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LDPC Decoding Over Nonbinary Queue-Bad
Burst Noi Channels
Pedro Melo,Cecilio Pimentel,Senior Member,IEEE,and
Fady Alajaji,Senior Member,IEEE
Abstract—Iterative decoding bad on the sum-product algorithm(SPA) is examined for nding low-density parity check(LDPC)codes over a dis-crete nonbinary queue-bad Markovian burst noi channel.This channel model is adopted due to its analytical tractability and its recently demon-strated capability in accurately reprenting correlatedflat Rayleigh fad-ing channels under antipodal signaling and either hard or soft output quantization.SPA equations are derived in clod form for this model in terms of its parameters.It is then numerically obrved that potentially large coding gains can be realized with respect to the Shannon limit by exploiting channel memory as oppod to ignoring it via interleaving. Finally,the LDPC decoding performance under both matched and mis-matched decoding regimes is evaluated.It is shown that the Markovian model provides noticeable gains over channel interleaving and that it can effectively capture the underlying fading channel behavior when decoding LDPC codes.
Index Terms—Burst noi,channel interleaving,finite-state Markov channels(FSMCs),hard-and soft-decision demodulation,iterative decod-ing,low-density parity check(LDPC)codes,matched and mismatched decoding,modeling correlated Rayleigh fading channels,Shannon limit(SL).
I.I NTRODUCTION
A discrete(binary-input2q-ary output)burst noi channel model was recently introduced in[1],which is called the nonbinary noi discrete channel(NBNDC),to model fading channels with memory and soft-decision information.The channel’s2q-ary output process can be written as an explicit function of the binary input and2q-ary noi process.We refer to NBNDC-Q
B as the NBND
C with the nonbi-nary queue-bad(QB)M th-order Markov noi process with2q+2 independent parameters propod in[1].The NBNDC-QB has a small number of parameters(as,typically,q is not greater than three) and is mathematically tractable,featuring desirable statistical and information-theoretic properties(such as symmetry,Markovian noi structure,and clod-form expressions for its block distribution) unlike the classical burst-noi Gilbert–Elliott channel(GEC)and otherfinite-state Markov channel(FSMC)models in the literature (which typically exhibit a hidden Markovian noi , e[2]–[4]).The NBNDC-QB can effectively model(in terms of replicating channel capacity and noi autocorrelation function)a discrete fading channel(DFC)compod of a binary pha-shift keying modulator,a time-correlatedflat Rayleigh fading channel with Clarke’s autocorrelati
on function,and a q-bit(uniform)scalar soft-Manuscript received July1,2014;revid October17,2014,December8, 2014;accepted December27,2014.Date of publication January21,2015; date of current version January13,2016.This work was supported in part by the Natural Sciences and Engineering Rearch Council of Canada and in part by the Brazilian National Council for Scientific and Technologi-cal Development(CNPq).The review of this paper was coordinated by Dr.A.J.Al-Dweik.
P.Melo and C.Pimentel are with the Department of Electronics and Sys-tems,Federal University of Pernambuco,50711-970Recife-PE,Brazil(e-mail: cecilio@ufpe.br;).
F.Alajaji is with the Department of Mathematics and Statistics,Queen’s University,Kingston,ON K7L3N6,Canada(e-mail:fady@mast.queensu.ca). Color versions of one or more of thefigures in this paper are available online at ieeexplore.ieee.
Digital Object Identifier10.1109/TVT.2015.2395382
0018-9545©2015IEEE.Personal u is permitted,but republication/redistribution requires IEEE permission.
See www.ieee/publications_standards/publications/rights/index.html for more information.
quantized coherent demodulator[1].It also subsumes,as a special ca,(when q=1)the binary-input–binary-output QB channel[5] shown in[6]to accurately reprent(also in terms of capacity and noi autocorrelation function)the DFC under hard-decision demod-ulation.In[7],it is further demonstrated that the NBNDC-QB is a goodfit for the DFC in terms of signal-to-distortionfidelity under both channel-optimized quantization and scalar quantization with quence maximum a posteriori detection.
The aim of this correspondence is to examine the potential theo-retical channel coding gains achievable via the NBNDC-QB model vis-is its corresponding(ideally interleaved)memoryless counterpart as well as examine the NBNDC-QB’s modeling effectiveness of ap-proximating the behavior of correlated fading channels when decoding practical powerful low-density parity check(LDPC)codes.Several authors have studied the design of a belief propagation scheme for joint decoding and channel state estimation of LDPC codes over FSMC’s[8]–[14].The decoder us a factor graph with variable nodes related to the code constraints as well as to the FSMC structure.The works are typically concentrated on a special ca of a binary(binary-input binary-output)FSMC,which,unlike the NBNDC-QB model, do not accommodate nonbinary output alphabets for capturing the soft-decision information of underlying soft-output quantized fading channels.Moreo
ver,in the works,the channel type that corrupts the codeword is the same as that ud in the factor graph at the decoder (which results in a matched decoder).Other related works include the development of efficient iterative detection and decoding methods for coherent and noncoherent fading channels with memory(e[15]and [16]and the references therein).
The contribution of this correspondence is threefold.First,we spe-cialize the sum-product algorithm(SPA)to the NBNDC-QB channel, exploiting its mathematical tractability,to derive clod-form equa-tions for the messages pasd through the factor graph when decoding LDPC codes over this channel.Then,we study the potential coding gain in terms of the Shannon limit(SL)provided by the NBNDC-QB over the standard delay-prone approach of interleaving the channel to spread its error bursts over the t of received codewords and render it memoryless with respect to the decoder.Finally,we study the accuracy of the NBNDC-QB in approximating the DFC from a new perspective (not considered in[1],[8]–[14]).Specifically,we investigate the LDPC performance when the true underlying channel is the DFC while the decoder us an NBNDC-QB channel chon tofit the DFC by minimizing the Kullback–Leibler divergence rate between the channel noi sources as in[1].We show that this mismatch decoding is capable of outperforming the fully-interleaved(memoryless)DFC; this demonstrates that thefinite-memory NBNDC-QB channel with low
number of parameters can be effectively ud to exploit the fading statistical dependence of the DFC.Moreover,this study illustrates the practicality of modeling the DFC via the NBNDC-QB in terms of iterative channel decoding performance.
II.N ONBINARY N OISE D ISCRETE
C HANNEL-Q UEUE B ASE
D M ODEL
Let{X k}be the binary-input process and{Y k}be the2q-ary output process(over the alphabet{0,1,...,2q−1})of the NBNDC-QB channel.Its2q-ary noi process{Z k}is independent from the input process and is given by[1]Z k=[Y k−(2q−1)X k]/(−1)X k.The channel noi is an M th-order stationary ergodic Markov process with state S k=(Z k−1,Z k−2,...,Z k−M)and is generated bad on the following ball sampling mechanism.First,one of two parcels(an urn containing balls with2q different colors reprenting noi symbols and a queue of size M are lected with probability distribution {ε,1−ε}).If the urn is lected,the model generates a noi symbol Z k=j with probabilityρj,j∈{0,1,...,2q−1}.If the queue is lected,a noi symbol Z k is an entry from the queue randomly lected with a probability distribution that depends on M and a bias parameterα≥0.The en
tries of the queue are shifted to the right,and Z k becomes thefirst entry in the queue.The QB noi parameters are the distribution(ρ0,...,ρ2q−1)and the triplet(M,α,ε).The state stationary distribution vectorΠ=[πZ M]given by[1,eq.(18)](each state is indexed by an M-tuple z M=(z0,z1,...,z M−1))and the noi correlation coefficient is given by
Cor QB=
E[Z k Z k+1]−E[Z k]2
Var(Z k)
=
ε
M−1+α
1−(M−2+α)ε
M−1+α
(1)
where E[·]denotes the expected value,and Var(Z k)denotes the variance of Z k.Given a DFC withfixed parameters(signal-to-noi ratio(SNR),normalized maximum Doppler frequency f D T,soft-decision resolution q,and quantization stepδ),wefit the NBNDC-QB byfirst matching the1-D probability distributionρj=P DFC(y|x), where j=(y−(2q−1)x)/(−1)x,and P DFC(y|x)is given by [1,eq.(3)].The remaining QB parameters(M,α,ε)are obtained by minimizing the Kullback–Leibler divergence rate between the DFC and QB noi process(e details of the parameterization procedure in[1]).This minimization assures that the n-order prob-ability distributions P QB(z n)and P DFC(z n)of both process are statistically clo for large block lengths.While P QB(z n)has a clod-form expression in terms of the QB parameters,P DFC(z n)is obtained by computer simulations of time-correlated Rayleigh fading samples using the sum-of-sinusoids method[17].The samples at the output of the matchedfilter are compared with the thresholds of a uniform quantizer,where the optimum quantization stepδis lected to maximize a lower bound on the Shannon capacity of the underly-ing DFC.In Section IV,the similarity between the two channels is investigated in terms of the system’s end-to-end bit error rate (BER)when decoding LDPC codes.
III.L OW-D ENSITY P ARITY C HECK D ECODING A PPLIED TO THE N ONBINARY N OISE D ISC
RETE C HANNEL-Q UEUE B ASED
Let x=(x1,...,x N)be a codeword encoded by an(N,K)LDPC code.When this codeword is transmitted through an FSMC,the joint probability density function of the transmitted codeword,state -quence s=(s1,...,s N),and received word y=(y1,...,y N)is[12]
P(x,s,y)=P S(s1)
N−1
i=1
P S
i+1
|S i(s i+1|s i)
×
N
i=1
P Y
i
|X i,S i(y i|x i,s i)h(x)(2)
where h(x)is the characteristic function of an error-correcting code [18].From(2),it is possible to obtain the factor graph prented in Fig.1[12],[13].This graph may be decompod into two subgraphs, i.e.,one that involves variables and functions related to the code(the code graph)and one that involves variables and functions related to the channel dynamics(the channel graph).We denote the t of bits that participate in the code constraint m as N(m)and the t of code constraints in which bit n participates as M(n).We also denote as N(m)\n the t N(m)with bit n excluded and as M(n)\m the t M(n)with code constraint m excluded.
We prent next the SPA[18]messages pasd through the factor graph in Fig.1when the channel model ud at the decoder is the NBNDC-QB(the SPA messages for the GEC are treated in[12] and[13]).If the channel that corrupts a codeword is the DFC,it is
Fig.1.Factor graph ud to decode an LDPC over an FSMC.
assumed that it is modeled by means of an NBNDC-QB.The decoding procedure is as follows.
•Initialization :We obtain for n =1to N
αn (z M )=βn (z M )=πz M
πz M =
2q −1 =0ξ −1
m =0
(1−ε)ρ +m ε
M −1+α
M −1k =0
(1−ε)+
k ε
M −1+α
where ξ =
M −1
k =0
δz k , ,and
δi,j =
1,if i =j
0,otherwi .
Compute ¯y n =2q −1−y n ,and for n =1to N ,the log-likelihood (LLR)U n ’s are the messages from the channel subgraph to the associated bit nodes.It is known that the LLR messages offer implementation advantages in the messages exchanged in the code subgraph.The message U n is given by (4),i.e.,
U n =ln ⎛⎝
z M
P QB (y n |x n =0,s n =z M )πz M
z M
P QB (y n |x n =1,s n =z M )πz M
⎞⎠
=ln ⎛⎝ z M P QB (z n =y n |s n =z M
)πz M z M
P QB (z n =¯y n |s n =z M
)πz M
⎞⎠(3)
=ln ⎛⎜⎜⎝
(1−ε)ρy n +ε
M −1+α
z M
M −1
=1
δy n ,z +αδy n ,z 0 πz M
(1−ε)ρ¯y n +ε
M −1!+α
z M
M −1
=1
δ¯y n ,z +αδ¯y n ,z 0
πz M
⎞⎟
⎟⎠
(4)
where the derivation of (4)from (3)is bad on the QB noi gen-eration mechanism.Moreover,t ˜Z
m,n =U n for m ∈N (m ).•Iterative processing
—Processing in the code subgraph
1)For m =1to N −K and n ∈N (m ),the message {L m,n }pasd from the code constraint m to bit node n is calculated according to the “tanh”,
L m,n =2arctan ⎛
⎝
n ∈N (m )\n
tanh 12
Z m,n ⎞
⎠.
2)For n =1to N ,the message pasd from the code graph to the channel graph is V n = m ∈M (n )L m,n with probabilistic reprentation
v n (0)=e V n /(1+e V n ),
服饰图片v n (1)=1/(1+e V n ).
3)Variable node update
a)For n =1to N and for m ∈M (n ):˜
Z m,n =U n +
m ∈M (n )\m L m ,n
.—Processing in the channel subgraph
The messages pasd in the channel subgraph are 2qM -dimensional vectors r n =[r n (z M )],w n =[w n (z M )],αn =[αn (z M )],ρn =[ρn (z M )],βn =[βn (z M )],and γn =[γn (z M )].The entries of the vectors are as follows.
1)For n =1to N ,the messages r n (z M )are given by (5),shown at the bottom of the page.
2)The messages leaving a state node are
γn (z M )=αn (z M )r n (z M ),for n =1to N −1
ρn (z M )=βn (z M )r n (z M ),for n =2to N.
r n (z M
)=
1 x n =0
P QB (y n |x n ,z M )v n (x n )
=P QB (z n =y n |z M )v n (0)+P QB (z n =¯y n |z M )v n (1)=
M −1
=1
δy n ,z +αδy n ,z 0
ε
M −1+α
+(1−ε)ρy n
v n (0)+
M −1
=1
δ¯y n ,z +αδ¯y n ,z 0
ε
M −1+α
+(1−ε)ρ¯y n
v n (1)(5)
3)The messagesαn(z M),for n=2to N,andβn(z M),
for n=1to N−1,are given by(6)and(7),shown at
the bottom of the page.
4)For n=1to N−1:w n(z M)=αn(z M)βn(z M).
For n=1to N,the messages U n are given by(8),shown
at the bottom of the page.
There are veral ways to organize the message passing schedule when running SPA,as described in[13].We choo a schedule that performs one iteration on the code subgraph and then one iteration on the channel subgraph,giving an equal schedule time to each subgraph.1 In the channel s
ubgraph,allαn vectors arefirst pasd in a forward man-ner and all messages are stored in each state node.When the N th state
node is reached,allβ
n vectors are calculated in a backward manner,
yielding the w n and U n messages,so thatβ
n does not need to be
stored.
IV.R ESULTS
A.Shannon Limit
For a system using an error-correcting code with rate r=K/N, the optimal performance theoretically achievable or SL,estab-lished by the lossy joint source-channel coding ,e [19,Th.10.4.1]and[20,Sec.V-B]),yields the lowest channel SNR for which decoding can be realized at
a target end-to-end BER P e. To calculate SL for an NBNDC-QB channel with q=1,2wefirstfix its parameters(ε,α,M)and the remaining parameterρ1is given via [1,eq.(3)]in terms of SNR as the1-D channel error rate of the underlying DFC.Then,for a given target BER P e,we determine the SNR value that satisfies
C(ρ1)=r[1+P e log
2P e+(1−P e)log
2
(1−P e)]=:rR(P e)
where channel capacity C=C(ρ1)is calculated using[5,eq.(24)]. This means that if data are nt at a rate r<C/R(P e),then a probability of error as low as P e can be achieved.
1This schedule provides a good tradeoff between convergence speed and performance[13].
2We only evaluate SL for q=1,which corresponds to hard-decision demod-ulation in the underlying DFC,since the NBNDC does not admit a clod-form capacity expression as a function of its parameters for q>1
现在进行时的被动语态
[1].Fig.2.SL[end-to-end BER versus SNR(in decibels)]for NBNDC-QB chan-nels with Cor QB=0.5α=1,q=1,and M=2,4,6,8.Code rate r=1/2. To analyze the potential coding gain provided by the NBNDC-QB, we study the SL behavior for this channel under afixed correlation co-efficient(Cor QB).For a given value of ,M=2,4,6,8,we u α=1andfindεusing(1)such that Cor QB=0.5.The S
L curves(P e versus SNR)for the NBNDC-QB channels with increasing values of the memory order M are prented in Fig.2.We obrve impressive coding gains even for small values of M.For example,for the simplest model prented with M=2,a gain superior to4dB is obtained with respect to the SL of the binary symmetric channel(BSC),which corresponds to the perfectly interleaved channel.The gain reaches 7.5dB for M=8.
We next consider NBNDC-QB channels that approximate the DFC withfixed parameters.The QB parametersαandεare obtained by the minimization of the Kullback–Leibler divergence rate for lected val-ues of M[1].The SL curves obtained for NBNDC-QB channels with increasing values of M that approximate a DFC with f D T=0.005, SNR=1dB,and q=1are shown in Fig.3.We obrve a coding gain up to1.07dB for the ca M=22.
B.Matched and Mismatched Decoding of LDPC Codes
To evaluate the system’s end-to-end BER obtained by modeling the DFC through the NBNDC-QB,we implement the iterative LDPC decoder prented in Section III with,at most,200iterations.We u a regular LDPC code with parameters(N=15000,K=7500)with
αn(z M)=
s n−1P QB(s n=z M|s n−1)γn−1(s n−1)=
2q−1
i=0
αδz
M,i
+
M−1
=1
扫黑除恶专项行动δz
M,z M−
ε
M−1+α
+(1−ε)ρz
M
γn−1(z2,z3,...,z M,i)
(6)
βn(z M)=
s n+1P QB(s n+1|s n=z M)ρn+1(s n+1)=
2q−1
i=0
αδz
0,i
+
M−1
=1
δz
,i
ε
M−1+α
+(1−ε)ρi
ρn+1(i,z1,z2,...,z M−1)
(7)
U n=ln
⎛
⎜⎜
⎝
(1−ε)ρy
n
+ε
M−1+α
z M
膨胀玻化微珠M−1
=1
δy
n,z
+αδy
n,z0
w n(z M)
(1−ε)ρ¯y
n
+ε
M−1+α
z M
M−1
=1
δ¯y
n,z
+αδ¯y
n,z0
w n(z M)
⎞
⎟⎟
⎠(8)
Fig.3.SL [end-to-end BER versus SNR (in decibels)]for NBNDC-QB chan-nels that approximate a DFC with f D T =0.005,q =1,and SNR =1dB,for a code of rate r =1/
2.
Fig.4.End-to-end BER versus SNR (in decibels)performance of LDPC codes (N =15000,K =7500)under matched decoding over the BSC (q =1),DMC (q =2),NBNDC-QB channels (QB-QB sch
eme)with Cor QB =0.5,M =2,α=1,and q =1,2.
column degree d v =3.The parity-check matrix H is generated using the PEG algorithm [21].
The channel type that corrupts the codeword can be either an NBNDC-QB or a DFC.If the channel being ud is the NBNDC-QB,the decoder us the QB parameters in the channel subgraph,as the receiver is assumed to have knowledge of them,which results in a matched decoding regime.On the other hand,a mismatched decoding tup is obtained if the underlying channel is a DFC while the decoder us the QB model that fits the DFC (i.e.,it employs the QB decoder to decode data nt over the DFC).In this ca,the decoder is assumed to know the normalized Doppler frequency,the SNR,and q for which it is able to choo the appropriate QB parameters.The latter scheme is denoted by DFC-QB,whereas the former is denoted by QB-QB.We expect that QB parameters lected as in [1]to minimize the Kullback–Leibler divergence rate provide a good approximation to the corre-sponding DFC.
大西洋鳕鱼
We first consider the QB-QB scheme in two scenarios:In the first scenario,the QB noi model has parameters M =2,α=1,Cor QB =0.5,and q =1(hard decision)and q =2(soft decision),and in the cond scenario,the QB parameters are M =3,α=2,Cor QB =0.3,and q =1,2.The BER curves versus SNR for the two systems are shown in Figs.4and 5,respectively.The channels BSC and DMC cor-respond to perfectly interleaved channels,for q =1and q =2,respec-tively.In Fig.4,we obrve a coding gain (at BER equal to 10−4)
属鼠多大due
Fig.5.End-to-end BER versus SNR (in decibels)performance of LDPC codes (N =15000,K =7500)under matched decoding over the BSC (q =1),DMC (q =2),NBNDC-QB channels (QB-QB scheme)with Cor QB =0.3,M =3,α=2,and q =1,
2.
Fig.6.End-to-end BER versus SNR (in decibels)performance of LDPC codes (N =15000,K =7500)over the DFC with f D T =0.005and q =1.BER comparisons for the BSC (fully interleaved DFC),the DFC decoded by its QB model DFC-QB (mismatched decoding),and the QB-QB scheme (matched decoding).M =10.
TABLE I
QB P ARAMETERS FOR A DFC W ITH f D T =0.005AND q =
1
to only soft decision is around 2.1dB (compare the BSC curve with the DMC curve).The gain due to
memory is around 3dB (for hard deci-sion)and 3.6dB when we compare the DMC and NBNDC-QB with q =2.The total gain of this NBNDC-QB with q =2relative to the BSC is around 5.8dB.Similar coding gains are obrved in Fig.5,although they are less pronounced in Fig.4as the channel’s noi correlation is smaller.
Fig.6shows BER curves versus SNR for the transmission over a DFC with parameters f D T =0.005and q =1under mismatched decoding,where the decoder assumes that the channel in u is an NBNDC-QB with M =10.The values of QB channel parameters αand εare given in Table I for the considered range of SNR’s.The BER for the BSC (the fully interleaved DFC)and the matched decoder (QB-QB)are also shown for the purpo of comparison.We remark that,
