Accelerating Iterative Detection for Spatially Coupled Systems by Collaborative Training

0.15 MB·English

by  Keigo Takeuchi

IEEECOMMUNICATIONSLETTERS,VOL.,NO.,2012 1
Accelerating Iterative Detection for Spatially
Coupled Systems by Collaborative Training
Keigo Takeuchi, Member, IEEE
Abstract—Thisletterproposesanovelmethodforaccelerating When a long chain is used, many iterations are required
iterative detection for spatially coupled (SC) systems. An SC forpropagatingreliableinformationoverthewholechain.The
system is constructed by one-dimensional coupling of many
purposeofthisletteristoproposeanovelmethodofcoupling
subsystems, which are classified into training and propagation
2 for accelerating the convergence of iterations. A remarkable
parts.Anirregularstructureisintroducedintothesubsystemsin
1
thetrainingpartsothatinformationinthatpartcanbedetected effort oriented in the same direction was made by Truhachev
0
successfully. The obtained reliable information may spread over et al.: They proposed to divide a long chain into several
2
the whole system via the subsystems in the propagation part. In short chains and to re-connect the divided chains in a ladder-
t order to allow the subsystems in the training part to collabo-
c like[10]orloop-like[11]structure.Theconnectingpositions,
O rate, shortcuts between them are created to accelerate iterative along with the end positions, correspond to the training posi-
detection for that part. As an example of SC systems, SC code-
3 divisionmultiple-access(CDMA)systemsareconsidered.Density tions.Reliableinformationattheconnectingpositionsspreads
2 EvolutionforSCCDMAsystemsshowsthattheproposedmethod over the chains simultaneously. As a result, the propagation
can provide a significant reduction in the number of iterations of reliable information over the whole system is accelerated,
] for highly loaded systems, compared to conventional methods. compared to one long chain.
T
Index Terms—spatial coupling, code-division multiple-access, In this letter, we shall accelerate iterative detection in the
I
. small world, belief propagation, density evolution. training stage by creating shortcuts between distant training
s
c positions. Reliable informationat each training position prop-
[ I. INTRODUCTION agates to the other training positions through the shortcuts,

1 SPATIAL coupling has been proved to improve the whereas there is no collaboration between distant training
v belief-propagation(BP)performanceofconventionallow- positions for conventional SC systems. This collaborative
8 density parity-check (LDPC) codes up to the correspond- training accelerates iterative detection in the training stage.
6 ing maximum-a-posteriori(MAP) performance [1]. This phe- This letter is organized as follows: After summarizing the
1
nomenon has been observed in many other problems, such as notation and terminology used in this letter, in Section II we
6
. code-divisionmultiple-access(CDMA) systems [2]–[4], com- focus on SC CDMA systems as an example of SC systems,
0
pressed sensing [5]–[7], and models in statistical physics [8]. and explain how to connect training positions. Section III
1
2 See [3], [9] for a theoretical treatment of general systems. presents the density-evolution (DE) analysis of BP detection.
1 A spatially coupled (SC) system is constructed as a one- InSectionIV,comparisonsbetweencollaborativetrainingand
: dimensionalchain of L large subsystems. A slightly irregular non-collaborativetrainingare madein termsofthe numberof
v
i structure,whichresultsinarateloss,isintroducedatbothends iterations. Section V concludes this letter.
X of the chain so that information at both ends can be detected For integersl and L, (l)L is equalto l+iLfor an integeri
r successfully.BP detectionconsists oftwo stages:trainingand such that 0 l + iL L 1. The set of consecutive
a ≤ ≤ −
propagation stages. In the training stage, information at both integers i,i + 1,...,j (i < j) is written as [i : j]. The
ends is first detected by utilizing the irregularity. We refer to vector 1n{ denotes the n}-dimensional vector whose elements
positionsatwhichirregularityisimposedastrainingpositions. are all one. The Q-function Q() is the upper tail probability
·
In the propagation stage, on the other hand, the reliable of the standard real Gaussian distribution. The real Gaussian
information at both ends propagates toward the center of the distribution with mean m and covariance Σ is written as
chain.Order preservedin each largesubsystem enableserror- (m,Σ). In a graph, the degree of a node is defined as the
N
free propagation. Since the rate loss due to the irregularity numberofedgesconnectedtothenode.Thedistancebetween
at both ends vanishes as L , the best performance is two nodes is the number of edges in the shortest path that
→ ∞
achieved1 in that limit. This implies that a long chain should connects the two nodes.
be used to approach the MAP performance.
II. SYSTEMMODEL
Manuscript received ***,2012.
K. Takeuchi is with the Department of Communication Engineering and A. Spatially Coupled CDMA Systems
Informatics, the University of Electro-Communications, Tokyo 182-8585,
Japan(e-mail: [email protected]). We consider synchronousK-user SC CDMA systems over
Digital ObjectIdentifier *** therealadditivewhite Gaussiannoise (AWGN)channelas an
1Reliable information fails topropagate withafiniteprobability whenthe
example of SC systems. Note that it is possible to apply the
size of each subsystem is finite. Consequently, an infinitely long chain may
notbebestforfinite-sized systems. idea in this letter to the other SC systems. For simplicity in
0000–0000/00$00.00 (cid:13)c 2012IEEE
2 IEEECOMMUNICATIONSLETTERS,VOL.,NO.,2012
presentation,densespreadingsequencesareused,whereasthe 24
22 26
dense limit of sparse spreadingsequencesshould be taken for
20 28
rigorous DE [12], [13]. See [3] for the details.
22 24 26
Transmission over L symbol periods is considered. Let Nl 18 20 28 30
andN¯ =L−1 L−1N denotethespreadingfactorinsymbol 18 30
Pl=0 l
period l and the average spreading factor, respectively. The 16 16 0 0
received vector y =(yT,...,yT )T RLN¯ is given by
0 L−1 ∈ 14 2
y =Sx+w, w ∼N(0,σ2I). (1) 14 12 10 8 6 4 2
In (1), x = (xT,...,xT )T 1, 1 LK denotes the 12 4
0 L−1 ∈ { − }
data symbol vector with binary phase shift keying (BPSK), 10 6
8
inwhichx 1, 1 K consistsofthemthdatasymbolsof
m
all users. Furt∈he{rmo−re,}S represents the LN¯ LK spreading
matrix, constructed as × Fig.1. Regular bipartite graphwithL=32andW =2.
b S b S
0,0 0,0 0,L−1 0,L−1
 . ··· . 
S = .. .. . (2) In this letter, we refer to the bipartite graph that represents
  the regular base matrix B(reg) as the regular bipartite graph
bL−1,0SL−1,0 bL−1,L−1SL−1,L−1 L,W
··· B(reg). See Fig. 1 for the regular bipartite graph B(reg).
In (2), S are independent N K matrices that have L,W 32,2
{ l,m} l × Asanensembleofcollaborativetraining,weuseanensem-
independent entries taking 1/√N with probability 1/2.
Furthermore,b Risthe±(l,m)-ellementofthebasematrix ble based on a small-world (SW) network [14], which is a
B RL×L, wl,hmic∈h characterizes spatial coupling. Imposing model for elucidating the so-called SW phenomenon such as
L∈−1b2 =1 for all m normalizes the average power used “sixdegreesofseparation.”SWnetworksarehighlyclustered,
Pl=0 l,m asregularnetworksare,andhaveshortdistancesbetweenany
for transmitting each data symbol.
two nodes, as random networks do. These properties of SW
Theset =[0:L 1]ofallsymbolperiodsisdecomposed
L − networksaresuitableforacceleratingiterativedetectioninthe
into the training phase and the propagation phase
T ⊂ L training stage. In this letter, this SW-network-basedensemble
,whicharedisjointsubsetsof .LetN andN denote
P ⊂L L tr is referred to as “SW ensemble.”
the spreading factors in the training and propagation phases,
Let (reg)(m) denote the set of node m and nodes
respectively.LargeNtr isassumedtoobtainreliableestimates with diCstWance 2 fro⊂mLnodem in a regularbipartite graphwith
of the data symbols transmitted in the training phase. The
coupling width W. For example, (reg)(0) for variable nodes
average load α¯ is defined as C2
is shown by the gray nodes in Fig. 1. The SW ensemble is
α¯ = LK = α−1τ +α−1 1 τ −1, (3) obtained by modifying a regular bipartite graph at positions
|T|Ntr+|P|N n tr L (cid:16) − L(cid:17)o included in equally spaced c clusters {CW(reg)(iL/c)}ci=−01.
with τ = |T|. In (3), αtr = K/Ntr and α = K/N denote 1) Leti=0andgeneratetheregularbipartitegraphB(Lr,eWg).
the loads in the training and propagationphases, respectively. 2) Repeat the following for all variable nodes m
Under the BPSK assumption the average load (3) is equal to (reg)(iL/c) in the ith cluster: With probability p, re∈-
CW
the average sum rate, and tends toward α as L and connecteach edge that is connected to variable node m
→ ∞
τ/L 0. to a factor node l (reg)(jL/c) in the other
→ ∈ ∪j6=iCW
clusters uniformly and randomly.
B. Spatial Coupling & Collaborative Training 3) Ifi=c 1,terminatethealgorithm.Otherwise,goback
−
to Step 2) after i:=i+1.
In one-dimensional or circular coupling with coupling
width W, a circulant matrix is used as the base matrix B: In bipartite graphs obtained by the algorithm above, dif-
ferent factor nodes may have different degrees, whereas all
1
b = b , (4) variable nodes have degree 2W +1. We propose a heuristic
l,m √2W +1 (l−m)L
algorithm for determining the training phase with size
T
with (b ,...,b ) = (1T ,0,1T ). This regular base τ = . The proposed algorithm assigns factor nodes with
0 L−1 W+1 W |T|
matrix is written as B(reg). In order to present collaborative largest degrees to the training phase.
L,W
training,we shall introducea graphrepresentationofthe base 1) Let τ˜ = τ, = , and d = d , with d denoting
max max
T ∅
matrix B. the maximum degree of the factor nodes.
The L L base matrix B can be representedby a bipartite 2) If the number of factor nodes with degree d is greater
×
graph that consists of L factor nodes and L variable nodes. than or equal to τ˜, pick up τ˜ factor nodes from all
The factor nodes shown by squares correspond to the row factor nodes with degree d uniformly and randomly,
indicesofB,whereasthevariablenodesrepresentedbycircles add the correspondingindicesinto , and terminatethe
T
are associated with the column indices. If the element b is algorithm. Otherwise, go to the next step.
l,m
non-zero, there is an edge between factor node l and variable 3) Add the indices that represent all factor nodes with
d
L
node m. Otherwise, there is no edge between the two nodes. degree d into , and go back to Step 2) after τ˜ :=
T
TAKEUCHI:ACCELERATINGITERATIVEDETECTIONFORSPATIALLYCOUPLEDSYSTEMSBYCOLLABORATIVETRAINING 3
60 61 62 63 0 1 2 3 4 1
#iterations=20, 40,..., 280 from top to bottom
10-1
58 59 60 61 62 63 0 1 2 3 4 5 6
26 27 28 29 30 31 32 33 34 35 36 37 38
ER 10-2
B
28 29 30 31 32 33 34 35 36 10-3
Fig.2. Abipartitegraphobtainedfromthe(64,2,0.1,2,14)-SWensemble.
Theblackfactornodesshowthenodescorrespondingtothetrainingphases. 10-4
The variable nodes m∈/ C(reg)(0)∪C(reg)(32) and the associated factor 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2 2
nodesareomitted,sincetheycanberestoreduniquely. m/L
Fig. 3. BER versus m/L for the bipartite graph of coupling obtained
τ˜−|Ld| and d:=d−1. fromthe (64,2,0.1,2,14)-SW ensemble, shown inFig. 2.1/σ2 =10dB,
We refer to the ensemble2 that consists of all instances αtr=1.45,andα=1.98.
obtained by the two algorithms above as the (L,W,p,c,τ)-
SW ensemble. See Fig. 2 for an instance obtained from the
Proof: The proof is a generalization of the results in [3]
SW ensemble.
and is therefore omitted.
Remark 1. The degrees of the factor nodes provide an When a bipartite graph of coupling and the load α in the
tr
impact on the complexity of iterative detection algorithms. training phase are given, the DE equations (5) and (6) have a
One method for making a fair comparison between regular uniquefixed-pointfor smallSNR 1/σ2 or smallload α in the
and irregular bipartite graphs would be to consider sparsely propagation phase. On the other hand, there may be multiple
spreadCDMA systems in which,forfactornodeswith a large fixed-points for large SNR and large load. The BP threshold
degree, small row weights are assigned to the correspond- for fixed SNR is defined as the maximum load such that the
ing subsystems. However, this influence on the performance DE equations have a unique fixed-point.
vanishes in the dense limit. This argument implies that com-
Definition 1. When α , a bipartite graph of coupling, and
parisons presented in this letter are not necessarily unfair in tr
SNR are fixed, the BP threshold is defined as the supremum
terms of the complexity.
of α such that the DE equations (5) and (6) have a unique
c
III. DENSITY EVOLUTION fixed-point for all α<αc.
We consider an iterative detection algorithm based on BP TheoperationalmeaningoftheBPthresholdisthat,whenα
with the Gaussian approximation[15]. See [3] for the details. is smaller than the BP threshold, the BP-based iterative algo-
In order to evaluate the bit error rate (BER) via DE, we take rithm can eliminate multiple-access interference (MAI), and
the large-system limit in which K and N tend to infinity achieve performance close to the single-user bound. On the
l
{ }
with the ratios αl =K/Nl fixed for all l. other hand, the system is MAI-limited when α is larger than
theBPthreshold.When1/σ2 =10dB,theBPthresholdα
Theorem1. Inthelarge-systemlimit,theBERofeachelement BP
fortheuncoupledCDMAsystem isgivenbyα 1.73078.
of the mth data symbol vector xm in iteration i is given As L and W tend to infinity with W/L 0B,Pon≈the other
by Q(psirm(i)), in which {sirm(i)} are determined via the hand,the correspondingBP thresholdα(reg→) forthe circularly
coupled DE equations, BP
coupled CDMA system tends toward the optimal threshold
L−1 b2 α 1.98267 [3].
sirm(i)= X σ2l,(mi), (5) MAP ≈
l=0 l
IV. NUMERICAL COMPARISONS
L−1
σ2(i)=σ2+α b2 MMSE(sir (i 1)), (6) We shall make comparisons between regular coupling and
l l X l,m m − the SW ensemble. Bipartite graphs obtained from the SW
m=0
ensemble have distinctly different performance instance by
with sir (0) = 0 for all m. In (6), the function MMSE(x)
m instanceforsmall-sizedgraphs,sothatweusedgoodinstances
denotes the minimum mean-squared error (MMSE) of the
obtained by numerical sampling from the SW ensemble.
BPSK-inputAWGNchannelwithsignal-to-noiseratio(SNR)x.
Figure 3 shows the evolution of the BERs for the bipartite
2 The ensemble with L = 64, W = 2, and c = 2 contains all bipartite graph of coupling shown in Fig. 2. We find that the data
graphs obtained by re-connection of the edges in Fig. 2. However, their symbols in the two clusters (reg)(0) and (reg)(32) are
occurrence probabilities are non-uniform and determined on the basis of C2 C2
detected first. Their BERs are not constant, because of the
theprobability p,whereasinstancescontained inconventional ensemblesare
assumedtooccuruniformlyandrandomly. irregularity of the bipartite graph in the clusters. Then, the
4 IEEECOMMUNICATIONSLETTERS,VOL.,NO.,2012
1 V. CONCLUSIONS
(64,2,0.1,2,14)-SW
(128,2,0.1,4,28)-SW The SW ensemble of coupling has been proposed to
regular (L=64, 128)
accelerate iterative detection in the training stage for SC
10-1 α=1.98 minimum BER
CDMA systems. Instances picked up from the ensemble have
direct connectionsbetween distant training positions. The DE
R
E analysis has shown that the proposed method can provide a
B
ge 10-2 significant reduction in the number of iterations for highly
vera α=1.96 loaded systems, compared to the conventional method of
A
coupling. We conclude that irregular coupling in the training
10-3 phase can accelerate iterative detection for SC systems.
α=1.9
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