Network Coding over a Noisy Relay : a Belief Propagation Approach

0.09 MB·English

by  Sichao Yang

Network Coding over a Noisy Relay : a Belief
Propagation Approach
Sichao Yang and Ralf Koetter
Coordinated Science Laboratory
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
{syang8, koetter}@uiuc.edu
7 Abstract—In recent years, network coding has been investi- y1 =ϕ(x1)+n1,y2 =ϕ(x2)+n2,y =ϕ(x1⊕x2)+nr from
0 gated asa method toobtain improvements inwireless networks. threeindependentchannels.Thedestinationjointlydecodesx
0 A typical assumption of previous work is that relay nodes 1
andx bythesethreeobservations.Theauthorsin[2]compare
2 performingnetworkcodingcandecodethemessagesfromsources 2
n perfectly. On a simple relay network, we design a scheme to theirschemeto areferencescheme,in whichtherelayspends
a obtain network coding gain even when the relay node cannot half of her effort for helping s1 and half for helping s2 in
J perfectly decode its received messages. In our scheme, the a time division manner. The scheme using network coding
operation at the relay node resembles message passing in belief shows a significant improvement.
9
propagation,sendingthelogarithmlikelihoodratio(LLR)of the

network coded message to the destination. Simulation results
] s s s s
T demonstratethegainobtainedoverdifferentchannelconditions. 1 2 1 2
I The goal of this paper is not to give a theoretical result, but x1 x2 x1 x2
. to point to possible interaction of network coding with user
s
cooperationinnoisyscenario.Theextrinsicinformationtransfer
c
[ (EXIT)chart isshown tobeausefulengineeringtool toanalyze
the performance of joint channel coding and network coding in x1 x⊕2 x2 x1 x⊕2 x2
1 the network. x1 Lx1
v
2 I. INTRODUCTION
6 Since the seminal paper by Ahlswede et al. [1], network
0 codinghas been investigatedas a potentialtool for the design d d
1
of communication networks in order to let the data transmis-
0 (a) (b)
7 sion rateapproachthe capacitylimit. Somerecentwork[2–4]
0 studiestheapplicationofnetworkcodingtowirelessnetworks Fig.1. Networkcodingoveranoisyrelaybybelief propagation
/ as a way for providing users with cooperative diversity. All
s
c these papers show that network coding does have practical In [2], as well as other previous work such as [3,4], a
:
v benefits and can substantially improve wireless throughput. fundamentalassumptionisthattherelaynodeisabletodecode
i Our paper was motivated by a simple relay network pre- the source messages reliably. This assumption limits their
X
sented in [2]. The structure of the relay network from [2] is investigation to the cases that the channels from sources to
r
a shown in Figure 1.(a). There are two sources s1,s2 and one the relay have goodquality and the channelcodesapplied are
destinationd. Both sourcesbroadcasttheir codedmessages to strongenough.However,in reality,the relay may be far away
the relay and destination. The relay helps the transmission by fromthesourcessothatthechannelsfromsourcestotherelay
sending its observation of the sources to the destination. The are subject to high noise or severe fading.
different point-to-point channels are assumed to be Gaussian In this paper, we aim to investigate how network coding
andnon-interferingin[2].Thecrucialassumptionin[2]isthat gain may be achieved in a wireless networks even when the
the relay node is assumed to be able to decode the messages transmission to the relay can not be recovered perfectly. The
from both the sources reliably. basicideaofourschemeisthefollowing:Insteadofdecoding
We note that even in a single source and single relay the messages from the sources, the relay node mimicks a
network, the accurate capacity is still unknown. In [2], the message passing belief propagation setup, transmitting the
authors evaluate the performance of the following scheme in logarithm likelihood ratio (LLR) of the network coded mes-
the relay network shown in Figure 1.(a): LDPC codes are sage to the destination as shown in Figure 1.(b). We consider
applied in all point-to-point channels. Let x and x be the this approach a step towards reconciling network coding and
1 2
bitssentbythesources.Letϕ(x):{0,1}→{+1,−1}bethe user cooperation in noisy environment and show the network
standardBPSKmodulationmap.Therelaydecodesthesignal throughput improvement in our scheme. Moreover, we hope
from the source perfectly and it sends x=ϕ(x ⊕x ) to the thatourworkprovidesinitiativeinrethinkingthechannelcode
1 2
destination. Then, the channel can avail of three observations design rules for the cooperation in wireless networks.
Thepaperisorganizedasfollows.InSection II, the system Thedestinationreceivesthreemessages:y =ϕ(x )+N ,
1 1 sd
model and our scheme is introducedin details. In Section III, y2 = ϕ(x2) + Nsd, and yr = Lr + Nrd, where Lr is
the numerical results are presented. In Section IV, the EXIT transmitted as analog value. The destination run an iterative
chart over the system is studied for performance analysis. In decoding algorithm based on these three observations. The
Section V, conclusions are made and some future directions iterativemessagepassingbetweentwoconvolutionaldecoders
are discussed. are shown in Figure 3.
II. SYSTEMMODEL
y1
In this section, we introduce the system model in details. L(1) Decoder 1
A
We study the relay network shown in Figure 1, which has the
+
identical topology as the network studied in [2]. Two sources yr _
s1 and s2 are independent binary random sources with equal p −1 L(1)
probabilityfor0and1.AllthechannelsareGaussianchannels. E
All point-to-point channels are interference-free, so that the L(E2) _ p
relay is able to receive a network coded message in one time yr
+
slot and the destination is able to run an iterative decoding L(2)
A
algorithm synchronously.We further assume that the network Decoder 2
is symmetric: the two channels from the sources to the relay y2
have the same quality and the two channels from the sources Convolutional Decoder Relay Check Node
to the destination have the same quality.
Fig.3. Iterative decoding
We use the following notations for the noise and signal-to-
noise ratio (SNR) on different channels:
Theadditionaloperatorsbetweentheconvolutionaldecodes
Nsd, SNRsd: Noise and SNR on the channel stand for the relay check, of which the indicate function is
from the sources to the destination.
′
1, if x=x ⊕x .
Nsr, SNRsr: Noise and SNR on the channel T(x ,x′,x)= 1 2 (1)
from the sources to the relay. 1 2 (0, otherwise.
N , SNR : Noise and SNR on the channel
rd rd Clearly, if the quality of relay channels is extremely bad, the
from the relay to the destination.
decoder appears to be two separate convolutional decoders.
A systematic view of our scheme is shown in Figure 2. If the quality of relay channel and the relay information Lr
are extremely good, the decoder appears to be a simple turbo
u1 Channel x1 y1 decoder. However, note that, in contrast to classical turbo
Encoder
Relay node operations: codes, all code bits in the two convolutionally coded data
Nsr BCJRL1 LLR Nsd streams are coupled because x1+x′2 is known.
of Lr yr JDoeicnotldying III. SIMULATION RESULTS
Nsr BCJRL2 p x1⊕x′2 Nrd We study the performance of the scheme presented in sec-
u2 Channel y2 tionII bysimulation.In thesimulation,we use thesystematic
Encoder x2 rate 12 recursive convolutional codes in the original Turbo
Nsd codes [5] as the channel code, for which the code generator
is
Fig.2. Blockdiagram ofthesystem
1+D4
G(D)= 1, . (2)
We use convolutional codes as channel codes, which is 1+D+D2+D3+D4
(cid:18) (cid:19)
simple for the performance analysis and illuminative when
Weinvestigatethesystemperformanceunderdifferentchannel
we demonstrate our ideas. The relay node operation includes
conditions. The simulation results are shown in Figure 4
three steps:
and Figure 5. The figures show the bit error probability
Step 1: The BCJR algorithms are run to derive the LLR
(BER) of the system in a combination of different values of
for the messages from each sources. In Figure 2, L and L
1 2 SNRsr,SNRsd and SNRrd. In Figure 4, SNRsr is 5dB. In
denote the LLR of messages from s and s respectively.
1 2 Figure 5, SNRsr is 0dB. In both of the figures, the Y-axis
Step 2: Permute LLRs of a codeword from s , which
2 is the BER and X-axis is SNR . The different curves in a
sd
decrease the channel dependency of the three messages sent
figure stand for the different SNR , where SNR = −∞
rd rd
to the destination.
implies there is no relay. Obviously, if the channel condition
Step 3: Calculate the LLR of the network coded message
from sources to relay is better, more gain is obtained through
by
network coding. In Figure 5, when SNRsr is 0dB, i.e, the
eL1 +eL2 channelsarenotsogood,therearestillsignificantperformance
Lr =log 1+eL1+L2 improvement.
(cid:18) (cid:19)
110000 nel. Otherwise, a crucialassumptionin ten Brink’sanalysis is
SSNNRRssrr == 55 ddBB,, SSNNRRrrdd ==--¥¥ ddBB that the sequence of information is reasonably approximated
--1100 ddBB
--55 ddBB by observations y from a Gaussian channel.
00 ddBB
55 ddBB In this paper, the decoder has four different componentsas
1100--11 1100 ddBB
shown in Figure 3: two convolutionaldecoders and two relay
RR check nodes. In the following discussion, we study the EXIT
EE
BB charts of the two components marked in Figure 3. The EXIT
1100--22 chartsoftheothertwocomponentsarethesamebysymmetry.
Here we assume that both yr, the observation of the relay
check node, and y the observation of the convolutional
2
decoder are from Gaussian channels. That is,
1100--33
--66 --55 --44 --33 --22 --11 00 11 22 33
y =ϕ(x )+N
SSNNRR ((ddBB)) 2 2 sd
ssdd
Fig.4. Biterrorrate(BER)ofthesystemwithSNRsr=5dB yr =ϕ(x1⊕x2)+Nr
where N is the actually noise in the channels from the
sd
110000 SSNNRRssrr == 00 ddBB,, SSNNRRrrdd ==--¥¥ ddBB sources to the destination and Nr is assumed to be an
--1100 ddBB approximationof the noise on a concatenation of the channel
--55 ddBB
fromasourcetotherelayandthechannelfromtherelaytothe
00 ddBB
55 ddBB destination. We let X(1) and X(2) denote the randomsources
1100--11 1100 ddBB
from s and s .
1 2
RR The EXIT chart of the relay node is shown in Figure
EE
BB
6. Clearly, if the channel condition on the relay is bad,
1100--22 the information from the other decoder barely pass through.
However,ifthechannelconditionontherelayisgoodenough,
all the information from the other decoder passes through.
1100--33
--66 --55 --44 --33 --22 --11 00 11 22 33
SSNNRR ((ddBB))
ssdd
Fig.5. Biterrorrate(BER)ofthesystemwithSNRsr=0dB
IV. PERFORMANCE ANALYSIS BY EXITCHART
We briefly describe in this section a standard analysis tool
called an extrinsic information transfer (EXIT) chart to ease
the selection of system parameters, such as the channel code
in the system. An EXIT chart, first developed by Stephan
ten Brink [6], is a technique to aid the construction of good
iteratively-decoded error-correcting codes (in particular low-
density parity-check (LDPC) codes and Turbo codes). EXIT
charts were built on the concept of extrinsic information
developed in the Turbo coding community. For the EXIT Fig.6. EXITcharts oftherelaychecknode
analysis, each component of the decoder (for example a
convolutional decoder of a Turbo code, the LDPC parity- TheEXITchartofconvolutionaldecoderswereextensively
check nodes or the LDPC variable nodes) is modeled as a studied in many papers. For illustration of our way in analyz-
device mapping a sequence of random variables y and Li ing the system, we draw the EXIT chart of the convolution
to a new sequence of random variables Lo, where y is the decode with generator (2) in Figure 7.
channel observation and Li and Lo are interpreted as LLRs We analyze the system performance by the iteration of
for some random source Xi and Xo. For iterations between extrinsic information. As shown in Figure 3, the extrinsic
the components,the extrinsicinformationis usuallymeasured information of decoder 1 is I(X(1),L(1)). After permutation,
E
by mutual information I(Xi,Li) and I(Xo,Lo). it changesto be the inputof the relaycheck node.The output
A key assumption in EXIT chart analysis is that the mes- of the relay checknode is I(X(2),L(2)), which is the a-priori
A
sagestoandfromacomponentofthedecodercanbedescribed information of decoder 2. After the decoding process, the
by a sequence of single numbers, the a-priori information Li extrinsic information of decoder 2, I(X(2),L(E2)), is derived.
and extrinsic information Lo. This is for example true when InFigure8,the iterationofthe extrinsicinformationisshown
the sequenceof observationsy is from a binaryerasure chan- in the case that SNRsd =−5dB and SNRr =1dB.
EXIT charts in studying the performance of joint channel
coding and network coding schemes.
Wireless networks are subject to various crucial physical
limits such as channel capacity, power, and chip speed, The
ultimate goal is to build theory background and design engi-
neeringtoolsinfindingoptimalchannelcodes,i.e. codeswith
lower complexity and low error probability.By taking advan-
tage of cooperative diversity gain such as network coding,
some weak channel codes can be found for reliable commu-
nication in a wireless network though they may work outside
of a point-to-pointchannel capacity limit in the network.
Some problems are interesting to explore in the future. In
thesimplerelaynetworkpresentedinthispaper,ifthechannel
conditionsfromtwosourcestotherelayandthedestinationare
different,it’s notalways betterto donetworkcodingthanjust
Fig.7. EXITchartsoftheconvolutional decoder
to decode forward or amplify forward the source with better
channel condition. It’s important to design schemes which
adapt to the channel conditions. Furthermore, we design our
scheme in a tentative way by assuming that the signals from
the relay to the destination are analog. It will be interesting
to investigate how the quantization of the signals affects the
performance system by rate-distortion theory.
VI. ACKNOWLEDGEMENT
The authors gratefully acknowledge the useful discussion
with Lei Ying andProf. Bruce Hajek, University of Illinoisat
Urbana-Champaign.
REFERENCES
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all bits of a code. As shown in Figure 8, the iteration stops
at a point with almost 1 bit extrinsic information and two
curvesare veryclose. Therefore,in case thatSNR =−5dB
sd
and SNRr = 1dB, the convolutional code with generator (2)
is good channel code for such channels. However, if either
SNRsd or SNRr decreased, the two curves will intersect at
themiddleofthechart.Channelcodesstrongerthanthecodes
in (2) are required for reliable communication between the
sources and the destination.
V. CONCLUSIONS AND FUTUREWORKS
In the paper, we investigate the users’ network coding gain
when the relay is noisy in a simple relay network. We use