Spatially-Coupled MacKay-Neal Codes Universally Achieve the Symmetric Information Rate of Arbitrary Generalized Erasure Channels with Memory

0.27 MB·English

by  Masaru Fukushima

Spatially-Coupled MacKay-Neal Codes
Universally Achieve the Symmetric Information
Rate of Arbitrary Generalized Erasure Channels
with Memory
Masaru Fukushima, Takuya Okazaki and Kenta Kasai
Department of Communications and Computer Engineering,
5
Tokyo Institution of Technology
1
0 Email: {fukushima,osa,kenta}@comm.ce.titech.ac.jp
2

n
a
J
Abstract—This paper investigates the belief propagation de- of spatially-coupleregularcodesachievethe SIR of DEC and
7
coding of spatially-coupled MacKay-Neal (SC-MN) codes over PR2 channels, respectively, by increasing node degree.
2
erasurechannelswithmemory.WeshowthatSC-MNcodeswith In this paper, we show that SC-MN codes with bounded

bounded degree universally achieve the symmetric information
] degree universally achieve the symmetric information rate of
T rate(SIR)ofarbitraryerasurechannelswithmemory. Wemean
wide variety of erasure channels with memory. We mean by
by universality the following sense: the sender does not need
I
. to know the whole channel statistics but needs to know only universality the following sense: the sender does not need to
s
the SIR, while the receiver estimates the transmitted codewords know the whole channel statistics p(y|x) but needs to know
c
[ from channel statistics and received words. The proof is based only the SIR, while the receiver estimates the transmitted
on potential function.
codewords from channel statistics and received words. The
1 Index Terms—spatially-coupled codes, MacKay-Neal codes,
v LDPC codes, potential function proof is based on the powerful potential function method [5],
6 [6], [7].
3 I. INTRODUCTION
II. BACKGROUND
7
Felstro¨m and Zigangirov introduced convolutional LDPC
6 A. Generalized Erasure Channel (GEC)
codes [1]. Later the codes are called spatially-coupled (SC)
0
. codes.Lentmaieretal.confirmedthatregularSCLDPCcodes Denote a channel input and output as x = (x1,...,xn) ∈
1 achieve excellent BP decoding [2]. Kudekar et al. proved {0,1}n and y =(y1,...,yn) ∈ Yn, respectively. We assume
0 that SC codes achieve MAP threshold by BP decoding on that x ∈ {0,1}n is uniformly distributed. Let us assume that
5
1 the binary erasure channel (BEC) [3] and binary memoryless byintroducingsomeappropriatestatenodesσ =(σ1,...,σn)
: symmetric channels [4]. such that
v
i In [5], Takeuchi et al. introduced potential function to un- p(x|y)= p(x,σ|y),
X derstand how threshold saturation phenomenon happen. With
σ
X
r some modifications, Yedla et al. proved threshold saturation
a wecanfactorizep(x,σ|y)sothatitsfactorgraphisatree.By
of spatially-coupled LDPC codes over BEC in [6], [7].
Bayes rule, we have p(x,σ|y) ∝ p(y|σ,x)p(σ|x)p(x). Con-
Kasai et al. introduced SC MacKay-Neal (MN) codes, and
sidertheAPPdetectorimplementedbysum-productalgorithm
showed that these codes with finite maximum degree achieve
to calculate
capacityofBECbynumericalexperiment[8].Obataetal.[9]
and Okazaki et al. [10] proved respectively, (dl,dr =2,dr = p(xj =0|y),p(xj =1|y) =: µj(0),µj(1)
2) SC-MN codes and (dl,dr = 3,dg = 3) SC-MN codes for j =(cid:0) 1,2,...,n. We say the(cid:1)chan(cid:0)nel p(y|x) i(cid:1)s a gen-
achieve the capacity of BEC for any d > d , where each of
l r eralized erasure channel (GEC) if (µ (0),µ (1)) is one of
j j
d ,d ,d describes the degree of nodes in the Tanner graph
l r g {(1,0),(0,1),(1/2,1/2)}.ThecorrespondingLLRvaluesare
of the codes.
+∞, −∞ and 0. Some authors use 0, 1 and ?, instead.
Reliable transmissions over channels with memory are
B. LDPC codes over GEC
practically important, e.g., magnetic recording with ISI and
Rayleighfadingchannelwithmemory[11].Pfister andSiegel Next, consider x is uniformly distributed in an LDPC code
proved that carefully designed irregular LDPC codes can C. In precise,
achievethesymmetricinformationrate(SIR)ofdicodeerasure
1/#C (x∈C)
channels (DEC) under joint iterative decoding [12]. In [13] p(x)=
and [14], it was empirically observed that the BP threshold (0 (x∈/ C)
Let us denote the factor graph of p(y|σ,x)p(σ|x)p(x) by G. decodingroundby x(t). The recursionof density evolutionof
j
We divide G into two subgraphs. One is the factor graph MET-LDPC codes on BEC is given by
of p(y|σ,x)p(σ|x) and the other is the factor graph of p(x).
ν (y(t);Φ(x;ǫ)) µ (1−x(t))
They are corresponding to APP detector and LDPC decoder, x(t) = j , y(t+1) =1− j ,
respectively. Since the message from APP detector to the bit j νj(1;idR) j µj(1)
nodes is one of M, the messages used in the LDPC decoder where ν (x;Φ(x;ǫ)):= ∂ ν(x;Φ(x;ǫ)), µ (x):= ∂ µ(x)
also takes value in M. and idRjis the identity fu∂nxcjtion idR(x)=x.j ∂xj
Consider the density evolution of the BP decoding on G Then, the density evolution of (d ,d ,d ) MN codes is
l r g
in the limit of code length. We define φ(x;ǫ) as a function
which maps the erasure probability of messages from LDPC x(0) =1, x(t+1) =f(g(x(t));ǫ),
decoder to APP detector via bit nodes x, to the erasure
where
probability of messages from APP detector to from LDPC
decoder φ(x;ǫ), where ǫ is a parameter which defines the f(x;ǫ):= xdl−1,φ(xdg;ǫ) (1)
1 2
channel. From the definition, it follows that φ(x;ǫ) is non- g(x):=(cid:0)1−(1−x1)dr−(cid:1)1(1−x2)dg,
decreasing in x∈[0,1]. In this paper, we further assume that
φ(x;ǫ) is twice continuouslydifferentiablewith x and strictly (cid:0)1−(1−x1)dr(1−x2)dg−1 . (2)
increasingwith ǫ∈[0,1].Inthissetting,the densityevolution E. Spatially-Coupled MacKay-Neal Codes (cid:1)
for (d ,d ) regular LDPC codes over GEC with φ(x;ǫ) is
l r SC-MN codes of chain length L and of coupling width w
written as follows.
are defined by the Tanner graph constructed by the following
x(0) =1, process. First, at each section i ∈ Z, place rM/l bit nodes
of type 1 and M bits nodes of type 2. Bit nodes of type 1
x(t+1) =φ (1−(1−x(t))dr−1)dl;ǫ and 2 are of degree d and d , respectively. Next, at each
l g
(cid:0) ·(1−(1−x(t))dr−1)d(cid:1)l−1, section i∈Z, place M check nodes of degree dr+dg. Then,
connect edges uniformly at random so that bit nodes of type
where x(t) is the erasure probability of messages from bit 1 at section i are connected with check nodes at each section
nodes to parity-check nodes at the t-th decoding round. i∈[i,...,i+w−1]withd M/wedges,andbitnodesoftype
r
2 at section i are connected with check nodes at each section
C. Symmetric Information Rate
i ∈ [i,...,i + w − 1] with d M/w edges. Bits at section
g
SIR I(ǫ) is the mutual information is defined as follows i ∈/ [0,L−1]) are shortened. Bits of type 1 and 2 at section
i∈[0,L−1] are puncturedand transmitted,respectively.The
1
I(ǫ):= lim I(X;Y(ǫ)), rate of SC-MN codes RMN(d ,d ,d ,L,w) is given by
l r g
n→∞n
RMN(d ,d ,d ,L,w)
under the existence of limit, where capital letters represent l r g
random variables. In [15], [16], it is shown that the SIR is d 1+w−2 w (1−(i)dr+dg)
= r + i=0 w
calculated via φ(x;ǫ) as follows. d L
l P
d
1 = r (L→∞). (3)
I(ǫ)=1− φ(x;ǫ)dx=1−Φ(1;ǫ), d
l
Z0 F. Vector Admissible System and Potential Function
where Φ(x;ǫ) := xφ(x′;ǫ)dx′. Let R be the coding rate.
0 In this section, we define vector admissible systems and
Define SIR limit as ǫ such that I(ǫ) = R, and denote
potential functions.
R
it by ǫSIR(R). Uniqueness of ǫSIR(R) is again due to the
assumption that φ(x;ǫ) is increasing in ǫ. Definition 1. DefineX ,[0,1]d, andF :X×[0,1]→R and
G : X → R as functional satisfying G(0) = 0. Let D be a
D. MacKay-Neal Codes d×d positive diagonal matrix. Consider a general recursion
defined by
(d ,d ,d ) MN codes are multi-edge type (MET) LDPC
l r g x(ℓ+1) =f(g(x(ℓ));ǫ) (4)
used over GEC with φ(x;ǫ) codes [17] defined by pair of
multi-variables degree distributions (µ,ν) listed below. where f : X × [0,1] → X and g : X → X are defined
by F′(x;ǫ) = f(x;ǫ)D and G′(x) = g(x)D, where
d
ν(x;Φ(x;ǫ))= drlxd1l +Φ(xd2g;ǫ), µ(x)=xd1rxd2g. Fa ′v(exc;toǫ)r a,dm(i∂s∂Fsxi(b1xl)e,.s.y.s,te∂m∂Fx(inxf)). Then the pair (f,g) defines
Here,weslightlyextendedthedefinitionofdegreedistribution
1. f,g are twice continuously differentiable,
insuchawaythatthebitscorrespondingtothetermΦ(xdg;ǫ)
2 2. f(x;ǫ) and g(x) are non-decreasing in x and ǫ with
are transmitted through the GEC with φǫ. In the case of respect to (cid:22) 1,
BEC(ǫ), Φ(x,ǫ) = ǫx. We define the erasure probability
message sent from bit nodes along edges of type j at the t-th 1Wesayx(cid:22)y ifxi≤yi forall1≤i≤d
3. f(g(0);ǫ)=0 and F(g(0);ǫ)=0. III. PROOF OF ACHIEVING SIR
We say x is a fixed point if x=f(g(x);ǫ).
In this section, we will prove that (d ,d = 2,d = 2)
l r g
Definition 2 ([18, Def. 2]). We define the potential function and (dl,dr = 3,dg = 3) SC-MN codes, for any dl > dr,
U(x;ǫ) of a vector admissible system (f,g) by achieve the SIR of any GEC. First, we calculate the potential
threshold ǫ∗(d ,d ,d ) of MN codes, and show that the
U(x;ǫ),g(x)DxT −G(x)−F(g(x);ǫ). l r g
potential threshold equals to the SIR limit ǫSIR(d /d ). Then
r r
Definition 3 ([18, Def. 7]). Let F(ǫ),{x∈X \{0}|x= we apply Theorem 1 which proves that density evolution of
f(g(x);ǫ)} be a set of non-zero fixed points for ǫ ∈ [0,1]. SC-MN code has a unique fixed point 0 for ǫ smaller than
The potential threshold ǫ∗ is defined by SIR limit ǫSIR(d /d ).
r r
ǫ∗ ,sup{ǫ∈[0,1]|minx∈F(ǫ)U(x;ǫ)>0}. A. Potential Function at Trivial Fixed Point
L6]e.tFǫo∗srbǫestuhcrhesthhoaltdǫo∗sf<unǫc<ouǫp∗l,edwseydsteefimnedeenfienregdyigna[p18∆,ED(eǫf). inRDeceafilnlittihoen d3e.finWiteionneoefdptootenintivaelsttihgraetsehotlhde ǫs∗t(rdulc,tdurre,dgo)f
as F(ǫ) to calculate the potential threshold ǫ∗(d ,d ,d ). The
∆E(ǫ), max inf U(x;ǫ′). density evolution (1) of (d ,d ,d ) MN codeslatrfixegd point
ǫ′∈[ǫ,1]x∈F(ǫ′) l r g
(x ,x ;φ(x;ǫ)) can be rewritten as
Definition 4 ([18, Def. 9]). For a vector admissible system 1 2
(f,g), we define the SC system of chain length L and x =(1−(1−x )dr−1(1−x )dg)dl−1, (6)
1 1 2
coupling width w as x =φ({1−(1−x )dr(1−x )dg−1}dg;ǫ)
2 1 2
w−1 w−1
x(it+1) = w1 f w1 g(x(i+t)j−k);ǫi−k , ·{1−(1−x1)dr(1−x2)dg−1}dg−1. (7)
!
kX=0 Xj=0 First, observe that (x1 = 1,x2 = φ(1;ǫ)) ∈ F(ǫ) for ǫ ∈
ǫ, i∈{0,...,L−1}, [0,1]. We call these fixed points trivial. From (II-F) and the
ǫ =
i (0, i∈/ {0,...,L−1}. definitionofSIRlimit, thenextlemmaasserts thatthesign of
U 1,φ(1;ǫ);ǫ changes at the SIR limit ǫSIR dr .
If we define (f,g) as the density evolution for (dl,dr,dg) dl
MNcodesin(II-D)and(II-D),theSCsystemgivesthedensity Le(cid:0)mma 1. (cid:1) (cid:0) (cid:1)
evolution of SC-MN codes with chain length L and coupling d
r
width w. U 1,φ(1;ǫ);ǫ =1− −Φ(1;ǫ),
d
Next theorem asserts that if ǫ<ǫ∗ then fixed points of SC l
vector system converge towards 0 for sufficiently large w. (cid:0) (cid:1) >0 if ǫ<ǫSIR ddrl ,
Theorem 1 ([18, Thm. 1]). Consider the constant K U 1,φ(1;ǫ);ǫ =0 if ǫ=ǫSIR(cid:0)ddrl(cid:1), (8)
f,g <0 if ǫ>ǫSIR dr .
defined in [18, Lem. 11]. This constant value depends only (cid:0) (cid:1) (cid:0)dl(cid:1)
on (f,g). If ǫ < ǫ∗ and w > (dKf,g)/(2∆E(ǫ)), then the Proof:Thefirstpartisstraightforwardfrom(cid:0)(II-F(cid:1)).Thesecond
SC system of (f,g) with chain length L and coupling width part is obvious from the definition of SIR limit.
w has a unique fixed point 0.
B. Potential Function at Non-Trivial Fixed Point
Itcanbeseenthatthedensityevolution(f,g)of(d ,d ,d )
l r g
Next, for given x ∈ (0,1), solve (III-A) in terms of x .
MN codes over GEC(φ(x;ǫ)) is a vector admissible system 1 2
bychoosingF x;ǫ ,G(x) andD asbelow,sincethissystem Denote this by x2[x1].
(f,gF)(sxa;tiǫs)fi=es(cid:0)dtdhrexc(cid:1)d1oln+diΦti(oxnd2gin;ǫD),efinition 1. x2[x1]=1−(cid:18)(11−−xx11d)ld1−r1−1(cid:19)d1g.
l
G(x)=drx1+dgx2+(1−x1)dr(1−x2)dg −1, Note that for some x1, x2[x1] may fall outside [0,1] as one
can see from Fig. 2. Such pointsare excludedfrom the set of
d 0
D = r . fixed points. Then it follows that
0 d
g
(cid:18) (cid:19)
From Definition 2, the potential function U(x ,x ,ǫ) of (x ,x [x ])∈F(ǫ[x ]) iff (x [x ],ǫ[x ])∈[0,1]2, (9)
1 2 1 2 1 1 2 1 1
(d ,d ,d ) MN codes is given by
l r g
whereǫ[x ] istheuniquesolutionoftheequation(III-A)with
1
U(x1,x2;ǫ)) (5) x2 =x2[x1] holds. In precise, the equation is as follows.
=1−Φ({1−(1−x1)dr(1−x2)dg−1}dg;ǫ) x [x ]=φ({1−(1−x )dr(1−x [x ])dg−1}dg;ǫ[x ])
2 1 1 2 1 1
d
− dr{1−(1−x1)dr−1(1−x2)dg}dl ·{1−(1−x1)dr(1−x2[x1])dg−1}dg−1.
l
−(1−x )dr(1−x )dg · 1+ drx1 + dgx2 . Uniqueness is due to the assumption that φ(x;ǫ) is strictly
1 2
1−x1 1−x2 increasing in ǫ. We call such fixed points (III-B) non-trivial.
(cid:16) (cid:17)
0.20
BEC BEC
DEC 0.5 DEC
PR2 PR2
0.15
0.4
0.3
0.10
0.2
U(x0.0,5x [x ];ǫ[x ]) U(x ,x [x ];ǫ[x ])
1 2 1 1 1 2 1 1
0.1
0.00 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
x x
1 1
Fig.1. Potential functions U(x1,x2[x1];ǫ[x1])atnon-trivial fixedpointsof(4,2,2)MNcodesand(6,3,3)MNcodesfor3typesofgeneralized erasure
channels BEC(ǫ[x1]),DEC(ǫ[x1]),PR2(ǫ[x1])andPR3(ǫ[x1]).
For example of BEC(ǫ), ǫ [x ] can be written in an From[9, Lemma4]and [10, Lemma1, Lemma2],we have
BEC 1
explicit form. the following lemma.
ǫ [x ]:= x2[x1] .(10) Lemma 2. Consider transmissions over BEC(ǫ), i.e., we
BEC 1 {1−(1−x1)dr(1−x2[x1])dg−1}dg−1 have φ(x) = φBEC(x) = ǫ and φ[x1] = ǫ[x1]. Let
Obviously, trivial and non-trivial fixed-points cover the set U (x ,x [x ];ǫ [x ]) be the potential function of
BEC 1 2 1 BEC 1
F(ǫ), in precise, F(ǫ)=F (ǫ)∪F (ǫ). We denote the set of (d ,d ,d ) MN codes at the non-trivial fixed points. For any
t n l r g
trivial and non-trivial fixed-points respectively by F (ǫ) and d >d it holds that
t l r
F (ǫ).
n U (x ,x [x ];ǫ [x ])>0 for x ∈(0,1)
BEC 1 2 1 BEC 1 1
F (ǫ):= 1,φ(1;ǫ)
t for (d ,d )=(2,2) and (3,3).
r g
F (ǫ):= (x ,x [x ])|ǫ[x ]=ǫ,x ∈(0,1)
n (cid:8)(cid:0) 1 2 1(cid:1)(cid:9) 1 1
Lemma 3. For any GEC(φ(x;ǫ)), let U(x ,x [x ];ǫ[x ])
F (ǫ)∩F (ǫ) is an empty set. 1 2 1 1
t (cid:8)n (cid:9) be the potential function at the non-trivial fixed point
Let φ[x1] be the unique solution of (III-A) in terms of φ(·) (x1,x2[x1]) ∈ F(ǫ[x1]) for x1 ∈ (0,1) as defined in (III-B).
for given x1,x2 =x2[x1] and ǫ=ǫ[x1]. Then, it holds that
x [x ]
φ[x1]:= 1−(1−x1)dr(12−1x2[x1])dg−1 dg−1. (11) ProofU: (Fxr1o,mx2[(xI1II]-;Bǫ[)x,1]w)e≥UhBavEeC(x(x1,[xx2[]x,1ǫ][;xǫB])EC[∈x1])[.0,1]2.
2 1 1
Equivalently, we(cid:0)have (cid:1) From (III-B), we have ψ[x1] ∈ [0,1]. From this and using
φ[x1]=φ(ψ[x1];ǫ[x1]), (12) (III-B) and (III-B), we obtain ǫBEC[x1] ∈ [0,1]. This jus-
tifies that U (x ,x [x ];ǫ [x ]) is well-defined since
BEC 1 2 1 BEC 1
where
x ,x [x ],ǫ [x ]∈[0,1]. We have
1 2 1 BEC 1
ψ[x ]:= 1−(1−x )dr(1−x [x ])dg−1 dg. (13)
1 1 2 1 U(x ,x [x ];ǫ[x ])−U (x ,x [x ];ǫ [x ])
1 2 1 1 BEC 1 2 1 BEC 1
Notethatx2[x1],(cid:0)φ[x1]andψ[x1]donotdependon(cid:1)thechannel (=a)ǫ [x ]·(1−(1−x )dr(1−x [x ])dg−1)dg
GEC(φ(·;ǫ)),whileǫ[x ]does.From(III-B)and(III-B),itcan BEC 1 1 2 1
1
be seen that −Φ (1−(1−x )dr(1−x [x ])dg−1)dg;ǫ[x ]
1 2 1 1
ǫBEC[x1]=φ[x1]. (14) (=b)φ[x1](cid:0)·ψ[x1]−Φ ψ[x1];ǫ[x1] (cid:1)
Substituting x1,x2[x1] and φ[x1] into (II-F), at non-trivial (cid:0)ψ[x1] ′ (cid:1) ′
fixed points (x ,x [x ]) ∈ F(ǫ[x ]), we have the potential =φ[x1]·ψ[x1]− φ(x;ǫ[x1])dx
1 2 1 1
function as Z0
(c) ψ[x1] ′ ′
= φ[x ]−φ(x;ǫ[x ])dx
U(x ,x [x ];ǫ[x ]) (15) 1 1
1 2 1 1
Z0
=1−Φ(ψ[x1];ǫ[x1]) (d) ψ[x1] ′ ′
= φ(ψ[x ];ǫ[x ])−φ(x;ǫ[x ])dx
d 1 1 1
− r{1−(1−x1)dr−1(1−x2[x1])dg}dl Z0
d
l (e)
d x d x [x ] ≥0.
−(1−x )dr(1−x [x ])dg · 1+ r 1 + g 2 1 .
1 2 1
1−x1 1−x2[x1] The equality (a) is due to (III-B) and (1). The equality (b)
(cid:16) (cid:17)
is due to (III-B). In (c), we used the fact that ψ[x ] does
1
not depend on channel GEC(φ(·;ǫ)). The equality (d) is due IV. CONCLUSION AND FUTURE WORK
to (III-B). In (e), we used the fact that φ(ψ[x ];ǫ[x ]) ≥
1 1 We have shown that some SC-MN codes achieve the SIR
φ(x′;ǫ[x ])forx′ ∈ 0,ψ[x ] sinceφ(x;ǫ)isnon-decreasing
1 1 limit of any GEC(φ(x;ǫ)) via potential function. The future
in x. The equality is attained with φ(x;ǫ) = φ (x;ǫ) =
(cid:2) (cid:3) BEC works include an extension erasure multi-acess channels [19]
ǫ.
and to more general channels, e.g., PR2 channels with Gaus-
In Fig. 2, we plotted the potential functions at non-trivial
sian noise.
fixedpointsof(4,2,2)MNcodesand(6,3,3)MNcodesfor3
typesofgeneralizederasurechannelsBEC(ǫ[x1]),DEC(ǫ[x1]) REFERENCES
and PR2(ǫ[x ]). One can see that the potential function for
1
[1] A. J. Felstro¨m and K. S. Zigangirov, “Time-varying periodic convo-
BECislowerthanothercurvesforanyx ∈(0,1)asclaimed
1 lutional codes with low-density parity-check matrix,” IEEETrans. Inf.
in Lemma 3.
Theory,vol.45,no.6,pp.2181–2191, June1999.
[2] M.Lentmaier,D.V.Truhachev,andK.S.Zigangirov,“Tothetheoryof
low-density convolutional codes. II,” Probl. Inf. Transm. , no. 4, pp.
C. Potential Threshold Equals to SIR Limit
288–306,2001.
[3] S. Kudekar, T. Richardson, and R. Urbanke, “Threshold saturation via
Next theorem shows the potential threshold of some MN
spatial coupling: Whyconvolutional LDPCensembles perform sowell
codes is equal to the SIR limit. over the BEC,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 803–834,
Feb.2011.
Theorem 2. For any GEC(φ(x;ǫ)), the potentialthreshold ǫ∗ [4] ——, “Spatially coupled ensembles universally achieve capacity under
of(d ,d ,d )MNcodesisequaltoǫSIR(dr)foranyd >d , beliefpropagation,”IEEETrans.Inf.Theory,vol.59,no.12,pp.7761–
l r g dl l r 7813,2013.
(d ,d )=(2,2) and (3,3).
r g [5] K.Takeuchi, T.Tanaka, andT.Kawabata, “A phenomenological study
on threshold improvement via spatial coupling.” IEICE Transactions,
Proof: From Lemma 2 and Lemma 3, we have that for any
vol.95-A,no.5,pp.974–977, 2012.
x1 ∈(0,1) such that ǫ=ǫ[x1]∈ 0,ǫSIR(ddrl) , [6] tAh.reYsheodllda,sYat.-uYra.tiJoinan,foPr.cNoguupyleedn,vaenctdorHr.ecPufirsstieorn,s,“”AinsimPprolec.p7rothofInotf.
(cid:2) (cid:1) Symp.onTurboCodes andRelated Topics,Sept.2012,pp.51–55.
U(x ,x [x ];ǫ[x ])>0. (16)
1 2 1 1 [7] ——, “A simple proof of maxwell saturation forcoupled scalar recur-
sions,” IEEE Trans. Inf. Theory, vol. 60, no. 11, pp. 6943–6965, Nov
It follows that 2014.
[8] K.KasaiandK.Sakaniwa,“Spatially-coupled MacKay-Nealcodesand
∗ Hsu-Anastasopoulos codes,” IEICE Trans. Fundamentals, vol. E94-A,
ǫ =sup{ǫ∈[0,1]| min U(x ,x ;ǫ)>0}
(x1,x2)∈F(ǫ) 1 2 no.11,pp.2161–2168,Nov.2011.
[9] N. Obata, Y.-Y. Jian, K. Kasai, and H. D. Pfister, “Spatially-coupled
=sup{ǫ∈[0,1]| min U(x ,x ;ǫ)>0}
1 2 multi-edgetypeLDPCcodeswithboundeddegreesthatachievecapacity
(x1,x2)∈Ft(ǫ)∪Fn(ǫ) on the BEC under BP decoding,” in Proc. 2013 IEEE Int. Symp. Inf.
(=a)sup{ǫ∈ 0,ǫSIR(dr) | min U(x ,x ;ǫ)>0} Theory(ISIT),July2013,pp.2433–2437.
dl (x1,x2)∈Ft(ǫ)∪Fn(ǫ) 1 2 [10] Tno.ObiktanzoadkeisanodfdKeg.rKeeastawi,o“aScphaiteivaellyth-ceocuappleadcitMyaocfKBaEyC-N,”eianlPcorodce.s2w0i1t4h
(cid:2) (cid:1)
IEEEInt.Symp.Inf.Theory(ISIT),June2014,pp.506–510.
(=b)ǫSIR ddrl . [11] Afa.ctPo.rWgorartphhesn,”anIEdEWE.TSrtaarnks,.“IUnfn.ifiTehdeodreys,igvnolo.f4i7te,rnatoiv.e2,repcpe.iv8e4rs3–u8si4n9g,
Feb2001.
In (a), w(cid:0)e u(cid:1)sed (1). In (b), we used (1) and (III-C). [12] B.Kurkoski,P.Siegel,andJ.Wolf,“Jointmessage-passingdecodingof
Define the BP threshold ǫBP(d ,d ,dg,L,w) as the spre- LDPC codes and partial-response channels,” IEEE Trans. Inf. Theory,
l r vol.48,no.6,pp.1410–1422, June2002.
mum value of ǫ such that the SC system with chain length L [13] S. Kudekar and K. Kasai, “Threshold saturation on channels with
and coupling width w of (d ,d ,d ) MN codes over GEC(ǫ) memoryviaspatialcoupling,”inProc.2011IEEEInt.Symp.Inf.Theory
l r g
converges to zero. In precise, ǫBP(d ,d ,dg,L,w) is (ISIT),Aug.2011,pp.2568–2572.
l r [14] S. Sekido, K. Kasai, and K. Sakaniwa, “Threshold saturation on PR2
erasurechannelviaspatialcoupling,”inProc.Symp.onInf.Theoryand
sup{ǫ∈[0,1]| lim x(t) =0 for i=0,1,...,L−1}. itsApplications (SITA2011),Dec.2011.
t→∞ i [15] H. Pfister and P. Siegel, “Joint iterative decoding of ldpc codes for
channelswithmemoryanderasurenoise,”IEEEJ.Sel.Area.Commun.,
Then,from(II-E)andTheorem3and1wehavethefollowing
vol.26,no.2,pp.320–337, February2008.
theorem. [16] A. Ashikhmin, G. Kramer, and S. ten Brink, “Extrinsic information
transfer functions: modelanderasure channel properties,” IEEETrans.
Theorem 3. For any GEC(φ(x;ǫ)), the potentialthreshold ǫ∗ Inf.Theory,vol.50,no.11,pp.2657–2673, Nov2004.
of(d ,d ,d )MNcodesisequaltoǫSIR(dr)foranyd >d , [17] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge
l r g dl l r University Press,Mar.2008.
(d ,d )=(2,2) and (3,3).
r g [18] A. Yedla, Y.-Y. Jian, P. Nguyen, and H. Pfister, “A simple proof of
thresholdsaturationforcoupledvectorrecursions,”inProc.2012IEEE
lim lim ǫBP(d ,d ,dg,L,w)=ǫSIR(d /d), Information ThoeryWorkshop(ITW),Sept2012,pp.25–29.
l r r l
w→∞L→∞ [19] S. Kudekar and K. Kasai, “Spatially coupled codes over the multiple
d accesschannel,”inProc.2011IEEEInt.Symp.Inf.Theory(ISIT),Aug.
lim lim RMN(dl,dr,dg,L,w)= r. 2011,pp.2817–2821.
w→∞L→∞ dl
In words, some SC-MN codes universally achieve the SIR APPENDIX
limit of any GEC(φ(x;ǫ)) in the limit of large L and w. In Appendix, we give three GECs as example.
1.00 1.0
BEC BEC
DEC DEC
PR2 PR2
0.80 0.8
0.60 0.6
ǫ[x ] ǫ[x ]
1 1
0.40 0.4
0.20 0.2
0.00 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x x
1 1
Fig. 2. Plot of ǫ[x1] at non-trivial fixed points of (4,2,2) MN codes and (6,3,3) MN codes for 3 types of generalized erasure channels BEC(ǫ[x1]),
DEC(ǫ[x1]),PR2(ǫ[x1])andPR3(ǫ[x1]).
Example 1 (Binary Erasure Channel). For the bi- given as
nary erasure channel BEC(ǫ) with erasure probability ǫ,
4ǫ2
φ(x;ǫ),Φ(x;ǫ),I(ǫ) and ǫ[x1] are respectively given as φDEC(x;ǫ)= (2−x(1−ǫ))2
4ǫ2
φ (x;ǫ)=ǫ,
BEC Φ (x;ǫ)= ,
DEC
(1−ǫ)(2−(1−ǫ)x)
Φ (x;ǫ)=ǫx,
BEC
2ǫ2
I (ǫ)=1−ǫ, I (ǫ)=1− ,
BEC DEC
1+ǫ
ǫ[x ]=ǫ [x ],
1 BEC 1 ǫ[x ]=ǫ [x ],
1 DEC 1
where ǫBEC[x1] is the unique solution ǫBEC of the following where ǫDEC[x1] is the unique solution ǫDEC of the following
equation. equation.
x [x ]=φ ({1−(1−x )dr(1−x [x ])dg−1}dg;ǫ [x ]) x2[x1]=φDEC({1−(1−x1)dr(1−x2[x1])dg−1}dg;ǫDEC)
2 1 BEC 1 2 1 BEC 1
·{1−(1−x )dr(1−x [x ])dg−1}dg−1. ·{1−(1−x1)dr(1−x2[x1])dg−1}dg−1.
1 2 1
In the case of DEC(ǫ), ǫ [x ] can be written in an explicit
DEC 1
ǫ [x ] can be written in an explicit form (III-B). form.
BEC 1
The potential function is given as
1/2
(2−ψ[x ]) x2[x1]
ǫ [x ]:= 1 ψ[x1](dg−1)/dg
U (x ,x [x ];ǫ[x ]) (17) DEC 1 (cid:16) (cid:17)1/2
BEC 1 2 1 1 2−ψ[x ] x2[x1]
:=1−ǫBEC[x1]·({1−(1−x1)dr(1−x2[x1])dg−1}dg) 1 ψ[x1](dg−1)/dg
(cid:16) (cid:17)
− dr{1−(1−x )dr−1(1−x [x ])dg}dl In general, we do not need the explicit form of ǫ[x1] for
d 1 2 1 the purpose of this paper. The potential function is given as
l
−(1−x )dr(1−x [x ])dg follows.
1 2 1
d x d x [x ]
· 1+ r 1 + g 2 1 . UDEC(x1,x2[x1];ǫ[x1])
1−x 1−x [x ]
(cid:16) 1 2 1 (cid:17) :=1−ΦDEC({1−(1−x1)dr(1−x2[x1])dg−1}dg;ǫDEC[x1])
Example 2 (DicodeErasureChannel). Forthedicodeerasure
d
channel DEC(ǫ) with erasure probability ǫ, the output y ∈ − r{1−(1−x1)dr−1(1−x2[x1])dg}dl
{0,1,−1,?}n for given x∈{0,1}n is defined as follows. dl
−(1−x )dr(1−x [x ])dg
1 2 1
d x d x [x ]
xj −xj−1 w.p. 1−ǫ, · 1+ r 1 + g 2 1 .
y =
j 1−x 1−x [x ]
(? w.p. ǫ, (cid:16) 1 2 1 (cid:17)
Example 3 (Partial Response 2 Channel). For the PR2
wherex :=0.φ(x;ǫ),Φ(x;ǫ),I(ǫ)andǫ[x ]arerespectively channel PR2(ǫ) with erasure probability ǫ, the output y ∈
0 1
{0,1,2,3,4,?}n for given x∈{0,1}n is defined as follows.
xj +2xj−1+xj−2 w.p. 1−ǫ,
y =
j
(? w.p. ǫ,
wherex :=0.φ(x;ǫ),Φ(x;ǫ),I(ǫ)andǫ[x ]arerespectively
0 1
given as
4ǫ3 4−4(1−ǫ)x+(1−ǫ)x2
φ (x;ǫ)=
PR2 (4−2(1−ǫ2)x−(1−ǫ)ǫ2x2)2
(cid:0) (cid:1)
1 2−x
Φ (x;ǫ)=4ǫ − ,
PR2 2 4−2(1−ǫ2)x−(1−ǫ)ǫ2x2
(cid:16) (cid:17)
2ǫ3(ǫ+1)
I (ǫ)=1− ,
PR2 ǫ3+ǫ +2
2
ǫ[x ]=ǫ [x ],
1 PR2 1
where ǫ [x ] is the unique solution ǫ of the following
PR2 1 PR2
equation.
x [x ]=φ ({1−(1−x )dr(1−x [x ])dg−1}dg;ǫ )
2 1 PR2 1 2 1 PR2
·{1−(1−x )dr(1−x [x ])dg−1}dg−1.
1 2 1
The potential function is given as follows.
U (x ,x [x ];ǫ[x ])
PR2 1 2 1 1
:=1−Φ ({1−(1−x )dr(1−x [x ])dg−1}dg;ǫ [x ])
PR2 1 2 1 PR2 1
d
− r{1−(1−x )dr−1(1−x [x ])dg}dl
1 2 1
d
l
−(1−x )dr(1−x [x ])dg
1 2 1
d x d x [x ]
r 1 g 2 1
· 1+ + .
1−x 1−x [x ]
1 2 1
(cid:16) (cid:17)