An outer bound for 2-receiver discrete memoryless broadcast channels

1 An outer bound for 2-recei v er discrete memoryless broadcast channels Chandra Nair Department of Informati on Enginee ring The Chinese Uni versity of Hong Kong Abstract — An outer bound to the discrete memor yless broad- cast channel is presented. W e compare it to the known outer bounds and show that th e outer bound p resented is at least as tight as the existin g bounds. I . I N T R O D U C T I O N There has been a series of outer bounds presented to the capacity region of the bro adcast ch annel [2 ], [ 3], [4]. All the b ound s follow from the use of Fano’ s inequ ality and the Csiszar sum lemma[1]. In this note, we pre sent another ou ter bound along th ese lines that is at least as tigh t as the k nown bound s. I I . T W O R E C E I V E R B RO A D C A S T C H A N N E L W I T H P R I V AT E M E S S A G E S O N LY The following lemma presents an outer bound for the cap ac- ity region of the two r eceiver discrete mem oryless br oadcast channels. Lemma 1: Consider the set of all random variables U, V , W 1 , W 2 such that ( U, V , W 1 , W 2 ) → X → ( Y 1 , Y 2 ) for a Markov cha in. Fu rther assume tha t U and V are indepe ndent; and the distribution ( U, V , W 1 , W 2 , X , Y 1 , Y 2 ) satisfies the following e qualities: I ( U ; Y 1 | W 1 ) = I ( U ; Y 1 | V , W 1 ) I ( V ; Y 2 | W 2 ) = I ( V ; Y 2 | U, W 2 ) I ( U ; V | W 1 , W 2 , Y 1 ) = I ( U ; V | W 1 , W 2 , Y 2 ) I ( W 2 ; Y 1 | W 1 ) = I ( W 1 ; Y 2 | W 2 ) I ( W 2 ; Y 1 | U, W 1 ) = I ( W 1 ; Y 2 | U, W 2 ) (1) I ( W 2 ; Y 1 | V , W 1 ) = I ( W 1 ; Y 2 | V , W 2 ) I ( W 2 ; Y 1 | U, V , W 1 ) = I ( W 1 ; Y 2 | U, V , W 2 ) . Then th e set o f rate pair s R 1 , R 2 satisfying R 1 ≤ I ( U ; Y 1 | W 1 ) R 2 ≤ I ( V ; Y 2 | W 2 ) constitutes an outer bound to the capacity re gion of the discrete memory less br oadcast chann el. Pr oof: T he inequ alities follows immediately from Fano’ s inequality and the f ollowing id entifications: ˆ W 1 i = Y i − 1 1 ˆ W 2 i = Y n 2 i +1 U = M 1 V = M 2 . W e then set W 1 = ( ˆ W 1 , Q ) , W 2 = ( ˆ W 2 , Q ) , wh ere Q is an indep enden t r andom variable ch osen un iformly at rand om from the interval { 1 , ..., n } . The last four equalities are a direct application of the Csiszar sum lemm a [1] and the pro of is omitted. The fir st two equalities follow from Fano’ s ineq uality and the independ ence of M 1 and M 2 ; and the p roof is ag ain omitted. The third equality follows as follows: I ( U ; V | W 1 , W 2 , Y 1 ) − I ( U ; V | W 1 , W 2 , Y 2 ) = lim n →∞ 1 n n X i =1 I ( M 1 ; M 2 | Y i 1 , Y n 2 i +1 ) − I ( M 1 ; M 2 | Y i − 1 1 , Y n 2 i ) = lim n →∞ 1 n ( I ( M 1 ; M 2 | Y n 1 ) − I ( M 1 ; M 2 | Y n 2 )) = 0 The last step follows f rom Fano’ s inequality . Remark 1: W e note the following div ergence fr om the normal p resentation o f th e ou ter bou nds: the absence of a sum r ate co nstraint, as well as the presen ce of a num ber of equalities. W e will compare this bound to the following existing bound 1 for the same setting. Bound 1 : The unio n of rate pair s ( R 1 , R 2 ) that satisfy the following in equalities R 1 ≤ I ( U, W ; Y 1 ) R 2 ≤ I ( V , W ; Y 2 ) R 1 + R 2 ≤ min { I ( W ; Y 1 ) , I ( W ; Y 2 ) } + I ( U ; Y 1 | W ) + I ( V ; Y 2 | U, W ) R 1 + R 2 ≤ min { I ( W ; Y 1 ) , I ( W ; Y 2 ) } + I ( U ; Y 1 | V , W ) + I ( V ; Y 2 | W ) . 1 The equi v alenc e of the bounds can be observ ed from the fa ct that for the identi fications in [2] I ( U ; V | W, Y 1 ) = I ( U ; V | W, Y 2 ) , and this implies the bound presente d in [3]. 2 over all p ( u ) p ( v ) p ( w | u, v ) p ( x | u, v , w ) p ( y 1 , y 2 | x ) form s an outer b ound to the cap acity region . Claim 1: Th e region spe cified b y the lemma 1 is at least as tigh t as the regio n specified by Bound 1. Pr oof: W e need to show that any ( R 1 , R 2 ) satisfy ing the constraints of Lemma 1 is contain ed in th e region d escribed by Boun d 1. T o show th e inclusion , we set W = ( W 1 , W 2 ) . Observe that I ( V ; Y 2 | U, W 1 , W 2 ) = I ( V ; Y 2 | W 1 , W 2 ) − I ( U ; V | W 1 , W 2 ) + I ( V ; U | W 1 , W 2 , Y 2 ) Using the equ ality I ( V ; U | W 1 , W 2 , Y 2 ) = I ( V ; U | W 1 , W 2 , Y 1 ) it is easy to see that I ( U ; Y 1 | W 1 , W 2 ) + I ( V ; Y 2 | U, W 1 , W 2 ) = I ( U ; Y 1 | V , W 1 , W 2 ) + I ( V ; Y 2 | W 1 , W 2 ) . (2) Therefo re the two su m rate co nstraints in Lemma 1 are identical. Hence it suffices to prove that I ( U ; Y 1 | W 1 ) + I ( V ; Y 2 | W 2 ) ≤ I ( W 1 , W 2 ; Y 1 ) + I ( U ; Y 1 | W 1 , W 2 ) + I ( V ; Y 2 | U, W 1 , W 2 ) . (The other on e obtained by rep lacing I ( W 1 , W 2 ; Y 1 ) with I ( W 1 , W 2 ; Y 2 ) fo llows similarly . T o g et the symmetric expr es- sion, just use (2).) Observe that I ( U, W 1 , W 2 ; Y 1 ) + I ( V ; Y 2 | U, W 1 , W 2 ) = I ( W 1 ; Y 1 ) + I ( U ; Y 1 | W 1 ) + I ( W 2 ; Y 1 | U, W 1 ) + I ( V ; Y 2 | U, W 1 , W 2 ) = I ( W 1 ; Y 1 ) + I ( U ; Y 1 | W 1 ) + I ( W 1 ; Y 2 | U, W 2 ) + I ( V ; Y 2 | U, W 1 , W 2 ) = I ( W 1 ; Y 1 ) + I ( U ; Y 1 | W 1 ) + I ( V , W 1 ; Y 2 | U, W 2 ) = I ( W 1 ; Y 1 ) + I ( U ; Y 1 | W 1 ) + I ( W 1 ; Y 2 | U, V , W 2 ) + I ( V ; Y 2 | U, W 2 ) ( a ) = I ( W 1 ; Y 1 ) + I ( U ; Y 1 | W 1 ) + I ( W 1 ; Y 2 | U, V , W 2 ) + I ( V ; Y 2 | W 2 ) ≥ I ( U ; Y 1 | W 1 ) + I ( V ; Y 2 | W 2 ) , where ( a ) follows from the following: I ( V ; Y 2 | U, W 2 ) = I ( V ; Y 2 | W 2 ) . I I I . T W O R E C E I V E R B RO A D C A S T C H A N N E L W I T H C O M M O N M E S S AG E A S W E L L A S P R I V A T E M E S S AG E S The f ollowing outer bo und was p resented in [4] for the capacity region of the b roadcast chann el for two receiv ers with a co mmon message as well as priv ate messages. Bound 2 : [4] The cap acity r egion is a sub set of the New- Jer se y region, which can be o btained by tak ing the union of rate triples ( R 0 , R 1 , R 2 ) satisfying R 0 ≤ min I ( T ; Y 1 | W 1 ) , I ( T ; Y 2 | W 2 ) R 1 ≤ I ( U ; Y 1 | W 1 ) R 2 ≤ I ( V ; Y 2 | W ) R 0 + R 1 ≤ I ( T , U ; Y 1 | W 1 ) R 0 + R 1 ≤ I ( U ; Y 1 | T , W 1 , W 2 ) + I ( T , W 1 ; Y 2 | W 2 ) R 0 + R 2 ≤ I ( T , U ; Y 2 | W 2 ) R 0 + R 2 ≤ I ( V ; Y 2 | T , W 1 , W 2 ) + I ( T , W 2 ; Y 1 | W 1 ) R 0 + R 1 + R 2 ≤ I ( U ; Y 1 | T , V , W 1 , W 2 ) + I ( T , V , W 1 ; Y 2 | W 2 ) R 0 + R 1 + R 2 ≤ I ( V ; Y 2 | T , U, W 1 , W 2 ) + I ( T , U, W 2 ; Y 1 | W 1 ) R 0 + R 1 + R 2 ≤ I ( U ; Y 1 | T , V , W 1 , W 2 ) + I ( T , W 1 , W 2 ; Y 1 ) + I ( V ; Y 2 | T , W 1 , W 2 ) R 0 + R 1 + R 2 ≤ I ( V ; Y 2 | T , U, W 1 , W 2 ) + I ( T , W 1 , W 2 ; Y 2 ) + I ( U ; Y 1 | T , W 1 , W 2 ) for some p ( u ) p ( v ) p ( t ) p ( w 1 , w 2 | u, v , t ) p ( x | u, v , t, w 1 , w 2 ) p ( y 1 , y 2 | x ) . Further o ne can restrict X to be a determ inistic functio n of ( u, v , t, w 1 , w 2 ) a nd also assume that th e marginals of U, V , T are un iform. Similar to lemma 1 we can wr ite an outer boun d for th is case as well, and this region is a t least as tight as the New Jer se y outer boun d. Lemma 2: Consider the set of all ra ndom variables T , U, V , W 1 , W 2 such that ( T , U, V , W 1 , W 2 ) → X → ( Y 1 , Y 2 ) for a Markov chain. Further assume tha t T , U , and V are indepen dent; and the distribution ( U, V , W 1 , W 2 , X , Y 1 , Y 2 ) satisfies the following equalities: I ( T ; Y 1 | W 1 ) = I ( T ; Y 2 | W 2 ) I ( T ; Y 1 | W 1 ) = I ( T ; Y 1 | V , W 1 ) = I ( T ; Y 1 | U, W 1 ) = I ( T ; Y 1 | U, V , W 1 ) I ( T ; Y 2 | W 2 ) = I ( T ; Y 2 | V , W 2 ) = I ( T ; Y 2 | U, W 2 ) (3) = I ( T ; Y 2 | U, V , W 2 ) I ( U ; Y 1 | W 1 ) = I ( U ; Y 1 | V , W 1 ) = I ( U ; Y 1 | T , W 1 ) = I ( U ; Y 1 | T , V , W 1 ) I ( V ; Y 2 | W 2 ) = I ( V ; Y 2 | U, W 2 ) = I ( V ; Y 2 | T , W 2 ) = I ( V ; Y 2 | T , U, W 2 ) , I ( B 1 ; B 2 | A, W 1 , W 2 , Y 1 ) = I ( B 1 ; B 2 | A, W 1 , W 2 , Y 2 ) (4) holds for a ll A ⊆ { T , U, V } , B 1 ⊆ { T , U } , B 2 ⊆ { T , V } , and I ( W 2 ; Y 1 | A, W 1 ) = I ( W 1 ; Y 2 | A, W 2 ) (5) holds fo r all A ⊆ { T , U, V } . Then the set o f rate tup les ( R 0 , R 1 , R 2 ) satisfying R 0 ≤ min { I ( T ; Y 1 | W 1 ) , I ( T ; Y 2 | W 2 ) } R 1 ≤ I ( U ; Y 1 | W 1 ) R 2 ≤ I ( V ; Y 2 | W 2 ) constitutes an outer bo und to the capacity region o f the discrete memory less br oadcast chann el. 3 Further, just as in Lemma 2 one can restrict X to be a deterministic func tion of ( u, v , t, w 1 , w 2 ) and also assum e that the m arginals of U, V , T are un iform. Pr oof: T = M 0 is the on ly new identification as compare d to Lemma 1. The argum ents f or this lemm a are similar to those of Lemma 1 and are om itted. Claim 2: Th e region presen ted b y Lemma 2 is at least as tight as the New-J erse y outer b ound. Pr oof: Again the argumen ts are similar to tho se of Claim 1 a nd are omitted. I V . C O N C L U S I O N An outer bound to the cap acity r egion to the two r eceiv er broadc ast chann el (with and without commo n info rmation) is deter mined. In both cases, this is at least as tight as the currently b est kn own bo unds. A C K N O W L E D G E M E N T The author wishes to tha nk Prof. Bruce Hajek as this work benefitted tremen dously from the discussions between the autho r and Prof. Hajek, d uring Prof. Hajek’ s visit to the Chinese Un iv ersity of Ho ng Kong in Janu ary 200 8. R E F E R E N C E S [1] I. Csiz ´ ar and J. K ¨ o rner , “Broadcast channels with confidential messages, ” IEEE T ran s. Info. Theory , vol. IT -24, pp. 339–348, May , 1978. [2] C. Nair and A. El Gamal, “ An outer bound to the capacity region of the broadca st channel, ” IE EE T rans. Info. Theory , vol. IT -53, pp. 350–355, January , 2007. [3] Y . Liang and G. Kramer, “Rate regions for relay broadc ast channels, ” IEEE T ransac tions on Informat ion Theory , vol . 53, no. 10, pp. 3517– 3535, 2007. [4] Y . Liang, G. Kramer , and S. Shamai, “Capacity outer bounds for relay broadca st channels, ” Procee dings of IEEE Inf. 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