The Capacity Region of a Class of Discrete Degraded Interference Channels

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1 The Capacity Region of a Class of Discrete Degraded Interference Channels Nan Liu Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 74 nkancy@umd.edu ulukus@umd.edu Abstract We provide a single-letter characterization for the capacity region of a class of discrete degraded interference channels (DDICs. The class of DDICs nsidered includes the discrete additive degraded interference channel (DADIC studied by Benzel ]. We show that for the class of DDICs studied, ender operation does not increase the capacity region, and therefore, the capacity region of the class of DDICs is the same as the capacity region of the rresponding degraded broadcast channel. I. INTRODUCTION In wireless mmunications, where multiple transmitter and receiver pairs share the same medium, interference is unavoidable. How to best manage interference ming from other users and how not to cause too much interference to other users while maintaining the quality of mmunication is a challenging question and of a great deal of practical interest. To be able to understand the effect of interference on mmunications better, interference channel (IC has been introduced in ]. The IC is a simple network nsisting of two pairs of transmitters and receivers. Each pair wishes to mmunicate at a certain rate with negligible probability of error. However, the two mmunications interfere with each other. To best understand the management of interference, we need to find the capacity region of the IC. However, the problem of finding the capacity region of the IC is essentially open except in some special cases, e.g., a class of deterministic ICs 3], discrete additive degraded interference channels (DADICs ], strong ICs 4], 5], ICs with statistically equivalent outputs 6] 8]. In this paper, we nsider a class of discrete degraded interference channels (DDICs. In a DDIC, only the bad receiver faces interference, while the good receiver has the ability to dede both messages and thus, behaves like the receiver of a multiple access channel. It is this fact that makes the DDIC easier to analyze as mpared to the IC, where both receivers are faced with interference. We provide a single-letter characterization for the capacity region of a class of DDICs. The class of DDICs includes the DADICs studied by Benzel ]. We show that for the class of DDICs studied, ender operation does not increase the capacity region, and therefore, the capacity region of the This work was presented at Allerton 6, and submitted to IEEE Trans. on Information Theory. This work was supported by NSF Grants CCR 3-3, CCF and CCF class of DDICs is the same as the capacity region of the rresponding degraded broadcast channel, which is known. II. SYSTEM MODEL A discrete memoryless IC nsists of two transmitters and two receivers. Transmitter has message W to send to receiver. Transmitter has message W to send to receiver. Messages W and W are independent. The channel nsists of two input alphabets, X and X, and two output alphabets, Y and Y. The channel transition probability is p(y, y x, x. In this paper, our definition of degradedness is in the stochastic sense, i.e., we say that an IC is DDIC if there exists a probability distribution p (y y such that p(y x, x = p(y x, x p (y y ( y Y for all x X, x X and y Y. However, we note that for any DDIC, we can form another DDIC (physically degraded by p(y, y x, x = p(y x, x p (y y ( which has the same marginals, p(y x, x and p(y x, x, as the original DDIC. Since the receivers do not operate in an IC, similar to the case of the broadcast channel 9, Problem 4.], the capacity region is only a function of the marginals, p(y x, x and p(y x, x, and the rate pairs in the capacity region can be achieved by the same achievability scheme for different ICs with the same marginals. Hence, the capacity results that we obtain for DDICs which satisfy ( will be valid for any DDIC that has the same marginals, p(y x, x and p(y x, x. Thus, without loss of generality, from now on, we may restrict ourselves to studying DDICs that satisfy (. A DDIC is characterized by two transition probabilities, p (y y and p(y x, x. For notational nvenience, let T denote the Y Y matrix of transition probabilities p (y y, and T x denote the Y X matrix of transition probabilities p(y x, x, for all x X. Throughout the paper, n will denote the probability simplex n (p, p,, p n p i =, p i, i =,,, n i= (3

2 and J n will denote the representation of the symmetric group of permutations of n objects by the n n permutation matrices. The class of DDICs we nsider in this paper satisfies the following nditions: T is input symmetric. Let the input symmetry group be G. For any x, x X, there exists a permutation matrix G G, such that T x = GT x (4 3 H(Y X = x, X = x = η, independent of x, x. 4 p(y x, x satisfies x p(y x, x = X Y, x X, y Y (5 5 Let p x,x be the Y dimensional vector of probabilities p(y x, x for a given x, x. Then, there exists an x X, such that a x,x p x,x : a x,x =, a x,x x,x x (,x G b x p x, x : b x =, b x, G G The definition of an input symmetric channel is given in, Section II.D]. For mpleteness, we repeat it here. For an m n stochastic matrix T (an n input, m output channel, the input symmetry group G is defined as (6 G = G J n : Π J m, T G = ΠT (7 i.e., G is the set of permutation matrices G such that the lumn permutations of T with G may be achieved with rresponding row permutations. T is input symmetric, if G is transitive, i.e., any element of,,, n can be mapped to every other element of,,, n by some member of G. G being a transitive subgroup means that the output entropy of channel T is maximized when the input distribution is chosen to be the uniform distribution, i.e., max p n H(T p = H(T u (8 where u denotes the uniform distribution in n. This is because, for any p n, if we let q = G G G Gp, then we have ( H(T q = H = H ( G G G T Gp G G G Π G T p (9 ( G G G H (Π G T p ( = H(T p ( where ( follows from the fact that G G, and ( follows from the ncavity of the entropy function. Note that for any G G, G q = q (3 by the fact that G is a group. Since G is also transitive, q = u. Condition implies that for any p(x, H(Y X = x does not depend on x. Combined with ndition, ndition further implies that H(Y X = x does not depend on x either. These two facts will be proved and utilized in other proofs later. A sufficient ndition for ndition 3 to hold is that the vectors p(y X = x, X = x for all (x, x X X are permutations of each other. This is true for instance when the channel from Y to Y is additive ]. By ndition 4, we can show that when X takes the uniform distribution, Y will also be uniformly distributed. Combined with ndition, ndition 4 implies that when X takes the uniform distribution, H(Y is maximized, irrespective of p(x. In ndition 5, the first line of (6 denotes the set of all nvex mbinations of vectors p x,x for all x, x X X, while the send line denotes all nvex mbinations, and their permutations with G G, of vectors p x, x for all x X, but for a fixed x X. Therefore, this ndition means that all nvex mbinations of p x,x may be obtained by a mbination of nvex mbinations of p x, x for a fixed x, and permutations in G. The DADICs nsidered in ] satisfy nditions -5, as we will show in Section VI-A. The aim of this paper is to provide a single-letter characterization for the capacity region of DDICs that satisfy nditions -5, and we will follow the proof technique of ] with appropriate generalizations. III. THE OUTER BOUND (CONVERSE When we assume that the enders are able to fully operate, i.e., both enders know both messages W and W, we get a rresponding degraded broadcast channel with input x = (x, x. The capacity region of the rresponding degraded broadcast channel serves as an outer bound on the capacity region of the DDIC. The capacity region of the degraded broadcast channel is known 9], ], ], and thus, a single-letter outer bound on the capacity region of the DDIC is (R, R : R I(X, X ; Y U p(u,p(x,x u R I(U; Y ] (4 where denotes the closure of the nvex hull operation, and the auxiliary random variable U, which satisfies the Markov chain U (X, X Y Y, has cardinality

3 bounded by U min ( Y, Y, X X. More specifically, for DDICs that satisfy ndition 3, (4 can be written as p(u,p(x,x u Let us define T (c as T (c = (R, R : R H(Y U η max p(up(x, x u H(Y U = c U min ( Y, Y, X X R I(U; Y ] (5 I(U; Y (6 where the entropies are calculated acrding to the distribution p(u, x, x, y, y = p(up(x, x up(y x, x p (y y (7 Using ndition 3, we can show that η c log Y. T (c is ncave in c ], 3], and therefore, (5 can also be written as (R, R : R c η η c log Y R T (c IV. AN ACHIEVABLE REGION (8 Based on 7, Theorem 4], the following region is achievable, (R, R : R I(X ; Y X p(x,p(x R I(X ; Y ] (9 which rresponds to the achievability scheme that the bad receiver treats the signal for the good receiver as pure noise, and the good receiver dedes both messages as if it is the receiver in a multiple access channel. For DDICs that satisfy ndition 3, (9 reduces to p(x,p(x (R, R : R H(Y X η ] R H(Y H(Y X ( We note that ( remains an achievable region if we choose p(x to be the uniform distribution. Furthermore, by choosing p(x as the uniform distribution, we have p(y = p(y x, x p(x X x,x ( = p(x p(y x, x X x x ( = Y (3 where (3 uses ndition 4. Thus, when p(x is chosen as the uniform distribution, p(y results in a uniform distribution as well. Let us define τ as τ = max H(T p (4 p Y Using the fact that the DDIC under nsideration satisfies ndition, i.e., it satisfies (8, we have that when p(x is uniform, and nsequently p(y is uniform, H(Y = τ (5 Hence, choosing p(x to be the uniform distribution in (, yields the following as an achievable region, (R, R : R H(Y X = x η X p(x x R τ ] H(Y X = x X x (6 Due to ndition, for any p(x = p and any x, x X, there exists a permutation matrix G G such that H(Y X = x = H(T x p (7 = H(GT x p (8 = H(T x p (9 = H(Y X = x (3 which means that for any p(x, H(Y X = x does not depend on x. Furthermore, for any p(x = p and any x, x X, there exist permutation matrices G G and Π, of order Y and Y respectively, such that H(Y X = x = H(T T x p (3 = H(T GT x p (3 = H(ΠT T x p (33 = H(T T x p (34 = H(Y X = x (35 where (33 follows from the fact that G G. (35 means that for any p(x, H(Y X = x does not depend on x either. Hence, the achievable region in (6 can further be written as (R, R : R H(Y X = x η p(x R τ H(Y X = x ] (36

4 for any x X. Since we will use ndition 5 later, we choose to write the region of (36 as (R, R : R H(Y X = x η p(x where x is given in ndition 5. Let us define F (c as F (c = min p(x H(Y X = x = c R τ H(Y X = x ] (37 H(Y X = x (38 where the entropies are calculated acrding to the distribution p(y, y, x x = p(x p(y x, x p (y y (39 In (38, we can write min instead of inf by the same reasoning as in 4, Section I]. Note that F (c is not a function of x because of (3 and (35. Again, by ndition 3, we can show that η c log Y. Hence, the achievable region in (37 can be written as, (R, R : R c η η c log Y R τ F (c ] (4 which by, Facts 4 and 5], can further be written as (R, R : R c η η c log Y R τ envf (c (4 where envf ( denotes the lower nvex envelope of the function F (. V. THE CAPACITY REGION In this section, we show that the achievable region in (4 ntains the outer bound in (8, and thus, (8 and (4 are both, in fact, single-letter characterizations of the capacity region of DDICs satisfying nditions -5. To show this, it suffices to prove that T (c τ envf (c, η c log Y (4 Let us fix a c η, log Y ]. Let p (u, p (x, x u be the distributions that achieve the maximum in (6, i.e., H(Y U = c (43 I(U; Y = T (c (44 Using ndition 5, for each u U, there exists a p u (x = p u and a permutation matrix G u G, such that x,x p (x, x U = up x,x = G u T x p u (45 Thus, we have H(Y U = u = H (G u T x p u = H (T x p u (46 (46 means that p u is in the feasible set of the optimization in (38 when c = H(Y U = u. Hence, We have F (H (Y U = u H (T T x p u (47 H(Y U = u = H (T G u T x p u (48 = H (Π u T T x p u (49 = H (T T x p u (5 F (H (Y U = u (5 where (48, (49 and (5 follow from (45, the fact that G G, and (47, respectively. Thus, H(Y U = u u P (U = uh(y U = u (5 P (U = uf (H (Y U = u (53 P (U = uenvf (H (Y U = u (54 u ( envf P (U = uh (Y U = u (55 u = envf (H (Y U (56 = envf (c (57 where (53 follows from (5, (54 follows from the definition of env, and (55 follows from nvexity of envf (. Finally, for η c log Y, we have T (c = I(U; Y (58 = H(Y H(Y U (59 τ envf (c (6 where (6 follows from (57 and the definition of τ in (4. Therefore, we nclude that the single-letter characterization of the capacity region of DDICs satisfying nditions -5 is (4, and also (8. To achieve point (R, R on the boundary of the capacity region, if R and R are such that R = c η, R = τ F (c (6 for some η c log Y, transmitters and generate random debooks acrding to p (x, which is the minimizer of F (R + η, and p (x, which is the uniform distribution, respectively, and transmit the dewords rresponding to the realizations of their own messages. Receiver performs successive deding, in the order of message, and then message. Receiver dedes its own message treating interference from transmitter as pure noise. To achieve point (R, R on the capacity region, where R and R do not satisfy (6, time-sharing should be used. Furthermore, we note that for these DDICs, ender operation cannot increase the capacity region.

5 VI. EXAMPLES In this section, we will provide three examples of DDICs for which nditions -5 are satisfied. The first example is the channel model adopted in ], for which the capacity region is already known. In the send and third examples, the capacity regions are previously unknown, and using the results of this paper, we are able to determine the capacity regions. A. Example A DADIC is defined as ] where Y = X X V (6 Y = X X V V (63 X = X = Y = Y = S =,,, s (64 and denotes modulo-s sum, and V and V are independent noise random variables defined over S with distributions p i = (p i (, p i (,, p i (s, i =, (65 Since Y = Y V, matrix T is circulant, and thus input symmetric, Section II.D]. Hence, ndition is satisfied. It is straightforward to check that nditions -5 are also satisfied. For example, when s = 3, we have p ( p ( p ( T = p ( p ( p ( (66 p ( p ( p ( and the input symmetry group for T is G = G =, G =, G = (67 which is transitive, i.e., G, G 3, G, G 3, 3 G, 3 G. From (6, we write p ( p ( p ( T = p ( p ( p ( (68 p ( p ( p ( p ( p ( p ( T = p ( p ( p ( (69 p ( p ( p ( p ( p ( p ( T = p ( p ( p ( (7 p ( p ( p ( Conditions -4 are satisfied because T = G T, T = G T (7 η = H(V (7 x p(y x, x = p ( + p ( + p ( = (73 Next, we check ndition 5. a x,x p x,x : a x,x =, a x,x x,x x,x p ( p ( p ( = a p ( + b p ( + c p ( : p ( p ( p ( a + b + c =, a, b, c (74 (75 because even though (74 is a nvex mbination of 9 vectors, due to vectors repeating themselves in the lumns of T, T and T, the set, in fact, nsists of nvex mbinations of only 3 vectors. On the other hand, for x =, ( G b x p x, x : b x =, b x, G = G (76 p ( p ( p ( = a p ( + b p ( + c p ( : p ( p ( p ( a + b + c =, a, b, c (77 because (76 is the nvex mbinations of the lumns of T, with the unitary permutation. Thus, a x,x p x,x : a x,x =, a x,x x,x x (,x = G b x p x, x : b x =, b x, G = G ( G b x p x, x : b x =, b x, G G and ndition 5 is satisfied. B. Example (78 (79 Next, we nsider the following DDIC. We have X = X = Y =, Y = 3, and p(y x, x is characterized by Y = X X V (8 where V is Bernoulli with p. p (y y is an erasure channel with parameter α, i.e., the transition probability matrix is α T = α α (8 α Thus, the channel is such that the bad receiver cannot receive all the bits that the good receiver receives. More specifically, α proportion of the time, whether the bit is a or is unregnizable, and thus denoted as an erasure e.

6 It is easy to see that T is input symmetric because the input symmetry group ] ] G =, (8.8 cba bca is transitive. Conditions -5 are satisfied because p(y x, x is the same as in Example in Section VI-A. C. Example 3 Let a, b, c, d, e, f be non-negative numbers such that a+b+ c = and d+e+f = /. We have X = 4, X = Y = 3, and Y = 6. The DDIC is described as d e f e f d T = d f e f e d (83 e d f f d e T = (84 T = (85 T = (86 It is straightforward to see that T is input symmetric because the input symmetry group G = G =, G =, G =, G 3 =, G 4 =, G 5 = (87 is transitive. Conditions -4 are satisfied because T = G T, T = G T (88 η = a log a b log b c log c (89 x p(y x, x = a + b + c = (9 To show ndition 5, we use Figure. The set on the first line of (6 in ndition 5 is the nvex mbination of the following six points, a a c b b c b, c, a, a, c, b (9 c b b c a a abc.4 acb cab Fig.. Explanation of ndition 5 in example 3..4 bac resulting in all the points within the hexagon in Figure. The three sets ( G b x p x, x : b x =, b x, G = G = µ b + µ c + µ 3 a + µ 4 b : 4 µ i =, µ i (9 i= and ( G b x p x, x : b x =, b x, G = G = µ a + µ b + µ 3 c + µ 4 c : 4 µ i =, µ i (93 i= and ( G b x p x, x : b x =, b x, G = G = µ c + µ a + µ 3 b + µ 4 a : 4 µ i =, µ i (94 i= rrespond to the points in the three shaded areas, abc, cba, bca, cab], acb, abc, bca, cab], and bac, cab, abc, bca], respectively. Since the three shaded areas ver the entire hexagon, and G, G, G G, ndition 5 is satisfied..

7 VII. CONCLUSION We provide a single-letter characterization for the capacity region of a class of DDICs, which is more general than the class of DADICs studied by Benzel ]. We show that for the class of DDICs studied, ender operation does not increase the capacity region, and the best way to manage the interference is through random debook design and treating the signal for the good receiver as pure noise at the bad receiver. REFERENCES ] R. Benzel, The capacity region of a class of discrete additive degraded interference channels, IEEE Trans. on Information Theory, vol. 5, pp. 8 3, March 979. ] C. E. Shannon, Two-way mmunication channels, Proc. 4th Berkeley Symp. Math. Stat. Prob., pp , 96. 3] A. El Gamal and M. Zahedi, The capacity region of a class of deterministic interference channels, IEEE Trans. on Information Theory, vol. 8, no., pp , March 98. 4] H. Sato, The capacity of the Gaussian interference channel under strong interference, IEEE Trans. on Information Theory, vol. 7, pp , November 98. 5] M. Costa and A. El Gamal, The capacity region of the discrete memoryless interference channel with strong interference, IEEE Trans. on Information Theory, vol. 33, no. 5, pp. 7 7, September ] A. B. Carleial, Interference channels, IEEE Trans. on Information Theory, vol. 4, pp. 6 7, January ] H. Sato, The two-user mmunication channels, IEEE Trans. on Information Theory, vol. 3, pp , May ] R. Ahlswede, Multi-way mmunication channels, in Proc. nd Int. Symp. Inform. Theory, H. A. Sci., Ed., Tsahkadsor, Armenian S.S.R., 97, pp ] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley- Interscience, 99. ] H. Witsenhausen and A. Wyner, A nditional entropy bound for a pair of discrete random variables, IEEE Trans. on Information Theory, vol., no. 5, pp , September 975. ] T. M. Cover, Broadcast channels, IEEE Trans. on Information Theory, vol. 8, no., pp. 4, January 97. ] R. G. Gallager, Capacity and ding for degraded broadcast channels, Problemy Peredaci Informaccii, vol., no. 3, pp. 3 4, ] R. Ahlswede and J. Korner, Source ding with side information and a nverse for degraded broadcast channels, IEEE Trans. on Information Theory, vol., pp , November ] H. Witsenhausen, Entropy inequalities for discrete channels, IEEE Trans. on Information Theory, vol., no. 5, pp. 6 66, September 974.