Parallel Gaussian Networks with a Common State-Cognitive...

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Parallel Gaussian Networks with a CommonState-Cognitive Helper

Ruchen Duan,Member, IEEE, Yingbin Liang,Member, IEEE, Ashish Khisti,Member, IEEEandShlomo Shamai (Shitz),Fellow, IEEE

Abstract—State-dependent parallel networks with a commonstate-cognitive helper is studied, in whichK transmitters wishto send K messages to their corresponding receivers overKstate-corrupted parallel channels, and a helper who knows thestate information noncausally wishes to assist these receiversto cancel state interference. Furthermore, the helper alsohasits own message to be sent simultaneously to its correspondingreceiver. Since the state information is known only to the helper,but not to other transmitters, transmitter-side state cognitionand receiver-side state interference are mismatched. Our focusis on the high state power regime, i.e., the state power goesto infinity. Three (sub)models are studied. Model I serves asa basic model, which consists of only one transmitter-receiver(with state corruption) pair in addition to a helper that assiststhe receiver to cancel state in addition to transmitting its ownmessage. Model II consists of two transmitter-receiver pairsin addition to a helper, and only one receiver is interferedby a state sequence. Model III generalizes model I to includemultiple transmitter-receiver pairs with each receiver corruptedby independent state. For all models, inner and outer boundson the capacity region are derived, and comparison of the twobounds yields characterization of either full or partial boundaryof the capacity region under various channel parameters.

Index Terms—Capacity region, channel state, parallel channel,helper, dirty paper coding, Gel’fand-Pinsker scheme, noncausalstate information.

I. I NTRODUCTION

State-dependent network models, in which receivers areinterfered by random state sequences, have recently beenintensively studied. In many such models studied before, someor all of the transmitters know the states that interfere theirtargeted receivers noncausally, and can hence exploit the stateinformation in encoding of messages as in [1], [2]. In this

The material in this paper was presented in part at the IEEE InformationTheory Workshop, Seville, Spain, September 2013, and in part at the IEEEInternational Symposium on Information Theory, Honolulu,HI, USA, June-July 2014.

The work of R. Duan and Y. Liang was supported by the NationalScience Foundation under Grant CCF-12-18451 and by the National ScienceFoundation CAREER Award under Grant CCF-10-26565. The workof A.Khisti was supported by the Canada Research Chair’s Program. The work ofS. Shamai(Shitz) was supported by the Israel Science Foundation (ISF), andthe European Commission in the framework of the Network of Excellence inFP7 Wireless COMmunications NEWCOM++.

R. Duan is with Samsung Semiconductor Inc., San Diego, CA 92121USA (email: [email protected]). Y. Liang is with the Department ofElectrical Engineering and Computer Science, Syracuse University, Syracuse,NY 13244 USA (email: [email protected]). Ashish Khisti is with the De-partment of Electrical and Computer Engineering, University of Toronto,Toronto, ON, M5S3G4, Canada (email: [email protected]). ShlomoShamai (Shitz) is with the Department of Electrical Engineering, Technion-Israel Institute of Technology, Technion city, Haifa 32000, Israel (email:[email protected]).

way, the state interference can be efficiently or even fullycancelled at receivers. For example, the state-dependent broad-cast channel has been studied in, e.g., [3]–[7], in which thetransmitter knows the state noncausally and can exploit suchinformation to select the codeword to be sent in the channel.In[8], the state-dependent relay channel is studied, in whichthesource node knows the state and can use such information forencoding. In [3], [9], the multiple access channel (MAC) withthe receiver being corrupted by one state variable is studied. Insuch a model, both transmitters are assumed to know the statesequence noncausally, and can use the state information toindependently encode their own messages. Similarly, in [10],the state-dependent cognitive MAC is studied, in which onetransmitter knows both messages as well as the state, and canhence use state information to encode both messages. As asimilar situation, the state-dependent interference channel isstudied in [11]–[14], in which the state information is knownat both transmitters and can hence be exploited for encodingtheir messages, respectively.

A common nature that the above models share is that foreach message to be transmitted, at least one transmitter inthe system knows both the message and the state, and canincorporate the state information in encoding of the messageso that state interference at the corresponding receiver canbe cancelled. However, in practice, it is often the case thattransmitters that have messages intended for receivers do notknow the state, whereas some third-party nodes know the state,but do not know the message. In such a mismatched case, stateinformation cannot be exploited in encoding of messages, butcan still serve to cancel the state interference at receivers. Anumber of previously studied models capture such mismatchedproperty. For example, in [15], a transmitter sends a messageto a state-dependent receiver, and a helper knows the statenoncausally and can help the transmission. Lattice coding isdesigned in [15] for the helper to assist state cancelation atthe receiver, and is shown to be optimal under certain channelconditions. In [16], [17], the state-dependent relay channel isstudied, and the case with the state noncausally known only atthe relay is the mismatched scenario. Furthermore, in [18],thestate-dependent MAC channel is studied with the state knownat only one transmitter. In such a case, the other transmitter’smessage cannot be encoded with the information of the state.In [19]–[21], the MAC is corrupted by two states that arerespectively known at the two transmitters. In such a case,neither message can be encoded with the full information ofthe state.

In this paper, we focus on the mismatched scenarios, where

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the state is known at only a helper which does not knowmessages. Furthermore, we are interested in the followingissues that are not captured in the previously studied mod-els: (1) when there are multiple state-dependent transmitter-receiver links, how should the helper trade off among helpingmultiple state-interfered receivers; (2) when the helper hasits own message intended for a separate receiver (not state-dependent), how should the helper trade off between sendingits own message and assisting state-dependent receivers; and(3) under what channel conditions, the above two tradeoffs areoptimal (i.e., achieve the boundary of the capacity region).

More specifically, we study a class of state-dependentparallel networks with a common state-cognitive helper (seeFig. 2). In our model,K transmitters wish to sendK messagesrespectively toK receivers overK parallel channels, andthe receivers are corrupted by states. The channel state isknown to neither the transmitters nor the receivers, but to ahelper noncausally. The helper hence assists these transmitter-receiver pairs to cancel state interference. Furthermore,thehelper also has its own message to be sent simultaneouslyto its corresponding receiver. Since the state informationis known only to the helper, but not to the correspondingtransmitters, transmitter-side state cognition and receiver-sidestate interference are mismatched. Our goal is to investigatesuch a mismatched scenario in high state power regime, i.e.,as the power of the state sequences go to infinity. This is themost challenging scenario with state interference, because thestate cannot be cancelled by simple reversion to achieve non-zero rate. More sophisticated schemes must be designed tocancel state interference. Such a model also suggests to exploitthe state cognition for improving communication rates otherthan the traditional message cognition studied in the contextof cognitive channels and networks.

A. Practical Implication of the Model

Fig. 1: A practical example for the parallel Gaussian networkswith a common state-cognitive helper.

The model we study is well justified in practical wirelessnetworks, and implies a new perspective of interference can-celation. We illustrate the idea via a simple example (see

Fig. 1). Consider a cellular network that incorporates device-to-device (D2D) communications. It is typical that the cellularbase station causes interference to D2D transmissions. In fact,the base station itself knows such interference noncausally,because the interference is the signal that the base stationsendsto cellular receivers. Thus, the interference can be viewedasthe noncausal state sequence (denoted asSn in Fig. 1). Thebase station is then able to exploit such interference (i.e., state)information and send a help signal (denoted byXn

0 in Fig. 1)to assist D2D users to cancel the interference. Although thehelp signalXn

0 may also cause interference to the cellularreceiver, our results in this paper show that small powerof Xn

0 can cancel large interferenceSn (even with infinitepower). Thus, the network can still have substantial gain inthroughputs. We further note that although the base stationmay employ Han-Kobayashi scheme [22] for interferencecancelation, this requires that the base station share codebooksof cellular users with D2D receivers, and is not feasible inpractice.

In this paper, we are interested in high state-power regime,which is well justified by the above example, because the basestation typically transmit at a large power. In such a regime,the cellular user can easily decode the stateSn (that carriesthe base station’s information to the cellular user) and canthen subtract it from its received signal. Thus,Sn does notappear in the cellular user’s output in our models describedinSection II. Furthermore, the helper signalXn

0 by nature can notonly cancel the state interference but also convey additionalinformation to the cellular user. Hence, in our models, weallow Xn

0 to carry information for the cellular user to capturethe tradeoff between the above two roles that such a signalcan play.

Such a simple scenario can be further extended. For ex-ample, there can be a number of D2D transceiver pairsthat are interfered by cellular transmissions. Typically,D2Dtransceivers are not located together, and cellular interferenceto different D2D transceivers are independent. This is becausethe base station can transmit independent signals to cellularusers located in different areas via directional antennas or viadifferent relay nodes. In such cases, as we describe above,the base station (which knows the interference noncausally)can serve as a helper to assist all D2D users to cancel theinterference. Our model described in Section II captures sucha more general scenario.

B. Main Contributions

In this paper, we study three (sub)models of the state-dependent parallel networks with a common helper whichcharacterize the main features of the motivated practicalmodel. Model I serves as a basic model, which consists ofonly one state-corrupted receiver (K = 1) and a helper thatassists this receiver to cancel state interference in addition totransmitting its own message. We dispense the state signal atthe helper’s corresponding receiver, because the state actuallycontains messages for this receiver and is hence decodablewith large power. Instead, we require the helper to transmitits own message using the helper signal, which indicates the

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trade-off between transmitting message and assisting otherinterfered receivers. Same settings are also applied for modelII and model III. Our study of this model provides necessarytechniques to deal with state in the mismatched context forstudying more complicated models II and III. In fact, thismodel can be viewed as the state-dependent Z-interferencechannel, in which the interference is only at receiver 1 causedby the helper. In contrast to the state-dependent Z-interferencechannel studied previously in [23], which assumes that stateinterference at both receivers are known to both (correspond-ing) transmitters, our model assumes that state interference isknown noncausally only to the helper, not to the correspondingtransmitter 1.

The challenge to design capacity-achieving schemes formodel I lies in: (1) due to the mismatched structure, it isdifficult for the helper to fully cancel the state at the receiver,and (2) the helper needs to resolve the tension betweentransmitting its own message and helping receiver 1 to cancelits interference. In this paper, we design an adapted dirty papercoding scheme for state cancelation in the mismatched context,in which correlation between the state variable and the state-cancelation variable is a design parameter, and can be chosento optimize the rate region. This is in contrast to classicaldirtypaper coding [2], in which such a correlation parameter isfixed for fully canceling the state. We further design a layeredcoding scheme, in which the adapted dirty paper coding issuperposed onto the helper’s transmission of its own message.In particular, the design parameters in superposition enable thehelper to judiciously trade off between transmitting its ownmessage and helping to cancel the state.

Based on such a layered coding scheme, we derive achiev-able regions for both the discrete memoryless and Gaussianchannels. We further derive an outer bound for the Gaussianchannel in high state power regime. By comparing the innerand outer bounds, we characterize the boundary of the ca-pacity region either fully or partially for all Gaussian channelparameters in high state power regime. Our result also impliesthat the capacity region is strictly inside the capacity regionof the corresponding channel without state [24] due to infinitestate power. This is in contrast to the results for Costa typeof dirty paper channels, for which dirty paper coding achievesthe capacity of the corresponding channels without state.

We then further study model II, which consists of twotransmitter-receiver pairs in addition to the helper, and onlyone receiver is interfered by a state sequence. Here, thechallenge lies in the fact that the helper inevitably causesinterference to receiver 2 while assisting receiver 1 to cancelthe state. For this model, we start with the scenario withthe helper fully assisting the receivers without transmittingits own message. We first derive an outer bound on thecapacity region. We then develop a two-layer dirty papercoding scheme with one layer helping receiver 1 to cancelstate via dirty paper coding, and with the other layer of dirtypaper coding canceling the interference caused by the helperin assisting receiver 1. By comparing inner and outer bounds,we characterize two segments of the capacity region boundary.One segment corresponds to the case, in which our schemeachieves the point-to-point channel capacity for receiver2

and certain positive rate for receiver 1. This implies that thehelper is able to assist receiver 1 without causing interferenceto receiver 2 effectively. The other segment corresponds tothecase, in which our scheme achieves the best single-user rateforreceiver 1 with assistance of the helper, while receiver 2 treatsthe helper’s signal as noise. Such a scheme is guaranteed byour outer bound to be the best to achieve the sum capacityunder certain channel parameters. We further extend theseresults to the scenario with the helper sending its own messagein addition to assisting the two receivers.

We finally study model III, in which a common helperassists multiple transmitter-receiver pairs with each receivercorrupted by an independently distributed state sequence.Wenote that this model is more general than model I, but doesnot include model II as a special case. This is because modelIII has each receiver (excluding the helper) being corruptedby an infinitely powered state sequence, and hence neverreduces to the model II, in which receiver 2 is not corruptedby a state sequence. This also leads to different technicalchallenges to characterize the capacity for model III due tothecompound state interference. The same technical challengeisalso reflected in the studies [25]–[27] of the state-dependentcompound channel, for which the capacity is not known ingeneral. As for model II, we also start with the scenario, inwhich the helper fully assists other users without sending itsown message. We first derive a useful outer bound, whichcaptures the sum rate limit due to the common helper. Wethen derive an inner bound based on a time-sharing scheme,in which the helper alternatively assists receivers. Somewhatinterestingly, such a time-sharing scheme achieves the sumcapacity under many channel parameters, although each indi-vidual transmitter may not be able to achieve its individualbest rate. This is because these transmitters effectively havelarger power during their transmissions in the time-sharingscheme so that the transmission rate matches the outer boundon the sum rate. We also characterize the full capacity regionunder certain channel parameters. We then extend our resultsto the general scenario with the helper also transmitting itsown message.

The rest of the paper is organized as follows. In Section II,we describe the channel model. In Sections III, IV, and V,we present our results for models I, II, and III, respectively.Finally, in Section VII, we conclude the paper with a fewremarks.

II. CHANNEL MODEL

In this paper, we investigate the state-dependent parallelnetwork with a common state-cognitive helper (see Figure 2),in which K transmitters wish to sendK messages to theircorresponding receivers over state-corrupted parallel channels,and a helper who knows the state information noncausallywishes to assist these receivers to cancel state interference.Furthermore, the helper also has its own message to be sentsimultaneously to its corresponding receiver.

More specifically, each transmitter (say transmitterk) hasan encoderfk : Wk → Xn

k , which maps a messagewk ∈ Wk

to a codewordxnk ∈ Xn

k for k = 1, . . . ,K. The K in-puts xn

1 , . . . , xnK are transmitted overK parallel channels,

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Fig. 2: Parallel Gaussian channel model with a common state-cognitive helper.

respectively. Each receiver (say receiverk) is interfered byan i.i.d. state sequenceSn

k for k = 1, . . . ,K, which is knownat none of transmitters1, . . . ,K and receivers1, . . . ,K. Acommon helper is assumed to know all state sequencesSn

k

for k = 1, . . . ,K noncausally. Thus, the encoder at the helper,f0 : W0 × {Sn

1 , . . . ,SnK} → Xn

0 , maps a messagew0 ∈ W0

and the state sequences(sn1 , . . . , snK) ∈ Sn

1 × . . . × SnK

to a codewordxn0 ∈ Xn

0 . The entire channel transitionprobability is given byPY0|X0

∏Kk=1 PYk|X0,Xk,Sk

. There areK + 1 decoders with each at one receiver,gk : Yn

k → Wk,maps a received sequenceynk into a messagewk ∈ Wk fork = 0, 1, . . . ,K.

Remark 1. Without state interference, our model becomes theK + 1-user Z-interference channel, in which the signal of thehelper interferences all remainingK receivers.

The average probability of error for a length-n code isdefined as

P (n)e =

1

|W0||W1| . . . |WK |

|W0|∑

w0=1

|W1|∑

w1=1

· · ·|Wk|∑

wk=1

Pr{(w0, w1, . . . , wK) 6= (w0, w1, . . . , wK)}. (1)

A rate tuple (R0, R1, . . . , RK) is achievable if there ex-ists a sequence of message setsW(n)

k with |W(n)k | =

2nRk for k = 0, 1, . . . ,K, and encoder-decoder tuples(f

(n)0 , f

(n)1 , . . . , f

(n)K , g

(n)0 , g

(n)1 ,

. . . , g(n)K ) such that the average error probabilityP (n)

e → 0 asn → ∞. Thecapacity regionis defined to be the closure of theset consisting of all achievable rate tuples(R0, R1, . . . , RK).

In this paper, we study the following three Gaussian channelmodels.

In model I,K = 1, i.e., the helper assists one transmitter-receiver pair. The channel outputs at receiver 0 and 1 for onesymbol time are given by

Y0 = X0 +N0, (2a)

Y1 = X0 +X1 + S1 +N1. (2b)

In model II, K = 2, in which one helper assists twotransmitter-receiver pairs, and only one receiver is interfered

by a state sequence. The channel outputs at receivers 0, 1 and2 for one symbol time are given by

Y0 = X0 +N0, (3a)

Y1 = X0 +X1 + S1 +N1, (3b)

Y2 = X0 +X2 +N2. (3c)

In model III,K is general, in which a common helper assistsmultiple transmitter-receiver pairs with each receiver corruptedby an independently distributed state sequence. This modelismore general than model I, but does not include model II as aspecial case (due to infinite state power). The channel outputsat receivers 0 and receivers1, . . . ,K for one symbol time aregiven by

Y0 = X0 +N0, (4a)

Yk = X0 +Xk + Sk +Nk, for k = 1, . . . ,K (4b)

In the above three models, the noise variablesN0, N1 . . . , NK

and the state variablesS1, . . . , SK are Gaussian distributedwith distributionsN0, . . . , NK ∼ N (0, 1) andSk ∼ N (0, Qk)for k = 1, . . . ,K, and all of the variables are indepen-dent and are i.i.d. over channel uses. The channel inputsX0, X1, . . . , XK are subject to the average power constraints1n

∑ni=1 X

2ki 6 Pk for k = 0, 1, . . . ,K.

We are interested in the regime of high state power, i.e.,asQk → ∞ for k = 1, . . . ,K. Our goal is to design helperstrategies in order to cancel the high power state interferenceand to further characterize the capacity region in this regime.

III. M ODEL I: K = 1

Model I with K = 1 is a basic model, in which thehelper assists one transmitter-receiver pair. Understanding thismodel will help the study of the general parallel network.In this section, we first develop outer and inner bounds onthe capacity region, and then characterize the boundary of thecapacity region based on these bounds.

A. Outer Bound

In this subsection, we provide an outer bound on thecapacity region in high state power regime.

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Proposition 1. For the Gaussian channel of model I, an outerbound on the capacity region for the regime whenQ1 → ∞consists of rate pairs(R0, R1) satisfying:

R1 61

2log(1 + P1) (5a)

R0 +R1 61

2log(1 + P0). (5b)

The bound (5a) onR1 follows simply from the capacity ofthe point-to-point channel between transmitter 1 and receiver1 without signal and state interference. The bound (5b) on thesum rate is limited only by the powerP0 of the helper, anddoes not depend on the powerP1 of transmitter 1. Intuitively,this is becauseP0 is split for transmission ofW0 and forhelping transmission ofW1 by removing state interference,and henceP0 determines a trade-off betweenR0 and R1.On the other hand, improving the powerP1, although mayimprove R1, can also cause more interference for receiver1 to decode the auxiliary variable for canceling state andinterference. Thus, the balance of the two effects determinesthatP1 does not affect the sum rate.

Proof. The proof is detailed in Appendix A.

We further note that although the sum-rate upper bound (5b)can be achieved easily by keeping transmitter 1 silent (i.e., R0

achieves the sum rate bound withR1 = 0), we are interested incharacterizing the capacity region (i.e., the trade-off betweenR0 andR1) rather than a single point that achieves the sum-rate capacity. In the next section, we characterize such optimaltrade-off based on the sum-rate bound.

Remark 2. The outer bound in Proposition 1 is strictlyinside an achievable rate region of the corresponding channelwithout state interference (i.e., the Z-interference channel)[24]. This implies that the capacity region of our model isstrictly inside that of the corresponding channel without state.This suggests that state interference does cause performancedegradation for systems with mismatched state cognition andinterference in high state power regime. This is in contrasttothe results on Costa-type dirty paper channels [2], for whichdirty paper coding achieves the capacity of the correspondingchannels without state.

B. Inner Bound

The major challenge in designing an achievable schemearises from the mismatched property due to transmitter-sidestate cognition and receiver-side state interference, i.e., stateinterference to receiver 1 is known noncausally only to thehelper, not to the corresponding transmitter 1. Since we studythe regime with large state power, transmitter 1 can sendinformation to receiver 1 only if the helper assists to cancelthe state. Thus, the helper needs to resolve the tension betweentransmitting its own message to receiver 0 and helping receiver1 to cancel its interference. A simple scheme of time-sharingbetween the two transmitters in general is not optimal.

We design a layered coding scheme as follows. The helpersplits its signal into two parts in a layered fashion: one(represented byX ′

0 in Proposition 2) for transmitting its own

message and the other (represented byU in Proposition 2) forhelping receiver 1 to remove both state and signal interference.In particular, the second part of the scheme applies a single-bindirty paper coding scheme, in which transmission ofW1 andtreatment of state interference for decodingW1 are performedseparately by transmitter 1 and the helper. In traditional (multi-bin) dirty paper coding as in [2], the bin number carries theinformation of the message, and the index within each bincarries the information about the state. In our model, the helperknows only the state, not the message (of transmitter 1), andhence can only encode the state into the index within a singlebin. For the state-dependent Gaussian channel, the single-bindirty paper coding can be understood as quantization of thestate interference. Such a scheme was also used by otherstudies, e.g., [28, Theorem 1]. Based on such a scheme, weobtain the following achievable rate region for the discretememoryless channel, which is useful for deriving an innerbound for the Gaussian channel.

Proposition 2. For the discrete memoryless channel of modelI, an inner bound on the capacity region consists of rate pairs(R0, R1) satisfying:

R0 6 I(X ′0;Y0) (6a)

R1 6 I(X1;Y1|U) (6b)

R1 6 I(X1U ;Y1)− I(U ;S1X′0) (6c)

for some distributionPS1PX′

0PU|S1X

′

0PX0|US1X

′

0PX1

PY0|X0

PY1|S1X0X1.

Proof. The proof is detailed in Appendix B.

Based on Proposition 2, we have the following simpler innerbound.

Corollary 1. For the discrete memoryless channel of model I,an inner bound on the capacity region consists of rate pairs(R0, R1) satisfying:

R0 6 I(X ′0;Y0) (7a)

R1 6 I(X1;Y1|U) (7b)

for some distributionPS1PX′

0PU|S1X

′

0PX0|US1X

′

0PX1

PY0|X0

PY1|S1X0X1that satisfies

I(U ;Y1) > I(U ;S1X′0). (8)

Proof. The region follows from Proposition 2 because (6c) isredundant due to the condition (8).

The inner bound in Corollary 1 corresponds to an intuitiveachievable scheme based on successive cancelation. Namely,the condition guarantees that receiver 1 decodes the auxiliaryrandom variableU first, and then removes it from its outputand decodes the message, which results in the bound (7b). Inparticular, cancelation ofU leads to cancelation of signal andstate interference at receiver 1.

We next derive an inner bound for the Gaussian channel ofmodel I based on Corollary 1.

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Proposition 3. For the Gaussian channel of model I, in theregime whenQ1 → ∞, an inner bound on the capacity regionconsists of rate pairs(R0, R1) satisfying:

R0 61

2log

(

1 +βP0

βP0 + 1

)

(9a)

R1 61

2log

(

1 +P1

1 + (1− 1α)2βP0

)

(9b)

for some real constantsα > 0 and 0 6 β 6 1 that satisfyα 6

2βP0

βP0+P1+1 .

Proof. Proposition 3 follows from Corollary 1 by choosing thejoint Gaussian distribution for random variables as follows:

U = X ′′0 + α(S1 +X ′

0), X0 = X ′0 +X ′′

0

X ′0 ∼ N (0, βP0), X

′′0 ∼ N (0, βP0)

X1 ∼ N (0, P1)

whereX ′0, X ′′

0 , X1 andS1 are independent,α > 0, 0 6 β 6 1,and β = 1− β.

We note that in Proposition 3, the parameterα capturescorrelation between the state variableS1 and the auxiliaryvariableU for dealing with the state, and can be chosen tooptimize the rate region. This is in contrast to the classicaldirty paper coding [2], in which such correlation parameteris fixed for state cancelation. Therefore, although Corollary1 may provide a smaller inner bound than that given inProposition 2, it can be shown that two inner bounds areequivalent for our chosen auxiliary random variables and inputdistribution after optimizing overα.

C. Capacity Region

In this section, we characterize the boundary points of thecapacity region for the Gaussian channel of model I basedon the inner and outer bounds given in Propositions 3 and 1,respectively. We divide the Gaussian channel into three casesbased on the conditions on the power constraints: (1)P1 >

P0 + 1; (2) P0 − 1 6 P1 < P0 + 1 and (3)0 6 P1 < P0 − 1.For each case, we optimize the dirty paper coding parameterα in the inner bound in Proposition 3 to find achievable ratepoints that lie on the sum-rate upper bound (5b) in order tocharacterize the boundary points of the capacity region.

Case 1: P1 > P0 + 1. The capacity region is fullycharacterized in the following theorem.

Theorem 1. For the Gaussian channel of model I, in theregime whenQ1 → ∞, if P1 > P0 + 1, the capacity regionconsists of the rate pairs(R0, R1) satisfying

R0 +R1 61

2log(1 + P0). (10)

Proof. Let P1 be the actual power for transmittingW1. Thenthe inner bound (9b) onR1 is optimized whenα = 2βP0

βP0+P1+1.

By settingP1 = βP0+1, the inner bound given in Proposition3 matches the outer bound given in Proposition 1, and henceis the capacity region.

The capacity region of case 1 is illustrated in Fig. 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R0 (bit/use)

R1(b

it/us

e)

Fig. 3: The capacity region for case 1 withP0 = 1.5 andP1 = 3.

Theorem 1 implies that whenP1 is large enough, the powerof the helper limits the system performance. Furthermore,sinceP1 for transmission ofW1 causes interference to receiver1 to decode the auxiliary variable for canceling state andinterference, beyond a certain value, increasingP1 does notimprove the rate region any more. Theorem 1 also suggeststhat in order to achieve different points on the boundary ofthe capacity region (captured by the parameterβ), differentamounts of powerP1 should be applied.

Case 2:P0−1 6 P1 < P0+1. We summarize the capacityresult in the following theorem.

Theorem 2. Consider the Gaussian channel of model I inthe regime whenQ1 → ∞, and P0 − 1 6 P1 < P0 + 1. IfP1 > 1, the rate points(R0, R1) on the lineA-B (see Fig. 4(a) and Fig. 5 (a)) are on the capacity region boundary. Morespecifically, the pointsA andB are characterized as:

Point A :

(

1

2log(1 + P0), 0

)

Point B :

(

1

2log(1 +

P0 − P1 + 1

P1),1

2logP1

)

If P1 < 1 the rate pointA (see Fig. 4 (b) and Fig. 5 (b)) ison the capacity region boundary, and is characterized as:

Point A :

(

1

2log(1 + P0), 0

)

Proof. We first setα = 2βP0

βP0+P1+1 , and then substituteα into(9b) and obtain the following inner bound:

R0 61

2log

(

1 +βP0

βP0 + 1

)

(13a)

R1 61

2log

(

1 +4βP0P1

4βP0 + (P1 + 1− βP0)2

)

. (13b)

WhenP1 > 1, by settingβ = P1−1P0

, we obtain an achievable

rate point B given by(

12 log(1 +

P0−P1+1P1

), 12 logP1

)

, whichis also on the outer bound. It is also clear that the point Agiven by

(

12 log(1 + P0), 0

)

is achievable by settingβ = 0,which is also on the outer bound. Thus, the lineA−B is onthe boundary of the capacity region due to time sharing.

For this case, ifP1 > 1, i.e., P1 is larger than the noisepower, inner and outer bounds match over the line A-B as

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

R0(bit/use)

R1(b

it/us

e)

Inner bound

A

E

C

B

Outer bound

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

R0(bit/use)

R1(b

it/us

e)

E

C

Inner Bound

A

Outer Bound

(a) P1 > 1 with P0 = 1.5 andP1 = 1.8 (b) P1 < 1 with P0 = 0.5 andP1 = 0.8

Fig. 4: Inner and outer bounds for case 2 withP0 < P1, which match partially on the boundaries

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R0(bit/use)

R1(b

it/us

e)

Outer bound

C

A

BInner bound

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

R0(bit/use)

R1(b

it/us

e)

Inner Bound

Outer Bound

A

E

C

(a) P1 > 1 with P0 = 2 andP1 = 1.8 (b) P1 < 1 with P0 = 0.8 andP1 = 0.5

Fig. 5: Inner and outer bounds for case 2 withP0 > P1, which match partially on the boundaries

illustrated in Fig. 4 (a) and Fig. 5 (a), and thus optimal trade-off betweenR0 andR1 is achieved over the points on the lineA-B. If P1 < 1, the inner and outer bounds match only at therate point A as illustrated in Fig. 4 (b) and Fig. 5 (b), whichachieves the sum-rate capacity. We further note that Fig. 4 isdifferent from Fig. 5 in the outer bound. Fig. 5 corresponds tothe case withP0 ≥ P1, and hence the capacity region is alsoupper bounded by the point-to-point capacity ofR1. Such abound is redundant in Fig. 4 which corresponds to the casewith P0 < P1, becauseP0 is not large enough to perfectlycancel state and signal interference at receiver 1. However,in case 3, we show that this point-to-point capacity ofR1 isachievable simultaneously with a certain positiveR0.

Case 3:0 6 P1 < P0 − 1. We first summarize the capacityresults in the following theorem.

Theorem 3. Consider the Gaussian channel of model I in theregime whenQ1 → ∞, andP1 < P0 − 1. If P1 > 1, the ratepoints (R0, R1) on the lineA-B (see Fig. 6 (a)) are on theboundary of the capacity region. More specifically, the points

A andB are characterized as:

Point A :

(

1

2log(1 + P0), 0

)

Point B :

(

1

2log(1 +

P0 − P1 + 1

P1),1

2logP1

)

And the rate points(R0, R1) on the lineD-E (see Fig. 6 (a))are on the boundary of the capacity region. The pointsD andE are characterized as:

Point D :

(

1

2log(

P0 + 1

P1 + 2),1

2log(1 + P1)

)

Point E :

(

0,1

2log(1 + P1)

)

If P1 < 1, then pointA (see Fig. 6 (b)) is on the capacityregion boundary. The pointA is characterized as:

Point A :

(

1

2log(1 + P0), 0

)

And the rate points(R0, R1) on the lineD-E (see Fig. 6 (b))are on the boundary of the capacity region. The pointsD and

8

E are characterized as:

Point D :

(

1

2log(

P0 + 1

P1 + 2),1

2log(1 + P1)

)

Point E :

(

0,1

2log(1 + P1)

)

Proof. For case 3, the inner bound boundary given in Propo-sition 3 is characterized by segment I consisting of rate pointssatisfying:

R0 61

2log

(

1 +βP0

1 + βP0

)

(18a)

R1 61

2log(1 + βP0) (18b)

for 0 6 β 6P1+1P0

; and segment II consisting of rate pointssatisfying

R0 61

2log

(

1 +βP0

1 + βP0

)

(19a)

R1 61

2log(1 + P1) (19b)

for P1+1P0

6 β 6 1. Segment I is obtained by settingα =2βP0

βP0+P1+1 , and segment II is obtained by settingα = 1.For segment I, ifP1 > 1, the line A-B is on the boundary

of the capacity region as shown in Fig. 6 (a). IfP1 < 1, onlypointA is on the capacity boundary as shown in Fig. 6 (b). Forsegment II, it is clear that the point-to-point channel capacityfor R1 is achievable. Furthermore, by settingβ = P1+1

P0

, thepointD is achievable. Thus, the lineD−E as shown in Fig. 6(a) and (b) is on the boundary of the capacity region.

Similarly to case 2, the inner and outer bounds matchpartially over the sum rate bound, i.e., the two bounds matchover the line A-B (see Fig. 6 (a)) ifP1 > 1 and match at onlythe point A (see Fig. 6 (b)) ifP1 < 1. However, differentlyfrom case 2, the inner and outer bounds also match whenR1 = 1

2 log(1 + P1) over the line D-E (see Fig. 6 (a) and(b)). This is because the powerP0 of the helper in this caseis large enough to fully cancel state and signal interference sothat transmitter 1 is able to reach its maximum point-to-pointrate to receiver 1 without interference. Furthermore, the helperis also able to simultaneously transmit its own message at acertain positive rate.

IV. M ODEL II: K = 2

In this section, we consider the Gaussian channel of modelII with K = 2, in which only receiver 1 is interfered by aninfinite power state. We first study the scenario, in which thehelper devotes to help two users without transmitting its ownmessage, i.e.,W0 = φ. We then extend the result to the moregeneral scenario, in which the helper also has its own messagedestined for the corresponding receiver in addition to helpingthe two users, i.e.,W0 6= φ.

A. Scenario with Dedicated Helper (W0 = φ)

In this subsection, we study the scenario in which onlyreceiver 1 is corrupted by a state sequence, and the helper(without transmission of its own messages) fully assists tocancel such state interference. Here, the challenge lies inthefact that the helper needs to assist receiver 1 to remove the stateinterference, but such signal inevitably causes interferenceto receiver 2. In this subsection, we first derive an outerbound, and then derive an inner bound based on the helperusing a layered coding scheme with one layer assisting thestate-interfered receiver, and the other layer canceling theinterference in order to address the challenge mentioned above.We then characterize segments of the capacity boundary andthe sum capacity under certain channel parameters.

We first derive a useful outer bound for Model II.

Proposition 4. For the Gaussian channel of model II withW0 = φ, in the regime whenQ1 → ∞, an outer bound onthe capacity region consists of rate pairs(R1, R2) satisfying:

R1 6min

{

1

2log(1 + P0),

1

2log(1 + P1)

}

(20a)

R2 61

2log(1 + P2) (20b)

R1 +R2 61

2log(1 + P0 + P2). (20c)

Proof. The proof is detailed in Appendix C.

We note that (20a) represents the best single-user rate ofreceiver 1 with the helper dedicated to help it as shownin Proposition 1, (20b) is the point-to-point capacity forreceiver 2, and (20c) implies that although the two transmitterscommunicate over parallel channels to their correspondingreceivers, due to the shared common helper, the sum rate isstill subject to a certain rate limit.

We next describe our idea to design an achievable scheme.We first note that although receiver 2 is not interfered by thestate, the signal that the helper sends to assist receiver 1 to dealwith the state still causes unavoidable interference to receiver2. A natural idea to optimize the transmission rate to receiver2 is simply to keep the helper silent. In this case, without thehelper’s assistance, receiver 1 gets zero rate due to infinitestate power. Here, we design a novel scheme, which enablesthe point-to-point channel capacity for receiver 2 and a certainpositive rate for receiver 1 simultaneously. Consequently, thehelper is able to assist receiver 1 without causing interferenceto receiver 2. In our achievable scheme, the signal of thehelper is split into two parts, represented byU andV as inProposition 5. Here,U is designed to help receiver 1 to cancelthe state while treatingV as noise, andV is designed to helpreceiver 2 to cancel the interference caused byU . Since thereis no state interference at receiver 2,U is decoded only atreceiver 1. Based on such an achievable scheme, we obtainthe following achievable region.

Proposition 5. For the discrete memoryless channel of modelII with W0 = φ, an achievable region consists of the rate pair

9

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R0(bit/use)

R1(b

it/us

e)

B

Outer bound

Inner bound

E DC

A0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

R0(bit/use)

R1(b

it/us

e)

Outer Bound

Inner Bound

C

DE

A

(a) P1 > 1 with P0 = 3 andP1 = 1.8 (b) P1 < 1 with P0 = 1.5 andP1 = 0.5

Fig. 6: Inner and outer bounds for case 3, which match partially on the boundaries.

(R1, R2) satisfying

R1 6 I(X1;Y1U) (21a)

R1 6 I(X1U ;Y1)− I(U ;S1) (21b)

R2 6 I(X2;Y2V ) (21c)

R2 6 I(X2V ;Y2)− I(V ;US1) (21d)

for some distributionPS1UV X0X1X2= PS1

PUV X0|S1PX1

PX2,

whereU and V are auxiliary random variables.

Proof. The proof is detailed in Appendix D.

A straight-forward but more convenient subregion of theabove inner bound is as follows.

Corollary 2. For the discrete memoryless channel of model IIwith W0 = φ, an achievable region consists of the rate pair(R1, R2) satisfying

R1 6 I(X1;Y1|U), (22a)

R2 6 I(X2;Y2|V ), (22b)

for some distributionPS1UV X0X1X2= PS1

PUV X0|S1PX1

PX2,

whereU and V are auxiliary random variables satisfyinf that

I(U ;Y1) > I(U ;S1), (23a)

I(V ;Y2) > I(V ;US1). (23b)

Proof. The region follows from Proposition 5 because (21b)and (21d) are redundant due to conditions (23a) and (23b)

Following from the above achievable region, we obtainan achievable region for the Gaussian channel by setting anappropriate joint input distribution.

Proposition 6. For the Gaussian channel of model II withW0 = φ, in the regime whenQ1 → ∞, an inner bound onthe capacity region consists of rate pairs(R1, R2) satisfying:

R1 61

2log

(

1 +P1

(1 − 1α)2P01 + P02 + 1

)

(24a)

R2 61

2log

(

1 +P2

1 + (β−1)2P02P01

P02+β2P01

)

(24b)

whereP01, P02 > 0, P01 +P02 6 P0, 0 < α 62P01

1+P0+P1

, andP 202 + 2βP01P02 > β2P01(P02 + P2 + 1).

Proof. The region follows from Corollary 2 by choosingjointly Gaussian distribution for random variables as follows:

U = X01 + αS1, V = X02 + βX01

X0 = X01 +X02

X01 ∼ N (0, P01), X02 ∼ N (0, P02)

X1 ∼ N (0, P1), X2 ∼ N (0, P2)

whereX01, X02, X1, X2 andS1 are independent.

Comparing the inner and outer bounds given in Propositions6 and 4, respectively, we characterize two segments of theboundary of the capacity region, over which the two boundsmeet.

Theorem 4. Consider the Gaussian channel of model II withW0 = φ, in the regime whenQ1 → ∞, the rate points on theline A-B (see Fig. 7) are on the capacity region boundary.More specifically, if12 (1+P0+P1) >

P 2

0

P0+P2+1 , pointsA andB are characterized as

Point A :

(

0,1

2log(1 + P2)

)

Point B :(1

2log

(

1 +4P1P

20

(1 + P0 + P1)2(1 + P0 + P2)− 4P1P20

)

,

1

2log(1 + P2)

)

.

If 12 (1 + P0 + P1) <

P 2

0

P0+P2+1 , pointsA and B are charac-terized as

Point A :

(

0,1

2log(1 + P2)

)

Point B :

(

1

2log

(

1 +P1(P0 + P2 + 1)

P0 + (P0 + 1)(P2 + 1)

)

,1

2log(1 + P2)

)

.

Furthermore, the rate points on the lineC-D (see Fig. 7) arealso on the capacity region boundary. IfP1 > P0 + 1, thepointsC andD are characterized as

Point C :

(

1

2log(1 + P0),

1

2log

(

1 +P2

P0 + 1

)

)

Point D :

(

1

2log(1 + P0), 0

)

,

as illustrated in Fig. 7(a).

10

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

R1(bits/use)

R2(b

its/u

se)

Outer BoundInner Bound

BA

C

D

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

R1(bits/use)

R2(b

its/u

se)


A

C

B

D

(a) P1 > P0 + 1 0 6 P1 6 P0 − 1

Fig. 7: Segments of the capacity boundary for the Gaussian channel of model II

If P1 6 P0 − 1, the pointsC andD is characterized as

Point C :

(

1

2log(1 + P1),

1

2log

(

1 +P2

P1 + 2

)

)

Point D :

(

1

2log(1 + P1), 0

)

,

as illustrated in Fig. 7(b).

Proof. We first show that the lineA-B is achievable. The pointA is achievable by keeping the helper silent. To show that thepoint B is achievable, we setβ = 1 in Proposition 6, andhence the achievable rateR2 in (24b) reaches the point-to-point channel capacity, and the conditionP 2

02 + 2βP01P02 >

β2P01(P02+P2+1) becomesP01 6P 2

0

P0+P2+1 . We setP01 =P 2

0

P0+P2+1 .

If 12 (1+P0+P1) >

P 2

0

P0+P2+1 , we have 2P01

1+P0+P1

6 1. Thus,settingα = 2P01

1+P0+P1

, (24a) and (24b) imply that the point Bis achievable.

If 12 (1 + P0 + P1) 6

P 2

0

P0+P2+1 , we have 2P01

1+P0+P1

> 1.By settingα = 1, (24a) and (24b) imply that the point B isachievable.

We next show that the lineC-D is on the capacity boundary.As implied by Theorems 1 and 3, the only possible cases

that the outer bound (20a) (i.e. the maximum rateR1 withthe helper fully assisting receiver 1) can be achieved are whenP1 6 P0 − 1 andP1 > P0 + 1.

If P1 > P0 + 1, setting the actual transmission power oftransmitter 1 asP1 = P0 + 1, P01 = P0, α = P0

1+P0

andβ = 0, then (24a) and (24b) imply that the rate point C isachievable. This point also achieves the sum capacity. It isobvious that pointD is achievable, and hence the points onthe lineC-D are on the capacity boundary.

If P1 6 P0 − 1, by settingβ = 0, α = 1 and P01 =P0 = P1 + 1 (where P0 is the actual transmission power ofthe helper), then (24a) and (24b) imply that the rate point Cis achievable. In particular, the actual power the helper uses isP1+1 rather thanP0, because largerP0 does not help receiver1 to decode more, but increases interference to receiver 2. Itis clear that the pointD is achievable. Hence, the points onthe lineC-D are on the capacity boundary.

The capacity result for the lineA-B in Theorem 4 indicatesthat our coding scheme effectively enables the helper to assistreceiver 1 without causing interference to receiver 2. Hence,R2 achieves the corresponding point-to-point channel capacity,while transmitter 1 and receiver 1 communicate at a certainpositive rateR1 with the assistance of the helper.

The capacity result for the lineC-D in Theorem 4 canbe achieved based on a scheme, in which the helper assistsreceiver 1 to deal with the state and receiver 2 treats thehelper’s signal as noise. Such a scheme is guaranteed to bethe best by the outer bound if receiver 1’s rate is maximized.

Remark 3. Theorem 4 implies that ifP1 > P0 + 1, the sumcapacity is achieved by the pointC as illustrated in Fig. 7(a).

Corollary 3. For the Gaussian channel of model II withW0 =φ, in the regime whenQ1 → ∞, if P1 > P0 + 1, the sumcapacity is given by12 log(1 + P0 + P2).

B. Scenario with Non-Dedicated Helper (W0 6= φ)

In this subsection, we study the scenario, in which the helperalso has its own message to transmit in addition to assistingthe state-corrupted receivers, i.e.,W0 6= φ. The results wepresent below extend those in the preceding subsection forthe scenario withW0 = φ. The proof techniques are similarand hence are omitted.

We first provide an outer bound for the Gaussian channel,which generalizes Propositions 1 and 4.

Proposition 7. For the Gaussian channel of model II withW0 6= φ, an outer bound on the capacity region for the regime

11

whenQ1 → ∞ consists of rate pairs(R1, R2) satisfying:

R0 61

2log(1 + P0) (29a)

R1 6min

{

1

2log(1 + P0),

1

2log(1 + P1)

}

(29b)

R2 61

2log(1 + P2) (29c)

R0 +R1 +R2 61

2log(1 + P0 + P2). (29d)

We next present an achievable region for the discrete mem-oryless channel which generalizes Proposition 5 and Corollary2.

Proposition 8. For the discrete memoryless channel of modelII with W0 6= φ, an achievable region consists of the ratetuples(R0, R1, R2) satisfying

R0 6 I(X00;Y0) (30a)

R1 6 I(X1;Y1|U) (30b)

R2 6 I(X2;Y2|V ) (30c)

for some distribution PS1UV X00X0X1X2=

PX00PS1

PUV X0|S1X00PX1

PX2, where U and V are auxiliary

random variables that satisfy

I(U ;Y1) > I(U ;S1X00) (31a)

I(V ;Y2) > I(V ;US1X00). (31b)

In the above proposition, the variableX00 is an auxiliaryrandom variable representing the helper’s own informationforits intended receiver.

We then obtain an achievable region for the Gaussian chan-nel by setting an appropriate joint distribution in Proposition8.

Proposition 9. For the Gaussian channel of model II withW0 6= φ, an inner bound on the capacity region for the regimewhenQ1 → ∞ consists of rate tuples(R0, R1, R2) satisfying:

R0 61

2log

(

1 +P00

P01 + P02 + 1

)

(32a)

R1 61

2log

(

1 +P1

(1− 1α)2P01 + P02 + 1

)

(32b)

R2 61

2log

1 +P2

1 + (β−1)2P02(P00+P01)P02+β2(P00+P01)

. (32c)

whereP00, P01, P02 > 0, P00 + P01 + P02 6 P0, 0 < α 62P01

1+P01+P02+P1

, and P 202 + 2βP02(P01 + P00) > β2(P00 +

P01)(P02 + P2 + 1).

Proof. The region follows from Proposition 8 by choosingjointly Gaussian distribution for random variables as follows:

U = X01 + α(S1 +X00), V = X02 + β(X00 +X01)

X0 = X00 +X01 +X02

X00 ∼ N (0, P00), X01 ∼ N (0, P01), X02 ∼ N (0, P02)

X1 ∼ N (0, P1), X2 ∼ N (0, P2)

whereX00, X01, X02, X1, X2 andS1 are independent.

By comparing the inner and outer bounds, we obtain seg-ments on the capacity region boundary as generalization ofTheorem 4.

Theorem 5. Consider the Gaussian channel of model II withW0 6= φ, in the regime whenQ1 → ∞, for any given achiev-able rateR0, correspondingly a certain powerP00, the ratepoints on the lineA-B are on the capacity region boundary.More specifically, if12 (1 +P1 +P0 −P00) >

P 2

0

P0+P2+1 −P00,the pointsA andB are characterized as:

Point A :(1

2log

(

1 +P00

P0 − P00 + 1

)

, 0,1

2log(1 + P2)

)

Point B :(1

2log

(

1 +P00

P0 − P00 + 1

)

,

1

2log

(

1 +P1

(1+P0−P00+P1)2(1+P0+P2)

4(P2

0−P00(P0+P2+1))

− P1

)

,1

2log(1 + P2)

)

If 12 (1 + P1 + P0 − P00) 6

P 2

0

P0+P2+1 − P00, the pointsA andB are characterized as:

Point A :(1

2log

(

1 +P00

P0 − P00 + 1

)

, 0,1

2log(1 + P2)

)

Point B :(1

2log

(

1 +P00

P0 − P00 + 1

)

,

1

2log

(

1 +P1(P0 + P2 + 1)

P0 + (P0 + 1)(P2 + 1)

)

,1

2log(1 + P2)

)

Furthermore, the rate points on the lineC-D characterizedbelow is on the capacity region boundary. IfP1 > P0−P00+1,the pointsC andD are characterized as:

Point C :(1

2log

(

1 +P00

P0 − P00 + 1

)

,

1

2log(1 + P0 − P00),

1

2log

(

1 +P2

P0 + 1

)

)

Point D :(1

2log

(

1 +P00

P0 − P00 + 1

)

,1

2log(1 + P0 − P00), 0

)

If P1 6 P0 − P00 − 1, the pointsC andD are characterizedas:

Point C :(1

2log

(

1 +P00

P0 − P00 + 1

)

,

1

2log(1 + P1),

1

2log

(

1 +P2

P1 + 2

)

)

Point D :(1

2log

(

1 +P00

P0 − P00 + 1

)

,1

2log(1 + P1), 0

)

Theorem 5 implies the following characterization of the sumcapacity.

Corollary 4. For the Gaussian channel of model II withW0 6=φ, in the regime whenQ1 → ∞, for a given0 6 P00 6 P0,if P1 > P0 −P00 +1, the sum capacity is given by12 log(1 +P0 + P2).

V. M ODEL III: G ENERAL K

In this section, we consider the Gaussian channel of modelIII with K > 2, in which there are multiple receivers with eachinterfered by an independent state. We first study the scenario,in which the helper dedicates to help two users withouttransmitting its own message, i.e.,K = 2 and W0 = φ.We then extend the result to the more general scenario, inwhich the helper dedicates to help more than two users withouttransmitting its own message, i.e.,K > 2 and W0 = φ.

12

Finally, we study the case, in which the helper also has its ownmessage destined for the corresponding receiver in addition tohelpingK (K > 2) users, i.e.,K > 2 andW0 6= φ.

A. Scenario with Two State-Corrupted Receivers and Dedi-cated Helper (K = 2, W0 = φ)

In this subsection, we study the scenario withK = 2, whichis more instructional. The case withK > 2 is relegated toSection V-B. We note that model III is more general thanmodel I, but does not include model II as a special case,because model II has one receiver that is not corrupted bystate, but each receiver (excluding the helper) in model IIIiscorrupted by an infinitely powered state sequence. Hence formodel III, the challenge lies in the fact that the helper needsto assist multiple receivers to cancel interference causedbyindependent states. In this subsection, we first derive an outerbound on the capacity region, and then derive an inner boundbased on a time-sharing scheme for the helper. Somewhatinterestingly, comparing the inner and outer bounds concludesthat the time-sharing scheme achieves the sum capacity undercertain channel parameters, and we hence characterize seg-ments of the capacity region boundary corresponding to thesum capacity under these channel parameters.

We first derive an outer bound on the capacity region.

Proposition 10. For the Gaussian channel of model III withK = 2 and W0 = φ, in the regime whenQ1, Q2 → ∞,an outer bound on the capacity region consists of rate pairs(R1, R2) satisfying:

R1 61

2log(1 + P1) (37)

R2 61

2log(1 + P2) (38)

R1 +R2 61

2log(1 + P0). (39)

Proof. The proof is detailed in Appendix E.

We note that although the two transmitters transmit overparallel channels, the above outer bound suggests that theirsum rate is still subject to a certain constraint determinedbythe helper’s power. This implies that it is not possible forone common helper to cancel the two independent high-powerstates simultaneously (i.e., using the common resource). Thisfact also suggests that a time-sharing scheme, in which thehelper alternatively assists each receiver, can be desirable toachieve the sum rate upper bound (i.e., to achieve the sumcapacity).

We hence design the following time-sharing achievablescheme. The helper splits its transmission duration into twotime slots with the fractionγ of the total time durationfor assisting receiver 1 and the fraction1 − γ for assistingreceiver 2. Each transmitter transmits only during the timeslot that it is assisted by the helper, and keeps silent whilethehelper assisting the other transmitter. We note that the powerconstraints for transmitters 1 and 2 in their correspondingtransmission time slots areP1

γand P2

1−γ, respectively.

Now at each transmission slot, the channel consists ofone transmitter-receiver pair with the receiver corruptedby

a infinite-power state, and one helper that assists the receiverto cancel the state interference. Such a model is a special caseof model I studied in Section III (with the helper not having itsown message). Hence, following from Proposition 3, we canwrite the achievable rate as a function ofP andP0 respectivelyat the transmitter and the helper as follows:

R(P, P0) :=

12 log(1 + P0), P > P0 + 112 log(1 +

4P0P4P0+(P0−P−1)2 ),

P0 − 1 6 P 6 P0 + 112 log(1 + P ), P 6 P0 − 1.

(40)

By employing the time-sharing scheme between the helperassisting one receiver and the other alternatively, we obtainthe following achievable region.

Proposition 11. For the Gaussian channel of model III withK = 2 andW0 = φ, in the regime withQ1, Q2 → ∞, an innerbound on the capacity region consists of rate pairs(R1, R2)satisfying:

R1 6 γR

(

P1

γ, P0

)

(41a)

R2 6 (1− γ)R

(

P2

1− γ, P0

)

(41b)

where 0 6 γ 6 1 is the time-sharing coefficient, and thefunctionR(·, ·) is defined in(40).

We note that following from (40), the best possible single-user rate is12 log(1+P0), which can be achieved ifP > P0+1.This best rate may not be possible ifP is not large enough.Interestingly, in a time-sharing scheme, both transmitters cansimultaneously achieve the best single user rate1

2 log(1+P0)over their transmission fraction of time, because both of theirpowers get boosted over a certain fraction of time, althoughneither power is larger thanP0 + 1. In this way, the sum rateupper bound (39) can be achieved. Such a fact also suggeststhat in the helper-power limited regime, letting the helpersimultaneously assist all users typically does not achievefurther multiplexing gain. The following theorem characterizesthe sum capacity of the channel for the scenario describedabove.

Theorem 6. For the Gaussian channel of model III withK = 2 and W0 = φ, in the regime withQ1, Q2 →∞, if P1 + P2 > P0 + 1, then the sum capacity equals12 log(1 + P0). The rate points that achieve the sum capac-ity (i.e. on the capacity region boundary) are characterizedas (R1, R2) =

(

γR(P1

γ, P0), (1− γ)R( P2

1−γ, P0)

)

for γ ∈(

max(1− P2

P0+1 , 0),min( P1

P0+1 , 1))

.

Proof. The proof is detailed in Appendix F.

The above theorem implies the following characterizationof the full capacity region under certain parameters.

Corollary 5. For the Gaussian channel of model III withK = 2 and W0 = φ, in the regime withQ1, Q2 → ∞, if

13

P1, P2 > P0+1, then the capacity region consists of the ratepair (R1, R2) satisfyingR1 +R2 6

12 log(1 + P0).

We next provide channel examples to understand the outerand inner bounds respectively in Proposition 10 and 11, and inthe sum capacity in Theorem 6. It can be seen that the powerconstraints fall into four cases, among which we consider thefollowing three cases: case 1.P1 > P0, P2 > P0; case 2.P1 > P0, P2 < P0; and case 3.P1 < P0, P2 < P0 by notingthat case 4 is opposite to case 2 and is omitted due to symmetryof the two transmitters.

• Case 1:P1 > P0, P2 > P0. We consider an examplechannel withP0 = 1, P1 = 1.8 andP2 = 1.5. Fig. 8 (a)plots the inner and outer bounds on the capacity region.In particular, the two bounds meet over the line segmentB-C, which corresponds to the rate points(R1, R2) =(

γR(P1

γ, P0), (1− γ)R( P2

1−γ, P0)

)

for γ ∈(

max(1 −P2

P0+1 , 0),min( P1

P0+1 , 1))

as characterized in Theorem 6.All these rate points achieve the sum capacity. It canalso be seen that although neither transmitter achievesthe best possible single-user rate, the sum capacity can beachieved due to the time-sharing scheme. We also notethat, in this case, if the conditions in Corollary 5 aresatisfied, the full capacity region is characterized.

• Case 2:P1 > P0, P2 6 P0. We consider an examplechannel withP0 = 2, P1 = 2.5 and P2 = 0.8. Fig. 8(b) plots the inner and outer bounds on the capacityregion. Similarly to case 1, the two bounds meet overthe line segment B-C as characterized in Theorem 6,and the points on such a line segment achieve the sumcapacity. Differently from case 1, transmitter 2 achievesits point-to-point channel capacity indicated by the pointA in Figure 8 (b). This is consistent with the single userrate provided in (40) for the case withP2 6 P0 − 1.

• Case 3:P1 < P0, P2 < P0. We consider an examplechannel withP0 = 4, P1 = 3 and P2 = 3. Figure8 (c) plots the inner and outer bounds on the capacityregion. The points on the line segment B-C achieve thesum capacity as characterized in Theorem 6, and thepoints A and D respectively achieve the point-to-pointcapacity for two transceiver pairs. This is consistent withthe single-user rate provided in (40) for the case withP1, P2 6 P0 − 1.

B. Scenario with Multiple State-Corrupted Receivers andDedicated Helper (K > 2, W0 = φ)

In this subsection, we study the scenario, withK > 2state-corrupted receivers, in which the helper is dedicated toassist these receivers, i.e.,W0 = φ. The results we presentbelow extend those in the preceding subsection for the scenariowith K = 2. The proof techniques are similar and hence areomitted.

We first provide an outer bound on the capacity region asfollows, which is an extension of Proposition 10.

Proposition 12. For the Gaussian channel of model III withK > 2 andW0 = φ, in the regime whenQ1, . . . , QK → ∞,

an outer bound on the capacity region consists of rate tuples(R0, . . . , RK) satisfying:

Rk 61

2log(1 + Pk) for k = 1, . . . ,K

K∑

k=1

Rk 61

2log(1 + P0).

In order to derive an inner bound, we employ the followingtime-sharing scheme. We divide the entire transmission timeinto K slots, and during each slot the helper assists onereceiver. We provide the corresponding achievable region asfollows as an extension of Proposition 11.

Proposition 13. For the Gaussian channel of model III withK > 2 andW0 = φ, in the regime whenQ1, . . . , QK → ∞,an inner bound consists of rate tuples(R1, . . . , RK) satisfy-ing:

Rk 6 γkR

(

Pk

γk, P0

)

, for k = 1, . . . ,K

for someγk that satisfy

K∑

k=1

γk = 1, and γk > 0 for k = 1, . . . ,K,

whereR(·, ·) is the function defined in(40).

By comparing the inner and outer bounds, we obtain thefollowing result on the capacity as an extension of Theorem6.

Theorem 7. For the Gaussian channel of model III withK > 2 and W0 = φ, in the regime withQ1, . . . , Qk → ∞,the sum capacity equals12 log(1 + P0). The rate points thatachieve the sum capacity on the capacity region boundary arecharacterized as

Rk 6γk

2log(1 + P0) k = 1, . . . ,K

where

K∑

k=1

γk = 1, γk > 0

Pk

γk> P0 + 1 for k = 1, . . . ,K.

C. Scenario with Non-Dedicated Helper (W0 6= φ)

In this subsection, we study the scenario, in which the helperalso has its own message for the corresponding receiver inaddition to assisting the state-corrupted receivers, i.e.,W0 6= φ.The results we present below extend those in the precedingsubsection for the scenario withW0 = φ as well as the resultsin Section III for model I. The proof techniques combine thosein Sections V-A and III, and some are omitted.

We first provide an outer bound on the capacity region asfollows, which is an extension of Proposition 10.

Proposition 14. For the Gaussian channel of model III withW0 6= φ, in the regime whenQ1, . . . , QK → ∞, an

14

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

R1(bits/use)

R2(b

its/u

se)


B

C

P0=1

P1=1.8

P2=1.5

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

R1(bits/use)

R2(b

its/u

se)


P0=2

P1=2.5

P2=0.8

A

C

B

(a) P1 > P0, P2 > P0 (b) P1 > P0, P2 < P0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

R1(bits/use)

R2(b

its/u

se)


D

C

B

A

P0=4

P1=3

P2=3

(c) P1 < P0, P2 < P0

Fig. 8: Inner and outer bounds for the Gaussian channel of model III with K = 2 andW0 = φ, which match partially.

outer bound on the capacity region consists of rate tuples(R0, . . . , RK) satisfying:

Rk 61

2log(1 + Pk) for k = 0, . . . ,K

K∑

k=0

Rk 61

2log(1 + P0).

Proof. The proof is detailed in Appendix G.

In order to derive an inner bound, we employ the followingtime-sharing scheme. We divide the entire transmission timeinto K slots, and during each slot the helper assists onereceiver and simultaneously transmits its own message. Weprovide the corresponding achievable region as follows as anextension of Proposition 11.

Proposition 15. For the Gaussian channel of model III withW0 6= φ, in the regime whenQ1, . . . , QK → ∞, an innerbound consists of rate tuples(R0, . . . , RK) satisfying:

R0 6

K∑

k=1

γk

2log

(

1 +βkP0

βkP0 + 1

)

Rk 6 γkR

(

Pk

γk, βkP0

)

for k = 1, . . . ,K

for some parameters that satisfy

K∑

k=1

γk = 1, γk > 0,

βk + βk = 1, βk > 0 for k = 1, . . . ,K,

whereR(·, ·) is the function defined in(40).

By comparing the inner and outer bounds, we obtain thefollowing result on the capacity as an extension of Theorem6.

Theorem 8. For the Gaussian channel of model III withW0 6=φ, in the regime withQ1, . . . , Qk → ∞, the sum capacityequals 1

2 log(1 + P0). The rate points that achieve the sumcapacity on the capacity region boundary are characterizedas

R0 6

K∑

k=1

γk

2log

(

1 +βkP0

βkP0 + 1

)

Rk 6γk

2log(1 + βkP0) for k = 1, . . . ,K

15

whereK∑

k=1

γk = 1, γk > 0

Pk

γk> βkP0 + 1

βk + βk = 1, βk > 0, for k = 1, . . . ,K.

VI. FURTHER DISCUSSION

In this paper, we focus on state-dependent Gaussian modelsin the large state-power regime, i.e.,Q → ∞. In this case,the state cannot be canceled by direct reversion as the helperhas only finite amount of power. Thus, we design achievableschemes that precode the state into helper signals via single-bin dirty paper coding so that the state-interfered receiver cancancel the state interference. Two outer bounds are useful forus to understand the optimality of our achievable schemes. Oneis the capacity of the corresponding channel models withoutstate, which captures how the transmitters’ power limits thetransmission rate if the state can be fully cancelled. We furtherdevelop a new outer bound in this paper, which captures howthe helper’s power limits the transmission rate. Based on theseouter bounds, we characterize under what channel parametersthe developed achievable schemes are optimal.

Such a study also provides some useful understanding forfurther exploring these models in the finite state-power regime.Since the state power is finite, direct reversion can help topartially or fully cancel the state. Hence, a natural achievablescheme is to combine direct reversion with single-bin dirtypaper coding that we apply in this paper to cancel the stateinterference. For outer bounds, other than outer bounds usedin this paper that capture impact of the transmitters’ powerand the helper’s power on the transmission rate, it is desirableto derive more refined outer bounds that can capture how thestate power limits the transmission rates. Such outer bounds(together with the inner bounds) will suggest under whatchannel parameters, only partially canceling state interferenceis already optimal.

In this paper, we study two types of models, i.e., model IIwith two users respectively being/not being corrupted by state,and model III (including model I as a special case) with allusers being corrupted by state. Since the state power is infinite,models II and III do not contain each other as a special case.Although these two models capture different communicationscenarios, they can be unified by a general model described asfollows. Suppose there are two groups of transmitter-receiverpairs. All receivers in group 1 are corrupted by independentstates, whereas all receivers in group 2 are not corrupted bystate. Similarly to models II and III, a helper transmitter knowsthe states noncausally and wishes to help all receivers in addi-tion to transmitting its own information. It is clear that such amodel includes both model II and model III as special cases.Our analysis for models II and III can be naturally extended tostudy the above general model. First, the achievable schemecan be designed by integrating our achievable schemes formodels II and III together. More specifically, for the receiversin group 1, time-sharing can be employed for the helper to

assist one transmitter-receiver pair at a time. Then, betweenthe two groups, superposition coding as for model II can beemployed to design the achievable scheme. In order to designouter bounds, for each pair of receivers with one from group1 and one from group 2, the outer bound that we derive formodel II is applicable and serves as one upper bound. Then,combining all such upper bounds together provides an outerbound for the more general model. Based on such inner andouter bounds, one can analyze under what channel parametersthe capacity region of the general model can be characterized.

VII. C ONCLUSION

In this paper, we proposed and studied three models ofparallel communication networks with a state-cognitive helper.For each model, there is unique challenge to design capacity-achieving schemes for the helper to trade off among multiplefunctions. For model I, we designed an adapted dirty papercoding together with superposition coding for the helper totrade off between assisting to cancel the state and transmittingits own message. We showed that such a scheme achieves thefull capacity region or segments on the capacity region bound-ary for all channel parameters. For model II, we designed amulti-layer scheme, such that the helper assists receiver 1tocancel the infinite-power state while simultaneously eliminat-ing its interference to receiver 2. Such a scheme achieves twosegments on the capacity region. Over one segment, the helperis capable to fully cancel the interference that it causes toreceiver 2, and simultaneously assists receiver 1 to achieve acertain positive rate. In the second segment, the sum capacityis obtained with the helper dedicated to help receiver 1. Formodel III, we employed a time-sharing scheme such thatthe helper alternatively assists each receiver, and we showedthat such a scheme achieves the sum capacity for certainchannel parameters. These results suggest that with only stateinformation of parallel users, a helper can still assist theseusers to cancel infinite-power state in an optimal way. Weanticipate that the techniques that we develop in this paperwillbe useful for studying various other multi-user state-dependentmodels with state-cognitive helpers.

As we mentioned in Section I-A, large state interference andstate-cognitive helpers are well justified in practical wirelessnetworks, and such a model implies a promising perspectiveof interference cancelation in wireless networks. Hence, theschemes developed here can be potentially used to greatlyimprove the throughput of wireless networks.

APPENDIX APROOF OFPROPOSITION1

The first bound follows easily from the single-user ratebound of receiver 1 as follows.

nR1 6 I(W1;Yn1 ) + nǫn

6 I(W1;Yn1 Sn

1Xn0 ) + nǫn

= I(W1;Yn1 |Sn

1Xn0 ) + nǫn

6 h(Y n1 |Sn

1Xn0 )− h(Y n

1 |W1Sn1X

n1 X

n0 ) + nǫn

= h(Xn1 +Nn

1 )− h(Nn1 ) + nǫn

6n

2log(1 + P1) (42)

16

We then bound the sum rate as follows. For the messageW0,based on Fano’s inequality, we have

nR0 6 I(W0;Yn0 ) + nǫn (43)

= h(Y n0 )− h(Y n

0 |W0) + nǫn,

whereǫn → 0 asn → ∞.For the messageW1, based on Fano’s inequality, we have

nR1 6 I(W1;Yn1 ) + nǫn (44)

= h(Y n1 )− h(Y n

1 |W1) + nǫn

6 h(Y n1 )− h(Y n

1 |W1Xn1 ) + nǫn

= h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 ) + nǫn

6 h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 |W0Yn0 ) + nǫn

Summation of (43) and (44) yields

n(R0 +R1) 6 h(Y n0 ) + h(Y n

1 )

− h(Y n0 , Xn

0 + Sn1 +Nn

1 |W0) + nǫn

= h(Y n0 ) + h(Y n

1 )

− h(Xn0 +Nn

0 , Xn0 + Sn

1 +Nn1 |W0) + nǫn

(45)

Since the two receivers perform decoding independently,the capacity region of the channel depends on only themarginal distributions of(X0, Y0) and (X0, X1, S, Y1). It isclear that settingN1 = N0 does not change the two marginaldistributions respectively involvingY0 andY1, and hence doesnot affect the capacity region. Thus,

n(R0 +R1)

6h(Y n0 ) + h(Y n

1 )

− h(Xn0 +Nn

1 , Xn0 + Sn

1 +Nn1 |W0) + nǫn

6h(Y n0 ) + h(Y n

1 )− h(Sn1 , X

n0 +Nn

1 |W0) + nǫn

6h(Y n0 ) + h(Y n

1 )− h(Sn1 )− h(Nn

1 ) + nǫn

6n

2log(1 + P0)

+n

2log

(

1 +P0 + 2

√P0Q1 + P1 + 1

Q1

)

+ nǫn (46)

As Q1 → ∞, the second term of the above bound goes to0, and we have

R0 +R1 61

2log(1 + P0). (47)

APPENDIX BPROOF OFPROPOSITION2

We use random codes and fix the following joint distribu-tion:

PS1X′

0UX0X1Y0Y1

=PS1PX′

0PU|S1X

′

0PX0|US1X

′

0

· PX1PY0|X0

PY1|X0X1S1.

Let T nǫ (PS1X

′

0UX0X1Y0Y1

) denote the strongly jointǫ-typical set (see, e.g., [29, Sec. 10.6] and [30, Sec. 1.3]for definition) based on the above distribution. For a givensequencexn, let T n

ǫ (PU|X |xn) denote the set of sequencesun

such that(un, xn) is jointly typical based on the distributionPXU .

1) Codebook Generation

• Generate2nR i.i.d. codewordsun(t) according toP (un) =

∏n

i=1 PU (ui) for the fixed marginalprobabilityPU as defined, in whicht ∈ [1, 2nR].

• Generate2nR0 i.i.d codewordsx′n0 (w0) according

to P (x′n0 ) =

∏n

i=1 PX′

0(x′

0i) for the fixed marginalprobabilityPX′

0as defined, in whichw0 ∈ [1, 2nR0].

• Generate2nR1 i.i.d. codewordsxn1 (w1) according

to P (xn1 ) =

∏n

i=1 PX1(x1i) for the fixed marginal

probabilityPX1as defined, in whichw1 ∈ [1, 2nR1].

2) Encoding

• Encoder at the helper: Givenw0, map w0 intox

′n0 (w0). For each x

′n0 (w0), select t such that

(un(t), sn1 , x′n0 (w0)) ∈ T n

ǫ (PS1PX′

0PU|S1X

′

0).

If un(t) cannot be found, set t = 1.Then map (sn1 , u

n(t), x′n0 (w0)) into

xn0 = f

(n)0 (x

′n0 (w0), s

n1 , u

n(t)). Based on therate distortion type of argument [29, Sec. 10.5]or the Covering Lemma [31, Sec. 3.7], it can beshown that suchun(t) exists with high probabilityfor largen if

R > I(U ;S1X′0). (48)

• Encoder 1: Givenw1, mapw1 into xn1 (w1).

3) Decoding

• Decoder 0: Given yn0 , find w0 such that(x

′n0 (w0), y

n0 ) ∈ T n

ǫ (PX′

0Y0). If no or more

than onew0 can be found, declare an error. Itcan be shown that the decoding error is small forsufficient largen if

R0 6 I(X ′0;Y0). (49)

The proof for the above bound (and the similarbounds in the sequel) follows the standard tech-niques as given in [29, Sec. 7.7], and hence isomitted.

• Decoder 1: Givenyn1 , find a pair(t, w1) such that(un(t), xn

1 (w1), yn1 ) ∈ T n

ǫ (PUX1Y1). If no or more

than one such pair can be found, then declare anerror. It can be shown that decoding is successfulwith small probability of error for sufficiently largen if the following conditions are satisfied

R1 6I(X1;Y1|U), (50)

R 6I(U ;Y1|X1), (51)

R1 + R 6I(UX1;Y1). (52)

We note that (51) corresponds to the decoding error forthe indext, which is not the message of interest. Hence, thebound (51) can be removed. Hence, combining (48), (49), (50),and (52), and eliminatingR, we obtain the desired achievableregion.

17

APPENDIX CPROOF OFPROPOSITION4

The single rate bounds follow from Proposition 1 and thepoint-to-point channel capacity. For the sum rate bound, basedon Fano’s inequality, we have

n(R1 +R2)

6I(W1;Yn1 ) + I(W2;Y

n2 ) + nǫn

=h(Y n1 )− h(Y n

1 |W1) + h(Y n2 )− h(Y n

2 |W2) + nǫn(a)=h(Y n

1 )− h(Y n1 |W1X

n1 ) + h(Y n

2 )− h(Y n2 |W2X

n2 ) + nǫn

=h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 )

+ h(Y n2 )− h(Xn

0 +Nn2 ) + nǫn

6h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 |Xn0 +Nn

1 )

+ h(Y n2 )− h(Xn

0 +Nn2 ) + nǫn

where (a) follows from the fact thatXn1 is a function ofW1,

andXn2 is a function ofW2, and they are independent from

Xn0 , state and noise. As argued in Appendix A, settingNn

1 =Nn

2 does not change the capacity region. Thus,

n(R1 +R2)

6h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 , Xn0 +Nn

1 ) + h(Y n2 ) + nǫn

=h(Y n1 )− h(Sn

1 , Xn0 +Nn

1 ) + h(Y n2 ) + nǫn

=h(Y n1 )− h(Sn

1 )− h(Xn0 +Nn

1 |Sn1 ) + h(Y n

2 ) + nǫn

6h(Y n1 )− h(Sn

1 )− h(Xn0 +Nn

1 |Sn1 , X

n0 ) + h(Y n

2 ) + nǫn(b)=h(Xn

0 +Xn1 + Sn

1 +Nn1 )− h(Sn

1 )− h(Nn1 )

+ h(Xn0 +Xn

2 +Nn1 ) + nǫn

6n

2log 2πe(P1 + P0 + 2

√

P0Q1 +Q1 + 1)− n

2log(2πeQ1)

+n

2log 2πe(P0 + P2 + 1)− n

2log(2πe) + nǫn

=n

2log

(

P1 + P0 + 2√P0Q1 +Q1 + 1

Q1

)

+n

2log(P0 + P2 + 1) + nǫn

→n

2log(P0 + P2 + 1) as Q1 → ∞

where (b) follows from the fact thatXn0 andSn

1 are indepen-dent fromNn

1 .

APPENDIX DPROOF OFPROPOSITION5

We use random codes and fix the following joint distribu-tion:

PS1UV X0X1X2Y1Y2=PV US1

PX0|V US1

· PX1PX2

PY1|X0X1S1PY2|X0X2

. (53)

Let T nǫ (PS1UV X0X1X2Y1Y2

) denote the strongly jointǫ-typicalset based on the above distribution.

1) Codebook Generation

• Generate2nR1 i.i.d. codewordsun(t) accordingto P (un) =

∏n

i=1 PU (ui) for the fixed marginalprobabilityPU as defined, in whicht ∈ [1, 2nR1 ].

• Generate2nR2 i.i.d. codewordsvn(k) accordingto P (vn) =

∏n

i=1 PV (vi) for the fixed marginalprobabilityPV as defined, in whichk ∈ [1, 2nR2].


to P (xn1 ) =

∏n

i=1 PX1(x1i) for the fixed marginal

probabilityPX1as defined, in whichw1 ∈ [1, 2nR1].


to P (xn2 ) =

∏ni=1 PX2

(x2i) for the fixed marginalprobabilityPX2

as defined, in whichw2 ∈ [1, 2nR2].

2) Encoding

• Encoder at the helper: Givensn1 , find t, such that(un(t), sn1 ) ∈ T n

ǫ (PS1U ). Such un(t) exists withhigh probability for largen if

R1 > I(U ;S1). (54)

• For each t selected, select k, such that(vn(k), un(t), sn1 ) ∈ T n

ǫ (PV US1). Such vn(k)

exists with high probability for largen if

R2 > I(V ;S1U). (55)

• Map (sn1 , un, vn) into xn

0 = f(n)0 (un(t), vn(k), sn1 ).



3) Decoding

• Decoder 1: Given yn1 , find (w1, t) such that(xn

1 (w1), un(t), yn1 ) ∈ T n

ǫ (PX1UY1). If no or more

than onew1 can be found, declare an error. One canshow that the decoding error is small for sufficientlargen if

R1 6 I(X1;Y1U) (56)

R1 + R1 6 I(X1U ;Y1). (57)

• Decoder 2: Givenyn2 , find (w2, k) such that(xn

2 (w2), vn(k), yn2 ) ∈ T n

ǫ (PX2V Y2). If no or more

than onew2 can be found, declare an error. One canshow that the decoding error is small for sufficientlargen if

R2 6 I(X2;Y2V ), (58)

R2 + R2 6 I(X2V ;Y2). (59)

Combining (54)-(59), and eliminatingR1 andR2, we obtainthe desired achievable region.

APPENDIX EPROOF OFPROPOSITION10

The bounds onR1 andR2 follow from the point-to-pointchannel capacity. For the sum rate bound, based on the Fano’s

18

inequality, we have

n(R1 +R2)

6I(W1;Yn1 ) + I(W2;Y

n2 ) + nǫn

=h(Y n1 )− h(Y n

1 |W1) + h(Y n2 )− h(Y n

2 |W2) + nǫn(a)=h(Y n

1 )− h(Y n1 |W1X

n1 ) + h(Y n

2 )− h(Y n2 |W2X

n2 ) + nǫn

=h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 ) + h(Y n2 )

− h(Xn0 + Sn

2 +Nn2 ) + nǫn

6h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 |Xn0 +Nn

1 ) + h(Y n2 )

− h(Xn0 + Sn

2 +Nn2 |Xn

0 +Nn2 , X

n0 + Sn

1 +Nn1 )

+ h(Xn0 +Nn

1 )− h(Xn0 +Nn

1 ) + nǫn

where (a) follows from the fact thatXn1 is a function ofW1,

Xn2 is a function ofW2, and they are independent fromXn

0 ,Sn1 , Sn

2 , Nn1 andNn

2 . Since receivers 1 and 2 decode basedon the marginal distributions only, settingNn

1 = Nn2 does not

affect the channel capacity. Therefore,

n(R1 +R2)

6h(Y n1 )− h(Xn

0 + Sn1 +Nn

1 , Xn0 + Sn

2 +Nn1 , X

n0 +Nn

1 )

+ h(Y n2 ) + h(Xn

0 +Nn1 ) + nǫn

=h(Y n1 )− h(Sn

1 , Sn2 , X

n0 +Nn

1 )

+ h(Y n2 ) + h(Xn

0 +Nn1 ) + nǫn

=h(Y n1 )− h(Sn

1 )− h(Sn2 )− h(Xn

0 +Nn1 |Sn

1 , Sn2 )

+ h(Y n2 ) + h(Xn

0 +Nn1 ) + nǫn

6h(Y n1 )− h(Sn

1 )− h(Sn2 )− h(Xn

0 +Nn1 |Sn

1 , Sn2 , X

n0 )

+ h(Y n2 ) + h(Xn

0 +Nn1 ) + nǫn

(b)=h(Y n

1 )− h(Sn1 )− h(Sn

2 )− h(Nn1 )

+ h(Y n2 ) + h(Xn

0 +Nn1 ) + nǫn

(c)=

n∑

i=1

[h(Y1i|Y i−11 )− h(S1i)− h(S2i)− h(N1i)

+ h(Y2i|Y i−12 ) + h(X0i +N1i|X i−1

0 +N i−11 )] + nǫn

6

n∑

i=1

[h(Y1i)− h(S1i)− h(S2i)− h(N1i)

+ h(Y2i) + h(X0i +N1i)] + nǫn

=n∑

i=1

[h(X0i +X1i + S1i +N1i)− h(S1i)

− h(S2i)− h(N1i) + h(X0i +X2i + S2i +N1i)

+ h(X0i +N1i)] + nǫn (60)

We then derive the items respectively. The first term in (60)can be derived as

n∑

i=1

h(X0i +X1i + S1i +N1i)

(d)

61

2

n∑

i=1

log 2πe(E(X0i +X1i + Si +Ni)2)

61

2

n∑

i=1

log 2πe(

E[X20i] + E(X0iSi)

+ E[S2i ] + E[X2

1i] + E[N2i ]))

(e)

6n

2log 2πe

(

1

n

n∑

i=1

E[X20i] +

2

n

n∑

i=1

E(X0iSi)

+1

n

n∑

i=1

E[S2i ] +

1

n

n∑

i=1

E[X21i] +

1

n

n∑

i=1

E[N2i ])

)

(f)

6n

2log 2πe

(

P0 +Q+ P1 + 1 +2

n

n∑

i=1

E(X1iSi)

)

6n

2log 2πe

(

P0 + P1 +Q+ 1 + 2√

P0Q)

(61)

where (d) follows from the fact that the Gaussian distributionmaximizes the entropy given the variance of the randomvariable, (e) follows from the concavity of the logarithmfunction and Jensen’s inequality, and(f) follows from thepower constraints. Similarly, we have

n∑

i=1

h(X0i +X2i + S1i +N1i)

6n

2log 2πe(P2 + P0 + 2

√

P0Q2 +Q2 + 1)

n∑

i=1

h(X0i +N1i) 6n

2log(P0 + 1)

And hence, we have

n(R1 +R2)

6n

2log 2πe(P1 + P0 + 2

√

P0Q1 +Q1 + 1)

− n

2log(2πeQ1)−

n

2log(2πeQ2)−

n

2log(2πe)

+n

2log 2πe(P2 + P0 + 2

√

P0Q2 +Q2 + 1)

+n

2log 2πe(P0 + 1) + nǫn

6n

2log

(

P1 + P0 + 2√P0Q1 +Q1 + 1

Q1

)

+n

2log

(

P2 + P0 + 2√P0Q2 +Q2 + 1

Q2

)

+n

2log(P0 + 1) + nǫn

→n

2log(P0 + 1) as Q1 → ∞, Q2 → ∞

where (b) follows from the fact thatXn0 , Sn

1 and Sn2 are

independent fromNn1 .

19

APPENDIX FPROOF OFTHEOREM 6

The proof contains two parts: 1. we first show that ifP1 + P2 > P0 + 1, then the sum capacity can be obtained; 2.we further characterize the time allocation parametersγ thatachieves the sum capacity.

1. For a givenP0, we consider the following two cases.a). If the power constraint satisfiesP1 + P2 = P0 + 1,

by applying Proposition 11, and by settingγ = P1

P1+P2

, thepoint(R1, R2) = ( P1

2(P1+P2)log(1+P0),

P2

2(P1+P2)log(1+P0))

is achievable, which achieves the sum rate outer bound inProposition 10.

b). If P1+P2 > P0+1, we set the actual transmission powerP1 andP2 of transmitters 1 and 2 to satisfyP1+ P2 = P0+1,P1 6 P1 and P2 6 P2. Then following a), the sum capacityis obtained.

2. In order for each transmitter to achieve the sum capacityduring its own transmission slot, (40) together with (41a) and(41b) imply that

P1

γ> P0 + 1 (62)

P2

1− γ> P0 + 1. (63)

It is clear that (62) implies

γ 6P1

P0 + 1,

and (63) implies

γ > 1− P2

P0 + 1.

Considering0 ≤ γ ≤ 1, we obtain the desired bounds onγ.

APPENDIX GPROOF OFPROPOSITION14

The individual rate can be bounded by the point-to-pointchannel capacity. We then bound the sum rate. By followingthe Fano’s inequality, we have

K∑

k=1

nRk

6

K∑

k=1

I(Wk;Ynk )

=

K∑

k=1

[h(Y nk )− h(Y n

k |Wk)]

(a)=

K∑

k=1

[h(Y nk )− h(Y n

k |WkXnk )]

=

K∑

k=1

[h(Y nk )− h(Xn

0 + Snk +Nn

k )]

6

K∑

k=1

[h(Y nk )− h(Xn

0 + Snk +Nn

k |Xn0 +Nn

1 ,

Xn0 + Sn

k−1 +Nnk−1, . . . , X

n0 + Sn

1 +Nn1 )]

+ h(Xn0 +Nn

1 )− h(Xn0 +Nn

1 )

=

K∑

k=1

h(Y nk )

− h(Xn0 + Sn

1 +Nn1 , . . . , X

n0 + Sn

K +NnK , Xn

0 +Nn1 )

+ h(Xn0 +Nn

1 )

where (a) follows from thatXnk is function ofWk, and they are

independent fromXn0 , state and noise. Because the decoders

decode based on the marginal distribution only, we can setNn

k = Nn1 for k = 1, . . . ,K, therefore,

K∑

k=1

nRi

6

K∑

k=1

h(Y nk )− h(Sn

1 , . . . , SnK , Xn

0 +Nn1 ) + h(Xn

0 +Nn1 )

=

K∑

k=1

[h(Y nk )− h(Sn

k )]

− h(Xn0 +Nn

1 |Sn1 , . . . , S

nK) + h(Xn

0 +Nn1 )

6

K∑

k=1

[h(Y nk )− h(Sn

k )]

− h(Xn0 +Nn

1 |Sn1 , . . . , S

nK , Xn

0 ) + h(Xn0 +Nn

1 )

(b)=

K∑

k=1

[h(Xn0 +Xn

k + Snk +Nn

k )− h(Snk )]

− h(Nn1 ) + h(Xn

0 +Nn1 )

6

K∑

k=1

[

n

2log 2πe

(

Pk + P0 + 2

n∑

i=1

E(X0iSki) +Qk + 1

)

− n

2log(2πeQk)

]

− n

2log(2πe) +

n

2log 2πe(P0 + 1)

6

K∑

k=1

n

2log

(

Pk + P0 + 2√P0Qk +Qk + 1

Qk

)

+n

2log(P0 + 1)

→n

2log(P0 + 1) asQk → ∞

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Ruchen Duan(S’11, M’15) received the B.E. degree from Beijing Universityof Posts and Telecommunications, Beijing, China in 2010, and the Ph.D.degree in Electrical Engineering from Syracuse Universit in 2015. Shereceived the Syracuse University Fellowship Award for years 2010-2012. Sheis now a senior engineer at Samsung Semiconductor Inc.. Ruchen’s researchinterests focus on information theory and wireless communications.

Yingbin Liang (S’01, M’05) received the Ph.D. degree in Electrical En-gineering from the University of Illinois at Urbana-Champaign in 2005. In2005-2007, she was working as a postdoctoral research associate at PrincetonUniversity. In 2008-2009, she was an assistant professor atthe Departmentof Electrical Engineering at the University of Hawaii. Since December 2009,she has been on the faculty at Syracuse University, where sheis an associateprofessor. Dr. Liang’s research interests include information theory, wirelesscommunications and networks, and machine learning.

Dr. Liang was a Vodafone Fellow at the University of Illinoisat Urbana-Champaign during 2003-2005, and received the Vodafone-U.S. FoundationFellows Initiative Research Merit Award in 2005. She also received the M.E. Van Valkenburg Graduate Research Award from the ECE department,University of Illinois at Urbana-Champaign, in 2005. In 2009, she received theNational Science Foundation CAREER Award, and the State of Hawaii Gover-nor Innovation Award. More recently, her paper received the2014 EURASIPBest Paper Award for the EURASIP Journal on Wireless Communications andNetworking. She is currently serving as an Associate Editorfor the ShannonTheory of the IEEE Transactions on Information Theory.

Ashish Khisti (M’08) is an Associate Professor at the University of Torontoand holds a Canada Research Chair in Wireless Networks. He obtained hisBASc degree from the Engineering Sciences program (Electrical EngineeringOption) at the same university in 2002, and his SM and PhD degrees fromthe Massachusetts Institute of Technology (MIT) in Electrical Engineeringand Computer Science in 2004 and 2009 respectively. His research interestsinclude Information Theory, Physical Layer Security and Error Control Codingfor Multimedia Applications. He also actively consults telecommunicationcompanies. He is a recipient of the HP-IRP award from Hewlett-Packard andan Ontario Early Researcher Award. He presently serves as a Associate Editorin Shannon Theory for the IEEE Transactions on Information Theory.

Shlomo Shamai (Shitz)(S’80, M’82, SM’88, F’94) received the B.Sc., M.Sc.,and Ph.D. degrees in electrical engineering from the Technion—Israel Instituteof Technology, in 1975, 1981 and 1986 respectively.

During 1975-1985 he was with the Communications Research Labs, in thecapacity of a Senior Research Engineer. Since 1986 he is withthe Departmentof Electrical Engineering, Technion—Israel Institute of Technology, where heis now a Technion Distinguished Professor, and holds the William FondillerChair of Telecommunications. His research interests encompasses a widespectrum of topics in information theory and statistical communications.

Dr. Shamai (Shitz) is an IEEE Fellow, a member of the Israeli Academyof Sciences and Humanities and a foreign member of the US NationalAcademy of Engineering. He is the recipient of the 2011 Claude E. ShannonAward and the 2014 Rothschild Prize in Mathematics/Computer Sciences andEngineering.

He has been awarded the 1999 van der Pol Gold Medal of the UnionRadio Scientifique Internationale (URSI), and is a co-recipient of the 2000IEEE Donald G. Fink Prize Paper Award, the 2003, and the 2004 jointIT/COM societies paper award, the 2007 IEEE Information Theory SocietyPaper Award, the 2009 and 2015 European Commission FP7, Network ofExcellence in Wireless COMmunications (NEWCOM++, NEWCOM#) BestPaper Awards, the 2010 Thomson Reuters Award for International Excellencein Scientific Research, the 2014 EURASIP Best Paper Award (for theEURASIP Journal on Wireless Communications and Networking), and the2015 IEEE Communications Society Best Tutorial Paper Award. He is alsothe recipient of 1985 Alon Grant for distinguished young scientists and the2000 Technion Henry Taub Prize for Excellence in Research. He has servedas Associate Editor for the Shannon Theory of the IEEE Transactions onInformation Theory, and has also served twice on the Board ofGovernorsof the Information Theory Society. He has served on the Executive EditorialBoard of the IEEE Transactions on Information Theory

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