoppingset.Assumethatbα 6=bβ holdsforanyα,β ∈S(α6= k,L (cid:18) f(1) f(2) ... f(kL) (cid:19)
Let H be a parity check matrix obtained by lifting up
B∗(l,r,L)withtheliftupfactorM.Themaximalcorrectable
burst length W (H) of the permuted SC-code satisfies the
max
following inequalities:
(L−1)M <W (H)<(L+1)M. (19)
max
(Proof) Assume that S = {i ,i ,...,i } ⊆ [1,n] is a
1 2 u
stopping set of B(l,r,L). The BSP maps S to
Fig.2. Thestructureofconventionalbasematrixandpermutedbasematrix. S∗ ={f−1(i1),f−1(i2),...,f−1(iu)}.
Note that S∗ is also a stopping set of B∗(l,r,L) because
Thesecondrowoftwolinenotation,i.e.,thebijectivefunction B∗(l,r,L){f−1(i1),f−1(i2),...,f−1(iu)} contains a row of weight
f on [1,kL], is defined by 1 as well. This means that there is one-to-onecorrespondence
between stopping sets in B(l,r,L) and those in B∗(l,r,L).
(f(1) f(2) ··· f(kL))=(a1 a2 ··· ak), (13) Theorem2 indicatesthata non-emptyirreduciblestoppingset
where a (i∈[1,k]) is given by consists of two indices in the same block. Assume that a pair
i α,β ∈ [1,kL] is such a pair of indices. From a definition
a = ( 1 1+k 1+2k ··· 1+(L−1)k ) of the matrix A in (18), it is clear that α and β belong
1
. to the same column in A. The definition of f in (13) thus
.
. leads to the inequality |f−1(α)−f−1(β)| ≥ L that implies
a = ( k k+k k+2k ··· k+(L−1)k ). (14) the length of irreducible stopping sets in B∗(l,r,L) is larger
k
than or equal to L+1. Note that the equality holds when α
The permutation σk,L can be seen as a block interleaver and β are consecutive. From the definition of the span (6),
of interleaving depth k. Applying σk,L to the base matrix we thus have Span(B∗(l,r,L)) = L + 1 and this implies
vBe(rls,ior,nLo)f a=ba(sbe1,m.a.t.r,ixbkBL)∗,(lw,re,Lo)bt=ain(bfa(1c)o,.lu.m.,nbfp(keLrm))u.ted tWhimsathxe(Bor∗e(ml,ris,Lob))ta=ineLd.. By using Theorem 1, the claim o(cid:3)f
For example, when k =2,L=3, we have
The inequalities of Theorem 3 indicates that the maximal
a1 =( 1 3 5 ), a2 =( 2 4 6 ) (15) correctable burst length of the permuted (l,r,L) SC-codes is
1 2 3 4 5 6 proportionalto the number of sections L. The inequality (10)
σ2,3 =(cid:18) 1 3 5 2 4 6 (cid:19). (16) indicates that the maximal correctable burst length does not
depend on L for the case of the conventional (l,r,L) SC-
Applying σ2,3 to B(3,6,3), the permuted base matrix is codes.Thisresultclearlyshowstheadvantageofthepermuted
obtained as SC-codesoverthe conventional(i.e.,non-permuted)SC-codes
1 0 0 1 0 0 with respect to the maximal correctable burst length.
1 1 0 1 1 0
 
B∗(3,6,3)= 1 1 1 1 1 1 . (17) C. Maximal correctable burst ratio
 0 1 1 0 1 1 
  In this subsection, we focus on the maximal correctable
 0 0 1 0 0 1 
  burst ratio λmax of the permuted (l,r,L) SC-codes.
Let us define k×L matrix A by Bydividingbothsidesin(19)bythecodelengthkLM,we
can obtain following inequalities for the maximal correctable
a
1 burst ratio:
A= .. . (18)
. L−1 L+1
 ak  kL <λmax < kL . (20)
 
It is easy to see that i-th column of A corresponds the block From (20), it is clear that λmax converges to 1/k when L→
T which is defined by (8). This implies that column vectors ∞.Ontheotherhand,thedesignrateR(l,r,L)ofthe(l,r,L)
i
in B(l,r,L) belonging to the same block are rearranged in SC-codesconvergesto1−1/kasLgoestoinfinity.Fromthese
B∗(l,r,L) as apart as possible. This property enhances the results, we have
span of the base matrix.
lim (λ +R(l,r,L))=1 (21)
max
L→∞
B. Bounds on maximal correctable burst length
that indicates asymptotic optimality of permuted (l,r,L) SC-
By liftingupthepermutedbase matrixB∗(l,r,L),we can codes in terms of burst erasure correction with erasure-BP.
obtain a parity check matrix of permuted (l,r,L) SC-codes. Note that no binary linear code of length n with design rate r
The next theorem provides upper and lower bounds on the can correct burst erasures of length larger than n(1−r).
maximal correctable burst length of permuted (l,r,L) SC-
codes. This is the main contribution of this work. V. NUMERICAL RESULTS
Theorem 3 (Bounds on maximal correctable burst length) We have seen that the maximal correctable burst ratio of
Let B∗(l,r,L) be the permuted base matrix defined above. permuted SC-codes can be approximated by λ ≃ 1/k
max
0.8 700
Band Splitting Permuted (3,6,32) SC-codes
0.7 600 Randomly Permuted (3,6,32) SC-codes
o
ati 0.6
R 500
st
Bur 0.5 ncy 400
ble 0.4 que
a e 300
ect 0.3 Fr
orr 200
al C 0.2 ConvPeenrtmiountaeld ( 3(3,6,6,L,L) )SSCC--ccooddeess ((ulopwpeerr bboouunndd))
xim 0.1 BP threshold 100
a 1 - R(3,6,L)
M
0 0
0 40 80 120 160 200 240 280 0 0.1 0.2 0.3 0.4 0.5 0.6
Number of sections Maximal Correctable Burst Ratio
Fig.3. Relation betweenLandλmax of(l,r,L)=(3,6,L)SC-codes Fig. 4. Histogram ofλmax of(l,r,L)=(3,6,32) SC-codes (M =40,
1000samples)
when L is large enough. We will here show the relationship
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max
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max
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ACKNOWLEDGEMENT
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This work was supportedby JSPS Grant-in-Aidfor Scien-
vol.57,no.5,pp.1320-1328, May.2009
tific Research (B) Grant Number 25289114.
[16] G.Sridharan,A.Kumarasubramanian, A.Thangaraj,andS.Bhashyam,
“Optimizing burst erasure correction ofLDPCcodes by interleaving,”
inProc.IEEEInt.Symp.Inf.Theory,pp.1143-1147, Jul.2008
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