A Proof of Threshold Saturation for Spatially-Coupled LDPC Codes on BMS Channels

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by  Santhosh Kumar

A Proof of Threshold Saturation for
Spatially-Coupled LDPC Codes on BMS Channels
Santhosh Kumar†, Andrew J. Young†, Nicolas Macris‡, and Henry D. Pfister†
Department of Electrical and Computer Engineering, Texas A&M University†
School of Computer and Communication Sciences, E´cole Polytechnique Fe´de´rale de Lausanne‡
3
1 Abstract—Low-density parity-check (LDPC) convolutional limituniformlyoverthisclasswhendv/dc isfixedanddv,dc
0 codes have been shown to exhibit excellent performance un- are increased.
2 der low-complexity belief-propagation decoding [1], [2]. This The idea of threshold saturation via spatial coupling has
n pcohuepnloimnge.noTnheisunnodwerltyeirnmgedprtinhcriepslheoldbehsaintudratthioisn avpiapesaprsatitaol recently started a small revolution in coding theory, and
a be very general and spatially-coupled (SC) codes have been SC codes have now been observed to universally approach
J successfully applied in numerous areas. Recently, SC regular the capacity regions of many systems [8], [9], [10], [11],

5 LDPC codes have been proven to achieve capacity universally, [12], [13], [14], [15]. For spatially-coupled systems with
2 overtheclassofbinarymemorylesssymmetric(BMS)channels,
suboptimal component decoders, such as message-passing
under belief-propagation decoding [3], [4].
decodingofcode-divisionmultipleaccess(CDMA)[16],[17]
] In [5], [6], potential functions are used to prove that the
T or iterative hard-decisiondecoding of SC generalized LDPC
BP threshold of SC irregular LDPC ensembles saturates, for
I the binary erasure channel, to the conjectured MAP threshold codes [18], the threshold saturates instead to an intrinsic
.
s (known as the Maxwell threshold) of the underlying irregular threshold defined by the suboptimal component decoders.
c ensembles. In this paper, that proof technique is generalized SC hasalso ledto new resultsfor K-SAT,graphcoloring,
[
to BMS channels, thereby extending some results of [4] to
and the Curie-Weiss model in statistical physics [19], [20],
irregular LDPC ensembles. We also believe that this approach
1
[21]. For compressive sensing, SC measurement matrices
can be expanded to cover a wide class of graphical models
v
whose message-passing rules are associated with a Bethe free were introduced in [22], shown to give large improvements
1
1 energy. with Gaussian approximated BP reconstruction in [23], and
1 Index Terms—convolutional LDPC codes, density evolution, finally proven to achieve the information-theoretic limit
entropyfunctional,potentialfunctions,spatialcoupling,thresh-
6 in [24]. Recent results based on spatial coupling are now
old saturation.
.
1 too numerous to cite thoroughly.
0 A different proof technique, based on potential functions,
3 I. INTRODUCTION is used in [5] to prove that the BP threshold of spatially-
1
Low-density parity-check (LDPC) convolutional codes coupled(SC)irregularLDPCensemblessaturatestothecon-
:
v were introduced in [7] and shown to have outstanding jecturedMAPthreshold(knownastheMaxwellthreshold)of
Xi performance under belief-propagation (BP) decoding in [1], the underlyingirregularensembles.Thistechniqueis closely
[2], [8]. The fundamental principle behind this phenomenon relatedtotheanalysisin[19],[20]fortheCurie-Weissmodel,
r
a is described by Kudekar, Richardson, and Urbanke in [3] the heuristic approach in [25], and the continuum approach
and coined threshold saturation via spatial coupling. For used to prove threshold saturation for compressed sensing
the binary erasure channel (BEC), they prove that spatially in [24]. In this paper, the proof technique based on potential
coupling a collection of (d ,d )-regular LDPC ensembles functions (in [5], [6]) is extended to BMS channels. This
v c
produces a new (nearly) (d ,d )-regular ensemble whose extends the results of [4] by proving threshold saturation
v c
BP threshold approaches the MAP threshold of the original to the Maxwell threshold for BMS channels and a wide
ensemble. Recently, a proof of threshold saturation (to the class of SC irregular LDPC ensembles. The main result is
“area threshold”) has been given for (d ,d )-regular LDPC summarizedinthefollowingtheoremwhoseproofcomprises
v c
ensemblesonbinarymemorylesssymmetric(BMS)channels the majority of this paper.
when d /d is fixed and d ,d are sufficiently large [4]. Theorem 1: Consider the SC LDPC(λ,ρ) ensemble de-
v c v c
This result implies that SC-LDPC codes achieve capacity fined in Section IV-A and a family of BMS channels c(h)
universallyovertheclassofBMSchannelsbecausethe“area thatisorderedbydegradationandparameterizedbyentropy,
threshold” of regular LDPC codes approaches the Shannon h. For any BMS channel c(h) with h less than the potential
thresholdh∗,thereexistsasufficientlylargecouplingparam-
This material is based upon work supported in part by the National
ScienceFoundation(NSF)underGrantNo.0747470.TheworkofN.Macris eter w0 such that, for all w > w0, the SC density evolution
was supported by Swiss National Foundation Grant No. 200020-140388. convergesto the perfect decoding solution.
Anyopinions,findings,conclusions,andrecommendationsexpressedinthis
Many observations, formal proofs, and a large variety
materialarethoseoftheauthorsanddonotnecessarilyreflecttheviewsof
thesesponsors. of applications systems bear evidence to the generality of
threshold saturation. In particular, the approach taken in this Moreover, the space of symmetric probability measures is
papercanbeseenasanalyzingtheaverageBethefreeenergy closedunderthese binaryoperations[28]. Ina moreabstract
of the SC ensemble in the large system limit [26], [27]. sense, the measure space M along with the multiplication
Therefore,it is temptingto conjecturethatthis approachcan operator ∗ forms a commutative unital associative algebra
be applied to more general graphical models by computing and this algebraic structure is induced on the space of
the average Bethe free energy of the corresponding SC symmetric probability measures. There is also an intrinsic
system. connection between the algebras defined by each operator
and one consequence is the duality (or conservation) result
II. PRELIMINARIES
in Proposition 4. For the multiplicative identities in these
A. Measure Algebras algebras, e(cid:16) = ∆0 and e(cid:24) = ∆∞, we define x∗0 = e∗ and
We call a Borel measure x symmetric if observethe followingrelationshipsunderthe dualoperation:
e−α/2x(dα)= e−α/2x(dα), ∆ (cid:24)x=∆ ∆ (cid:16)x=∆ .
0 0 ∞ ∞
Z−E ZE
for all Borel sets E ⊂R where one of the integrals is finite. In general, however, these operators do not associate
Let M = M(R) be the space of finite signed symmetric
BorelmeasuresontheextendedrealnumbersR.Inthiswork, x1(cid:16)(x2(cid:24)x3)6=(x1(cid:16)x2)(cid:24)x3
the primary interest is on convex combinations and differ- x1(cid:24)(x2(cid:16)x3)6=(x1(cid:24)x2)(cid:16)x3,
encesofsymmetricprobabilitymeasuresthatinheritmanyof
nor distribute
their properties from M. Let X ⊂M be the convex subset
ofsymmetricprobabilitymeasures.Also,let X ⊂Mbethe
d x (cid:16)(x (cid:24)x )6=(x (cid:16)x )(cid:24)(x (cid:16)x )
subset of differences of symmetric probability measures: 1 2 3 1 2 1 3
x (cid:24)(x (cid:16)x )6=(x (cid:24)x )(cid:16)(x (cid:24)x ).
X ,{x −x |x ,x ∈X}. 1 2 3 1 2 1 3
d 1 2 1 2
B. Partial Ordering by Degradation
Intheinterestofnotationalconsistency,xisreservedforboth
finitesignedsymmetricBorelmeasuresandsymmetricprob- Degradation is an important concept that allows one to
ability measures, and y, z denote differences of symmetric compare some LLR message distributions. The order im-
probability measures. posed by degradation is indicative of relating probability
In this space, there are two important binary operators, (cid:16) measures through a communication channel [28, Definition
and(cid:24),thatdenotethevariable-nodeandcheck-nodedensity 4.69].
evolution operations for log-likelihood ratio (LLR) message Definition 2: For x∈X and f :[0,1]→R, define
distributions,respectively.Thewildcard∗isusedtorepresent
either operator in statements that apply to both operations. I (x), f tanh α x(dα).
For example, the shorthand x∗n is used to denote f 2
Z
(cid:0)(cid:12) (cid:0) (cid:1)(cid:12)(cid:1)
x∗···∗x. For x ,x ∈X, x is said to(cid:12)be degrad(cid:12)ed with respect to x
1 2 1 2
n (denoted x1 (cid:23) x2), if If(x1) ≥ If(x2) for all concave non-
increasing f. Furthermore, x is said to be strictly degraded
For a polynomial λ(α) |= {znde=g}0(λ)λnαn with real coeffi- with respect to x (denoted x1 ≻x ) if x (cid:23)x and there is
cients, we define 2 1 2 1 2
P a concave non-increasing f so that If(x1)>If(x2).
deg(λ) Degradation defines a partial order, on the space of sym-
λ∗(x), λ x∗n.
n metric probability measures, with maximal element ∆ and
0
nX=0 minimal element ∆∞. In this paper, the notation for real
Now, we give an explicit integral characterization of the intervalsisoverloadedand,forexample,a half-openinterval
operators (cid:16) and (cid:24). For x1,x2 ∈M, and any Borel set E ⊂ of measures is denoted by
R, define
(x ,x ],{x′ ∈X |x ≺x′ (cid:22)x }.
1 2 1 2
(x (cid:16)x )(E), x (E−α)x (dα),
1 2 1 2
Z This partial ordering is also preserved under the binary
(x (cid:24)x )(E), x 2tanh−1 tanh(E2) x (dα). operations as follows.
1 2 Z 1 tanh(α2)!! 2 Proposition 3 ([28, Lemma 4.80]): Let x1,x2 ∈ X be
ordered by degradation x (cid:23)x . For any x ∈X, one has
Associativity,commutativity,andlinearityfollow.Therefore, 1 2 3
for measures x , x , x ∈M and scalars α , α ∈R,
1 2 3 1 2 x ∗x (cid:23)x ∗x .
1 3 2 3
x ∗(x ∗x )=(x ∗x )∗x ,
1 2 3 1 2 3
This ordering is our primary tool in describing relative
x ∗x =x ∗x ,
1 2 2 1 channel quality, and thresholds. For further information see
(α x +α x )∗x =α (x ∗x )+α (x ∗x ).
1 1 2 2 3 1 1 3 2 2 3 [28, pp. 204-208].
C. Entropy Functional for Symmetric Measures It is also easy to see that the functional M (·) takes the
k
following product form under the operator (cid:24),
Entropy is a fundamental quantity in information theory
andcommunication,anditisdefinedasthelinearfunctional, M (x (cid:24)x )=M (x )M (x ).
H:M→R, given by the integral k 1 2 k 1 k 2
Also, if y ,y ∈ X are the differences of symmetric prob-
1 2 d
H(x), log 1+e−α x(dα). ability measures, then y (R) = y (R) = (y ∗y )(R) = 0.
2 1 2 1 2
Z This leads to the following result.
The entropy functional is th(cid:0)e primary(cid:1) functional used in Proposition 7: For y ,y ∈ X , the entropy functional
1 2 d
our analysis. It exhibits an order under degradation and, for reduces to
symmetric probability measures x ≻x , one has
1 2 ∞ (log2)−1
H(y )=− M (y ),
H(x )>H(x ). 1 k 1
1 2 2k(2k−1)
k=1
X
The restriction to symmetric probability measures (x ∈ X) ∞ (log2)−1
also implies the bound H(y1(cid:24)y2)=− Mk(y1)Mk(y2).
2k(2k−1)
k=1
0≤H(x)≤1. X
In particular,
The operators (cid:16) and (cid:24) admit a number of relationships
∞ (log2)−1
undertheentropyfunctional.Thefollowingresultswillprove H(y (cid:24)y )=− M (y )2 ≤0.
1 1 k 1
invaluable in the ensuing analysis. Proposition 4 provides 2k(2k−1)
k=1
X
an important conservation result (also known as the duality
with equality iff y =0 almost everywhere.
rule for entropy) and Proposition 5 extends this relation to 1
The above proposition implies an important upper bound on
encompass differences of symmetric probability measures.
theentropyfunctionalfordifferencesofsymmetricprobabil-
Proposition 4 ([28, pp. 196]): For x ,x ∈X, one has
1 2 ity measures under either operator, (cid:16) or (cid:24).
H(x1(cid:16)x2)+H(x1(cid:24)x2)=H(x1)+H(x2). (1) Proposition 8: If x1,x′1,x2,x3 ∈X with x′1 (cid:23)x1, then
Proposition 5: For x , x , x , x ∈X, one finds that |H((x′ −x )∗(x −x ))|≤H(x′ −x ).
1 2 3 4 1 1 2 3 1 1
H(x (cid:16)(x −x ))+H(x (cid:24)(x −x ))=H(x −x ), Proof: We show the result for the operator (cid:24). The
1 3 4 1 3 4 3 4
extension to (cid:16) follows from the duality rule for entropy for
H((x −x )(cid:16)(x −x ))+H((x −x )(cid:24)(x −x ))=0.
1 2 3 4 1 2 3 4 differences of symmetric probability measures, Proposition
Proof: Consider the LHS of the first equality, 5. From Proposition 7
H(x (cid:16)(x −x ))+H(x (cid:24)(x −x )) |H((x′ −x )(cid:24)(x −x ))|
1 3 4 1 3 4 1 1 2 3
=H(x1(cid:16)x3)+H(x1(cid:24)x3)−H(x1(cid:16)x4)−H(x1(cid:24)x4) ≤ ∞ (log2)−1 |M (x′ −x )||M (x −x )|
=H(x )+H(x )−H(x )−H(x ) (Proposition 4) 2k(2k−1) k 1 1 k 2 3
1 3 1 4
k=1
X
=H(x3−x4). (=a)− ∞ (log2)−1 M (x′ −x )|M (x −x )|,
The second equality follows by expanding the LHS and 2k(2k−1) k 1 1 k 2 3
k=1
applying the first equality twice. X
(b) ∞ (log2)−1
Due to the symmetry of the measures the entropy func- ≤ − M (x′ −x )
2k(2k−1) k 1 1
tional has an equivalent series representation.
k=1
X
Proposition 6: If x∈M, then =H(x′ −x ),
1 1
H(x)=x R − ∞ (log2)−1 tanhα 2kx(dα). where (a) follows from Mk(x′1) ≤ Mk(x1) and (b) follows
2k(2k−1) 2 since 0≤M (x ),M (x )≤1.
(cid:0) (cid:1) Xk=1 Z (cid:16) (cid:17) Corollary k9: 2For xk,x3′,x ,x ,x ∈ X with x′ (cid:23) x , one
Proof:Fora sketch of the proof,see [29, Lemma3]. 1 1 2 3 4 1 1
has
Define the linear functional
M (x), tanhα 2kx(dα). |H((x′1−x1)∗(x2−x3)∗x4)|≤H(x′1−x1).
k
2
Z (cid:16) (cid:17) Proof:ThisfollowsfromProposition8 byreplacing x2,
Since−x2k isaconcavedecreasingfunctionovertheinterval x with x ∗x and x ∗x , respectively.
3 2 4 3 4
[0,1], for symmetric probability measures x (cid:23) x , by
1 2
D. Directional Derivatives
Definition 2,
The main result in this paper is derived using potential
M (x )≤M (x ).
k 1 k 2 theoryand differentialrelations. One can avoid the technical
Moreover, for x∈X, challengesofdifferentiationintheabstractspaceofmeasures
byfocusingondirectionalderivativesoffunctionalsthatmap
0≤Mk(x)≤1. measures to real numbers.
Definition 10: Let F : M → R be a functional on M. III. SINGLE SYSTEM
The directional derivative of F at x in the direction y is LetLDPC(λ,ρ)denotetheLDPCensemblewithvariable-
nodedegreedistributionλandcheck-nodedegreedistribution
F(x+δy)−F(x)
d F(x)[y] , lim , ρ. The edge-perspective degree distributions λ,ρ have an
x
δ→0 δ equivalent representation in terms of the node-perspective
whenever the limit exists. degree distributions L, R, namely,
Thisdefinition is naturallyextendedto higher-orderdirec- L′(α) R′(α)
λ(α)= , ρ(α)= .
tional derivatives using L′(1) R′(1)
dnF(x)[y ,...,y ],d (···d (d F (x)[y ])[y ]···)[y ], This differential relationship is crucial in developing an ap-
x 1 n x x x 1 2 n
propriate potential functional for the LDPC(λ,ρ) ensemble.
Density evolution (DE) characterizes the asymptotic per-
and vectors of measures using, for x=[x ,··· ,x ],
1 m
formance of the LDPC(λ,ρ) ensemble by describing the
F(x+δy)−F(x) evolution of message distributions with iteration. For this
d F(x)[y], lim ,
x ensemble, the DE update is compactly described by
δ→0 δ
whenever the limit exists. Similarly, we can define higher- ˜x(ℓ+1) =c(cid:16)λ(cid:16)(ρ(cid:24)(˜x(ℓ))),
order directional derivatives of functionals on vectors of where ˜x(ℓ) is the variable-node output distribution after ℓ
measures.
iterations of message passing [30], [28].
The utility of directional derivatives for linear functionals
We now develop the necessary definitions for the single-
is evident from the following Lemma. system potential framework.Includedare the potential func-
Lemma 11: Let F : M → R be a linear functional and tional, fixed points, stationary points, the directional deriva-
∗ be an associative, commutative,and linearbinaryoperator. tive of the potential functional, and thresholds.
Then, for x,y,z∈M, we have Definition 13: Considera family of BMS channelswhose
LLR distributions c(h) : [0,1] → X are ordered by degra-
dxF(x∗n)[y]=nF(x∗(n−1)∗y), dation and parameterized by their entropy H(c(h)) = h.
d2F(x∗n)[y,z]=n(n−1)F x∗(n−2)∗y∗z . The BP threshold channel of such a family is defined to be
x cBP ,c(hBP), where hBP ,
(cid:16) (cid:17)
Proof:A binaryoperation ∗ thatis associative,commu- sup h∈[0,1]|x≻c(h)(cid:16)λ(cid:16)(ρ(cid:24)(x))∀x∈(∆ ,∆ ] .
∞ 0
tative, and linear admits a binomial expansion of the form
De(cid:8)finition 14: For a family of BMS channels, c(h),(cid:9)the
n MAP threshold is given by hMAP ,
n
(x+δy)∗n = δi x∗(n−i)∗y∗i.
Xi=0 (cid:18)i(cid:19) inf h∈[0,1]|linm→i∞nf n1E[H(Xn |Yn(c(h)))]>0 .
n o
Then, the linearity of F implies that Definition 15: The potential functional (or the average
Bethe free energy), U : X ×X → R, of the LDPC(λ,ρ)
F((x+δy)∗n)−F (x∗n) ensemble is
=δnF(x∗(n−1)∗y)+ n δiF(x∗(n−i)∗y∗i). U(x;c), RL′′((11))H R(cid:24)(x) +L′(1)H ρ(cid:24)(x)
Xi=2 −L′(1)(cid:0)H x(cid:24)(cid:1)ρ(cid:24)(x) −H(cid:0)c(cid:16)L(cid:1)(cid:16) ρ(cid:24)(x) .
Dividing by δ and taking the limit gives Remark 16: This p(cid:0)otential fu(cid:1)nctiona(cid:0)l is ess(cid:0)entially(cid:1)(cid:1)the
negative of the replica-symmetric free energies calculated
dxF(x∗n)[y]=nF(x∗(n−1)∗y). in [29], [26], [31]. When applied to the binary erasure
channel, it is a constant multiple of the potential function
Ananalogousargumentshowstheresultforthesecond-order defined in [5]. An example of U(x;c) is shown in Fig. 1.
directional derivative. Definition 17: The fixed-point potential, P : X → R, of
Remark 12: Ingeneral,applyingTaylor’stheoremtosome the LDPC(λ,ρ) ensemble is
mapping T : X → X requires advanced mathematical
P(x),U(x;f(x)),
machinery. However, in our problem, the entropy functional
and its interplay with the operators (cid:16) and (cid:24) impose a where f :X →X satisfies f(x)(cid:16)λ(cid:16)(ρ(cid:24)(x))=x.
polynomial structure on the functions of interest, obviating Remark 18: Forregularcodes,onecanusedualityrulefor
the needforFre´chetderivatives.Therefore,Taylor’stheorem entropytoshowthatthefixed-pointpotentialdefinedaboveis
becomes quite simple for parameterized linear functionals exactlyequaltothenegativeoftheGEXITintegralfunctional
φ:[0,1]→R of the form denoted by A in [4, Lemma 26]. A similar relationship with
EBP GEXIT integrals is believed to be true in general, but
φ(t)=F (x +t(x −x )).
1 2 1 the authors are not aware of a proof for this.
·10−2
Definition 23: For a channel c ∈ X, the energy gap is
defined as
) 3
h() h=hBP ∆E(c), inf U(x;c).
C x∈X\V(c)
S
B Lemma 24: Thefollowingrelationsareimportantinchar-
2
′h); acterizing the thresholds. Suppose h1 >h2. Then
(
C a) c(h )≻c(h )
N h=hMAP 1 2
G 1 b) U(x;c(h1))<U(x;c(h2)), if x6=∆∞.
W
c) V(c(h ))⊆V(c(h ))=⇒X \V(c(h ))⊇X \V(c(h ))
A 1 2 1 2
(B ∆E d) ∆E(c(h1))<∆E(c(h2))
U 0 Definition 25: Thepotentialthresholdchannelofafamily,
c(h), of BMS channels is given by c∗ ,c(h∗), where
0 0.2 0.4 0.6 0.8 1
h∗ ,sup{h∈[hBP,1]|∆E(c(h))>0}
h′
Fig.1. Potentialfunctionalfor(λ,ρ)=(x2,x5)onthebinarysymmetric Note that, if h<h∗, from Lemma 24d, ∆E(c(h))>0.
channel (BSC), with h∈{0.40,0.416,0.44,0.469,0.48}. The x-input is Lemma 26: From [29], [26], [31], the following holds
chosentobethebinaryAWGNchannel (BAWGNC)withentropy h′. for degree distributions without odd-degree checks on any
Definition 19: For x∈X, BMSchannelandanydegreedistributiononthebinary-input
AWGN channel:
a) x is a fixed point of density evolution if
a) liminf1E[H(Xn|Yn(c(h)))]≥− inf U(x;c(h)),
x=c(cid:16)λ(cid:16) ρ(cid:24)(x) , n→∞ n x∈X
b) hMAP ≤h∗.
b) x is a stationary point of the p(cid:0)otentia(cid:1)l if, for all y∈Xd, Proof: Since the potential functional is the negative of
the replica-symmetric free energies in [29], [26], [31], the
d U(x;c)[y]=0.
x
main result of these papers translates directly into the first
Lemma 20: For x,c ∈ X and y ∈ Xd, the directional result.Forthesecondresult,considerany h>h∗.Below,we
derivative of the potential functionalwith respect to x in the willarriveataconclusionthat h≥hMAP, whichestablishes
direction y is the desired result. Note that from Lemma 24d, ∆E(c(h))<
0. From the first part of this lemma,
d U(x;c)[y]
x
=L′(1)H x−c(cid:16)λ(cid:16) ρ(cid:24)(x) (cid:16) ρ′(cid:24)(x)(cid:24)y . liminf1E[H(Xn|Yn(c(h)))]
n→∞ n
Proof: The(cid:0)d(cid:2)irectional de(cid:0)rivative(cid:1)(cid:3)for(cid:2)each of the(cid:3)(cid:1)four ≥− inf U(x;c(h))≥− inf U(x;c(h))
x∈X x∈X\V(c)
terms is calculated following the procedure outlined in the
=−∆(E(c(h)))>0.
proof of Lemma 11. The first three terms are
d H R(cid:24)(x) [y]=R′(1)H ρ(cid:24)(x)(cid:24)y , Hence, h≥hMAP.
x
Remark 27: Thisimpliesthatthe MAP thresholdis upper
d H ρ(cid:24)(x) [y]=H ρ′(cid:24)(x)(cid:24)y ,
x (cid:0) (cid:1) (cid:0) (cid:1) boundedbythepotentialthresholdforthesinglesystem.Itis
dxH x(cid:24)(cid:0)ρ(cid:24)(x)(cid:1)[y]=H(cid:0)ρ(cid:24)(x)(cid:24)y(cid:1)+H x(cid:24)ρ′(cid:24)(x)(cid:24)y conjecturedthat, in general,these two quantitiesare actually
(cid:0) (cid:1) (=a)H(cid:0)ρ(cid:24)(x)(cid:24)y(cid:1) +H(cid:0)ρ′(cid:24)(x)(cid:24)y (cid:1) equal. Some progress has been made towards proving this
for special cases [32].
−H x(cid:16) ρ′(cid:24)(x)(cid:24)y ,
(cid:0) (cid:1) (cid:0) (cid:1)
IV. COUPLEDSYSTEM
where (a) follows from Prop(cid:0)ositio(cid:2)n 5, ρ′(cid:24)(x(cid:3))(cid:1)(cid:24) y is the
The potential theory from Section III is now extended to
difference of probability measures multiplied by the scalar
spatially-coupled systems. Vectors of measures are denoted
ρ′(1). Since the operators (cid:16) and (cid:24) do not associate, one
by underlines (e.g., x) with [x] = x . Functionals operating
i i
must exercise care in analyzing the last term,
onsingledensitiesare distinguishedfromthoseoperating on
d H c(cid:16)L(cid:16) ρ(cid:24)(x) [y] vectors by their input (i.e., U(x;c) vs. U(x;c)). Also, for
x
=L′(1)H c(cid:16)λ(cid:16)(ρ(cid:24)(x)) (cid:16) ρ′(cid:24)(x)(cid:24)y . vectors x and x′, we write x (cid:23) x′ if xi (cid:23) x′i for all i, and
(cid:0) (cid:0) (cid:1)(cid:1) x≻x′ if x(cid:23)x′ and x ≻x′ for some i.
i i
Consolidating the f(cid:0)o(cid:2)ur terms gives t(cid:3)he d(cid:2)esired resul(cid:3)t.(cid:1)
A. Spatial Coupling
Corollary 21: If x ∈ X is a fixed point of density
evolution, then it is also a stationary point of the potential The ideas underlying spatial coupling now appear to be
functional. quite general. The local coupling in the system allows the
Definition 22: For a channel c ∈ X, define the basin of effect of the perfect information, provided at the boundary,
attraction for ∆ as to propagate throughout the system. In the large system
∞
limit, these coupled systems show a significant performance
V(c), x∈X |x′ ≻c(cid:16)λ(cid:16)(ρ(cid:24)(x′))∀x′ ∈(∆ ,x] .
∞ improvement.
(cid:8) (cid:9)
The (λ,ρ,N,w) SC ensemble is defined as follows. A
collection of 2N variable-node groups are placed at all
positions in N = {1,2,...,2N} and a collection of
v
2N +(w−1) check-node groups are placed at all positions
in N = {1,2,...,2N + (w −1)}. For notational conve-
c
nience, the rightmost check-node group index is denoted by
N ,2N+(w−1). The integer M is chosen large enough 0 1 2 i0 NwNw+1
w
··· l l l ··· l ··· l l ···
so that i) ML , ML′(1)R /R′(1) are natural numbers for
i j
∆ x x x x ∆
1 ≤ i ≤ deg(L), 1 ≤ j ≤ deg(R), and ii) ML′(1) is ∞ 1 2 i0 Nw ∞
divisible by w.
Fig. 2. This figure depicts the entropies of x1,···,xNw in a typical
At each variable-node group, ML nodes of degree i are iteration. Theblueline(solid)correspondstothetwo-sidedsystemandthe
i
placed for 1 ≤ i ≤ deg(L). Similarly, at each check-node redline(dashed)totheone-sidedsystem.Thedistributionsoftheone-sided
system are always degraded with respect to the two-sided system, hence a
group, ML′(1)R /R′(1) nodes of degree i are placed for
i higher entropy. Thedistributions outside theset {1,···,Nw}arefixedto
1 ≤ i ≤ deg(R). At each variable-node and check-node ∆∞ forboththesystems.
groupthe ML′(1) socketsare partitionedinto w equal-sized
system,(4)ismoremathematicallytractableandeasilyyields
groups using a uniform random permutation. Denote these
a coupledpotentialfunctional.Assuch,we adoptthesystem
partitions, respectively, by Pv and Pc at variable-node
i1,j i2,j characterizedby(4)andrefertoitasthe two-sidedspatially-
and check-node groups, where 1 ≤ i ≤ 2N, 1 ≤ i ≤ N
1 2 w coupled system.
and1≤j ≤w. TheSCsystem isconstructedbyconnecting
The two-sided spatially-coupled system is initialized with
the sockets from Pv to Pc using uniform random
i,j i+j−1,j
permutations. This construction leaves some sockets of the
x(0) =∆ , 1≤i≤N .
check-node groups at the boundaries unconnected and these i 0 w
sockets are assigned the binary value 0 (i.e., the socket and
This symmetric initialization, uniform coupling coefficients,
edge are removed). These 0 values form the seed that gets
and symmetric boundary conditions (i.e., seed information)
decoding started.
induce symmetry on all the message distributions. In partic-
B. Spatially-Coupled Systems ular,thetwo-sidedsystem is fullydescribedbyonlyhalf the
Let ˜x(ℓ) be the variable-node output distribution at node distributions because
i
i after ℓ iterations of message passing. Then, the input
x(ℓ) =x(ℓ) ,
distribution to the i-th check-node group is the normalized i 2N+w−i
sum of averaged variable-node output distributions
forall ℓ. As densityevolutionprogresses, the perfectbound-
1 w−1 ary information from the left and right sides propagates in-
x(ℓ) = ˜x(ℓ) . (2)
i w i−k ward. This propagationinduces a nondecreasingdegradation
kX=0 ordering on positions 1,...,⌈Nw/2⌉ and a nonincreasing
Backward averaging follows naturally from the setup and is ordering on positions ⌈(N +1)/2⌉,...,N [4, Def. 44].
w w
essentially the transpose of the forward averaging for the The nondecreasing ordering by degradation introduces a
check-node output distributions. This model uses uniform degraded maximum at i , N +⌊w⌋, and this maximum
0 2
coupling over a fixed window, but in a more general setting allows one to define a modified recursion that upper bounds
windowsizeandcoefficientweightscouldvaryfromnodeto the two-sided spatially-coupled system.
node. By virtue of the boundary conditions, ˜x(iℓ) = ∆∞ for Definition 28 ([4]): The one-sided spatially-coupled sys-
i ∈/ Nv and all ℓ, and from the relation in (2), this implies tem is a modification of (4) defined by fixing the values of
x(iℓ) =∆∞ for i∈/ Nc and all ℓ. positions outside Nc′ , {1,2,...,i0}, where i0 is defined
Generalizing[4,Eqn.12]toirregularcodesgivesevolution as above. As before, the boundary is fixed to ∆ , that is
∞
of the variable-node output distributions, x(ℓ) = ∆ for i 6∈ N and all ℓ. More importantly, it also
i ∞ c
fixes the values x(ℓ) =x(ℓ) for i <i≤N and all ℓ.
1 w−1 1 w−1 i i0 0 w
˜x(ℓ+1) =c(cid:16)λ(cid:16) ρ(cid:24) ˜x(ℓ) . (3) The density evolution update for the one-sided system is
i w Xj=0 w Xk=0 i+j−k! identicalto(4)forthefirsti0terms,{x(1ℓ),x(2ℓ),...,xi(0ℓ)}.But,
  fortheremainingterms,itsimplyrepeatsthedistribution x(ℓ)
Making a change of variables, the variable-node output i0
distribution evolution in (3) can be rewritten in terms of and this implies x(iℓ) = x(i0ℓ) for i0 < i ≤ Nw at every step.
check-node input distributions Theone-sidedandtwo-sidedsystemsareillustratedinFig.2.
Lemma 29: For the one-sided spatially-coupled system,
w−1 w−1
1 1 the fixed point resulting from the density evolution satisfies
x(ℓ+1) = c (cid:16)λ(cid:16) ρ(cid:24) x(ℓ) , (4)
i w i−k w i−k+j 
Xk=0 Xj=0 (cid:16) (cid:17) x (cid:23)x , 1≤i≤N
i i−1 w
 
where c = c when i ∈ N and c = ∆ otherwise.
i v i ∞
While (3) isa morenaturalrepresentationfortheunderlying Proof: See [4, Section IV-D].
Nw 1 2N 1 w−1
U(x;c),L′(1) H R(cid:24)(x ) +H ρ(cid:24)(x ) −H x (cid:24)ρ(cid:24)(x ) − H c(cid:16)L(cid:16) ρ(cid:24)(x ) (5)
R′(1) i i i i  w i+j 
Xi=1(cid:20) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:21) Xi=1 (cid:16) Xj=0 (cid:17)
 
Nw 1 w−1 1 w−1
d U(x;c)[y]=L′(1) H x − c (cid:16)λ(cid:16) ρ(cid:24)(x ) (cid:16) ρ′(cid:24)(x )(cid:24)y (6)
x i i−k i−k+j i i
 w w  
i=1 k=0 j=0
X X X (cid:2) (cid:3)
   
d2U(x;c)[y,z]=
x
Nw 1 w−1 1 w−1 ρ′′(cid:24)(x )
L′(1)ρ′′(1) H c (cid:16)λ(cid:16) ρ(cid:24)(x ) (cid:16) i (cid:24)y (cid:24)z
"w i−k w i−k+j ! ρ′′(1) # i i
i=1 k=0 j=0
X X X
Nw ρ′′(cid:24)(x ) Nw ρ′(cid:24)(x ) 
−L′(1)ρ′′(1) H x (cid:24) i (cid:24)y (cid:24)z −L′(1)ρ′(1) H i (cid:24)y (cid:24)z (7)
i ρ′′(1) i i ρ′(1) i i
i=1 (cid:18)(cid:20) (cid:21) (cid:19) i=1 (cid:18) (cid:19)
X X
− L′(1)λ′(1)ρ′(1)2 Nw min{i+(w−1),Nw} H 1 w−1c (cid:16) λ′(cid:16) w1 wj=−01ρ(cid:24)(xi−k+j)! (cid:16) ρ′(cid:24)(xm) (cid:24)z (cid:16) ρ′(cid:24)(xi) (cid:24)y
w w i−k Pλ′(1) ρ′(1) m ρ′(1) i
!
Xi=1m=max{Xi−(w−1),1} Xk=0 h i h i
C. Spatially-Coupled Potential potentialfunctionalatthisshiftandthedirectionalderivative
Definition 30: The potential functional for a spatially- alongthe shiftdirection[S(x)−x].Lemma36establishesan
coupled system U :XNw ×X →R is given by (5). important bound on the second-order directional derivative
of the potential. Theorem 37 proves threshold saturation.
Lemma 31: Foraspatially-coupledsystem,thedirectional
derivativeofthepotentialfunctionalwithrespecttox∈XNw Definition 33: The shift operator S : XNw → XNw is
evaluated in the direction y∈XNw is given by (6). defined pointwise by
d
Proof: The proof is similar to the single system case [S(x)] ,∆ , [S(x)] ,x , 2≤i≤N .
1 ∞ i i−1 w
and is omitted for brevity.
Lemma 34 ([5, Lemma 4]): Let x ∈ XNw be such that
Lemma 32: For a spatially-coupled system, the second-
x =x , for i ≤i≤N . Then, the change in the potential
order directional derivative of the potential functional with i i0 0 w
functionalfor a spatially-coupledsystem associated with the
respect to x evaluated in the direction [y,z] ∈ XNw ×XNw
d d shift operator is bounded by
is given by (7).
Proof: We skip the proof for brevity. U(S(x);c)−U(x;c)≤−U(x ;c).
i0
V. APROOF OFTHRESHOLDSATURATION Proof: Due to boundary conditions, xi = xi0 for i0 ≤
i≤N ,theonlytermsthatcontributetoU(S(x);c)−U(x;c)
We now prove threshold saturation for the spatially- w
are given in (8). As the last w values, that is x for 2N ≤
coupled LDPC(λ,ρ) ensemble. Consider a spatially-coupled i
system with potential functional U : XNw ×X → R as in i≤LeNmwm,aar3e5e:qIufaxl t≻o ∆xi0, w,e(h∆ave,.th.e.,d∆esir)edisraesfiuxlte.d point
Definition 30, and a parameterization φ:[0,1]→R ∞ ∞ ∞
of the one-sided spatially-coupled system, then
φ(t)=U(x+t(x′−x);c), d U(x;c)[S(x)−x]=0,
x
where x is a non-trivial fixed point of the one-sided SC and x ∈/ V(c) (i.e., x not in basin of attraction of ∆ ).
system resultingfrom DE. The path endpoint x′ is chosen to Pi0roof: See Appeni0dix. ∞
imposeasmallperturbationonx.Forallchannelsbetterthan Lemma 36: Letx ∈XNw beavectorofsymmetricprob-
1
the potential threshold, c ≺ c(h∗), and any non-trivial fixed ability measures, and let x ∈ XNw be a vector of coupled
2
point,itcanbeshownthattheone-sidedSCpotentialstrictly check-node inputs ordered by degradation, [x ] (cid:23) [x ] ,
2 i 2 i−1
decreases along this perturbation. However, since a fixed generated by coupling a vector of variable-node outputs
pointis also a stationarypointof thepotentialfunctional,all ˜x ∈X2N
2
variationsinthepotentialuptothesecond-ordercanbemade w−1
1
arbitrarily small by choosing a large coupling parameter w. [x2]i = [˜x2]i−k.
w
Thus, one obtains a contradiction to the existence of a non- k=0
X
trivialfixedpointfromthesecond-orderTaylorexpansionof The second-order directional derivative of U(x ;c) with
1
φ(t). respect to x evaluated along [S(x )−x ,S(x )−x ] can
1 2 2 2 2
These ideas are formalized below. A right shift is cho- be absolutely bounded with
sen for the perturbation. This shift operator is defined in K
Definition 33 and Lemmas 34, 35 discuss its effect on the d2x1U(x1;c)[S(x2)−x2,S(x2)−x2] ≤ wλ,ρ,
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
L′(1)
U(S(x);c)−U(x;c)=− H R(cid:24)(x ) −L′(1)H ρ(cid:24)(x ) +L′(1)H x (cid:24)ρ′(cid:24)(x ) (8)
R′(1) Nw Nw Nw Nw
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
w−1 w−1
1 1
+H c(cid:16)L(cid:16) ρ(cid:24)(x ) −H c(cid:16)L(cid:16) ρ(cid:24)(x )
2N+j 1+j
 w   w 
(cid:16) Xj=0 (cid:17) (cid:16) Xj=0 (cid:17)
   
where the constant degradedwithrespecttothetwo-sidedsystem,andtherefore,
the only fixed point of the two-sided system is also ∆ .
K ,γL′(1) 2ρ′′(1)+ρ′(1)+2λ′(1)ρ′(1)2 ∞
λ,ρ
VI. CONCLUSIONS
is independent of N a(cid:0)nd w. (cid:1)
In this paper, the proof technique based on potential
Proof: See Appendix.
functions (in [5], [6]) is extended to BMS channels. This
Theorem 37: Consider a family of BMS channels c(h)
extends the results of [4] by proving threshold saturation to
that is ordered by degradation and parameterized by the
the Maxwell threshold for BMS channels and the class of
entropy, h. For a spatially-coupled LDPC (λ,ρ) ensemble
SC irregular LDPC ensembles. In particular, for any family
with a coupling window w > K /(2∆E(c(h))), and a
λ,ρ of BMS channels c(h) that is ordered by degradation and
channel c(h) with h < h∗, the only fixed point of density
parameterized by entropy, h, a potential threshold h∗ is
evolution is ∆ .
∞ defined that satisfies hMAP ≤ h∗, where hMAP defines the
Proof: Consider a one-sided spatially-coupled system.
MAP threshold of the underlying LDPC(λ,ρ) ensemble.
Fix w > K /(2∆E(c(h))). Suppose x ≻ ∆ is a fixed
λ,ρ ∞ The main result is that, for sufficiently large w, the SC
pointof density evolution.Let y=S(x)−x and φ:[0,1]→
DE equation converges to the perfect decoding fixed point
R be defined by
foranyBMSchannelc(h)withh<h∗. Theapproachtaken
φ(t)=U(x+ty;c(h)), inthispapercanbeseenasanalyzingtheaverageBethefree
energy of the SC ensemble in the large system limit [26],
it is important to note that, for all t ∈ [0,1], x+ty = (1−
[27].
t)x+tS(x) is a vector of probability measures. By linearity
This result reiterates the generality of the threshold sat-
ofthe entropyfunctionalandbinaryoperators (cid:16)and(cid:24),φ is
uration phenomenon, which is now evident from the many
a polynomial in t, and thus infinitely differentiable over the
observationsand proofsthat span a wide variety of systems.
entireunitinterval.Thesecond-orderTaylorseriesexpansion
We also believe that this approach can be extended to more
about t=0 evaluated at t=1 provides
general graphical models by computing the average Bethe
φ(1)=φ(0)+φ′(0)(1−0)+ 1φ′′(t )(1−0)2, (9) free energy of the corresponding SC system.
2 0
Acknowledgment: The authors thank Ru¨diger Urbanke,
for some t ∈ [0,1]. The first and second derivatives of φ
0 ArvindYedla,andYung-YihJianforanumberofveryuseful
are characterized by the first- and second-order directional
discussions during the early stages of this research.
derivatives of U:
APPENDIX
U(x+(t+δ)y;c(h))−U(x+ty;c(h))
φ′(t)= lim In the appendix, for a vector of measures x , x is used
δ→0 δ i i,j
to denote [x ] .
=d U(x+ty;c(h))[y], i j
x+ty
A. Proof of Lemma 35
and similarly,
Since x is a fixed point of the one-sided spatially coupled
φ′′(t)=d2 U(x+ty;c(h))[y,y]. system,
x+ty
w−1 w−1
Substituting and rearranging terms in (9) provides x = 1 c (cid:16)λ(cid:16) 1 ρ(cid:24)(x )
i i−k i−k+j
w w 
1d2 U(x+t y;c(h))[y,y] k=0 j=0
2 x+t0y 0 X X
=U(S(x);c(h))−U(x;c(h))−dxU(x;c(h))[S(x)−x] for 1 ≤ i ≤ i0, and [S(x) −x]i = 0 for i0 < i≤ Nw.
The first result follows from applying these relations to the
=U(S(x);c(h))−U(x;c(h)) (Lemma 35)
directionalderivative,giveninLemma31.Now,observethat
≤−U(x ;c(h)) (Lemma 34)
i0 w−1 w−1
≤−∆E(c(h)). (Lemma 35 and Definition 22) x = 1 c (cid:16)λ(cid:16) 1 ρ(cid:24)(x )
i0 w i0−k w i0−k+j 
Taking an absolute value and applying the second-order k=0 j=0
X X
directional derivative bound from Lemma 36 gives w−1  w−1 
1 1
∆E(c(h))≤ Kλ,ρ =⇒ w ≤ Kλ,ρ , (cid:22) w ci0−k(cid:16)λ(cid:16)w ρ(cid:24)(xi0)
2w 2∆E(c(h)) kX=0 Xj=0
(cid:22)c(cid:16)λ(cid:16) ρ(cid:24)(x ) . 
a contradiction. Hence the only fixed point of the one-sided i0
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