1 - interference channels

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60 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-24, NO. 1, -JANUARY 1978

Interference ChannelsAYDANO B. CARLEIAL, MEMBER, IEEE

Abstract-An interference channel is a communication mediumshared by M sender-receiver pairs. Transmission of informationfrom each sender to its corresponding receiver interferes with thecommunications between the other senders and their receivers.This corresponds to a frequent situation in communications, anddefines an M-dimensional capacity region. In this paper, we obtaingeneral bounds on the capacity region for discrete memorylessinterference channels and for linear-superposition interferencechannels with additive white Gaussian noise. The capacity regionis determined in special cases.

I. INTRODUCTION

T HE SITUATION often occurs where severalsender-receiver pairs share a common communicationchannel so that transmission of information from onesender to its corresponding receiver interferes with com-munications between the other senders and their receivers.In radio communications, for example, since the electro-magnetic spectrum is a limited resource, frequency bandsare often simultaneously used by several radio links thatare not completely isolated. A communication channel thatis shared in this manner is called an interference chan-nel.In this paper, we are concerned with the capacity ofmemoryless interference channels for simultaneous com-munications. A definition of the interference channel interms of a more general concept, the interference network,is given in Section II. A broadcast channel [l], for example,is an interference network with a single sender, or inputterminal. A multiple-access channel [2], [3] is an interfer-ence network with a single receiver, or output terminal.General bounds on the capacity region of interferencechannels are presented in Section III of the paper. Boundsfor channels with degraded output signals appear in Sec-tion IV. The capacity region is easily determined when theoutput signals are statistically equivalent. Section V pre-sents the binary and the Gaussian examples representativeof discrete and continuous-alphabet interference channels,respectively.Sections VI and VII are dedicated to the Gaussian in-terference channel. An achievable rate region obtainedthrough signal-superposition techniques is compared withthe region that results from simple time-division or fre-quency-division multiplexing. The capacity region of thisManuscript received October 15, 1976; revised April 26, 1977. Thispaper was presented at the 1976 IEEE International Symposium on In-formation Theory, Ronneby, Sweden, June 21-24. This work was sup-ported in part by the USAF Office of Scientific Research, under ContractF44620-73-C-0065.The author was with the Department of Electrical Engineering,Stanford University, Stanford, CA 94305. He is now with Instituto dePesquisas Espaciais (INPE) (the Brazilian space agency), SBo Jo& dosCampos, &o Paulo 12200, Brazil.

channel does not shrink monotonically when the intensityof the interfering signals is increased. We have previouslyshown [4] that i nterference can be completely eliminatedif it is sufficiently strong.Long proofs omitted in this paper can be found in [5].Some of the results presented here were also independentlyobtained by Sato [6], [7] and Bergmans [a].II. DEFINITIONS AND PRELIMINARIES

Definition 1: An interference network, also referred toas an M-to-N network, is a communication network withM senders, or input terminals, respectively, with alphabetsx1x2, - - * ,XM; N receivers, or output terminals, withalphabets Y 1,Y 2, - - - ,YN, respectively; and a collectionof conditional probability measures on the set of outputsignals, given the input signals.A discrete-time, memoryless, M-to-N network has

I II xii=lconditional distributions

on the output lettersYE fi Yj.

j=lIn this paper, we make the assumption that synchroniza-tion among senders is possible.Definition 2: An interference channel is an M-to-Mnetwork (i.e., an interference network with the samenumber M of senders and receivers) where a one-to-onecorrespondence exists between senders and receivers suchthat each sender communicates information only to itscorresponding receiver.

Thus the interference channel is defined as an inter-ference network that is utilized in a prescribed specializedway. By way of analogy, a channel with feedback is a sim-ilarly restricted version of a two-way channel.An interference channel has M principal links (betweencorresponding terminals) and M(M - 1) interferencelinks. Within the form of utilization prescribed in Defi-nition 2 (see Fig. l), there are only M communication ratesof direct interest. An M-dimensional capacity region isnaturally defined.Definition 3: The capacity region @ of an interferencechannel is the closure of the set of rate vectors R =(RI,&, -. - ,RM) for which jointly reliable communicationsare possible over the M principal links, with independentinformation sources at the input terminals.

001%9448/78/0100-0060$00.75 0 1978 IEEE


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CARLEIAL: INTERFERENCE CHANNELS 61

SOURCES ENCODERS CHANNEL DECODERSFig. 1. Interference channel.

An M-to-N network is utilized in a more general way ifeach sender is allowed to communi cate a separate mes sageto each of the 2N - 1 combinations of receivers, definingan M(2N - 1)dimensional capacity region [5]. We assumethat all information sources connected to the network areindependent so that capacity regions are well-defined.Problems involving interdependent sources have beenstudied with the introduction of new dimensions [3], buta general formulation has not yet been established. An-other specialized use of the M-to-N network, where eachtransmitter sends a messa ge Ui, i = 1,2, * - * ,M, to all Nreceivers, was studied by Ahlswede [9] and Ulrey [lo]. Theyshowed that this problem is essentially one of simultaneousmultiple-access communic ations, and determined thecorresponding M-dimensional capacity region.Since the N receivers of an interference network areisolated amo ng themselves and cannot collaborate in de-coding signals, it follows that if two such networks have thesame respective marginal conditional probability distri-butions pyllx(- ] x), . - - ,pyNl x(- 1 ) for every choice ofinputs x, then their MN c orresponding individual linksare capable of the same communication performance. Ifthe same code is used over both networks, identical de-coders at corresponding receivers have identical errorprobabilities. In this sense, dependencies among the out-puts, given the inputs, are irrelevant, a fact that has beenpointed out for broadcast channels [l]. Interference net-works with the same marginals are thus consideredequivalent. In particular, equivalent networks have thesame capacity region.

III. BOUNDS ON THE CAPACITY REGIONAn M-to-M network can be viewed as a combination ofM broadcast channels or, alternatively, as a combinationof M multiple-access channels. The latter view leads tocrude inner and outer bounds for the capacity region of aninterference channel. These bounds are usually weak, butrelatively easy to compute. They are sketched in Fig. 2 foran example with M = 2.Let px stand for the class of all joint distributions pxon the inputs such that Xi,Xz, -a * ,XM ar e independent.For i = 1,2 a - M, define the principal-link capacities

G = sup I(X,,Yi JX1X p - - * Xi-lXi+r . . . X,).PX

Fig. 2. Simple boun ds on the capacity region of an interferencechannel.

Theorem 1: The capacity region @ of an interferencechannel satisfies %I C @ C %c, with the definitions: ?31k Region bounded by the (M - l)-dimensional hyper-plane containing the rate vectors (Ci,O, a-- ,O), (O&s,* - - ,O), * * * ,(O,O, - - * ,CM); .1RsC Region (parallelepiped)formed by all rate vectors dominated by (Ci,Cs, s-s,CM).

Proof: Consider the ith multiple-access channel, i.e.,the subnetwork formed by the M sender s and the ith re-ceiver. If Ci = 0, the ith dimension of @can be eliminated.If Ci > 0, we know that, given an arbitrary 0 < vi < CL,there exist codes achieving rates Rii = Ci - vi and Rji > 0,for j # i, over this multiple-access channel [3]. Using onesuch code over the interference channel and ignoringoutput signals other than Y;, R = (O,O, -a ,Ci - vi, . - - ,O)is achieved. The argument holds for i = 1,2, - - - ,M. Since@ s closed and the 17~s re arbitrarily small, we can use atime-sharing argument to conclude that .fRzC @.Conversely, again consider the ith multiple-accesschannel defined above. By the converse theorem for itscapacity region [3], reliable communic ation over its ith li nkis not possible at rates exceeding Ci. But this result wouldbe contradicted if any rate vector R with Ri > Ci wereachievable over the interference channel. The argumentholds for i = 1,2, - . - ,M, proving that no point outside theclosed region & belongs to @. Q.E.D.The bounds descr ibed in the remainder of this sectionare in general better than those given in Theorem 1.

A. Inner BoundsConsider a memoryl ess interference channel with two

sender-receiver pairs, discrete alphabets Xl,%&, Y 1, andYs, and conditional distributions pyIy2~x1x2. Given twopositive integers ml and m2, let 21 = (1,2,. - - ,ml) and Zz= (1,2, * * * ,m2] and define independent random variables21 and 22 on these sample spaces, with arbitrarily chosenprobability distributions pzl and pzz. These randomvariables represent, at least in an intuitive sense, portionsof the input i nformation (respectively contained in X1 andX2) that will be decoded at both receivers. Now define thejoint ensemble ZiX1ZsXsYiYs by arbitrarily specifying


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62 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. m24, NO. 1, JANUARY 1978conditional distributions pxllzl and pxz~zz on the inputsignals and writing

PZlXlZ2X2YlY2 = PzlPz2Px1(z1Px2~z2PYlY2Ix1x2~Under these conditions, define the region Rmlmz(21X1,22X2) as the set. of all rate pairs R = (R&z) suchthat it is possible to write RI = RI0 + RI1 and Rz = Rzo +Rzz with nonnegative R 10,R20, R 11, and RZ satisfying thefollowing four conditions.1) At least one pair of the following two pairs of ine-qualities holds:

or

2) Same as 1) with Y2 replacing Y,.3) Rn I QXd112122).4) Rz2 5 W2;YzIWz).The following theorem defines a family of achievableregions. A long but straightforward proof, using results onjointly typical sequences [ 111, [ 121, s given in [5]. It is givenin abbreviated form in the Appendix.Theorem 2: For a given pair of positive integers ml , m2,define

72mlm2 = convex hull of UYiTR,,,,(Z~XJ2X2)where the union is over all distributions pzl,pz2,p~~l~~,p~~l~~. Then take the union over all positive integervalues of ml,m2, to define

9? = closure of UBm,,, .The region R is contained in the capacity region of theinterference channel.

Remarks : If ml 5 ml and rn2 I m2, it is clear that3 mlmz C 9m1rnz~t foll ows that 9? s convex. Computationof these regions may be quite difficult in practice. Theresult can be generalized to M > 2, but its enunciationbecomes awkward.B. Outer Bounds

Theorem 3: The capacity region P of the interferencechannel is contained in .%o A Closure of the convex hull ofthe union, over all distributions px E PX on the inde-pendent inputs of the sets of rate vectors R that satisfy,for i = 1,2,. . . ,M,Ri _< (Xi;Yi IX1X2.. . Xi-lXi+~. . . XM).

This bound is intuitive from consideration of results formultiple-access channels. The formal proof given in [5] isomitted here because the treatment is completely analo-gous to that of [16, pp. 76-821 and [3]. An important im-provement to the bound was obtained by Sate [6] whonoted that the conditionRI + R2 I Z(X1X2;Y1Y2)

can be added to the above in the M = 2 case. In general, 2M- M - 1 new inequalities can be written, because the sumof the rates over several principal links should not exceedthe conditional mutual information between their jointinputs and outputs, given the remainder of the inputs. Thismutual information depends on p ylx, whereas we haveseen that the capacity region depends only on the mar-ginals pyi 1X, i = 1,2, . . . ,M. Thus an outer bound on @ sobtained by intersecting the regions defined by all p YIXthat yield the correct marginals , as pointed out by Sato [6].It turns out that, for some channels with strong interfer-ence, the new inequalities do not improve the bound -Roas given in Theorem 3.

IV. DEGENERATE~NTERFERENCE~HANNELSA straightforward result is the determination of thecapacity region of an interference channel with s tatisticallyequivalent output signals. We say that the output signalsare statistically equivalent if the output alphabets areidentical (or isomorphic ) and, for each choice of inputs x,

p~1l~t-l~) = PY~IX(- ~X) = --- = PY~IX(-I~). The Mmultiple-access channels that compri se the network areall identical in this case. Therefore any decoding operationproducing oi from Yi (at the ith receiving terminal) canequivalently be performed from the output available at anyother receiving terminal, for example the first. It followsthat the capacity region of the interference channel equalsthe capacity region of any one of its multiple-access com-ponents, which is already known [3]. Thus, we have thefollowing theorem.Theorem 4: The capacity region @ of an interferencechannel with statistically equivalent output signals is theclosure of the convex hull of the union over Px of the setsof rate vectors R which satisfyRI 5 1(X1;Y4X2X3**XM). . .RM 5 1(Xn(r;Y4X1Xy**XM-1)RI + R2 5 I(X1X2;Y1)X3X4...XM). . .R2+RS+..- + RM _


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CARLEIAL: INTERFERENCE CHANNELS 63are degraded if, for some ordering of its M sender-receiverpairs, there exist conditional probability distributionsp~~+~fly~, k = 1,2,. . . ,M - 1, such that Ys,Y3/, .. . ,YMare, respectively , statistically equivalent to Yz, Ya, . . . , YMand

k = 1,2,. . . ,M - 1for each choice of inputs x. This concept of degradationis analog ous to that defined for broadcast c hannels [l], [13].It mea ns that; if j > i, then Yj is a degra ded ver sion of Yiin the sense that a signal Yj statistically equivalent to Yjcan be obtained as the output of a cascaded processingchannel with Yi as input. Transmission over this pro-cessing channel depends on X only through Yi.

Theorem 5: Consider an interference channel withoutput signals Yi, Y2, . - . , Y, degraded in this order, andlet @ denote its capacity region. Define a modified inter-ference network with the same input alphabets and withall output signals statistically equivalent to Yi, and let @yldenote the capacity region of the corresponding interfer-ence channel. Similarly, define a modified network withoutputs statistically equivalent to YM, and let e y,+,denotethe capacity region of its interference channel. Then @ ,,,,C @ C @yl, and @yl and ey,+, are given by Theorem 4.

Proof: It is possible to append cascaded processingchannel s to the first M - 1 output terminal s of the originalinterference network, and thereby reduce it to a networkwhere all output signals are equivalent to YM. Therefore,@yIMC e. Now consider the modified network that has allits output signals equivalent to Yi. The region @y, con-tains @ since it is possible to append cascaded processingchannels to the last M - 1 output terminals of this networkin such a way as to reduce it to a network equivalent to theoriginal interference network. Q.E.D.

If we restrict our attention to a proper subset of the Msender-receiver pairs, it is clear that the correspondingoutput signals Yj are still degraded, and therefore thestatement of Theorem 5 applies to the projection of @ onthe subspace comprising the dimensions that correspondto the subset. An inner bound and an outer bound on thisprojection are thus obtained, and bounds on @ can be in-ferred from these. This is a useful observation because, ingeneral, these bounds neither contain nor are containedin the bounds obtained by direct applicati on of the theo-rem to the interference channel as a whole.More generally, the degradation property may holdwithin proper subsets of an interference channelssender-receiver pairs, but not be true for the channel asa whole. The subsets need not form disjoint classes, sincedegradation is not an equivalence relation (it is not sym-metric). The preceding results now apply to the projectionsof e on subspaces comprising dimensions included in asubset. This in turn indirectly provides bounds on 6.

V. EXAMPLES OF INTERFERENCE CHANNELSA. Binary Interference Channel

Interference channels with two senders and two re-ceivers and binary alphabets Xi = X2 = Yi = Y2 = {O,l)can be character ized by eight transition probabilities, forexample PY~~x~x~(~(~~,~~) and PY~~x~x~(~I~~J for(3ti,zz) E (0,1)2. Results obtained for a given binary inter-ference channel apply to all channels with the same tran-sition probabilities under a relabeling of input symbols oroutput symbols. Jelineks canonical decomposition forbinary two-way channels [14] can be applied to binaryinterference channels. The result is that each link is rep-resented by two cascaded stages: the first, called a purealternating channel, contains onl y the effect of interfer-ence; the second, a pure noise stage, does not de pend onthe distribution of the interfering input signal. Analysi sof binary i nterference channel s is surprisi ngly difficult, andthe capacity region is known only in the trivial cases ofchannel s without interference and channel s with statisti-cally equivalent output signals.Let us briefly consider channels with symmetry , or bi-nary symmetric interference channels. Consider a channelwith tr ansition probabilities as given in Table I, with fourparameters 0 2 ei,ti*,tz,ez* 5 1. One interpretation for thischannel is that transmission over the principal links takesplace over binary symmetri c channels (BSCs) with r e-spective crossover parameters ti,tg when the input symbolsare identical, but the parameters are chang ed to ti*,tz*,respectively, if the input symbols disagree. It can be veri-fied that the following less apparent interpretation alsoapplies and is, therefore, equivalent: with probability 1 -ul,Y1 reproduces X 1 over a BSC with crossover parameterfill, but it may instead (with probability ~1) reproduce theother input signal, X2, over a BSC with parameter $21;similarly Ys reproduces X2 over a BSC with parameter 622,with probability 1 - us, or it reproduces Xi over a BSCwith parameter 612.We may think of u] and u2 as inter-ference probabilities. The two formulat ions are identicalif we set

61 = (1 - udk+ Ulfi21q* = (1 - u& + Ul(l - 821)

t2* = (1 - &J&g + ug(1 - 612).It does not suffice to consider only the channels with

tl,cl*,t2,~2* 2 l/2, b ecause not all casescan be reduced to thisform by relabeling input or output symbols. It is easy tosee, however, that ti,cs < l/2 may be assumed without lossof generality. Table I is representative of the class of binarysymmetric interference channels where the interferenceeffect appears in both links as a result of a match or amismatch of input symbols. Another configuration ispossible where the value of the interfering signal (0 or 1)by itself dictates the BSC parameter (t or t*) of the inter-fered link. Thus three other different classes of binary


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64 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-24, NO. 1, JANUARY 1978

TABLE ITRANSITIONPROBABILITIESFORABINARYSYMMETRIC

INTERFERENCECHANNEL

00

01

10

11

Y20 1

1 -E2 E2E*2 1-E;

1-c; 9

E2 l-E2

92 IX,3

symmetric interference channels are obtainable fromTable I: one by exchanging the third and fourth rows of thetable for pyI~xIx2; another, by exchanging the second andfourth rows of the table for py21x1x2; and still another, bymaking both exchanges.B. Gaussian Interference Channel

An important continuous-alphabet channel is the lin-ear-superposition additive-white-Gaussian-noise (AWGN)interference channel, called the Gaussian interferencechannel or AWG N interference channel, for simplicity . Itsinput and output alphabets are the set of real numbers,and the outputs are linear combinati ons of the inputs, plusGaussian noise, i.e.,

or Y = CX + V. The signal-transmission coefficients cijare known constants for a given channel. We also definethe power-transmi ssion coefficients aij A cij2 and thematrix A formed by these coefficients. The noise termsVl,V2,"' , VM are zero-mean Gaussian random variableswith variances Ni,Ns, . . - ,Nn/l (noise powers). The Vj areindependent of the input signal vector X, but need not beindependent among themselves. The conditional proba-bility densities p yj 1 (. 1 ) that result on the outputs areGaussian densities, with respective means (cijcsj . . . .cMj) x and variances Nj, j = 1,2, . . * ,M. In accordance withour assumption of memorylessness, noise at each receiveris a white random process, i.e., a discrete-ti me sequenceof independent copies of Vj. Average power constraintsPlP2," - ,PM on the input signals require that

I 5 Pi, i = 1,2, . . . ,M

for some block length n. A Gaussian interference channelcan be specified by its power-transmission matrix A, itspower-constraint vector P, and its noise-power vectorN.A scaling transformation of a Gaussi an interferencechannel is a transformation on its parameters and inputsignals such that resulting changes in the output signalscan be removed by constant gain factors (attenuation oramplification) at the output terminals. Two Gaussian in-terference channels related by a scaling transformation areequivalent in the sense that they are capable of exactl y thesame communicat ion performance. A channel that has allnoise powers equal to unity and all diagonal elements ofA (or C) also equal to unity (i.e., Ni = 1, aii = 1, i = 1,2,. . . ,M) is said to have the standard form. An arbitrarychannel given by (A,P,N) can be transformed into anequivalent standard-form channel (A,P,l) by scaling asfollows:

aij = aijNi/(aiiNj)

Nj = 1.Codewords for the original channel must be scaled by

and the additive noise is normalized byVj = Vj/* .

Since each equivalence class of Gaussian i nterferencechannels has a standard-form representative, it is suffi-cient to deal with channels in standard form. It is easilyverified that a standard-form AWG N interference channel(i) has statistically equivalent output signals if and onlyif C&j = 1 for all i,j E {1,2, - *. ,MJ; (ii) has degraded outputsignals if and only if aikakj = ai; for all i&j E (1,2, * * * ,MJ.If we restrict PX to include only distributions with vari -ances axL2 s Pi, Theorem 4 gives the capacity region forthe case (i) above; it coincides with the multiple-accesschannels capacity region [15]. Fig. 3 shows an outer boundon @ or PI = 5, P2 = 4, a21 = 1.25, aI2 = 0.80, an exampleof case (ii), obtained by intersecting 930 (defined with theabove restricti on on ?+) with the special bound of Theo-rem 5. A tighter outer bound for this degraded case wasobtained by Sato [7] using an argument where the inputterminals are combined to form a broadcast channel.

VI. ACHIEVABLEREGIONS FORTHE AWGNCHANNELA. Four Rate Vectors

Consider a standard-form AWGN interference channelwith two sender-receiver pairs, average power constraintsPi and P2, and power-transmi ssion coefficients al2 and azlover the two interference links. From these parameters,


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CARLEIAL: INTERFERENCE CHANNELS 65Gaussian n-sequences. Let Pei, Fez, and Fe, respectively,denote the average probabilities of error for the two re-ceivers and for the system as a whole over the ensembl e ofcodes and messages. The coding t heorem for simpleAWGN channels can be invoked to assert that, for suffi-ciently large block l ength, P,i and pes can be made as smallas desired, provided the communication rates are not ex-cessive, by performing maximum-likelihood decoding forXi and X2, respectively, at the first and second receivingterminals, treating the interference as noise. Under theseconditions, pe can be made as small as desired since it isbounded by the sum of P,i and Fez. The conditions on therates are

Fig. 3. Special outer bound for the capacity region of a Gaussi an in-terference channel with degraded output signals, with PI = 5, P2 = 4,a12 = 0.80, a21 = 1.25. Rates are in nats/transmis sion, and C1 = 0.900,C2 = 0.805,S1 = 1.200.we define six useful positive quantities (rates):

cs+og(l+P?)

RI < ;log [l + Pil(1 + c~~~P~)],Rz < ; log [l + Psl(1 + c~~~P~ ,].

The expressions on the right can be made as close to D1and 02 as desired by appropriately choosing PI and P2.On the other hand, the subset of codes in the ensembl e thatdo not satisfy both power constraints has vanishingprobability for increasing n, because PI < PI and P2

1 - interference channels

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