Table Of ContentMulti-Edge type Unequal Error Protection LDPC
Codes
H. V. Beltra˜o Neto, W. Henkel V. C. da Rocha Jr.
Jacobs University Bremen Department of Electronics and Systems
Campus Ring 1 Federal University of Pernambuco
D-28759 Bremen, Germany P.O. Box 7800 Recife, Brazil, 50711-970
Email: h.beltrao, w.henkel @jacobs-university.de Email: [email protected]
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Abstract—Irregularlow-densityparitycheck(LDPC)codesare a fact also observed in [4]. In [5], the authors observed that
0
2 particularly well-suited for transmission schemes that require this vanishingUEP gradationof an iteratively decodedLDPC
unequalerrorprotection(UEP)ofthetransmitteddataduetothe
codeisdependentonthealgorithmusedtoconstructtheparity
n different connection degrees of its variable nodes. However, this
checkmatrix,andsuggestedthattheconnectivitybetweenthe
a UEP capability is strongly dependent on the connection profile
J among the protection classes. This paper applies a multi-edge classesisthekeyfactortobeobservediftheUEPcapabilities
1 type analysis of LDPC codes for optimizing such connection should be held as the number of iterations grows.
3 profile according to the performance requirements of each Herein, we propose an optimization algorithm for the con-
protectionclass.ThisallowstheconstructionofUEP-LDPCcodes
nectivity profile between the different protection classes of
] where the difference between the performance of the protection
T LDPC codes in order to not only keep the UEP capability of
classes can be adjusted and with an UEP capability that does
I not vanish as the number of decoding iterations grows. a code for a moderate to large number of decodingiterations,
.
s but also to adjust the performance of the protection classes
c I. INTRODUCTION as required for different applications. This is achieved by
[
In communicationsystems where source bits with different means of a multi-edge type (MET) analysis [6], [7] of the
1 sensitivitiestoerrorsarebeingtransmitted,itisoftenwasteful LDPCcodes.Themulti-edgeanalysisenablesustodistinguish
v oreveninfeasibleto provideuniformprotectionforthewhole betweenthemessagesexchangedduringtheiterativedecoding
5
data stream. In this scenario, the common strategy is the use among the different protection classes within one codeword.
8
of schemes with unequal error protection (UEP) capabilities. Thus,we can controlthe amountof informationthatthe most
9
5 There are mainly three strategies to achieve UEP on trans- protectedclassesreceivefromthelessprotectedonesandvice
. mission systems: bit loading, multilevel coded modulation, versa.Ifthemostprotectedclassesreceivealotofinformation
1
0 and channel coding [1]. In this paper, we will focus on the fromthelessprotectedones,itsperformancewillbedecreased
1 latter, more specifically on low-density parity-check (LDPC) while the one of the less protected classes will be enhanced.
1 codes that provide inherent unequal error protection within a Our main goal is to show how this exchange of performance
:
v codeword. amongtheprotectionclassescanbecontrolledandoptimized.
i Irregular LDPC codes [2] can inherently provide unequal This paper is organized as follows. In Section II, we
X
error protection due to the different connection degrees of describe the multi-edge type analysis of UEP-LDPC codes.
r
a the coded bits. The connection degrees of the variable and Section III discusses the asymptotic analysis of multi-edge
check nodes of such codes are defined by the polynomials type UEP-LDPC codes and the optimization algorithm used
λ(x) = dvmax λ xi−1 and ρ(x) = dcmax ρ xi−1, where to optimize the connection profile between the protection
i=2 i i=2 i
d and d are the maximum variable and check node classes. In Section IV, we show the results of the developed
vmax P cmax P
degrees of the code. From now on, we will refer to irregular optimization method for a chosen example. Finally, some
LDPCcodeswherethevariablenodesaredividedintodisjoint concluding remarks are drawn in Section V.
setscalledprotectionclassesasunequalerrorprotectionLDPC
II. MULTI-EDGETYPE UNEQUAL ERROR PROTECTION
codes (UEP-LDPC).
LDPC CODES
Aflexibleoptimizationoftheirregularityprofileofirregular
LDPC codes based on a hierarchical optimization of the A. Multi-edge LDPC codes
variable node degree distribution was proposed in [3], where Multi-edge type LDPC codes [6] are a generalization of
the authors interpret the UEP properties of an LDPC code irregular and regular LDPC codes. Diverting from standard
as different local convergencespeeds, i.e., the most protected LDPC ensembles where the graph connectivity is constrained
bits are assigned to the bits in the codeword which converge only by the node degrees, in the multi-edge setting, several
to their right value in the smallest number of iterations. This edge classes can be defined and every node is characterized
assumptionis made in orderto cope with the observationthat by the number of connections to edges of each class. Within
the UEP gradation vanishes as the number of iterations grow, this framework, the code ensemble can be specified through
two multinomialsassociated to variableandchecknodes.The
two multinomials are defined by [7]
L(r,x)= L rbxd, (1)
b,d
X
R(x)= R xd, (2)
d
whereb,d,r,andxarevectorsXwhichareexplainedasfollows.
First,letm denotethenumberofedgetypesusedtorepresent
e
the graph ensemble and m the number of different received Fig. 1. Multi-edge graph with two different edge types and one received
r
distribution.
distributions. The number m represents the fact that the
r
differentbitscangothroughdifferentchannelsandthus,have
different received distributions. Each node in the ensemble
protectionclass are all of the same type. For example,in Fig.
graph has associated to it a vector x = (x ,...,x ) that
1 me 1thefirst4variablenodescanbeseenasoneprotectionclass
indicates the different types of edges connected to it, and a
since they are connected only to type 1 edges (depicted by
vector d = (d ,...,d ) referred to as edge degree vector
1 me solid lines), and the last 3 variable nodes compound another
which denotes the number of connections of a node to edges
protectionclass,sincetheyareonlyconnectedtotype2edges
of type i, where i (1,...,m ).
∈ e (depictedbythedashedlines).Itisworthnotingthatopposed
For the variable nodes, there is additionally the vector
tothe variablenodes,the checknodesadmitconnectionswith
r = (r ,...,r ) which represents the different received dis-
1 mr edges of different types simultaneously as can be inferred
tributions1, and the vector b=(b ,...,b ) that indicates the
0 mr fromFig.1.Inthefollowing,wewilldividethevariablenodes
number of connections to the different received distributions
into m protection classes (C ,C ,...,C ) with degrading
(b isusedtoindicatethepuncturingofavariablenode).Inthe e 1 2 me
0 levels of protection.
sequel,weassumethatbhasexactlyoneentrysetto1andthe
rest set to zero. This simply indicates that each variable node
has access to only one channel observation at a time. We use B. Edge perspective notation
xd to denote me xdi andrb to denote mr rbi. Finally, the
i=1 i i=0 i The connection between the protection classes occurs
coefficients L and R are non-negative reals that represent
Qb,d d Q through the check nodes since they can have different types
the fraction of variable nodes of type (b,d) and check nodes
of edges attached to them. Consider irregular LDPC codes
of type (d) within a codeword, respectively. Furthermore, we
with node perspective variable and check node multi-edge
have the additional notations defined in [7]
multinomials L(r,x) = L rbxd and R(x) = R xd,
b,d d
dL(r,x) respectively.
Lri(r,x)= dri , (3) In order to implementPthe optimization algorithmP, it will
be more convenient to work with the edge, instead of the
dL(r,x)
Lxi(r,x)= dx , (4) node perspective. We now define edge perspective multi-
i edge multinomials. Let ρ(j) denote the fraction of type j
d
dR(x) (j = 1,...,m ) edges connected to check nodes of type d.
R (x)= . (5) e
xi dxi Thefractionρ(j) iscalculatedwithrespecttothetotalnumber
d
Note that in a valid multi-edge ensemble, the number of of edges of type j and is thus given by
connections of each edge type should be the same at both
d R
variable and check nodes sides. This gives rise to the socket ρ(j) = j d . (7)
count equality constraint which can be written as d Rxj(1)
Lxi(1,1)=Rxi(1), i=1,...,me, (6) Similarly, let λ(ij) denote the fraction (computed with respect
where 1 denotes a vector with all entries equal to 1, with tothetotalnumberoftypej edges)oftypej edgesconnected
length being clear from the context. tovariablenodesofdegreei.Thisgivesusthefollowingedge
perspective multinomial
Unequal error protection LDPC codes can be included in
a multi-edge framework in a straightforward way. This can
be done by distinguishing between the edges connected to λ(j)(x)= λ(ij)xi−1, (8)
different protection classes within a codeword. According to i
X
this strategy, the edges connected to variable nodes within a
where λ(j) represents the fraction of class j edges connected
i
1Inthemulti-edge framework, onecanconsiderthatthedifferent variable tovariablenodeswithdegreei.Inthenextsection,wewilluse
nodetypes mayhavedifferent received distributions, i.e.,theassociated bits Eqs.(7)and(8)inthederivationoftheoptimizationalgorithm
may be transmitted through different channels. In this work, we consider
for the connection profile among the protection classes of an
that the variable nodes have access solely to one observation and that the
transmissionismadethroughanAWGNchannel. UEP-LDPC code.
III. CHECK NODE PROFILE OPTIMIZATION Let’s now deal with the more general case of irregular
codes. For irregular codes, the incoming densities to a node
A. Asymptotic Analysis
are not necessarily equal due to varying degrees. Therefore,
Our main objective is, given the overallvariable and check
the overall mutual information at the output of the variable
node degree distributions of an UEP-LDPC code, to optimize
and check nodes is averaged over the different degrees. Also
theconnectionprofilesbetweenthedifferentprotectionclasses
note that, in the irregular case, for optimizing the connection
inordertocontroltheamountofprotectionofeachclasswhile
profile between the protection classes, we need to consider
preservingthe UEP capability of the code after a moderate to
the case where check nodeswith differentedge degree vector
high number of decoding iterations. In order to simplify the
d are allowed. This posed, the mutual information between
optimizationalgorithm,we supposethatthe LDPCcodeto be
the received channel values and the messages sent from the
optimized is check-regular, i.e., all the check nodes have the
check to variable nodes and from variable to check nodes at
same degree.
iteration l within the protectionclass C , computedby means
Despite of having the same degree, each check node may j
of density evolution using the Gaussian approximation [10],
have a different number of edges belonging to each one of
are given by
the m classes. Consider for example a check node with an
e
dassioscitahteednuemdgbeerdoefgrceoennveeccttioornds t=o th(de1p,rdo2t,e.c.ti.o,ndmclea)s,swihaenrde ICj =dvmaxλ(j)J 4/σ2+(i 1)[J−1(ICj )]2 ,
iim=e1di = dcmax. If we then consider a code with me = 3 v,l Xi=2 i (cid:18)q − c,l−1 (cid:19)(12)
protection classes, each check node may be connected to d
1
P
edges of class 1, d2 edges of class 2, and d3 edges of class dcmax
3. This posed, one can compute the evolution of the iterative ICj =1 ρ(j)
c,l − d
decoding by means of density evolution. Xs=1 d:Xdj=s
We assume here standard belief propagation decoding of
LDPC codes where the messages exchanged between the J (d 1)J−1(1 ICj)2+ d J−1(1 ICi)2 .
variable and check nodes are log-likelihood ratios having a · j − − v,l i − v,l
s i6=j
X
symmetric Gaussian distribution (variance equals twice the (13)
mean). Before dealing with the subtle case of irregular multi-
edge type LDPC codes, let’s make for now two simplifying Combining Eqs. (12) and (13), one can summarize the
assumptions: 1) All variable nodes have the same degree d . densityevolutionasafunctionofthemutualinformationofthe
v
2) All check nodes have the same edge degree vector d. previous iteration, the mutual information contribution from
Under these assumptions, we can generalize the results the other classes, noise variance, and degree distributions
obtained in [8] for regular LDPC codes and compute the ICj =F(λ(j)(x),ρ(j),σ2,ICj ,ICi ), (14)
mutual information between the output message of a class l d l−1 l−1
Cj, degree-dv variable node and the observed channel value2 where i (1,...,me) and i = j. By means of Eq. (14), we
∈ 6
at iteration l, which is given by canpredicttheconvergencebehaviorofthedecodingandthan
optimize the degree distribution ρ(j) under the constraint that
IvC,lj =J 4/σ2+(dv−1)[J−1(IcC,lj−1)]2 , (9) the mutual information shall be indcreasing as the number of
(cid:18)q (cid:19) iterations grow, i.e.,
where σ2 is the channel noise variance, ICj is the mutual
information received from the neighborincg,l−c1heck nodes at F(λ(j)(x),ρ(dj),σ2,ICj,ICi)>ICj. (15)
iteration l 1 and the J(.) function is defined as in [9]
− At this point, it is worth noting that, as pointed out in [4]
J(σ)=1 ∞ e(ξ−2σσ22/2)2 log [1+e−ξ]dξ . (10) aonfdco[5n]n,etchteioUnEaPmcoanpgabthileitipersootefcatiocnodceladsespees,ndi.eo.n, itfhethaemmouonstt
− √2πσ · 2
Z−∞ protectedclassiswellconnectedtotheleastprotectedone,the
Similarly,themutualinformationsenttothevariablenodesof performanceoftheformerwilldecreasewhiletheperformance
class Cj from a check node with an associated vector d can of the latter will be improved. For example, suppose a code
be expressed as with 2 protection classes and d =4. The possible values
cmax
ICj =1 for d = (d1,d2) are (0,4), (1,3), (2,2), (3,1), and (4,0). On
c,l the one hand, if a code has a majority of check nodes with
d=(4,0),the firstprotectionclass will beveryisolated from
−J (dj −1)(J−1(1−IvC,lj))2+ di(J−1(IvC,il))2. class2whichwillleadtoanenhancedperformancedifference
s Xi6=j betweenthe two classes. On the otherhand,if a largeamount
(11) of the check nodes are of type d = (2,2), one can expect
the protection classes to be very connected, which favors the
2For the sake ofsimplicity, through the rest oftext wewill simply speak
overallperformancebutmitigatestheUEPcapabilityofacode
of “mutual information of some variable” when we refer to the mutual
information betweenthis variable anditsobservation throughthechannel. at a moderate to large number of decoding iterations. This
indicates that for controlling the UEP capability of an LDPC already been performedand the results of these optimizations
code and to prevent this characteristic from vanishing as the areusedasconstraintsinthecurrentoptimizationprocess.The
number of decoding iterations grows, one has to control the optimizationalgorithmcanbewritten,forgivenλ(x),σ2,m ,
e
amount of check nodes of each type, i.e., optimize ρ(j). d , and maxρ(j) for j =1,...,m as shown in Fig. 2.
d cmax d e
B. Optimization Algorithm
1) Compute λ(j)(x)
tioTnhperoalfigloeribtehtmwedeensctrhibeevdahrieoruesaipmrosteacttoiopntimcliazsinsegsthperecsoennnteocn- 23)) IWnihtiialelizoapttiiomnizda(mtji)ionn=fadilcumreax
aUEP-LDPCcode,i.e.,ρ(j).Initially,thealgorithmcomputes a) minimize the average check node degree
d
the variable node degree distribution of each class λ(j)(x) dcmax s· ρ(j) under the
s=d(j) d:dj=s d
based on λ(x), dcmax, and the number of edges on each fPollowimnignconPstraints,
class.Thealgorithmthenproceedssequentiallyoptimizingthe C : dcmax ρ(j) =1,
connection profile to the check nodes of one class at a time, 1 s=d(j) d:dj=s d
min
proceedingfromthe less protectedclass to the mostprotected C :P ρP(j) ∈[0,maxρ(j)],
one.Thisschedulingisdoneinordertocontroltheamountof 2 d:dj=s d d
messages coming up from the less protected classes that are C3 :PF(λ(j)(x),ρ(dj),σ2,x,ICj,ICi)>x,
∀x∈[0,1) and i6=j,
forwarded to the more protected ones.
sinSgilnecoebwjeectiavreefuusninctgiolnin,ewaer pchroogsreamitmtoinbge(tLhPe)mwinitihmioznaltyiona C4 :ρ∀(jj)′ >is fijxaendd. d:djm′ in ≤dj′ ≤dcmax,
of the average check node degree within the class being d
optimized, i.e., it minimizes the average number of edges of b) d(j) =d(j) −1
min min
such a class connectedto the check nodes. This minimization End (While)
aims at diminishing the amount of unreliable messages (i.e.,
theonescomingupfromthelessprotectedvariablenodes)that Fig.2. Checknodeprofileoptimization algorithm.
flows through a check node. In addition to it, we control the
proportionofchecknodesoftypedintroducingtheparameter Notethattheoptimizationcanbesolvedbylinearprogram-
lmimaixtsρd(tjh)ewprhoipcohrtiisonaonfucphpecekr nbooduensdoofntypedd:d:j=djsρ=(djs),foi.re.s,=it (mCi3n)gasriencleintehaercionstthfeunpcatiroanmaentedrsthρe(djc)o.nTsthraeinctosn(sCtr1a)i,n(tC2(C),4a)nids
P the previousoptimizationconstraint.Once we have optimized
0,...,d , thus regulating the degree of connection among
the protcemctaixon classes. For example, setting maxρ(j) = 0.35 the check node profile, the code can be realized through the
d construction of a parity check matrix following the desired
means that the maximum fraction of check nodes with edge
profile.
degree vector d (for any d) will be 0.35.
The optimization is then performed for each class Cj by IV. SIMULATION RESULTS
minimizing its average check node degree for a decreasing
In this section, simulation results for multi-edgetype UEP-
d(j) from d to 1, where d(j) is the minimum number
min cmax min LDPCcodeswithoptimizedchecknodeconnectionprofileare
of edges of class j connected to a check node. At this point,
presented.WedesignedUEP-LDPCcodesoflengthn=4096
one can argue that since our goal is to minimize the average
with m = 3 protection classes, rate 1/2, and d = 30
connection degree within a protection class, we should thus e vmax
following the algorithm of [3]. The proportionsof the classes
set d(j) = 1. The problem with this strategy is that it
min are chosen such that C1 contains 20% of the information
would shorten the degree of freedom for the optimization of
bits and C contains 80%. The third protection class C
2 3
the next class, e.g., suppose the optimization of a two-class
contains all parity bits. Therefore, we are mainly interested
code with d(m2)in = 3 and dcmax = 5. Once we proceed to in the performances of classes C1 and C2. The optimized
the optimization of class two, the coefficients ρ(2) , ρ(2) , variable and check node degree distribution for the UEP-
(2,3) (1,4)
and ρ(2) are determined and consequently fixed for the next LDPC code are given by λ(x) = 0.2130x + 0.0927x2 +
(0,5)
optimization step, i.e., the optimization of class 1, we will 0.2511x3+0.2521x17+0.0965x18+0.0946x29andρ(x)=x8,
haveasvariablesonlythecoefficientsρ(1) ,ρ(1) ,andρ(1) . respectively.
(3,2) (4,1) (5,0)
In order to have a low-complexity systematic encoder,
Notethatinthiscase,ifwehadsetd(2) =1,therewillbeno
min we construct parity check matrices in lower triangular form
degree of freedom for optimizing class 1 since the only non-
[11]. This approach also leads to a simplification in our
optimized d would be d=(5,0) which would be determined
optimization procedure, i.e., given that the parity bits are in
by dcmax ρ(j) =1, i.e., the sum of all fraction of
edgPessm=dum(sjt)inbPe edq:duja=lstodone. itnhealelsoswperrotterciatendgucllaarssfoCr3ma,nwdethsattartthetyheshooputlidmbizeatoiorgnanfirzoemd
The iterative procedure is successful, when a solution ρ(j) the less protected information bits class C , since all the
d 2
is found which convergesfor the given σ2 and d(j) >0. We connections between the variable nodes of class C and
min 3
assume that the optimizations for classes Cj′,j′ < j have the check nodes are completely determined by the lower
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TABLEI
LOCALVARIABLEDEGREEDISTRIBUTIONS.THECOEFFICIENTSλ(ij) 100
REPRESENTTHEFRACTIONOFEDGESCONNECTEDTOVARIABLESNODES
OFDEGREEiWITHINTHECLASSCj.
C1 C2 C3 10−1
λ(1)=0.00197 λ(2)=0.23982 λ(3)=0.93901
4 3 2
λ(1)=0.57263 λ(2)=0.76018 λ(3)=0.06099
18 4 3
λ(119)=0.21085 10−2
λ(1)=0.21455
30 ER C1 maxρ(2) = 0.35
B d
OPTIMIZEDCHECKNODEPROTFAILBELFEORII3PROTECTIONCLASSES.THE 10−3 C2 maxρd(2) = 0.35
COEFFICIENTSρ(sj)REPRESENTρ(Dj)WITHD:dj =s. C1 maxρd(2) = 0.55
C2 maxρ(2) = 0.55
C1 C2 C3 10−4 d
maxρ(d2)=0.35 ρ(21)=0.08977 ρ(22)=0.35 ρ(13)=0.00024 C1 maxρd(2) = 0.75
ρρ(3(411))==00..1394693171 ρρ(3(422))==00..3350 ρρ(2(333))==00..9036807979 10−5 C2 maxρd(2) = 0.75
ρ(1)=0.36475 0 0.2 0.4 0.6 0.8 1 1.2
7
maxρ(d2)=0.55 ρ(31)=0.25248 ρ(32)=0.45 ρ(13)=0.00024 Eb/N0 (dB)
ρ(1)=0.54860 ρ(2)=0.55 ρ(3)=0.93877
4 4 2
ρ(1)=0.19892 ρ(3)=0.06099 Fig. 3. Classes bit error rate of the optimized multi-edge unequal error
7 3
maxρ(d2)=0.75 ρ(31)=0.14027 ρ(32)=0.25 ρ(13)=0.00024 protection LDPCcodes fordifferent values ofmaxρ(d2).
ρ(1)=0.74809 ρ(2)=0.75 ρ(3)=0.93877
4 4 2
ρ(1)=0.11164 ρ(3)=0.06099
7 3
ACKNOWLEDGMENT
This work is funded by the German Research Foundation
triangular form construction algorithm. Table I summarizes (DFG).
the classes’ variable degree distributions λ(j)(x). We applied
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