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Capacity Bounds for a Class of Diamond Networks Shirin Saeedi Bidokhti Gerhard Kramer Lehrstuhl fu¨r Nachrichtentechnik Lehrstuhl fu¨r Nachrichtentechnik Technische Universita¨t Mu¨nchen, Germany Technische Universita¨t Mu¨nchen, Germany [email protected] [email protected] broAabdsctraasctt—comApcolansesnotfisdimamodoenlldednebtywotwrkosianrdeepsteunddieendtwbhite-rpeiptehse. Relay1 X1n Nimepwrouvpepperreavnioduslowboubnodusn.dTshaereudpepreirvebdouonndthies cinaptahceityforwmhicohf W V2n MAC Yn Wˆ a max-min problem, where the maximization is over a coding Source Encoder Decoder Sink p(y|x1,x2) 4 dTfoihrsterthipberuoGtoifoanutescasihnandniqtmhuueelgtmiepinnleeirmdaeilziszacetrsiiopbntoiouisnndopvirneogrbltaeenmchan(u1iqx9iu8li1eas)r,oyafncOdhzaKanranonewlg. V1n Relay2 X2n 1 and Liu for the Gaussian diamond network (2011). The bounds 0 areevaluatedforaGaussianmultipleaccesschannel(MAC)and Fig. 1: Problem setup 2 thebinaryadderMAC,andthecapacity isfoundforinteresting ranges of the bit-pipe capacities. n a I. INTRODUCTION techniqueof[4]. Ourupperboundappliesto thegeneralclass J 3 The diamond network was introduced in [1] as a simple of discrete memoryless MACs, and strictly improves the cut- 2 multi-hopnetworkwithbroadcastandmultipleaccesschannel set bound. In Section IV, we improve the achievable rates (MAC) components. The two-relay diamond network models of [3] by communicating a common piece of information ] thescenariowhereasourcecommunicateswithasinkthrough fromthe sourceto bothrelaysusingsuperpositioncodingand T two relay nodes that do not have information of their own to Marton’s coding. Finally, we study our bounds for networks I . communicate. The underlying challenge may be described as with a Gaussian MAC (Section V) and a binary adder MAC s c follows. In order to fully utilize the MAC to the receiver, we (SectionVI).Forbothexamples,wefindconditionsonthebit- [ would ideally like full cooperation between the relay nodes. pipe capacities such that the upper and lower bounds meet. 1 On the other hand, in order to communicate the maximum II. PRELIMINARIES v amount of information and better use the diversity that is 5 offered by the relays, we would like to send independent A. Notation 3 informationtotherelaynodesoverthebroadcastchannel.The We use standard notation for random variables, e.g., X, 1 6 problemof findingthe capacityof this networkis unresolved. probabilities, e.g., pX(x) or p(x), entropies, e.g., H(X) and . Lower and upper boundson the capacity are given in [1]. We H(X Y), and mutual information, e.g., I(X;Y). We denote 1 | remark that the problem is solved over linear deterministic the sequence X ,...,X by Xn. Sets are denoted by script 0 1 n 4 relay networks, and the capacity of Gaussian relay networks letters and matrices are denoted by bold capital letters. 1 has been approximated within a constant number of bits [2]. : In this paper, we study a class of diamond networks where B. Model v i the broadcast channel is modelled by two independent bit- Consider the diamondnetworkin Fig. 1. A sourcecommu- X pipes. This problem was initially studied in [3] where lower nicatesamessageofrateRtoasink.Thesourceisconnected r and upper bounds were derived on the capacity. The best to two relays via noiseless bit-pipes of capacities C and C , a 1 2 known upper bound for the problem is the cut-set bound, and the relays communicate with the receiver over a MAC. which does not fully capture the tradeoff between cooper- The source encodes a message W with nR bits into ation and diversity. Recently, [4] studied the network with sequence Vn, which is available at encoder 1, and sequence 1 a Gaussian MAC and proved a new upper bound which Vn, whichis availableatencoder2.Vn andVn are suchthat 2 1 2 constrains the mutual information between the MAC inputs H(Vn) nC and H(Vn) nC . Each relay i, i = 1,2, 1 ≤ 1 2 ≤ 2 to improve the cut-set bound. The bounding technique in maps its received sequence Vn into a sequence Xn which i i [4] is motivated by [5] that treats the Gaussian multiple is sent over the MAC. The MAC is characterized by its description problem. Unfortunately, neither result seems to input alphabets and , output alphabet , and transition 1 2 X X Y apply to discrete memoryless channels. probabilitiesp(y x ,x ),foreachx ,x .Fromthe 1 2 1 1 2 2 This paper is organized as follows. We state the problem receivedsequenc|e,thesinkdecodesa∈nXestimat∈eWXˆ ofW. We setup in Section II. In Section III, we prove a new upper are interestedin characterizingthe highestrate R thatpermits bound on the achievable rates by generalizing the bounding arbitrarily small positive error probability Pr(Wˆ =W). 6 III. AN UPPERBOUND nC1+nC2+nI(X1,Q,X2,Q;YQ UQ) ≤ | The idea behind our upper bound is motivated by [4], +nI(X ;U X )+nI(X ;U X ). 1,Q Q 2,Q 2,Q Q 1,Q | | [5]. The proposed bound is applicable not only to Gaussian Step (a) follows because we have the following two Markov channels, but also to general discrete memoryless channels, chains: and it strictly improvesthe cut-set boundas we show via two examples. (XnXnUnYi 1) (X X U ) Y (8) 1 2 − − 1i 2i i − i In the network of Fig. 1, the cut-set bound gives (XnXnUi 1) (X X ) U . (9) 1 2 − − 1i 2i − i R C +C (1) 1 2 We thus have the following upper bound. ≤ R C +I(X ;Y X ) (2) ≤ 1 2 | 1 Theorem 1. The rate R is achievable only if there exists R C +I(X ;Y X ) (3) 2 1 2 a joint distribution p(x ,x ) for which inequalities (10)-(14) ≤ | 1 2 R I(X X ;Y). (4) hold for every auxiliary channel p(ux ,x ,y). ≤ 1 2 | 1 2 However, the bound in (1) disregards the potential correla- R C +C (10) 1 2 ≤ tion between the inputs of the MAC. A tighter boundon R is R C +I(X ;Y X ) (11) 2 1 2 given in the following multi-letter form: ≤ | R C +I(X ;Y X ) (12) 1 2 1 nR H(Vn,Vn) ≤ | ≤ 1 2 R I(X1X2;Y) (13) = H(Vn)+H(Vn) I(Vn;Vn) ≤ 1 1 − 1 2 2R C1+C2+I(X1X2;Y U) nC +nC I(Xn;Xn). ≤ | ≤ 1 2− 1 2 +I(X1;U X2)+I(X2;U X1) (14) | | It is observed in [4] that I(Xn;Xn) can be written in the 1 2 Remark 1. The boundof Theorem 1 is in the form of a max- following form for any random sequence Un. min problem, where the maximization is over p(x ,x ) and 1 2 I(Xn;Xn) = I(XnXn;Un) I(Xn;Un Xn) the minimization is over p(ux1,x2,y). 1 2 1 2 − 1 | 2 | I(Xn;Un Xn)+I(Xn;Xn Un) Remark 2. For a fixed auxiliary channel p(ux ,x ,y) and − 2 | 1 1 2| | 1 2 a fixed MAC p(y x ,x ), the upper bound in Theorem 1 is Therefore, we have | 1 2 concave in p(x ,x ). See [6]. 1 2 nR nC +nC I(XnXn;Un) ≤ 1 2− 1 2 Remark3. Therighthandside(RHS)of (14)maybewritten +I(X1n;Un|X2n)+I(X2n;Un|X1n). (5) as Itmaynotbeclearhowtosingle-letterizetheabovebound. C +C +I(X X ;YU) I(X ;X )+I(X ;X U). (15) 1 2 1 2 1 2 1 2 We proceed as follows. First, note that − | We study the upper bound of Theorem 1 with a Gaussian nR ≤ I(X1nX2n;Yn)≤I(X1nX2n;YnUn). (6) MAC and a binary adder MAC in Sections V and VI respectively. Combining inequalities (5) and (6), we have 2nR nC +nC +I(XnXn;Yn Un) IV. A LOWER-BOUND ≤ 1 2 1 2 | +I(Xn;Un Xn)+I(Xn;Un Xn). (7) The achievable scheme that we present here is based on 1 | 2 2 | 1 [3], but we further allow a common message at both relaying Define Ui from X1i,X2i,Yi through the channels nodes.Thisisdonebyratesplitting,superpositioncodingand opfUiU|X1wiXe2ihYaiv(ueit|hxe1if,oxl2loi,wyiin)g, ich=ai1n,o2f,.i.n.e,qnu.alWitiieths:this choice Marton’scodingandissummarizedinthefollowingtheorem1. i Theorem 2. The rate R is achievable if it satisfies the 2nR following conditions for some pmf p(u,x ,x ,y), where 1 2 nC +nC +I(XnXn;Yn Un) p(u,x ,x ,y) = p(u,x ,x )p(y x ,x ), and U with ≤ 1 2 1 2 | 1 2 1 2 | 1 2 ∈ U +I(Xn;Un Xn)+I(Xn;Un Xn) min 1 2 +2, +4 . 1 | 2 2 | 1 |U|≤ {|X ||X | |Y| } n C +C I(X ;X U) nC +nC + I(XnXn;Y UnYi 1) 1 2− 1 2| ≤ 1 2 1 2 i| − C2+I(X1;Y X2U)  |  + n I(X1n;UXii|=X12nUi−1)+ n I(X2n;Ui|X1nUi−1) R≤minCI12((1XC+1X+I(CX;2Y2+;)YI(|XX11XU2);Y|U)−I(X1;X2|U)) (16) i=1 i=1 1 2 (a)nCX+nC + n I(X XX;Y U ) Sketchofproof: The innerboundin Theorem2 is found 1 2 1i 2i i i using the following achievable scheme. ≤ | i=1 X n n 1 ThemethodtoderivethistheoremwasdiscussedingeneraltermsbyG. + I(X1i;Ui X2i)+ I(X2i;Ui X1i) Kramer and S. Shamai after G. Kramer presented the paper [3] at ITW in | | 2007. i=1 i=1 X X a) Codebook construction: Fix the joint pmf and p (u,x ,x 2) = p(2)(u,x ,x ). Then we have P (u,x ,x ). Let R=R +R and let the foUlloXw1Xin2g|Qchain1of 2in|equalities. 1 2 UX1X2 1 2 12 ′ R′ =R1+R2 I(X1;X2 U) δǫ. (17) fℓ(C⋆) − | − <αf (C(1))+(1 α)f (C(2)) Generate 2nR12 sequences un(m ) independently, each in ℓ − ℓ 12 2C⋆ I(X ;X UQ) an i.i.d manner according to lPU(ul). For each sequence C⋆+−I(X ;1Y X2| UQ) stdtQuoieiontqlin(uoPmaenllXnla1Py12cll)|Xe)Uy,i2)(np|gxUdiaen1ie(nr,dxplsee|e2urpn,(allexdt|)enuneadln((n)emti.dl)nyFt(,2loiyine,r,)aRme2ec1aahnccR)shhi,eQ2nxqssinueanee(qnqmnuauicnee.einn.sidcc,.eiemx.msdun1xa(n)mnmn2)(nm(a1emtnh2r1n,a21eamt2)rc,,ac1mparo)ecirc2cd(k)jocion(ro2icdgnnnoidtnRntlioyg′-- ≤minCIC212C((⋆⋆U2⋆++CX−⋆II1+IX((XX(I2X(211;X;;Y1YY;1|XQX||XX2)2|122;UUUYQQQ|U)))Q)−I(X1;X2|UQ)) typQicbal). IEnndceoxdisnugc:h1Tpoaic1ros2mbmy1umn′ic∈2at{e1m1,2e.s.s.a,g22enRW′}.=(m12,m′), ≤minC12(⋆2+C⋆I+(XI(2X;Y1X||X21;UYQ|U)Q)−I(X1;X2|UQ)) I(UX X ;Y) communicate (m ,m ) with relay 1 and (m ,m ) with 1 2 relay 2 (we assum1e2 (xn11(m12,m1),xn2(m12,m2)1)2is th2e m′th (a)f (C⋆).  jointly typical pair for m12). Relays 1 and 2 then send ≤ ℓ xn1(m12,m1) and xn2(m12,m2) over the MAC, respectively. Step (a) follows by renaming (U,Q) a U and comparing the c) Decoding: Upon receiving yn, the receiver looks for lefthandwiththelowerboundcharacterizationofTheorem2. (mˆ ,mˆ ) for which the following tuple is jointly typical: So we have a contradiction, and the proposition is proved. 12 ′ (un(mˆ ),xn(mˆ ,mˆ ),xn(mˆ ,mˆ ),yn) n. V. THEGAUSSIAN MAC 12 1 12 1 2 12 2 ∈Tǫ The output of the Gaussian MAC is given by d) Error Analysis: The error analysis is straightforward and we omit it for the sake of brevity. The above scheme Y =X1+X2+Z performs reliably if where Z (0,1), and the transmitters have average R12+R1 <C1 (18) p1owenr coEn(sX∼tr2ainN)ts PP1,.P2; i.e., n1 ni=1E(X12,i) ≤ P1 and R12+R2 <C2 (19) n Wei=s1pecial2iz,ie T≤heo2rem 1 to obtaPin an upper bound on the R<I(UX1X2;Y) (20) achPievable rate R. To simplify the bound, pick U to be a R +R I(X ;X U)<I(X X ;Y U) (21) noisy version of Y in the form of U = Y +Z , where Z 1 2 1 2 1 2 N N − | | R <I(X ;Y X ,U)+I(X ;X U) (22) is a Gaussian noise with zero mean and variance N (to be 2 2 1 1 2 | | optimized later). Inequalities (10)-(14) are upper bounded as R <I(X ;Y X ,U)+I(X ;X U). (23) 1 1 | 2 1 2| follows using maximum entropy lemmas: Conditions(17)-(23),togetherwithR 0,R 0,R 0, 1 ≥ 2 ≥ 12 ≥ R C1+C2 (24) characterize an achievable rate. Eliminating R ,R ,R , we ≤ 12 1 2 1 arrive at Theorem 2. Cardinality boundsare derived using the R C2+ log 1+P1(1 ρ2) (25) ≤ 2 − standard method via Caratheodory’s theorem [7]. R C + 1log(cid:0)1+P (1 ρ2)(cid:1) (26) 1 2 Proposition 1. The lower bound of Theorem 2 is concave in ≤ 2 − C1,C2. R 1log 1+P(cid:0)1+P2+2ρ P(cid:1)1P2 (27) ≤ 2 Proof: We prove the statement for the case of C1 = (a) C1+(cid:16)C2+12log 1+P1+pP2+2ρ(cid:17)√P1P2 CW2e=exCpr.eTsshethseamloewaerrgubmouenndthoofldTshfeoorrtehme m2oirnegteernmesralofcatshee. 2R ≤ +21log (1(+1+NN++P(cid:0)1P(11+−Pρ22+))2(ρ1√+NP1+PP2)2((11+−Nρ2)))(cid:1)!. (28) maximization problem maxp(u,x1,x2)fℓp(C,p(u,x1,x2)) and To obtain inequali(cid:16)ty (a) above, write the RHS of(cid:17)(14) as wedenoteitbyf (C).Inthisformulation,fp(C,p(u,x ,x )) ℓ ℓ 1 2 C +C +h(Y U) h(YU X X )+h(U X ) is just the minimum term on the right hand of (16). 1 2 | − | 1 2 | 1 +h(U X ) h(U X X ). The proof of the theorem is by contradiction. Suppose | 2 − | 1 2 that the lower bound is not concave in C; i.e., there exist The negative terms are easy to calculate because of the values C(1),C(2), and α, 0 α 1, such that C⋆ = Gaussian nature of the channel and the choice of U. The ≤ ≤ αC(1) + (1 α)C(2) and f (C⋆) < αf (C(1)) + (1 positive terms are bounded from above using the conditional ℓ ℓ − − α)f (C(2)). Let p(1)(u,x ,x ) (resp. p(2)(u,x ,x )) be the version of the maximum entropy lemma [8]. It remains to ℓ 1 2 1 2 probability distribution that maximizes fp(C(1),p(u,x ,x )) solve a max-min problem (max over ρ and min over N). So ℓ 1 2 (resp. fp(C(2),p(u,x ,x ))). Let p (1) = α and p (2) = the rate R is achievable only if there exists some ρ > 0 for ℓ 1 2 Q Q 1 α, and define p (u,x ,x 1) = p(1)(u,x ,x ) which for every N >0 inequalities (24)-(28) hold. − UX1X2|Q 1 2| 1 2 such that 1.15 ρ 1+ 1 1 , 1.1 ≤ 4P1P2 − 4P1P2 R q C1+C2 q ≤ R C + 1log 1+P (1 ρ2) ≤ 2 2 1 − 1.05 R C + 1log 1+P (1 ρ2) ≤ 1 2 (cid:0) 2 − (cid:1) R 1log 1+P +P +2ρ√P P ≤ 2 (cid:0)1 2 (cid:1)1 2 1 2R ≤ C1+(cid:0)C2+ 12log 1+P1+P2+2(cid:1)ρ√P1P2 R) 1log 1 , ate (0.95 or −2 (cid:16)1−ρ2(cid:17) (cid:0) (cid:1) R 1+ 1 1 ρ 1, 0.9 Lower Bound (Gaussian Dist.) 4P1P2 − 4P1P2 ≤ ≤ Lower Bound (Mixture of Two Gaussian Dist.) Rq ≤ C1+Cq2 R C + 1log 1+P (1 ρ2) New Upper Bound ≤ 2 2 1 − 0.85 Cut−Set Bound R ≤ C1+ 21log(cid:0)1+P2(1−ρ2)(cid:1) R 1log 1+P +P +2ρ√P P . ≤ 2 (cid:0)1 2 (cid:1)1 2 0.8 For a lower boun(cid:0)d, we use Theorem 2(cid:1)and choose 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 (U,X ,X )tobejointlyGaussianwithmean0andcovariance Link Capacity (C) matrix1K2 . Fig. 2 shows the upper and lower bound as UX1X2 a function of C for a symmetric network with C =C =C Fig. 2: Upper and lower bounds on R as functions of C for 1 2 and P =P =1. The dash-dotted curve in Fig. 2 shows the Gaussian MAC with P =P =1. 1 2 1 2 rates achievable using the scheme of Section IV with a joint Gaussian distribution. It is interesting to see that the obtained lower bound is not concave in C. This does not contradict We choose N to be (see [4, eqn. (21)]) Proposition 1 because Gaussian distributions are sub-optimal. The improved dashed curve shows rates that are achievable using a mixture of two Gaussian distributions. + 1 N =(cid:18) P1P2(cid:18)ρ −ρ(cid:19)−1(cid:19) . (29) P1F=orPs2ym=mPet,riwcedisapmecoinfydanertewgoimrkesowfhCerefoCr1w=hicCh2t=heClowanedr p andupperboundsmeetandthuscharacterizethecapacity.This is summarized in Theorem 3. See [6] for the proof. Let us first motivate this choice. It is easy to see that Theorem 3. Fora symmetric Gaussiandiamondnetwork, the inequality (28) is simply upper bound in Theorem 1 meets the lower bound if C ≤ 1log(1+2P), C 1log(1+4P), or 4 ≥ 2 I(X X ;Y) I(X ;X )+I(X ;X U) (30) 1 2 1 2 1 2 − | 1 1+2P(1+ρ(1)) 1 1+2P(1+ρ(2)) log C log (33) 4 1 ρ(1)2 ≤ ≤ 4 1 ρ(2)2 evaluated for the joint Gaussian distribution p(x ,x ) with 1 2 − − covariance matrix where (1+2P)+ 12P2+(1+2P)2 P ρ√P P ρ(1) = − (34) 1 1 2 . (31) 6P ρ√P P P p 1 2 2 (cid:20) (cid:21) 1 1 ρ(2) = 1+ . (35) 4P2 − 2P The choice (29) makesU satisfy the Markovchain X U r 1 X for the regime where − − For example, the upper and lower bounds match in Fig. 2 2 (where P = 1) for C 0.3962, 0.4807 C 0.6942, and ≤ ≤ ≤ C 1.1610. 1 ≥ P P ρ 1 0 (32) 1 2 ρ − − ≥ Remark 4. There is a close connectionbetween the boundof (cid:18) (cid:19) p Corollary 1 andthatof[4]. The upperboundin [4] is tighter thanCorollary1 in certainregimesofoperation.Forexample and thus minimizes the RHS of (28). Otherwise, U = Y is in Fig. 2, the upper bound of [4] matches the lower bound chosen which results in a redundant bound. The upper bound for all C 0.6942. The reason seems to be that the methods is summarized in Corollary 1. ≤ used in [4] provide a better single letterization of the upper Corollary 1. Rate R is achievable only if there exists ρ 0 bound for the Gaussian MAC. ≥ Y U 0 1−α 1.57 Cut−Set Bound α New Upper Bound 1 0 1.55 Lower Bound 2 1 12α 1 ate (R)1.53 R 2 1−α 1.51 1.49 Fig. 3: Channel p(ux ,x ,y) 1 2 | 1.47 VI. THEBINARY ADDERCHANNEL 0.73 0.75 0.77 0.79 0.81 0.83 0.85 0.87 0.89 Link Capacity (C) Consider the binary adder channel defined by = 0,1 , 1 X { } = 0,1 , = 0,1,2 , and Y = X +X . The best 2 1 2 kXnown{uppe}r bYound{for this}channel is the cut-set bound. We Fig. 4: Lower and upper bounds on R as functions of C for specialize Theorem 1 to derive a new upper bound on the the symmetric binary adder MAC. achievable rate. For simplicity, we assume C =C =C. 1 2 Definep(ux ,x ,y)tobeasymmetricchannelasshownin 1 2 Fig.3,withpa|rameterαtobeoptimized.FromTheorem1,the Corollary 2. Rate R is achievable only if there exists some rateRisachievableonlyifforsomepmfp(x ,x )inequalities p, 0 p 1, such that 1 2 ≤ ≤ 2 (10)-(14) hold for the aforementioned choice of U and for R 2C every α; i.e., we have to solve a max-min problem(max over ≤ R C+h (p) p(x ,x ), min over α). ≤ 2 (39) 1 2 R h (p)+1 p 2 In the rest of this Section we manipulate the above bound ≤ − 2R 2C+2h (p) p. 2 to formulate a simplified upperbound.First, loosen the upper ≤ − bound by looking at the min-max problem (rather than the Fig. 4 plots this bound (Corollary 2) and compares it with max-min problem). Fix α. For a fixed channel p(ux ,x ,y), the cut-set bound and the lower bound of Theorem 2 for 1 2 the upper bound is concave in p(x ,x ) (see Rema|rk 2). The different values of C. It turns out that the lower and upper 1 2 concavity,togetherwith the symmetryof the problemand the bounds meet for C .75 and C .7929. See [6] for the ≤ ≥ auxiliary channel imply the following lemma. proof. Lemma1. Forafixedα,theoptimizingpmfisthatofadoubly VII. ACKNOWLEDGMENT symmetric binary source. TheworkofS.SaeediBidokhtiwassupportedbytheSwiss NationalScienceFoundationfellowshipno.146617.Thework Using Lemma 1, p(x ,x ) may be assumed to be a doubly 1 2 of G. Kramer was supported by an Alexander von Humboldt symmetric binary source with cross-over probability p. The Professorship endowed by the German Ministry of Education bounds in (10)-(14) reduce to and Research. R 2C R ≤ C+h (p) REFERENCES 2 R ≤ h2(p)+1 p (36) [1] B. E. Schein, Distributed coordination in network information theory. ≤ − (a) PhDDissertation, MIT,2001. 2R 2C+2h2(p) p+I(X1;X2 U). [2] A. Avestimehr, S. Diggavi, and D. 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