On Index Coding in Noisy Broadcast Channels with Receiver Message Side Information

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by  Behzad Asadi

1
On Index Coding in Noisy Broadcast Channels with
Receiver Message Side Information
Behzad Asadi, Lawrence Ong, and Sarah J. Johnson
Abstract—This letter investigates the role of index coding in [9]. Multiplexing coding can achieve the capacity region of
thecapacityofAWGNbroadcastchannelswithreceivermessage broadcast channels where the receivers know some of the
sideinformation. Wefirstshow that indexcodingisunnecessary
messagesdemandedbytheotherreceivers,andwanttodecode
4 where there are two receivers; multiplexing coding and super-
all messages [10]. This scheme can also achieve the capacity
1 position coding are sufficient to achieve the capacity region. We
0 nextshowthat,formorethantworeceivers,multiplexingcoding region of a more general scenario where a noisy version of
2 and superposition coding alone can be suboptimal. We give an messagesisavailableatthereceivers,whoalsoneedtodecode
examplewherethesetwocodingschemesalonecannotachievethe all messages [11].
n capacityregion,butaddingindexcodingcan.Thisdemonstrates
The transmitter of the broadcast channel can utilize the
a that, in contrast to the two-receiver case, multiplexing coding
J structure of the side information available at the receivers
cannot fulfill the function of index coding where there are three

9 or more receivers. to accomplish compression by XORing its messages. This
2 is performed such that the receivers can recover their re-
I. INTRODUCTION quested messages using XORed messages and their own side
]
T We consider additive white Gaussian noise broadcast chan- information. As an example, consider a broadcast channel
I nels (AWGN BCs) with receiver message side information, where M1 and M2 are the two requested messages by two
s. where the receivers know parts of the transmitted messages a receivers, and each receiver knows the requested message of
c priori.Codingschemescommonlyusedforthesechannelsare the other receiver.Then the transmitter only needsto transmit
[
superposition coding, multiplexing coding and index coding. Mx = M1⊕M2, which is the bitwise XOR of M1 and M2
1 In this letter, we show that index coding is redundant where with zero padding for messages of unequal length. Modulo-
v there are only two receivers, but can improve the rate region addition can also be used instead of the XOR operation [12].
4
where there are more than two receivers. This coding scheme is called index coding. It is also called
0
network coding [13], as any index coding problem can be
4 A. Background
7 formulated as a network coding problem [14].
Superposition coding is a layered transmission scheme in
. Separate index and channel coding is a suboptimal scheme
1 which the codewords of the layers are linearly superimposed
0 in broadcast channels with receiver message side information.
to form the transmitted codeword. Specifically, consider an
4 The separation, where the receiver side information is not
AWGNchannelwherethetransmittersendsmessagesM and
1 1 consideredduringthechanneldecodingoftheXORofthetwo
v: M2. A two-layer transmitted codeword is messages, leads to a strictly smaller achievable rate region in
i u(n)(M )+v(n)(M ), two-receiver broadcast channels [9], [10]. Separate index and
X 1 2
channel coding has been shown to achieve within a constant
r whereu(n) =(u1,u2,...,un)andv(n) =(v1,v2,...,vn)are gap of the capacity region of three-receiverAWGN BCs with
a
the codewords of the layers. This scheme was proposed for
only private messages [15].
broadcastchannels[1]andisalso usedinotherchannels,e.g.,
Using a combination of index coding, multiplexing coding
interferencechannels[2]andrelaychannels[3].Superposition
and superposition coding, Wu [16] characterized the capacity
coding can achieve the capacity region of degraded broadcast
region of two-receiver AWGN BCs for all possible message
channels[4],[5].Inbroadcastchannelswithreceivermessage
and side information configurations. In this setting, the first
side information, superposition coding is used in conjunction
receiver requests {M ,M ,M } and knows M , and the
1 3 5 4
with multiplexing coding [6] and index coding [7].
second receiver requests {M ,M ,M } and knows M . The
2 3 4 5
In multiplexing coding, two or more messages are bijec-
transmitted codeword of the capacity-achieving transmission
tively mapped to a single message, and a codebook is then
scheme is
constructed for this message. For instance, suppose a trans-
mitter wants to send messages M1 ∈ {1,2,...,2nR1} and u(n)(M1)+v(n)(Mmx), (1)
M2 ∈{1,2,...,2nR2}. The single message Mm = [M1,M2] where M =[M ,M ,M ⊕M ].
mx 2 3 4 5
is first formedfromM and M , where [·] denotesa bijective
1 2 The combination of these coding schemes can also achieve
map. Then codewords are generated for M , i.e., x(n)(m )
m m the capacity region of some classes of three-receiver less-
where mm ∈ {1,2,...,2n(R1+R2)}. This coding scheme is noisyandmore-capablebroadcastchannelswhere(i)onlytwo
alsocallednestedcoding[8]orphysical-layernetworkcoding
receivers possess side information, and (ii) the only message
requested by the third receiver is also requested by the other
This work is supported by the Australian Research Council under grants
FT110100195andDE120100246. two receivers (i.e, a common message) [17].
2
Y(n)
B. Contributions 1 Receiver1 Wˆ
1
In this work, we first show that, in two-receiver AWGN
BCs with receiver message side information, index coding Z(n) K
1 1
need not necessarily be applied prior to channel coding if Y(n)
2 Receiver2 Wˆ
multiplexing coding is used. To this end, we show that index 2
X(n)
coding is a redundantcoding scheme in (1), and multiplexing M Transmitter
coding and superposition coding are sufficient to achieve the Z...2(n) K2
capacity region. We then derive the capacity region of a
three-receiver AWGN BC. Prior to this letter, the best known Y(n)
L ReceiverL Wˆ
achievable region for this channel was within a constant gap L
of the capacity region [15]. In this channel, index coding
Z(n) K
proves to be useful in order to achieve the capacity region; L L
superposition coding and multiplexing coding cannot achieve Fig.1. TheAWGNbroadcastchannelwithreceivermessagesideinformation,
the capacity region of this channelwithout index coding.Our whereWi⊆Misthesetofmessagesdemandedbyreceiveri,andKi⊂M
isthesetofmessagesknowntoreceiver iapriori.
result indicates that index coding cannot be made redundant
by multiplexing coding in broadcast channels with receiver
message side information where there are more than two Proof:Wecanuseonlymultiplexingcodingandsuperpo-
receivers. sitioncodingtoachievethecapacityregionofthetwo-receiver
AWGN BC with the generalmessage setting givenin (2), i.e.,
II. AWGN BC WITHSIDE INFORMATION
x(n) =u(n)(M )+v(n)(M ), (3)
In an L-receiver AWGN BC with receiver message side 1 m
information, as depicted in Fig. 1, the signals received whereM =[M ,M ,M ,M ].Theonlydifferencebetween
m 2 3 4 5
by receiver i, Y(n) i = 1,2,...,L, is the sum of the (1)and(3)isthatM ⊕M inM isreplacedwith[M ,M ].
i 4 5 mx 4 5
transmitted codeword, X(n), and an i.i.d. noise sequence, This scheme achieves the same rate region as (1), i.e., the
Zi(n) i = 1,2,...,L, with normal distribution, Zi ∼ capacity region. This is because from the standpoint of each
N (0,Ni). The transmitted codeword has a power constraint receiver,theamountofuncertaintytoberesolvedinM4⊕M5
of n E X2 ≤nP and is a function of source messages, isthesameasthatin[M ,M ].Thisuncertaintytoberesolved
l=1 l 4 5
MP= {M1,(cid:0)M2,(cid:1)...,MK}. The messages {Mj}Kj=1 are inde- in M4⊕M5 (or [M4,M5]) is M5 by receiver 1, and M4 by
pendent, and each message, Mj, is intended for one or more receiver 2.
receivers at rate Rj. This channel is stochastically degraded, Thisresult indicatesthatmultiplexingcodingcan fulfillthe
and without loss of generality, we can assume that receiver 1 function of index coding in two-receiver AWGN BCs with
isthe strongestandreceiverL isthe weakestin the sense that receiver message side information.
N1 ≤N2 ≤···≤NL. Remark 1: For two-receiver discrete memoryless broadcast
To model the request and the side information of each channels with receiver message side information, we con-
receiver, we define two sets corresponding to each receiver; jecture that index coding is not necessary. Using the same
thewantsset,Wi,isthesetofmessagesdemandedbyreceiver reasoningasabove,wecanalwaysreplaceitwithmultiplexing
i andthe knowsset, Ki, isthe set ofmessagesthatareknown coding without affecting the achievable rate region.
to receiver i.
IV. WHERE INDEX CODING IMPROVES THERATE REGION
III. WHERE INDEXCODING IS NOT REQUIRED
In this section, we demonstrate that multiplexing coding
In this section, we consider the general message setting in
cannot fulfill the role of index coding in AWGN broadcast
two-receiverAWGN BCs with receivermessage side informa-
channels with receiver message side information where there
tion,wheretheknowsandwantssetsofthereceiversaregiven
are more than two receivers. To this end, we establish the
by
capacityregionofathree-receiverAWGNBCwherethewants
Receiver 1: W1 ={M1,M3,M5},K1 ={M4}, and knows sets of the receivers are
(2)
Receiver 2: W2 ={M2,M3,M4},K2 ={M5}. Receiver 1: W1 ={M1},K1 =∅,
The capacity region of this channel has been derived by Receiver 2: W ={M },K ={M }, (4)
2 2 2 3
Wu[16].Itisachievableusingacombinationofindexcoding,
Receiver 3: W ={M },K ={M },
3 3 3 2
multiplexing coding, and superposition coding, as in (1).
For the special case where M = M = M = 0, and show that multiplexing coding and superposition coding
1 2 3
Oechtering et al. [10] have shown that multiplexing coding cannot achieve the capacity region of this channel without
alone can achieve the capacity region. In spirit of Oechtering index coding.
et al., we now show that index coding is also not necessary
A. Using Index Coding Prior to Multiplexing Coding and
for the general case (2).
Superposition Coding
Theorem 1: Multiplexing coding and superposition coding
are sufficient to achieve the capacity region of two-receiver In this subsection, we establish the capacity region of the
AWGN BCs with receiver message side information. broadcast channel of interest, stated as Theorem 2.
3
Theorem 2: The capacity region of the three-receiver Therequirementsforachievabilityconcerningthe decoding
AWGN BC with the configuration given in (4) is the closure at receivers 2 and 3 are the same as (5b) and (5c). This is
of the set of all rate triples (R ,R ,R ), each satisfying because the unknown information rates of both M and M
1 2 3 x m
are the same from the standpoint of each of these receivers.
αP
R1 <C(cid:18)N (cid:19), (5a) So, as far as these two receivers are concerned, multiplexing
1 coding gives the same result as index coding.
(1−α)P
R <C , (5b) However,decodingv(n) atreceiver1 while treating u(n) as
2 (cid:18)αP +N (cid:19)
2 noise requires
(1−α)P
R3 <C(cid:18)αP +N3(cid:19), (5c) R2+R3 <C(cid:18)(α1P−+αN)P(cid:19), (9)
for some 0≤α≤1, where C(x)= 1log (1+x). 1
2 2 for achievability which is not a redundant condition consid-
Here, we provethe achievabilitypartof Theorem2; the proof
of the converse is presented in the appendix. ering (5b) and (5c). The difference between (7) and (9) is
Proof: (Achievability) Index coding and superposition because receiver 1 needs to decode the correct v(n) over the
codingareemployedtoconstructthetransmissionschemethat set of 2nmax{R2,R3} candidates for the former, but over the
achievesthecapacityregion.Thecodebookofthisschemecon- setof2n(R2+R3) candidatesforthelatter.Afterdecodingv(n),
tains two subcodebooks.The first subcodebookincludes2nR1 receiver 1 decodes m1 which requires (5a) for achievability.
i.i.d. codewords, u(n)(m1) where m1 ∈ {1,2,...,2nR1}, When the messages are directly fed to multiplexing coding
and U ∼ N (0,αP) for an 0 ≤ α ≤ 1. The second sub- and superposition coding, an extra condition, given in (9), is
codebook includes 2nmax{R2,R3} i.i.d. codewords, v(n)(mx) required for achievability and, as a result, this scheme cannot
where mx = m2 ⊕ m3, mx ∈ {1,2,...,2nmax{R2,R3}}, achievethecapacityregion.Evenifreceiver1doesnotdecode
V ∼ N (0,(1−α)P), and V is independent of U. Using v(n) and treats it as noise, it can only decode u(n) at rates up
superpositioncoding,thetransmittedcodewordoverthebroad- to R <C αP , which is strictly smaller than (5a).
cast channel is given by Rece1iver 1 (cid:16)ca(n1−aαls)oP+uNse1(cid:17)simultaneous decoding [18, p. 88] to
x(n) =u(n)(M1)+v(n)(Mx). (6) dfoercoadcehimev1abwilhitiyc;hirneqthuiisrecsa(s5ea,)thaendexRtr1a+coRn2d+itioRn3o<nCthe(cid:16)NsPu1m(cid:17)
Theachievabilityoftheregionin(5a)-(5c)usingthetransmis-
rate prevents this scheme from achieving the capacity region.
sion scheme in (6) can be verified by considering two points
Note that alternative message combinations for multiplex-
during the decoding. First, receivers 2 and 3 consider u(n)
ing coding and superposition coding are also possible. The
as noise. Since receiver 2 knows M a priori, and receiver 3
3 proofof theirsuboptimalityisstraightforwardbuttediousand
knowsM apriori,weobtain(5b)and(5c)astherequirements
2 repetitive.
forachievability.Second,receiver1decodesm whiletreating
x
u(n) as noise. This requires
V. CONCLUSION
(1−α)P
max{R2,R3}<C(cid:18)αP +N (cid:19), (7) In this work, we first showed that multiplexing coding can
1
fulfillthefunctionofindexcodingintwo-receiverAWGNBCs
forachievability.However,consideringtheinequalitiesin(5b)
with receiver message side information. We next established
and (5c), this condition is redundant and can be dropped.
the capacity region of a three-receiver AWGN BC, where
Receiver 1 then removes v(n) from its received signal and
superposition coding and multiplexing coding alone cannot
decodes m , which yields (5a) in the achievable region.
1 achieve the capacity region unless index coding is also used.
B. NotUsingIndexCodingPriortoMultiplexingCodingand This shows that index coding cannot be discharged by mul-
Superposition Coding tiplexing coding in broadcast channels with receiver message
side information where there are more than two receivers.
Inthissubsection,wecharacterizetheachievablerateregion
for the broadcastchannelof interest when M =M ⊕M in
x 2 3 APPENDIX
(6) is replaced with M =[M ,M ]. This employs the same
m 2 3
XOR-multiplexingsubstitutionshowntobeoptimalinthetwo- Based on the proofs for the AWGN BC without side
receivercase.Thismeansthatthemessagesaredirectlyfedto information[4],[18],weprovetheconversepartofTheorem2
multiplexing coding and superposition coding. The codebook usingFano’sinequalityandtheentropypowerinequality(EPI).
of this transmission scheme also contains two subcodebooks Wealsousethefactthatthecapacityregionofastochastically
in which only the second subcodebook is different from degradedbroadcastchannelwithoutfeedbackisthesameasits
the scheme using index coding. The second subcodebook of equivalentphysicallydegradedbroadcastchannel[18, p. 444]
this scheme includes 2n(R2+R3) i.i.d. codewords, v(n)(mm) where the channel input and outputs form a Markov chain,
where mm = [m2,m3], mm ∈ {1,2,...,2n(R2+R3)}, V ∼ X →Y1 →Y2 →···→YL, i.e.,
N (0,(1−α)P)foran0≤α≤1,andV isindependentofU.
Y =X +Z ,
The transmitted codeword of this scheme using superposition 1 1
coding is given by Yi =Yi−1+Z˜i i=2,3,...,L, (10)
x(n) =u(n)(M )+v(n)(M ). (8) where Z˜ ∼N (0,N −N ) for i=2,3,...,L.
1 m i i i−1
4
Proof: (Converse) Based on Fano’s inequality, we have Finally, for R , we have
1
nR =H(M )
H M |Y(n) ≤nǫ , (11) 1 1
(cid:16) 1 1 (cid:17) n =H(M1 |Y1(n),M2,M3)+I(M1;Y1(n) |M2,M3)
H(cid:16)M2 |Y2(n),M3(cid:17)≤nǫ′n, (12) ≤nǫn+h(Y1(n) |M2,M3)−h(Y1(n) |M1,M2,M3)
H M3 |Y3(n),M2 ≤nǫ′n′, (13) (≤g)nǫ + nlog2πe(αP +N )−h(Y(n) |M ,M ,M )
(cid:16) (cid:17) n 2 1 1 1 2 3
where ǫn, ǫ′n and ǫ′n′ tend to zero as n→∞. For the sake of (=l)nǫ + nlog2πe(αP +N )− nlog2πeN , (17)
simplicity we use ǫn instead of ǫ′n and ǫ′n′ as well in the rest. n 2 1 2 1
The rate R2 is upper bounded as where (g) is the result of substituting from (15) and
n
nR2 =H(M2) h(Z˜2(n) |M2,M3)=h(Z˜2(n))= 2 log2πe(N2−N1),
=H(M |Y(n),M )+I(M ;Y(n),M )
2 2 3 2 2 3 into the conditional EPI for Y(n) =Y(n)+Z˜(n). In (17), (l)
(=a)H(M |Y(n),M )+I(M ;Y(n) |M ) is due to 2 1 2
2 2 3 2 2 3
=H(M2 |Y2(n),M3)+h(Y2(n) |M3)−h(Y2(n) |M2,M3) h(Y1(n) |M1,M2,M3)
n
(≤b)nǫn+h(Y2(n) |M3)−h(Y2(n) |M2,M3) =h(Y1(n) |X(n))=h(Z1(n))= 2 log2πeN1.
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