The Two User Gaussian Compound Interference Channel

The Tw o User Gaussian Comp ound In terference Channel Adnan Ra ja, Vino d M. Prabhak aran and Pra m o d Visw ana th Octob er 29, 2018 Abstract W e int ro duce the t w o u ser ﬁnite state comp ound Gaussian inte rfer en ce c han- nel and c haracterize its capacit y region to within one bit. The main con tributions in vo lve b oth no v el inner and outer b ounds. The inner b ound is m ultilev el sup er - p osition cod ing but the deco ding of the level s is op p ortu nistic, dep ending on the c hannel state. The genie ai ded outer b ound is motiv a ted by the t ypical er r or ev ents of th e ac hiev a b le sc heme. 1 In tro duc tion The fo cus of this pap er is the commun ication scenario depicted in Fig ure 1. Two transmitter- receiv er pairs comm unicate reliably in the face of in terference. The discrete time complex baseband mo del is: y 1 [ m ] = h 11 x 1 [ m ] + h 21 x 2 [ m ] + z 1 [ m ] , (1) y 2 [ m ] = h 12 x 1 [ m ] + h 22 x 2 [ m ] + z 2 [ m ] . (2) Here m is the time index, y k is the signal at receiv er k while x k is the signal sen t out b y the transmitter k (w ith k = 1 , 2). The noise sequences { z 1 [ m ] , z 2 [ m ] } m are memoryless complex Gaussian with zero mean and unit v ariance. The transmitters are sub ject to a v erage pow er constraints: N X m =1 | x k [ m ] | 2 ≤ N P k , k = 1 , 2 , ∀ N ≥ 1 . (3) The complex parameters { h k ℓ , ℓ = 1 , 2 , k = 1 , 2 } mo del the ch annel co eﬃcien ts b etw e en the pairs of transmitters and receiv ers. They do not v ary with time but the tr a nsmitters and receiv ers hav e diﬀerent inf ormation ab out them: 1 • Receiv er k is exactly aw are of the t w o c hannel coeﬃcien ts h 1 k , h 2 k ; this mo dels c oher ent comm unication. • T ransmitters are only c o ars e ly aw are of the c hannel co eﬃcien ts: the transmitters kno w that the c hannel co eﬃcien ts b elong to a ﬁnite set. Sp eciﬁcally , bo t h the transmitters kno w that ( h 1 k , h 2 k ) ∈ A k , k = 1 , 2 . (4) This mo dels p oten tial partial feedbac k to the tra nsmitters r ega rding the c hannel co eﬃcien ts. A more general comp ound channel mo del a llo ws for all four channe l parameters to join tly tak e on diﬀeren t choices : ( h 11 , h 12 , h 21 , h 22 ) ∈ A . (5) Ho w ev er, since the receiv ers do not co op erate in the in terference c hannel, it turns out that the setting in Equation (5 ) is no more general t ha n the one in Equation (4). This is explored in Section 8. + ❤ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❘             ✒ ✻ ❄ ✲ ✲ ✲ ✲ x 1 x 2 h 22 h 11 z 2 z 1 h 21 h 12 y 1 y 2 + ❤ Figure 1: The t wo user Ga uss ian interferenc e c hannel. The k ey problem of in terest is the c haracterization of the capacit y regio n: the set of rate pairs at whic h ar bitrarily reliable comm unication b etw ee n the tw o transmitter- receiv er pairs. The “comp ound” asp ect of the channel is in insisting that the receiv ers b e able to deco de the messages of interes t with arbitrar ily high probability , no matt er whic h of the ﬁnite states the c hannel co eﬃcien ts take on. Our main result is a c haracterization of the capa city region up to one bit. 2 A sp ecial instance of the problem studied here is the classical tw o user G aussian in terference c hannel: in a recen t w ork, Etkin, Tse and W a ng [4] show e d that a sing le sup erposition co ding sc heme (a sp eciﬁc c hoice among the broad class of sc hemes ﬁrst iden tiﬁed by Ha n and Kobay ashi [1]) ac hiev es p erformance within one bit of the capacity region. The transmiss ion inv olv ed splitting the data in to t w o parts – o ne public and the other private – and linearly sup erposing them. T he idea is that the public dat a stream is decoded by b oth the receiv ers while the priv ate data stream only by the receiv er of in terest. The k ey iden tity of t he prop osed sup erp osition sc heme is the f ollo wing: the p o w er allo cated to the pr iv ate stream is such that it app ears at exactly the same lev el as the bac kground noise at the uninte nded receiv er (the idea is that since the priv ate data stream is b eing treated as noise at the unin tended receiv er, there is no extra incen tiv e to reduce its level eve n further than that of the additive noise). A no v el outer b ound dev elop ed in [4] show ed that this simple sup erpo sition sc heme is within one bit of the capacit y region. Implemen tation of the sp eciﬁc sup erp osition sc heme pro posed ab o v e requires eac h transmitter to be a w are o f the in terference lev el it is causing to the uninte nded receiv er. In the context of the comp ound c hannel b eing studied here, the transmitter is not aw are o f the in terference lev el; this p oses an obstacle to adopting the idea of a ppropriately choosing the p o w er lev el of the priv ate data stream. One p ossibilit y could b e to set the p o w er level of the priv ate data stream based on the strongest in terfering link lev el (among the set o f p ossible c hoices) – this would ensure that it is only receiv ed b elow noise lev el when the in terfering link lev el tak es o n the other p ossible choices . Ho w ev er, this a ppro ac h migh t b e to o p essimis tic and its closeness to optimalit y is unclear. W e circum ve nt this problem by prop osing t he following no ve l tw ist to the general sup erposition co ding sche me. Our main idea is b est describ ed when the in terference links ( h 12 and h 21 in F igure 1) tak e on only t w o p ossible v alues and the direct links are ﬁxed (i.e., the sets A 1 and A 2 ha v e cardinality of tw o, cf. Equation (4)). W e no w sup erp ose thr e e data streams at eac h transmitter. Tw o of them, public and priv ate, ar e as earlier: all receiv ers in all c hannel states deco de the public message while only the receiv er of in terest deco des the priv a te message (no matter t he c hannel state, aga in). The no v elty is in the third data stream that w e will call semi-public : this data stream is decoded b y the uninten ded r ece ive r only when the in terference link is the stronger of the t w o c hoices (and treated as noise ot herwise). As suc h, this data stream is neither fully priv a te nor public (the unin tended receiv er either treats it as noise or deco des it based on the channe l state) and the nomenclature is c hosen to highligh t this feature. The p ow er split rule is the follow ing: the p o w er of the priv a te stream is set suc h that at the higher of the interference link lev els, it is receiv ed at the unin tended receiv er at the same level as the additive noise. The p ow er of the semi-public data stream is set suc h that it is receiv ed at the unin tended receiv er at the same lev el as the additive noise o nly when the in terference link leve l is a t the low er o f the tw o possible choice s. The rationale is that the semi-public data stream is not deco ded only when the interference link lev el 3 is at the low er of the t wo p ossible choices , and th us it can tr a nsmit hig her p o wer t han if its pow er is restricted b y the higher o f the in terference link lev els. This approac h scales naturally when the interference link lev els can take on more than t w o p ossible c hoices (the n umber of splits of the data stream is one more tha n the cardinalit y of the set of p ossible c hoices). W e deriv e no ve l outer b ounds to sho w tha t our simple ac hiev able sche me is within one bit of the capacit y region. Our o uter b ounds are genie aided and are based on the clues pro vided b y the typic al err or events in the achiev able sche me. This a ppro ac h sheds op erational insigh t into the nature of the out er b ounds ev en in the noncomp ound v ersion (th us eliminating the “guessw ork” inv olve d in the deriv a tion, cf. Section IV of [5]). The paper is orga nized as follo ws: we start with a simple t wo-state comp ound in t er- ference c hannel. In this setting, b oth the direct and in terference link lev els can tak e on only one of t w o p ossible v alues (so the sets A 1 and A 2 ha v e cardinalit y t w o). Using a somewhat abstract setting (describ ed in Section 2) t ha t features the Ga uss ian problem of in terest as a sp ecial case, w e presen t our main results (b oth inner and outer b ounds) for this t wo-state comp ound interferenc e c hannel. Our deﬁnition of the a bstract setting is motiv a ted b y that c hosen in [5] and could b e view ed as a natural comp ound vers ion of the inte rference c hannel studied b y T elatar and Tse [5]. This is do ne in Section 3. W e discuss the insigh ts garnered from these results in the con text of the simpler noncom- p ound in terference c hannel in Section 6. Next, w e a r e ready to set up the mo del and describe the solution the more general ﬁnite state interferenc e c hannel; w e do this ﬁrst in the abstract setting (Section 7) follow ed by sp ecializing to the Gaussian scenario of in terest (Section 8). 2 Mo del Consider the t w o-user, t w o-state compound memoryless in terference c hannel depicted in Figure 2. There are t w o transmitters whic h w an t to reliably comm unicate indep enden t messages to t wo corresponding receiv ers. The input to the c hannel from the ﬁrst tra ns- mitter at an y discrete time X 1 ∈ X 1 passes through a degraded discrete memoryless broadcast c hannel: the t wo outputs of the degraded broadcast c hannel are S 1 α ∈ S 1 and (the degraded v ersion) S 1 β ∈ S 1 . Similarly , at a n y time, the input to the channel from transmitter 2 X 2 ∈ X 2 pro duces S 2 α ∈ S 2 and a degraded v ersion S 2 β ∈ S 2 of it. The c hannel to an y one of the tw o receiv ers is decided by the state of that receiv er: here there are only t w o states α and β . Once the state is decided, it is ﬁxed for the entire durat io n of comm unicatio n. When the ﬁrst receiv er is in state α , the output at any time is Y 1 α = f 1 α ( X 1 , S 2 α ) ∈ Y 1 . (6) Similarly , when the ﬁrst receiv er is in state β , the output at any time is Y 1 β = f 1 β ( X 1 , S 2 β ) ∈ Y 1 . (7) 4 Y 1 α q q q q q ✻ ❄ ✲ ❄ ❄ ❄ ✲ ✻ ✻ ✻ ✲ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✲ ✲ ✲ ✲ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ✲ X 2 X 1 f 1 α f 1 β p S 2 β | S 2 α p S 2 α | X 2 p S 1 α | X 1 p S 1 β | S 1 α f 2 β Y 2 α f 2 α Y 2 β Y 1 β q Figure 2: A tw o-state comp ound c hannel mo del. Here f 1 α and f 1 β are deterministic functions suc h that for ev ery x 1 ∈ X 1 , s 2 ∈ S 2 , a nd η = α, β , the following function is inv ertible: f 1 η ( x 1 , . ) : S 2 → Y 1 . Lik ewise, the outputs of user 2 under the t w o p ossible states the c hannel to it can ta k e are deﬁned using similar deterministic functions f 2 α and f 2 β . W e allo w eac h receiv er to b e in p oten tially diﬀeren t states, and they are b oth aw are of the state they are in. A pa ir of comm unication rates ( R 1 , R 2 ) is said to be achievable if f or ev ery ǫ > 0, there are blo c k length n encoders, enc k : { 1 , . . . , M k } → X n k , M k ≥ 2 n ( R k − ǫ ) , k = 1 , 2 , (8) and deco ders dec k η : Y n k → { 1 , . . . , M k } , k = 1 , 2 , η = α , β , (9) suc h that 1 M 1 M 2 X m 1 ,m 2 Pr  dec k ,η ( Y n k η ) = m k , k = 1 , 2 , η = α, β | X n k = enc k ( m k ) , k = 1 , 2  ≥ 1 − ǫ. (10) W e are in terested in the capacity region C , whic h is the set of all achiev able ( R 1 , R 2 ) pairs. W e can mak e a few observ a t ions: 5 • The channe l describ ed here can b e tho ugh t of as a natural generalization of that studied in [5]. • An imp ortan t sp ecial case o ccurs when the c hannels p S kα | X k and p S kβ | S kα are deter- ministic for b oth k = 1 , 2. This c hannel is a comp ound v ersion of the deterministic c hannel considered by El Ga mal and Costa [2] with the in terference in state β being a deterministic function of the in terference in state α . • The comp ound Gaussian in terference channe l, with the cardinalit y of b oth the sets A 1 and A 2 restricted to 2 (in the nota t io n intro duced in Section 1), is a sp ecial instance of the mo del in Figure 2. W e start with a comp ound Gaussian interference c hannel with ( h 11 , h 21 ) ∈ { ( h 11 α , h 21 α ) , ( h 11 β , h 21 β ) } , ( h 22 , h 12 ) ∈ { ( h 22 α , h 12 α ) , ( h 22 β , h 12 β ) } . F urther, without loss of generalit y , w e can assume that | h 21 α | ≥ | h 21 β | , (11) | h 12 α | ≥ | h 12 β | . (12) With the following assignmen t, we see that the mo del in Figure 2 can capture t he Gaussian mo del in F ig ure 1: S 1 α = h 12 α X 1 + Z 2 , (13) S 1 β = h 12 β h 12 α S 1 α + 1 −     h 12 β h 12 α     2 ! 1 / 2 Z ′ 2 , (14) S 2 α = h 21 α X 2 + Z 1 , (15) S 2 β = h 21 β h 21 α S 2 α + 1 −     h 21 β h 21 α     2 ! 1 / 2 Z ′ 1 , (16) Y 1 α = f 1 α ( X 1 , S 2 α ) = h 11 α X 1 + S 2 α , (17) Y 1 β = f 1 β ( X 1 , S 2 β ) = h 11 β X 1 + S 2 β , (18) Y 2 α = f 2 α ( X 2 , S 1 α ) = h 22 α X 2 + S 1 α , (19) Y 2 β = f 2 β ( X 2 , S 1 β ) = h 22 β X 2 + S 1 β . (20) Here Z 1 , Z ′ 1 , Z 2 and Z ′ 2 are indep enden t complex Gaussian random v ariables with unit v ariance. 3 Main Res ult Our main r esults o n the 2- state comp ound in terference c hannel a re the following. 6 • W e ﬁrst sho w t he p erformance of an ac hiev able sc heme and hence characterize an inner-b ound; • next, we giv e an outer-b ound to the capa city region and quan tify the gap b et w een the outer-b ound and the achie v a ble sc heme; • sp ecializing to the compo und deterministic interferenc e c hannel, we completely c har- acterize the capacit y region; • sp ecializing to the comp ound Gaussian in terference c hannel, w e characterize the capacit y region up to a gap of 1 bit (at all op erating SNR v a lues and all channel parameter v alues). 3.1 Inner-b ound: Ac hiev able Sc heme The ac hiev able sc heme is c haracterized by P , the set of random v ariables ( Q, X 1 , U 1 α , U 1 β , X 2 , U 2 α , U 2 β ) , (21) suc h that the following Marko v chain is satisﬁed: U 1 β − U 1 α − X 1 − Q − X 2 − U 2 α − U 2 β . Alternativ ely , t he joint probabilit y distribution function factors as p ( q , x 1 , , u 1 α , u 1 β , x 2 , u 2 α , u 2 β ) = p ( q ) p ( x 1 | q ) p ( u 1 α | x 1 ) p ( u 1 β | u 1 α ) p ( x 2 | q ) p ( u 2 α | x 2 ) p ( u 2 β | u 2 α ) . (22) Our ac hiev able sc heme is a multile ve l sup erp osition co ding one and can b e view ed as a generalization of t he tw o-level sup erp osition co ding sc heme of Chong et al. [3]. The random co ding metho d can b e in tuitiv ely describ ed as follo ws, using the “cloud-cente r” analogy from Co v er and Thomas (Section 1 4.6.3, [8]); a formal statemen t a nd its pro of follo w later. The random v aria bles U 1 β and U 2 β are used to generate the outermost co de b o oks (with rate R 1 β and R 2 β , respectiv ely) for the t w o users. These messages enco ded via these co de b o oks are deco ded b y b oth receiv ers and, as suc h, can b e in terpreted as public information. Next, the ra ndom v ariables U 1 α and U 2 α are used to generate the next lev el of co de b o oks (with rate R 1 α and R 2 α , r esp ectiv ely). The messages enco ded via these co de b o oks are decoded b y the r ece ive r with stronger inte rference (i.e. Rx k α ) but treated as noise b y the receiv er with we aker interferenc e ( i.e. Rx k β ); as suc h, these messages can b e view ed as semi-public information. Finally , the messages enco ded via the inner most co de b o oks (rates R 1 p and R 2 p ) are only deco ded b y the receiv er of in terest; th us this constitutes private informat io n. Giv en P ∈ P , w e deﬁne the six-dimensional region R (6) in ( P ) , { ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) : satisfying (24) − (55) } . (23) 7 R 1 p ≤ I ( Y 1 β ; X 1 | U 1 α , U 2 β , Q ) = γ 11 (24) R 2 β + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 α , Q ) = γ 12 (25) R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 1 β , U 2 β , Q ) = γ 13 (26) R 2 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 β , Q ) = γ 14 (27) R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 2 β , Q ) = γ 15 (28) R 2 β + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | Q ) = γ 16 (29) R 1 p ≤ I ( Y 1 α ; X 1 | U 1 α , U 2 α , Q ) = δ 11 (30) R 2 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 α , U 2 β , Q ) = δ 12 (31) R 2 β + R 2 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 α , Q ) = δ 13 (32) R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 | U 1 β , U 2 α , Q ) = δ 14 (33) R 2 α + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 β , U 2 β , Q ) = δ 15 (34) R 2 β + R 2 α + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 β , Q ) = δ 16 (35) R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 | U 2 α , Q ) = δ 17 (36) R 2 α + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 2 β , Q ) = δ 18 (37) R 2 β + R 2 α + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | Q ) = δ 19 (38) R 2 p ≤ I ( Y 2 β ; X 2 | U 2 α , U 1 β , Q ) = γ 21 (39) R 1 β + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | U 2 α , Q ) = γ 22 (40) R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 | U 2 β , U 1 β , Q ) = γ 23 (41) R 1 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | U 2 β , Q ) = γ 24 (42) R 2 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 | U 1 β , Q ) = γ 25 (43) R 1 β + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | Q ) = γ 26 (44) R 2 p ≤ I ( Y 2 α ; X 2 | U 2 α , U 1 α , Q ) = δ 21 (45) R 1 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 α , U 1 β , Q ) = δ 22 (46) R 1 β + R 1 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 α , Q ) = δ 23 (47) R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 | U 2 β , U 1 α , Q ) = δ 24 (48) R 1 α + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 β , U 1 β , Q ) = δ 25 (49) R 1 β + R 1 α + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 β , Q ) = δ 26 (50) R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 | U 1 α , Q ) = δ 27 (51) R 1 α + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 1 β , Q ) = δ 28 (52) R 1 β + R 1 α + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | Q ) = δ 29 (53) R 1 p + R 1 α + R 1 β ≥ 0 (54) R 2 p + R 2 α + R 2 β ≥ 0 . (55) 8 W e deﬁne the t w o-dimensional regio n, R in ( P ) , { ( R 1 , R 2 ) : R 1 = R 1 p + R 1 α + R 1 β , R 2 = R 2 p + R 2 α + R 2 β , ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) ∈ R (6) in ( P ) } . (56) In ot her words R in ( P ) is the pro jection of the six-dimensional p olytop e R (6) in ( P ). One approac h to take the pro jection, is to do the F ourier-Mot zkin elimination, as done for the basic superp osition co ding sc heme in the contex t of the regular (noncomp ound) interfer- ence c hannel [3 ]. D oing this explicitly is rather cum b ersome as the inequalities here a re m uc h more in num b er than the inequalities that w ere handled by Chong et al. in [3]. Theorem 1 The c ap ac ity r e g ion C sa tisﬁ e s C ⊇ [ P ∈P R in ( P ) . (57) Pr o o f : A formal description of the achie v a ble sc heme a nd the pro of of this theorem are a v ailable in Section 4.1. ✷ P articularizing, w e restrict ourselv es to a subset of P deﬁned as follows. Give n r andom v ariables ( Q, X 1 , X 2 ) suc h that X 1 and X 2 are conditionally indep enden t when conditioned on Q , we deﬁne ra ndom v ariables U 1 α and U 1 β whic h tak e v alues in S 1 , and U 2 α and U 2 β whic h take v a lues in S 2 . They ar e join t ly distributed with ( Q, X 1 , X 2 ) according to the conditional distribution p ( u 1 α , u 1 β , u 2 α , u 2 β | q , x 1 , x 2 ) = p S 1 α | X 1 ( u 1 α | x 1 ) p S 1 β | S 1 α ( u 1 β | u 1 α ) p S 2 α | X 2 ( u 2 α | x 2 ) p S 2 β | S 2 α ( u 2 β | u 2 α ) . (58) Note t ha t, conditioned on Q , w e ha ve the follo wing t w o Mark o v c hains, with the sets of random v ariables in v olved in the t w o c hains b eing conditionally indep enden t. U 1 β − U 1 α − X 1 − S 1 α − S 1 β U 2 β − U 2 α − X 2 − S 2 α − S 2 β . Our ch oice is motiv ated by the c hoice in the pap er by T elatar and Tse [5]. Ev ery mem b er of this family is uniquely determined b y joint random v ariables ( Q, X 1 , X 2 ) suc h that X 1 − Q − X 2 is a Marko v c hain. W e will henceforth denote the corresp onding r egio ns R (6) in ( P ) b y R (6) in ( Q, X 1 , X 2 ) and R in ( P ) b y R in ( Q, X 1 , X 2 ). W e now hav e t he natura l result: 9 Corollary 2 C ⊇ [ Q,X 1 ,X 2 R in ( Q, X 1 , X 2 ) , (59) wher e the union is over al l ( Q, X 1 , X 2 ) such that X 1 − Q − X 2 is a Markov chain. Pr o o f : F ollo ws directly from Theorem 1. ✷ W e observ e that the F ourier-Motzkin eliminatio n pro cedure to implemen t the pro jec- tion op eration in obtaining R in ( Q, X 1 , X 2 ) w ould yield only a ﬁnite set of inequalities. F urther, t he righ t hand sides of these inequalities w ould b e linear functions of p ( q ) a nd for a ﬁxed Q = q 0 the righ t hand sides form a closed set of ﬁnite dimensions. Th us, by Carath ` eo dory’s theorem, w e can conclude that the cardinality of Q can tak en to b e ﬁnite without loss of generalit y in the union in Equation (59). 3.2 Outer-b ound Theorem 3 F o r e v ery ( Q, X 1 , X 2 ) such that X 1 − Q − X 2 is a Markov chain , ther e is a r e g i o n R out ( Q, X 1 , X 2 ) ⊆ R 2 + such that the fol lowing ar e true: (i) C ⊆ [ Q,X 1 ,X 2 R out ( Q, X 1 , X 2 ) , (60) wher e the union is over al l ( Q, X 1 , X 2 ) such that X 1 − Q − X 2 is a Markov chain . (ii) I f ( R 1 , R 2 ) ∈ R out ( Q, X 1 , X 2 ) , then ( R 1 − ∆ 1 , R 2 − ∆ 2 ) ∈ R in ( Q, X 1 , X 2 ) , wher e ∆ 1 ( Q, X 1 , X 2 ) = max( I ( X 2 ; S 2 α | U 2 α ) , I ( X 2 ; S 2 β | U 2 β )) , (61) ∆ 2 ( Q, X 1 , X 2 ) = max( I ( X 1 ; S 1 α | U 1 α ) , I ( X 1 ; S 1 β | U 1 β )) , (62) in which the r andom va ri a b les ar e jo i n tly distribute d a c c or di ng to (58) and the chan- nel c onditional distributions. Pr o o f : P art(i) is prov ed in Section 5.1. P art(ii) is prov ed in Section 5.2. ✷ The se t R out ( Q, X 1 , X 2 ) is deﬁned in Section 5 .1 . Our deﬁnition is motiv ated by the external r epr esen tation of R in ( Q, X 1 , X 2 ) that w e obtain in Section 4.2. 10 3.3 Sp ecial Cases Our mo del captures tw o imp ortant sp ecial cases: • the comp ound deterministic in terference c hannel; • the comp ound G aussian inte rference c hannel, as discussed in Section 2. Thus our results apply to these cases (readily f o r the determin- istic c hannel, and with an appropriate approximation result to the contin uous alphab et Gaussian c hannel). Moreo v er, the structure aﬀorded by these sp ecial cases allow s us to deriv e further insigh t in to the nature of the general results deriv ed earlier. 3.3.1 Comp ound Deterministic In terference Channel In this instance, the capacity region is exactly describ ed. Corollary 4 F or the deterministic c o m p ound interfer enc e ch annel, the inne r b ound in The or em 1 is the c ap acity r e gio n . Pr o o f : The pro of is elemen tary . When the channel is deterministic, we see that the gap claimed b y Theorem 3 ∆ 1 ( Q, X 1 , X 2 ) = ∆ 2 ( Q, X 1 , X 2 ) = 0 . (63) This completes the pro of. ✷ 3.3.2 2 -state Comp ound Gaussian In terference Channel F or the Gaussian v ersion, w e can c haracterize the capacity to within one-bit. Corollary 5 F or the 2 -state c omp o und Gaussian interfer enc e channel, the achievable r e g i o n of The or em 1 is within at most one bit of the c ap acity r e gion. Pr o o f : F or the Gaussian channel, eac h of the mutual info rmation t erms in the expressions for ∆ 1 ( Q, X 1 , X 2 ) and ∆ 2 ( Q, X 1 , X 2 ) can b e upp er b ounded b y 1 bit. T o see this, note that S 1 α = h 1 α X 1 + N 1 α and U 1 α = h 1 α X 1 + N ′ 1 α , where N 1 α and N ′ 1 α are indep enden t and iden tically distributed memoryless complex Gaussian random v a r iables. Hence I ( X 1 ; S 1 α | U 1 α ) = h ( S 1 α | U 1 α ) − h ( N 1 α ) ≤ h ( S 1 α − U 1 α ) − h ( N 1 α ) = 1 . 11 Similarly , I ( X 1 ; S 1 β | U 1 β ) ≤ 1 , (64) I ( X 2 ; S 2 α | U 2 α ) ≤ 1 , (65) I ( X 2 ; S 2 β | U 2 β ) ≤ 1 . (66) ✷ Additionally , w e can use Ga uss ian co de b o oks to get to within one bit of the capacit y . Corollary 6 F or the 2 -state c om p ound Gaussian interfer enc e chan nel, C ⊆ R out ( Q ∗ , X ∗ 1 , X ∗ 2 ) , (67) wher e Q ∗ = 1 , X ∗ 1 ∼ C N (0 , P 1 ) , X ∗ 2 ∼ C N (0 , P 2 ) . This implies that R in ( Q ∗ , X ∗ 1 , X ∗ 2 ) is within one- b it of the c ap a city r e gion C of the 2-state Gaussia n c omp ound interfer enc e channel. Pr o o f : See Section 5.3. ✷ 4 An Ac hie v a ble S c heme W e will presen t a natural, and no v el, ac hiev able sc heme ﬁrst. W e will ev aluate the set of reliable comm unication rates using this strategy and hence ch ara cterize an inner b ound to the capacity region; this will complete the pro of o f Theorem 1. Next, w e will see some imp ortan t geometric prop erties of the ac hiev able rate region. 4.1 Pro of Of Theorem 1 Our co ding sc heme is a natural generalization of the sche me of Chong et a l. [3]. Since there are t w o p ossible states for b oth receiv ers, eac h encoder now sends tw o sets of com- mon information, with the receiv ers opp ortunistically decoding the common information (dep ending on the state). w e c ho ose the random v aria bles corresp onding to the t w o sets of common informa t ion in a degraded manner, fo llo wing the same ordering of degra dedne ss of the interference s under the t w o states (c.f. X k − S k α − S k β , k = 1 , 2) . (68) Fix a P ∈ P . 12 Co debo ok Generation Generate a co dew o rd Q n of length n , generating eac h elemen t i.i.d. a ccording to Π n i =1 p ( q i ). F or the co dew ord Q n , generate 2 nR 1 β indep enden t co dew ords U n 1 β ( j 1 ) , j 1 ∈ { 1 , 2 , · · · , 2 nR 1 β } , (69) generating eac h elemen t i.i.d. according to Π n i =1 p ( u 1 β i | q i ). F or each of the co dew ords U n 1 β ( j 1 ), generate 2 nR 1 α indep enden t co dew ords U n 1 α ( j 1 , k 1 ) , k 1 ∈ { 1 , 2 , · · · , 2 nR 1 α } , (70) generating eac h elemen t i.i.d. according to Π n i =1 p ( u 1 αi | u 1 β i , q i ). F or eac h of the co dew ords U n 1 α ( j 1 , k 1 ), g ene rat e 2 nR 1 p indep enden t co dew ords X n 1 ( j 1 , k 1 , l 1 ) , l 1 ∈ { 1 , 2 , · · · , 2 nR 1 p } , (71) generating eac h elemen t i.i.d. according to Π n i =1 p ( x 1 | u 1 αi , u 1 β i , q i ). Similarly generate co de b o o ks U n 2 β ( j 2 ) , j 2 ∈ { 1 , 2 , · · · , 2 nR 2 β } , (72) U n 2 α ( j 2 , k 2 ) , k 2 ∈ { 1 , 2 , · · · , 2 nR 2 α } , (73) X n 2 ( j 2 , k 2 , l 2 ) , l 2 ∈ { 1 , 2 , · · · , 2 nR 2 p } . (74) Enco ding T ransmitter Tx 1 sends X n 1 ( j 1 , k 1 , l 1 ) to communicate the message index ed b y ( j 1 , k 1 , l 1 ). T ransmitter Tx 2 sends X n 2 ( j 2 , k 2 , l 2 ) to comm unicate the message indexed by ( j 2 , k 2 , l 2 ). Deco ding The receiv ers do join t typical set deco ding. Let A ( n ) ǫ (Ω) denote the se t of join tly t ypical sequence s ω n where Ω is the probability space con taining the en tire collection of random v ariables. Receiv er Rx 1 β determines a unique  ˆ j 1 , ˆ k 1 , ˆ l 1  and an y ˆ j 2 suc h that  Q n , U n 1 β  ˆ j 1  , U n 1 α  ˆ j 1 , ˆ k 1  , X n 1  ˆ j 1 , ˆ k 1 , ˆ l 1  , U n 2 β  ˆ j 2  , Y n 1 β  ∈ A ( n ) ǫ ( Q, U 1 β , U 1 α , X 1 , U 2 β , Y 1 β ) . It declares an error if it fails to ﬁnd suc h a c hoice. 13 Receiv er Rx 1 α determines a unique  ˆ j 1 , ˆ k 1 , ˆ l 1  and a ny  ˆ j 2 , ˆ k 2  suc h that,  Q n , U n 1 β  ˆ j 1  , U n 1 α  ˆ j 1 , ˆ k 1  , X n 1  ˆ j 1 , ˆ k 1 , ˆ l 1  , U n 2 β  ˆ j 2  , U n 2 α  ˆ j 2 , ˆ k 2  , Y n 1 β  ∈ A ( n ) ǫ ( Q, U 1 β , U 1 α , X 1 , U 2 β , U 2 α , Y 1 α ) . It declares an error if it fails to ﬁnd suc h a c hoice. Similar deco ding is done by receiv ers Rx 2 β and R x 2 α . F rom the a nalysis of the probabilit y of error, we sho w in App endix A that the rate v ector ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) is achiev able if it satisﬁes the followin g conditions: R 1 p ≤ I ( Y 1 β ; X 1 | U 1 α , U 2 β , Q ) , if R 1 p > 0 , (75) R 2 β + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 α , Q ) , if R 1 p > 0 , (76) R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 1 β , U 2 β , Q ) , if R 1 α + R 1 p > 0 , (77) R 2 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 β , Q ) , if R 1 α + R 1 p > 0 (78) R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 2 β , Q ) , if R 1 β + R 1 α + R 1 p > 0 , (79) R 2 β + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | Q ) , if R 1 β + R 1 α + R 1 p > 0 , (80) R 1 p ≤ I ( Y 1 α ; X 1 | U 1 α , U 2 α , Q ) , if R 1 p > 0 , (81) R 2 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 α , U 2 β , Q ) , if R 1 p > 0 , (82) R 2 β + R 2 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 α , Q ) , if R 1 p > 0 , (83) R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 | U 1 β , U 2 α , Q ) , if R 1 α + R 1 p > 0 , (84) R 2 α + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 β , U 2 β , Q ) , if R 1 α + R 1 p > 0 , (85) R 2 β + R 2 α + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 1 β , Q ) , if R 1 α + R 1 p > 0 , (86) R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 | U 2 α , Q ) , if R 1 β + R 1 α + R 1 p > 0 , (87) R 2 α + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | U 2 β , Q ) , if R 1 β + R 1 α + R 1 p > 0 , (88) R 2 β + R 2 α + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 α ; X 1 , U 2 α | Q ) , if R 1 β + R 1 α + R 1 p > 0 , (89) R 2 p ≤ I ( Y 2 β ; X 2 | U 2 α , U 1 β , Q ) , if R 2 p > 0 , (90) R 1 β + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | U 2 α , Q ) , if R 2 p > 0 , (91) R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 | U 2 β , U 1 β , Q ) , if R 2 α + R 2 p > 0 , (92) R 1 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | U 2 β , Q ) , if R 2 α + R 2 p > 0 , (93) R 2 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 | U 1 β , Q ) , if R 2 β + R 2 α + R 2 p > 0 , (94) R 1 β + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 β ; X 2 , U 1 β | Q ) , if R 2 β + R 2 α + R 2 p > 0 , (95) 14 R 2 p ≤ I ( Y 2 α ; X 2 | U 2 α , U 1 α , Q ) , if R 2 p > 0 , (96) R 1 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 α , U 1 β , Q ) , if R 2 p > 0 , (97) R 1 β + R 1 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 α , Q ) , if R 2 p > 0 , (98) R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 | U 2 β , U 1 α , Q ) , if R 2 α + R 2 p > 0 , (99) R 1 α + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 β , U 1 β , Q ) , if R 2 α + R 2 p > 0 , (100) R 1 β + R 1 α + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 2 β , Q ) , if R 2 α + R 2 p > 0 , (101) R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 | U 1 α , Q ) , if R 2 β + R 2 α + R 2 p > 0 , (102) R 1 α + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | U 1 β , Q ) , if R 2 β + R 2 α + R 2 p > 0 , (103) R 1 β + R 1 α + R 2 β + R 2 α + R 2 p ≤ I ( Y 2 α ; X 2 , U 1 α | Q ) , if R 2 β + R 2 α + R 2 p > 0 , (104) R 1 p ≥ 0 , (105) R 1 α ≥ 0 , (106) R 1 β ≥ 0 , (107) R 2 p ≥ 0 , (108) R 2 α ≥ 0 , (109) R 2 β ≥ 0 . (110) Note t ha t (75)-(80) are the deco dabilit y conditions at Rx 1 β ; ( 8 1)-(89) are the deco d- abilit y conditio ns at Rx 1 α ; (90)-(95) are the deco dabilit y conditions at Rx 2 β ; (96)-(104) are the deco dabilit y conditions at Rx 2 α and (105)-(1 10 ) are stating the fact that the rates are nonnegativ e real n um b ers. Deﬁne ˜ R (6) in ( P ) , { ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) : satisﬁes (75)-(110) } , (111) and its pro jection onto the tw o dimension space ( R 1 , R 2 ) by ˜ R in ( P ). Lemma 7 R in ( P ) ⊆ ˜ R in ( P ) . Pr o o f : See App endix B. ✷ Th us, we ha ve show n that the capacit y r egio n C satisﬁes C ⊇ [ P ∈P R in ( P ) . (112) 15 In particular, restricting to a subfamily of P , where giv en ra ndom v ariables ( Q, X 1 , X 2 ) suc h that X 1 − Q − X 2 is a Mark ov chain and ( U 1 α , U 1 β , U 2 α , U 2 β ) are deﬁned by (58), we get C ⊇ [ ( Q,X 1 ,X 2 ) R in ( Q, X 1 , X 2 ) . (113) This completes the pro of of Theorem 1. ✷ 4.2 Geometric Prop erties Of R in ( Q, X 1 , X 2 ) W e hav e noted earlier that it is tedious to characterize R in ( Q, X 1 , X 2 ) explicitly . Nev- ertheless, w e w ould lik e to deriv e some useful insigh ts into the geometric prop erties of R in ( Q, X 1 , X 2 ). These will prov e use ful in deriving the outer b ound. W e b egin b y noting that R in ( Q, X 1 , X 2 ) is a close d and b ounded con v ex region. (In fact, w e kno w that it is a p olyhedron.) The extr ema l r epr ese n tation the or em o f classical Con v ex set theory (see Theorem 1 8.8, [7]) states that “an n -dimensional closed con v ex set in R n is the intersec tion of the closed half- spaces tang en t to it”. Th us, R in ( Q, X 1 , X 2 ) =  ( R 1 , R 2 ) : aR 1 + bR 2 ≤ c ∗ ( a, b | ( Q, X 1 , X 2 )) , ∀ ( a, b ) ∈ R 2  . (114) Here, c ∗ ( a, b | ( Q, X 1 , X 2 )) is the supp o rt function (Section 13, [7]) of R in ( Q, X 1 , X 2 ) and is deﬁned as the solution o f the following linear progra m, Max aR 1 + bR 2 , (115) s . t . ( R 1 , R 2 ) ∈ R in ( Q, X 1 , X 2 ) . Since R in ( Q, X 1 , X 2 ) is the pro jection of the six-dimensional region R (6) in ( Q, X 1 , X 2 ), the linear program (115) is equiv alent to the follo wing linear program. Max aR 1 p + aR 1 α + aR 1 β + bR 2 p + bR 2 α + bR 2 β , (116) s . t . ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) ∈ R (6) in ( Q, X 1 , X 2 ) . The dual of the linear program in Equation (116) sheds imp ortan t geometric infor- mation. Let us denote the dual- v ariables asso ciated with the inequalities ( 2 4)-(29) b y ν 11 , . . . , ν 16 , with (30)-(38) b y µ 11 , . . . , µ 19 , with (39)-(44) b y ν 21 , . . . , ν 26 , with (45)-(53) b y µ 21 , . . . , µ 29 and with (54)-(55) b y ω 1 and ω 2 . Deﬁne Λ ( a,b ) ⊂ R 32 b y , Λ ( a,b ) ,  { ν 1 i } 6 1 , { µ 1 i } 9 1 , { ν 2 i } 6 1 , { µ 2 i } 9 1 , { ω i } 2 1  : satisfying (11 8) − (124)  . (117) 16 6 X i =1 ν 1 i + 9 X i =1 µ 1 i − ω 1 = a (118) 6 X i =3 ν 1 i + 9 X i =4 µ 1 i + ( µ 22 + µ 25 + µ 28 ) + ( µ 23 + µ 26 + µ 29 ) − ω 1 = a (119) 6 X i =5 ν 1 i + 9 X i =7 µ 1 i + ( µ 23 + µ 26 + µ 29 ) + ( ν 22 + ν 24 + ν 26 ) − ω 1 = a (120) 6 X i =1 ν 2 i + 9 X i =1 µ 2 i − ω 2 = b (121) 6 X i =3 ν 2 i + 9 X i =4 µ 2 i + ( µ 12 + µ 15 + µ 18 ) + ( µ 13 + µ 16 + µ 19 ) − ω 2 = b (122) 6 X i =5 ν 2 i + 9 X i =7 µ 2 i + ( µ 13 + µ 16 + µ 19 ) + ( ν 12 + ν 14 + ν 16 ) − ω 2 = b (123) µ ij , ν ij , ω i ≥ 0 . (124) F or an y λ ∈ Λ ( a,b ) deﬁne c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) , 2 X j =1 6 X i =1 ν j i γ j i + 9 X i =1 µ j i δ j i ! . (125) The dual linear pro gram is min c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) , (126) suc hthat λ ∈ Λ ( a,b ) . By the stro ng dualit y theorem c ∗ ( a, b | ( Q, X 1 , X 2 )) = min n c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) | λ ∈ Λ ( a,b ) o . (127) Therefore, { ( R 1 , R 2 ) : aR 1 + bR 2 ≤ c ∗ ( a, b | ( Q, X 1 , X 2 )) } = \ λ ∈ Λ ( a,b ) n ( R 1 , R 2 ) : aR 1 + bR 2 ≤ c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) o . (128) Using (11 4) and (128), R in ( Q, X 1 , X 2 ) can now b e describ ed as, R in ( Q, X 1 , X 2 ) = { ( R 1 , R 2 ) : aR 1 + bR 2 ≤ c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) , ∀ λ ∈ Λ ( a,b ) , ∀ ( a, b ) ∈ R 2 } . (129) 17 The set of linear inequalities (c.f. (129)) that is used to describ e R in ( Q, X 1 , X 2 ) is very large and man y of the ineq ualities migh t be redundan t. The follow ing result, Lemma 8, tries to c haracterize some of these redundan t inequalities. Let Λ ′ ( a,b ) b e a subset of Λ ( a,b ) deﬁned b y Λ ′ ( a,b ) ,  λ ∈ Λ ( a,b ) : ω 1 = 0 , ω 2 = 0  . (130) Lemma 8 R in ( Q, X 1 , X 2 ) = { ( R 1 , R 2 ) : R 1 ≥ 0 , R 2 ≥ 0 aR 1 + bR 2 ≤ c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) , ∀ λ ∈ Λ ′ ( a,b ) , ∀ a ≥ 0 , b ≥ 0 } . (131) Pr o o f : Ev ery inequality used to deﬁne R in ( Q, X 1 , X 2 ) in ( 1 29) is describ ed b y pa ram- eters ( a, b ) and λ ∈ Λ ( a,b ) . Note that this set o f inequalities includes the follo wing tw o inequalities: − R 1 ≤ 0 (132) − R 2 ≤ 0 . (133) Consider an y inequalit y , ot her than the t w o sp ecial ones ab o ve , described b y ( a, b ) and λ ∈ Λ ( a,b ) , suc h that λ 6∈ Λ ′ ( a,b ) : aR 1 + bR 2 ≤ c ( in ) ( λ,a,b ) . (134) Deﬁne (˜ a, ˜ b ) , ( a + ω 1 , b + ω 2 ) . (135) Consider ˜ λ ∈ Λ ′ (˜ a, ˜ b ) , obt a ined b y replacing ω 1 and ω 2 in λ by 0 . No w w e hav e c ( in ) ( λ,a,b ) = c ( in ) ( ˜ λ, ˜ a , ˜ b ) . (136) Therefore ( a + ω 1 ) R 1 + ( b + ω 2 ) R 2 = ˜ aR 1 + ˜ bR 2 ≤ c ( in ) ( ˜ λ, ˜ a , ˜ b ) = c ( in ) ( λ,a,b ) . (137) The ab o v e inequalit y , along with R 1 ≥ 0 and R 2 ≥ 0, implies (13 4). Therefore w e ha ve that (134) is redundant. Th us w e hav e pro ve d that inequalities that are c haracterized by a λ 6∈ Λ ′ ( a,b ) are redundan t a nd can b e remo v ed. It a lso follo ws from (118)-(124) that Λ ′ ( a,b ) is an empt y set if either a or b is less than 0. Thus we only need to consider inequalities c har acterize d b y ( a, b ), where a ≥ 0 and b ≥ 0. This completes the pro of. ✷ W e also state the following prop osition, that will b e used in pro ving the outer b ound. 18 Prop osition 1 F or any λ ∈ Λ ′ ( a,b ) , ( µ 11 + µ 12 + µ 13 ) + ( ν 11 + ν 12 ) = ( µ 22 + µ 23 + µ 25 + µ 26 + µ 28 + µ 29 ) (138) ( µ 14 + µ 15 + µ 16 ) + ( µ 22 + µ 25 + µ 28 ) + ( ν 13 + ν 14 ) = ( ν 22 + ν 24 + ν 26 ) (139 ) ( µ 21 + µ 22 + µ 23 ) + ( ν 21 + ν 22 ) = ( µ 12 + µ 13 + µ 15 + µ 16 + µ 18 + µ 19 ) (140) ( µ 24 + µ 25 + µ 26 ) + ( µ 12 + µ 15 + µ 18 ) + ( ν 23 + ν 24 ) = ( ν 12 + ν 14 + ν 16 ) . (141) Pr o o f : The pro of follo ws directly from (118)-(1 2 3 ). ✷ 5 Outer Bo und Our goal in this Section is to show t hat, if ( R 1 , R 2 ) is achie v a ble then there exist random v ariables ( Q, X 1 , X 2 ), where X 1 − Q − X 2 is a Mark o v c hain, and a region R out ( Q, X 1 , X 2 ) , { ( R 1 , R 2 ) : R 1 ≥ 0 , R 2 ≥ 0 aR 1 + bR 2 ≤ c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) , ∀ λ ∈ Λ ′ ( a,b ) , ∀ a ≥ 0 , b ≥ 0 } , (142) suc h that ( R 1 , R 2 ) ∈ R out ( Q, X 1 , X 2 ) . (143) The term c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) is deﬁned in Section 5.1 . Note that our deﬁnition of R out ( Q, X 1 , X 2 ) is inspired b y the characterization of R in ( Q, X 1 , X 2 ) that w e ha v e obtained through Lemma 8. F urther, quan tifying the diﬀerence b et w een c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) and c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) will giv e us the gap b et we en the inner and the outer b ounds. 5.1 Pro of Of Theorem 3 (i) Supp ose there is a sequenc e of enco ders at rat es ( R 1 , R 2 ), sequence d b y the blo ck length n , and deco ders with probability of error going to 0 as n → ∞ . F ix the blo c k length n and consider the corresponding co de b o ok. Let X n 1 , X n 2 , S n 1 , S n 2 , Y n 1 , Y n 2 b e the ra ndo m v ariables induced by the channe l and enco ders for uniformly distributed messages, in- dep enden t across the t w o users. W e deﬁne random v aria bles U n 1 α whic h is obtained b y passing X n 1 through an independen t copy of the c hannel p S 1 α | X 1 , a nd U n 1 β b y pass ing the U n 1 α so obtained through an indep ende nt cop y of rhe c hannel p S 1 β | S 1 α . Similarly , w e also deﬁne U n 2 α and U n 2 β from X n 2 and indep enden t copies o f p S 2 α | X 2 and p S 2 β | S 2 α . 19 Consider any non-negativ e pair ( a, b ) and an y λ ∈ Λ ′ ( a,b ) . Since the probability of error go es t o 0 as n → ∞ , by F ano’s inequality there exists a sequence ǫ n → 0 suc h that n ( aR 1 + bR 2 − ( a + b ) ǫ n ) (1) ≤ 6 X i =1 ν 1 i ! I ( X n 1 ; Y n 1 β ) + 9 X i =1 µ 1 i ! I ( X n 1 ; Y n 1 α ) + 6 X i =1 ν 2 i ! I ( X n 2 ; Y n 2 β ) + 9 X i =1 µ 2 i ! I ( X n 2 ; Y n 2 α ) (2) ≤ 6 X i =1 ν 1 i I ( X n 1 ; Y n 1 β , V n 1 iβ ) + 9 X i =1 µ 1 i I ( X n 1 ; Y n 1 α , V n 1 iα ) + 6 X i =1 ν 2 i I ( X n 2 ; Y n 2 β , V n 2 iβ ) + 9 X i =1 µ 2 i I ( X n 2 ; Y n 2 α , V n 2 iα ) . (144) Note that in step (1), we split up a and b according to (118) and (121 ) , considering deco ders under diﬀeren t states. In step (2), w e consider genies whic h pro vide diﬀeren t side-information V ’s to the deco ders. Consider, f o r instance, the term ν 11 I ( X n 1 ; Y n 1 β , V n 11 β ). W e will c ho ose the side-information V n 11 β in such a wa y that w e can form a corresp on- dence b et wee n this term and the term contributed to the inner b ound b y the right hand side of t he constrain t (24). In particular, w e c ho ose the genie provide d side-information V n 11 β to matc h the error- ev en t corresp onding to (24). More sp eciﬁcally , w e note that the corresp onding error - ev en t is when receiv er 1 in state β cor r ectly deco des the other user’s common information U 2 β , a nd its o wn common information ( U 1 β , U 1 α ), but mak es an error in deco ding its priv ate mess age. Hence, the genie provides the side -info rmation ( U n 1 α , U n 1 β , U n 2 β ) which can b e shrunk to V n 11 β = ( U n 1 α , U n 2 β ) b ecause of the Mark ov relation- ship betw een X n 1 , U n 1 α , a nd U n 1 β . No w, w e expand the term I ( X n 1 ; Y n 1 β , V n 11 β ) to get (145). W e can rep eat these t w o step s for ev ery term in ( 144): the (expanded) upp er b ounds on all t he terms are given in (145)-(174). I ( X n 1 ; Y n 1 β , V n 11 β ) , I ( X n 1 ; Y n 1 β , U n 1 α , U n 2 β ) = H ( Y n 1 β | U n 1 α , U n 2 β ) − H ( S n 2 β | U n 2 β ) + H ( U n 1 α ) − H ( U n 1 α | X n 1 ) (145) I ( X n 1 ; Y n 1 β , V n 12 β ) , I ( X n 1 ; Y n 1 β , U n 1 α ) = H ( Y n 1 β | U n 1 α ) − H ( S n 2 β ) + H ( U n 1 α ) − H ( U n 1 α | X n 1 ) (146) I ( X n 1 ; Y n 1 β , V n 13 β ) , I ( X n 1 ; Y n 1 β , U n 1 β , U n 2 β ) = H ( Y n 1 β | U n 1 β , U n 2 β ) − H ( S n 2 β | U n 2 β ) + H ( U n 1 β ) − H ( U n 1 β | X n 1 ) (147) I ( X n 1 ; Y n 1 β , V n 14 β ) , I ( X n 1 ; Y n 1 β , U n 1 β ) = H ( Y n 1 β | U n 1 β ) − H ( S n 2 β ) + H ( U n 1 β ) − H ( U n 1 β | X n 1 ) (148) I ( X n 1 ; Y n 1 β , V n 15 β ) , I ( X n 1 ; Y n 1 β , U n 2 β ) = H ( Y n 1 β | U n 2 β ) − H ( S n 2 β | U n 2 β ) (149) I ( X n 1 ; Y n 1 β , V n 16 β ) , I ( X n 1 ; Y n 1 β ) = H ( Y n 1 β ) − H ( S n 2 β ) (150) 20 I ( X n 1 ; Y n 1 α , V n 11 α ) , I ( X n 1 ; Y n 1 α , U n 1 α , U n 2 α ) = H ( Y n 1 α | U n 1 α , U n 2 α ) − H ( S n 2 α | U n 2 α ) + H ( U n 1 α ) − H ( U n 1 α | X n 1 ) (151) I ( X n 1 ; Y n 1 α , V n 12 α ) , I ( X n 1 ; Y n 1 α , U n 1 α , U n 2 β ) = H ( Y n 1 α | U n 1 α , U n 2 β ) − H ( S n 2 α | U n 2 β ) + H ( U n 1 α ) − H ( U n 1 α | X n 1 ) (152) I ( X n 1 ; Y n 1 α , V n 13 α ) , I ( X n 1 ; Y n 1 α , U n 1 α ) = H ( Y n 1 α | U n 1 α ) − H ( S n 2 α ) + H ( U n 1 α ) − H ( U n 1 α | X n 1 ) (153) I ( X n 1 ; Y n 1 α , V n 14 α ) , I ( X n 1 ; Y n 1 α , U n 1 β , U n 2 α ) = H ( Y n 1 α | U n 1 β , U n 2 α ) − H ( S n 2 α | U n 2 α ) + H ( U n 1 β ) − H ( U n 1 β | X n 1 ) (154) I ( X n 1 ; Y n 1 α , V n 15 α ) , I ( X n 1 ; Y n 1 α , U n 1 β , U n 2 β ) = H ( Y n 1 α | U n 1 β , U n 2 β ) − H ( S n 2 α | U n 2 β ) + H ( U n 1 β ) − H ( U n 1 β | X n 1 ) (155) I ( X n 1 ; Y n 1 α , V n 16 α ) , I ( X n 1 ; Y n 1 α , U n 1 β ) = H ( Y n 1 α | U n 1 β ) − H ( S n 2 α ) + H ( U n 1 β ) − H ( U n 1 β | X n 1 ) (156) I ( X n 1 ; Y n 1 α , V n 17 α ) , I ( X n 1 ; Y n 1 α , U n 2 α ) = H ( Y n 1 α | U n 2 α ) − H ( S n 2 α | U n 2 α ) (157) I ( X n 1 ; Y n 1 α , V n 18 α ) , I ( X n 1 ; Y n 1 α , U n 2 β ) = H ( Y n 1 α | U n 2 β ) − H ( S n 2 α | U n 2 β ) (158) I ( X n 1 ; Y n 1 α , V n 19 α ) , I ( X n 1 ; Y n 1 α ) = H ( Y n 1 α ) − H ( S n 2 α ) (159) I ( X n 2 ; Y n 2 β , V n 21 β ) , I ( X n 2 ; Y n 2 β , U n 2 α , U n 1 β ) = H ( Y n 2 β | U n 2 α , U n 1 β ) − H ( S n 1 β | U n 1 β ) + H ( U n 2 α ) − H ( U n 2 α | X n 2 ) (160) I ( X n 2 ; Y n 2 β , V n 22 β ) , I ( X n 2 ; Y n 2 β , U n 2 α ) = H ( Y n 2 β | U n 2 α ) − H ( S n 1 β ) + H ( U n 2 α ) − H ( U n 2 α | X 2 n ) (161) I ( X n 2 ; Y n 2 β , V n 23 β ) , I ( X n 2 ; Y n 2 β , U n 2 β , U n 1 β ) = H ( Y n 2 β | U n 2 β , U n 1 β ) − H ( S n 1 β | U n 1 β ) + H ( U n 2 β ) − H ( U n 2 β | X n 2 ) (162) I ( X n 2 ; Y n 2 β , V n 24 β ) , I ( X n 2 ; Y n 2 β , U n 2 β ) = H ( Y n 2 β | U n 2 β ) − H ( S n 1 β ) + H ( U n 2 β ) − H ( U n 2 β | X n 2 ) (163) I ( X n 2 ; Y n 2 β , V n 25 β ) , I ( X n 2 ; Y n 2 β , U n 1 β ) = H ( Y n 2 β | U n 1 β ) − H ( S n 1 β | U n 1 β ) (164) I ( X n 2 ; Y n 2 β , V n 26 β ) , I ( X n 2 ; Y n 2 β ) = H ( Y n 2 β ) − H ( S n 1 β ) (165) I ( X n 2 ; Y n 2 α , V n 21 α ) , I ( X n 2 ; Y n 2 α , U n 2 α , U n 1 α ) = H ( Y n 2 α | U n 2 α , U n 1 α ) − H ( S n 1 α | U n 1 α ) + H ( U n 2 α ) − H ( U n 2 α | X n 2 ) (166) I ( X n 2 ; Y n 2 α , V n 22 α ) , I ( X n 2 ; Y n 2 α , U n 2 α , U n 1 β ) = H ( Y n 2 α | U n 2 α , U n 1 β ) − H ( S n 1 α | U n 1 β ) + H ( U n 2 α ) − H ( U n 2 α | X n 2 ) (167) I ( X n 2 ; Y n 2 α , V n 23 α ) , I ( X n 2 ; Y n 2 α , U n 2 α ) = H ( Y n 2 α | U n 2 α ) − H ( S n 1 α ) + H ( U n 2 α ) − H ( U n 2 α | X n 2 ) (168) I ( X n 2 ; Y n 2 α , V n 24 α ) , I ( X n 2 ; Y n 2 α , U n 2 β , U n 1 α ) = H ( Y n 2 α | U n 2 β , U n 1 α ) − H ( S n 1 α | U n 1 α ) + H ( U n 2 β ) − H ( U n 2 β | X n 2 ) (169) I ( X n 2 ; Y n 2 α , V n 25 α ) , I ( X n 2 ; Y n 2 α , U n 2 β , U n 1 β ) = H ( Y n 2 α | U n 2 β , U n 1 β ) − H ( S n 1 α | U n 1 β ) + H ( U n 2 β ) − H ( U n 2 β | X n 2 ) (170) I ( X n 2 ; Y n 2 α , V n 26 α ) , I ( X n 2 ; Y n 2 α , U n 2 β ) = H ( Y n 2 α | U n 2 β ) − H ( S n 1 α ) + H ( U n 2 β ) − H ( U n 2 β | X n 2 ) (171) I ( X n 2 ; Y n 2 α , V n 27 α ) , I ( X n 2 ; Y n 2 α , U n 1 α ) = H ( Y n 2 α | U n 1 α ) − H ( S n 1 α | U n 1 α ) (172) I ( X n 2 ; Y n 2 α , V n 28 α ) , I ( X n 2 ; Y n 2 α , U n 1 β ) = H ( Y n 2 α | U n 1 β ) − H ( S n 1 α | U n 1 β ) (173) I ( X n 2 ; Y n 2 α , V n 29 α ) , I ( X n 2 ; Y n 2 α ) = H ( Y n 2 α ) − H ( S n 1 α ) . (174) 21 Con tin uing with our outer b ound deriv ation, from (144), n ( aR 1 + bR 2 − ( a + b ) ǫ n ) (a) ≤ 6 X i =1 ν 1 i H ( Y n 1 β | V n 1 iβ ) + 9 X i =1 µ 1 i H ( Y n 1 α | V n 1 iα ) + H ( U n 1 α ) { ( ν 11 + ν 12 ) + ( µ 11 + µ 12 + µ 13 ) } + H ( U n 1 β ) { ( ν 13 + ν 14 ) + ( µ 14 + µ 15 + µ 16 ) } − H ( S n 1 α ) { ( µ 23 + µ 26 + µ 29 ) } − H ( S n 1 β ) { ( ν 22 + ν 24 + ν 26 ) } − H ( U n 1 α | X n 1 ) { ( ν 11 + ν 12 ) + ( µ 11 + µ 12 + µ 13 ) } − H ( U n 1 β | X n 1 ) { ( ν 13 + ν 14 ) + ( µ 14 + µ 15 + µ 16 ) } − H ( S n 1 α | U n 1 α ) { ( µ 21 + µ 24 + µ 27 ) } − H ( S n 1 β | U n 1 β ) { ( ν 21 + ν 23 + ν 25 ) } − H ( S n 1 α | U n 1 β ) { ( µ 22 + µ 25 + µ 28 ) } + 6 X i =1 ν 2 i H ( Y n 2 β | V n 2 iβ ) + 9 X i =1 µ 2 i H ( Y n 2 α | V n 2 iα ) + H ( U n 2 α ) { ( ν 21 + ν 22 ) + ( µ 21 + µ 22 + µ 23 ) } + H ( U n 2 β ) { ( ν 23 + ν 24 ) + ( µ 24 + µ 25 + µ 26 ) } − H ( S n 2 α ) { ( µ 13 + µ 16 + µ 19 ) } − H ( S n 2 β ) { ( ν 12 + ν 14 + ν 16 ) } − H ( U n 2 α | X n 2 ) { ( ν 21 + ν 22 ) + ( µ 21 + µ 22 + µ 23 ) } − H ( U n 2 β | X n 2 ) { ( ν 23 + ν 24 ) + ( µ 24 + µ 25 + µ 26 ) } − H ( S n 2 α | U n 2 α ) { ( µ 11 + µ 14 + µ 17 ) } − H ( S n 2 β | U n 2 β ) { ( ν 11 + ν 13 + ν 15 ) } − H ( S n 2 α | U n 2 β ) { ( µ 12 + µ 15 + µ 18 ) } (175) (b) = 6 X i =1 ν 1 i H ( Y n 1 β | V n 1 iβ ) + 9 X i =1 µ 1 i H ( Y n 1 α | V n 1 iα ) + H ( U n 1 α ) { ( µ 22 + µ 25 + µ 28 ) } − H ( U n 1 β ) { ( µ 22 + µ 25 + µ 28 ) } − H ( S n 1 α | U n 1 β ) { ( µ 22 + µ 25 + µ 28 ) } − H ( U n 1 α | X n 1 ) { ( µ 22 + µ 25 + µ 28 ) + ( µ 23 + µ 26 + µ 29 ) } − H ( U n 1 β | X n 1 ) { ( ν 22 + ν 24 + ν 26 ) − ( µ 22 + µ 25 + µ 28 ) } − H ( S n 1 α | U n 1 α ) { ( µ 21 + µ 24 + µ 27 ) } − H ( S n 1 β | U n 1 β ) { ( ν 21 + ν 23 + ν 25 ) } + 6 X i =1 ν 2 i H ( Y n 2 β | V n 2 iβ ) + 9 X i =1 µ 2 i H ( Y n 2 α | V n 2 iα ) + H ( U n 2 α ) { ( µ 12 + µ 15 + µ 18 ) } − H ( U n 2 β ) { ( µ 12 + µ 15 + µ 18 ) } − H ( S n 2 α | U n 2 β ) { ( µ 12 + µ 15 + µ 18 ) } − H ( U n 2 α | X n 2 ) { ( µ 12 + µ 15 + µ 18 ) + ( µ 13 + µ 16 + µ 19 ) } − H ( U n 2 β | X n 2 ) { ( ν 12 + ν 14 + ν 16 ) − ( µ 12 + µ 15 + µ 18 ) } − H ( S n 2 α | U n 2 α ) { ( µ 11 + µ 14 + µ 17 ) } − H ( S n 2 β | U n 2 β ) { ( ν 11 + ν 13 + ν 15 ) } (176) 22 (c) ≤ 6 X i =1 ν 1 i H ( Y n 1 β | V n 1 iβ ) + 9 X i =1 µ 1 i H ( Y n 1 α | V n 1 iα ) − H ( U n 1 α | X n 1 ) { ( µ 22 + µ 25 + µ 28 ) + ( µ 23 + µ 26 + µ 29 ) } − H ( U n 1 β | X n 1 ) { ( ν 22 + ν 24 + ν 26 ) } − H ( S n 1 α | X n 1 ) { ( µ 21 + µ 24 + µ 27 ) } − H ( S n 1 β | X n 1 ) { ( ν 21 + ν 23 + ν 25 ) } + 6 X i =1 ν 2 i H ( Y n 2 β | V n 2 iβ ) + 9 X i =1 µ 2 i H ( Y n 2 α | V n 2 iα ) − H ( U n 2 α | X n 2 ) { ( µ 12 + µ 15 + µ 18 ) + ( µ 13 + µ 16 + µ 19 ) } − H ( U n 2 β | X n 2 ) { ( ν 12 + ν 14 + ν 16 ) } − H ( S n 2 α | X n 2 ) { ( µ 11 + µ 14 + µ 17 ) } − H ( S n 2 β | X n 2 ) { ( ν 11 + ν 13 + ν 15 ) } (177) (d) = 6 X i =1 ν 1 i H ( Y n 1 β | V n 1 iβ ) + 9 X i =1 µ 1 i H ( Y n 1 α | V n 1 iα ) − H ( S n 1 α | X n 1 ) 9 X i =1 µ 2 i ! − H ( S n 1 β | X n 1 ) 6 X i =1 ν 2 i ! + 6 X i =1 ν 2 i H ( Y n 2 β | V n 2 iβ ) + 9 X i =1 µ 2 i H ( Y n 2 α | V n 2 iα ) − H ( S n 2 α | X n 2 ) 9 X i =1 µ 1 i ! − H ( S n 2 β | X n 2 ) 6 X i =1 ν 1 i ! . (178) Here, • to get inequality (a ), w e used (145)-( 1 74 ) in (144) and collected the terms t o gether; • for equalit y (b), w e used Prop osition 1 along with the facts H ( U n 1 α ) = H ( S n 1 α ) , (179) H ( U n 1 β ) = H ( S n 1 β ) , ( 1 80) H ( U n 2 α ) = H ( S n 2 α ) (181) H ( U n 2 β ) = H ( S n 2 β ); ( 1 82) • inequalit y (c) follo ws fro m the fact tha t conditioning reduces en trop y . In particular, H ( U n 1 α ) − H ( U n 1 β ) − H ( S n 1 α | U n 1 β ) = − H ( U n 1 β | S n 1 α ) ≤ − H ( U n 1 β | X n 1 ) (183) H ( U n 2 α ) − H ( U n 2 β ) − H ( S n 2 α | U n 2 β ) = − H ( U n 2 β | S n 2 α ) ≤ − H ( U n 2 β | X n 2 ); (184) • for equalit y (d) w e used H ( U n 1 α | X n 1 ) = H ( S n 1 α | X n 1 ) , H ( U n 1 β | X n 1 ) = H ( S n 1 β | X n 1 ) H ( U n 2 α | X n 2 ) = H ( S n 2 α | X n 2 ) , H ( U n 2 β | X n 2 ) = H ( S n 2 β | X n 2 ) . 23 No w w e single-letterize using the c hain rule along with the fact that the c hannel is mem- oryless and conditioning reduces en t rop y . aR 1 + bR 2 − ( a + b ) ǫ n ≤ 1 n n X q =1 ( 6 X i =1 ν 1 i H ( Y 1 β ( q ) | V 1 iβ ( q )) + 9 X i =1 µ 1 i H ( Y 1 α ( q ) | V 1 iα ( q )) − H ( S 1 α ( q ) | X 1 ( q )) 9 X i =1 µ 2 i ! − H ( S 1 β ( q ) | X 1 ( q )) 6 X i =1 ν 2 i ! + 6 X i =1 ν 2 i H ( Y 2 β ( q ) | V 2 iβ ( q )) + 9 X i =1 µ 2 i H ( Y 2 α ( q ) | V 2 iα ( q )) − H ( S 2 α ( q ) | X 2 ( q )) 9 X i =1 µ 1 i ! − H ( S 2 β ( q ) | X 2 ( q )) 6 X i =1 ν 1 i !) . (1 85) aR 1 + bR 2 − ( a + b ) ǫ n ≤ 6 X i =1 ν 1 i H ( Y 1 β | V 1 iβ , Q ) + 9 X i =1 µ 1 i H ( Y 1 α | V 1 iα , Q ) − 9 X i =1 µ 2 i ! H ( S 1 α | X 1 , Q ) − 6 X i =1 ν 2 i ! H ( S 1 β | X 1 , Q ) + 6 X i =1 ν 2 i H ( Y 2 β | V 2 iβ , Q ) + 9 X i =1 µ 2 i H ( Y 2 α | V 2 iα , Q ) − 9 X i =1 µ 1 i ! H ( S 2 α | X 2 , Q ) − 6 X i =1 ν 1 i ! H ( S 2 β | X 2 , Q ) , c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) , (186) where we set ( Q, U 1 β , U 1 α , X 1 , S 1 β , S 1 α , U 2 β , U 2 α , X 2 , S 2 β , S 2 α ) to b e j o in t random v ariables suc h that Q is uniformly distributed ov er { 1 , 2 , . . . , n } a nd, Pr ( U 1 β , U 1 α , X 1 , S 1 α , S 1 β , U 2 β , U 2 α , X 2 , S 2 α , S 2 β | Q = q ) = Pr ( U 1 β ( q ) , U 1 α ( q ) , X 1 ( q ) , S 1 α ( q ) , S 1 β ( q ) , U 2 β ( q ) , U 2 α ( q ) , X 2 ( q ) , S 2 α ( q ) , S 2 β ( q )) , (187) for 1 ≤ q ≤ n . Since the messages are independent f or the tw o users, so are X 1 ( q ) and X 2 ( q ). Therefore, ( Q, X 1 , X 2 ) satisﬁes the Mark ov c hain X 1 − Q − X 2 . F urther b ecause of our c hoice of ( U 1 β ( q ) , U 1 α ( q ) , U 1 β ( q ) , U 1 α ( q )), the random v ariables satisfy the 24 condition (58). Hence the ra ndo m v ar ia bles ( Q, X 1 , U 1 α , U 1 β , X 2 , U 2 α , U 2 β ) b elong to the sub-family of P t ha t w e describ ed earlier, whose elemen ts are deﬁned by ( Q, X 1 , X 2 ). W e are no w ready to formally deﬁne R out ( Q, X 1 , X 2 ): R out ( Q, X 1 , X 2 ) , { ( R 1 , R 2 ) : R 1 ≥ 0 , R 2 ≥ 0 aR 1 + bR 2 ≤ c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) , ∀ λ ∈ Λ ′ ( a,b ) , ∀ a ≥ 0 , b ≥ 0 } . W e ha v e prov ed t ha t if ( R 1 , R 2 ) is ac hiev able, then ( R 1 , R 2 ) ∈ [ Q,X 1 ,X 2 R out ( Q, X 1 , X 2 ) . (188) This completes the pro of. ✷ 5.2 Pro of Of Theorem 3 (ii) F or a giv en ( Q, X 1 , X 2 ) suc h tha t X 1 − Q − X 2 is a Mark ov c hain, w e need to quantify the gap b et wee n R out ( Q, X 1 , X 2 ) and R in ( Q, X 1 , X 2 ), whic h are deﬁned b y Equations (131) and (142) resp ectiv ely . In order to do t his, we quan tify t he ga p b et w een c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) and c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ). c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) − c ( in ) ( λ,a,b ) ( Q, X 1 , X 2 ) = ( H ( S 1 α | U 1 α , Q ) − H ( S 1 α | X 1 , Q )) 9 X i =1 µ 2 i ! + ( H ( S 1 β | U 1 β , Q ) − H ( S 1 β | X 1 , Q )) 6 X i =1 ν 2 i ! + ( H ( S 2 α | U 2 α , Q ) − H ( S 2 α | X 2 , Q )) 9 X i =1 µ 1 i ! + ( H ( S 2 β | U 2 β , Q ) − H ( S 2 β | X 2 , Q )) 6 X i =1 ν 1 i ! = I ( S 1 α ; X 1 | U 1 α , Q ) 9 X i =1 µ 2 i ! + I ( S 1 β ; X 1 | U 1 β , Q ) 6 X i =1 ν 2 i ! + I ( S 2 α ; X 2 | U 2 α , Q ) 9 X i =1 µ 1 i ! + I ( S 2 β ; X 2 | U 2 β , Q ) 6 X i =1 ν 1 i ! ≤ b max( I ( S 1 α ; X 1 | U 1 α , Q ) , I ( S 1 β ; X 1 | U 1 β , Q )) + a max( I ( S 2 α ; X 2 | U 2 α , Q ) , I ( S 2 β ; X 2 | U 2 β , Q )) ≤ a ∆ 1 ( Q, X 1 , X 2 ) + b ∆ 2 ( Q, X 1 , X 2 ) . (189) 25 Here ∆ 1 ( Q, X 1 , X 2 ) and ∆ 2 ( Q, X 1 , X 2 ) a re deﬁned as follo ws: ∆ 1 ( Q, X 1 , X 2 ) , max( I ( S 2 α ; X 2 | U 2 α ) , I ( S 2 β ; X 2 | U 2 β )) , ∆ 2 ( Q, X 1 , X 2 ) , max( I ( S 1 α ; X 1 | U 1 α ) , I ( S 1 β ; X 1 | U 1 β )) . This completes the pro of of Theorem 3. ✷ 5.3 Pro of Of Corollary 6 Consider the 2-state comp o und Ga uss ian interference c hannel. F or this sp ecial case, w e ha v e the following result that identiﬁe s the Gaussian co de b o oks to b e suﬃcien t. Lemma 9 R out ( Q, X 1 , X 2 ) ⊆ R out ( Q ∗ , X ∗ 1 , X ∗ 2 ) (190) wher e Q ∗ = 1 , X ∗ 1 ∼ C N (0 , P 1 ) , X ∗ 2 ∼ C N (0 , P 2 ) . W e note for easy reference t ha t R out ( Q, X 1 , X 2 ) is deﬁned in Equation ( 142). Pr o o f : It suﬃces t o sho w that c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) ≤ c ( out ) ( λ,a,b ) ( Q ∗ , X ∗ 1 , X ∗ 2 ) , where c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) is as deﬁned in (186 ) . c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) = 6 X i =1 ν 1 i h ( Y 1 β | V 1 iβ , Q ) + 9 X i =1 µ 1 i h ( Y 1 α | V 1 iα , Q ) − 9 X i =1 µ 2 i ! h ( S 1 α | X 1 , Q ) − 6 X i =1 ν 2 i ! h ( S 1 β | X 1 , Q ) + 6 X i =1 ν 2 i h ( Y 2 β | V 2 iβ , Q ) + 9 X i =1 µ 2 i h ( Y 2 α | V 2 iα , Q ) − 9 X i =1 µ 1 i ! h ( S 2 α | X 2 , Q ) − 6 X i =1 ν 1 i ! h ( S 2 β | X 2 , Q ) The terms h ( S 1 α | X 1 , Q ) , h ( S 1 β | X 1 , Q ) , h ( S 2 α | X 2 , Q ) and h ( S 2 β | X 2 , Q ) are the diﬀeren tia l en tropies of complex Gaussian no ise with kno wn v ariance and are readily handled. Let 26 us no w turn to the term h ( Y 1 β | V 11 β , Q ): h ( Y 1 β | V 11 β , Q ) = h ( Y 1 β | U 1 α U 2 β , Q ) (191) = X q p ( q ) h ( Y 1 β | U 1 α U 2 β , Q = q ) (192) ( a ) ≤ X q p ( q ) log  P 1 q | h 12 α | 2 P 1 q + 1 + 1 + | h 21 β | 2 P 2 q | h 21 β | 2 P 2 q + 1  (193) ( b ) ≤ log  P 1 | h 12 α | 2 P 1 + 1 + 1 + | h 21 β | 2 P 2 | h 21 β | 2 P 2 + 1  (194) = h ( Y ∗ 1 β | U ∗ 1 α U ∗ 2 β , Q ∗ ) . (195) Here, • in step (a), we denoted E  | X k | 2 | Q = q  = P k q , k = 1 , 2 (196) and used the fact that conditio na l diﬀerential en tropy is maximized with the Gaus- sian distribution under a co v ariance constraint (Lemma 1 [6]); • in step(b), w e used Jensen’s inequalit y . A similar argumen t follow s for the other terms. T o conclude, w e hav e show n that c ( out ) ( λ,a,b ) ( Q, X 1 , X 2 ) ≤ c ( out ) ( λ,a,b ) ( Q ∗ , X ∗ 1 , X ∗ 2 ) . (197) This completes the pro of. ✷ Finally , w e can readily see the pro of of Coro llary 6. This is b ecause, [ ( Q,X 1 ,X 2 ) R out ( Q, X 1 , X 2 ) = R out ( Q ∗ , X ∗ 1 , X ∗ 2 ) , (198) as a direct consequence o f Lemma 9. 6 Discuss ion: Insigh ts On The Non-Comp ound In- terferenc e Channel In this section we consider the non-comp ound in terference channel mo del in tro duced in [5]; t his is a sp eciﬁc instance of our mo del and is obtained by setting α = β . Our results, when sp ecialized to this instance prov ide a n alternativ e deriv atio n of the r esults of Chong 27 et al. [3 ] and Tse and T elatar [5]. Below we brieﬂy sk etc h our results with an aim to compare and con trast the diﬀeren t pro ofs. The g oal is not only to giv e b etter insight in to existing results, but also to g iv e an idea on how our new pro of tec hnique scales more naturally to the 2-state comp ound interference c hannel (and in general to the n - state comp ound in terference c hannel that w e will describ e in the next section). W e ﬁrst describe the ac hiev able sc heme and the inner b ound. F ollo wing tha t , w e will describe the outer-b ound, fo cusing on contrasts b et we en t he diﬀerent a pproac hes. 6.1 Ac hiev able Sc heme The sp ecial case of the noncomp ound v ersion is obtained b y setting S k β = S k α = S k (199) and, corresp ondingly , U k β = U k α = U k (200) for k = 1 , 2. W e also set R k α = 0 . (201) W e rename R k β as T k and R k p as S k to b e consisten t with the no t ation of Chong et al. [3]. The sup erp osition ac hiev able sc heme can no w b e describ ed b y joint random v ariables P = ( Q, U 1 , X 1 , U 2 , X 2 ) (202) with the join t distribution factoring as p ( q ) p ( x 1 | q ) p ( x 2 | q ) p ( u 1 | x 1 q ) p ( u 2 | x 2 q ) . (203) F rom Section 4.1, it f o llo ws that any rate v ector ( S 1 , T 1 , S 2 , T 2 ) that satisﬁes, S 1 ≤ I ( Y 1 ; X 1 | U 1 , U 2 , Q ) , if S 1 > 0 , (204) T 2 + S 1 ≤ I ( Y 1 ; X 1 U 2 | U 1 , Q ) , if S 1 > 0 , (205) T 1 + S 1 ≤ I ( Y 1 ; X 1 | U 2 , Q ) , if T 1 + S 1 > 0 , (206) T 2 + T 1 + S 1 ≤ I ( Y 1 ; X 1 U 2 | Q ) , if T 1 + S 1 > 0 , (207) S 2 ≤ I ( Y 2 ; X 2 | U 2 , U 1 , Q ) , if S 2 > 0 , (208) T 1 + S 2 ≤ I ( Y 2 ; X 2 , U 1 | U 2 , Q ) , if S 2 > 0 , (209) T 2 + S 2 ≤ I ( Y 2 ; X 2 | U 1 , Q ) , if T 2 + S 2 > 0 , (210) T 1 + T 2 + S 2 ≤ I ( Y 2 ; X 2 , U 1 | Q ) , if T 2 + S 2 > 0 , (211) S 1 , T 1 , S 2 , T 2 ≥ 0 , (212) 28 is ac hiev able. Deﬁne the 4-dimensional region ˜ R (4) in ( P ) , { ( S 1 , T 1 , S 2 , T 2 ) : satisﬁes (204)-(212) } (213) and its pro jection on the 2- dimensional space ˜ R in ( P ) , { ( R 1 , R 2 ) : R k = S k + T k , k = 1 , 2 } . (214) On the other hand, deﬁne R (4) in ( P ) , { ( S 1 , T 1 , S 2 , T 2 ) : satisﬁes (216)-(225) } . (215) S 1 ≤ I ( Y 1 ; X 1 | U 1 , U 2 , Q ) (216) T 2 + S 1 ≤ I ( Y 1 ; X 1 U 2 | U 1 , Q ) (217) T 1 + S 1 ≤ I ( Y 1 ; X 1 | U 2 , Q ) (218) T 2 + T 1 + S 1 ≤ I ( Y 1 ; X 1 U 2 | Q ) (219) S 2 ≤ I ( Y 2 ; X 2 | U 2 , U 1 , Q ) (220) T 1 + S 2 ≤ I ( Y 2 ; X 2 , U 1 | U 2 , Q ) (221) T 2 + S 2 ≤ I ( Y 2 ; X 2 | U 1 , Q ) (222) T 1 + T 2 + S 2 ≤ I ( Y 2 ; X 2 , U 1 | Q ) (223) S 1 + T 1 ≥ 0 , (224) S 2 + T 2 ≥ 0 . (225) Let its pro jection on the 2-dimensional space R in ( P ) , { ( R 1 , R 2 ) : R k = S k + T k , k = 1 , 2 } . (226) In Theorem 2 of [3], the authors explic itly ev a lua ted the constraints t hat deﬁne this set and described it as the “compact v ersion” of the Han-Kobay ashi region [1] (which r esults from a somewhat diﬀeren t co ding strategy , as compared to the sup erp osition co ding one). Ho w ev er, w e kno w from Lemma 7 that R in ( P ) ⊆ ˜ R in ( P ) . (227) W e thus conclude the alternate pro of of [3](T heorem 2). Our approac h diﬀers from the approac h of Chong et al. [3] in t w o w ays : • It is instructiv e to observ e the similarities and diﬀerences b et w een the 4-dimensional ac hiev able region ˜ R (4) in ( P ) to the o ne in [3](Lemma 3). First, the inequalities in v olve d are t he same. Ho w evb er, sev eral of these constraints are ina ctive when the b oundary 29 conditions on the data rates bite. W e can immediately conclude that our ac hiev able region ( ˜ R (4) in ( P )) is in general a sup erset of the r egio n in [3]. This is somewhat surprising since the enco ding metho d in b oth cases is sup erp osition co ding. The diﬀerences result due to o ur careful consideration of the error ev en ts in the deco ding pro cess. • Chong et al. [3] describ ed the 2-dimensional region explicitly b y carrying out the somewhat tedious algorit hmic pro cedure of F ourier-Motzkin elimination. F urther, they show ed t ha t a p oten tially bigger region (the compact description region) is ac hiev able b y time-sharing b et w een t w o other sc hemes deﬁned b y ( Q, ∅ , X 1 , U 2 , X 2 ) and ( Q, U 1 , X 1 , ∅ , X 2 ). In our approac h, w e entire ly av o id describing the 2-dimensional region explicitly . F urther, w e sho w ed that there is no need to time-share b et wee n an y other sc hemes, to ac hiev e R in ( P ). 6.2 Outer B ound F or a giv en P = ( Q, U 1 , X 1 , U 2 , X 2 ), the inner-b ound region in Chong et al. [3] is describ ed b y sev en linear inequalities in v olving R 1 and R 2 . In [5], T elatar and Tse pic k ed a sp eciﬁc c hoice of ( U 1 , U 2 ) giv en b y p ( u 1 , u 2 | q , x 1 , x 2 ) = p S 1 | X 1 ( u 1 | x 1 ) p S 2 | X 2 ( u 2 | x 2 ) . (228) In deriving the outer b ound, T elatar a nd Tse [5] ga v e extra information to the receiv ers (the so-called “genie-aided” a pproac h) t o handle the sev en inequalities. The rationale to what side information the genie should pro vide to handle the diﬀeren t linear inequalities w as somewhat sp eculativ e (cf. Section IV [5]). Our approac h a v oids an explicit represen tation of the inner-b ound. This higher lev el description allow ed us (cf. Section 4.2 ) to show that an y inequalit y in v olv ed in the pro- jected region can b e obtained by linear com binatio n of the inequalities (21 6)-(223) . F ur- ther, each inequality in (216)-(223) arises fro m a typic al err or ev ent consideration. W e no w ha v e the op erational insight into what side informa t io n to giv e when. W e demon- strate this pro cess in the instance of Equation (216). This inequality mus t b e satisﬁed to ensure that the R ece ive r 1 deco des its o wn priv ate message, o n the condition that it can deco de b oth the public messages correctly . This suggests t ha t corresp o nding to this inequalit y , w e may give the side information ( U n 1 , U n 2 ). A similar argumen t handles eac h of the o ther inequalities (216) - (223). 7 N -state Comp ound In t erference Channel In this section w e consider the natural extension of the 2-state compo und in terference c hannel to an N -state comp ound in terference channel. Our earlier results (b oth inner 30 and outer b ounds) a lso generalize naturally to the more general N -state mo del. 7.1 Mo del The N -state comp ound in terference c hannel is depicted in Figure 3. Eac h receiv er can b e in one of t he N p ossible states denoted b y α 1 , α 2 , . . . , α N . p S 1 α N | S 1 α ( N − 1) q ✲ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ✻ ✻ ✲ ✲ ✻ ✻ ✲ ✲ ❄ ✲ ❄ ❄ ❄ ✻ ✲ ✲ ✻ ❄ ✲ ❄ X 2 X 1 p S 2 α 1 | X 2 p S 1 α 1 | X 1 f 1 α 1 f 1 α 2 f 2 α 1 f 2 α 2 f 2 α N p S 2 α 2 | S 2 α 1 p S 1 α 2 | S 1 α 1 Y 1 α 1 Y 1 α 2 f 1 α N Y 1 α N Y 2 α 1 Y 2 α 2 Y 2 α N p S 2 α N | S 2 α ( N − 1) q Figure 3: The N -state comp ound in terference c hannel model. 7.2 Results W e can characterize the inner b ound and outer b ounds to the capacit y region in a w a y similar to the 2-state comp o und ch annel. Inner Bound Our co ding sc heme is N + 1-lev el sup erp osition co ding. This is m uc h along the lines of the 3-lev el sup erp osition co ding emplo y ed for the 2-state comp ound in terference channe l. The co ding sc heme is c haracterized by join tly distributed random v ariables ( Q, X 1 , U 1 α 1 , . . . , U 1 α N , X 2 , U 2 α 1 , . . . , U 2 α N ) (229) whic h satisfy the Mark ov c hain U 1 α N − . . . − U 1 α 1 − X 1 − Q − X 2 − U 2 α 1 − . . . − U 2 α N . (230) As earlier, w e restrict ourselv es to a subfamily of the join tly distributed random v ariables uniquely determined b y ( Q, X 1 , X 2 ) in the following wa y: 31 Giv en ( Q, X 1 , X 2 ), we pick ra ndom v a riables { U k α n , n = 1 , . . . , N , k = 1 , 2 } , (231) suc h that they hav e the same join t distribution as { S k α n , n = 1 , . . . , N , k = 1 , 2 } , (232) but are indep enden t of them. Using these random v aria bles , w e generate t he ( N + 1)-lev el superp osition random code b o oks for each user with rates ( R 1 α N , . . . , R 1 α 1 , R 1 p ) a nd ( R 2 α N , . . . , R 2 α 1 , R 2 p ) r es p ec- tiv ely . The deco ding at eac h receiv er is join tly typical set deco ding. It is similar to the deco ding describ ed for the 2-state. Eac h receiv er t r ies to deco de fully all of its own messages, but only partially decodes the other (in terfering) user. This strategy can b e seen as a n opp ortunistic strategy where the extent of the interferenc e t ha t the receiv er deco des dep ends up on the lev el of in terference it sees. The remainder description of the achie v a ble rate regio n follows the same dev elopmen t pattern as for the 2-stat e comp ound channel. It w ould b e impractical (in terms of the length of the descriptions) to explicitly detail this description. As such , w e brieﬂy itemize the main p oin ts in the a c hiev able region description b elow . • W e ﬁrst hav e an achie v able ra t e region ˜ R 2( N +1) in ( Q, X 1 , X 2 ) in 2( N + 1) dimensions along the same lines as (111) (we ha v e a v oided the explicit description of the linear inequalities describing the region due to the tedium and length inv olv ed in do ing so). As earlier, let ˜ R in ( Q, X 1 , X 2 ) b e the pr o jection onto the 2-dimensional space ( R 1 , R 2 ) where, R 1 = R 1 α N + . . . + R 1 α 1 + R 1 p , R 2 = R 2 α N + . . . + R 2 α 1 + R 2 p . (233) W e ha v e that ˜ R in ( Q, X 1 , X 2 ) is achiev able. • W e next deﬁne R 2( N +1) in ( Q, X 1 , X 2 ) as a g eneralization of (23) and deﬁne its pro jec- tion o n to the t wo dimensional space R in ( Q, X 1 , X 2 ). Lemma 7 can b e appropriately generalized to sho w that R in ( Q, X 1 , X 2 ) ⊆ ˜ R in ( Q, X 1 , X 2 ) , (234) th us pro ving t ha t R in ( Q, X 1 , X 2 ) is also achiev able. • W e next characterize an extremal represen tation of R in ( Q, X 1 , X 2 ), using a n appro- priate generalization of (131) to the N - state mo del). In other w ords, w e represen t it as a n intersec tion of hy p erplanes, where the inequalit y used to deﬁne the h y- p erplane can b e obtained a s a linear combin atio n of the inequalities used to deﬁne R 2( N +1) in ( Q, X 1 , X 2 ). 32 Outer Bound An outer- bound R out ( Q, X 1 , X 2 ) can b e deriv ed with an extremal represen tation that is similar to the corresp onding one for the inner b ound R in ( Q, X 1 , X 2 ) (this step is a natural generalization of (142)). In deriving the outer b ound, w e use appropriate genie- aided tec hniques (that in v olv e pro viding suitable side information to the receiv er). Again, what side information is shared is decided based on the typical error ev en ts whic h lead to the correspo nding inequality in the inner b ound. Gap Finally , we c haracterize the gap b et wee n the outer and inner b ounds to the capacity region for the n -state comp ound c hannel, in m uc h the same w ay as we did for the 2-state comp ound channel. This is stated formally b elo w. Theorem 10 F o r the N -state c omp ound interfer enc e channel of Figur e 3, if ( R 1 , R 2 ) is in the outer b ound to the c ap acity r e gion , then ( R 1 − ∆ 1 , R 2 − ∆ 2 ) is achievab l e , whe r e ∆ 1 ( Q, X 1 , X 2 ) = max 1 ≤ n ≤ N I ( X 2 ; S 2 α n | U 2 α n ) , (235 ) ∆ 2 ( Q, X 1 , X 2 ) = max 1 ≤ n ≤ N I ( X 1 ; S 1 α n | U 1 α n ) . (236 ) Sp ecializing to the deterministic v ersion, w e can see that this gap is zero and hence the capacity region is characterize d exactly . Sp ecializing to the Gaussian vers ion, we can see that this gap is no more than one bit. This completes the extension to the N -state comp ound channel scenario. 7.3 Discussion A few commen ts on the structure and prop erties of the ac hiev able sc heme are in order here. • Note that the structure of the ac hiev able sc heme (or the p o w er split in the Gaussian sc heme), whic h is characteriz ed b y t he joint random v ar ia bles ( Q, X 1 , U 1 α 1 , . . . , U 1 α N , X 2 , U 2 α 1 , . . . , U 2 α N ) , (237) dep ends only o n the in terfer enc e s tates and not on the d e terministic functions f k α n . The functions f k α n ho w ev er ma y still help in determining the a ctual achiev able ra te region. 33 W e highligh t this p oint by considering the case when eac h of the degraded in terfer- ence c hannels in our mo del are identit y , i.e., S k α 2 = S k α 1 = . . . = S k α N , k = 1 , 2 . (238) F or this mo del the “ comp oundness” of the channel is only due to the functions f k α n . Indeed, only tw o lev els o f superp osition co ding suﬃce, m uch as in the noncomp ound v ersion of the problem. • Let us assume S 1 α 2 = S 1 α 1 . (239) Then our ac hiev able sc heme sets U 1 α 2 = U 1 α 1 . (240) This implies that the lev el of the code bo ok corresponding to U 1 α 2 is “degenerate” and that w e migh t as w ell set R 1 α 2 = 0 . (241) Supp ose, ho w ev er that f 1 α 2 6 = f 1 α 1 (242) and hence the tw o receiv er stat es Y 1 α 1 and Y 1 α 2 are not the same. While the receiv er in either state adopts the same deco ding tec hnique (with resp ect to the lev el o f in terference it deco des), the higher dimensional constraints on the rate v ector, as imp osed by the deco ding condition for eac h state, are diﬀeren t. Nev ertheless, we see that for the Gaussian case one of these states is alw a ys w o r se than the other and th us w ould b e t he critical b ottlenec k in determining the ac hiev able rates; this is done next. 8 The Comp ound Gaussian In te rference Channel 8.1 Mo del The scalar Gaussian interferenc e c hannel is deﬁned by the complex c hannel parameters ( h 11 , h 21 , h 12 , h 22 ). The ﬁnite state comp ound channel v ersion of it allow s these parameters to tak e v alues in a ﬁnite set A . A =  ( h 11 , h 21 , h 12 , h 22 ) 1 , ( h 11 , h 21 , h 12 , h 22 ) 2 , . . . , ( h 11 , h 21 , h 12 , h 22 ) |A|  . (243) Deﬁne A k , { ( h 1 k , h 2 k ) | ( h 11 , h 21 , h 12 , h 22 ) ∈ A} , k = 1 , 2 , (244) 34 Observ e t ha t the channe ls fro m the tw o transmitters to the rece ive r k are deﬁned solely b y the parameters ( h 1 k , h 2 k ). Therefore the set A k is the set of states that the receiv er k can tak e. No w deﬁne A ′ as A ′ , A 1 × A 2 = { ( h 11 , h 21 , h 12 , h 22 ) | ( h 11 , h 21 ) ∈ A 1 , ( h 12 , h 22 ) ∈ A 2 } . (245) In ot her w ords A ′ allo ws for al l c ombi n ations of t he p ossible states for b oth the receiv ers. Let C ( A ) denote the capacity region of the comp ound c hannel deﬁned b y the set A . W e ha v e the following pro p osition: Prop osition 2 C ( A ) = C ( A ′ ) . Pr o o f : Note that A ⊆ A ′ . Thus it is clear tha t an y sc heme that w orks for the comp ound c hannel A ′ also w orks for the compound c hannel A . How ev er, since the tw o r ece ive rs do not co op erate, only the mar ginal c hannels to each r eceiv er decides the deco dability of a n y comm unication sc heme. W e no w conclude that a sc heme that w orks fo r the comp ound c hannel A also w orks for the comp ound c hannel A ′ . This completes the pro of. ✷ In the light of this observ atio n, we can, without loss of generality , consider ﬁnite state comp ound channels whose state set A decomp oses as A 1 × A 2 . In Section 2 we saw that the case where the cardina lity of A 1 and A 2 is restricted to 2 is captured by the 2 -state comp ound in terference c ha nnel of Figur e 2. Analo g ously , the general case where |A 1 | and |A 2 | are ﬁnite (with cardinalit y no more than N ) is captured b y the N -state comp ound in terference c hannel of Figure 3. W e see this formally b elo w. The k ey p oin t is the inﬁnitely divisible nature of Gaussian statistics. This asp ect w as used to show that the scalar Gaussian broadcast channel is alwa ys sto c hastically degraded. In a similar v ein, the comp ound scalar Gaussian in terference c hannel can alw ays b e supp osed to ha v e degraded in terference states. W e b egin by no ting that if |A 1 | 6 = |A 2 | , (246) then w e can add redundan t duplicate copies t o one of the sets, so tha t |A 1 | = |A 2 | . (247) Therefore, without of loss o f generality , we supp ose this is true.: |A 1 | = |A 2 | = N . (248) Then, or der the ﬁnite sets A 1 and A 2 suc h that | h 21 α 1 | ≥ | h 21 α 2 | ≥ . . . ≥ | h 21 α N | , | h 12 α 1 | ≥ | h 12 α 2 | ≥ . . . ≥ | h 12 α N | . 35 Next, w e do the f ollo wing substitution to reduce the ﬁnite state G a uss ian in terference c hannel to the mo del o f Figure 3. S 1 α 1 = h 12 α 1 X 1 + Z 1 α 1 , (249) S 1 α n = h 12 α n h 12 α ( n − 1) S 1 α ( n − 1) +   1 −      h 12 α n h 12 α ( n − 1)      2   1 / 2 Z 1 α n , n = 2 , . . . , N , (250) S 2 α 1 = h 21 α 1 X 2 + Z 2 α 1 , (251) S 2 α n = h 21 α n h 21 α ( n − 1) S 2 α ( n − 1) +   1 −      h 21 α n h 21 α ( n − 1)      2   1 / 2 Z 2 α n , n = 2 , . . . , N . (252) Y 1 α n = f 1 α n ( X 1 , S 2 α n ) = h 11 α n X 1 + S 2 α n , 1 ≤ n ≤ N , (253) Y 2 α n = f 2 α n ( X 2 , S 1 α n ) = h 22 α n X 2 + S 1 α n , 1 ≤ n ≤ N . (254) Here Z k α n s are indep enden t complex Gaussian random v ar iables with unit v ariance. Note t ha t the function f k α n captures the direct link gains h 11 and h 22 . The c hannels p ( S k α n | S k α ( n − 1) ) capture the cross link gains h 12 and h 21 as w ell as the a dditiv e no ise. 8.2 Main Result W e are no w ready to summarize the main result o f this pap er. Theorem 11 F o r the ﬁnite state c o mp ound Gaussian interfer enc e chann e l, multilevel sup erp osition c o ding with Gaussian c o de b o oks and opp ortunistic de c o ding dep ending on the i n terfer e n c e state is within one bit of the c ap acity r e gion. Pr o o f : W e ha ve sho wn earlier in this section that an y ﬁnite state Gaussian in terference c hannel is captured as a sp ecial case of the mo del in Figure 3. Sp ecializing the result of Theorem 10 t o the Gaussian case, w e ha v e that the multilev el sup erposition co ding is within one bit to the capacity . F urther, it suﬃce s to only consider Gaussian co de b o oks in the superp osition co de (along the same lines as Coro lla ry 6). ✷ 8.3 Discussion A few remarks are in order now. 36 1. While we hav e restricted b oth the direct-link gains ( h 11 and h 22 ) a nd the cross-link gains ( h 12 and h 21 ) to a ﬁnite set of v alues so far, it turns out that w e can relax this assumption for the direct link gain. In particular, supp ose ( h 11 , h 21 ) , ( h ′ 11 , h 21 ) ∈ A 1 , | h 11 | < | h ′ 11 | . (255) These corresp ond t o tw o states of the receiv er 1, whic h diﬀer only in the direct link gain, but ha v e the same cross link gain. As observ ed in the previous section, for either o f the tw o states, the receiv er adopts the same deco ding metho d. F urther, since w e hav e restricted o urselv es to Gaussian code b o oks, w e see that the p erfor- mance is restricted only by the state that has the w eak er of the t w o direct links: in this case, it is the one with parameters ( h 11 , h 21 ) T o summarize: If a rate ve ctor is achie v able for the state ( h 11 , h 21 ), then it is also achiev - able for ( h ′ 11 , h 21 ). Therefore, at an y r ece ive r, for a ﬁxed cross-link v alue the direct-link gain at the corresp onding rece iv er can take on v alues in a set. This set could b e inﬁnite. The receiv er state that has the w eake st direct link is the b ottlenec k. 2. A t an y rece ive r, the set of unique v alues the cross-link gains t a k e is assumed to b e ﬁnite. This ﬁniteness assumption app ears to b e critical since the n um b er of lev els required in the superp osition co ding at the in terfering tra nsmitter dep ends on it. If t he cardinality of t he set is N , then the n um b er of lev els in the sup erp osition co ding is N + 1. It is an o pen question to determine if a ﬁnite num b er of lev els of sup erposition coding suﬃce to handle a con tin uum o f interferenc e cross -link gains. A Analysis Of Probabi lit y Of Error In the follow ing we consider the deco dabilit y conditions at receiv er Rx 1 β only . A v ery similar analysis applies to the other receiv er-state pairs. Due to the symmetry of the random co de b o ok generation, the pro ba bility of error a v eraged ov er the ensem ble of random random co de b o oks, do es not dep end on whic h co dew ord w as sen t. Hence, without loss of generality , w e can assume that the mes sages indexed b y ( j 1 , k 1 , l 1 ) = ( 1 , 1 , 1) , ( j 2 , k 2 , l 2 ) = ( 1 , 1 , 1) , (256) w ere sen t by the t wo transmitters resp ectiv ely . Let us deﬁne the f o llo wing ev en t E j k lm =  Q n , U n 1 β ( j ) , U n 1 α ( j, k ) , X n 1 ( j, k , l ) , U n 2 β ( m ) , Y n 1 β  ∈ A ( n ) ǫ ( Q, U 1 β , U 1 α , X 1 , U 2 β , Y 1 β )  . 37 Letting P ( n ) e denote the pr o babilit y of deco ding error at Rx 1 β w e ha v e P ( n ) e = P  ( ∪ m E 111 m ) c [ ∪ ( j,k ,l ) 6 =(1 , 1 , 1) E j k lm  (257) ≤ P (( ∪ m E 111 m ) c ) | {z } ( a ) + X l 6 =1 P ( E 11 l 1 ) | {z } ( b ) + X l 6 =1 ,m 6 =1 P ( E 11 lm ) | {z } ( c ) + X k 6 =1 ,l P ( E 1 kl 1 ) | {z } ( d ) + X k 6 =1 ,l,m 6 =1 P ( E 1 kl m ) | {z } ( e ) + X j 6 =1 ,k ,l P ( E j k l 1 ) | {z } ( f ) + X j 6 =1 ,k ,l,m 6 =1 P ( E j k lm ) | {z } ( g ) . (258) The ﬁnal inequalit y used the union b ound. Let us consider each term in (258) and study the conditions needed to mak e it go to 0 asymptotically (in n ). • It is straigh tforward to see that ( a ) go es to 0 as n → ∞ . • No w consider ( b ). W e b egin b y noting that l ∈ { 1 , . . . , 2 nR 1 p } . Therefore if R 1 p = 0 then ( b ) = 0. Else, ( b ) ≤ 2 nR 1 p 2 − n ( I ( Y 1 β ; X 1 | U 1 α ,U 2 β ,Q ) − 5 ǫ ) . (259) Therefore for ( b ) to g o to 0 as n → ∞ , w e mus t ha v e R 1 p ≤ I ( Y 1 β ; X 1 | U 1 α , U 2 β , Q ) , if R 1 p > 0 . (260) • Similarly , ( c ) is 0 if R 1 p = 0 or R 2 β = 0 . Else, it m ust b e that R 2 β + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 α , Q ) . (261) It is imp ortan t to note that if R 2 β = 0, but R 1 p > 0 then, (261) is redundan t b ecause of (26 0). Therefore fo r ( c ) to go to 0 as n → ∞ (a ssuming that ( b ) go es t o 0 to o), w e m ust hav e, R 2 β + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 α , Q ) , if R 1 p > 0 . (262) Similarly for ( d ) , ( e ) , ( f ) and ( g ), w e m ust hav e R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 1 β , U 2 β , Q ) , if R 1 α + R 1 p > 0 , (263) R 2 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | U 1 β , Q ) , if R 1 α + R 1 p > 0 , (264) R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 | U 2 β , Q ) , if R 1 β + R 1 α + R 1 p > 0 , (265) R 2 β + R 1 β + R 1 α + R 1 p ≤ I ( Y 1 β ; X 1 , U 2 β | Q ) , if R 1 β + R 1 α + R 1 p > 0 , (266) resp ectiv ely . 38 B Pro of Of L emma 7 Consider an y ( R 1 , R 2 ) ∈ R in ( P ). Then there exists an ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) ∈ R (6) in ( P ) (267) suc h that, R 1 = R 1 p + R 1 α + R 1 β and R 2 = R 2 p + R 2 α + R 2 β . (268) W e will ﬁnd a ( ˜ R 1 p , ˜ R 1 α , ˜ R 1 β , ˜ R 2 p , ˜ R 2 α , ˜ R 2 β ) ∈ ˜ R (6) in ( P ) (269) suc h that, ˜ R 1 p + ˜ R 1 α + ˜ R 1 β = R 1 p + R 1 α + R 1 β , ˜ R 2 p + ˜ R 2 α + ˜ R 2 β = R 2 p + R 2 α + R 2 β , b y the follo wing algor it hmic pro cedure. Step 1a) F or k = 1 , 2, if R k β < 0 then, R k β ← 0 , R k α ← R k α + R k β . (270) Step 1b) F or k = 1 , 2, if R k α < 0 then, R k α ← 0 , R k p ← R k p + R k α . (271) Step 2a) F or k = 1 , 2, if R k p < 0 then, R k p ← 0 , R k α ← R k α + R k p . (272) Step 2b) F or k = 1 , 2, if R k α < 0 then, R k α ← 0 , R k β ← R k β + R k α . (273) First up, w e note that at eac h step we are ensuring that R 1 p + R 1 α + R 1 β and R 2 p + R 2 α + R 2 β sta y inv a r ian t. Next, note tha t if R 1 p , R 1 α , R 1 β , R 2 p , R 2 α and R 2 β are all nonnegativ e to b egin with, then it is easy to see that ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) ∈ ˜ R (6) in ( P ) (274) and hence ( R 1 , R 2 ) ∈ ˜ R in ( P ) . (275) 39 Claim 1 A t the end of Step 1b , the new ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) stil l r emains in R (6) in ( P ) and satisﬁes R 1 α , R 1 β , R 2 α , R 2 β ≥ 0 . (276) Pr o o f : Consider Step 1a. Note t hat in this step w e are p oten tially increasing R k β , but the rest of the comp onen ts either remain the same or decrease. Also note that in this step, w e are k eeping R k β + R k α in v ariant. Therefore, w e o nly need to ensure that the inequalities among (24 )-(53) t ha t ha ve R k β , but not R k α are not violated. This can b e v eriﬁed to b e true, b ecause of the p olymatroidal nature of eac h blo c k o f the inequalities in (24)-(53). The argumen t is similar for Step 1b. ✷ Claim 2 A t the end of Step 2b , the new ( R 1 p , R 1 α , R 1 β , R 2 p , R 2 α , R 2 β ) is in ˜ R (6) in ( P ) . Pr o o f : Note that in Step 2a, the only comp onen t that p otentially increases is R k p , a nd so w e migh t b e violating one of the following constrain ts: (24),(2 5),(30)-(32),(39),(40) and (45)-(47). Ho w ev er, b y setting R k p = 0, these violated constraints no longer matter for ˜ R (6) in ( P ). The argumen t is similar for St ep 2b. Note that at the end o f Step 2b, w e ha v e ensured that all the comp onen ts are nonnegative. ✷ References [1] T. S. Han a nd K. Kobay ashi, “A new ach iev able rate region for the in terference c hannel,” IEEE T r ans. Inform. The ory , vol. 27, pp. 4 9 –60, Jan. 1981. [2] A. A. El Gamal and M. H. M. Costa, “The capacity r egio n of a class of deterministic in terference channels ,” IEEE T r a ns. Inform. The ory , v ol. 28(2), pp. 343-346 , Mar. 1982. [3] H. F. Chong, M. Mota ni, H. K . Garg , a nd H. E. Gamal, “On the Han-Kobay ashi region for the interference c hannel,” submitte d to the IEEE T r ans. Inform. The ory , 2006. [4] R. H. Etkin, D. Tse, and H. W ang, “Gaussian interfere nce c hannel capacit y to within one bit,” subm i tt e d to the IEEE T r ans. Inform. The ory , 200 7. [5] E. T elatar and D. Tse, “Bounds on the capacity region of a class of interferenc e c hannels,” in Pr o c. I EEE In ternational Symp osium o n Information The ory 2007 , Nice, F rance, Jun. 2007. [6] J. A. Thomas, “F eedbac k can at most double Gaussian m ultiple access channe l ca- pacit y ,” IEEE T r ans. Inform. The ory , V ol. 33(5), pp. 7 11-716, Sept. 1987 . 40 [7] R. T yrrell Ro c k afellar, “Con ve x Analysis,” Princ eton Unive rs ity Pr ess , 197 0. [8] T. M. Co v er and J. A. Thomas, “Elemen ts of Information Theroy ,” Wiley Series in T ele c om munic ations , 19 91. 41

The Two User Gaussian Compound Interference Channel

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