Optimal Independent Encoding Schemes for Several Classes of Discrete Degraded Broadcast Channels

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1 Optimal Independent Encoding Schemes for Several Classes of Discrete Degraded Broadcast Channels Bike Xie, Student Member, IEEE and Richard D. Wesel, Senior Member, IEEE arxiv: v1 [cs.it] 25 Nov 2008 Abstract Let X Y Z be a discrete memoryless degraded broadcast channel (DBC) with marginal transition probability matrices T YX and T ZX. Denote q as the distribution of the channel input X. For any given q, and H(Y X) s H(Y), where H(Y X) is the conditional entropy of Y given X and H(Y) is the entropy of Y, define the function F T Y X,T ZX (q,s) as the infimum of H(Z U), the conditional entropy of Z given U with respect to all discrete random variables U such that a) H(Y U) = s, and b) U and Y,Z are conditionally independent given X. This paper studies the function F, its properties and its calculation. This paper then applies these results to several classes of DBCs including the broadcast Z channel, the inputsymmetric DBC, which includes the degraded broadcast group-addition channel, and the discrete degraded multiplication channel. This paper provides independent encoding schemes and demonstrates that each achieve the boundary of the capacity region for the corresponding class of DBCs. This paper first represents the capacity region of the DBC X Y Z with the function F T Y X,T ZX. Secondly, this paper shows that the OR approach, an independent encoding scheme, achieves the optimal boundary of the capacity region for the multi-user broadcast Z channel. This paper then studies the inputsymmetric DBC, introduces the permutation approach, an independent encoding scheme, for the inputsymmetric DBC and proves its optimality. As a consequence, the group-addition approach achieves the optimal boundary of the capacity region for the degraded broadcast group-addition channel. Finally, this paper studies the discrete degraded broadcast multiplication channel and shows that the multiplication approach achieves the boundary of the capacity region for the discrete degraded broadcast multiplication channel. Keywords Degraded broadcast channel, independent encoding, broadcast Z channel, input-symmetric, groupaddition channel, multiplication channel. I. Introduction In the 70 s, Cover [1], Bergmans [2] and Gallager [3] established the capacity region for degraded broadcast channels. The optimal transmission strategy to achieve the boundary of the capacity region for degraded broadcast channels is generally a joint encoding scheme. Particularly, the data sent to the user with the most degraded channel is encoded first. This work was supported by the Defence Advanced Research Project Agency SPAWAR Systems Center, San Diego, California under Grant N The authors are with the Electrical Engineering Department, University of California, Los Angeles, CA USA ( xbk@ee.ucla.edu; wesel@ee.ucla.edu). 1

2 Given the encoded bits for that user, an appropriate codebook for the second most degraded channel user is selected, and so forth. The joint encoding scheme is potentially too complex to implement. Fortunately, combining independently encoded streams, one for each user, can achieve the optimal boundary of the capacity region for some broadcast channels including broadcast Gaussian channels [4], broadcast binary-symmetric channels [2] [5] [6] [7], discrete additive degraded broadcast channels [8] and two-user broadcast Z channels [9] [10]. Shannon s entropy power inequality (EPI) [11] gives a lower bound on the differential entropy of the sum of independent random variables. In Bergmans s remarkable paper [4], he applied EPI to establish a converse showing the optimality of the independent encoding scheme given by [1] [2] for broadcast Gaussian channels. Mrs. Gerber s Lemma [12] provides a lower bound on the entropy of a sequence of binary-symmetric channel outputs. Wyner and Ziv obtained Mrs. Gerber s Lemma and applied it to establish a converse showing that the independent encoding scheme for broadcast binary-symmetric channels suggested by Cover [1] and Bergmans[2] achieves the optimal boundary of the capacity region [5]. EPI and Mrs. Gerber s Lemma play the same significant role in proving the optimality of the independent encoding schemes for broadcast Gaussian channels and broadcast binarysymmetric channels. Witsenhausen and Wyner studied a conditional entropy bound for the channel output of a discrete channel and applied the results to establish an outer bound of the capacity region for degraded broadcast channels [6] [7]. For broadcast binary-symmetric channels, this outer bound coincides with the capacity region. This paper borrows ideas from Witsenhausen and Wyner [7] to study a conditional entropy bound for the channel output of a discrete degraded broadcast channel and represent the capacity region of discrete degraded broadcast channels with this conditional entropy bound. This paper simplifies the expression of the conditional entropy bound for broadcast binary-symmetric channels and broadcast Z channels. For broadcast Z channels, the simplified expression of the conditional entropy bound demonstrates that the independent encoding scheme provided in [9], which is optimal 2

3 for two-user broadcast Z channels, is also optimal for multi-user broadcast Z channels. The input-symmetric channel is introduced by Witsenhausen and Wyner [7] and studied in [13] and [14]. This paper extends the definition of the input-symmetric channel to the definition of the input-symmetric degraded broadcast channel. This paper introduces the permutation approach, an independent encoding scheme, for the input-symmetric degraded broadcast channel and proves its optimality. The discrete additive degraded broadcast channel [8] is a special case of the input-symmetric degraded broadcast channel, and the optimal approach for the discrete additive degraded broadcast channel [8] is also a special case of the permutation approach. The degraded broadcast group-addition channel is a class of input-symmetric degraded broadcast channels. The permutation approach for the degraded broadcast group-addition channel turns out to be the group-addition approach. Finally, this paper studies the discrete degraded broadcast multiplication channel and shows that the optimal transmission strategy is just the multiplication approach, which is also an independent encoding scheme. This paper is organized as follows. Section II defines the conditional entropy bound F ( ) for the channel output of a discrete degraded broadcast channel and represents the capacity region of the discrete degraded broadcast channel with the function F. In Section III, we establish a number of theorems concerning various properties of F. Section IV evaluates F ( ) and indicates the optimal transmission strategy for the discrete degraded broadcast channel. Section V proves the optimality of the OR approach, an independent encoding scheme, for the multi-user broadcast Z channel. Section VI introduces the input-symmetric degraded broadcast channel, provides the permutation approach, an independent encoding scheme, for the input-symmetric degraded broadcast channel and proves its optimality. Section VII studies the discrete degraded broadcast multiplication channel and provides the multiplication approach which achieves the capacity region for the discrete degraded broadcast multiplication channel. 3

4 II. The Conditional Entropy Bound F ( ) Let X Y Z be a discrete memoryless degraded broadcast channel where X {1,2,,k}, Y {1,2,,n} and Z {1,2,,m}. Let T YX be an n k stochastic matrix with entries T YX (j,i) = Pr(Y = j X = i) and T ZX be an m k stochastic matrix with entries T ZX (j,i) = Pr(Z = j X = i). Thus, T YX and T ZX are the marginal transition probability matrices of the degraded broadcast channel. Let vector q in the simplex k of probability k-vectors be the distribution of the channel input X. For H(Y X) s H(Y), where H(Y X) is the conditional entropy of Y given X and H(Y) is the entropy of Y, define the function FT Y X,T ZX (q,s) as the infimum of H(Z U), the conditional entropy of Z given U, with respect to all discrete random variables U such that a) H(Y U) = s; b) U and Y,Z are conditionally independent given X, i.e., the sequence U,X,Y,Z forms a Markov chain U X Y Z. We will use FT Y X,T ZX (q,s), F (q,s) and F (s) interchangeably. Theorem 1: FT Y X,T ZX (q,s) is monotonically nondecreasing in s and the infimum in its definition is a minimum. Hence, FT Y X,T ZX (q,s) can be taken as the minimum H(Z U) with respect to all discrete random variables U such that a) H(Y U) s; b) U and Y,Z are conditionally independent given X. The proof of Theorem 1 will be given in Section III. Theorem 2: The capacity region for the discrete memoryless degraded broadcast channel X Y Z is the closure of the convex hull of all rate pairs (R 1,R 2 ) satisfying 0 R 1 I(X;Y), (1) R 2 H(Z) F T Y X,T ZX (q,r 1 +H(Y X)), (2) 4

5 for any q k, where I(X;Y) is the mutual information of between X and Y, H(Y X) is the conditional entropy of Y given X and H(Z) is the entropy of Z with respect to the channel input s distribution q. Proof: The capacity region of the degraded broadcast channel is known in [1] [3] [15] as co {(R 1,R 2 ) : R 1 I(X;Y U),R 2 I(U;Z)}, (3) p(u),p(x u) where co denotes the closure of the convex hull operation, and U is the auxiliary random variable which satisfies the Markov chain U X Y Z and U min( X, Y, Z ). Rewrite (3) and we have co p(u),p(x u) = co p X =q k = co p X =q k {(R 1,R 2 ) : R 1 I(X;Y U),R 2 I(U;Z)} p(u,x) with p X =q p(u,x) with p X =q {(R 1,R 2 ) : R 1 I(X;Y U),R 2 I(U;Z)} (4) {(R 1,R 2 ) : R 1 H(Y U) H(Y X),R 2 H(Z) H(Z U)} (5) = co { (R1,R 2 ) : R 1 s H(Y X),R 2 H(Z) FT Y X,T ZX (q,s) } (6) p X =q k = co, p X =q k { (R1,R 2 ) : 0 R 1 I(X;Y),R 2 H(Z) F T Y X,T ZX (q,r 1 +H(Y X)) } (7) where p X is the vector expression of the distribution of channel input X. Some of these steps are justified as follows: (4) follows from the equivalence of p(u),p(x u) and p X =q k p(u,x) with p X =q ; 5

6 (6) follows from the definition of the conditional entropy bound F ; (7) follows from the nondecreasing property of F (s) in Theorem 1. Note that for a fixed distribution p X = q of the channel input X, the items I(X;Y), H(Z) and H(Y X) in (7) are constants. This theorem provides the relationship between the capacity region and the conditional entropy bound F for a discrete degraded broadcast channel. It also motivates the further study of F. III. Some Properties of F ( ) In this section, we will borrow ideas from [7] to establish several properties of the conditional entropy bound F ( ). In [7], Witsenhausen and Wyner defined a conditional entropy bound F( ) for a pair of discrete random variables and provided some properties of F( ). The definition of F( ) is restated here. Let X Z be a discrete memoryless channel with the m k transition probability matrix T, where the entries T(j,i) = Pr(Z = j X = i). Let q be the distribution of X. For any q k, and 0 s H(X), the function F T (q,s) is the infimum of H(Z U) with respect to all discrete random variables U such that H(X U) = s and the sequence U,X,Z is a Markov chain. By definition, F T (q,s) = FI,T (q,s), where I is an identity matrix. Since F ( ) is the generalization of F( ), most of the properties of F ( ) in this section are the generalization of the properties of F( ) in [7]. For any choice of l > 0, w = [w 1,,w l ] T l and p j k,j = 1,,l, let U be a l-ary random variable with distribution w, and let T XU = [p 1,,p l ] be the transition probability matrix from U to X. We can compute l p = p X = T XU w = w j p j (8) ξ = H(Y U) = η = H(Z U) = j=1 l w j h n (T YX p j ) (9) j=1 l w j h m (T ZX p j ) (10) j=1 6

7 where h n : n R is the entropy function, i.e., h n (p 1,,p n ) = p i logp i. 1 Thus the choices of U satisfying conditions a) and b) in the definition of FT Y X,T ZX (q,s) corresponds to the choices of l,w and p j for which (8) (9) yields p = q and ξ = s. Let S = {(p,h n (T YX p),h m (T ZX p)) k [0,logn] [0,logn] p k }. Since k is (k 1)-dimensional, k [0,logn] [0,logn] is a (k+1)-dimensional convex polytope. The mapping p (p,h n (T YX p),h m (T ZX p)) assigns a point in S for each p k. Because this mapping is continuous and the domain of the mapping, k, is compact and connected, the image S is also compact and connected. Let C be the set of all (p,ξ,η) satisfying (8) (9) and (10) for some choice of l, w and p j. By definition, the set C is the convex hull of the set S. Thus, C is compact, connected and convex. Lemma 1: C is the convex hull of S, and thus C is compact, connected and convex. Lemma 2: i) Every point of C can be obtained by (8) (9) and (10) with l k+1. In other words, one only need to consider random variables U taking at most k +1 values. ii) Every extreme point of the intersection of C with a two-dimensional plane can be obtained with l k. The proof of Lemma 2 is the same as the proof of a similar lemma for F( ) in [7]. The details of the proof is given in Appendix A. Let C = {(ξ,η) (q,ξ,η) C } be the projection of the set C on the (ξ,η)-plane. Let C q = {(ξ,η) (q,ξ,η) C } be the projection on the (ξ,η)-plane of the intersection of C with the two-dimensional plane p = q. By definition, C = q k C q. Also, C and C q are compact and convex. By definition, FT Y X,T ZX (q,s) is the infimum of all η, for which C q contains the point (s,η). Thus F T Y X,T ZX (q,s) = inf{η (q,s,η) C} = inf{η (s,η) C q }. (11) 1 All logarithms are natural in this paper. 7

8 Lemma 3: For any fixed q as the distribution of X, the domain of F T Y X,T ZX (q,s) in s is [H(Y X),H(Y)], i.e.,[ k q ih n (T YX e i ),h n (T YX q)], where e i is a vector, for which the i th entry is 1 and all other entries are zeros. Proof: For any Markov chain U X Y, by the Data Processing Theorem [16], H(Y U) H(Y X) and the equality is achieved when the random variable U = X. One also has H(Y U) H(Y) and the equality is achieved when U is a constant. Thus, the domain of F T Y X,T ZX (q,s) in s is [H(Y X),H(Y)] for a fixed distribution of channel input X. Since q is the distribution of X, H(Y X) = k q ih n (T YX e i ) and H(Y) = h n (T YX q). Q.E.D. Theorem 3: ThefunctionF T Y X,T ZX (q,s)isdefinedonthecompactconvexdomain{(q,s) q k, k q ih n (T YX e i ) s h n (T YX q)} and for each (q,s) in this domain, the infimum in its definition is a minimum, attainable with U taking at most k +1 values. Proof: By Lemma 3, the function F is defined on the compact domain {(q,s) q k, k q ih n (T YX e i ) s h n (T YX q)}. This domain is convex because k is convex, the entropy function h n (T YX q) is concave in q and k q ih n (T YX e i ) is linear in q. For each (q,s) in this domain, the set {η (s,η) C q } is non-empty. It is in fact a compact interval since C q is compact. Therefore, F T Y X,T ZX (q,s) = inf{η (s,η) C q } = min{η (s,η) C q } = min{η (q,s,η) C}. (12) By Lemma 2 i), this minimum is attained with U taking at most k +1 values. Q.E.D. By Lemma 2 ii), the extreme points of C q can be attained by convex combinations of at most k points of S. Thus, every linear function of (ξ,η) could attain its minimum with U taking at most k value since every linear function of (ξ,η) achieves its minimum over C q at an extreme point of the compact set C q. Lemma 4: The function FT Y X,T ZX (q,s) is jointly convex in (q,s). Proof: FT Y X,T ZX (q,s) is jointly convex in (q,s) because C is a convex set. In particular, 8

9 the domain of F is convex by Theorem 3. For any tow points (q 1,s 1 ) and (q 2,s 2 ) in the domain, and for any 0 θ 1, FT Y X,T ZX (θq 1 +(1 θ)q 2,θs 1 +(1 θ)s 2 ) =min{η (θq 1 +(1 θ)q 2,θs 1 +(1 θ)s 2,η) C} min{θη 1 +(1 θ)η 2 (q 1,s 1,η 1 ),(q 2,s 2,η 2 ) C} =θft Y X,T ZX (q 1,s 1 )+(1 θ)ft Y X,T ZX (q 2,s 2 ). Therefore, F T Y X,T ZX (q,s) is jointly convex in (q,s). Q.E.D. Now we give the proof of Theorem 1. Since Theorem 3 has shown that the infimum in the definition of F is a minimum, it suffices to show that F (s) = FT Y X,T ZX (q,s) is monotonically nondecreasing in s. For any fixed q, the domain of s is [H(Y X),H(Y)]. On the one hand, F (q,h(y X)) = min{h(z U) p X = q,h(y U) = H(Y X)} min{h(z U) p X = q,u = X} = H(Z X). (13) On the other hand, for any s [H(Y X),H(Y)], F (q,s) = min{h(z U) p X = q,h(y U) = s} min{h(z U,X) p X = q,h(y U) = s} (14) = H(Z X), (15) where (14) follows from H(Z U) H(Z U,X) and (15) follows from the conditional independence between Z and U given X. Equation (13) and (15) imply that for any s 9

10 [H(Y X),H(Y)], F (q,s) F (q,h(y X)). (16) Combining (16) and the fact that F (q,s) is convex in s for any fixed q, we have F (q,s) is monotonically nondecreasing in s. Q.E.D. The proof of Theorem 1 also gives an endpoint of F (s), F (q,h(y X)) = H(Z X), (17) which is achieved when U = X. The following theorem will provide the other endpoint, F (q,h(y)) = H(Z), (18) which is obtained when U is a constant. Theorem 4: For H(Y X) s H(Y), an lower bound of F (s) is F (s) s+h(z) H(Y). (19) If F (s) is derivative in s, then Proof: 0 df (s) ds 1. (20) I(U;Z) I(U;Y) (21) H(Z) H(Z U) H(Y) H(Y U) H(Z U) H(Y U)+H(Z) H(Y) F (s) s+h(z) H(Y). (22) Some of these steps are justifies as follows: 10

11 H(Z) F*(q,s) slope 1 H(Z X) slope 0 H(Y X) H(Y) s Fig. 1 The graph of the function F (s) = F T Y X,T ZX (q,s). (21) follows from the Data Processing Theorem [16]; (22) follows from the definition of F (s). When the random variable U is a constant, H(Y U) = H(Y) and H(Z U) = H(Z). Thus, the equality of F (s) s+h(z) H(Y) is attained when s = H(Y). If F (s) is derivative, then for any H(Y X) s H(Y), df (s) ds df (s) ds 1. (23) s=h(y) The first inequality follows from the convexity of F (s) and the second inequality follows from F (s) s + H(Z) H(Y) and F (H(Y)) = H(Z). df (s) ds 0 because F (s) is monotonically nondecreasing. The graph of the function F (s) = F T Y X,T ZX (q,s) is shown in Fig. 1. Q.E.D. For a fixed vector q as the distribution of X, by Theorem 2, finding the maximum of R 2 +λr 1 is equivalent to finding the minimum of F (q,s) λs. Theorem 4 indicates that forevery λ > 1, theminimum off (q,s) λsisattainedwhen s = H(Y)andF (s) = H(Z), i.e., U is a constant. Thus, the non-trivial range of λ is 0 λ 1. 11

12 The following theorem is the key to the applications in Section V and is an extension and generalization of Theorem 2.4 in [7]. Let X = (X 1,,X N ) be a sequence of channel inputs to the degraded broadcast channel X Y Z. The corresponding channel outputs are Y = (Y 1,,Y N ) and Z = (Z 1,,Z N ). Thus, the sequence of the channel outputs (Y i,z i ), i = 1,,N, areconditionallyindependent witheachothergiventhechannel inputs X. Note that the channel outputs (Y i,z i ) do not have to be identically or independently distributed since X 1,,X N could be correlated and have different distributions. Denote q i as the distribution of X i for i = 1,,N. Thus, q = q i /N is the average of the distribution of the channel inputs. For any q k, define F T (N) Y X,T(N) ZX (q,ns) be the infimum of H(Z U) with respect to all random variables U and all possible channel inputs X such that H(Y U) = Ns, the average of the distribution of the channel inputs is q and U X Y Z is a Markov chain. Theorem 5: For all N = 1,2,, and all T YX,T ZX, q, and H(Y X) s H(Y), one has F T (N) Y X,T(N) ZX (q,ns) = NF T Y X,T ZX (q,s). (24) Proof: We first prove that F T (N) Y X,T(N) ZX (q,ns) NF T Y X,T ZX (q,s). Since Ns = H(Y U) = = N H(Y i Y 1,,Y i 1,U) (25) N s i, (26) 12

13 where s i = H(Y i Y 1,,Y i 1,U) and (25) follows from the chain rule of entropy [15], H(Z U) = = N H(Z i Z 1,,Z i 1,U) (27) N H(Z i Z 1,,Z i 1,Y 1,,Y i 1,U) (28) N H(Z i Y 1,,Y i 1,U) (29) N FT Y X,T ZX (q i,s i ) (30) N N NFT Y X,T ZX ( q i /N, s i /N) (31) = NF T Y X,T ZX (q,s). (32) Some of these steps are justified as follows: (27) follows from the chain rule of entropy [15]; (28) holds because conditional entropy decreases when the conditioning increases; (29) follows from the fact that Z i and Z 1,,Z i 1 are conditionally independent given Y 1,,Y i 1 ; (30) follows from the definition of F if considering the Markov chain (U,Y 1,,Y i 1 ) X i Y i Z i ; (31) is implied by applying Jensen s inequality to the convex function F. By the definition of F T (N) Y X,T(N) ZX (q, Ns), Equation (32) implies that F T (N) Y X,T(N) ZX (q,ns) NF T Y X,T ZX (q,s). (33) On the other hand, in the case that U is composed with N independently identically distributed (i.i.d.) random variables (U 1,,U N ), and each U i X i achieves p Xi = q, 13

14 H(Y i U i ) = s and H(Z i U i ) = F T Y X,T ZX (q,s), one has H(Y U) = Ns and H(Z U) = NFT Y X,T ZX (q,s). Since F T (N) Y X,T(N) ZX is defined by taking the minimum, F T (N) Y X,T(N) ZX (q,ns) NF T Y X,T ZX (q,s). (34) Combining (33) and (34), one has F T (N) Y X,T(N) ZX (q,ns) = NF T Y X,T ZX (q,s). Q.E.D. Theorem 5 indicates that if using the degraded broadcast channel X Y Z for N times, and for a fixed q as the average of the distribution of the channel inputs, the conditional entropy bound F T (N) Y X,T(N) ZX (q, Ns) is achieved when the channel is used independently and identically for N times, and single use of the channel at each time achieves the conditional entropy bound FT Y X,T ZX (q,s). IV. Evaluation of F ( ) In this section, we evaluate F (s) = FT Y X,T ZX (q,s) via a duality technique, which is also used for evaluating F( ) in [7]. This duality technique also provides the optimal transmission strategy for the degraded broadcast channel X Y Z to achieve the maximum of R 2 +λr 1 for any λ 0. Theorem 3 shows that F T Y X,T ZX (q,s) = min{η (s,η) C q } = min{η (q,s,η) C}. (35) Thus, the function F T Y X,T ZX (q,s) is determined by the lower boundary of C q. Since C q is convex, its lower boundary can be described by the lines supporting its graph from the below. The line with slope λ in the (ξ,η)-plane supporting C q has the equation η = λξ +ψ(q,λ), (36) whereψ(q,λ)istheη-interceptofthetangentlinewithslopeλforthefunctionf T Y X,T ZX (q,s). 14

15 Thus, ψ(q,λ) = min{f (q,ξ) λξ H(Y X) ξ H(Y)} (37) = min{η λξ (ξ,η) C q } (38) = min{η λξ (q,ξ,η) C}. (39) For H(Y X) s H(Y), the function F (s) = F T Y X,T ZX (q,s) can be reconstructed by F (s) = max{ψ(q,λ)+λs < λ < }. (40) Theorem 1 shows that the graph of F (s) is supported at s = H(Y X) by a line of slope 0, and Theorem 4 shows that the graph of F (s) is supported at s = H(Y) by a line of slope 1. Thus, for H(Y X) s H(Y), F (s) = max{ψ(q,λ)+λs 0 λ 1}. (41) Let L λ be a linear transformation (q,ξ,η) (q,η λξ). It maps C and S onto the sets C λ = {(q,η λξ) (q,ξ,η) C}, (42) and S λ = {(q,h m (T ZX q) λh n (T YX q)) q k }. (43) The lower boundaries of C λ and S λ are the graphs of ψ(q,λ) and φ(q,λ) = h m (T ZX q) λh n (T YX q) respectively. Since C is the convex hull of S, and thus C λ is the convex hull of S λ, ψ(q,λ) is the lower convex envelope of φ(q,λ) on k. In conclusion, ψ(,λ) can be obtained by forming the lower convex envelope of φ(,λ) for each λ and F (q,s) can be reconstructed from ψ(q,λ) by (41). This is the dual approach to 15

16 the evaluation of F. Theorem 2 represents the capacity region for a degraded broadcast channel by the function F (q,s). Since ψ(q,λ) and F (q,s) can be constructed by each other from (37) and (41), the capacity region could also be determined by ψ(q,λ) as shown below: for any λ 0, q k max{r 2 +λr 1 p X = q} = max{h(z) F (q,s)+λs λh(y X)} q k = (H(Z) λh(y X) min{f (q,s) λs}) q k = (H(Z) λh(y X) ψ(q,λ)) (44) q k We have shown the relationship among F, ψ and the capacity region for the degraded broadcast channel. Now we state a theorem which provides the relationship among F (q,s), ψ(q, λ), φ(q, λ), and the optimal transmission strategies for the degraded broadcast channel. Theorem 6: i) For any 0 λ 1, if a point of the graph of ψ(,λ) is the convex combination of l points of the graph of φ(,λ) with arguments p j and weights w λ, j = 1,,l, then F T Y X,T ZX ( j w j p j, j w j h n (T YX p j )) = j w j h m (T ZX p j ). (45) Furthermore, for a fixed channel input distribution q = j w jp j, the optimal transmission strategy to achieve the maximum of R 2 +λr 1 is determined by l,w j and p j. In particular, the optimal transmission strategy is U = l, Pr(U = j) = w j and p X U=j = p j, where p X U=j denotes the conditional distribution of X given U = j. ii)for a predetermined channel input distribution q, if the transmission strategy U = l, Pr(U = j) = w j and p X U=j = p j achieves max{r 2 + λr 1 j w jp j = q}, then the point (q,ψ(q,λ)) is the convex combination of l points of the graph of φ(,λ) with arguments p j 16

17 and weights w λ, j = 1,,l. The proof is given in Appendix B. Note that if for some pair (q,λ), ψ(q,λ) = φ(q,λ), then the corresponding optimal transmission strategy has l = 1, which means, U is a constant. Thus, the line η = λξ +ψ(q,λ) supporting the graph of F (q, ) at its endpoint (h n (T YX q),h m (T ZX q)). A. Broadcast binary-symmetric channel For the broadcast binary-symmetric channel X Y Z with T YX = 1 α 1 α 1,T ZX = 1 α 2 α 2, (46) α 1 1 α 1 α 2 1 α 2 where 0 < α 1 < α 2 < 1/2, one has φ(p,λ) = φ((p,1 p) T,λ) = h((1 α 2 )p+α 2 (1 p)) λh((1 α 1 )p+α 1 (1 p)), (47) where h(x) = xlogx (1 x)log(1 x) is the entropy function. Taking the second derivative of φ(p,λ) with respect to p, we have φ (1 2α 2 ) 2 (p,λ) = (α 2 p+(1 α 2 )(1 p))((1 α 2 )p+α 2 (1 p)) λ(1 2α 1 ) 2 + (α 1 p+(1 α 1 )(1 p))((1 α 1 )p+α 1 (1 p)), (48) which has the sign of ρ(p,λ) = ( 1 α 1 1 2α 1 p)( 1 α 1 1 2α 1 +p)+λ( 1 α 2 1 2α 2 p)( 1 α 2 1 2α 2 +p). (49) For any 0 λ 1, min p ρ(p,λ) = λ 4(1 2α 2 ) 1 2 4(1 2α 1 ) 2. (50) 17

18 p p 0 p 1/2 1-p 1 p Fig. 2 Thus, for λ (1 2α 2 ) 2 /(1 2α 1 ) 2, φ (p,λ) 0 for all 0 p 1, and so ψ(p,λ) = φ(p,λ). In this case, the optimal transmission strategy achieving the maximum of R 1 also achieves the maximum of R 2 + λr 1, and thus the optimal transmission strategy has l = 1, which means, U is a constant. Note that φ(1/2+p,λ) = φ(1/2 p,λ). For λ < (1 2α 2 ) 2 /(1 2α 1 ) 2, φ(p,λ) has negative second derivative on an interval symmetric about p = 1/2. Let p λ = argmin p φ(p,λ) with p λ 1/2. Thus p λ satisfies φ p (p λ,λ) = 0. By symmetry, the envelope ψ(,λ) is obtained by replacing φ(p,λ) on the interval (p λ,1 p λ ) by its minimum over p, which is shown in Fig. 2. Therefore, the lower envelope of φ(p,λ) is φ(p λ,λ), for p λ p 1 p λ ψ(p,λ) = φ(p, λ), otherwise. (51) For the predetermined distribution of X, p X = q = (q,1 q) T with p λ < q < 1 p λ, (q,ψ(q,λ)) is the convex combination of the points (p λ,ψ(p λ,λ)) and (1 p λ,ψ(1 p λ,λ)). Therefore, by Theorem 6, F (q,s) = h 2 (T ZX (p λ,1 p λ ) T ) = h(α 2 + (1 2α 2 )p λ ) for s = h 2 (T YX (p λ,1 p λ ) T ) = h(α 1 + (1 2α 1 )p λ ), and 0 p λ q or, equivalently, 18

19 1 q p λ 1. This defines F (q, ) on its entire domain [h(α 1 ),h(α 1 + (1 2α 1 )q)], i.e., [H(Y X),H(Y)]. For the predetermined distribution of X, q = (q,1 q) T with q < p λ or q > 1 p λ, one has φ(q,λ) = ψ(q,λ), which means that a line with slope λ supports F (q, ) at point s = H(Y) = h(α 1 + (1 2α 1 )q, and thus the optimal transmission strategy has l = 1, which means U is a constant. V. Broadcast Z Channels The Z channel, shown in Fig. 3(a), is a binary asymmetric channel which is noiseless when symbol 1 is transmitted but noisy when symbol 0 is transmitted. The capacity of the Z channel was studied in [17]. The Broadcast Z channel is a class of discrete memoryless broadcast channels whose component channels are Z channels. A two-user broadcast Z channel with marginal transition probability matrices T YX = 1 α 1,T ZX = 1 α 2, (52) 0 1 α α 2 where 0 < α 1 α 2 < 1, is shown in Fig 3(b). The two-user broadcast Z channel is stochastically degraded and can be modeled as a physically degraded broadcast channel as shown in Fig. 4, where α = (α 2 α 1 )/(1 α 1 ) [9]. The OR approach for broadcast Z channels is an independent encoding scheme in which the transmitter first independently encodes users information messages into binary codewords and then broadcasts the binary OR of these encoded codewords. The OR approach achieves the whole boundary of the capacity region for the two-user broadcast Z channel [9] [10]. In this section, we will show that the OR approach also achieves the whole boundaries of the capacity regions for multiuser broadcast Z channels. 19

20 1 0 X (a) Y 1 0 X (b) Y Z Fig. 3 The broadcast Z channel 1 X Y Z Fig. 4 The degraded version of the broadcast Z channel A. F for the broadcast Z channel For the broadcast Z channel X Y Z shown in Fig. 3(b) and Fig. 4 with T YX = 1 α 1,T ZX = 1 α 2, (53) 0 β 1 0 β 2 where 0 < α 1 α 2 < 1, β 1 = 1 α 1, and β 2 = 1 α 2, one has φ(p,λ) = φ((1 p,p) T,λ) = h(pβ 2 ) λh(pβ 1 ). (54) Taking the second derivative of φ(p,λ) with respect to p, we have φ (p,λ) = β2 2 λβ1 2, (55) (1 pβ 2 )pβ 2 (1 pβ 1 )pβ 1 20

21 (p, ) (p, ) p 0 1 p Fig. 5 The graphs of φ(,λ) and ψ(,λ) for the broadcast Z channel which has the sign of ρ(p,λ) = pβ 1 β 2 (1 λ)+λβ 1 β 2. (56) Let β = β 2 /β 1. For the case of β λ 1, φ (p,λ) 0 for all 0 p 1. Hence, φ(p,λ) is convex in p and thus φ(p,λ) = ψ(p,λ) for all 0 p 1. In this case, the optimal transmission strategy achieving the maximum of R 1 also achieves the maximum of R 2 +λr 1, and the optimal transmission strategy has l = 1, i.e., U is a constant. Note that the transmission strategy with l = 1 is a special case of the OR approach in which the only codeword for the second user is an all one codeword. [ For the case of 0 λ < β, φ(p,λ) is concave in p on [0, β 2 λβ 1 β 1 β 2 (1 λ) β 2 λβ 1 β 1 β 2 ] and convex on (1 λ),1]. The graph of φ(,λ) in this case is shown in Fig. 5. Since φ(0,λ) = 0, ψ(, λ), the lower convex envelope of φ(, λ), is constructed by drawing the tangent through the origin. Let (p λ,φ(p λ,λ)) be the point of contact. The value of p λ is determined by φ p (p λ,λ) = φ(p λ,λ)/p λ, i.e., log(1 β 2 p λ ) = λlog(1 β 1 p λ ). (57) Let q = (1 q,q) T be the distribution of the channel input X. For q p λ, ψ(q,λ) is obtained as a convex combination of points (0,0) and (p λ,φ(p λ,λ)) with weights (p λ q)/p λ 21

22 U X Y Z 1-q/p q/p 1-p 1 p 1 Fig. 6 The optimal transmission strategy for the broadcast Z channel and q/p λ. By Theorem 6, it corresponds to s = [(p λ q)/p λ ]0+[q/p λ ]h(β 1 p λ ) = qh(β 1 p λ )/p λ and F (q,s) = q/p λ h(β 2 p λ ). Hence, for the broadcast Z channel, F T Y X,T ZX (q,qh(β 1 p)/p) = qh(β 2 p)/p (58) for p [q,1], which defines FT Y X,T ZX (q, ) on its entire domain [qh(β 1 ),h(qβ 1 )]. Also by Theorem 6, the optimal transmission strategy U X achieving max{r 2 +λr 1 j w jp j = q} is determined by l = 2, w 1 = (p λ q)/p λ, w 2 = q/p λ, p 1 = (1,0) T and p 2 = (1 p λ,p λ ) T. Sincetheoptimaltransmission strategyu X isazchannel asshowninfig.6, therandom variable X could also be constructed as the OR operation of two Bernoulli random variables withparameters(p λ q)/p λ and1 p λ respectively. Hence, theoptimaltransmission strategy for the broadcast Z channel is still an OR approach in this case. For q > p λ, ψ(q,λ) = φ(q,λ) and so the optimal transmission strategy has l = 1, i.e., U is a constant. Therefore, we provide an alternative proof to show that the OR approach achieves the whole boundary of the two-user broadcast Z channel. B. Multi-user broadcast Z channel Let X = (X 1,,X N ) be a sequence of channel inputs to the broadcast Z channel X Y Z satisfying (53). The corresponding channel outputs are Y = (Y 1,,Y N ) and Z = (Z 1,,Z N ). Thus, the sequence of the channel outputs (Y i,z i ), i = 1,,N, are conditionally independent with each other given the channel inputs X. Note that the 22

23 X Y 1 Y 2 Y 3 Y K-1 Y K... 1 / 1 / / Fig. 7 The K-user broadcast Z channel channel outputs (Y i,z i ) do not have to be identically or independently distributed since X 1,,X N could be correlated and have different distributions. Lemma 5: Consider the Markov chain U X Y Z with i Pr(X i = 0)/N = q, if H(Y U) N q p h(β 1p), (59) for some p [q,1], then The proof is given in Appendix C. H(Z U) N q p h(β 2p) (60) = N q p h(β 1pβ ), (61) Consider a K-user broadcast Z channel with marginal transition probability matrices T Yj X = 1 α j, (62) 0 β j where 0 < α 1 α K < 1, and β j = 1 α j for j = 1,,K. The K-user broadcast Z channel is stochastically degraded and can be modeled as a physically degraded broadcast channel as shown in Fig. 7. The OR approach for the K-user broadcast Z channel is to independently encode K users information messages into K binary codewords and broadcast the OR of these K encoded codewords. The j th user then successively decodes the messages 23

24 for User K, User K 1,, and finally for User j. The codebook for the j th user is designed by random coding technique according to the binary random variable X (j) with Pr{X (j) = 0} = q (j). Denote X (i) X (j) as the OR of X (i) and X (j). Hence, the channel input X is the OR of X (j) for all 1 j K, i.e., X = X (1) X (K). From the coding theorem for degraded broadcast channels [2] [3], the achievable region of the OR approach for the K-user broadcast Z channel is determined by R j I(Y j,x (j) X (j+1),,x (K) ) (63) = H(Y j X (j+1),,x (K) ) H(Y j X (j),x (j+1),,x (K) ) (64) ( K ) ( j K ) j 1 = q (i) h(β j q (i) ) q (i) h(β j q (i) ) (65) i=j+1 i=j = q t j h(β j t j ) q t j 1 h(β j t j 1 ) (66) where t j = j q(i) for j = 1,,K, and q = Pr(X = 0) = K q(i). Denote t 0 = 1. Since 0 q (1),,q (K) 1, one has 1 = t 0 t 1 t K = q. (67) We now state and prove that the achievable region of the OR approach is the capacity region for the multi-user broadcast Z channel. Fig. 8 shows the communication system for the K-user broadcast Z channel. X = (X 1,,X N ) is a length-n codeword determined by the messages W 1,,W K. Y 1,,Y K are the channel outputs corresponding to the channel input X. Theorem 7: If N Pr{X i = 0}/N = q, then no point (R 1,,R K ) such that R j q t j h(β j t j ) q t j 1 h(β j t j 1 ), j = 1,,K R d = q t d h(β d t d ) q t d 1 h(β d t d 1 )+δ, some d {1,,K},δ > 0 (68) 24

25 S 1 W 1 {1,...,M 1 } S 2... W 2 X(W 1,...,W K ) Encoder Z Z Z Y 1 Y 2 Y K S K W K {1,...,M K } DEC DEC DEC Wˆ 1 Wˆ 2 WˆK Fig. 8 The communication system for the multi-user broadcast Z channel is achievable, where the t j are as in (66) and (67). Proof (by contradiction): This proof borrows the idea of proving the converse of the coding theorem for broadcast Gaussian channels [2]. Lemma 5 plays the same role in this proof as the entropy power inequality does in the proof for broadcast Gaussian channels. We suppose that the rates of (68) is achievable, which means that the probability of decoding error for each receiver can be upper bounded by an arbitrarily small ǫ for sufficiently large N Pr{Ŵj W j Y j } < ǫ, j = 1,,K. (69) By Fano s inequality, this implies that H(W j Y j ) h(ǫ)+ǫlog(m j 1), j = 1,,K. (70) 25

26 Let o(ǫ) represent any function of ǫ such that o(ǫ) 0 and o(ǫ) 0 as ǫ 0. Equation (70) implies that H(W j Y j ), j = 1,,K, are all o(ǫ). Therefore, H(W j ) = H(W j W j+1,,w K ) (71) = I(W j ;Y j W j+1,,w K )+H(W j Y j,w j+1,,w K ) (72) I(W j ;Y j W j+1,,w K )+H(W j Y j ) (73) = H(Y j W j+1,,w K ) H(Y j W j,w j+1,,w K )+o(ǫ), (74) where (71) follows from the independence of the W j, j = 1,,K. From (68), (74) and the fact that NR j H(W j ), H(Y j W j+1,,w K ) H(Y j W j,w j+1,,w K ) N q t j h(β j t j ) N q t j 1 h(β j t j 1 ) o(ǫ). Next, using Lemma 5 and (75), we show in the Appendix D that (75) H(Y j W j+1,,w K ) N q t j h(β j t j ) o(ǫ), (76) and where Nδ should be added to the right side of (76) for j d. Therefore, it follows from (76), with j = N d, that H(Y K ) Nh(β K q)+nδ o(ǫ), (77) where q = t K = N Pr(X i = 0)/N. Since ǫ can be arbitrarily small for sufficient large N, o(ǫ) 0 as N. For sufficiently large N, H(Y K ) Nh(β K q)+nδ/2. However, it is 26

27 contradict to H(Y K ) = N H(Y K,i ) (78) N h(β K Pr(X i = 0)) (79) Nh(β K N Pr(X i = 0)/N) (80) = Nh(β K q). (81) Some of these steps are justified as follows: (78) follows from Y K = (Y K,1,,Y K,N ); (80) is obtained by applying Jensen s inequality to the concave function h( ); (81) follows from q = N Pr(X i = 0)/N. The desired contradiction has been obtained, so the theorem is proved. VI. Input Symmetric Degraded Broadcast Channels The input symmetric channel was first introduced in [7] and further studied in [13] [14]. The definition of the input symmetric channel is as follows: Let Φ n denote the symmetric group of permutations of n objects by the n n permutation matrices. An n input m output channel with transition probability matrix T m n is input symmetric if the set G T = {G Φ n Π Φ m, s.t. TG = ΠT} (82) is transitive, which means each element of {1,,n} can be mapped to every other element of {1,,n} by some permutation matrix in G T [7]. Extend the definition of the input symmetric channel to the input symmetric degraded broadcast channel as follows: Definition 1: Input Symmetric Degraded Broadcast Channel: A discrete memoryless de- 27

28 graded broadcast channel X Y Z with X = k, Y = n and Z = m is input symmetric if the set G TY X,T ZX = G TY X G TY X (83) = {G Φ k Π YX Φ n,π ZX Φ m, s.t. T YX G = Π YX T YX,T ZX G = Π ZX T ZX } (84) is transitive. Lemma 6: G TY X,T ZX is a group under matrix multiplication. Proof: Since G TY X,T ZX is a subset of Φ k, which is a group under matrix multiplication, it suffices to show that G TY X,T ZX is closed under matrix multiplication. Suppose G 1,G 2 G TY X,T ZX such that T YX G 1 = Π YX,1 T YX, T ZX G 1 = Π ZX,1 T ZX, T YX G 2 = Π YX,2 T YX and T ZX G 2 = Π ZX,2 T ZX. Thus, T YX G 1 G 2 = Π YX,1 Π YX,2 T YX, (85) and T ZX G 1 G 2 = Π ZX,1 Π ZX,2 T ZX. (86) Therefore, G 1 G 2 G TY X,T ZX. Q.E.D. Let l = G TY X,T ZX and G TY X,T ZX = {G 1,,G l }. Lemma 7: l G i = l k 11T, where l k Proof: Since G TY X,T ZX is a group, for all j = 1,,l, is an integer and 1 is an all one vector. l l G j ( G i ) = ( G i )G j = l G i. (87) Hence, l G i has k identical columns and k identical rows since G TY X,T ZX is transitive. Therefore, l G i = l k 11T. Q.E.D. 28

29 Definition 2: Smallest transitive subset of G TY X,T ZX : {G i1,,g ils } is a smallest transitive subset of G TY X,T ZX if l s j=1 G ij = l s k 11T, (88) where ls k is the smallest feasible integer. A. Some Examples The class of input symmetric degraded broadcast channels includes most of common discrete memoryless degraded broadcast channels. For example, the broadcast binarysymmetric channel with marginal transition probability matrices (46) is input symmetric since G TY X,T ZX = 1 0, (89) is transitive. Another interesting example is the broadcast symmetric binary-erasure channel with 1 a a 2 0 T YX = a 1 a 1,T ZX = a 2 a 2, (90) 0 1 a a 2 where 0 a 1 a 2 0. It is input symmetric since its G TY X,T ZX is the same as that of the broadcast binary-symmetric channel as shown in (89). Definition 3: Degraded Broadcast Group-addition Channel: A degraded broadcast channel X Y Z with X,Y,Z {1,,n} is a degraded broadcast group-addition channel if there exit two n-ary random variables N 1 and N 2 such that Y X N 1 and Z Y N 2 as shown in Fig. 9, where denotes identical distribution and denotes group addition. The class of the degraded broadcast group-addition channel includes the broadcast binarysymmetric channel and the discrete additive degraded broadcast channel [8] as special cases. Theorem 8: Degraded broadcast group-addition channels are input symmetric. 29

30 N1 N2 X Y Z Fig. 9 The degraded broadcast group-addition channel. Proof: For the degraded broadcast group-addition channel X Y Z with X,Y,Z {1,,n}, let G x for x = 1,,n, be 0-1 matrices with entries 1 if j x = i G x (i,j) = 0 otherwise for i,j = 1,,n. (91) G x forx = 1,,n,areactuallypermutationmatricesandhavethepropertythatG x1 G x2 = G x2 G x1 = G x1 x 2. Let (γ 0,,γ n 1 ) T be the distribution of N 1. Since Y has the same distribution as X N 1, one has n T YX = γ x G x. (92) x=1 Hence, T YX G x = G x T YX for all x = 1,,n. Similarly, we have T ZX G x = G x T ZX for all x = 1,,n, and so {G 1,,G n } G TY X,T ZX. (93) Since the set {G 1,,G n } is transitive by definition, G TY X,T ZX is also transitive and the degraded broadcast group-addition channel is input symmetric. Q.E.D. By definition, n j=1 G j = 11 T, and hence, {G 1,,G n } is a smallest transitive subset of G TY X,T ZX for the degraded broadcast group-addition channel. 30

31 B. Optimal input distribution and capacity region Consider the input symmetric degraded broadcast channel X Y Z with the marginal transition probability matrices T YX and T ZX. Recall that the set C is the set of all (p,ξ,η) satisfying (8) (9) and (10) for some choice of l, w and p j, j = 1,,l, the set C = {(ξ,η) (q,ξ,η) C } is the projection of the set C on the (ξ,η)-plane, and the set C q = {(ξ,η) (q,ξ,η) C } is the projection on the (ξ,η)-plane of the intersection of C with the two-dimensional plane p = q. Lemma 8: ForanypermutationmatrixG G TY X,T ZX and(p,ξ,η) C,onehas(Gp,ξ,η) C. Proof: Since (p,ξ,η) satisfying (8) (9) and (10) for some choice of l, w and p j, l w j Gp j = Gp (94) j=1 l w j h n (T YX Gp j ) = j=1 l w j h m (T ZX Gp j ) = l w j h n (Π YX T YX p j ) = ξ (95) j=1 j=1 j=1 l w j h m (Π YX T ZX p j ) = η. (96) Hence, (Gp,ξ,η) satisfying (8) (9) and (10) for the choice of l, w and Gp j. Q.E.D. Corollary 1: p k and G G TY X,T ZX, one has C Gp = C p, and so F (Gp,s) = F (p,s) for any H(Y X) s H(Y). Lemma 9: For any input symmetric degraded broadcast channel, C = C u, where u denotes the uniform distribution. Proof: For any (ξ,η) C, there exits a distribution p such that (p,ξ,η) C. Let G TY X,T ZX = {G 1,,G l }. By Corollary 1, (G j p,ξ,η) C for all j = 1,,l. By the convexity of the set C, l (q,ξ,η) = ( G j p,ξ,η) C, (97) j=1 31

32 where q = l j=1 G jp. Since G TY X,T ZX isagroup, forany permutationmatrix G G TY X,T ZX, G q = l G G j p = j=1 l G j p = q. (98) j=1 Hence, the i th entry and the j th entry of q are the same if G permutes i th row to j th row. Since the set G TY X,T ZX for an input symmetric degraded broadcast channel is transitive, all the entries of q are the same, and so q = u. This implies that (ξ,η) C u. Since (ξ,η) is arbitrarily taken from C, one has C C u. On the other hand, by definition, C C u. Therefore, C = C u. Q.E.D. Now we state and prove that the uniform distributed X is optimal for input symmetric degraded broadcast channels. Theorem 9: For any input symmetric degraded broadcast channel, its capacity region can be achieved by using the transmission strategies such that the broadcast signal X is uniformly distributed. As a consequence, the capacity region is co { (R 1,R 2 ) : R 1 s h n (T YX e 1 ),R 2 h m (T ZX u) FT Y X,T ZX (u,s),h n (T YX e 1 ) s log(n) }, where e 1 = (1,0,,0) T, n = Y, and m = Z. (99) Proof: Let q = (q 1,,q k ) T be the distribution of the channel input X for the input symmetric degraded broadcast channel X Y Z. Since G TY X is transitive, the columns 32

33 of T YX are permutations of each other. H(Y X) = = = k H(Y X = i) (100) k q i h n (T YX e i ) (101) k q i h n (T YX e 1 ) (102) = h n (T YX e 1 ), (103) which is independent with q. Let l = G TY X,T ZX and G TY X,T ZX = {G 1,,G l }. H(Z) = h m (T ZX q) (104) = h m (T ZX q) (105) = 1 l l h m (T ZX G i q) (106) h m (T ZX 1 l l G i q) (107) = h m (T ZX u), (108) where (107) follows from Fano s inequality. Since C = C u for the input symmetric degraded broadcast channel, F (q,s) F (u,s). (109) 33

34 Plugging (103), (108) and (109) into (6), the expression of the capacity region for the degraded broadcast channel, one has co { (R1,R 2 ) : R 1 s H(Y X),R 2 H(Z) FT Y X,T ZX (q,s) } (110) p X =q k co { (R1,R 2 ) : R 1 s h n (T YX e 1 ),R 2 h m (T ZX u) FT Y X,T ZX (u,s) } p X =q k (111) = co { (R 1,R 2 ) : R 1 s h n (T YX e 1 ),R 2 h m (T ZX u) FT Y X,T ZX (u,s) }, (112) which means that the capacity region can be achieved by using the transmission strategies such that the broadcast signal X is uniformly distributed. On the other hand, co { (R1,R 2 ) : R 1 s H(Y X),R 2 H(Z) FT Y X,T ZX (q,s) } (113) p X =q k co { (R 1,R 2 ) : p X = u,r 1 s H(Y X),R 2 H(Z) FT Y X,T ZX (u,s) } (114) = co { (R 1,R 2 ) : R 1 s h n (T YX e 1 ),R 2 h m (T ZX u) FT Y X,T ZX (u,s) }. (115) Hence, Equation (99) expresses the capacity region for the input symmetric degraded broadcast channels. Q.E.D. C. Permutation approach and its optimality Permutation approach is an independent encoding scheme which achieves the capacity region for input symmetric degraded broadcast channels. The block diagram of the permutationapproach is shown in Fig. 10. W 1 is the message for User 1 who sees the better channel T YX and W 2 is the message for User 2 who sees the worse channel T ZX. The permutation approach is first to independently encode these two messages into two codewords X 1 and 34

35 S 1 S 2 W 1 W 2 Encoder 1 Encoder 2 X 1 X 2 g X ( X ) 2 1 X T YX T ZX Y Z Successive Decoder Decoder 2 Wˆ 1 Wˆ 2 Fig. 10 The block diagram of the permutation approach Y Decoder 1 Xˆ,Wˆ 1 1 Decoder 2 ˆX 2 Fig. 11 The structure of the successive decoder for input symmetric degraded broadcast channels X 2 respectively. Let G s be a smallest subset of G TY X,T ZX. Denote k = X and l s = G s. We use random coding technique to design the codebook for User 1 according to the k-ary random variable X 1 with distribution p 1 and the codebook for User 2 according to the l-ary random variable X 2 with uniform distribution. Let G s = {G 1,,G ls }. Define the permutation function g x2 (x 1 ) = x if the permutation matrix G x2 maps the x th 1 column to the x th column, where x 2 {1,,l s } and x,x 1 {1,,k}. Hence, g x2 (x 1 ) = x if and only if the x th 1 row, x th column entry of G x2 is 1. The permutation approach is then to broadcast X which is obtained by applying the single-letter permutation function X = g X2 (X 1 ) on symbols of codewords X 1 and X 2. Since X 2 is uniformly distributed and l s j=1 G j = ls k 11T, the broadcast signal X is also uniformly distributed. User 2 receives Z and decodes the desired message directly. User 1 receives Y and successively decodes the message for User 2 and then for User 1. The structure of the successive decoder is shown in Fig. 11. Note that Decoder 1 in Fig. 11 is not a joint decoder even it has two inputs Y and ˆX 2. In particular, for the degraded group-addition channel with Y X N 1 and Z Y N 2, 35

36 Y y (- x ) ˆ2 Y Decoder 1 Xˆ,Wˆ 1 1 Decoder 2 ˆX 2 Fig. 12 The structure of the successive decoder for degraded group-addition channels the permutation function g x2 (x 1 ) is the group addition x 2 x 1. Hence the permutation approach for the degraded group-addition channel is the group-addition approach, which independently encodes two messages for two users and broadcast the group addition of these two encoded codewords. The successive decoder for the degraded group-addition channel is shown in Fig. 12, where ỹ = y ( ˆx 2 ). (116) From the coding theorem for degraded broadcast channels [2] [3], the achievable region of the permutation approach for the input symmetric degraded broadcast channel is determined by R 1 I(X;Y X 2 ) (117) = H(Y X 2 ) H(Y X) (118) = = = l s x 2 =1 l s x 2 =1 l s k H(Y X 2 = x 2 ) H(Y X = x) (119) h n (T YX G x2 p 1 ) x=1 k h n (T YX e x ) (120) x=1 h n (Π YX,x2 T YX p 1 ) x 2 =1 x=1 k h n (T YX e 1 ) (121) = h n (T YX p 1 ) h n (T YX e 1 ), (122) 36

37 and R 2 I(X 2 ;Z) (123) = H(Z) H(Z X 2 ) (124) = h m (T ZX u) = h m (T ZX u) l s x 2 =1 l s x 2 =1 h m (T ZX G x2 p 1 ) (125) h m (Π ZX,x2 T ZX p 1 ) (126) = h m (T ZX u) h m (T ZX p 1 ), (127) (128) where u is the k-ary uniform distribution, p 1 is the distribution of X, and e x is a 0-1 vector such that the x th entry is 1 and all other entries are 0. Hence, the achievable region is [ ] co {(R 1,R 2 ) : R 1 h n (T YX p 1 ) h n (T YX e 1 ),R 2 h m (T ZX u) h m (T ZX p 1 )} p 1 k (129) Define F(s) as the infimum of h m (T ZX p 1 ) with respect to all distributions p 1 such that h n (T YX p 1 ) = s. Hence the achievable region (129) can be expressed as { } (R 1,R 2 ) : R 1 s h n (T YX e 1 ),R 2 h m (T ZX u) env F(s),h n (T YX e 1 ) s h n (T YX u), (130) where env F(s) denotes the lower convex envelope of F(s). In order to show that the achievable region (130) is the same as the capacity region (99) for the input symmetric degraded broadcast channel, it suffices to show that env F(s) F (u,s) (131) 37

38 For any U X with uniformly distributed X, H(Z U) = u Pr(U = u)h(z U = u) (132) = u Pr(U = u)h m (T ZX p X U=u ) (133) u Pr(U = u) F(h m (T YX p X U=u )) (134) u Pr(U = u)env F(h m (T YX p X U=u )) (135) env F( u Pr(U = u)h m (T YX p X U=u )) (136) = env F(H(Y U)), (137) where p X U=u is the conditional distribution of X given U = u. Some of these steps are justified as follows: (134) follows from the definition of F(s); (136) follows from Fano s inequality. Therefore, by definition, env F(s) F (u,s). The results of this subsection may be summarized in the following theorem. Theorem 10: The permutation approach achieves the capacity region for input symmetric degraded broadcast channels, which is expressed in (129) (130) and (99). Corollary 2: The group-addition approach achieves the capacity region for degraded groupaddition channels. VII. Discrete Degraded Broadcast Multiplication Channels Definition 4: Discrete Degraded Broadcast Multiplication Channel: A discrete degraded broadcast channel X Y Z with X,Y,Z {0,1,,n} is a degraded broadcast group-addition channel if there exit two (n+1)-ary random variables N 1 and N 2 such that Y X N 1 and Z Y N 2 as shown in Fig. 13, where denotes ring multiplication. 38

39 N1 N2 X Y Z Fig. 13 The discrete degraded broadcast multiplication channel. By the definition of ring multiplication and group addition, the multiplication of zero and any element in {0,1,,n} is always zero and {1,,n} under the ring multiplication operation forms a group. Hence, the discrete degraded broadcast multiplication channel X Y Z has the channel structure as shown in Fig. 14. The sub-channel X Ỹ Z is a degraded broadcast group-addition channel with marginal distributions TỸ X and T Z X = T ZỸ T Ỹ X, where X, Ỹ, Z = {1,,n}. For the discrete degraded broadcast multiplication channel X Y Z, if the channel input X is zero, the channel outputs Y and Z are zeros for sure. If the channel input is a non-zero symbol, the channel output Y is zero with probability α 1 and Z is zero with probability α 2, where α 2 = α 1 + (1 α 1 )α. Therefore, the marginal transmission probability matrices for X Y Z are T YX = 1 α 11 T,T ZY = 1 α 1 T, (138) 0 (1 α 1 )TỸ X 0 (1 α )T ZỸ and T ZX = T ZY T YX = 1 α 11 T 1 α 1 T = 1 α 21 T, (139) 0 (1 α 1 )TỸ X 0 (1 α )T ZỸ 0 (1 α 2 )T ZỸ where 1 is an all one vector and 0 is an all zero vector. 39