Binary Fading Interference Channel with No CSIT

Transcription

1 Biary Fadig Iterferece Chael with No CSIT Alireza Vahid, Mohammad Ali Maddah-Ali, A. Salma Avestimehr, ad Ya Zhu arxiv: v3 [cs.it] 4 Mar 07 Abstract We study the capacity regio of the two-user Biary Fadig or Erasure Iterferece Chael where the trasmitters have o kowledge of the chael state iformatio. We develop ew ier-bouds ad outer-bouds for this problem. We idetify three regimes based o the chael parameters: weak, moderate, ad strog iterferece regimes. Iterestigly, this is similar to the geeralized degrees of freedom of the two-user Gaussia iterferece chael where trasmitters have perfect chael kowledge. We show that for the weak iterferece regime, treatig iterferece as erasure is optimal while for the strog iterferece regime, decodig iterferece is optimal. For the moderate iterferece regime, we provide ew ier ad outer bouds. The ier-boud is based o a modificatio of the a- Kobayashi scheme for the erasure chael, ehaced by timesharig. We study the gap betwee our ier-boud ad our outer-bouds for the moderate iterferece regime ad compare our results to that of the Gaussia iterferece chael. Derivig our ew outer-bouds has three mai steps. We first create a cotracted chael that has fewer states compared to the origial chael, i order to make the aalysis tractable. We the prove the Correlatio Lemma that shows a outerboud o the capacity regio of the cotracted chael also serves as a outer-boud for the origial chael. Fially usig the Coditioal Etropy Leakage Lemma, we derive our outerboud o the capacity regio of the cotracted chael. Idex Terms Iterferece chael, biary fadig, capacity, chael state iformatio, o CSIT, packet collisio. I. INTRODUCTION The two-user Iterferece Chael IC itroduced i [] is a caoical example to study the impact of iterferece i commuicatio etworks. There exists a extesive body of work o this problem uder various assumptios e.g., [3] []. I this work, we focus o a specific cofiguratio of this etwork, amed the two-user Biary Fadig Iterferece Chael BFIC as depicted i Fig. i which the chael gais at each time istat are i the biary field accordig to some Beroulli distributio. The iput-output relatio of this chael at time istat t is give by Y i [t] = G ii [t]x i [t] G īi [t]x ī [t], i =,, where ī = 3 i, G ii [t], G īi [t] {0, }, ad all algebraic operatios are i F. This model was first itroduced i []. A. Vahid is with the School of Electrical ad Computer Egieerig, Duke Uiversity, Durham, NC, USA. alireza.vahid@duke.edu. Mohammad Ali Maddah-Ali is with the Departmet of Electrical Egieerig, Sharif Uiversity of Techology, Tehra, Ira. maddah ali@sharif.edu. A. S. Avestimehr is with the School of Electrical ad Computer Egieerig, Uiversity of Souther Califoria, Los Ageles, CA, USA. avestimehr@ee.usc.edu. Ya Zhu is with Aerohive Networks Ic., Suyvale, CA, USA. zhuya79@gmail.com. Prelimiary parts of this work was preseted i the 04 Iteratioal Symposium o Iformatio Theory ISIT []. Copyright c 04 IEEE. Tx X [t] X [t] Tx G [t] G [t] G [t] G [t] Rx Rx Y [t] Y [t] Fig.. Two-user Biary Fadig Iterferece Chael BFIC. The motivatio for studyig the Biary Fadig or Erasure Iterferece Chael is twofold. First as demostrated i [3], it provides a simple yet useful physical layer abstractio for wireless packet etworks i which wheever a collisio occurs, the receiver ca store its received aalog sigal ad utilize it for decodig the packets i the future for example, by successive iterferece cacellatio techiques. I this cotext, the biary fadig model is motivated by a shadow fadig eviromet i which each lik is either o or off accordig to the shadow fadig distributio, ad the multiple access MAC is modeled such that if two sigals are trasmitted simultaeously ad the liks betwee the correspodig trasmitters ad the receiver are ot i deep fade, the a liear combiatio of the sigals is available to the receiver. The study of the BFIC i [3] has led to several codig opportuities that ca be utilized by the trasmitters to exploit the available sigal at the receivers for iterferece maagemet. Moreover, this model allows researchers to focus o other iterestig challeges i iterferece chaels such as spatial correlatio [4] ad locality of chael state kowledge [5], [6]. The secod motivatio for studyig the BFIC is that it ca be a first step towards uderstadig the capacity of the fadig iterferece chaels with o kowledge of the chael state iformatio at the trasmitters CSIT. This model was used i [7], [8] to derive the capacity of the oe-sided iterferece chael also kow as Z-Chael. Motivated by the determiistic approach [9], a layered erasure broadcast chael model was itroduced i [0] to approximate the capacity of fadig broadcast chaels. Oe ca view our Biary Fadig model as the model itroduced i [8], [0] with a sigle layer. I this work, we cosider the two-user BFIC uder the o chael state iformatio at the trasmitters assumptio. I this o CSIT model, the trasmitters are oly aware of the distributios of the chael gais but ot the actual realizatios. We develop ew ier-bouds ad outer-bouds for this problem ad we idetify three regimes based o the chael parameters: weak, moderate, ad strog iterferece

2 regimes. For the weak ad the strog iterfrece regimes, we show that the etire capacity regio is achieved by applyig poit-to-poit erasure codes with appropriate rates at each trasmitter, ad usig either treat-iterferece-as-erasure or iterferece-decodig at each receiver. For the moderate iterferece regime, we provide ew ier ad outer bouds. The ier-boud is based o a modificatio of the a-kobayashi scheme for the erasure chael, ehaced by time-sharig. I the moderate iterferece regime, the ier-bouds ad the outer-bouds do ot match. We provide some further isights ad compare this problem to the two-user static o-fadig Gaussia iterferece chael where trasmitters have perfect chael kowledge. To derive the outer-boud, we icorporate two key lemmas. The first lemma, the Coditioal Etropy Leakage Lemma, establishes how much iformatio is leaked from each trasmitter to the uiteded receiver. The secod lemma, the Correlatio Lemma, shows that if the chael gais are correlated uder a give set of coditios, the capacity regio caot be smaller tha the case of idepedet chael gais. Usig the Correlatio Lemma, we create a cotracted chael that has fewer states as opposed to the origial chael ad hece, the problem becomes tractable. The, usig the Coditioal Etropy Leakage Lemma, we derive a outer-boud o the capacity regio of the cotracted chael which i tur, serves as a outer-boud for the origial chael. The rest of the paper is orgaized as follows. I Sectio II, we formulate our problem. I Sectio III, we preset our mai results. Sectio IV is dedicated to derivig the outer-boud. We describe our achievability strategy i Sectio V. Sectio VI cocludes the paper ad describes future directios. II. PROBLEM SETTING We cosider the two-user Biary Fadig Iterferece Chael BFIC as illustrated i Fig.. The chael gai from trasmitter Tx i to receiver Rx j at time istat t is deoted by G ij [t], i, j {, }. We assume that the chael gais are either 0 or i.e. G ij [t] {0, }, ad they are distributed as idepedet Beroulli radom variables idepedet from each other ad over time. Furthermore, we cosider the symmetric settig where G ii [t] d Bp d ad G iī [t] d Bp c, for 0 p d, p c, ī = 3 i, ad i =,. We defie q d = pd ad q c = pc. At each time istat t, the trasmit sigal at Tx i is deoted by X i [t] {0, }, i =,, ad the received sigal at Rx i is give by Y i [t] = G ii [t]x i [t] G īi [t]x ī [t], i =,, 3 where all algebraic operatios are i F. Due to the ature of the chael gais, a total of 6 chael realizatios may occur at ay give time istat as give i Table I. The chael state iformatio CSI at time istat t is deoted by the quadruple G[t] = G [t], G [t], G [t], G [t]. 4 We use the followig otatios i this paper. We use capital letters to deote radom variables RVs, e.g. G ij [t] is a radom variable at time istat t, ad small letters deote the realizatios, e.g. g ij [t] is a realizatio of G ij [t]. For a atural umber k, we set Fially, we set G k = [G[], G[],..., G[k]]. 5 G t iix t i G t īix t ī 6 = [G ii[]x i[] G īi []X ī [],..., G ii[t]x i[t] G īi [t]x ī [t]]. I this paper, we cosider the o CSIT model for the available chael state iformatio at the trasmitters. I this model, we assume that trasmitters oly kow the distributio from which the chael gais are draw, but ot the actual realizatios of them. Furthermore, we assume that receiver i has istataeous kowledge of G ii [t] ad G īi [t] i.e. the icomig liks to receiver i, i =,. Cosider the sceario i which Tx i wishes to reliably commuicate message W i {,,..., Ri } to Rx i durig uses of the chael, i =,. We assume that the messages ad the chael gais are mutually idepedet ad the messages are chose uiformly. For each trasmitter Tx i, i =,, uder o CSIT assumptio, let message W i be ecoded as Xi usig the followig equatio X i [t] = f i,t W i, t =,,...,, 7 where f i,.. is the ecodig fuctio at trasmitter Tx i. Receiver Rx i is oly iterested i decodig W i, ad it will decode the message usig the decodig fuctio Ŵ i = ϕ i Y i, G ii, īi G. A error occurs whe Ŵi W i. The average probability of decodig error is give by λ i, = E[P [Ŵ i W i ]], i =,, 8 ad the expectatio is take with respect to the radom choice of the trasmitted messages W ad W. A rate tuple R, R is said to be achievable, if there exist ecodig ad decodig fuctios at the trasmitters ad the receivers respectively, such that the decodig error probabilities λ,, λ, go to zero as goes to ifiity. The capacity regio C p d, p c is the closure of all achievable rate tuples. I the ext sectio, we preset the mai results of the paper. III. MAIN RESULTS I this sectio, we preset our mai cotributios. We first eed to defie the symmetric sum-rate. Defiitio. For the two-user BFIC with o CSIT, the symmetric sum-rate is defied as R sym p d, p c = R + R such that R = R, R, R C p d, p c. 9 The followig theorems state our mai cotributios. Theorem. For the two-user BFIC with o CSIT the followig symmetric sum-rate, R sym p d, p c, is achievable:

3 3 TABLE I ALL POSSIBLE CANNEL REALIZATIONS; SOLID ARROW FROM TRANSMITTER Tx i TO RECEIVER Rx j INDICATES TAT G ij [t] =. ID ch. realizatio ID ch. realizatio ID ch. realizatio ID ch. realizatio For 0 p c p d +p d : For p d +p d p c p d : p d q c ; where p d + p c p d p c + p d p c Cδ ; Cδ := p dp c p d p c p d p c p d p c. 0 For p d p c : p d + p c p d p c. The followig theorem establishes the outer-boud. Theorem. The capacity regio of the two-user BFIC with o CSIT, C p d, p c, is cotaied i C p d, p c give by: For 0 p c p d R 0 R, R i p d i =, R i +βr ī βp d +p c p d p c where p d +p d : β = p d p c p d p c. For +p d p c p d : 0 R i p d i =, R, R R i + R ī p c R i + p d p c R ī p d p c p d + p c p d p c 3 For p d p c : R 0 R, R i p d i =, R i + R ī p d + p c p d p c 4 Fig.. The gap betwee the the ier-bouds ad the outer-bouds for the two-user BFIC with o CSIT. As we show later, the ier-bouds ad the outer-bouds for the two-user BFIC with o CSIT match whe 0 p c p d +p d or p d p c which correspod to the weak ad the strog iterferece regimes. owever, for the moderate iterferece p regime, i.e. d +p d p c p d, the bouds do ot meet, thus the capacity regio remais ope. The gap betwee the the ier-bouds ad the outer-bouds is plotted i Fig. where the bouds match i the white regio. For the weak ad the strog iterferece regimes, the capacity regio is obtaied by applyig poit-to-poit erasure codes with appropriate rates at each trasmitter, ad usig either treat-iterferece-as-erasure or iterferece-decodig at each receiver based o the chael parameters. More precisely for 0 p c < p d / + p d, the capacity regio is obtaied by treatig iterferece as erasure, while for p d p c, the capacity regio is obtaied by iterferece-decodig i.e.

4 ,0.6, , , , , a b Fig. 3. a Capacity regio for p d = ad p c = 0, 0.4, 0.5; ad b capacity regio for p d = 0.5 ad p c = 0, 0., 0.5. the itersectio of the capacity regios of the two multiple access chaels at receivers. For the moderate iterferece regime, the ier-boud is based o a modificatio of the a-kobayashi scheme for the erasure chael, ehaced by time-sharig. The detailed proof of the achievability strategy ca be foud i Sectio V. Fig. 3 illustrates the capacity regio for several values of p d ad p c. I Fig. 3a the capacity regio is depicted for p d = ad p c = 0, 0.4, 0.5. From Fig. 3b, we coclude that ulike the case i Fig. 3a, decreasig p c does ot ecessarily elarge the capacity regio. I fact, for p d = 0.5, we have C 0.5, 0.5 C 0.5, 0., C 0.5, 0. C 0.5, Usig the results of [], we have plotted the capacity regio of the two-user Biary Fadig IC for p d = p c = 0.5 uder three differet scearios i Fig. 4. Uder the delayed CSIT, we assume that trasmitters become aware of all chael realizatios with uit delay ad receivers have access to istataeous CSI, i.e. ad X i [t] = f i,t Wi, G t, t =,,...,, 6 Ŵ i = ϕ i Y i, G ; 7 ad uder istataeous CSIT model, we assume that at time istat t, trasmitters ad receivers have access to G t. As we ca see i Fig. 4, whe trasmitters have access to the delayed kowledge of all liks i the etwork, the capacity regio is strictly larger tha the capacity regio uder the o CSIT assumptio. To prove Theorem, we icorporate two key lemmas as discussed i Sectio IV-A. The first step i obtaiig the outer-boud is to create a cotracted chael that has fewer states compared to the origial chael. Usig the Correlatio Lemma, we show that a outer-boud o the capacity regio of the cotracted chael also serves as a outer-boud for the origial chael. Fially, usig the Coditioal Etropy Leakage Lemma, we derive this outer-boud. The ituitio for the Correlatio Lemma was first provided by Sato [4]: the capacity regio of all iterferece chaels that have the same margial distributios is the same. We take o CSIT istataeous CSIT delayed CSIT Fig. 4. Capacity regio of the two-user Biary Fadig IC for p d = p c = 0.5, with o CSIT, delayed CSIT, ad istataeous CSIT. this ituitio ad impose a certai spatial correlatio amog chael gais such that the margial distributios remai uchaged. For the moderate iterferece regime, we also icorporate the outer-bouds o the capacity regio of the oesided BFIC with o CSIT [8]. The rest of the paper is dedicated to the proof of our mai cotributios. We provide the coverse proof i Sectio IV, ad we preset our achievability strategy i Sectio V. IV. CONVERSE I this sectio, we provide the coverse proof of Theorem for the o CSIT assumptio. We icorporate two lemmas i order to derive the outer-boud that we describe i the followig subsectio. A. Key Lemmas Etropy Leakage Lemma: Cosider a broadcast chael as depicted i Fig. 5 where a trasmitter is coected to two receivers through biary fadig chaels. Suppose G i [t] is distributed as i.i.d. Beroulli radom variable i.e. G i [t] d Bp i where 0 p p, i =,. I this chael the received sigals are give as Y i [t] = G i [t]x[t], i =,, 8

5 5 Tx X[t] Rx Y [t] Y [t] Rx Fig. 5. A broadcast chael with biary fadig liks. where X[t] {0, } is the trasmit sigal at time istat t. Furthermore, suppose G 3 [t] d Bp 3, 0 p 3, such that Pr [G i [t] =, G j [t] = ] = 0, i j, i, j {,, 3}. 9 We further defie G T [t] = G [t], G [t], G 3 [t]. 0 The, for the chael described above, we have the followig lemma. Lemma. [Coditioal Etropy Leakage Lemma] For the chael described above with o CSIT, for p + p + p 3 ad ay iput distributio, we have Y G 3 X, G T p p Y G 3 X, G T. Proof. Let G [t] be distributed as Bp /p, ad be idepedet of all other parameters i the etwork. Let Y [t] = G [t]y [t], t =,...,. It is straightforward to see that Y t is statistically the same as Y t uder the o CSIT assumptio. For time istat t, where t, we have Y [t] Y t, G 3 X, G T a = Y [t] Y t, G 3 X, G T, G t b = p X[t] Y t, G 3 X, G T, G t, G [t] =, G 3 [t] = 0 c = p X[t] Y t, G 3 X, G T, G t, G 3[t] = 0 d = p X[t] Y t, G 3 X, G T, G t, G 3[t] = 0 e p X[t] Y t, Y t, G 3 X, G T, G t, G 3[t] = 0 f = p X[t] Y t, G 3 X, G T, G t, G 3[t] = 0 g = p 3 p Y [t] Y t, G 3 X, G T, G t p, G 3[t] = 0 h = p Y [t] Y t, G 3 X, G T, G t p i = p p Y [t] Y t, G 3 X, G T, 3 where a holds sice G t is idepedet of all other parameters i the etwork; b follows from Pr [G [t] = G 3 [t] = 0] = p p 3 ; 4 c holds sice the trasmit sigal is idepedet of the chael realizatios; d follows from the fact that Pr [ X[t], Y t, G 3 X, G T, G t, G 3[t] = 0 ] = Pr [ X[t], Y t, G 3 X, G T, G t, G 3[t] = 0 ]. 5 This equality holds sice startig from oe side a simple idex exchage of gives the other side ad X[t] is oblivious of the chael realizatios; e holds sice coditioig reduces etropy; f is true sice Y t t Y, G 3 X, G T, G t = 0; 6 g follows from the fact that Pr [G [t] = G 3 [t] = 0] = p p 3 ; 7 h is true sice Pr [G 3 [t] = 0] = p 3 ; ad i holds sice G t is idepedet of all other parameters i the etwork. Thus, summig all terms for t =,...,, we get Y [t] Y t, G 3 X, G T t= which implies p p t= Y [t] Y t, G 3 X, G T, 8 Y G 3 X, G T p p Y G 3 X, G T, 9 hece, completig the proof. Remark. I [], we drived the Etropy Leakage Lemma for the case where the trasmitter has the CSI with delay. We observe that, with delayed CSIT, the costat o the RS of would be smaller, meaig that the trasmitter ca further favor the stroger receiver. Correlatio Lemma: Cosider agai a biary fadig iterferece chael similar to the chael described i Sectio II, but where chael gais have certai correlatio. We deote the chael gai from trasmitter Tx i to receiver Rx j at time istat t by G ij [t], i, j {, }. We distiguish the RVs i this chael, usig. otatio e.g., X [t]. The iputoutput relatio of this chael at time istat t is give by Ỹ i [t] = G ii [t] X i [t] G īi [t] X ī [t], i =,. 30 We assume that the chael gais are distributed idepedet over time. owever, they ca be arbitrary correlated with each other subject to the followig costraits. Pr Gii [t] = = p d, Pr Gīi [t] = = p c, Pr Gii [t] =, G īi [t] = = Pr Gii [t] = Pr Gīi [t] =, i =,. 3 I other words, the chael gais correspodig to icomig liks at each receiver are still idepedet. Similar to the origial chael, we assume that the trasmitters i this BFIC have o kowledge of the CSI. We have the followig result. Lemma. [Correlatio Lemma] For ay BFIC that satisfies the costraits i 3, we have C p d, p c C p d, p c. 3 Proof. Suppose i the origial BFIC messages W ad W are ecoded as X ad X respectively, ad each receiver

6 6 ca decode its correspodig message with arbitrary small decodig error probability as. Now, we show that if we use the same trasmissio scheme i the BFIC that satisfies the costraits i 3, i.e. X i [t] = X i [t], t =,,...,, i =,, 33 the the receivers i this BFIC ca still decode W ad W. I the origial two-user biary fadig IC as described i Sectio II, Rx i uses the decodig fuctio Ŵ i = ϕ i Yi, G ii, G īi. Therefore, the error evet Ŵ i W i, oly depeds o the choice of W i ad margial distributio of the chael gais G ii ad G īi. Defie } {W, G, G s.t. Ŵ W E W = Ẽ W = { W, G, G the, the probability of error is give by p error = w Pr W = w Pr E w = R w a = R w, } s.t. Ŵ W, 34 Pr w, g, g 35 w,g,g E w Pr w, g, g = p error, w, g, g Ẽw where p error ad p error are the decodig error probability at Rx i the origial ad the BFIC satisfyig the costraits i 3 respectively; ad a holds sice accordig to 3, the joit distributio of G ad G is the same as G ad G ad the fact that, as metioed above, the error probability at receiver oe oly depeds o the margial distributio of these liks. Similar argumet holds for Rx. B. Derivig the Outer-bouds The key to derive the outer-boud is the proper applicatio of the two lemmas itroduced i Sectio IV-A. More precisely, we eed to fid a chael that satisfies the costraits i 3 such that the outer-boud o its capacity regio, coicides with the achievable regio of the origial problem. We provide the proof for three separate regimes. Regime I: 0 p c p d / + p d : The derivatio of the idividual bouds, e.g., R p d, is straightforward ad omitted here. We focus o where R + βr βp d + p c p d p c, 36 { } pd p c β = max,. 37 p d p c Due to symmetry, the derivatio of the other boud is similar. The first step is to defie the appropriate chael that satisfies the costraits i 3. The idea is to costruct a chael such that Gii [t] = wheever G iī [t] =, i =,. We costruct such chael with oly five states rather tha 6 states ad thus, we refer to it as cotracted chael. The five states are deoted by states A, B, C, D, ad E TABLE II TE CONTRACTED CANNEL FOR Regime I AND Regime II. ID chael realizatio ID chael realizatio A C Pr [state A] = p d p c Pr [state C] = q d p c B D Pr [state B] = p d p c Pr [state D] = q d p c with correspodig probabilities p d p c, p d p c, q d p c, q d p c, ad q d q c. These states are depicted i Table II with the exceptio of state E which correspods to the case where all chael gais are 0. ere, we have Pr G [t] = = j {A,B,C} Pr G [t] = = j {A,C} Pr State j = p d, Pr State j = p c, Pr G [t] =, G [t] = = Pr State A = p d p c, 38 thus, this chael satisfies the coditios i 3. From Lemma, we have C p d, p c C p d, p c, thus, ay outerboud o the capacity regio of the cotracted chael, provides a outer-boud o the capacity regio of the origial chael. We defie X A [t] = X [t] {state A occurs at time t}, 39 where {state A occurs at time t} is equal to whe at time t state A occurs. Similarly, we defie X B [t], X C [t], X D [t], X E [t], X A [t], X B [t], X C [t], X D [t] ad X E [t]. Therefore, we have X [t] = X A [t] X B [t] X C [t] X D [t] X E [t]. 40 Suppose i the cotracted chael, there exist ecoders ad decoders at trasmitters ad receivers respectively, such that each receiver ca decode its correspodig message with arbitrary small decodig error probability as ɛ 0. The derivatio of the outer-boud is give i 4 for ɛ 0 as ; ad 0 p c p d / + p d β = p d p c p d p c >. 4

7 7 We have R + β R ɛ a I X ; Ỹ G + βi X ; Ỹ G b = X D G + X B, X C X D, G + X A X A X B, X C, X D, G X D X, G X A X D, X, G + β X C G + β X B, X D X C, G + β X A X A X B, X D, X C, G β X C X, G β X A X C, X, G c X D G + X B, X C G + X A X A X C G G X D G X A X D, G + β + β X B, X D G + β X A X A G β X C G β X A X C, G d = X C G + X B X C, G + + β X A X A G β X A X C, G + β X B, X D G X A X D, G e X C G + + β X A X A G + β X D G + β X B X D, G X A X D, G f X C G + + β X A X A G + β X D G + β X B β X D, G g q d p c + + β p d p c + βq d p c + β p d p c β = βp d + p c p d p c, 4 where a follows from Fao s iequality ad data processig iequality; b follows from the defiitio of the cotracted chael ad the chai rule; c is true sice from Claim below, we have which results i I X ; X G = 0, 43 X A X D, X, G = X A X D, G, X A X C, X, G = X A X C, G ; 44 ad the fact that coditioig reduces etropy; d follows from the chai rule; e holds sice usig Lemma, we have use aalogy: X B Y, X C G 3 X, ad X A Y X B X C, G β X A X C, G 0; 45 ad g follows from the fact that etropy of a biary radom variable is maximized by i.i.d. Beroulli distributio with success probability of half. Dividig both sides by ad let, we get the desired result. Claim. Proof. I X ; X G = I X ; X G I W, X ; W, X G = I W ; W G + I W ; X W, G + I X ; W, X W, G = where the last equality holds sice I W ; W G = 0, 49 due to the fact that the messages ad the chael gais are mutually idepedet; I W ; X W, G = 0, 50 due to the fact that X = f W, G ; ad I X ; W, X W, G, 5 due to the fact that X = f W, G. Regime II: p d / + p d p c p d : I this regime for i =,, we borrow the outer-boud R i + p d p c p d p c R p d p c + p d p ī p d + p c q d, 5 c p d p c + p d p c from [8]. Thus, we focus o R i + R ī p c, i =,. 53 We use the same cotracted chael as for the case of Regime I. Let G S [t] be distributed as i.i.d. Beroulli RV ad ad for i =,, we defie G S [t] d B p d p c p d p c, 54 ad f follows sice from applyig Lemma, we have use aalogy: X A Y, X D G 3 X, ad X B Y X A X D, G β X B X D, G ; 46 X ia [t] = G S [t] X ia [t], ˆX ia [t] = G S [t] X ia [t], 55

8 8 We have R + R a ɛ X ; Ỹ G + I X ; Ỹ G I b = X A X A, X B, X C, X D G X A, X D X, G + X A X A, X B, X C, X D G X A, X C X, G c = X D G + X B, X C X D, G + X A X A X B, X C, X D, G X D X, G X A X, X D, G + X C G + X B, X D X C, G + X A X A X B, X D, X C, G X C X, G X A X C, X, G d = X D G + X B, X C G + X A X A X B, X C, X D, G X D G X A X D, G + X C G + X B, X D G + X A X A X B, X D, X C, G X C G X A X C, G where ɛ 0 as ; a follows from Fao s iequality ad data processig iequality; b holds due to the defiitio of the cotracted chael; c follows from the chai rule; d is true sice from Claim, we have I X ; X G = We use our defiitio i 55 for the rest of the proof. e = X C G + X B X C, G + X A X A X B, X C, X D, G X A X C, G ˆX A X B, X C, G + X D G + X B X D, G + X A X A X B, X D, X C, G X A X D, G ˆX A X B, X D, G f X C G + X A X A X B, X C, X D, G ˆX A X B, X C, G + X D G + X A X A X B, X D, X C, G ˆX A X B, X D, G 57 X C G + X A X A X B, X C, X D, G + ˆX A, ˆX A X A X A, X B, X C, X D, G ˆX A X B, X C, G + X D G + X A X A X B, X D, X C, G ˆX A X B, X D, G g X C G + X A X A X B, X C, X D, G + ˆX A ˆX A, X A X A, X B, X C, X D, G + X D G + X A X A X B, X D, X C, G h p c + ɛ, 58 where e follows from 55 ad the costructio of the sigals, we have thus, Pr [ ˆX A, X A, X C, G ] = Pr [ ˆX A, X B, X C, G ], ˆX A X A, X C, G = ˆX A X B, X C, G, ad similar statemet is true for X A ad X B ; f holds sice from Lemma, we have X B X C, G X A X C, G 0; 6 g holds sice ˆX A X A X A, X B, X C, X D, G h holds sice ˆX A X B, X C, G 0; 6 X C G q d p c, X D G q d p c, X A X A X B, X D, X C, G p d p c, X A X A X B, X C, X D, G p d p c, ˆX A ˆX A, X A X A, X B, X C, X D, G p c + p d p c p d. 63 Regime III: p d p c : I this regime, agai we have β =, ad the capacity regio would be equal to the itersectio of capacity regios of the two MACs formed at the receivers. Uder o CSIT assumptio, we do ot eed to create a cotracted chael. The derivatio of the outer-boud is easier compared to the other regimes. Basically, Rx i after decodig ad removig its correspodig sigal, has a stroger chael from Tx ī compared to Rx ī, ad thus it must be able to decode both W i ad W ī, i =,. Thus, we have R + R p d + p c p d p c. 64

9 9 V. ACIEVABILITY I this sectio, we provide the proof of Theorem. We show that for the weak ad the strog iterferece regimes, the etire capacity regio is achieved by applyig poit-to-poit erasure codes with appropriate rates at each trasmitter, usig either treat-iterferece-as-erasure or iterferece-decodig at each receiver, based o the chael parameters. For the moderate p iterferece regime, i.e. d +p d p c p d, the ier-boud is based o a modificatio of the a-kobayashi scheme for the erasure chael, ehaced by time-sharig. A. Weak ad Strog Iterferece Regimes Each trasmitter applies a poit-to-poit erasure radom code as described i []. O the other had, at each receiver we have two optios: iterferece-decodig; ad treat-iterferece-as-erasure. Whe a receiver decodes the iterferece alogside its iteded message, the achievable rate regio is the capacity regio of the multiple-access chael MAC formed at that receiver as depicted i Fig. 6. The MAC capacity at Rx, is give by R p d, R p c, 65 R + R q d q c. W W X X Tx Tx G G G G Rx Rx Fig. 6. The multiple-access chael MAC formed at Rx. As a result, Rx ca decode its message by iterferece decodig, if R ad R satisfy the costraits i 65. O the other had, if Rx treats iterferece as erasure, it basically igores the received sigal at time istats where G [t] =, ad p d q c fractio of the time, it receives the trasmit sigal of Tx. Thus, Rx ca decode its message by treat-iterfereceas-erasure, if R p d q c. Similarly, Rx ca decode its message by treat-iterfereceas-erasure, if R p d q c. Also, Rx ca decode its message by iterferece decodig, if R ad R are iside the capacity regio of the multiple-access chael MAC formed at Rx, i.e., R p c, R p d, R + R q d q c. Y Y 66 Therefore, the achievable rate regio by either treatiterferece-as-erasure or iterferece-decodig at each receiver is the covex hull of 67 o top of the ext page. I the remaiig of this sectio, we show that the covex hull of R matches the outer-boud of Theorem for the weak ad the strog iterferece regimes, i.e. For 0 p c p d C p d, p c = R 0 R, R i p d i =, R i +βr ī βp d +p c p d p c 68 where +p d : β = p d p c p d p c. 69 For p d p c : C p d, p c = R 0 R, R i p d i =, R i + R ī p d + p c p d p c 70 First, it is easy to verify that the regio described i 68, is the covex hull of the followig corer poits: R, R = 0, 0, R, R = p d, 0, R, R = 0, p d, R, R = p d, q d p c, R, R = q d p c, p d, R, R = p d q c, p d q c. 7 Now, whe 0 p c p d / + p d, the rate regio R as defied i 67 ad its covex hull are show i Fig. 7. As we ca ote from Fig. 7b, i this case the covex hull of R is ideed the covex hull of the corer poits i 7. O the other had, whe p d p c, the rate regio R as defied i 67 is depicted i Fig. 8, which is the covex hull of the first five corer poits i 7. I this case, the last poit i 7 is strictly iside the regio i Fig. 8, hece agai, R coicides with the covex hull of the corer poits i 7, ad this completes the proof of Theorem uder o CSIT assumptio for the weak ad strog iterferece regimes. B. Moderate Iterferece Regime I the remaiig of this sectio, a a-kobayashi K scheme is proposed ad the correspodig sum-rate is ivestigated. The mai focus would be the moderate iterferece p regime, i.e. d +p d p c p d. I this regime, either treatig iterferece as erasure or completely decodig the iterferece would be optimal. The result echoes that the sumcapacity might be a similar W curve as that of the o-fadig Gaussia iterferece chael [9]. To explore the power of K scheme uder this chael model, the codebooks of commo ad private messages are geerated based o rate-splittig, which is origially developed for MAC chaels [3]. I a MAC chael, each user ca split its ow message ito two sub-messages, ad the receiver I the view-poit of a MAC, each user is split ito so-called virtual users, which is equivalet to splittig the correspodig message istead.

10 0 R p d R = R, R R p c or R p d q c R + R q d q c }{{} decodability costrait at Rx ad R p c R p or R p d q c. 67 R + R q d q c }{{} decodability costrait at Rx a b Fig. 7. a Depictio of the rate regio R for 0 p c p d / + p d ; ad b its covex hull. Fig. 8. Depictio of the rate regio R for p d p c. I this case, corer poit R, R = p d q c, p d q c is strictly iside the covex hull of the first five poits i 7. decodes all messages ad sub-messages through a oiopeelig process. That is the receiver decodes oe message by treatig others as erasure, ad the removes the decoded cotributio from the received sigal. The receiver decodes the ext message from the remaiig sigal util o message is left. The key advatage of rate-splittig scheme for the MAC chael is that the etire capacity regio ca be achieved with sigle-user codebooks ad o time-sharig is required. We ow preset the modified K scheme for the two-user BFIC with o CSIT. Let B p deote Beroulli distributio with probability p takig value. Give δ [0, ], ay radom variable X with distributio B ca be split as X = maxx c, X p, where X c B δ ad δ Xp B δ. From the view poit of the radom codig, istead of usig X to geerate a sigle codebook, oe ca geerate two codebooks accordig to the distributios of X c ad X p respectively, ad the the chael iput is geerated by the max operator. It is easy to see that X X c = δ δ =: C δ. Therefore, we have I X c ; X = X X X c = C δ, I X p ; X X c = X X c = C δ. 7 Ituitively, C δ ca be viewed as the portio of the message carried by codebook geerated via X p. Note that C δ is cotiuous ad mootoically decreasig with respect to δ. Thus as δ goes from 0 to, the codig scheme cotiuously chages from X p oly scheme δ = 0 to X c oly scheme δ =. Although time-sharig does ot play a key role i achievig the capacity regio of the MAC, it does elarge achievable rate regio of the K scheme [4]. For this particular chael, we cosider a a-kobayashi scheme with time-sharig as follows. For ay δ [0, ], we geerate oe codebook accordig to distributio B δ for the commo message δ ad oe radom codebook accordig to distributio B δ for the private message. We deote these two codebooks by C c δ ad C p δ respectively. Similarly for ay δ [0, ], we ca geerate codebooks C c δ ad C p δ. I additio, let {Q[t]} be a i.i.d radom sequece with P Q[t] = = P Q[t] = = /, which geerates a particular time-sharig sequece. Before ay commuicatio begis, the time-sharig sequece is revealed to the trasmitters ad the receivers. The the two trasmitters commuicate their messages as follows. If Q[t] =, user ecodes its commo ad private messages accordig to codebooks C c δ ad C p δ respectively, ad uses the max operator to geerate the trasmit sigal. Meawhile user does the same thig except that it uses codebooks C c δ ad C p δ. If Q[t] =, the two users switch their codebooks. Equivaletly, we ca state the codig scheme i aother way: Give i.i.d. time-sharig radom sequece {Q[t]}, the two users ecode their commo ad private messages idepedetly accordig to i.i.d. sequeces {X c [t], X p [t]} ad {X c [t], X p [t]}. Give Q[t], the distributios of the other

11 sequeces are defied i Table III. For fixed δ, δ, we ca Q[t] X c [t] X p [t] X c [t] X p [t] B δ B B δ B δ δ δ B δ B B δ B δ δ δ TABLE III SUMMARY OF TE K SCEME WIT TIME-SARING compute the correspodig achievable sum-rate, i.e. R sum δ, δ := R δ, δ + R δ, δ, 73 ad the maximize over all possible values of δ, δ. To determie R sum δ, δ, it is sufficiet to make sure that both commo messages are decodable at both receivers ad each private message is decodable at its correspodig receiver. More specifically, we follow the similar procedure as [9, Sectio III] except that we explore time-sharig ad optimize the rate splittig. That is the rates of the commo messages are determied by the two virtual compoud-mac chaels at the two receivers, ad each private message is decoded oly at its correspodig receiver. We have followig theorem: Theorem 3. The followig sum-rate is optimal over δ, δ [0, ]: { pd p c if p c p d R sum = where p d + p c p d p c + p d p c C δ if +p d p d +p d < p c p d 74 Cδ := p dp c p d p c p d p c p d p c. 75 Remark. Treatig iterferece as erasure ca achieve a sum-rate as large as p d p c, while the sum-rate of iterferece-decodig scheme is p d + p c p d p c. I the weak iterferece regime where p c p d +p d, the proposed K scheme degrades to treatig-iterferece-as-erasure ad achieves the sum-capacity. I the moderate iterferece regime p where d +p d < p c p d, partially decodig iterferece ca outperform the other two schemes. Proof of Theorem 3: If p c p d +p d, the sum-rate i Theorem 3 is the sum-capacity ad ca be achieved by treatig iterferece as erasure, which correspods to the K scheme with δ = δ = 0. Thus, we p assume that d +p d < p c p d. Let G[t] = G [t], G [t] ad let R ic δ, δ ad R ip δ, δ deote achievable rates of the commo ad the private messages respectively. Sice all codebooks are geerated accordig to i.i.d. sequeces ad the chael is memoryless, we will drop all time idices i what follows. Sice at each receiver we decode private message last, we have R p δ, δ = I X p ; Y X c, X c, Q, G. For the commo messages, the achievable rates fall ito the itersectio of two virtual MACs at the two receivers. Takig symmetric properties of the chael ad the codig scheme ito accout, we coclude that ay positive rate pair for the commo messages satisfyig the followig coditios is achievable: R c δ, δ + R c δ, δ I X c, X c ; Y Q, G, 76 R c δ, δ I X c ; Y X c, Q, G, 77 R c δ, δ I X c ; Y X c, Q, G. 78 Sice p d p c, it is easy to see that 77 implies 78 as far as sum-rate is cocered. Therefore, we ca achieve the followig sum-rate: where R sum δ, δ M p δ, δ + mi M c δ, δ, M c δ, δ, 79 M p δ, δ = I X p ; Y X c, X c, Q, G, M c δ, δ = I X c, X c ; Y Q, G, M c δ, δ = I X c ; Y X c, Q, G. 80 The derivatio of M p δ, δ, M c δ, δ, ad M c δ, δ relies o the chai rule ad the basic equalities 7 as described below. Startig from M c δ, δ, we have M c δ, δ = I X c, X c ; Y Q, G = p d + p c p c p d Y X c, X c, Q, G. = p d + p c p c p d G X G X X c, X c, Q, G = p d + p c p c p d p d p c X X c, Q + p c p d X X c, Q + p d p c X X X c, X c, Q = p d + p c p c p d p d + p c p c p d C δ + C δ p d p c γδ, δ 8 where γδ, δ := X X X c, X c, Q = δ C δ + δ C δ + δ δ p, 4 8 ad p = δ δ δ. I additio, γδ δ, δ is upper bouded by C δ + C δ : γδ, δ X X c, Q + X X c, Q = C δ + C δ. 83 Next for M c δ, δ usig the chai rule ad 8, we get M c δ, δ = I X c, X c ; Y Q, G I X c ; Y Q, G = M c δ, δ I X c ; G X G X Q, G [ = M c δ, δ p d p c C ] δ + C δ = p c p c p c p d C δ + C δ + p d p c γδ, δ. 84

12 Similarly for M p δ, δ, we have M p δ, δ = I X p, X c, X c ; Y Q, G I X c, X c ; Y Q, G = I X p, X c ; Y Q, G + I X c ; Y X p, X c, Q, G M c δ, δ [ = p d p c + p c C ] δ + C δ M c δ, δ = p d p c p d C δ + C δ + p d p c γδ, δ. 85 Therefore, substitutig 8, 84, ad 85 ito 79, we ca get a expressio for R sum δ, δ. If we let δ =, the C δ = 0 ad γδ, δ = C δ. Furthermore, R sum max R sumδ, δ [0,] = max p dc δ + mi p d + p c p d p c p d + p c C δ, C δ [0,], p c p c + p c p d C δ = max mi p d + p c p d p c + p d p c C δ, C δ [0,] p c p c p d + p c p d C δ, 86 which achieves the maximum value of p d + p c p d p c + p d p c Cδ at C δ = Cδ. So it is sufficiet to show that the coverse is true. If M c δ, δ M c δ, δ, we have R sum δ, δ = M p δ, δ + M c δ, δ = p c p c p d + p d p c C δ + C δ. 87 If M c δ, δ < M c δ, δ, we have R sum δ, δ = M p δ, δ + M c δ, δ Q X c X p X c X p B δ B δ B B 0 δ B B 0 B δ B δ δ The evaluatio of achievable rate is similar to the previous case but it breaks the symmetric property i geeral. Defie followig short-had otatio: C δi := δ i δ i i =, 90 Ay rate pair satisfyig the followig coditios is achievable for the commo messages see Appedix A for the details. R c p c λ C δ p c λ C δ p c p d, R c p c λ C δ p c λ C δ p d p c, R c + R c p d + p c p d p c 9a 9b 9c maxλ C δ p d + λ C δ p c, λ C δ p c + λ C δ p d Rates of the private messages are give by R p I X p ; Y X c, X c, Q, Q, G = λ p d C δ, 9 R p I X p ; Y X c, X c, Q, Q, G = λ p d C δ. 93 Fially, we evaluate sum-rate with λ i = / ad C δ = C δ = C δ : R sum C δ = p d C δ + mip d + p c p d p c p d + p c C δ, p c p c + p c p d C δ 94 Some umeric results: For p d = ad for p c [/, ], we plotted the K rate optimized over δ i Fig. V-B. = p d + p c p d p c + p d p c p d p c C δ + C δ + p d p c γδ, δ p d + p c p d p c + p d p c C δ + C δ, 88 where 88 is due to 83. Now, let C δ3 = C δ + C δ. We have R sum max mi p d + p c p d p c + p d p c C δ3, C δ3 [0,] p c p c p d + p d p c C δ3. 89 Sum Rate bits For p d =, sum-rate vs p c R +R R +R withts Boud R +p c R Boud R +R p c Comparig 89 ad 86, they are i the same form except that 89 is over a larger domai. owever, 89 achieves its maximum whe C δ3 = Cδ, ad C δ with the assumptio p of d +p d < p c. Therefore, Theorem 3 holds. K Scheme with Time-Sharig: Let Q be a radom variable takig values i {, } with P Q = i = λ i for i =,. Codig scheme is geerated based o p c Fig. 9. The optimized a-kobayashi scheme for p d = ad for p c [/, ].

13 3 VI. CONCLUSION We studied the capacity regio of the two-user Biary Fadig Iterferece Chael with o CSIT. We showed that uder the weak ad the moderate iterferece regimes, the etire capacity regio is achieved by applyig poit-to-poit erasure codes with appropriate rates at each trasmitter, usig either treat-iterferece-as-erasure or iterferece-decodig at each receiver, based o the chael parameters. For the moderate iterferece regime, we devised a modified a-kobayashi scheme suited for discrete memoryless chaels ehaced by time-sharig. Our outer-bouds rely o two key lemmas, amely the Correlatio Lemma ad the Etropy Leakage Lemma. Rate bits / chael usage R +R p d +p c p d p c p d =.00,p c =0.60 ACKNOWLEDGEMENT The work of A. S. Avestimehr ad A. Vahid is i part supported by NSF Grats CAREER-09537, CCF-670, NETS-6904, AFOSR Youg Ivestigator Program Award, ONR award N , ad 03 Qualcomm Iovatio Fellowship. APPENDIX A PROOF OF 9 For the virtual MAC at receiver : I X c, X c ; Y Q, G = Y Q, G Y X c, X c, Q, G = p d + p c p d p c λ Y X c, X c, Q =, G + λ Y X c, X c, Q =, G = p d + p c p d p c λ C δ p d λ C δ p c 95 I X c ; Y X c, Q, G = I X c, X c ; Y Q, G I X c ; Y Q, G = I X c, X c ; Y Q, G λ I X c ; Y Q =, G + λ I X c ; Y Q =, G = I X c, X c ; Y Q, G λ p d p c C δ λ p d p c = p d + p c p d p c λ C δ p d λ C δ p c λ C δ p d p c λ p d p c = p c λ C δ p d λ C δ p c + λ C δ p d p c = p c λ C δ p c λ C δ p d p c 96 I X c ; Y X c, Q, G = I X c, X c ; Y Q, G I X c ; Y Q, G = I X c, X c ; Y Q, G λ I X c ; Y Q =, G λ I X c ; Y Q =, G = I X c, X c ; Y Q, G λ p c p d λ p c p d C δ = p d + p c p d p c λ C δ p d λ C δ p c λ p c p d λ p c p d C δ = p d λ C δ p d λ C δ p c + λ C δ p c p d = p d λ C δ p d λ C δ p c p d 97 Rate bits / chael usage δ a p d =0.50,p c =0.4 R +R p d +p c p d p c δ b Fig. 0. Numerical Results for the achievable K Rate For the virtual MAC at receiver : I X c, X c ; Y Q, G = Y Q, G Y X c, X c, Q, G = p d + p c p d p c λ Y X c, X c, Q =, G λ Y X c, X c, Q =, G = p d + p c p d p c λ C δ p c λ C δ p d 98 I X c ; Y X c, Q, G = I X c, X c ; Y Q, G I X c ; Y Q, G = p d + p c p d p c λ C δ p c λ C δ p d λ p c p d C δ λ p c p d = p d λ C δ p c λ C δ p d + λ p c p d C δ = p d λ C δ p d λ C δ p c p d 99

14 4 I X c ; Y X c, Q, G = I X c, X c ; Y Q, G I X c ; Y Q, G = p d + p c p d p c λ C δ p c λ C δ p d λ p d p c λ p d p c C δ = p c λ C δ p c λ C δ p d + λ p d p c C δ = p c λ C δ p c λ C δ p c p d 00 We reduce the umber of costraits by performig the followig comparisos. Comparig the two domiat faces: I X c, X c ; Y Q, G I X c, X c ; Y Q, G = p d + p c p d p c λ C δ p d λ C δ p c p d + p c p d p c λ C δ p c λ C δ p d = p c p d λ C δ λ C δ. 0 Thus the sig is determied by λ C δ λ C δ. Comparig sigle-user bouds for user s commo message: I X c ; Y X c, Q, G I X c ; Y X c, Q, G = p d λ C δ p d λ C δ p c p d p c λ C δ p c λ C δ p c p d = p d p c λ C δ 0. 0 So oly I X c ; Y X c, Q, G matters. Comparig sigle-user bouds for user s commo message: I X c ; Y X c, Q, G I X c ; Y X c, Q, G = p c λ C δ p c λ C δ p d p c p d λ C δ p d λ C δ p c p d = p c p d λ C δ So oly I X c ; Y X c, Q, G matters. I summary, the followig rates of commo messages are achievable: R c p c λ C δ p c λ C δ p c p d R c p c λ C δ p c λ C δ p d p c R c + R c p d + p c p d p c 04a 04b 04c maxλ C δ p d + λ C δ p c, λ C δ p c + λ C δ p d First, we evaluate cases where p d p c p d p c. I particular, Fig. 0 shows the achievable rate vs splittig parameter δ for p d, p c =, 0.6 ad /, 5/. [5] A. El-Gamal ad M.. Costa, The capacity regio of a class of determiistic iterferece chaels, IEEE Trasactios o Iformatio Theory, vol. 8, pp , Mar. 98. [6] E. C. va der Meule, Some reflectios o the iterferece chael, Commuicatios ad Cryptography, pp , 994. [7] A. B. Carleial, A case where iterferece does ot reduce capacity corresp., IEEE Trasactios o Iformatio Theory, vol., o. 5, pp , 975. [8] A. B. Carleial, Outer bouds o the capacity of iterferece chaels corresp., IEEE Trasactios o Iformatio Theory, vol. 9, o. 4, pp , 983. [9] R. Etki, D. Tse, ad. Wag, Gaussia iterferece chael capacity to withi oe bit, IEEE Trasactios o Iformatio Theory, vol. 54, pp , Dec [0] C. Suh ad D. Tse, Feedback capacity of the gaussia iterferece chael to withi bits, IEEE Trasactios o Iformatio Theory, vol. 57, o. 5, pp , 0. [] A. Vahid, C. Suh, ad A. S. Avestimehr, Iterferece chaels with ratelimited feedback, IEEE Trasactios o Iformatio Theory, vol. 58, o. 5, pp , 0. [] A. Vahid, M. A. Maddah-Ali, ad A. S. Avestimehr, Iterferece chael with biary fadig: Effect of delayed etwork state iformatio, i 49th Aual Allerto Coferece o Commuicatio, Cotrol, ad Computig, pp , 0. [3] A. Vahid, M. A. Maddah-Ali, ad A. S. Avestimehr, Commuicatio through collisios: Opportuistic utilizatio of past receptios, i Proceedigs IEEE INFOCOM, pp , 04. [4] A. Vahid ad R. Calderbak, Whe does spatial correlatio add value to delayed chael state iformatio?, i IEEE Iteratioal Symposium o Iformatio Theory ISIT, pp , 06. [5] A. Vahid ad R. Calderbak, Impact of local delayed csit o the capacity regio of the two-user iterferece chael, i IEEE Iteratioal Symposium o Iformatio Theory ISIT, pp. 4 45, 05. [6] A. Vahid ad R. Calderbak, Two-user erasure iterferece chaels with local delayed CSIT, IEEE Trasactios o Iformatio Theory, vol. 6, o. 9, pp , 06. [7] V. Aggarwal, L. Sakar, A. R. Calderbak, ad. V. Poor, Ergodic layered erasure oe-sided iterferece chaels, i IEEE Iformatio Theory Workshop, pp , IEEE, 009. [8] Y. Zhu ad D. Guo, Ergodic fadig Z-iterferece chaels without state iformatio at trasmitters, IEEE Trasactios o Iformatio Theory, vol. 57, o. 5, pp , 0. [9] A. S. Avestimehr, S. Diggavi, ad D. Tse, Wireless etwork iformatio flow: A determiistic approach, IEEE Trasactios o Iformatio Theory, vol. 57, Apr. 0. [0] D. N. Tse ad R. D. Yates, Fadig broadcast chaels with state iformatio at the receivers, IEEE Trasactios o Iformatio Theory, vol. 58, o. 6, pp , 0. [] A. Vahid, M. A. Maddah-Ali, ad A. S. Avestimehr, Capacity results for biary fadig iterferece chaels with delayed CSIT, IEEE Trasactios o Iformatio Theory, vol. 60, o. 0, pp , 04. [] P. Elias, The oisy chael codig theorem for erasure chaels, America Mathematical Mothly, pp , 974. [3] A. J. Grat, B. Rimoldi, R. L. Urbake, ad P. A. Whitig, Ratesplittig multiple access for discrete memoryless chaels, IEEE Trasactios o Iformatio Theory, vol. 47, o. 3, pp , 00. [4]. Te Su ad K. Kobayashi, A ew achievable rate regio for the iterferece chael, IEEE Trasactios o Iformatio Theory, vol. 7, o., pp , 98. REFERENCES [] A. Vahid, M. A. Maddah-Ali, ad A. S. Avestimehr, Biary Fadig Iterferece Chael with No CSIT, i IEEE Iteratioal Symposium o Iformatio Theory ISIT, pp , 04. [] R. Ahlswede, The capacity regio of a chael with two seders ad two receivers, The Aals of Probability, pp , 974. [3] T. a ad K. Kobayashi, A ew achievable rate regio for the iterferece chael, IEEE Trasactios o Iformatio Theory, vol. 7, o., pp , 98. [4]. Sato, Two-user commuicatio chaels, IEEE Trasactios o Iformatio Theory, vol. 3, o. 3, pp , 977.

15 5 Alireza Vahid received his Ph.D. ad M.Sc. degrees i Electrical ad Computer Egieerig both from Corell Uiversity, Ithaca, NY, i 05 ad 0 respectively. e obtaied his B.Sc. degree i Electrical Egieerig from Sharif Uiversity of Techology, Tehra, Ira, i 009. e is curretly a postdoctoral scholar at Iformatio Iitiative at Duke Uiversity, Durham, NC. is research iterests iclude etwork iformatio theory, wireless commuicatios, codig theory, ad data storage. Dr. Vahid received the 05 Outstadig PhD Thesis Research Award at Corell Uiversity. e also received the Director s Ph.D. Teachig Assistat Award i 00, Jacobs Scholar Fellowship i 009, ad Qualcomm Iovatio Fellowship i 03. Ya Zhu received his B.E. ad M.S. degrees from Tsighua Uiversity, Beijig, Chia, i 00 ad 005, respectively, ad his Ph.D. degree from Northwester Uiversity, Evasto, IL, i 00, all i electrical egieerig. From 00 to 05, e worked at Broadcom Ic. as a scietist, desig staff. e joied Calterah Ic. as a chief system architecturer sice 06. is research iterests iclude wireless commuicatios, iformatio theory, commuicatio etwork ad sigal processig. e is a co-recipiet of the 00 IEEE Marcoi Prize Paper Award i Wireless Commuicatios with D. Guo ad M. L. oig. Mohammad Ali Maddah-Ali S 03-M 08 received the B.Sc. degree from Isfaha Uiversity of Techology, ad the M.A.Sc. degree from the Uiversity of Tehra, both i electrical egieerig. From 00 to 007, he was with the Codig ad Sigal Trasmissio Laboratory CST Lab, Departmet of Electrical ad Computer Egieerig, Uiversity of Waterloo, Caada, workig toward the Ph.D. degree. From 007 to 008, he worked at the Wireless Techology Laboratories, Nortel Networks, Ottawa, ON, Caada. From 008 to 00, he was a post-doctoral fellow i the Departmet of Electrical Egieerig ad Computer Scieces at the Uiversity of Califoria at Berkeley. The, he joied Bell Labs, olmdel, NJ, as a commuicatio research scietist. Recetly, he started workig at Sharif Uiversity of Techology, as a faculty member. Dr. Maddah-Ali is a recipiet of NSERC Postdoctoral Fellowship i 007, a best paper award from IEEE Iteratioal Coferece o Commuicatios ICC i 04, the IEEE Commuicatios Society ad IEEE Iformatio Theory Society Joit Paper Award i 05, ad the IEEE Iformatio Theory Society Joit Paper Award i 06. A. Salma Avestimehr S 03-M 08-SM6 is a Associate Professor at the Electrical Egieerig Departmet of Uiversity of Souther Califoria. e received his Ph.D. i 008 ad M.S. degree i 005 i Electrical Egieerig ad Computer Sciece, both from the Uiversity of Califoria, Berkeley. Prior to that, he obtaied his B.S. i Electrical Egieerig from Sharif Uiversity of Techology i 003. e was a Assistat Professor at the ECE school of Corell Uiversity from 009 to 03. e was also a postdoctoral scholar at the Ceter for the Mathematics of Iformatio CMI at Caltech i 008. is research iterests iclude iformatio theory, the theory of commuicatios, ad their applicatios. Dr. Avestimehr has received a umber of awards, icludig the Commuicatios Society ad Iformatio Theory Society Joit Paper Award i 03, the Presidetial Early Career Award for Scietists ad Egieers PECASE i 0 for pushig the frotiers of iformatio theory through its extesio to complex wireless iformatio etworks, the Youg Ivestigator Program YIP award from the U. S. Air Force Office of Scietific Research i 0, the Natioal Sciece Foudatio CAREER award i 00, ad the David J. Sakriso Memorial Prize i 008. e is curretly a Associate Editor for the IEEE Trasactios o Iformatio Theory.