Sliding-Window Superposition Coding:Two-User Interference Channels

0.34 MB·

by  Lele Wang

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Sliding-Window Superposition Coding:
Two-User Interference Channels
Lele Wang, Young-Han Kim, Chiao-Yi Chen, Hosung Park, and Eren S¸as¸og˘lu
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0 Abstract
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n A low-complexity coding scheme is developed to achieve the rate region of maximum likelihood decoding for in-
a
terference channels. As in the classical rate-splitting multiple access scheme by Grant, Rimoldi, Urbanke, and Whiting,
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9 the proposed coding scheme uses superposition of multiple codewords with successive cancellation decoding, which can
be implemented using standard point-to-point encoders and decoders. Unlike rate-splitting multiple access, which is not
]
T rate-optimal for multiple receivers, the proposed coding scheme transmits codewords over multiple blocks in a staggered
I
manner and recovers themsuccessively over slidingdecoding windows, achieving the single-streamoptimal rateregion as
.
s
c wellasthemoregeneral Han–Kobayashi inner bound for thetwo-user interferencechannel. Thefeasibilityofthisscheme
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inpracticeisverifiedbyimplementingitusingcommercialchannelcodesoverthetwo-userGaussianinterferencechannel.

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3 I. INTRODUCTION
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0 For high data rates and massive connectivity, next-generation cellular networks are expected to deploy many small
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1 base stations. While such dense deployment provides the benefit of bringing radio closer to end users, it also increases
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7 the amount of interference from neighboring cells. Consequently, efficient and effective management of interference
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: is expected to become one of the main challenges for high-spectral-efficiency, low-power, broad-coverage wireless
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i communications.
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r Overthepastfewdecades,severaltechniquesatdifferentprotocollayers[1]–[3]havebeenproposedtomitigateadverse
a
effects of interference in wireless networks. One important conceptual technique at the physical layer is simultaneous
decoding[4, Section 6.2],[5]. In this decodingmethod,each receiver attemptsto recoverboth the intended anda subset
of the interfering codewordsat the same time. When the interference is strong [6], [7] and weak [8]–[11],simultaneous
The material in this paper was presented in part in the IEEEInternational Symposium on Information Theory (ISIT)2014, Honolulu, HI, and in
partintheIEEEGlobecom Workshops(GCWkshps)2014,Austin,TX.
L.Wangisjointly withtheDepartmentofElectrical Engineering, StanfordUniversity, Stanford,CA94305USAandtheDepartment ofElectrical
Engineering -Systems,TelAvivUniversity, TelAviv,Israel(email: [email protected]).
Y.-H.KimiswiththeDepartmentofElectrical andComputerEngineering, UniversityofCalifornia, SanDiego,LaJolla,CA92093USA(e-mail:
[email protected]).
C.-Y.CheniswithBroadcom Limited,190Mathilda Place, Sunnyvale, CA94086USA(email: [email protected]).
H. Park is with the School of Electronics and Computer Engineering, Chonnam National University, Gwangju 61186, Korea (e-mail:
[email protected]).
E.S¸as¸og˘luiswithIntelCorporation, SantaClara, CA95054USA(e-mail:[email protected]).
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decoding of random code ensembles achieves the capacity of the two-user interference channel. In fact, for any given
random code ensemble, simultaneous decoding achieves the same rates achievable by the optimal maximum likelihood
decoding [10], [12], [13]. The celebrated Han–Kobayashi coding scheme [14] also relies on simultaneous decoding as
a crucial component. As a main drawback, however, each receiver in simultaneous decoding (or maximum likelihood
decoding)has to employ some form of multiuser sequence detection, which usually has high computationalcomplexity.
This issue has been tackled recently by a few approaches based on emerging spatially coupled and polar codes [15],
[16], but these solutions involve very long block lengths.
For this reason, most practical communicationsystems use conventionalpoint-to-pointlow-complexitydecoding.The
simplestmethodistreatinginterferenceasnoise,inwhichonlystatisticalproperties(suchasthedistributionandpower),
rather than the actual codebook information, of the interfering signals, are used. In successive cancellation decoding,
similar low-complexity point-to-point decoding is performed in steps, first recovering interfering codewords and then
incorporatingthem as partof the channeloutputfor decodingof desired codewords.Successive cancellationdecodingis
particularlywellsuitedwhenthemessagesaresplitintomultiplepartsbyratesplitting,encodedintoseparatecodewords,
andtransmittedviasuperpositioncoding.Inparticular,whenthereisonlyonereceiver(i.e.,foramultipleaccesschannel),
this rate-splitting coding scheme with successive cancellation decoding was proposed by Rimoldi and Urbanke [17] for
the Gaussian case and Grant, Rimoldi, Urbanke, and Whiting [18] for the discrete case, and achieves the optimal rate
region of the polymatroidal shape (the pentagon for two senders). When there are two or more receivers—as in the
two-user interference channel or the compound multiple access channel—the rate-splitting multiple access scheme fails
to achieve the optimal rate region as demonstrated earlier in [19] for Gaussian codes and in Section III-B of this paper
(and [20]) for general codes.
A natural question is whether low-complexity point-to-point coding techniques, which could achieve capacity for
multipleaccessandsingle-antennaGaussianbroadcastchannels,arefundamentallydeficientfortheinterferencechannel,
and high-complexity simultaneous decoding would be critical to achieve the capacity in general. In this paper, we
develop a new coding scheme, called sliding-window superposition coding, that overcomes the limitations of low-
complexity decoding through a new diagonal superposition structure. The main ingredients of the scheme are block
Markov coding, sliding-window decoding (both commonly used for multihop relaying and feedback communication),
superposition coding, and successive cancellation decoding (crucial for low-complexity implementation using standard
point-to-pointcodes).Eachmessageisencodedintoa singlelongcodewordthataretransmitteddiagonallyovermultiple
blocks and multiple signal layers, which helps avoid the performance bottleneck for the aforementioned rate-splitting
multiple access scheme. Receivers recover the desired and interfering codewords over a decoding window spanning
multipleblocks.Successivecancellationdecodingisperformedwithineachdecodingwindowaswellasacrossasequence
of decoding windows for streams of messages. When the number and distribution of signal layers are properly chosen,
the sliding-window superposition coding scheme can achieve every rate pair in the rate region of maximum likelihood
decodingforthetwo-userinterferencechannelwithsinglestreams,providingaconstructiveanswertoourearlierquestion.
We develop a more complete theory behind the number and distribution of signal layers and the choice of decoding
orders, which leads to an extension of this coding scheme that achieves the entire Han–Kobayashi inner bound.
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For practical communication systems, the conceptual sliding-window superposition coding scheme can be readily
adapted to a coded modulation scheme using binary codes and common signal constellations. We compare this sliding-
window coded modulation scheme with two well-known coded modulation schemes, multi-level coding [21], [22] and
bit-interleavedcodedmodulation[23],[24].Weimplementthesliding-windowcodedmodulationschemeforthetwo-user
Gaussianchannelusingthe4GLTEturbocodeanddemonstrateitsperformanceimprovementovertreatinginterferenceas
noise.Followingearlierconferenceversions[20],[25]ofthispaper,severalpracticalimplementationsofsliding-window
superposition coding have been investigated [26], [27] and proposed to the 5G standards [28]–[33].
The rest of the paper is organized as follows. We first define the problem and the relevant rate regions in Section II.
Then, we explain the rate-splitting scheme and demonstrate its fundamental deficiency for the interference channel in
SectionIII.Weintroducethenewsliding-windowsuperpositioncodinginSectionIV,firstbydevelopingasimplescheme
thatachievesthecornerpointsofsimultaneousdecodingregion,andthenextendingittoachieveeverypointintheregion.
We devote Section V to sliding-window coded modulation and its application in a practical communication setting. In
Section VI, we present a more complete theory of the sliding-window superposition coding scheme with a discussion
on the number of superposition layers and alternative decoding orders. With further extensions and augmentations, we
develop a scheme that achieves the Han–Kobayashi inner bound [14] for the two-user interference channel with point-
to-point encoders and decoders in Section VII. We offer a couple of concluding remarks in Section VIII.
Throughoutthe paper, we closely follow the notation in [4]. In particular, for X p(x) and ǫ (0,1), we define the
∼ ∈
set of ǫ-typical n-sequences xn (or the typical set in short) [34] as
(n)(X)= xn : # i: x =x /n p(x) ǫp(x) for all x .
Tǫ | { i } − | ≤ ∈X
(cid:8) (cid:9)
We use Xn to denote the vector (X ,X ,...,X ). For n = 1,2,...,[n] = 1,2,...,n and for a 0,[2a] =
k k1 k2 kn { } ≥
1,2,...,2 a , where a is the smallest integer greater than or equal to a. The probability of an event is denoted
⌈ ⌉
{ } ⌈ ⌉ A
by P( ).
A
II. TWO-USERINTERFERENCECHANNELS
Consider the communication system model depicted in Fig. 1, whereby senders 1 and 2 wish to communicate
independentmessagesM andM to their respectivereceiversovera shared channelp(y ,y x,w). Here X and W are
1 2 1 2
|
channel inputs from senders 1 and 2, respectively, and Y and Y are channel outputs at receivers 1 and 2, respectively.
1 2
In network information theory, this model is commonly referred to as the two-user interference channel.
The Gaussian interference channel in Fig. 2 is an important special case with channel outputs
Y =g X +g W +N ,
1 11 12 1
(1)
Y =g X +g W +N ,
2 21 22 2
whereg denotesthechannelgaincoefficientfromsenderk toreceiverj,andN andN areindependentN(0,1)noise
jk 1 2
components. Under the average power constraint P on each input X and W, we denote the received signal-to-noise
ratios (SNRs) as S = g2 P and S = g2 P, and the received interference-to-noise ratios (INRs) as I = g2 P and
1 11 2 22 1 12
I =g2 P.
2 21
4
M1 Xn Y1n Mˆ1
Encoder 1 Decoder 1
p(y ,y x,w)
1 2
|
M2 Wn Y2n Mˆ2
Encoder 2 Decoder 2
Fig. 1: The interference channel with two sender–receiver pairs.
N
1
g
11
X Y
1
g
21
g
12
W g Y2
22
N
2
Fig. 2: The two-user Gaussian interference channel.
A (2nR1,2nR2,n) code n for the (two-user) interference channel consists of
C
two message sets [2nR1]:= 1,...,2⌈nR1⌉ and [2nR2],
• { }
two encoders, where encoder 1 assigns a codeword xn(m1) to each message m1 [2nR1] and encoder 2 assigns
• ∈
a codeword wn(m2) to each message m2 [2nR2], and
∈
two decoders, where decoder 1 assigns an estimate mˆ or an error message e to each received sequence yn and
• 1 1
decoder 2 assigns an estimate mˆ or an error message e to each received sequence yn.
2 2
The performance of a given code for the interference channel is measured by its average probability of error
n
C
P(n)( )=P (Mˆ ,Mˆ )=(M ,M ) ,
e Cn 1 2 6 1 2
(cid:8) (cid:9)
where the message pair (M1,M2) is uniformly distributed over [2nR1] [2nR2]. A rate pair (R1,R2) is said to be
×
achievable if there exists a sequence of (2nR1,2nR2,n) codes (Cn)∞n=1 such that limn→∞Pe(n)(Cn) = 0. A set of rate
pairs,typicallyreferredtoasarateregion,issaidtobeachievableifeveryratepairintheinteriorofthesetisachievable.
Thecapacityregionisthe closureofthe set ofachievablerate pairs(R ,R ), whichisthe largestachievablerate region
1 2
and captures the optimal tradeoff between the two rates of reliable communication over the interference channel. The
capacity region for the two-user interference channel is not known in general.
Let p = p(x)p(w) be a given product pmf on . Suppose that the codewords xn(m1),m1 [2nR1], and
X × W ∈
wn(m2),m2 ∈[2nR2],thatconstitutethecodebookaregeneratedrandomlyandindependentlyaccordingto ni=1pX(xi)
and ni=1pW(wi),respectively.Werefertothecodebooksgeneratedinthismannercollectivelyasthe(2nRQ1,2nR2,n;p)
Q
random code ensemble (or the p-distributed random code ensemble in short).
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Fixing the encoders as such, we now consider a few alternative decoding schemes. Here and henceforth, we assume
p=p(x)p(w) is fixed and write rate regions without p whenever it is clear from the context.
Treatinginterferenceasnoise(IAN).Receiver1recoversM bytreatingtheinterferingcodewordWn(M )asnoise
1 2
•
generated according to a given (memoryless) distribution p(w). In other words, receiver 1 performs point-to-point
decoding (either a specific decoding technique or a conceptual scheme) for the channel
p(yn xn)= p(wn)p(yn xn,wn)
1| 1|
Xwn
n n
= p (w )p (y x ,w )= p (y x ).
iY=1Xwi W i Y1|X,W 1i| i i Yi=1 Y1|X 1i| i
For example, if joint typicality decoding [35, Section 7.7] is used, the decoder finds mˆ such that (xn(mˆ ),yn)
1 1 1 ∈
(n)(X,Y ). Similarly, receiver 2 can recover M by treating Xn as noise. For the p-distributed random code
ǫ 1 2
T
ensemble, treating noise as interference achieves
R =R R
IAN 1,IAN 2,IAN
∩
where R and R denote the sets of all rate pairs (R ,R ) such that R I(X;Y ) and R I(W;Y ),
1,IAN 2,IAN 1 2 1 1 2 2
≤ ≤
respectively; see Fig. 3a.
Successive cancellation decoding (SCD). Receiver 1 recovers M by treating Xn as noise and then recovers M
2 1
•
based on Wn(M ) (and Yn). For example, in joint typicality decoding, the decoder finds a unique mˆ such that
2 1 2
(wn(mˆ ),yn) (n)(W,Y )andthenauniquemˆ suchthat(xn(mˆ ),wn(mˆ ),yn) (n)(X,W,Y ).Receiver2
2 1 ∈Tǫ 1 1 1 2 1 ∈Tǫ 1
operatesinasimilarmanner.Forthep-distributedrandomcodeensemble,successivecancellationdecodingachieves
R =R R ,
SCD 1,SCD 2,SCD
∩
where R consists of (R ,R ) such that
1,SCD 1 2
R I(W;Y ), R I(X;Y W),
2 1 1 1
≤ ≤ |
and similarly R consists of (R ,R ) such that
2,SCD 1 2
R I(X;Y ), R I(W;Y X).
1 2 2 2
≤ ≤ |
See Fig. 3b for an illustration of R .
SCD
Mixandmatch.Eachreceivercanchoosebetweentreatinginterferenceasnoiseandsuccessivecancellationdecoding.
•
This mix-and-match achieves
(R R ) (R R ). (2)
1,IAN 1,SCD 2,IAN 2,SCD
∪ ∩ ∪
The achievable rate region for mixing and matching is illustrated in Fig. 3c.
Simultaneous (nonunique) decoding (SND). Receiver 1 recovers both the desired message M and the interfering
1
•
message M simultaneously. It then keeps M as the message estimate and ignores the error in estimating M .
2 1 2
Receiver 2 operates in a similar manner. For example, in joint typicality decoding, receiver 1 finds a unique mˆ
1
such that (xn(mˆ1),wn(m2),y1n)∈Tǫ(n)(X,W,Y1) for some m2 ∈[2nR2], and receiver 2 finds a unique mˆ2 such
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that (xn(m1),wn(mˆ2),y2n)∈Tǫ(n)(X,W,Y2) for some m1 ∈[2nR1]. For the p-distributed randomcode ensemble,
simultaneous decoding achieves
R =R R ,
SND 1,SND 2,SND
∩
where R consists of (R ,R ) such that
1,SND 1 2
R I(X;Y ) (3)
1 1
≤
or
R I(W;Y X),
2 1
≤ | (4)
R +R I(X,W;Y ),
1 2 1
≤
and R is characterized by index substitution 1 2 and variable substitution X W in (3) and (4), i.e.,
2,SND
↔ ↔
R I(W;Y )
2 2
≤
or
R I(X;Y W),
1 2
≤ |
R +R I(X,W;Y ).
1 2 2
≤
Note that R can be written as
SND
R =(R R ) (R R )
SND 1,IAN 1,SD 2,IAN 2,SD
∪ ∩ ∪
=(R R ) (R R ) (R R ) (R R ), (5)
1,IAN 2,IAN 1,SD 2,IAN 1,IAN 2,SD 1,SD 2,SD
∩ ∪ ∩ ∪ ∩ ∪ ∩
where R is defined as the set of rate pairs (R ,R ) such that
1,SD 1 2
R I(X;Y W),
1 1
≤ |
R I(W;Y X), (6)
2 1
≤ |
R +R I(X,W;Y ),
1 2 1
≤
andR isdefinedsimilarlybymakingtheindexsubstitution1 2andvariablesubstitutionX W inR .
2,SD 1,SD
↔ ↔
As illustrated in Fig. 3d, R is in general strictly larger than the mix-and-match region in (2).
SND
It turns out no decoding rule can improve upon R . More precisely, given any codebook (xn(m ),wn(m )) ,
SND 1 2
{ }
the probability of decoding error is minimized by the maximum likelihood decoding (MLD) rule
n
mˆ =argmax p (y x (m ),w (m )),
1 m1 Xm2 iY=1 Y1|X,W 1i| i 1 i 2 (7)
n
mˆ =argmax p (y x (m ),w (m )).
2 m2 Xm1 iY=1 Y2|X,W 2i| i 1 i 2
The optimal rate region (or the MLD region) R (p) for the p-distributed random code ensembles is the closure of the
∗
set of rate pairs (R1,R2) such that the sequence of (2nR1,2nR2,n;p) random code ensembles satisfies
lim E[P(n)( )]=0,
n e Cn
→∞
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R2 R2
I(W;Y2|X)
R
R 2,SCD
1,IAN
R
2,IAN
I(W;Y2)
R
1,SCD
I(W;Y1)
R1 R1
I(X;Y1) I(X;Y2) I(X;Y1|W)
(a) RIAN is the intersection of the red-lined region R1,IAN and (b)RSCD istheintersectionofthered-linedregionR1,SCD and
the blue-lined region R2,IAN. the blue-lined region R2,SCD.
R2 R2
I(W;Y2|X) R2,IAN∪R2,SCD I(W;Y2|X)
R
2,SND
I(W;Y1|X)
I(W;Y2) R1,IAN∪R1,SCD I(W;Y2) R1,SND
I(W;Y1) I(W;Y1)
I(X;Y1|W)
R1 R1
I(X;Y2)I(X;Y1)I(X;Y1|W) I(X;Y2)I(X;Y1) I(X;Y2|W)
(c)Themix-and-matchregionistheintersectionofthered-lined (d)RSND istheintersectionofthered-linedregionR1,SND and
region R1,IAN ∪R1,SCD and the blue-lined region R2,IAN ∪ the blue-lined region R2,SND. RSND is identical to the MLD
R2,SCD. region R∗.
Fig. 3: Illustration of the MLD, IAN, SCD regions and their comparison.
where the expectation is with respect to the randomness in codebook generation. It is established in [13] that SND is
optimal for the p-distributed random code ensembles, i.e.,
R =R .
∗ SND
As shownin Fig. 3d,R =R is in generalstrictly largerthanthe mix-and-matchregionin (2), the gainofwhich
∗ SND
may be attributed to high-complexitymultiple sequence detection. The goal of this paper is to developa coding scheme
that achieves R using low-complexity encoders and decoders.
∗
III. RATESPLITTING FORTHEINTERFERENCECHANNEL
In order to improve upon the mix-and-match scheme in the previous section at comparable complexity, one can
incorporate the rate-splitting technique by Rimoldi and Urbanke [17] and Grant, Rimoldi, Urbanke, and Whiting [18].
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A. Rate-Splitting Multiple Access
Considerthemultipleaccesschannelp(y x,w)withtwoinputsX andW andthecommonoutputY .Itiswell-known
1 1
|
thatsimultaneousdecodingoftherandomcodeensemblegeneratedaccordingtop=p(x)p(w)achievesR (p)in(6).
1,SD
In the following, we show how to achieve this region via rate splitting with point-to-point decoders.
Suppose that the message M1 [2nR1] is split into two parts (M11,M12) [2nR11] [2nR12] while the message
∈ ∈ ×
M2 ∈ [2nR2] is not split. The messages m11 and m12 are encoded into codewords xn1 and xn2, respectively, which are
thensymbol-by-symbolmappedtothetransmittedsequencexn,thatis, x (m ,m )=x(x (m ),x (m )),i [n],
i 11 12 1i 11 2i 12
∈
for some functionx(x ,x ). The message m is mappedto wn. For decoding,the receiverrecoversmˆ , mˆ , and mˆ ,
1 2 2 11 2 12
successively, which is denoted as the decoding order
d : mˆ mˆ mˆ .
1 11 2 12
→ →
This rate-splitting scheme [17] with so-called homogeneous superposition coding [36] and successive cancellation
decoding in Fig. 4 can be easily implemented by low-complexity point-to-point encoders and decoders. Following the
standard analysis for random code ensembles generated by p(x )p(x )p(w), decoding is successful if
′ 1 ′ 2 ′
R <I(X ;Y ),
11 1 1
R <I(W;Y X ),
2 1 1
|
R <I(X ;Y X ,W)=I(X;Y X ,W).
12 2 1 1 1 1
| |
By setting R =R +R , it followsthatthe scheme achievesthe rate regionR (p) consistingof (R ,R ) suchthat
1 11 12 RS 1 2
R I(X ;Y )+I(X;Y X ,W),
1 1 1 1 1
≤ | (8)
R I(W;Y X ).
2 1 1
≤ |
By varying p(x )p(x ) and x(x ,x ), while maintaining p(x)= p(x )p(x )=p(x) and p(w)=
′ 1 ′ 2 1 2 ′ x1,x2:x(x1,x2)=x ′ 1 ′ 2 ′
p(w), which we compactly denote by p p, the rectangular regioPn (8) traces the boundary of rate region R (p).
′ 1,SD
≃
More precisely, we have the following identity; see Appendix A for the proof.
Lemma 1 (Layer-splitting lemma [18]).
R (p)= R (p).
1,SD RS ′
p[′ p
≃
M Xn
11 1
M12 X2n Xn p(y1 x,w) Y1n Mˆ11 →Mˆ2 →Mˆ12
|
M2 Wn
Fig. 4: Rate-splitting with successive cancellation for receiver 1.
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Remark 1. Simultaneous decoding of Mˆ ,Mˆ , and Mˆ cannot achieve rates beyond R (p) and therefore it does
11 12 2 1,SD
not improve upon (the union of) successive cancellation decoding for the multiple access channel.
B. Rate Splitting for the Interference Channel
Themainideaofratesplittingforthemultipleaccesschannelistorepresentthemessagesbymultiplepartsandencode
each into one of the superposition layers. Combined with successive cancellation decoding, this superposition coding
scheme transforms the multiple access channel into a sequence of point-to-point channels, over which low-complexity
encoders and decoders can be used. For the interference channel with multiple receivers, however, this rate-splitting
schemecan nolongerachievethe rateregionof simultaneousdecoding(cf.Remark1).The rootcauseof thisdeficiency
is not rate splitting per se, but suboptimal successive cancellation decoding. Indeed, proper rate splitting can achieve
rates better than no splitting when simultaneous decoding is used (cf. Han–Kobayashi coding).
To understandthelimitationsofsuccessivecancellationdecoding,we considerthe rate-splittingschemewith thesame
encoder structure as before and two decoding orders
d : mˆ mˆ mˆ ,
1 11 2 12
→ →
d : mˆ mˆ mˆ ,
2 11 12 2
→ →
as depicted in Fig. 5. Following the standard analysis, decoding is successful at receiver 1 if
M Xn
11 1
M12 X2n Xn p(y1 x,w) Y1n Mˆ11 →Mˆ2 →Mˆ12
|
Yn Mˆ Mˆ Mˆ
M2 Wn p(y2 x,w) 2 11 → 12 → 2
|
Fig. 5: Rate-splitting with successive cancellation in the two-user interference channel.
R <I(X ;Y ), (9a)
11 1 1
R <I(W;Y X ), (9b)
2 1 1
|
R <I(X;Y X ,W). (9c)
12 1 1
|
and at receiver 2 if
R <I(X ;Y ), (9d)
11 1 2
R <I(X;Y X ), (9e)
12 2 1
|
R <I(W;Y X). (9f)
2 2
|
10
By Fourier–Motzkin elimination, this scheme achieves the rate region consisting of (R ,R ) such that
1 2
R min I(X ;Y ), I(X ;Y ) +min I(X;Y X ,W), I(X;Y X ) , (10a)
1 1 1 1 2 1 1 2 1
≤ { } { | | }
R min I(W;Y X ), I(W;Y X) . (10b)
2 1 1 2
≤ { | | }
Remark 2 (Minofthesumvs. sumofthemin). We notea commonmisconceptionin theliterature,reportedalsoin [37]
(see the references therein), that the bounds on R and R in (9) would simplify to
11 12
R min I(X ;Y )+I(X;Y X ,W), I(X;Y ) , (11)
1 1 1 1 1 2
≤ { | }
which could be sufficient to achieve the MLD region R (p) in Section II. This conclusion is incorrect, since the bound
∗
in (10a)isstrictly smallerthan(11)ingeneral.In fact,therate regionin(10), evenaftertakingtheunionoverallp p
′
≃
is strictly smaller than R (p). In order to ensure reliable communication over two different underlying multiple access
∗
channels p(y x,w),i = 1,2, the message parts in the rate-splitting scheme have to be loaded at the rate of the worse
i
|
channel on each superposition layer, which in general incurs a total rate loss.
It turns out that this deficiency is fundamental and cannot be overcome by introducing more superposition layers
and different decoding orders (which include treating interference as noise d : mˆ mˆ and d : mˆ ). To be
1 11 12 2 2
→
more precise, we define the general (p,s,t,d ,d ) rate-splitting scheme. The message M is split into s independent
′ 1 2 1
parts M ,M ,...,M with rates R ,R ,...,R , respectively, and the message M is split into t independent
11 12 1s 11 12 1s 2
parts M ,M ,...,M at rates R ,R ,...,R , respectively. These messages are encoded by the random code
21 22 2t 21 22 2t
ensemblegeneratedaccordingto p = s p(x ) t p(w ) andthe correspondingcodewordsaresuperimposed
′ j=1 ′ j j=1 ′ j
(cid:0)Q (cid:1)(cid:0)Q (cid:1)
by symbol-by-symbol mappings x(x ,...,x ) and w(w ,...,w ). The receivers use successive cancellation decoding
1 s 1 t
with decoding orders d and d , where d is an ordering of elements in mˆ ,...,mˆ ,mˆ ,...,mˆ , k t, and
1 2 1 11 1s 21 2k
{ } ≤
d is an ordering of elements in mˆ ,...,mˆ ,mˆ , ...,mˆ , l s. The achievable rate region of this rate-splitting
2 11 1l 21 2t
{ } ≤
scheme is denoted by R (p,s,t,d ,d ). We establish the following statement in Appendix C.
RS ′ 1 2
Theorem 1. There exists an interference channel p(y ,y x,w) and some input pmf p=p(x)p(w) such that
1 2
|
RRS(p′,s,t,d1,d2)(R∗(p)
p[′ p
≃
for any finite s and t, and decoding orders d and d .
1 2
Remark 3. It can be easily checked that the first three regions in the decomposition of R in (5) are achievable by
∗
properly chosen (p,2,1,d ,d ) rate-splitting schemes. The fourth region R R is the bottleneck in achieving
′ 1 2 1,SD 2,SD
∩
the entire R with rate splitting and successive cancellation.
∗
IV. SLIDING-WINDOW SUPERPOSITION CODING
Inthissection,wedevelopanewcodingscheme,termedsliding-windowsuperpositioncoding(SWSC),thatovercomes
the limitation of rate splitting by encoding the message to multiple superposition layers across consecutive blocks.