A General Coding Scheme for Two-User Fading Interference Channels

0.14 MB·English

by  Lalitha Sankar

A General Coding Scheme for Two-User Fading
Interference Channels
Lalitha Sankar∗, Elza Erkip†, H. Vincent Poor∗
∗Dept. of Electrical Engineering, Princeton University, Princeton, NJ 08544. lalitha,[email protected]
†Dept. of Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201. [email protected]
Abstract—A Han-Kobayashi based achievable scheme is pre- Forone-sideduniformlyweak(UW)IFCs,inwhicheverysub-
sentedforergodicfadingtwo-userGaussianinterferencechannels channel is weak, we conjectured the optimality of ignoring
0 (IFCs) with perfect channel state information at all nodes
interferenceandseparablecodingin[14].Converseproofsfor
1 and Gaussian codebooks with no time-sharing. Using max-min
each of the above-mentioned sub-channels as well as for a
0 optimization techniques, it is shown that jointly coding across
2 all states performs at least as well as separable coding for the sub-class of uniformly mixed two-sided IFCs, comprised of
sub-classes of uniformly weak (every sub-channel is weak) and twocomplementaryUWandUSone-sidedIFCs,isdeveloped
n
a hybrid(mixofstrongandweaksub-channelsthatdonotachieve in [15].
J theinterference-freesum-capacity)IFCs.Fortheuniformlyweak For one-sided and two-sided IFCs, [15] identifies a sub-
IFCs, sufficient conditions are obtained for which the sum-rate
5 classofhybridIFCs(whichiscomplementarytoallpreviously
is maximized when interference is ignored at both receivers.
1 identifiedsub-classes;seeFig.1)comprisedofamixofstrong
I. INTRODUCTION and weak sub-channels or a mix of strong, weak, and mixed
]
T Gaussian interference channels (IFCs) model wireless net- sub-channels, respectively, for which the EVS conditions are
I works as a collection of two or more interfering transmit- not satisfied. Specifically, for one-sided IFCs, [15] unifies the
.
s receive pairs (links). Capacity results for two-user non-fading above-mentionedcapacityresultsusingaHK-basedachievable
c GaussianIFCsareonlyknownforspecificsub-classesofIFCs scheme that uses joint coding and no time-sharing such that
[
such as strong[1], [2], verystrong(a sub-classof strong)[3], across all sub-channels, the interfering transmitter sends a

1 one-sided weak [4], and very weak or noisy [5], [6], and [7]. commonand a private message. For the hybridsub-class, this
v
Outer boundsfor IFCs are developedin [8] and [9]. The best HK-based scheme is shown to achieve a sum-rate at least
6
known inner boundsare due to Han and Kobayashi (HK) [2]. as large as that achieved by separable coding in which only
8
7 Ergodic fading and parallel Gaussian IFCs (PGICs) are commonandprivatemessagesaresentinthestrongandweak
2 IFC models that include both the fading and interference sub-channels, respectively.
. characteristicsof the wireless medium. PGICs in which every In this paper, we develop the HK-based achievable scheme
1
0 sub-channelisstrongandone-sidedPGICsarestudiedin[10] using joint coding for two-sided IFCs. We demonstrate the
0 and[11],respectively;bothpaperspresentachievableschemes optimalityoftransmittingonlycommonmessagesforthetwo-
1 based on coding independently for each parallel sub-channel. sided EVS and US IFCs. The sum-capacity of two-sided UW
:
v For PGICs, [12] determines the conditions on the channel IFCs remains open; for the proposed HK-based joint coding
i coefficients and power constraints for which independent scheme we determine a set of sufficient conditions for which
X
transmission across sub-channels and treating interference as ignoring interference at both receivers and separable coding
r
a noise is optimal. maximize the sum-rate. Finally, we show that in general for
For ergodic fading Gaussian IFCs, henceforth referred to both the two-sided weak and the hybrid sub-classes, joint
simply as IFCs, we developed the sum-capacity and separa- coding of both private and common messages across all sub-
bility for specific sub-classes in [13] and [14]. In contrast to channels achieves at least as large a sum-rate as separable
the non-fadingcase, we provedthat ergodic fading IFCs with coding. Two-sided ergodic fading Gaussian IFCs are studied
a mix of weak and strong sub-channels that satisfy a specific in [16] usinga simplifiedformof the HK regionto determine
set of conditions can achieve the sum of the interference- the power policies that maximize a sum-rate inner bound. In
free capacities of the two intended links; we identified such contrast, we focus on the problem of separability and use
channels as the sub-class of ergodic very strong (EVS) IFCs. a max-min optimization technique to unify known and new
For this sub-class, we showed that jointly coding across all resultsforallsub-classes.Thepaperisorganizedasfollows.In
sub-channels(i.e.,transmittingthesamemessageineverysub- Section II, we present the channel models studied. In Section
channel) and requiring the receivers to decode the transmis- III, we summarize our main results. We conclude in Section
sions from both users achieves the capacity region. Further- IV.
more, in [13], we outlined the optimality of this achievable
coding scheme for a sub-class of uniformly strong (US) IFCs II. CHANNEL MODEL
in which every sub-channel is strong. The US and EVS sub- A two-user ergodic fading Gaussian IFC consists of two
classes overlap but in general are not the same (see Fig. 1). transmit-receive pairs, each pair indexed by k, for k = 1,2.
the standard information-theoreticdefinitions. Throughoutthe
EVS IFC:
sequel, we use the terms fading states and sub-channels
mix of weak and interchangeablyand refer to the ergodicfadingIFC as simply
strong sub-channels IFC. E[·] denotes expectation and C(x) denotes log(1+x)
Mixed where the logarithm is to the base 2.
IFCs: Hybrid IFC:
every US IFC: III. ACHIEVABLE SCHEME
non-EVS mix of
sub-ch every We consider an HK-based achievable scheme using Gaus-
mixed weak, strong, and
sub-ch. sian codebooks without time-sharing and joint encoding and
mixed
decoding across all sub-channels. We seek to determine the
UW: is strong
power fractions allocated to private and common messages at
weak sub-channels each transmitter that maximizes the sum-rate. Our motivation
stems from the fact that joint coding is optimal for EVS and
US IFCs [13] and achieves at least as large a sum-rate as
Two-user Ergodic Fading Two-sided IFCs separable coding for hybrid one-sided IFCs [15]. We outline
the achievable scheme below.
Fig. 1. A Venn diagram representation of the four sub-classes of ergodic Thus, transmitter k transmits the same message pair
fadingIFCs. (w ,w ) in every sub-channel where w and w are
kc kp kc kp
the common and private messages respectively. Each receiver
decodesbyjointlydecodingusingthereceivedsignalsfromall
In each use of the channel, transmitter k transmits the signal sub-channels. Let αk,H ∈[0,1] and αk,H =1−αk,H denote
Xk while receiver k receives Yk, k ∈K. For X=[X1 X2]T, the power fractions at transmitter k allocated to transmitting
the channel output vector Y =[Y Y ]T is given by theprivateandcommonmessages,respectively,insub-channel
1 2
H. The two transmitted signals in each use of sub-channelH
Y =HX+Z (1)
are
where Z = [Z Z ]T is a noise vector with entries that
are zero-mean, u1nit2variance, circularly symmetric complex X1(H)= α1,HP1(H)V1H+ α1,HP1(H)U1H (4a)
Gaussian noise variables and H is a random matrix of fading q q
gainswithentriesH ,forallm,k ∈{1,2},suchthatH X2(H)= α2,HP2(H)V2H+ α2,HP2(H)U2H (4b)
m,k m,k
denotesthe fadinggainbetweenreceivermandtransmitterk. where VkH anqd UkH, k = 1,2, arqe independent zero-mean
We assume the fading process {H} is stationary and ergodic unitvarianceGaussianrandomvariables,forallH.Weusethe
but not necessarily Gaussian. Note that the channel gains notation VkH and UkH to indicate that the random variables
Hm,k, for all m and k, are not assumed to be independent; are mutually independent for every instantiation of H, i.e.,
however, H is known instantaneously at all the transmitters independentcodebooksin each sub-channel.Let α denotea
H
and receivers. A one-sided fading Gaussian IFC results when vector of power fractions with entries αk,H, k =1,2.
either H1,2 =0 or H2,1 =0. A two-sided IFC can be viewed In [17, Theorem 4], the authors present the HK region
as a collection of two complementary one-sided IFCs, one achieved by superpositioncoding. Assuming no time-sharing,
with H1,2 =0 and the other with H2,1 =0. for the Gaussian signaling in (4) and with joint coding, one
Over n uses of the channel, the transmit sequences {Xk,i} can directlyextendthe analysisin [17, Theorem4] to ergodic
are constrained in power according to fadingGaussianIFCs.ThefollowingPropositionbasedon[17,
n Theorem 4] summarizes the resulting rate bounds.
|Xk,i|2 ≤nPk , for all k =1,2. (2) Proposition 1: A rate pair (R1,R2) is achievablefora HK
i=1 scheme with superposition coding and no time-sharing for
X
Since the transmitters know the fading states of the links on ergodic fading IFCs if
which they transmit, they can allocate their transmitted signal
R ≤B (α ,P (H)) (5)
powers according to the channel state information. We write k k H
P(H) with entries Pk(H) for all k to explicitly describe the R1+R2 ≤B3(αH,P(H)) (6)
power policy for the entire set of random fading states. For R +R ≤B (α ,P(H)) (7)
1 2 4 H
an ergodic fading channel, (2) then simplifies to R1+R2 ≤B5(αH,P(H)) (8)
E[Pk(H)]≤Pk for all k =1,2, (3) 2R1+R2 ≤B6(αH,P(H)) (9)
where the expectationin (3) is overthe distributionof H. We R1+2R2 ≤B7(αH,P(H)) (10)
denote the set of all feasible policies P (h), i.e., the power
where
policies whose entries satisfy (3), by P.
Our definitions of average error probabilities, capacity re- Bk = E C 1+α|Hj,Hk,|kH|2kP,jk|2(HPj)(H) , j,k =1,2,j 6=k
gions, and achievable rate pairs (R1,R2) for the IFC mirror h (cid:16) (cid:17)i (11)
B =E C |H1,1|2P1(H)+|H1,2|2α2,HP2(H) receivers that see weak and strong interference, respectively,
3 " 1+α2,H|H1,2|2P2(H) !# and is given by
(12) max min S (α∗ ,P (H)). (23)
m H
+E C |H2,2|2α2,HP2(H) (d) For UW IFPC(Hs,)∈thPems∈u{m2,-3r}ate is maximized by α∗ = 1 if,
" 1+α1,H|H2,1|2P1(H)!# for every P(H)∈P, H
B = B | (13)
4 3 indices1and2swapped |H |2 > 1+|H |2P (H) |H |2 (24)
2,2 2,1 1 1,2
B =E C α1,H|H1,1|2P1(H)+|H1,2|2α2,HP2(H) |H1,1|2 >(cid:16)1+|H1,2|2P2(H)(cid:17)|H2,1|2 (25)
5 " 1+α2,H|H1,2|2P2(H) !# and is given by (cid:16) (cid:17)
(14)
max S (1,P(H)). (26)
1
+E C α2,H|H2,2|2P2(H)+|H2,1|2α1,HP1(H) P(H)∈P
" 1+α1,H|H2,1|2P1(H) !# For a hybrid one-sided IFC, the achievable sum-rate is maxi-
mized by
α (H)∈(0,1] H is weak
B6 =E"C |H1,1|21P+1(αH2,)H+|H|H1,21,|22|P22α(2H,H)P2(H)!# anαd∗ki,sHg=ive(cid:26)n byk(17)0for this chHoiciesostfroαn∗g.. ,k =1,2, (27)
H
(15) Remark 3: The conditions in (24) and (25) hold for all
+E C |H1,1|2α1,HP1(H) ffaedasinibgleavperoawgeerdcpoonlisctireasintPin(H(3)),,ain.ed.,thpuosl,icaireesqusaitteisrfeysitnrgicttihvee
" 1+α2,H|H1,2|2P2(H)!# in defining the set of channel gains for which ignoring inter-
+E C α2,H|H2,2|2P1(H)+|H2,1|2α1,HP1(H) ferenceisoptimalforUWIFCs.However,theanalysisandthe
" 1+α1,H|H2,1|2P1(H) !# conditions (24) and (25) also hold for ergodic channels with
a per-symbolorequivalentlyper-fadingstate powerconstraint
B = B | (16) for which determining the largest values of the right-side of
7 6 indices1and2swapped
Theorem 2: The sum-capacity of ergodic fading IFCs is (24) and (25) is relatively easier.
lower bounded by Proof: Our proof relies on using the fact that
the maximization of the minimum of two functions, say
max min Sm(αH,P(H)) (17) f1(αH,P (H))andf2(αH,P (H))isequivalenttoaminimax
P(H)∈P,αk,H∈[0,1]m∈{1,2,3,4,5,6} optimization problem (see for e.g., [18, II.C]) for which the
where maximumsum-rateS∗isgivenbythefollowingthreecases.In
each case, the optimal P∗(H) and α∗ maximize the smaller
H
S1(αH,P(H))=B1(αH,P(H))+B2(αH,P(H)), ofthe two functionsand thereforemaximizebothforthe case
(18) when the two functions are equal. The three cases are
Sj(αH,P(H))=Bj+1(αH,P(H)), j =2,3,4 (19) Case 1: S∗ =f1(α∗H,P∗(H))<f2(α∗H,P∗(H))
S (α ,P(H))=(B (α ,P (H))+B (α ,P (H)))/2 (28a)
5 H 6 H 2 H
(20) Case 2: S∗ =f (α∗ ,P∗(H))<f (α∗ ,P∗(H))
2 H 1 H
S (α ,P(H))=(B (α ,P (H))+B (α ,P (H)))/2. (28b)
6 H 7 H 1 H
(21) Case 3: S∗ =f (α∗ ,P∗(H))=f (α∗ ,P∗(H))
1 H 2 H
(28c)
(a) For EVS IFCs, the sum-capacity S 0,P(wf)(H)
1
From (17), the sum-rate is the solution to a max-min opti-
is achieved by choosing α∗H = 0 fo(cid:16)r all H an(cid:17)d mization of f (·) = S (·) and f (·) = min S (·). We
P∗(H) = P(wf)(H) provided S 0,P(wf)(H) < 1 1 2 j>1 j
1 now consider each sub-class separately.
S 0,P(wf)(H) , for all j > 1, wher(cid:16)e P(wf)(H) (cid:17)is the Ergodicverystrong:Bydefinition,anEVSIFCresultswhen
j
the sum of the interference-free capacities of the two links
opt(cid:16)imalwaterfillin(cid:17)g policy for the two interference-freedirect
can be achieved. From (28), one special case of the max-min
links. (b) For US IFCs, the sum-capacity is achieved by
α∗ =0, for all H and is given by optimization in (17) corresponds to the EVS sub-class. This
H
results when
max min S (0,P(H)). (22)
P(H)∈Pm∈{1,2,3} m max S1(αH,P(H))=S1 0,P(wf)(H)
P(H),αH
(c) For UM IFCs, the sum-capacity is achieved by choosing (cid:16) (cid:17)
<S 0,P(wf)(H) , for all j >1, (29)
α∗ = 1 and α∗ = 0, j 6= k, where k and j are the j
k,H j,H
(cid:16) (cid:17)
Furthermore, when all sub-channels are strong, i.e., when
EVS IFCs: Pr[|H1,2| > |H2,2|] = 1, the bound S2(αH,P (H)) in (19)
Sep. sub-optimal can be rewritten as
w = w ,w = w
Mixed 1 1c 2 2c E C |H1,1|2P1(H)+|H1,2|2P2(H)
IFCs:
(w ,w ) Hybrid IFCs: US IFCs: h (cid:16) −E C 1+α2,H|H1,2|2P2(cid:17)(iH)
1p 2c
o(wr1c,w2p) (awchkp. ,swckhce)m,ke:= H1,K2 Sopept..: swu1b=-owp1tc., +hE"(cid:16)C 1+|Hα21,2,H|2|αH22,H,1P|22P(1H((cid:17))Hi)!#. (33)
w =w
UW: ach. 2 2c UsingtheUScondition,onecanverifythatforeverychoiceof
scheme: HK P(H), S2(αH,P(H)) is maximized by α∗1,H = α∗2,H = 0,
( ) i.e., w =w , k =1,2. Since S (·) is obtained from S (·)
w ,w ,k =1,2 k k,c 3 2
kp kc by swapping the indices, the above choice also maximizes
S (·). Transmitting only common messages at both transmit-
3
ters results in multiple-access regions at both receivers; one
Fig.2. Two-sidedergodicfadingIFCs:overview ofknownresults. canusethepropertiesofthesemultiple-accessregionstoshow
that the remaining sum-rate bounds are at least as much as
theminimumofS (0,·),j =1,2,3,suchthatthemaximum
j
where we have used the fact that the ergodic capacities achievablesum-rateisgivenby(22).Theouterboundanalysis
of the two interference-free links are maximized by the in [15, Theorem 3] helps establish that (22) is the US sum-
optimal single-user waterfilling policies [19], denoted by capacity.
P(wf)(H)withentriesP(wf)(H ).NotethatS (0,P (H)) Uniformly mixed: Without loss of generality, assume
k k,k 2
and S (0,P (H)) are the multiple-access sum-capacities at Pr[|H |2 > |H |2] = 1 and Pr[|H |2 < |H |2] =
3 2,1 1,1 1,2 2,2
receivers 1 and 2, respectively, such that 1, i.e., receivers 1 and 2 experience weak and strong in-
terference, respectively. Comparing with the US case, we
S (0,P (H))= S (0,P(H))| . (30)
2 3 swapindices1and2 choose α∗ = 0. Furthermore, is is straightforward to
1,H
We now show that (29) simplifies to the requirement verify that S2(αH,P(H)) is maximized by α∗2,H = 1 while
S3 is independent of α1,H for α∗1,H = 0. For j ≥ 4,
S1 0,P(wf)(H) <min S2 0,P(wf)(H) , Sj α∗1,H =0,α2,H,P (H) is in general maximized by a
(cid:16) (cid:17) (cid:16) (cid:16)S3 0,P(wf)(cid:17)(H) (31) aαn2d,(cid:0)Hfo6=r 1a.nyEvPalu(aHti)n,goanllef(cid:1)ucnanctiovnersifayt thαa∗1t,HS,α(∗2·,)H==S(0,(·1)),
1 2
(cid:0) (cid:1)
(cid:16) (cid:17)(cid:17) S (·) = S (·), and S (·) > min(S (·),S (·)), j = 5,6.
The proof follows trivially by expanding the terms 4 3 j 2 3
S 0,P(wf)(H) for all j > 1 and comparing them to Thus, the max-min optimization simplifies to (23). Finally,
j
using outer bounds developed for two complementary one-
S (cid:16)0,P(wf)(H)(cid:17). We illustrate this for S (0,P (H)) as sided IFCs (see [15, Theorem 5]), we can show that (23) is
1 4
foll(cid:16)ows. (cid:17) the sum-capacity.
Uniformlyweak:Forthissub-classofchannels,itisstraight-
S (0,P (H))=E C |H |2P(wf)(H) (32a)
4 1,2 2 forward to see that the conditions for Case 1 in (29) will not
+hE (cid:16)C |H |2P(wf)(H(cid:17)i) be satisfied (as otherwise the sub-class would be EVS), and
2,1 1 thus, α∗ 6= 0 for k = 1,2. For the case with one-sided
> E Ch |H(cid:16) |2P(wf)(H) (cid:17)i (32b) interferekn,Hce, in [15, Th. 4], we show that transmitting only
2,2 2
private messages at the interfering transmitter maximizes the
+hE C(cid:16) |H2,1|2P1(wf)(H(cid:17))i sum-rate and is in fact sum-capacity optimal. However, for
≥S 0h,P(cid:16)(wf)(H) (cid:17)i (32c) the two-sided case, the choice of α∗k,H = 1 for all k, i.e.,
3 w =w for all k, does not necessarily maximize the sum-
k k,p
where (32b) follows from(cid:16)simplifying ((cid:17)31) by expanding the rate.ConsiderthefunctionS2(αH,P(H))in(19).From(12),
multiple-access sum-capacity terms and (32b) follows from it can be rewritten as
using chain rule for mutual information. We note that (31)
is the EVS condition developed in [13, Th. 7] (see also [15, S2(αH,P(H))=E C |H1,1|2P1(H)+|H1,2|2P2(H)
Theorem 2]). The sum-capacity follows from noting that the
sumofthecapacitiesoftwointerference-freelinksisanouter +C |H2,1|2α1,HhP1(cid:16)(H)+|H2,2|2α2,HP2(H) (cid:17)
boUunndifoornmtlhyesItrFoCngs:uOmn-ceacpaancivtye.rifyin a straightforwardman- −C 1+α(cid:16)2,H|H1,2|2P2(H) −C 1+α1,H|H2,1|2P(cid:17)1(H) .
ner that S1(αH,P (H)) is maximized α∗1,H = α∗2,H = 0. (cid:16) (cid:17) (cid:16) (cid:17)i(34)
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We have presented a Han-Kobayashi based achievable
1992,Nov.1997.
scheme for two-sided ergodic fading Gaussian IFCs. Relying
on converse results in [15], we have shown the optimality
of transmitting only common messages for the sub-classes of
EVSandUSIFCs.Forthehybridsub-classes,wehaveshown
that the proposed joint coding scheme does at least as well
as separable coding by exploiting the strong sub-channels to
reduce interference in the weak sub-channels. In contrast to
the one-sided UW sub-class for which ignoring interference
and separable coding are optimal, we have argued here that
in generaljoint coding is requiredfor the two-sided UW sub-
class. While the sum-capacityoptimalschemeisunknownfor
thissub-class,wehavedevelopedasetofsufficientconditions
under which ignoring interference and separable coding are
optimal. Our results are summarizedby the Venn diagrams in
Fig. 2.