Capacity of a Class of Cognitive Radio Channels: Interference Channels with Degraded Message Sets

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1 Capacity of a Class of Cognitive Radio Channels: Interference Channels with Degraded Message Sets Wei Wu, Sriram Vishwanath and Ari Arapostathis Abstract This paper is motivated by two different scenarios. The first is a cognitive radio system where a cognitive radio knows a dumb radio s message and the second is a sensor network in a correlated field where sensors possessing a nested message structure assist one another s in information transmission. Both scenarios are modeled using the framework of discrete memoryless interference channels with degraded message sets (IFC-DMS, a setting where one of the two transmitters in an interference channel knows both the messages to be conveyed to the receivers. Both inner and outer bounds are provided in this paper for a class of IFC-DMS channels. The case of the Gaussian interference channels with degraded message sets is also investigated. In this case, achievability and converse arguments are presented for a class of weak interference channels, resulting in a characterization of this class capacity region. Index Terms Cognitive radio, Network information theory, Interference channel, Dirty-paper coding (a Data collection through IFC I. INTRODUCTION The interference channel (IFC is a basic building block of most wireless networks, and is thus considered a fundamentally important channel from both a theoretical and a practical perspective. However, the capacity region of this channel remains an open problem, with only some special cases being solved to date. Our goal in this paper is to investigate the capacity of a class of IFCs where one transmitter has full knowledge of the other transmitter s message in both the discrete memoryless and Gaussian cases. We term this class of channels interference channels with degraded message sets. IFCs with degraded message sets arise in many fairly important scenarios in wireless networks. The first is the cognitive radio channel introduced in []. In this model, a cognitive transmitter gains full knowledge of another dumb transmitter s message. Each transmitter has a separate receiver associated with it. In this setting, the cognitive transmitreceive pair exploit the cognitive transmitter s knowledge of both messages to improve overall system performance. The second motivation for this problem lies in sensor networks as illustrated in Figure (a. In this setting, Sensor A has better This research was supported in part by NSF grants CCF-04488,CCF , ECS-0807 and ECS-05448, in part by THECB ARP , in part by the Office of Naval Research through the Electric Ship Research and Development Consortium, and a grant from Freescale Semiconductor Corporation. Wei Wu was also supported by the Hemphill- Gilmore Student Endowed Fellowship through the University of Texas at Austin. The preliminary result of this paper was presented in part at the Conference on Information Scienceand Systems, Princeton, NJ, March, 006. The authors are with Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 787, USA ( {wwu,sriram,ari}@ece.utexas.edu. Concurrent work in [] also analyzes this channel. Fig.. (b Cascaded with degraded broadcast channel Two applications of IFCs with degraded message sets sensing capability than Sensor B and thus can detect both events, while Sensor B can only detect one of them. In this setting, we assume that each sensor is aware of the capabilities of the other sensor and that the collected data needs to be sent to different receivers. A third and final motivation for this problem arises from the network shown in Figure (b, in which the nodes B and C are constrained to fully decode the messages received from A. If the channel from A to B and C is a degraded broadcast channel, then the resulting channel from B and C to the receivers resembles an IFC with degraded message sets. A central aspect of this channel is that the degraded structure of the messages allows the two transmitters to cooperate. Cooperation among transmitters to improve achievable rates has received considerable attention [3] [5], particularly in an IFC setting [6] [7]. Our goal in this paper is twofold: first, to develop a cooperative encoding scheme that, for a class of discrete memoryless channels, achieves the capacity of the channel; second, to characterize the capacity region of a class of Gaussian IFC with degraded message sets with weak interference.

2 We begin this paper by discussing the discrete memoryless case, where we find inner and outer bounds on the capacity of a class of IFCs with degraded message sets. Further, we determine conditions under which these two bounds meet. Next, we proceed to analyze the Gaussian IFC with degraded message sets. For a class of these channels where the inherent interference structure is weak, we determine the capacity region. The rest of the paper is organized as follows. In Section II, basic notations and definitions are introduced. The main results are presented in Section III, including the capacity region of a class of discrete memoryless IFCs and Gaussian weak IFCs with degraded message sets. Numerical results of the Gaussian case are shown in Section IV. Detailed proofs for Section III are given in Section V. Finally we conclude the paper with Section VI. II. NOTATIONS AND PRELIMINARIES A. Channel Model and Definitions We adopt the following notational conventions. Random variables (RVs are denoted by capital letters, and their realizations by the respective lower case letters. X n m denotes the random vector (X m,...,x n, and X n denotes the random vector (X,...,X n. We use the notation X Y Z to denote that X and Z are conditionally independent given Y. recently by Maric, Yates and Kramer in [6], each transmitter has not only its own private message W t, but also the common message W 0 shared by all the transmitters. Thus the message set for Transmitter t is M t {W 0,W t }. In this paper, we consider IFCs with degraded message sets (IFC-DMS. For a two-user IFC-DMS, the message set of one transmitter is a strict subset of the other. For example, Figure corresponds to the message sets, {W } M M {W,W }, ( for which the capacity region is denoted as C T IFC to indicate that Transmitter knows both messages. Note that Receiver t desires to decode Message W t alone. Also note that M t is the set of messages available to Transmitter t, which is in general a superset of W t. Similarly, C T IFC denotes the capacity region of an IFC with Transmitter knowing both messages, namely {W } M M {W,W }. In recent works in the literature (e.g., [7], an IFC-DMS has also been referred to as an IFC with unidirectional cooperation one transmitter knows the other s message and thus can enhance the achievable rate region. The definition of degradedness of message sets can be further generalized to a K-user IFC: there are K messages W t, t,...,k, and W t is to be decoded by receiver t while Transmitter t has a set of messages M t. The message sets are degraded if there exists a permutation {σ k,k,...,k} of {,,...,K}, such that M σ M σk. In general, the capacity region of an IFC is an open problem and is only known for certain classes of IFCs, which include the so-called strong interference channels that satisfy Fig.. The model of an IFC with degraded message sets A two-user IFC (X, X, Y, Y,p is a channel with two input alphabets X, X, output alphabets Y, Y, and transition probability p(y,y x,x. It is assumed that the channel is memoryless, namely n p(y n,y n x n,x n p(y,i,y,i x,i,x,i. Transmitter t sends a message W t having M t bits, to Receiver t, in n channel usages at rate R t M t/n bits per usage. A (R,R,n,P e,,p e, code is defined as any code achieving the rate pair (R,R with block size n and decoding error probability P (n e,t, t,. The capacity region C IFC is the closure of the set of rate pairs (R,R, for which the receivers can decode their messages with error probability P (n e,t 0 for t,, as the block size n. In the classic IFC framework described above, each transmitter has its own message set M t {W t }, where W t {,,..., nrt } denotes the private message to Receiver t. For an IFC with common information, a model proposed I(X ;Y X I(X ;Y X I(X ;Y X I(X ;Y X, for all product distributions on the inputs X and X. The capacity region in this case coincides with the capacity region of a compound IFC which is the union of two compound multiple access channels (MACs, as discovered by Ahlswede [8]. Maric, Yates and Kramer find the capacity region of strong IFCs with common information [6], and with degraded message sets [7]. An achievable region for IFCs with degraded message sets in a more general setting can be found in []. B. Gaussian IFCs One of our main interests in this paper is the Gaussian IFC, in which the alphabets of inputs and outputs are real numbers and the outputs are linear combinations of input signals and white Gaussian noise. A Gaussian IFC is defined by Y X + ax + Z Y bx + X + Z, where a, b are real numbers, and Z, Z are independent, zeromean, unit-variance Gaussian random variables. Furthermore, ( (3

3 3 the transmitters are subject to average power constraints: N lim E[X N N tn] P t, t,. n The capacity region of the standard Gaussian IFC has been characterized in the following cases: (i when a b 0 (trivial; (ii either a 0, b or a, b 0; and (iii if a and b, in which case the strong interference conditions in ( are satisfied. The capacity of an IFC with strong interference is the set of (R,R satisfying (see [9], [0] 0 R log( + P (4a 0 R log( + P (4b 0 R + R log(p + a P + (4c 0 R + R log(b P + P +. (4d Note on terminology: when either 0 a or 0 b is satisfied, we say that the Gaussian IFC satisfies the weak interference condition. Achievable rate regions [0] [] and outer bounds [], [3] [5] are known for this scenario, but a characterization of the region is yet to be obtained. A recent outer bound by Kramer in [5] is given by (R,R satisfying (4a (4b, and R + R [ log (P + (P + a ] P + min(a,p + R + R log [ (P + (P + b P + min(b,p + Let the capacity region of a Gaussian IFC-DMS (i.e., Figure 3 be denoted by C T. In this paper, we characterize the capacity region for the class of Gaussian weak interference channels, C T, with b for any real valued a. ]. (5 A. Achievable regions and outer bounds for discretememoryless IFC-DMS In this subsection, we provide achievable regions and outer bounds for general discrete memoryless IFC-DMS and then specialize to a class of IFC-DMS, whose capacity region is established. We first present an achievable region for general discrete memoryless IFC-DMS. Definition 3.: Define R in to be the convex hull of all rate pairs (R,R satisfying R I(V ;Y I(V ;U,X R I(U,X ;Y, over all probability distributions p(x,x,u,v,y,y that factor as p(u,x p(v u,x p(x v,u,x p(y,y x,x. The following proposition gives the achievable region of IFC with Transmitter knowing both messages as in Figure using the Gel fand-pinsker coding scheme [6]. Proposition 3.: The capacity region of the discrete memoryless IFC-DMS in ( satisfies R in C T IFC. The proof of Proposition 3. follows that of the Gel fand- Pinsker coding scheme in [6], thus it is omitted. Note that (U,X are considered as the random parameters for the channel between X and Y. An outer bound is stated next. Definition 3.: Define R o to be the convex hull of all rate pairs (R,R satisfying R I(X ;Y X R I(U,X ;Y R + R I(X ;Y U,X + I(U,X ;Y, over all probability distributions p(x,x,u,y,y that factor as p(u,x p(x u,x p(y,y x,x. (6 Theorem 3.: The capacity region of the discrete memoryless IFC-DMS in ( satisfies Fig. 3. The Gaussian interference channel with degraded message sets. III. MAIN RESULTS In this section, we first obtain inner and outer bounds for the discrete-memoryless IFC-DMS, and then determine the capacity region of the Gaussian IFC-DMS with weak interference. The proof of Theorem 3. is provided in Sec- Proof: tion V-A. C T IFC R o. Both Proposition 3. and Theorem 3. hold for the general IFCs. However, as seen, the achievable region obtained in Proposition 3. does not, in general, meet the outer bound in Theorem 3.. Next we investigate the scenarios under which the capacity region of IFC-DMS can be obtained under additional assumptions.

4 4 Definition 3.3: Define the rate region R to be the convex hull of all rate pairs (R,R satisfying R I(X ;Y U,X R I(U,X ;Y, over all probability distributions p(x,x,u,y,y that factor as p(u,x p(x u,x p(y,y x,x. It is not difficult to see that R is a subset of R o, namely, R R o. As shown in Figure 4, for a fixed auxiliary random variable U, since I(X ;Y U,X I(X ;Y X, the rate region defined by (6 in R o corresponds to the area OABCD in Figure 4 while the rate region defined by (7 in R corresponds to the shaded rectangle OABE in Figure 4. (7 To show R o R, it is enough to show the point C is in R (due to the convexity of R. Under Assumption 3., the R -coordinate of the point C, R,C, satisfies R,C I(U,X ;Y + I(X ;Y U,X I(X ;Y X I(U,X ;Y + H(Y U,X H(Y X I(X ;Y + I(U;Y X I(U;Y X I(X ;Y. Let U be a constant, (R,R ( I(X ;Y X,I(X ;Y R, thus C R. Since A,B,C,D R, the polygon OABCD is a subset of R, therefore R o R. Next we show R is also achievable under a further assumption on the channel together with Assumption 3., thus the capacity region is established. R R +R I(U,X ;Y +I(X ;Y U,X Assumption 3.: For an IFC, I(X ;Y I(X ;Y (9 is satisfied over all input distributions to the channel p(x,x. I(U, X ; Y I(X ; Y O A B E C D I(X ;Y U, X I(X ;Y X Fig. 4. Comparison between rate regions R o and R. Under the following assumption on the channel, we can show that R o R. Assumption 3.: I(U;Y X I(U;Y X (8 is satisfied for all auxiliary random variables U, such that the probability distribution p(x,x,u,y,y can factor as p(u,x p(x u,x p(y,y x,x. Proposition 3.3: Under Assumption 3., R o R, thus the capacity region of the discrete memoryless IFC-DMS satisfying Assumption 3. satisfies C T IFC R. Proof: Because the auxiliary random variable U is over a similar set of probability distributions, it is enough to compare rate regions defined by inequalities in (6 and (7. As shown in Figure 4, because I(X ;Y U,X I(X ;Y X, clearly R R o. R Theorem 3.4: The capacity region of discrete memoryless IFC-DMS satisfying both Assumption 3. and Assumption 3. is C T IFC R. Proof: Proof is given in Section V-B. B. Gaussian IFC-DMS with weak interference Next we investigate Gaussian IFC-DMS with weak interference when b. Note for a general IFC that the two receivers cannot cooperate, the capacity region is the same as the one with the same marginal output p(y x,x, p(y x,x. The same result holds for Gaussian IFC-DMS as stated by the following lemma. Lemma 3.5: The capacity region of a Gaussian IFC-DMS given by (3 when b, is the same as that of a Gaussian IFC-DMS defined as where Ỹ Y + ax Ỹ by + X + Z, Y X + Z, (0 and Z is Gaussian distributed with variance b and independent of Z, i.e., given X, X Ỹ Ỹ. ( Proof: This follows directly from the arguments in [7] (0, Pg. 454 or [3]. Before determining its capacity region of Gaussian IFC- DMS, we first present an outer bound tailored for Gaussian IFC-DMS with weak interference.

5 5 Lemma 3.6: The capacity region of weak interference Gaussian IFC-DMS with b satisfies C T R. Proof: The proof is given in Section V-C. Theorem 3.7: The capacity region C T of the Gaussian IFC-DMS with Transmitter knowing both messages, when b, is the set of all rate pairs (R,R such that, for 0 α, R log ( + αp ( R ( log + hσht + b αp ( log + P b + b ( αp P + P + b (3 αp In (3, h is the vector [b ], and Σ is a covariance with diagonal elements equaling ( αp and P respectively. Proof: The proof is given in Section V-D. By swapping the parameters of two transmitters, the capacity region C T can be obtained as the following corollary. The DPC achievable region for the encoding sequence W, W is { R DPC (R,R : R log( + P, + ( αp R log( + P b + b ( αp P + P, } for 0 α. (6 We consider a symmetric Gaussian IFC with P P 6 and a b 0.3. The rate units are bits per channel use. In Figure 5, we compare the capacity region of Gaussian IFC-DMS C T with the dirty-paper coding regions R DPC and R DPC, and the outer bound of Gaussian IFCs in (5. We observe that the capacity region of Gaussian IFC-DMS is strictly larger than the outer bound of Gaussian IFCs, and the gap between these two indicates the performance improvement by allowing encoders to cooperate partially. The point at α 0 corresponds to full cooperation between two encoders in Gaussian IFC-DMS to transmit the message W, and it meets the capacity region of Gaussian BCs. The corresponding rate for W is equal to.97 bps, while Gaussian IFC without cooperation can only achieve.404 bps. Comparison of rate regions Corollary 3.8: The capacity region of the Gaussian IFC with Transmitter knowing both messages, C T, when a, is the set of all rate pairs (R,R such that, for 0 β, R log( + P + a ( βp P + a P + a βp R log( + βp. (4 R outer bound T C R DPC R DPC IV. NUMERICAL RESULTS In this section, we use numerical results to compare the capacity region of Gaussian IFC-DMS with achievable rate regions of Gaussian BCs and the outer bound of Gaussian IFCs. By allowing full cooperation between two transmitters, Gaussian IFCs become Gaussian multi-antenna (MIMO BC channels. According to the existing literature [8], dirtypaper coding (DPC [9] optimizes the capacity region of the Gaussian MIMO BC channel [0] []. If, in the dirty-paper coding strategy, W is encoded first and W second, the rates achieved are given by: { R DPC (R,R : R log( + αp, R ( + log P b + b ( αp P + P + b, αp } for 0 α. ( R Fig. 5. The capacity region of Gaussian IFC-DMS with P P 6, a b 0.3, achievable rate regions of Gaussian BC, and the outer bound of Gaussian IFC in [5]. A. Proof of Theorem 3. V. PROOFS Theorem 3. can be proved by adapting Marton s BC outer bound. For a (R,R,n,P (n (n e,,p e, code with decoding error P (n e,i 0, as n, we define the auxiliary random variable U by U i (W,Y i,y n,i+,x i,x n,i+. (7

6 6 Applying Fano s inequality [7], for each message W t, t,, we have H(W t Y n t nr t P (n e,t + H(P (n e,t nε (n t, (8 where ε (n t 0, as P (n e,t 0. Moreover, because Transmitter has no information about the message W, X n is independent of W, and the following relation holds H(W W,X n H(W. (9 To prove the converse, we need the following lemma: Lemma 5. ( []: For any random variable T, the following equality holds, I(Y n,i+;y,i Y i,t I(Y i ;Y,i Y n,i+,t. First we prove the outer bounds for R and R in (6. We have and nr I(W ;Y n X n (0 I(W ;Y,i Y i,x n [ H(Y,i X,i H(Y,i W,X n,y i,x,i ] I(X,i ;Y,i X,i, nr I(W ;Y n ( I(W ;Y,i Y,i+ n I(W,Y n,i+;y,i I(U i,x,i ;Y,i, where (0 and ( are from Fano s inequality in (8. Next, we prove the outer bound for the sum rate R + R in (6. We have n(r + R I(W ;Y n W,X n + I(W ;Y n (a [ I(W ;Y,i W,X n,y i + I(W ;Y,i Y n,i+ ] (b [ I(W,Y,i+;Y n,i W,X n,y i I(Y n,i+;y,i W,W,X n,y i + I(W,Y n,i+,x n ;Y,i I(X n ;Y,i W,Y n,i+ I(Y n,i+;y,i ]. (c Note (a is due to Fano s inequality in (8 and the conditional entropy relation in (9; the first two terms in (c are from the first term in (b and the third, fourth and fifth terms are from the second term in (b. Since mutual information is nonnegative, by dropping the second, fourth and fifth terms in (c, we obtain n(r + R [ I(W,Y n,i+;y,i W,X n,y i + I(W,Y,i+,X n n ;Y,i ] [ I(W ;Y,i W,Y i,y,i+,x n n + I(Y,i+;Y n,i W,Y i,x n + I(W,Y i,y,i+,x n n ;Y,i I(Y i ;Y,i W,Y,i+,X n n ] (d [ I(W ;Y,i U i,x,i + I(U i,x,i ;Y,i ] (e [ I(X,i ;Y,i U i,x,i + I(U i,x,i ;Y,i ]. (f In this calculation, the second and fourth terms in (d are equal from Lemma 5.; (e is obtained by using the auxiliary random variable U i defined in (7; and (f is true because (W,U i (X,i,X,i (Y,i,Y,i, for all i n. B. Proof of Theorem 3.4 Here we provide a proof for Theorem 3.4, i.e., the achievability of R for IFC-DMS satisfying Assumption 3. and Assumption 3.. Combining (8 and (9, it is easy to see the IFC-DMS must satisfy I(U,X ;Y I(U,X ;Y, (3

7 7 for all probability distributions p(x,x,u,y,y that factor as p(u,x p(x u,x p(y,y x,x. Under (3, we have the following coding scheme based on superposition coding [7]: Code Generation: Fix p(u,x, and generate nr independent codewords of length n at random according to the distribution n p(u i,x,i, for message w {,..., nr }. For each codeword (U n (w,x n (w, generate nr independent codewords X n (w,w according to n p(x,i u i (w,x,i (w, with w {,..., nr }. Encoding: Encoder transmits X n (W. Since Encoder knows both messages, it sends X n (W,W. Decoding: Receiver determines the unique Ŵ such that (U n (Ŵ,X n (Ŵ,Y n is jointly typical, and Receiver determines the unique (Ŵ,Ŵ such that (X n (Ŵ,Ŵ,X n (Ŵ,U n (Ŵ is jointly typical. It is easy to see that the probability of error at Receiver tends to zero, as n, if R I(U,X ;Y, and Receiver can decode W successfully, as n, provided R I(U,X ;Y. Under (3, R I(U,X ;Y I(U,X ;Y. Thus, Receiver can decode W as long as Receiver can do so. With the error probability of W tending to zero, as n, the error probability of W at Receiver goes to zero, provided R I(X ;Y U,X. The above analysis shows that both receivers can decode with the total probability of error tending to zero, if (7 is satisfied. Hence there exists a sequence of good codes with error probability tending to 0. C. Proof of Lemma 3.6 According to the result in Lemma 3.5, when b, the capacity region of any Gaussian IFC-DMS is equal to that of a Gaussian IFC-DMS satisfying that, given X X Y Y. (4 Thus it is enough to prove the outer bound for the Gaussian IFC-DMS satisfying (4. Afterward, the key is to identify the auxiliary random variable. For any (R,R,n,P (n e, P (n e,i 0, as n, we have nr h(w (n,p e, code with decoding error I(W ;Y n (5a I(W ;Y n W,X n (5b I(W ;Y,i W,Y i,x n, (5c where (5a is due to Fano s inequality (8; (5b follows from (9; (5c is due to the chain rule for mutual information. Defining U i (W,Y i,x i, we obtain nr I(W ;Y,i U i,x,i,x,i+ n [ h(y,i U i,x,i,x,i+ n h(y,i U i,x,i,x,i+,w n ] [ h(y,i U i,x,i h(y,i U i,x,i,x n,i+,w,x,i ] (5d [ h(y,i U i,x,i h(y,i U i,x,i,x,i ] (5e [ I(X,i ;Y,i U i,x,i ], where (5d follows from the fact that entropy decreases by adding conditionals, and (5e holds since W (X,i,X,i Y,i. The bound of R in (7 can be derived as follows: nr h(w I(W ;Y n (6a I(W ;Y,i Y i [ h(y,i h(y,i Y i,w,x i ], (6b where (6a follows from Fano s inequality in (8. Due to (4, given X, Y,i Y i Y i because conditioning on (X i,y i, Y i is independent of other random variables (e.g., Y,i, or, I(Y,i ;Y i W,X i I(Y,i ;Y i W,X i. Thus the conditional entropy in (6b satisfies h(y,i Y i,w,x i h(y,i Y i,w,x i h(y,i U i,x,i (7 Combining (6b and (7, we obtain the result in Lemma 3.6. D. Proof of Theorem 3.7 Achievability: The proof of achievability of this rate utilizes the dirty-paper coding strategy. First, we generate a codebook of nr codewords according to N(0,Σ, where Σ is the covariance between Transmitters and. Transmitter devotes

8 8 a fraction ( α of its power P to the transmission of W, while Transmitter devotes its entire power P to this effort. This yields a covariance of the form [ ] ( αp γ Σ (8 γ P The effective interference seen by Receiver is a combination of the signals communicated from both Transmitters and. Since Transmitter knows the exact realization of the message w W, it has non-causal side information on the interference and can completely cancel it out, achieving a rate R log ( + αp, (9 by using a Gaussian codebook with codewords that are correlated with the interference. At Receiver, this Gaussian codebook for W is perceived as additive interference, hence achieving a rate R ( log + hσht + b. (30 αp Maximizing R over γ ( αp P (i.e., keeping Σ positive definite, we find that R attains its maximum when γ ( αp P. and (3 can be achieved. Converse: Since for Gaussian IFC-DMS with b, both Lemma 3.5 and Lemma 3.6 hold. Thus it remains to prove the optimality of Gaussian input for the Gaussian IFC-DMS redefined as Y X + ax + Z Y b(x + Z + X + Z, (3 where Z is a Gaussian distributed random variable with variance b and independent of Z. For this, we need following lemmas: Lemma 5. (Lemma in [3]: Let X,X,...,X k be an arbitrary set of zero-mean random variables with covariance matrix K. Let S by any subset of {,,...,k} and S be its complement. Then where h(x S X S h(x S X S, (X,X,...,X k N(0,K. Lemma 5.3: Let X,X be arbitrarily distributed zeromean random variables and X,X be zero-mean Gaussian distributed random variables with the same covariance matrix as X,X. Then ( E[X X ] E [ (E[X X] ] E [ (X ] Proof: Using Cauchy inequality and the properties of conditional expectation, E[X X ] E[XX ] E [ E[XX X] ] E [ E[X X]X ] ( E [ (E[X X] ] E [ (X ], and the proof of the lemma is complete. Let X,X,U be Gaussian distributed random variables with the same covariance with X,X,U. First we note that and on the other hand, h(y U,X h(y U,X,X h(y X,X log(πe, h(y U,X h(x + ax + Z U,X h(x + Z U,X h(x + Z log( πe( + P. Without loss of generality, we assume that h(y U,X log( πe( + αp, (3 for some α [0,]. Thus, we obtain I(X ;Y U,X h(y U,X h(y U,X,X log( + αp. (33 On the other hand, by Lemma 5., the conditional entropy is upper bounded by the Gaussian variables with the same covariance matrix, thus, h(y U,X h(x + Z X h(x + Z X log (πe ( + Var(X X, (34 where Var( denotes the conditional covariance. Combining (34 with (3, we obtain the bound Hence, Var(X X αp. (35 Var(X X E[(X ] E [ (E[X X ] ] together with (35, yields E [ (E[X X ] ] ( αp. (36 Combining Lemma 5.3 and (36, Therefore, E[X X ] ( αp P. h(y h(bx + X + Z (πevar(bx log + X + Z (πe( log + b P + P + b E[X X ] (πe( log + b P + P + b ( αp P. (37

9 9 Next, we need to bound h(y X,U. According to (3, Y is a degraded version of Y conditioning on X. By the entropy power inequality (EPI [7], h(y X,U h(by X,U + h(z b h(y X,U + πe( b πe( + b αp, which yields h(y X,U (πe( log + b αp. (38 Finally, we combine (37 and (38, to obtain I(X,U;Y h(y h(y X,U log ( + b P + P + b ( αp P + b αp. (39 Since (33 and (39 are similar to ( and (3, the optimality of Gaussian inputs is established for the Gaussian IFC-DMS redefined in (3 and its capacity region is obtained. By Lemma 3.5, the capacity region of the original Gaussian IFC-DMS is equal to the region defined by ( and (3 as well. The proof is complete. VI. CONCLUSIONS AND FUTURE WORK In this paper, we investigated the capacity region of two-user IFC-DMS, for which one transmitter knows both messages. For the general discrete memoryless IFC setting, we have found achievable regions and outer bounds, which meet under additional assumptions (e.g., Assumption 3. and Assumption 3.. For the Gaussian IFC setting, we have determined the capacity region of those channels with weak interference. Possible extensions to this work include: i Lossy functions of the message (W made available to Transmitter rather than the perfect message. This might possibly yield a better outer bound for the noncooperative IFCs. ii Extending this approach to IFC with more than two users in the system. [8] R. Ahlswede, The capacity region of a channel with two senders and two receivers, Annals of Probability, vol., no. 5, pp , 974. [9] H. Sato, The capacity of the Gaussian interference channel under strong interference, IEEE Trans. Inform. Theory, vol. 7, pp , Nov. 98. [0] T. S. Han and K. Kobayashi, A new achievable rate region for the interference channel, IEEE Trans. Inform. Theory, vol. 7, pp , Jan. 98. [] M. H. Costa, On the Gaussian interference channel, IEEE Trans. Inform. Theory, vol. 3, pp , Sep [] I. Sason, On achievable rate regions for the Gaussian interference channel, IEEE Trans. Inform. Theory, vol. 50, pp , Jun [3] H. Sato, Two-user communication channels, IEEE Trans. Inform. Theory, vol. 3, pp , May 977. [4] A. Carleial, Outer bounds on the capacity of interference channels, IEEE Trans. Inform. Theory, vol. 9, pp , Jul [5] G. Kramer, Outer bounds on the capacity of Gaussian interference channels, IEEE Trans. Inform. Theory, vol. 50, pp , Mar [6] S. I. Gel fand and M. S. Pinsker, Coding for channel with random parameters, Probl. Peredachi Inform. (Probl. Inform. Trans., vol. 9, no., pp. 9 3, 980. [7] T. M. Cover and J. A. Thomas, Elements of information theory, ser. Wiley Series in Telecommunications. New York: John Wiley & Sons Inc., 99, a Wiley-Interscience Publication. [8] H. Weingarten, Y. Steinberg, and S. Shamai, The capacity region of the Gaussian MIMO broadcast channel, accepted by IEEE Trans. Inform. Theory, 005. [9] M. H. M. Costa, Writing on dirty paper, IEEE Trans. Inform. Theory, vol. 9, no. 3, pp , May 983. [0] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels, IEEE Trans. Inform. Theory, vol. 49, no. 0, pp , Oct [] P. Viswanath and D. N. C. Tse, Sum capacity of the vector Gaussin broadcast channel and uplink-downlink duality, IEEE Trans. Inform. Theory, vol. 49, pp. 9 9, Aug [] I. Csiszar and J. Korner, Information Theory: Coding Theorems for Discrete Memoryless Systems. Budapest: Akademiai Kiado, 997. [3] J. A. Thomas, Feedback can at most double Gaussian multiple access channel capacity, IEEE Trans. Inform. Theory, vol. 33, pp. 7 76, Sep REFERENCES [] N. Devroye, P. Mitran, and V. Tarokh, Achievable rates in cognitive radio channels, vol. 5, no. 5, pp , May 006. [] A. Jovicic and P. Viswanath, Cognitive radio: An information-theoretic perspective, in Proceedings of IEEE Int. Symp. Inf. Theory, July 006, pp [3] N. Jindal, U. Mitra, and A. Goldsmith, Capacity of ad-hoc networks with node cooperation, in Proceedings of IEEE Int. Symp. Inf. Theory, 004, p. 7. [4] A. Host-Madsen, A new achievable rate for cooperative diversity based on generalized writing on dirty paper, in Proceedings of IEEE Int. Symp. Inf. Theory, June 003, p. 37. [5] C. Ng and A. Goldsmith, Transmitter cooperation in ad-hoc wireless networks: Does dirty-paper coding beat relaying? in Proceedings of IEEE Information Theory Workshop, Oct [6] I. Maric, R. D. Yates, and G. Kramer, The strong interference channel with common information, in Proceedings of 43th Allerton Conference on Communications, Control and Computing, Monticello, IL, Sep [7], The strong interference channel with unidirectional cooperation, in Proceedings of the UCSD Workshop on Information Theory and its Applications, San Diego, CA, Feb. 006.