EXIT Chart Analysis of Block Markov Superposition Transmission of Short Codes

0.15 MB·English

by  Kechao Huang

EXIT Chart Analysis of Block Markov
Superposition Transmission of Short Codes
Kechao Huang and Xiao Ma Daniel J. Costello, Jr.
Dept. of ECE, Sun Yat-sen University Dept. of EE, University of Notre Dame
Guangzhou 510006, GD, China Notre Dame 46556, Indiana, USA
Email: [email protected] Email: [email protected]
5
1 Abstract—To be considered for a 2015 IEEE Jack Keil Wolf memorybetweentransmissions.Hence,BMSTcodestypically
0 ISIT Student Paper Award. In this paper, a modified extrin- have very simple encoding algorithms. To decode BMST
2 sic information transfer (EXIT) chart analysis that takes into codes, a sliding window decoding algorithm with a tunable
account the relation between mutual information (MI) and bit-
decoding delay can be used, as with SC-LDPC codes. The
n error-rate (BER) ispresentedto studytheconvergence behavior
a of block Markov superposition transmission (BMST) of short constructionofBMSTcodesisflexible[17],inthesensethatit
J codes (referred to as basic codes). We show that the threshold appliestoallcoderatesofinterestintheinterval(0,1).Further,

1 curveofBMSTcodesusinganiterativeslidingwindowdecoding BMST codes have near-capacity performance (observed by
3 algorithmwithafixeddecodingdelayachievesalowerboundin simulation) in the waterfall region of the bit-error-rate (BER)
] rthegeihonig,hdsuiegntaol-etror-onroipserorpaatigoat(iSoNn,Rt)hreegthiornes,hwohldilseoinf BthMeSloTwcSoNdeRs cruve and an error floor (predicted by analysis) that can be
T become slightly worse as the encoding memory increases. We controlled by the encoding memory.
I also demonstrate that the threshold results are consistent with On an additive white Gaussian noise channel (AWGNC),
. finite-length performance simulations. the well-known extrinsic information transfer (EXIT) chart
s
c analysis [18] can be used to obtain the threshold of LDPC-
[ I. INTRODUCTION
BC ensembles. In [19], a novel EXIT chart analysis was

1 Spatially coupled low-density parity-check (SC-LDPC) used to evaluate the performanceof protograph-basedLDPC-
v codes are constructed by coupling together a series of BC ensembles, and a similar analysis was used to find the
9 L disjoint Tanner graphs of an underlying LDPC block thresholds of q-ary SC-LDPC codes with sliding window
7 code (LDPC-BC) into a single coupled chain and can be decoding in [9]. Unlike LDPC codes, the asymptotic BER
0 viewed as a type of LDPC convolutional code [1]. It was of BMST codes with window decoding cannot be better
0 shown in [2,3] that the belief propagation (BP) decoding than a corresponding genie-aided lower bound [11]. Thus,
0
thresholds of SC-LDPC code ensembles are numerically in- conventional EXIT chart analysis cannot be applied directly
.
2 distinguishable from the maximum a posteriori (MAP) de-
to BMST codes. In this paper, we propose a modified EXIT
0 coding thresholds of their underlying LDPC-BC ensembles. chart analysis, that takes into account the relation between
5
Subsequently, it was proven analytically that SC-LDPC code mutual information (MI) and BER, to study the convergence
1
ensemblesexhibitthresholdsaturationonmemorylessbinary-
: behavior of BMST codes and to predict the performance in
v input symmetric-outputchannels under BP decoding[4]. Due the waterfall region of the BER curve. We also show that the
Xi to theirexcellentperformance,SC-LDPC codeshavereceived modifiedEXITchartanalysisofBMST codesis supportedby
a great deal of attention recently (see, e.g., [5–10] and the
finite-length performance simulations.
r
a references therein).
The concept of spatial coupling is not limited to LDPC II. SC-LDPC CODES VS.BMST CODES
codes. Block Markov superposition transmission (BMST) of
short codes [11,12], for example, is equivalent to spatial In this section, both SC-LDPC codes and BMST codes are
coupling of the subgraphs that specify the generator matrices describedintermsofmatricesforthepurposeofshowingtheir
of the short codes. From this perspective, BMST codes are similarities (dualities) and differences.
similar to braided block codes [13], staircase codes [14], and
spatially coupled turbo codes [15]. A BMST code can also A. Protograph-Based SC-LDPC Codes
be viewed as a serially concatenated code with a structure A protograph-based SC-LDPC code ensemble can be con-
similar to repeat-accumulate-like codes [16]. The outer code structed from a protograph-based LDPC-BC code ensemble
is a short code, referred to as the basic code (not limited to using the edge spreading technique [3], described here in
repetition codes), that introduces redundancy, while the inner termsof the base(parity-check)matrix representationof code
code is a rate-one block-oriented feedforward convolutional ensembles.LetB bea(N K) N basematrixrepresenting
code (instead of a bit-oriented accumulator) that introduces an LDPC-BC ensemble w−ith d×esign rate R = K/N. A
terminatedSC(convolutional)basematrixBSC withcoupling
This work was partially supported by the 973 Program (No.
width (syndrome former memory) m and coupling length L
2012CB316100), the China NSF (No. 61172082 and No. 91438101), and
theU.S.NSF(No.CCF-1161754). can be constructed by applying the edge spreading technique
to B, resulting in which has constantweight m+1in each row. Now assuming
B that we want to construct a rate R = k/n code, we select a
B0 B basiccodewithak n generatormatrixG.LetΠi (0 i
1 0 × ≤ ≤
 ... B1 ...  Tmh)enbeeamch+n1onrzaenrdoomenltyryseAleic,jteidnnA×isnrpeperlamcuetdatwiointhmaatkricesn.
BSC = Bm ... ... B0 , (1) kmatrinx GallΠ-zejr−oi manadtreixa,chrezseurlotinegntriny ianBAMiSsTrepcloadceedowf iltehn×gththe
 Bm ...... B...1  (GL×B+MSmT)nofatnhde dBiMmeSnTsicoondLekis. gTihveenrebsyulting generator matrix
 B  GBMST =
where the m+ 1component submatricems B0,B1,...,Bm, GΠ0 GGΠΠ10 G·Π·· 1 G·Π·· m GΠm
each of size (N K) N, satisfy m B = B. The graph  ... ... ... ... .
i
lifting operation−is then×applied to iBP=0SC by replacing each  GΠ0 GGΠΠ01 G·Π·· 1 G·Π·· m GΠm
mnountzaetiroonemntartyriixn1BanSdCeawcihthzearoraenndtorymilnyBseSleCctwedithMth×eMM×peMr- The rate of the BMST coLdekis L 
all-zero matrix,resulting in a terminatedSC-LDPC code with RBMST = = R, (4)
(L+m)n L+m
constraint length v = (m+1)MN, where M (typically a
s
large integer) is the lifting factor. The resulting SC-LDPC whichisslightlylessthantherateR=k/nofthebasiccode.
parity-checkmatrixHSC ofsize(L+m)(N K)M LNM However, similar to SC-LDPC codes, this rate loss becomes
− ×
is given by vanishingly small as L .
→∞
HSC = Thoughany code (linear or nonlinear)with a fast encoding
algorithm and an efficient soft-in soft-out (SISO) decoding
H0(0) algorithmcanbetakenasthebasiccode,inthispaperwefocus
 H1...(1) HH01((12)) ...  ocondtehe(RuCs)eoorfathseinMgle-fpoaldritCy-acrhteescikan(SpProCd)uccotdoefaasrtehpeetbiatisoinc
Hm(m) Hm(m... +1) ...... HH0(L1(L−)1) , crbceooasdsdpieece,coctrrioevdsaeuenllyti.iSsnLPgteChteinnGcoga0dievbB.eeMnTthhSbeeTy-KkRC××NncogdgeeennoeerrraaattoorBrmMmaaSttrTrii-xxSPGoCfaocnfoRtdheCe,
 .. .. 
 . .  G=diag G0, ,G0 , (5)
 H (L+m 1) { ··· }
 m − 
where the blank spaces in HSC correspond to zeros and the wherediag G0, ,G0 isablocMkdiagonalmatrixwithG0
submatrices H (t) have size (N K)M NM, i,t. The { ··· } | {z }
i on the diagonal, n=NM, and k=KM.
− × ∀
designrateoftheterminatedSC-LDPCcodeensembleisgiven
by C. Similarities and Differences
(L+m)(N K) L+m
From the previous two subsections, we see that both SC-
RSC =1 − =1 (1 R), (2)
− LN − L − LDPC codes and BMST codes can be derived from a small
which is slightly less than the design rate R = K/N of matrix by replacing the entries with properly-defined sub-
the uncoupled LDPC-BC ensemble due to the termination. matrices. We also see that the generator matrix GBMST of
However,thisratelossbecomesvanishinglysmallasL . BMST codes is similar in form to the parity-check matrix
→∞ HSC ofSC-LDPC codes.SC-LDPC codesintroducememory
B. BMST Codes by spatially coupling the parity-check matrices of the under-
lying LDPC-BCs, while BMST codes introduce memory by
In contrast to SC-LDPC codes, it is convenientto describe
spatially coupling the generator matrices of the basic code.
BMST codes using generator matrices. To describe a BMST
Thus, BMST codes can be viewed as a type of spatially
codeensemblewithcouplingwidth(encodingmemory)mand
coupled code. Similar to SC-LDPC codes, where increasing
coupling length L, we start with an L (L+m) matrix
× the lifting factor M improves waterfall region performance,
1 1 1 increasingtheCartesianproductorderM ofBMSTcodesalso
···
1 1 1 improves waterfall region performance. But the error floors,
···
A= ... ... ... ... , (3) which are solely determinedby the encodingmemory m (see
 1 1 1  Section III-A), cannot be lowered by increasing M.
 
 1 ·1·· 1
 ···  III. PERFORMANCE ANALYSIS OF BMST CODES
 
1If the nonzero entry Bi,j > 1, it is replaced by a summation of Bi,j Throughout the paper, we consider binary phase-shift key-
nonoverlapping randomlyselected permutation matrices ofsizeM×M. ing(BPSK)modulationoverthebinary-inputAWGNC.Inthis
MI between the channel observation and the corresponding
codeword bit is referred to as the channel MI. The analysis
assumes that the interleavers Π (0 i m) are arbitrarily
i
≤ ≤
large and random.
BMSTcodeensemblescanberepresentedbyaForney-style
factorgraph,also knownas anormalgraph[20],whereedges
represent variables and vertices (nodes) represent constraints.
All edges connected to a node must satisfy the specific
constraint of the node. A full-edge connects to two nodes,
t L L
while a half-edge connects to only one node. A half-edge is
Fig.1. Exampleofawindowdecoderwithdecodingdelayd=2operating also connected to a special symbol, called a “dongle”, that
on the normal graph of a BMST code ensemble with m = 2 at times t = denotescouplingtootherpartsofthetransmissionsystem(say,
0(solidblue),andt=1(dottedred).Foreachwindowposition/timeinstant, the channel or the information source) [20]. There are three
thefirstdecoding layeriscalled thetargetlayer.
types of nodes in the normal graph of BMST codes.2
section, we first discuss the problem that prevents the use
• Node + : All edges (variables) connected to node +
of conventional EXIT chart analysis for BMST codes, and
mustsumtozero.Themessageupdatingruleatnode +
then we provide a modified EXIT chart analysis to study the
is similar to that of the check node in the factor graph
convergencebehaviorofBMSTcodeswithwindowdecoding.
of a binary LDPC code. The only difference is that the
messagesonthehalf-edgesareobtainedfromthechannel
A. Genie-Aided Lower Bound on BER
observations.
Let pb = fBMST(γb) be the BER performance function • Node = : All edges (variables) connected to node =
corresponding to a BMST code with encoding memory (cou-
musttakethesame(binary)value.Themessageupdating
pling width) m and coupling length L, where p is the BER
b rule at node = is the same as that of the variable node
and γb , Eb/N0 is in dB. Let pb = fBasic(γb) be the BER in the factor graph of a binary LDPC code.
performancefunctionofthe basiccode.By assuminga genie-
• Node C : All edges (variables) connected to node C
aided decoder, we have [11]
must satisfy the constraint specified by the basic code.
m
fBMST(γb) fBasic γb+10log10(m+1) 10log10 1+ , The message updating rule at node C can be derived
≥ − L
(cid:16) (cid:16) ((cid:17)6(cid:17)) accordingly, where the messages on the half-edges are
where the term 10log (m+1) depends on the encoding associated with the information source.
10
memorymandtheterm10log10(1+m/L)isduetotherate ThenormalgraphofaBMSTcodeensemblecanbedivided
loss. In other words, a maximum coding gain over the basic into layers, where each layer typically consists of a node of
code of 10log10(m+1) dB in the low BER (high signal-to- type C ,anodeoftype = ,andanodeoftype + .Similarto
noise ratio (SNR)) region is achieved for large L. Intuitively, SC-LDPC codes, an iterative sliding window decoding EXIT
thisboundcanbe understoodbyassumingthata codewordin chart analysis algorithm with decoding delay d working over
thebasiccodeistransmittedm+1timeswithoutinterference. a subgraph consisting of d + 1 consecutive layers can be
implemented to study the convergence behavior of BMST
B. A Modified EXIT Chart Analysis
codes.3Thefirstlayerinanywindowiscalledthetargetlayer.
Todescribedensityevolution,itisconvenienttoassumethe
An example of a window decoder with decoding delay d=2
all-zerocodewordistransmittedandtorepresentthemessages
operatingonthenormalgraphofaBMSTcodeensemblewith
as log-likelihood ratios. The threshold of protograph-based
m=2isshowninFig.1.InourmodifiedEXITchartanalysis,
LDPC codes can be obtained based on a protograph-based
the convergence check at node C is performed as follows.
EXIT chart analysis [9,19] by determining the minimum
value of the SNR Eb/N0 such that the MI between the Algorithm 1: Convergence Check at Node C
a posteriori message at a variable node and an associated • LetIA denotetheaprioriMIandIE denotetheextrinsic
codeword bit (referred to as the a posteriori MI for short) MI. Then the a posteriori MI IAP is given by
goes to 1 as the number of iterations increases, i.e., the BER
at the variable nodestendsto zero as the numberof iterations IAP =J( [J−1(IA)]2+[J−1(IE)]2), (7)
tends to infinity. However, as shown in (6), the high SNR wheretheJ()andpJ−1()functionsaregivenin[21],I
A
performanceofBMSTcodeswithwindowdecodingcannotbe · ·
istheaprioriMI,andI istheextrinsicMI.Supposethat
E
better thanthe correspondinggenie-aidedlowerbound,which
theaposterioriMIisGaussian.AsshowninSectionIII-
means that the a posteriori MI of BMST codes does not tend
C of [18], an estimate of the BER p is then given by
est
to 1 as the number of iterations tends to infinity. Thus, the
−1
conventional EXIT chart analysis cannot be applied directly pest =Q J (1 IAP)/2 , (8)
−
to BMST codes.
(cid:0) (cid:1)
For convenience,the MI between the a prioriinputand the 2Formoredetails onthenormal realization ofBMSTcodes, werefer the
readerto[11,12].
corresponding codeword bit is referred to as the a priori MI,
3As with SC-LDPC codes, the decoding delay d must be chosen several
the MI between the extrinsic output and the corresponding
times as large as the encoding memory m in order to achieve good perfor-
codeword bit is referred to as the extrinsic MI, and the mance.
0.8
where Shannonlimit
BMST-RCN=2m=10d=30
Q(x)= 1 ∞exp t2 dt. (9) 0.7 BBBMMMSSSTTT---RRSPCCCNNN===484mmm===11534ddd===134592
√2π −2
Zx (cid:26) (cid:27) 0.6
• IftheestimatedBERpest islessthanthepreselectedtar-
getBER,alocaldecodingsuccessisdeclared;otherwise, 0.5
a local decoding failure is declared.
e
at0.4
GivenachannelparameterEb/N0, thechannelMI isgiven R
by 0.3
E
b
Ich =J 8RBMST . (10) 0.2
r N0 !
0.1
The modified EXIT chart analysis algorithm of BMST codes
IncreasingL
can now be described as follows.
0
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Algorithm 2: EXIT Chart Analysis of BMST Codes with Thresholdσ∗
ch
Window Decoding
(a)
• Initialization: All messages over those half-edges (con-
0.8
nected to the channel) at nodes + are initialized as I Shannonlimit
ch BMST-RCN=2m=10d=30
according to (10), all messages over those half-edges 0.7 BBMMSSTT--RRCCNN==48mm==1134dd==3492
(connected to the information source) at nodes C are BMST-SPCN=4m=5d=15
initialized as 0, and all messages over the remaining 0.6
(inter-connected) full-edges are initialized as 0. Set a
0.5
maximum number of iterations Imax >0.
• Sliding window decoding: For each window position, ate0.4
R
the d+1 decoding layers perform MI message process-
ing/passing layer-by-layer according to the schedule 0.3 IncreasingL
+ Π = C = Π + . 0.2
→ → → → → →
After a fixed number of iterations Imax, make a con- 0.1
vergence check at node C using Algorithm 1. If a
local decoding failure is declared, then window decod- −01 .5 −1 −0.5 0 0.5 1 1.5 2 2.5 3
ing terminates; otherwise, a local decoding success is Threshold(Eb/N0)∗(dB)
declared, the window position is shifted, and decoding (b)
continues. A complete decoding success for a specific
Fig.2. AWGNCBPthresholds intermsof(a)standard deviation σ∗ and
channel parameter Eb/N0 and target BER is declared if (b)SNR(Eb/N0)∗ (dB)forseveralfamiliesofBMSTcodeensemblecshwith
and only if all target layers declare decoding successes. increasing coupling length L,m≤L≤1000, forapreselected target BER
Now we can denote the iterative decoding threshold of10−7.
∗
(Eb/N0) of BMST code ensembles for a preselected target monotonically with increasing L. However, in both plots, we
BER as the minimum value of the channel parameter Eb/N0 can see that the gap to capacity decreases as L increases.
whichallowsthedecoderofAlgorithm1tooutputadecoding Example 2: For the coupling length L = 1000, we cal-
success, in the limit of large code lengths (i.e., M ). culated BP thresholds for several families of BMST code
→∞
ensembles with different preselected target BERs. The cal-
IV. NUMERICAL RESULTS ∗
In thesimulationsto computethe windowdecodingthresh- culated thresholds in terms of the SNR (Eb/N0) versus the
preselected target BERs together with the lower bounds are
oldsofBMST codes,we set a maximumnumberofiterations
shown in Fig. 3, where we observe that
Imax =1000.
Example 1: In Fig. 2, we display the thresholds of several 1) For a fixed encoding memory m, the thresholds remain
families of BMST code ensembles with increasing coupling constant at a value near capacity. Once a critical target
BERisreached,however,thethresholdsdegraderapidly
length L, m L 1000, for a preselected target BER of
10−7. The dec≤oding≤delay4 is set to d = 3m. Fig. 2(a) plots as the target BER decreases further.
the thresholds in terms of the standard deviation σ∗ of the 2) For a high target BER (roughly above 10−3), the
ch
threshold increases slightly as the encoding memory
noiseagainsttheensemblecoderateRBMST.Weobservethat,
as L increases, the rate also increases while the threshold m increases, due to errors propagating to successive
σ∗ remains constant. The same thresholds are depicted in decoding windows.
Ficgh. 2(b) in terms of the SNR (Eb/N0)∗. Since Eb/N0 takes 3) For a small decodingdelay (say d=m), the thresholds
into account the code rate, the thresholds (Eb/N0)∗ improve do not achieve the lower bounds even in the high SNR
region.
4Thethresholddoesnotimprovefurtherbeyondadecodingdelayd=3m. 4) Foralargerdecodingdelay(sayd=3m),thethresholds
100 10−1 Shannonlimitofrate1/2 ensemble. Using the modified EXIT chart analysis, we can
BMST-RCm=4 d=12 predicttheperformanceofBMSTcodesinthewaterfallregion
10−1 BMST-RCm=8 d=8
BMST-RCm=8 d=24 of the BER curve. Finally, we showed that the EXIT chart
10−2 10−2 BLLooMwwSeeTrr-bbRooCuunnmddff=oorr1mm0d===4830 analysis results are consistent with finite-length performance
Lowerboundform=10 simulations.
M=5000m=4 d=12
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