Slepian-Wolf Coding over Cooperative Networks

0.15 MB·English

by  Mohammad Hossein Yassaee

Slepian-Wolf Coding over Cooperative Networks
Mohammad Hossein Yassaee, Mohammad Reza Aref
Information Systems and Security Lab (ISSL)
EE Department, Sharif University of Technology, Tehran, Iran
E-mail: [email protected], [email protected]
Abstract—We present sufficient conditions for multicasting a ratesforMultipleAccessRelayChannelandmultisource,mul-
set of correlated sources over cooperative networks. We propose tirelay and multidestination networks, respectively. Compress
jointsource-Wyner-Zivencoding/sliding-windowdecoding scheme, and Forward (CF) strategy was generalized to relay networks
9 in which each receiver considers an ordered partition of other with one source and one destination by severalauthorsin [8],
0 nodes. Subject to this scheme, we obtain a set of feasibility [9]. Also, Avestimehr, et.al in [10], [11] proposed a quantize-
0 constraintsforeachorderedpartition.Weconsolidatetheresults
map scheme for Gaussian relay networks with multicast de-
2 ofdifferentorderedpartitionsbyutilizingaresultofgeometrical
mands which achieves the cut-set bound within a constant
approach to obtain the sufficient conditions. We observe that
n numberofbits.TheirschemeisbasedonWyner-Zivencoding
these sufficient conditions are indeed necessary conditions for
a Arefnetworks.Asaconsequenceofthemainresult,weobtainan at relays and a distinguishability argument at receivers.
J
achievablerateregionfornetworkswithmulticastdemands.Also, In this paper, we propose a joint Source-Wyner-Ziv encod-
5 wededuceanachievabilityresultfortwo-wayrelaynetworks,in ing/sliding window decoding scheme for Slepian-Wolf cod-
1 which two nodes want to communicate over a relay network. ing over cooperative networks. Our scheme results in the

operational separation between source and channel coding.
T] I. INTRODUCTION In addition, this scheme does not depend on the graph of
I We consider the problem of reliable transmission of dis- networks, so the result can easily be applied to any arbitrary
s. cretememorylesscorrelatedsources(DMCS)overcooperative network (In general for multi-user networks which are char-
c networks in which each node can simultaneously encode a acterized by a conditional probability distribution, it is not
[ message, relay the messages of other nodes and decode the always possible to describe networkswith a graph).We show
1 messages. The main goal of this paper is to find sufficient thatthesufficientconditions,arealsonecessaryconditionsfor
v conditions to the following problem: the Slepian-Wolf coding over arbitrary Aref networks. As an
8 Given a set of sources U ={U :a ∈A} observed at another consequence of the proposed scheme, we obtain an
1 nodes A = {a ,··· ,a } ⊆AV (V a=j {1j,··· ,N} is the set achievable rate region based on CF strategy. Moreover, one
1 M
2 of nodes in the network) respectively and a set of receivers can easily check that our achievable rate for relay networks
2 at nodes B = {b ,··· ,b } ⊆ V which is not necessarily subsumes the achievable rates
1. disjoint from A, w1hat conKditions must be satisfied to enable obtained for deterministic and Gaussian relay networks
0 us to reliably multicast U to all nodes in B? in [11]. Finally, we apply the main result and prove an
A
9 In addition to this problem,we are interested in the special achievability theorem for the two-way relay network, which
0 case of reliable transmission of independent sources (mes- is consisted of two transmitters communicating over a relay
v: sages) over cooperative networks with multicast demands. networks.
i In particular, we consider the problem of finding a feasible
X
rate region for two-way relay networks as a special case of II. PRELIMINARIES AND DEFINITIONS
r cooperative networks with two transmitters and two receivers
a We denote discrete random variables with capital letters,
with multicast demands.
e.g., X, Y, and their realizations with lower case letters x, y.
The problem of Slepian-Wolf coding over multi-user chan-
A random variable X takes values in a set X. We use |X| to
nels has been considered for some special networks. In
denote the cardinality of a finite discrete set X, and p (x)
[1], Tuncel obtained a necessary and sufficient condition for X
to denote the probability density function (p.d.f.) of X on X.
multicasting a source over a broadcast channel with side
For brevity we may omit the subscript X when it is obvious
information at each receiver. He proposed a joint source-
fromthecontext.We denotevectorswithboldfaceletters,e.g.
channel coding scheme that achieves operational separation
x, y. In addition,we let Xi =(X ,··· ,X ). We use Tn(X)
betweensourcecodingandchannelcoding.In[2],anecessary 1 i ǫ
to denote the set of ǫ-strongly typical sequences of length n,
and sufficient condition for multicasting a set of correlated
w.r.t.densityp (x)onX.Further,weuseTn(Y|x)todenote
sources over acyclic Aref networks [3] has been derived. X ǫ
thesetofalln-sequencey suchthat(x,y) arejointlytypical,
Also the problem of multicasting of correlated sources over
w.r.t. p (x,y). We denote the vectors in the jth block by a
networks was studied in network coding literature [4], [5]. XY
subscript[j]. Fora givensetS, we defineX ={X :i∈S}
Finding the achievable rate region of multi-relay networks S i
and R = R .
isoneoftheinterestingproblemsinShannontheory.Basedon S i∈S i
DecodeandForwardstrategy,[6]and[7]proposedachievable We conPsider the problem of reliable multicasting of
the DMCS U to the subset B of nodes, where trans-
A
mission is over discrete memoryless cooperative network
ThisworkwaspartiallysupportedbyIranian-NSFundergrantNo.84.5193-
2006 p(y1,··· ,yN|x1,··· ,xN) with input alphabet and output
1
alphabetXv andYv ateachnodev ∈V,respectively.Aformal H(US|UA\S) = mH(USm|UAm\S)
definition of the problem is given below.
1
Definition 1: WesaythatthesetofDMCS,U canreliably = (I(Um;Yn|Um )+H(Um|Um Yn))
A m S bi A\S S A\S bi
be transmitted over discrete memoryless cooperative network
1
to all nodes in B, if there exist positive integers (m,n) and a ≤ mI(USm;YWnC|UAm\S)+ǫ
sequence of encoding functions (=a) 1H(Yn |Um )+ǫ
f(m) :Um×Yt−1 →X for t=1,··· ,n m WC A\S
v,t v v v (=b) 1 n H(Y |Um Yi−1X )+ǫ
atallnodesv ∈V,wherefornon-sourcenodeswe letUv =∅ mX WC,i A\S WC WC,i
and a set of decoding functions defined at each node b ; i=1
i n
1
g(m,n) :Um×Yn →Um ≤ mXH(YWC,i|XWC,i)+ǫ
bi bi bi A i=1
fsourchaltlhbaitt∈heBpraosbmab,inlitygoPtro(ginb(imfin,nit)y(Uwbmiit,hYbmnni)g6=oeUsAto)voannei.shes (≤=c) mnnHH((YYWC,Q||XXWC,Q,)Q+)ǫ+ǫ
m WC,Q WC,Q
III. SUMMARY OFMAIN RESULTS
(d)
→ H(Y |X ) (6)
In this section, we provide a summary of our main results. WC WC
The following theorem is the main result of the paper. where (a) follows because Yn is a function of Um, (b)
Theorem 1: ThesetofDMCSUA canreliablybetransmit- follows from definition 1, (c)WiCs obtained by introduAcing a
ted over cooperative network, if there exist auxiliary random
standard time-sharing random variable Q and (d) follows, by
variables YˆV such that for each S ⊆A, we have allowingm,n→∞andsettingYV =YV,Q andXV =XV,Q.
H(US|UA\S)<bim∈Bin\S V⊇mWin⊇S:[I(XW;YbiYˆWC\{bi}|XWC) netNwoowrk,wleinecaornsdideteerrmtwinoistsicpecfiinailtec-fiaseelds onfetwaordketearnmdinAisrteicf
bi∈WC network. For linear deterministic finite-field network, it is
−I(YW;YˆW|XVYbiYˆWC\{bi})] (1) shown in [11] that the product uniform distribution achieves
simultaneously the maximum of RHS of (4) for all W ⊆ V.
where the joint p.d.f. of random variables factors as
InArefnetwork,itisshownthattheRHSof(4)onlydepends
p(uA)[ p(xv)p(yˆv|xv,yv)]p(yV|xV). (2) onthemarginaldistributions,i.e.,p(xv). Hence,lemma1and
vY∈V (3) together imply the following theorem:
Proof: We sketch the proof in the next section.
Theorem 2: A set of correlated sources can reliably be
Remark 1: Theconstraint(1) separatessourcecodingfrom
multicast over a deterministic network, if for each S ⊆A the
channel coding in the operational separation sense [1]. The
constraint (3) is satisfied. Moreover, this constraint is indeed
LHS of (1) represents the rate of Slepian-Wolf coding, while
necessary for two classes of deterministic networks, namely
the RHS of (1) provides an achievable flow through a cut
linear deterministic finite-field network and Aref network.
Λ=(W,WC) over the cooperative network.
Now, we concentrate on finding an achievable rate region
In the rest of this section, we consider some consequences
for cooperative networks. Let R be the rate of message of
of Theorem 1. First, assume that each channel output is a v
the node v. The next theorem gives an achievable rate region
deterministicfunctionofallchannelinputs,i.e.,y =g (x ).
v v V for cooperative network.
Setting Yˆ =Y in Theorem 1, we conclude that the reliable
v v Theorem 3: An N-tuple (R ,R ,··· ,R ) is contained in
transmission of DMCS over deterministic network is feasible 1 2 N
the achievable rate region of cooperative network with mul-
if there exists a product distribution p(x ) such that:
v v ticast demands at each node bi ∈ B, if for each S ⊆ V the
H(U |U )< min minQH(Y |X ) (3) following constraint holds:
S A\S WC WC
bi∈B\S Vb⊇i∈WW⊇CS: R < min min I(X ;Y Yˆ |X )−
Itrnanthsmeifsosilolonwoinfgcolerrmelmatae,dwseouprcroevsidoevear dceotnevremrsineisftoicr creoloiapbelre- S bi∈B\S Vb⊇i∈WW⊇CS:(cid:2) W bi WC\{bi} WC
ative network. I(YW;YˆW|XVYbiYˆWC\{bi}) + (7)
oveLremamdaet1er:mIifniastsiectnoeftwDoMrkC,SthUenAtchaenrereelxiaisbtlsyabejominutltpi.cda.sft. where [x]+ = max{x,0} and the joint p.d.f. of (xV,(cid:3)yV,yˆV)
factors as p(x )p(yˆ |x ,y )]p(y |x ).
p(xV) such that v∈V v v v v V V
Proof:QLetT bethelargestsubsetofV suchthattheRHS
of (1) is nonnegativesubject to each S ⊆T (Note that if two
H(US|UA\S)< min min H(YWC|XWC) (4) subsets T1,T2 have this property, then T1 ∪T2 also has this
bi∈B\S Vb⊇i∈WW⊇CS: property, so such T is unique). Now let A = T in Theorem
Proof: By Fano’s inequality, we have: 1. Assume U (v ∈A) have uniform distribution over the set
v
1
∀S ⊆V,b ∈B\S : H(Um|Um Yn)≤ǫ (5) Uv and be mutually independent. Substituting Rv = H(Uv)
i m S A\S bi in Theorem 1 yields that U can reliably be multicast, if (7)
T
For each (W,bi) such that S ⊆ W ⊆ V and bi ∈ WC, we holds. Hence (R1,··· ,RN) is achievable (Note that Rv =0
have: for each node v ∈TC).
Remark 2: Consider a relay network with node 1 as a Codebook generation at node v: Fix δ > 0 such that
transmitter which has no channel output, i.e., Y1 = ∅, N −2 |Tǫn(Uv)|<2n(H(Uv)+δ). To eachelementofTǫn(Uv), assign
relay nodes {2,··· ,N − 1} and node N as a destination a number wv ∈[2,2n(H(Uv)+δ)] using a one-to-one mapping.
which has no channel input, i.e., X = ∅. Substituting Moreover, we assign one to each non-typical sequence u .
N v
R = ··· = R = 0 in Theorem 3 gives the following We denote the result by u (w ). For channel coding repeat
2 N v v
achievable rate (R ) for relay network. independently the following procedure V times. We denote
CF
the resulting kth codebook by C (k).
v
RCF = min I(XS;YˆSC\{N}YN|XSC)− Choose2n(H(Uv)+I(Yv;Yˆv|Xv)+2δ) codewordsxv(wv,zv),each
1∈SS,⊆NV∈:SC (cid:2) drawn uniformly and independently from the set Tǫn(Xv)
I(YS;YˆS|XVYNYˆSC\{N}) + (8) where zv ∈ [1,2n(I(Yv;Yˆv|Xv)+δ)]. For Wyner-Ziv cod-
(cid:3) ing, for each xv(wv,zv) create 2n(H(Uv)+δ) lists Lv(wv′)
It can be shown that this rate subsumes the achievable rate of with 2n(I(Yv;Yˆv|Xv)+δ) codewords each drawn uniformly
[9, Theorem 3]. and independently from the set Tn(Yˆ |x ) where w′ ∈
Remark 3: Consider a two-way relay network with nodes ǫ v v v
1 and N as two transmitters, each demanding the message [1,2n(H(Uv)+δ)]. We denote the codewords of Lv(wv′) by
of the other node, and N −2 relay nodes {2,··· ,N −1}. yˆv(wv′,zv′|xv) where zv′ ∈[1,2n(I(Yv;Yˆv|Xv)+δ)].
Substituting R = ··· = R = 0 and Yˆ = Yˆ = ∅ Encoding at node v: Divide the nB-length source stream
2 N−1 1 N unB into B vectors (u : 1 ≤ j ≤ B) where u =
in Theorem 3 gives the following achievable rate region for v v,[j] v,[j]
(u ,··· ,u ). We say that channel encoder re-
two-way relay network. v,(j−1)n+1 v,jn
ceives m = (m ,··· ,m ), if for 1 ≤ j ≤ B, u
k =1,N : Rk = k∈SSm,⊆k¯iV∈n:SC (cid:2)I(XS;YˆSC\{k¯}Yk¯|XSC)− winaBs a+ss2igVvne−d3tobmlov,cv[k,1[]js]w∈h[e1r,e2vinn,([BHb]l(oUcvk)+bδ,)]w.eEnucseodtihnegcpoedrefbovorm,o[jks]
I(YS\{k};YˆS\{k}|XVYk¯YˆSC\{k¯}) + (9) Cv(b mod V). For 1≤b≤B+2V −3, define:
(cid:3)
where ¯1=N and N¯ =1.
m ,V ≤b≤B+V −1
Remark 4: Suppose the channel output of relay nodes be w = v,[b−V+1]
v,[b] (cid:26) 1 ,otherwise
a function of channel inputs, i.e., ∀v ∈ V\{1,N} : y =
v
g (x ).Setyˆ =y in(8),wededucethatthecut-setboundis
v V v v In block 1, a default codeword, x (1,1) is transmitted. In
achievablefor productdistribution.This is a generalizationof v
block b > 1, by knowing z from Wyner-Ziv coding at
[11, Theorem4.2.3]whichstates thatcut-setboundis achiev- v,[b−1]
the end of block b−1 (described below), node v transmits
able under product distribution for deterministic network.
x (w ,z ).
Remark 5: In [10], [11], authorsshow that by quantization v v,[b] v,[b−1]
at noise level, Gaussian relay network achieves the cut-set Wyner-Ziv coding: At the end of block b, node v knows
bound within 5|V| bits. It can be shown using [11, Appendix (xv,[b−1],yv,[b−1]) and wv,[b] (note that mv is available non-
A.5]andquantizationatthenoiselevelthattheachievablerate causally at node v), considers the list Lv(wv,[b]) and declares
ofRemark2achievesthecut-setboundwithin 3|V| −1bits. that zv,[b−1] =zv is received if zv is the smallest index such
A similar result holds for two-way Gaussian r(cid:4)el2ay n(cid:5)etwork. that (yˆv,[b−1](wv,[b],zv|xv,[b−1]),xv,[b−1],yv,[b−1]) are typi-
cal. Since Lv(wv,[b]) contains 2n(I(Yv;Yˆv|Xv)+δ) codewords,
IV. PROOF OF THEOREM1 such z exists with high probability (See Table I which
v
We prove Theorem 1 in three steps. In subsection IV-A, describes encoding for network with four nodes).
weproposeajointsource-Wyner-Zivencoding/slidingwindow Decoding at node bi: Let C(bi) = [L1,··· ,Lℓ]
decoding scheme. For encoding, each node first compresses be an ordered partition of the set V−bi = V\{bi}.
its observation using Wyner-Ziv coding, then jointly maps its We propose a sliding window decoding with respect to
source sequence and compressed observation to a codeword. C(bi). Define sv,[b] = (wv,[b],zv,[b−1]). Suppose that
Inthedecodingpartofthescheme,eachreceiverconsidersan (sL1,[b−1],sL2,[b−2],··· ,sLℓ,[b−ℓ]) were decoded correctly
ordered partition of other nodes to decode jointly the sources at the end of block b − 1. The node bi, declares that
andthecompressedobservationsofothernodes.Weprovidea (sL1,[b],··· ,sLℓ,[b−ℓ+1]) = (sˆL1,··· ,sˆLℓ) was sent, if for
each 1≤k ≤ℓ+1,
set of sufficientconditionsfor reliable transmission of DMCS
over cooperative networks. In subsection IV-B, by applying
a result of geometrical approach [9], we unify the results if 1≤b−k+1: x (sˆ ),yˆ (sˆ |x )
of subsection IV-A under different ordered partitions. The “ Lk Lk Lk−1 Lk−1 Lk−1,[b−k+1]
result of this section, yields Theorem 1 with an additional ,x ,yˆ ,y ,x ∈Tn
Lk,[b−k+1] Lk−1,[b−k+1] bi,[b−k+1] bi,[b−k+1]” ǫ
set of constraints corresponding to reliable decoding of the
if V ≤b−k+1≤V +B: (u (wˆ ),
compressed observations of other nodes. In subsection IV-C, Lk Lk
u (w ),u )∈Tn (10)
we show that without loss of generality, we can neglect these Lk Lk,[b−k+1] bi,[b−k+1] ǫ
constraints. This completes the proof.
whereLk =∪k−1L ,L =L =∅ands =(w ,z ).
A. Joint Source-Wyner-Ziv coding/Sliding Window Decoding Note that at thej=e1ndjof b0lock Bℓ++1 V +ℓ−2L,kthe veLctkor Lmk
A
We transmit m = nB length source over cooperative is decoded. Since each (uv,[j] :v ∈A,1≤j ≤ B) is typical
network in B +2V −3 blocks of length n where V is the with high probability, we find the source sequence unB with
A
cardinality of V. small probability of error.
TABLEI
ENCODINGSCHEMEFORMULTICASTOVERNETWORKWITHV ={1,2,3,4}(THEENCODINGSCHEMEOFOTHERNODESISSIMILAR)
Node Block1 Block2 Block3 Block4 Block5 Block6 Block7
u1(m1[1]) u1(m1[2])
1 x1(1,1) x1(1,z1[1]) x1(1,z1[2]) x1(m1[1],z1[3]) x1(m1[2],z1[4]) x1(1,z1[5]) x1(1,z1[6])
yˆ1(1,z1[1]|x1[1]) yˆ1(1,z1[2]|x1[2]) yˆ1(m1[1],z1[3]|x1[3]) yˆ1(m1[2],z1[4]|x1[4]) yˆ1(1,z1[5]|x1[5]) yˆ1(1,z1[6]|x1[6]) yˆ1(1,z1[7]|x1[7])
u2(m2[1]) u2(m2[2])
2 x2(1,1) x2(1,z2[1]) x2(1,z2[2]) x2(m2[1],z2[3]) x2(m2[2],z2[4]) x2(1,z2[5]) x2(1,z2[6])
yˆ2(1,z2[1]|x2[1]) yˆ2(1,z2[2]|x2[2]) yˆ2(m2[1],z2[3]|x2[3]) yˆ2(m2[2],z2[4]|x2[4]) yˆ2(1,z2[5]|x2[5]) yˆ2(1,z2[6]|x2[6]) yˆ2(1,z2[7]|x2[7])
Error Probability Analysis: We bound the probability of the proposed encoding scheme, to prove lemma 2. But in
error in (10) as follows: general, since the ordered partitions corresponding to each
receiver for reliable decoding are different, it is not possible
P = P ∃(sˆ ,··· ,sˆ )∈ to obtain a same offset encoding scheme for all destinations.
e L1 Lℓ
∅6=SX⊆V−biWX⊆S (cid:16) TanhyisinmfaokrmesatciloenarinwhthyetfiherstenVco−di1ngbslocchkesm.e does not transmit
NS(1W) ×···×NS(ℓW) :(sˆL1,··· ,sˆLℓ) satisfies (10) (11) Remark 7: Intheerroranalysis,weonlycomputethe error
(cid:17) corresponding to block V +ℓ−1 ≤ b ≤ V +B, for which
where N(k) is the following set: all ℓ consecutive blocks (b−ℓ+1,··· ,b) contain sources’
S,W
information.However,itcanbeshownthattheconstraintsare
NS(k,W) ={sLk :∀t∈Sk and t′ ∈Wk,st 6=st,[b−k+1] otobtbalioncekdsfrtohmatedroronroatnahlayvseisionffoortmheartibolnocakbsowuthtihchecsoorurrecsepso,nids
wt′ 6=wt′,[b−k+1]andsSkC =sSkC,[b−k+1],wWkC =wWkC,[b−k+1]} dominated by (14).
where S = S ∩L , W = W ∩L , SC = SC ∩L and
k k k k k k
WC =WC ∩L . B. Unified Sufficient Condition
k k
The probability inside the summation (11) represents the In this subsection, we provide a set of sufficient conditions
probability of error corresponding to incorrect decoding of thatdonotdependonaspecifiedorderedpartition.Todothis,
sS such that wS\W was decoded correctly. Denote this prob- weneedthefollowinglemmawhichwaspartiallystatedin[9]
ability by Pe,S,W. We compute it in equation (12) shown asa resultofgeometricalpropertiesofachievablerateregions
at the top of the next page, in which (a) follows, because obtained from sequential decoding:
ℓ ≤ V and the codebook generation of any V consecutive Lemma 3: LetF bethecollectionofallorderedpartitions
Z
blocksareindependent.Moreover,the codebookgenerationis of a set Z. For each C=[L ,··· ,L ]∈F , define
1 ℓ Z
independentof source stream and the sourcesare i.i.d., so the
source sequences are generated independently in consecutive RC ={(R1,··· ,R|Z|)∈R|Z| :∀S ⊆Z
blocks.(b)followsfromthefactthatx (s )andyˆ (s |x )were ℓ+1
Tdrnaw(Yˆn|uxni)f,orrmeslpyeacntidveilnyd.ependentlyfrotmtthesetstTǫnt(Xtt)and RS ≥Xk=1H(Y˜Sk−1X˜Sk|X˜SkCY˜SkC−1X˜LkY˜Lk−1Z˜)} (15)
ǫ t t
Note that each (z : t ∈ S) and (w : t′ ∈ W) take
t t′ then for any joint distribution p(x˜ ,y˜ ,z˜), the following
2n(I(Yt;Yˆt|Xt)+δ) and 2n(H(Ut′)+δ) values, respectively. This identity holds: Z Z
fact, (11) and (12) together imply that for reliable decoding,
for each (S,W) such that W ⊆S, we must have: RC ={(R1,··· ,R|Z|)∈R|Z| :∀S ⊆Z
ℓ+1 C[∈FZ
XI(Yt;Yˆt|Xt)≤XH(XtYˆt)−X`H(UWk|UWkCULkUbi) RS ≥H(Y˜SX˜S|X˜SCY˜SCZ˜)} (16)
t∈S t∈S k=1 Proof: The proof is omitted due to the space limitation.
+H(XSkYˆSk−1|XSkCYˆSkC−1XLkYˆLk−1YbiXbi)´ (13) Now consider the RHS of (14). Since the random variables
NotethattheRHSof(13)takestheminimumvalueforW = (U ) and (X ,Yˆ ,Y ) are independent, the RHS of (14)
A V V V
S. Hence we proved the following lemma: can be expressed in the form of (15) with Z = V ,
Lemma 2: The set of DMCS UA can reliably be multicast X˜ = (X ,U ), Y˜ = Yˆ and Z˜ = (Y ,X ,U ). For e−abcih
overcooperativenetworkto the subsetB ofnodes,if foreach t t t t t bi bi bi
bi ∈B,thereisanorderedpartitionC(bi) ofV\{bi}suchthat v ∈V, define Rv =H(XvYˆv)−I(Yv;Yˆv|Xv) and let
for each S ⊆V , the following constraint holds:
−bi ℓ+1 R(bi) =(R1,··· ,Rbi−1,Rbi+1,··· ,RV)
XH(XtYˆt)−I(Yt;Yˆt|Xt)≥X`H(USk|USkCULkUbi) Lemma 2 states that UA can be multicast over the network,
t∈S +H(XSkYˆSk−1|XSkCYˆSk=kC−11XLkYˆLk−1YbiXbi)´ (14) RifCfo(bri)e.aAchppblyiitnhgerleememxiast3s,Cw(ebic)on∈clFudVe−tbhiastuscuhchthCat(biR)(ebxi)is∈ts
iff :
whererandomvariables(x ,y ,yˆ )aredistributedaccording
V V V
to (2). ∀b ∈B, S ⊆V :
Remark 6: If there is only one destination, one can use i −bi
offset encoding scheme [6], [7] which has less delay than RS(bi) ≥H(YˆSXS|XSCYˆSCYbiXbi)+H(US|USCUbi) (17)
ℓ+1
P (=a) P (x (sˆ ),yˆ (sˆ |x ),x ,yˆ ,y ,x )∈Tn
e,S,W Lk Lk Lk−1 Lk−1 Lk−1,[b−k+1] Lk,[b−k+1] Lk−1,[b−k+1] bi,[b−k+1] bi,[b−k+1] ǫ
(sˆL1X,···,sˆLℓ) kY=1h (cid:0) (cid:1)
∈NS(1W) ×···×NS(ℓW)
×P (u (wˆ ),u (w ),u )∈Tn
Lk Lk Lk Lk,[b−k+1] bi,[b−k+1] ǫ
(cid:0) (cid:1)i
(=b) ℓ |N(p)|ℓ+1 |Tǫn(XSk,YˆSk−1|xSkC,yˆSkC−1,xLk,yˆLk−1,ybi,xbi)| × |Tǫn(UWk|uWkCuLkubi)| (12)
pY=1 SW kY=1(cid:0) t∈Sk|Tǫn(Xt)| t′∈Sk−1|Tǫn(Yˆt′|xt′)| t∈Wk|Tǫn(Ut)| (cid:1)
Q Q Q
Note that we can write (17) in the following form which V. CONCLUSIONS
will be used in the subsection IV-C to complete the proof
This paper obtained sufficient conditions for multicasting
of Theorem 1:
a set of correlated sources over a cooperative network. The
∀S ⊆A\{bi}: sufficient conditions resulted in an operational separation
H(US|UA\S)≤ Wm⊇inS RW(bi)−H(YˆWXW|XWCYˆWC\{bi}Ybi) between source and channel coding. It was shown that these
bi∈WC sufficient conditions are also necessary for the Aref network.
C.∀SFi⊆naAl RCe\s{ublit}:RS(bi)−H(YˆSXS|XSCYˆSC\{bi}Ybi)≥0 (18) Anestwaosrkpewciaasldcearsiev,edananadchtiheevarbesleulrtawteasresgpieocnififeodr tcooothpeerraetliavye
networkandtwo-wayrelaynetwork.Moreover,itwaspartially
This subsection claims that for each b , we can reduce the
i shownthattheseachievablerateregionssubsumesomerecent
constraints of (18) to the first term of it. We prove this by
achievable rate regions which were derived using Wyner-Ziv
induction on |V |. If |V | = 1, there is nothing to prove.
−bi −bi coding.
Now suppose the induction assumption is true for all k <
|V |. For each Z ⊆ V which contains b and each S ⊆
−bi i VI. ACKNOWLEDGEMENT
Z\{b }, let
i The authors wish to thank M. B. Iraji and B. Akhbari for
hZ(bi)(S)=RS(bi)−H(YˆSXS|XZ\SYˆZ\(S∪{bi})Ybi) comments that improved the presentation.
Assume there is a subset T of AC\{bi} such thath(Vbi)(T)< REFERENCES
0. For each W ⊆V observe that,
−bi [1] E.Tuncel. “Slepian-Wolfcodingoverbroadcastchannels”. IEEETrans.
hV(bi)(W∪T)=hV(bi)(T)+RW(bi\)T− Inform.Theory,52(4):1469–1482 ,2006.
H(YˆWXW|XWC\TYˆWC\(T∪{bi})Ybi) [2] oSv.eBr.AKroerfadnaetawnodrkDs”..VainsuPderovacn..IE“EBEroaIndtc.asStymanpd.SInlefopriamn.-WThoelofrmyu(lItSicITas)t,
<RW(bi\)T −H(YˆWXW|XWC\TYˆWC\(T∪{bi})Ybi) [3] 2M0.08R,.pAp.r1e6f.56“-I1n6fo6r0m.ation flow in relay networks”. Ph.D dissertation,
≤RW(bi)−H(YˆWXW|XWCYˆWC\{bi}Ybi) [4] ST.taHnfoo,rdR.UKniove.,ttCerA,.MOc.tM19ed8a0r.d, M. Effros, J. Shi, and D. Karger. “A
=h(bi)(W) (19) random linear network coding approach to multicast”. IEEE Trans.
V
Inform.Theory,52(10):4413–4430 ,2006.
Using(19),thefirsttermof(18)canbesimplifiedasfollows:
[5] M. Bakshi and M. Effros. “On achievable rates for multicast in the
presenceofsideinformation”. inProc.IEEEInt.Symp.Inform.Theory
H(U |U ) ≤ min h(bi)(W) (ISIT),2008,pp.1661-1665.
S A\S V [6] L. Sankar, G. Kramer and N. B. Mandayam. “Offset encoding
V⊃W⊇S:
bi∈WC for multiple-access relay channels”. IEEE Trans. Inform. Theory,
53(10):3814–3821, 2007.
(=a) min h(bi)(W ∪T) [7] L.-L.Xie and P. R. Kumar. “Multisource, multidestination, multirelay
V
V⊃W⊇S: wireless networks”. IEEE Trans. Inform. Theory, 53(10):3586–3595,
bi∈WC 2007.
(b) [8] G. Kramer, M. Gastpar, and P. Gupta. “Cooperative strategies and
≤ min h(bi) (W\T) capacity theorems for relay networks”. IEEE Trans. Inform. Theory,
V\T
Vb⊃i∈WW⊇CS: 51(9):3037–3063 ,2005.
[9] M. H. Yassaee and M. R. Aref. “Generalized compress-and-forward
= min h(bi) (W) (20) strategy forrelay networks”. in Proc.IEEEInt. Symp. Inform. Theory
V\T
V\T⊃W⊇S: (ISIT),2008,pp.2683-2687.
bi∈WC [10] A. S. Avestimehr, S. Diggavi and D. Tse “Approximate capacity of
where (a) followsfrom (19), because S ⊂W∪T and b ∈/ T gaussian relay networks”. in Proc. IEEE Int. Symp. Inform. Theory
i
and (b) follows from the first inequality in (19). (ISIT),2008,pp.474-478.
[11] A.S.Avestimehr. “Wireless network information flow:adeterministic
Now by induction assumption, the last term of (20) corre-
approach”. Ph.Ddissertation, Berkeley Univ,CA.Oct2008.
spondstothefeasibilityconstraintsofreliabletransmissionof
U to node b overcooperativenetworkwith the set of nodes
A i
V\T. Hence U can reliably be transmitted to node b over
A i
original network. This proves our claim. Now it is easy to
see that the first term of (18) is equivalent to (1), that proves
Theorem 1.