Interference Channels with Correlated Receiver Side Information

1 Interfer ence Channels with Correlated Recei v er Side Information Nan Liu 1 , D. G ¨ und ¨ uz 1 , 2 , A. Goldsmith 1 and H. V . Poor 2 1 Dept. of Electrical Engineer ing, Stanford Uni v ., Stanf ord, CA 94305, USA 2 Dept. of Electrical Engineering, Prince ton Uni v ., Prince ton, NJ 08544, USA { nanliu@st anford.edu, dgunduz@princeton.edu , andrea@stanf ord.edu, poor@princet on.edu } . Abstract The problem of join t sour ce-chann el coding in transmitting indep endent sour ces over in terferen ce channels with correlated receiv er side information is studied. When ea ch rec eiv er has side informa tion correlated with its own desired source, it is shown th at sour ce-chan nel code separation is optimal. When each receiver has side informa tion correlated with the in terfering source, sufficient condition s for re liable tran smission are p rovided based on a joint source-ch annel cod ing sche me using th e superposition encoding a nd partial decod ing idea of Han and K obayashi. When th e r eceiver side in forma tion is a d eterministic f unction of the interfer ing source, source-ch annel code separ ation is again sh own to b e op timal. As a spe cial c ase, for a class of Z- interferen ce channels, when the side info rmation of the receiv er facing interference is a deterministic function of the interfering source, necessary and sufficient condition s for reliable tr ansmission are p rovided in the fo rm of single letter expre ssions. As a bypr oduct of these jo int source-chan nel co ding results, th e cap acity region of a class of Z-chann els with d egraded m essage sets is also p rovided. Index terms: Interference channel, joint source-channel coding , receiv er si de inform ation, source- channel s eparation theorem This research w as supported in part by the U.S. National Science Foundation und er Grants ANI-03-38807, CCF-07-28208, and CNS- 06-25637 , the D ARP A ITMANET program under Grant 1105741 -1-TFIND, and the U.S. Army Research Office under MURI award W911NF-05-1-0246 . October 24, 2018 DRAFT I . I N T RO D U C T I O N The wireless medium i s shared by m ultiple communi cation systems o perating si multaneously , which leads to interference among users transm itting over the sam e frequency band . In the simpl e scenario of two transmi tter-r ecei ver pairs, the int erference channel [1] models two simultaneous transm issions interfering with each ot her . In the classical interference channel model, the sources intended for each receiv er are i ndependent of each other , and the receiv ers decode based only on their own recei ved si gnals. On the oth er hand, in appl ications such as senso r networks, it is reasonable to assume that the receivers have access to their own correlated observations about th e underlying so urce sequences as well. These correlated observations at the recei vers can be exploited to im prove the system p erformance. Even in the absence of s ide information, a finite letter expression for the capacity region of an inter- ference chann el in the general case is unkn own. W e hav e the capacity region in the case of i nterference channels with statistically equiv alent ou tputs [2]–[4], discrete additiv e degraded i nterference channels [5], a class of determini stic interference channels [6], strong interference channels [7]–[11], a class of degraded interference channels [12], and more recently for a class of Z-interference channels [13]. The best k nown achiev able rate region is due to Han and K obayashi [9], a si mplification of wh ich is giv en in [14 ]. In a point-to-poin t scenario, the av ailability of correlated sid e i nformation at the receiv er is con- sidered in [15]. It is shown t hat the source-channel separation theorem applies in this sim ple setting and, moreover , th at Slepi an-W olf source coding followed by channel codi ng is opt imal. W ith th e a vailability of side information at the receiver , we can transmit the sou rce reli ably ove r a channel with smaller capacity than th e one required when t here is no recei ver side inform ation. Howe v er , i t is k nown that the source-channel separation theorem do es not generalize t o multi -user channels [1], [16], and necessary and s uffi cient condit ions for reliable transmissi on i n the case of correlated sources and correlated receiver side in formation are not known in general. In [17], necessary and sufficient conditions are characterized for broadcasting a common source to multiple recei vers with diffe rent correlated side information . An alternativ e achiev ability schem e for the setup of [17] is giv en i n [18]. In [19], the results of [17] are extended to b roadcast channels with degraded message sets in which the recei vers have access to parts of the underlying messages. A va ilability of m essages or message parts at the recei vers of broadcast channels from the channel coding perspecti ve is studied also in [20]–[22]. In [23], broadcasting a pair of correlated sources with correlated receiv er side informati on is stud ied. 2 The interference channel wi th correlated sources is consid ered i n [24], and a sufficient condi tion for reliable transmissi on is given. In [25 ], an interference channel with ind ependent sources, i n wh ich each recei ver has access to side information correlated with the int erfering transmitter’ s source, is considered. Necessary and suffi cient conditio ns for this setup are characterized under t he strong source-channel interference conditions, which generalize the usual strong int erference conditio ns b y considering correlated side in formation as well. Th e result of [25] shows t hat int erference cancellation is optimal e ven when the underlying channel interference is not strong, as long as the overall sou rce- channel i nterference is. In t his paper , we extend the scenario stu died in [25] to more general interference channels. W e first consider the case in which each recei ver has si de i nformation correlated wi th the source sequence it wants to decode. W e prove the optim ality of source-channel cod e separation in t his situation ; that is, the optim al performance can b e achiev ed by first compressing each of the sources using Slepian-W olf coding with respect to th e correlated receiver si de informati on, and then transmittin g t he compressed bits over the channel using an optimal interference channel code. Next, we consider the s cenario in which each receiver has side informatio n correlated with t he interfering transmit ter’ s source. As an example of such a mo del and to illustrate t he benefits o f side information about the int erfering sou rce, consider the extreme case in which each recei ver has access to t he mess age of the interfering t ransmitter . Note that th is setup is equiv alent to the restricted two- way channel model of Shanno n, wh ose capacity is characterized in [1]. In this case, each receive r can excise t he int erference from the und esired transmitt er , since its message i s exactly known at the recei ver . Here, we consider the more general case of arbitrary correlation between the receiver sid e information and the i nterfering so urce, and prop ose a joint source-channel codin g scheme similar to that of Han and K obayashi [9] taking th e si de informati on into account . L ater , we cons ider t he case in which the s ide informati on is a deterministic function of the interfering source, and s how that sou rce- channel code separation is again optimal. Finally , we consid er a special class of interference channels called Z-interference channels, i n which only one receive r faces in terference. Further focus ing on a special class of Z-interference channels satisfyi ng certain cond itions (which will be stated later), and the case in whi ch the side information is a determinist ic functi on of the interfering source, we are able to characterize necessary and sufficient condit ions for reliable transmi ssion in the form of singl e letter expressions. This setting also const itutes an example for which the general suffic iency conditions we provide are also necessary , proving their tightness for certain special cases. The rest of the paper is organized as follows. In Section II we present the system model. In Section 3 III we prove th e optimali ty of s ource-channel code separation when the s ide i nformation is correlated with the desired s ource. The case in which t he side information is correlated wi th the interfering source i s considered in Section IV. In Section IV -A, we provide s uffi cient conditions for reliable transmissio n, while in Section IV -B, we prove the optimali ty of s ource-channel code separation when the s ide i nformation is a deterministic functi on of the interfering so urce. In Section IV -C we show that, for a special source and channel model, the suffi cient conditi ons for reliabl e transmiss ion proposed in Section IV -A are also necessary , and hence we give a single letter characterization of t he necessary and sufficient condit ions for th is m odel. In Section V we characterize the capacity region of a class of Z-channels with degraded message sets . This is followed by conclus ions in Section VI. I I . S Y S T E M M O D E L An i nterference channel is compos ed of two transmitter-recei ver pairs. The underlying d iscrete memoryless channel is characterized by the transition prob ability p ( y 1 , y 2 | x 1 , x 2 ) from finite input alphabet X 1 × X 2 to finite ou tput alph abet Y 1 × Y 2 . Transmitter k has access to the source sequence { U k ,i } ∞ i =1 , k = 1 , 2 . Consider side information sequences { V k ,i } ∞ i =1 , where the source and th e side information sequences are independent and identi cally distributed (i.i.d.) and are drawn according to joint distribution p ( u 1 , v 1 ) p ( u 2 , v 2 ) over a finite alphabet U 1 × V 1 × U 2 × V 2 ; that is, the two source-side information pairs are independent of each other . For k = 1 , 2 , Transmitter k o bserves U n k and wishes to transmit it noisel essly to Recei ver k over n uses of the channel 1 . Th e encodin g function at Tra nsmitter k is f n k : U n k → X n k . W e assume that the si de information V n π ( k ) is av ailable at receiv er k , where π ( · ) is a permut ation of { 1 , 2 } . Depending o n the scenario, we will specify whether the side i nformation is correlated with the d esired source or w ith the interfering source. The decodin g functi on at recei ver k reconstructs its estim ate ˆ U k from its channel output and side information vector u sing the decoding fun ction g n k : Y n k × V n π ( k ) → U n k . 1 Here we use t he notation U n k = ( U k, 1 , . . . , U k,n ) , and similar notation for other length- n sequences. 4 P S f r a g r e p l a c e m e n t s X 1 X 2 Y 1 Y 2 U 1 U 2 ˆ U 1 ˆ U 2 V 1 V 2 Tx 1 Tx 2 Rx 1 Rx 2 p ( u 1 , v 1 ) p ( u 2 , v 2 ) p ( y 1 , y 2 | x 1 , x 2 ) Fig. 1. Interference channel model in which the receiv ers hav e access to side information correlated with t he source they want to recei ve. The probabilit y of error for this system is defined as P n e = Pr { ( U n 1 , U n 2 ) 6 = ( ˆ U n 1 , ˆ U n 2 ) } , = X ( u n 1 ,u n 2 ) ∈U n 1 ×U n 2 p ( u n 1 , u n 2 ) P n ( ˆ U n 1 , ˆ U n 2 ) 6 = ( u n 1 , u n 2 )   ( U n 1 , U n 2 ) = ( u n 1 , u n 2 ) o . Definition 1: W e say t hat a source pair ( U 1 , U 2 ) can b e reliably transmitted over a given interference channel if there exist a s equence of encoders and decoders ( f n 1 , f n 2 , g n 1 , g n 2 ) such that P n e → 0 as n → ∞ . In the foll owing sections, we consider two cases in particular . In t he first case, each receive r has side information correlated with its desired source, i.e., π ( k ) = k , k = 1 , 2 . In t he second case, each recei ver has side information correlated with the interfering so urce, i.e., π (1) = 2 and π (2 ) = 1 . In both cases, we want to exploit the a v ailability of correlated side information at the recei vers. In the first case, each transm itter needs to transmit l ess information to its int ended recei ver due to the a vailability of correlated side informati on. In th e latter case, the side inform ation is used to mitigate the effects of i nterference. For no tational con venience, we drop the subscripts on probabi lity distributions u nless the arguments of th e distributions are not lower case versions of th e correspondi ng random variables. I I I . S I D E I N F O R M A T I O N C O R R E L A T E D W I T H T H E D E S I R E D S O U R C E In this section, we consi der an interference channel in which each rec eiv er has s ide i nformation correlated with the source i t wants to decode, i. e., receive r k has access to side information V k (see Fig. 1). For this special case, we prove that the source-channel separation theorem appl ies; that is , it is o ptimal for the transm itters first t o appl y Slepian-W olf source codin g to compress their sources 5 P S f r a g r e p l a c e m e n t s X 1 X 2 Y 1 Y 2 U 1 U 2 ˆ U 1 ˆ U 2 V 1 V 2 Tx 1 Tx 2 Rx 1 Rx 2 p ( u 1 , v 1 ) p ( u 2 , v 2 ) p ( y 1 , y 2 | x 1 , x 2 ) Fig. 2. Interference channel model in which the receiv ers have access t o side information correlated with the source of the interfering transmitter . conditioned on the si de information at the correspondin g receiver , and th en to transmit the comp ressed bits over the channel us ing an optimal int erference channel code. Note that, in the general case, we do not ha ve a si ngle-letter characterization of the capacity region of the interference channel, yet we can still prove the optimality of source-channel code separation. In the proof, we use the n -lett er expression for the capacity region, which was also used in [26] to prove t he opt imality of source-channel code separation for a multipl e access channel with recei ver side i nformation and feedback. The m ain result of th is section is t he fol lowing theorem. Theor em 1: Sources U 1 and U 2 can be transmitted reliably to their respective receivers over the discrete m emoryless i nterference channel p ( y 1 , y 2 | x 1 , x 2 ) wi th side inform ation V k at receiver k , k = 1 , 2 , if ( H ( U 1 | V 1 ) , H ( U 2 | V 2 )) ∈ int ( C ) (1) where int ( · ) denotes the int erior , and C denotes th e capacity re gion of th e underlying int erference channel. Con v ersely , if ( H ( U 1 | V 1 ) , H ( U 2 | V 2 )) / ∈ C , then sources U 1 and U 2 cannot be transm itted reliably . Pr oof: A proo f of Theorem 1 i s given in Appendix I. I V . S I D E I N F O R M A T I O N C O R R E L A T E D W I T H T H E I N T E R F E R I N G S O U R C E In this section we consider the case in which Receiv er 1 has access t o V 2 while Receiv er 2 has access to V 1 , i. e., each recei ver has side i nformation abou t the interfering transm itter’ s source (see Fig. 2). W e in vestigate how the side inform ation about the interference h elps in decodin g the desired information. 6 A. Sufficient Conditions for R eliable T ransmission W e first provide sufficient cond itions for reliable transmi ssion of the sources. In the spirit of the Han-K obayashi scheme for the classical interference channel, we propose a joint source-channel coding scheme that requires th e recei vers to decode part of the interference with the help of thei r side information. In the Han-K obayashi s cheme, each transmitter splits its message into two p ieces to allow the non -intended receiv er to decode part of the interference. In our s cheme, each t ransmitter enables a quanti zed version of its source to be decoded by bo th recei vers, w here the unint ended recei ver uses i ts correlated sid e inform ation as well as the channel outpu t to decode the i nterference corresponding to t his quantized part. Suffi cient condi tions for reliable t ransmission in t his setup are giv en in t he following theorem. Theor em 2 : Sources U 1 and U 2 can be transmitted reliably o ver the int erference chann el p ( y 1 , y 2 | x 1 , x 2 ) with side information V 1 at Receiv er 2 and V 2 at Receiv er 1 if there exist random variables W 1 and W 2 such that H ( U 1 ) I ( U 1 ; W 1 | Q ) , in which th e code words are generated i.i.d. with di stribution p ( w 1 | q ) . This codebook is denoted by C 1 w . For each p ossible so urce o utput u n 1 , count the nu mber of codewords i n C 1 w that are joi ntly ty pical with u n 1 . If there are at least L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ code words in C 1 w jointly typical with u n 1 , choose one uniformly at random, and call it w n 1 ( u n 1 ) . If there are fewer t han L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ code words of C 1 w jointly typical with u n 1 , randomly choose one codeword from C 1 w to be w n 1 ( u n 1 ) . The reason why we require t he number of codewords jointly t ypical with u n 1 to be lar ge is to benefit the probabilit y of error calculation later on i n the proof. In a simil ar fashion, we generate C 2 w . Define F ( u n 1 , u n 2 ) as t he e vent that the number of w n 1 ∈ C 1 w jointly t ypical with u n 1 is larger than L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ and the number of w n 2 ∈ C 2 w jointly t ypical with u n 2 is l ar ger than L 2 2 − nI ( U 2 ; W 2 | Q ) − 2 nǫ . Next, we wi ll show that Pr { F c ( U n 1 , U n 2 ) } ≤ 3 ǫ, (58) where “ c ” denotes the compl ement. For each ( q n , u n 1 , u n 2 ) ∈ T n ǫ ( QU 1 U 2 ) , define the random variable ν ( i, u n 1 ) as follows: ν ( i, u n 1 ) is 1 if the i -th codew o rd of C 1 w is jointly typ ical with u n 1 and 0 otherwise. Then, 2 − nI ( U 1 ; W 1 | Q ) − nǫ ≤ E [ ν ( i, u n 1 ) | q n ] = Pr { ν ( i, u n 1 ) = 1 | q n } ≤ 2 − nI ( U 1 ; W 1 | Q )+ nǫ (59) V [ ν ( i, u n 1 ) | q n ] ≤ E 2 [ ν ( i, u n 1 ) | q n ] ≤ E [ ν ( i, u n 1 )] (60) 18 where E and V denote the expectation and variance, respectiv ely . Further define random var iable N ( u n 1 ) as the number of codew ords in C 1 w that are joint ly typical w ith u n 1 , i .e., N ( u n 1 ) = L 1 X i =1 ν ( i, u n 1 ) . (61) Then, from (59) and (60), we hav e L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ ≤ E [ N ( u n 1 ) | q n ] = L 1 X i =1 E [ ν ( i, u n 1 ) | q n ] ≤ L 1 2 − nI ( U 1 ; W 1 | Q )+ nǫ (62) V [ N ( u n 1 ) | q n ] = L 1 X i =1 V [ ν ( i, u n 1 ) | q n ] ≤ E [ N ( u n 1 ) | q n ] . (63) Hence, we hav e Pr  N ( u n 1 ) ≤ L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ | q n  = Pr  E [ N ( u n 1 ) | q n ] − N ( u n 1 ) ≥ E [ N ( u n 1 ) | q n ] − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ | q n  (64) ≤ Pr  E [ N ( u n 1 ) | q n ] − N ( u n 1 ) ≥ L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ | q n  (65) ≤ Pr    E [ N ( u n 1 ) | q n ] − N ( u n 1 )   ≥ L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ   q n  (66) ≤ V [ N ( u n 1 ) | q n ] ( L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ ) 2 (67) ≤ E [ N ( u n 1 ) | q n ] ( L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ ) 2 (68) ≤ L 1 2 − nI ( U 1 ; W 1 | Q )+ nǫ ( L 1 2 − nI ( U 1 ; W 1 | Q ) − nǫ − L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ ) 2 (69) ≤ ǫ (70) where (65) and (69) follows from (62), (67) follows from Chebyshev’ s inequality , (68) follows from (63), and (70) is true when n i s large enoug h. The s ame analys is appli es for u n 2 . Hence, we hav e proved that Pr { F c ( u n 1 , u n 2 ) | q n } = Pr  N ( u n 1 ) ≤ L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ or N ( u n 2 ) ≤ L 2 2 − nI ( U 2 ; W 2 | Q ) − 2 nǫ | q n  ≤ 2 ǫ (71) 19 for all ( q n , u n 1 , u n 2 ) ∈ T n ǫ ( QU 1 U 2 ) and all sufficiently lar ge n . This means that Pr { F c ( U n 1 , U n 2 ) } = X q n ,u n 1 ,u n 2 Pr { F c ( U n 1 , U n 2 ) | ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } · Pr { ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } (72) = X ( q n ,u n 1 ,u n 2 ) ∈ T n ǫ ( QU 1 U 2 ) Pr { F c ( U n 1 , U n 2 ) | ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } · Pr { ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } + X ( q n ,u n 1 ,u n 2 ) / ∈ T n ǫ ( QU 1 U 2 ) Pr { F c ( U n 1 , U n 2 ) | ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } · Pr { ( U n 1 , U n 2 , Q n ) = ( u n 1 , u n 2 , q n ) } (73) ≤ 2 ǫ + Pr { ( Q n U n 1 , U n 2 ) / ∈ T n ǫ ( QU 1 U 2 ) } (74) ≤ 3 ǫ (75) where (74) fol lows from (71 ), and (75) follows when n is lar g e enough from t he asymptotic equipar - tition property (AEP) [29]. This means that with large probabili ty , the number of sequences jointly typical with U n 1 and U n 2 in codebooks C 1 w and C 2 w are larger than L 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ and L 2 2 − nI ( U 2 ; W 2 | Q ) − 2 nǫ , respectively . Thi s fact will be used in the probabil ity of error calculation. Codebook generation : For each possib le u n 1 sequence, generate one x n 1 sequence in an i.i. d. fashion, conditioned on w n 1 ( u n 1 ) , u n 1 and q n , according to p ( x 1 | u 1 , w 1 , q ) . This x n 1 sequence is denoted by x n 1 ( u n 1 , w n 1 ( u n 1 )) . The coll ection of all x n 1 sequences will be deno ted as th e codebook C 1 x . Similarly , we generate the codebook C 2 x . Encoding : When Tra nsmitter 1 observes the sequence u n 1 , i t transm its x n 1 ( u n 1 , w n 1 ( u n 1 )) . Similarly for T ransm itter 2. Decoding : Recei ver 1 finds the uniq ue pair ( u n 1 , w n 2 ) , u n 1 ∈ U n 1 , w n 2 ∈ C 2 w , such that ( u n 1 , w n 1 ( u n 1 ) , x n 1 ( u n 1 , w n 1 ( u n 1 )) , w n 2 , y n 1 , v n 2 ) are jointly typical and declares the first component of the pair as the transmitted source. If there are m ore t han on e pair , and the first component of the pairs are the same, then t he decoder declares the transm itted source to be the first com ponent. If there are mo re than one pair , and the first component o f the pairs are not the same, an error is declared. Also, if no such pair exists, an error is declared. Sim ilarly for Receiv er 2. Pr obabil ity o f err or calculation : Denote by E ( u n 1 , w n 2 ) the event ( u n 1 , w n 1 ( u n 1 ) , X n 1 ( u n 1 , w n 1 ( u n 1 )) , w n 2 , Y n 1 , V n 2 ) ∈ T n ǫ ( U 1 W 1 X 1 W 2 Y 1 V 2 | q n ) for ( u n 1 , w n 2 ) ∈ U n 1 × C 2 w . Further denote by G ( u n 1 , u n 2 ) the 20 e vent ( u n 1 , u n 2 , w n 1 ( u n 1 ) , w n 2 ( u n 2 )) ∈ T n ǫ ( U 1 U 2 W 1 W 2 | q n ) . Then, the probabilit y of error at Recei ver 1 conditi oned on Q n = q n , denoted by P 1 e , i s given by Pr { E c ( U n 1 , w n 2 ( U n 2 )) o r [ ( u n 1 ,w n 2 ): u n 1 6 = U n 1 E ( u n 1 , w n 2 )    (76) ≤ Pr ( E c ( U n 1 , w n 2 ( U n 2 )) o r F c ( U n 1 , U n 2 ) or G c ( U n 1 , U n 2 ) or [ ( u n 1 ,w n 2 ): u n 1 6 = U n 1 E ( u n 1 , w n 2 ) ) (77) ≤ Pr { E c ( U n 1 , w n 2 ( U n 2 )) o r F c ( U n 1 , U n 2 ) or G c ( U n 1 , U n 2 ) } + Pr    [ ( u n 1 ,w n 2 ): u n 1 6 = U n 1 E ( u n 1 , w n 2 )     E ∩ F ∩ G    (78) ≤ Pr { F c ( U n 1 , U n 2 ) } + Pr { G c ( U n 1 , U n 2 ) | F } + Pr { E c ( U n 1 , w n 2 ( U n 2 )) | F ∩ G } + E    X ( u n 1 ,w n 2 ): u n 1 6 = U n 1 Pr { E ( u n 1 , w n 2 ) | E ∩ F ∩ G }    , (79) where we hav e used th e s hort-hand E , F a nd G t o denote ev ents E ( U n 1 , w n 2 ( U n 2 )) , F ( U n 1 , U n 2 ) and G ( U n 1 , U n 2 ) , respectively . The first term in (79) is boun ded by 3 ǫ as shown by (75). From the achi e vability results o f multi- terminal rate-distortion theory [31], the second term i n (79) is bounded by ǫ for sufficiently large n . The thi rd term in (79) is bound ed by ǫ for suffic iently large n based on the AEP [29 ]. Hence, from now o n, we will concentrate on the fourth term in (79). The fourth term in (79) may be up per boun ded by the sum of the following four terms, which w ill be denot ed by A 1 , A 2 , A 3 , and A 4 , respectively: A 1 △ = E          X u n 1 6 = U n 1 w n 1 ( u n 1 ) 6 = w n 1 ( U n 1 ) Pr { E ( u n 1 , w n 2 ( U n 2 )) | E ∩ F ∩ G }          (80) A 2 △ = E                X u n 1 6 = U n 1 w n 1 ( u n 1 ) 6 = w n 1 ( U n 1 ) w n 2 6 = w n 2 ( U n 2 ) Pr { E ( u n 1 , w n 2 ) | E ∩ F ∩ G }                (81) 21 A 3 △ = E          X u n 1 6 = U n 1 w n 1 ( u n 1 ) = w n 1 ( U n 1 ) Pr { E ( u n 1 , w n 2 ( U n 2 )) | E ∩ F ∩ G }          (82) and A 4 △ = E                X u n 1 6 = U n 1 w n 1 ( u n 1 ) = w n 1 ( U n 1 ) w n 2 6 = w n 2 ( U n 2 ) Pr { E ( u n 1 , w n 2 ) | E ∩ F ∩ G }                . (83) First, we upper bound A 1 . Define the set B 1 = { u n 1 ∈ U n 1 : u n 1 6 = U n 1 , w n 1 ( u n 1 ) 6 = w n 1 ( U n 1 ) , ( u n 1 , w n 1 ( u n 1 )) ∈ T n ǫ ( U 1 W 1 | Y n 1 V n 2 w n 2 ( U n 2 ) q n ) } . (84) Then, we hav e E  |B 1 |   E ∩ F ∩ G  ≤ 2 nH ( U 1 | Y 1 ,V 2 ,W 2 ,Q )+ nǫ 2 nH ( W 1 | U 1 ,Y 1 ,V 2 ,W 2 ,Q )+ nǫ 2 − nH ( W 1 | U 1 ,Q )+ nǫ . (85) Hence, we may write A 1 = E    X u n 1 ∈B 1 Pr { E ( u n 1 , w n 2 ( U n 2 )) | E ∩ F ∩ G }    (86) ≤ E  |B 1 | max u n 1 ∈B 1 Pr { E ( u n 1 , w n 2 ( U n 2 )) | E ∩ F ∩ G }  (87) = E  |B 1 | max u n 1 ∈B 1 Pr { X n 1 ( u n 1 , w n 1 ( u n 1 )) ∈ T n ǫ ( X 1 | u n 1 w n 1 ( u n 1 ) w n 2 ( U n 2 ) Y n 1 V n 2 q n ) | E ∩ F ∩ G }  (88) ≤ E  |B 1 | max u n 1 ∈B 1 2 nH ( X 1 | U 1 ,W 1 ,W 2 ,Y 1 ,V 2 ,Q )+ nǫ 2 − nH ( X 1 | U 1 ,W 1 ,Q )+ nǫ   E ∩ F ∩ G  (89) ≤ 2 nH ( U 1 ) 2 − nI ( U 1 ,W 1 ,X 1 ; Y 1 ,V 2 | W 2 ,Q )+5 nǫ (90) ≤ 2 nH ( U 1 ) 2 − nI ( X 1 ; Y 1 ,V 2 | W 2 ,Q )+5 nǫ (91) where (91) follows because t he distri bution in (13) sati sfies the Markov chain relatio nship ( U 1 , W 1 ) → ( X 1 , W 2 , Q ) → ( V 2 , Y 1 ) . Next, we upper bound A 2 . Define the set B 2 = { u n 1 ∈ U n 1 , w n 2 ∈ C 2 w : u n 1 6 = U n 1 ,w n 1 ( u n 1 ) 6 = w n 1 ( U n 1 ) , w n 2 6 = w n 2 ( U n 2 ) , ( u n 1 , w n 1 ( u n 1 ) , w n 2 ) ∈ T n ǫ ( U 1 W 1 W 2 | Y n 1 V n 2 q n ) } . (92) Then, we hav e E {|B 2 |} ≤ 2 nH ( W 2 | Y 1 ,V 2 ,Q )+ nǫ 2 − nH ( W 2 | Q )+ nǫ ( L 2 − 1 ) 2 nH ( U 1 | W 2 ,Y 1 ,V 2 ,Q )+ nǫ 2 nH ( W 1 | U 1 ,W 2 ,Y 1 ,V 2 ,Q )+ nǫ 2 − nH ( W 1 | U 1 ,Q )+ nǫ . (93) 22 Similarly t o (86)-(90), we m ay write A 2 = E    X ( u n 1 ,w n 2 ) ∈B 2 Pr { E ( u n 1 , w n 2 ) | E ∩ F ∩ G }    (94) ≤ 2 nH ( U 1 ) L 2 2 − nI ( U 1 ,W 1 ,X 1 ,W 2 ; V 2 ,Y 1 | Q )+7 nǫ (95) = 2 nH ( U 1 ) L 2 2 − nI ( X 1 ,W 2 ; V 2 ,Y 1 | Q )+7 nǫ (96) where (96) follows from th e same reason as (91). Next, we up per boun d A 3 . Define the set B 3 = { u n 1 ∈U n 1 : u n 1 6 = U n 1 , w n 1 ( u n 1 ) = w n 1 ( U n 1 ) , u n 1 ∈ T n ǫ ( U 1 | w n 1 ( U n 1 ) Y n 1 V n 2 w n 2 ( U n 2 ) q n ) } . (97) Then, we hav e E  |B 3 |   E ∩ F ∩ G  ≤ 2 nH ( U 1 | W 1 ,Y 1 ,V 2 ,W 2 ,Q )+ nǫ 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ L 1 (98) which follows from the fa ct that we always choos e random ly from at least L 1 2 nI ( U 1 ; W 1 | Q ) − 2 nǫ choices to get w n 1 ( u n 1 ) . Similarly to (86)-(90), we may write A 3 = E    X u n 1 ∈B 3 Pr { E ( u n 1 , w n 2 ( U n 2 )) | E ∩ F ∩ G }    (99) ≤ 2 nH ( U 1 ) L 1 2 − nI ( U 1 ,X 1 ; Y 1 ,V 2 | W 1 ,W 2 ,Q )+5 nǫ (100) ≤ 2 nH ( U 1 ) L 1 2 − nI ( X 1 ; Y 1 ,V 2 | W 1 ,W 2 ,Q )+5 nǫ (101) where (101) follows from t he same reason as (91). Finally , we upp er boun d A 4 . Define the set B 4 = { u n 1 ∈ U n 1 , w n 2 ∈ C 2 w : u n 1 6 = U n 1 ,w n 1 ( u n 1 ) = w n 1 ( U n 1 ) , w n 2 6 = w n 2 ( U n 2 ) , ( u n 1 , w n 2 ) ∈ T n ǫ ( U 1 W 2 | w n 1 ( U n 1 ) Y n 1 V n 2 q n ) } . (102) Then, we hav e E  |B 4 |   E ∩ F ∩ G  ≤ 2 nH ( W 2 | Y 1 ,V 2 ,W 1 ,Q )+ nǫ 2 − nH ( W 2 | Q )+ nǫ ( L 2 − 1) 2 nH ( U 1 | W 1 ,W 2 ,Y 1 ,V 2 ,Q )+ nǫ 1 2 − nI ( U 1 ; W 1 | Q ) − 2 nǫ L 1 . (103) Similarly t o (86)-(90), we m ay write A 4 = E    X ( u n 1 ,w n 2 ) ∈B 4 [ E ( u n 1 , w n 2 ) | E ∩ F ∩ G ]    (104) ≤ L 2 L 1 2 nH ( U 1 ) 2 − nI ( U 1 ,X 1 ,W 2 ; Y 1 ,V 2 | W 1 ,Q )+7 nǫ (105) ≤ L 2 L 1 2 nH ( U 1 ) 2 − nI ( X 1 ,W 2 ; Y 1 ,V 2 | W 1 ,Q )+7 nǫ (106) 23 where (106) follows from t he same reason as (91). W e have sim ilar probabil ity of error calculation s at Recei ver 2. Since P n e ≤ E Q n [ P 1 e + P 2 e ] , (107) for thi s achiev abil ity scheme, as long as the following equations are satisfied, H ( U 1 ) ≤ I ( X 1 ; V 2 , Y 1 | W 2 , Q ) , (108) H ( U 1 ) − log L 1 ≤ I ( X 1 ; V 2 , Y 1 | W 1 , W 2 , Q ) , (109) H ( U 1 ) + log L 2 ≤ I ( W 2 , X 1 ; V 2 , Y 1 | Q ) , H ( U 1 ) + log L 2 − lo g L 1 ≤ I ( W 2 , X 1 ; V 2 , Y 1 | W 1 , Q ) , (110) H ( U 2 ) ≤ I ( X 2 ; V 1 , Y 2 | W 1 , Q ) , H ( U 2 ) − log L 2 ≤ I ( X 2 ; V 1 , Y 2 | W 1 , W 2 , Q ) , (111) H ( U 2 ) + log L 1 ≤ I ( W 1 , X 2 ; V 1 , Y 2 | Q ) , H ( U 2 ) + log L 1 − lo g L 2 ≤ I ( W 1 , X 2 ; V 1 , Y 2 | W 2 , Q ) , (112) log L 1 ≥ I ( U 1 ; W 1 | Q ) and (113) log L 2 ≥ I ( U 2 ; W 2 | Q ) , (114) for some p ( q ) , p ( w 1 , x 1 | u 1 , q ) , and p ( w 2 , x 2 | u 2 , q ) , the probabil ity of error is arbitrarily small for suffi ciently large n . By Fourier-Motzkin eli mination, we obtain the sufficient condit ions given in Theorem 2. A P P E N D I X I I I P R O O F O F L E M M A 1 W e first start with the p roof of achie vability . Fix distributions p ( s 1 s ) , p ( x 1 | s 1 s ) , p ( s 2 s ) and p ( x 2 | s 2 s ) . For codebook at Transmitter k , k = 1 , 2 , we generate an in ner codebook of 2 N R ks i.i.d. code words of length N with p robability Q N i =1 p ( s k s,i ) . Then, for each codew ord of the inner codebook, we generate an outer codebook of 2 N R kp i.i.d. codew o rds o f length N wit h p robability Q N i =1 p ( x k ,i | s k s,i ) . For W k s = w k s and W k p = w k p , T ransmitter k sends the w k p -th codew ord of the w k s -th outer codebook. For decodi ng, Receiv er 1 finds the code word in al l possi ble outer codebo oks that is jointly typical with th e receiv ed sequence and the w 2 s -th codeword o f the inner codeboo k of T ransmitter 2. Similarly for Recei ver 2. T he probability of error analy sis follows from standard arguments [29], and we can 24 show that the probabi lity of error can be drive n to zero as N → ∞ , as long as the rates satisfy the following conditions: R 1 p ≤ I ( X 1 ; Y 1 | S 1 s , S 2 s ) , (115) R 1 s + R 1 p ≤ I ( X 1 ; Y 1 | S 2 s ) , (116) R 2 p ≤ I ( X 2 ; Y 2 | S 1 s , S 2 s ) and (117) R 2 s + R 2 p ≤ I ( X 2 ; Y 2 | S 1 s ) . (118) For each n , similarly to [1, Theorem 5], by treating the interference channel p n ( y n 1 , y n 2 | x n 1 , x n 2 ) , which is a product channel of p ( y 1 , y 2 | x 1 , x 2 ) , as a memoryless channel, we conclude that the rates satisfying the following conditions are achiev able for any n : R 1 p ≤ 1 n I ( X n 1 ; Y n 1 | S n 1 s , S n 2 s ) , (119) R 1 s + R 1 p ≤ 1 n I ( X n 1 ; Y n 1 | S n 2 s ) , (120) R 2 p ≤ 1 n I ( X n 2 ; Y n 2 | S n 1 s , S n 2 s ) and (121) R 2 s + R 2 p ≤ 1 n I ( X n 2 ; Y n 2 | S n 1 s ) , (122) i.e., any rate quadruplet ( R 1 s , R 1 p , R 2 s , R 2 p ) ∈ G n is achiev able. By the definition of the capacity region, the limiting points of G n are also achi e vable, and thus, we hav e prov ed the achiev ability of all the p oints in C I . W e next prove the con verse. F o r any  2 nR 1 s , 2 nR 1 p , 2 nR 2 s , 2 nR 2 p , n  code, denote its input to the channel as random variables X n 1 and X n 2 and t he output of the channel as random variables Y n 1 , Y n 2 . Arbitrarily choose M 1 s △ = 2 nR 1 s n -letter sequences u 1 s 1 , u 1 s 2 , · · · , u 1 s M 1 s all in X n 1 , and M 2 s △ = 2 nR 2 s n -letter sequences u 2 s 1 , u 2 s 2 , · · · , u 2 s M 2 s all in X n 2 . Form a one-to-one correspond ence between W 1 s , W 2 s and S n 1 s , S n 2 s , respectively b y p n ( S n 1 s = u n | W 1 s = w 1 s ) =    1 i f u n = u 1 s w 1 s , w 1 s = 1 , 2 , · · · , M 1 s 0 ot herwise (123) p n ( S n 2 s = u n | W 2 s = w 2 s ) =    1 i f u n = u 2 s w 2 s , w 2 s = 1 , 2 , · · · , M 2 s 0 ot herwise (124) 25 By Fano’ s inequality [29], we hav e nR 1 p = H ( W 1 p ) = H ( W 1 p | W 1 s , W 2 s ) (125) = I ( W 1 p ; Y n 1 | W 1 s , W 2 s ) + H ( W 1 p | Y n 1 , W 1 s , W 2 s ) (126) ≤ I ( W 1 p ; Y n 1 | W 1 s , W 2 s ) + H ( W 1 p | Y n 1 , W 2 s ) (127) ≤ I ( W 1 p ; Y n 1 | W 1 s , W 2 s ) + nδ ( P n e ) (128) ≤ I ( X n 1 ; Y n 1 | W 1 s , W 2 s ) + nδ ( P n e ) (129) = I ( X n 1 ; Y n 1 | S n 1 s , S n 2 s ) + nδ ( P n e ) (13 0) where δ ( x ) in (128) is a non-negativ e function approaching zer o as x → 0 , (129) follows from data pro- cessing inequali ty [29] because the dist ributions f actor as p ( w 1 p ) p ( w 1 s ) p ( x n 1 | w 1 p , w 1 s ) p ( w 2 p ) p ( w 2 s ) p ( x n 2 | w 2 p , w 2 s ) p ( y n 1 | x n 1 , x n 2 ) and satisfy the Marko v cha in relationship ( W 1 p , W 1 s ) → ( X n 1 , W 2 s ) → Y n 1 , and (130) follows from the definitions of the sequences S n 1 s and S n 2 s in (123) and (124), respecti vely . W e also ha ve nR 1 s + nR 1 p = H ( W 1 s , W 1 p ) = H ( W 1 s , W 1 p | W 2 s ) (131) = I ( W 1 s , W 1 p ; Y n 1 | W 2 s ) + H ( W 1 s , W 1 p | Y n 1 , W 2 s ) (132 ) ≤ I ( W 1 s , W 1 p ; Y n 1 | W 2 s ) + nδ ( P n e ) (133) ≤ I ( X n 1 ; Y n 1 | W 2 s ) + nδ ( P n e ) (134) = I ( X n 1 ; Y n 1 | S n 2 s ) + nδ ( P n e ) (135) where (134) follows from the s ame reason as (129), and (135) follows from the same reason as (130). Similarly , we have nR 2 p ≤ I ( X n 2 ; Y n 2 | S n 1 s , S n 2 s ) + nδ ( P n e ) (136) nR 2 s + nR 2 p ≤ I ( X n 2 ; Y n 2 | S n 1 s ) + nδ ( P n e ) . (137) Hence, we hav e proved that for all n , ( R 1 s − δ ( P n e ) , R 1 p − δ ( P n e ) , R 2 s − δ ( P n e ) , R 2 p − δ ( P n e )) ∈ G n . (138) Since the region C I as defined in (33) contain s G n for every n [1, Theorem 5], we hav e ( R 1 s − δ ( P n e ) , R 1 p − δ ( P n e ) , R 2 s − δ ( P n e ) , R 2 p − δ ( P n e )) ∈ C I (139) 26 for all n . For codes wh ere P n e → 0 as n → ∞ , we have ( R 1 s , R 1 p , R 2 s , R 2 p ) ∈ C I (140) since C I is closed [1, T heorem 5 ]. Thi s concludes the con verse part o f the proof. A P P E N D I X I V P R O O F O F T H E O R E M 3 The achiev ability part of the proof is straightforward. If (34) holds, then there exists a rate quadruplet ( R 1 s , R 1 p , R 2 s , R 2 p ) in the interior of C such that H ( V k ) ≤ R k s and H ( U k | V k ) ≤ R k p for k = 1 , 2 . T ransm itter k first compresses V k into index W k s with rate H ( V k ) , and then U k | V k = v k into index W k p ( v k ) into rate H ( U k | V k ) , for all v k in the typical set. Then the indices can be transmitted reliably over the channel since ( R 1 s , R 1 p , R 2 s , R 2 p ) is in t he capacity re gion of the und erlying interference channel wi th mess age side in formation W 1 s at Receiv er 2 and W 2 s at Receive r 1. T o prove the con verse, we write nH ( U 1 | V 1 ) = H ( U n 1 | V n 1 ) = H ( U n 1 | V n 1 , V n 2 ) (141) = I ( U n 1 ; Y n 1 | V n 1 , V n 2 ) + H ( U n 1 | Y n 1 , V n 1 , V n 2 ) (142) ≤ I ( U n 1 ; Y n 1 | V n 1 , V n 2 ) + H ( U n 1 | Y n 1 , V n 2 ) (143) ≤ I ( U n 1 ; Y n 1 | V n 1 , V n 2 ) + nδ ( P n e ) (144) ≤ I ( X n 1 ; Y n 1 | V n 1 , V n 2 ) + nδ ( P n e ) (145) where (144) fol lows from Fano’ s in equality and δ ( x ) is a non-negativ e function approaching zero as x → 0 , and (145) follows from the data p rocessing inequ ality , in other words, from the Markov chain relationship ( U n 1 , V n 1 ) → ( X n 1 , V n 2 ) → Y n 1 . W e can also write nH ( V 1 ) + nH ( U 1 | V 1 ) = nH ( U 1 , V 1 ) (146) = nH ( U 1 ) (147) = H ( U n 1 ) (148) = H ( U n 1 | V n 2 ) (149) = I ( U n 1 ; Y n 1 | V n 2 ) + H ( U n 1 | Y n 1 , V n 2 ) (150) ≤ I ( U n 1 ; Y n 1 | V n 2 ) + nδ ( P n e ) (151) ≤ I ( X n 1 ; Y n 1 | V n 2 ) + nδ ( P n e ) (152) 27 where (147) follows because V 1 is a deterministic functi on of U 1 , and (151) follows from Fano’ s inequality , and (152) follows from the same reasoni ng as applied to (14 5). Similarly , we hav e nH ( U 2 | V 2 ) ≤ I ( X n 2 ; Y n 2 | V n 1 , V n 2 ) + nδ ( P n e ) and (153) nH ( V 2 ) + nH ( U 2 | V 2 ) ≤ I ( X n 2 ; Y n 2 | V n 1 ) + nδ ( P n e ) . (154) Hence, from (145), (152), (15 3) and (154), we have ( H ( V 1 ) − δ ( P n e ) , H ( U 1 | V 1 ) − δ ( P n e ) , H ( V 2 ) − δ ( P n e ) , H ( U 2 | V 2 ) − δ ( P n e )) ∈ G n (155) which by the same reasoning as applied to (139 ) and (140), for codes where P n e → 0 as n → ∞ , we hav e ( H ( V 1 ) , H ( U 1 | V 1 ) , H ( V 2 ) , H ( U 2 | V 2 )) ∈ C I (156) which concludes the proof. A P P E N D I X V P R O O F O F L E M M A 2 Due to the fact that the proof of this lemm a is very simi lar to the proof of the capacity region in [13], we om it certain detail s. For notational con venience, denote the channel of p ( y 1 | x 1 ) as ¯ V 1 and the channel p ( y 2 | x 1 , x 2 ) as ¯ V 2 , where ¯ V 1 ( a | b ) = Pr { Y 1 = a | X 1 = b } , (157) and ¯ V 2 ( c | b, d ) = Pr { Y 2 = c | X 1 = b, X 2 = d } . (158) A. Con verse Result The con verse result deri ved in this subsectio n is valid for any Z-interference channel satisfying Condition 1. The tool t hat we are usi ng comes from the fol lowing l emma. Lemma 3: [30, p age 314 , eqn (3.34)] For any n , and any random variables Y n and Z n and W , we ha ve H ( Z n | W ) − H ( Y n | W ) = n X i =1 ( H ( Z i | Y i − 1 , Z i +1 , Z i +2 , · · · , Z n , W ) − H ( Y i | Y i − 1 , Z i +1 , Z i +2 , · · · , Z n , W )) . (159) 28 Since the rate triplets ( R 1 s , R 1 p , R 2 p ) is achiev abl e, t here exist two sequences of codebooks 1 and 2, denoted by C n 1 and C n 2 , o f rate R 1 s + R 1 p and R 2 p , and probabi lity of error l ess than ǫ n , wh ere P n e → 0 as n → ∞ . Let us define X n 1 and X n 2 be uniformly distributed o n codebooks 1 and 2, respectiv ely . Let Y n 1 be connected via ¯ V n 1 to X n 1 , Y n 2 be connected via ¯ V n 2 to X n 1 and X n 2 . W e start t he conv erse with Fano’ s inequality [29], nR 1 p = H ( W 1 p ) (160) ≤ I ( W 1 p ; Y n 1 ) + nδ ( P n e ) (161) ≤ I ( W 1 p ; Y n 1 | W 1 s ) + nδ ( P n e ) (162) = H ( Y n 1 | W 1 s ) − H ( Y n 1 | W 1 s , W 1 p , X n 1 ) + nδ ( P n e ) (163) = H ( Y n 1 | W 1 s ) − H ( Y n 1 | X n 1 ) + nδ ( P n e ) (164) = H ( Y n 1 | W 1 s ) − n X i =1 H ( Y 1 i | X 1 i ) + nδ ( P n e ) (165) where (162) fol lows from t he fact that W 1 s and W 1 p are independent, (163) follows from the fac t that without lo ss of g enerality , we may consider determi nistic encoders, (164) follows from the Markov chain relations hip ( W 1 s , W 1 p ) → X n 1 → Y n 1 , and (165) follows from the m emoryless nature of ¯ V n 1 . W e also ha ve nR 1 s + nR 1 p = H ( W 1 p , W 1 s ) (166) ≤ I ( W 1 p , W 1 s ; Y n 1 ) + nδ ( P n e ) (167) ≤ I ( X n 1 ; Y n 1 ) + nδ ( P n e ) (168) ≤ n X i =1 I ( X 1 i ; Y 1 i ) + nδ ( P n e ) (169) where (168) follows from t he data processing i nequality [29]. Furthermore, we have nR 2 p = H ( W 2 p ) = H ( W 2 p | W 1 s ) (170) ≤ I ( W 2 p ; Y n 2 | W 1 s ) + nδ ( P n e ) (171) ≤ I ( X n 2 ; Y n 2 | W 1 s ) + nδ ( P n e ) (172) = H ( Y n 2 | W 1 s ) − H ( Y n 2 | X n 2 , W 1 s ) + nδ ( P n e ) (173) ≤ n X i =1 H ( Y 2 i ) − H ( Y n 2 | X n 2 , W 1 s ) + nδ ( P n e ) (174) ≤ nτ − H ( Y n 2 | X n 2 , W 1 s ) + nδ ( P n e ) (175) 29 where (170) follows from the independence of W 2 p and W 1 s , (172) follows from the Markov chain relationship W 2 p → ( X n 1 , W 1 s ) → Y n 2 , (174) follows from the fact that condition ing reduces entropy , and (175) follows from the definition of τ in (35). Let us define ano ther channel, ˆ V 2 : X 1 → Y 2 , as ˆ V 2 ( t | x 1 ) = V 2 ( t | x 1 , ¯ x 2 ) , (17 6) where ¯ x 2 is an arbitrary element in X 2 . Further , let us define another sequence of random variables, T n , whi ch is connected via ˆ V n 2 , the m emoryless channel ˆ V 2 used n times, to X n 1 , i.e., T i → X 1 i → T { i } c , X 1 { i } c , X n 2 , Y n 1 , Y n 2 . Also define ¯ x n 2 as th e n -sequence wit h ¯ x 2 repeated n ti mes. It i s easy to see that H ( Y n 2 | X n 2 , W 1 s ) = X x n 2 ∈C n 2 2 nR 1 s X w =1 1 2 nR 1 s 1 2 nR 2 p H ( Y n 2 | X n 2 = x n 2 , W 1 s = w ) (177) = 2 nR 1 s X w =1 1 2 nR 1 s H ( Y n 2 | X n 2 = ¯ x n 2 , W 1 s = w ) (178) = 2 nR 1 s X w =1 1 2 nR 1 s H ( T n | W 1 s = w ) (179) = H ( T n | W 1 s ) (180) where (178) follows from the fact th at the channel u nder consideration satisfies con dition 1, and (179) follows from the definition o f T n . By apply ing Lemma 3, we hav e H ( T n | W 1 s ) − H ( Y n 1 | W 1 s ) = n X i =1 H ( T i | Y i − 1 1 , T i +1 , T i +2 , · · · , T n , W 1 s ) − H ( Y 1 i | Y i − 1 1 , T i +1 , T i +2 , · · · , T n , W 1 s ) . (181) Furthermore, si nce conditio ning reduces entropy , we can writ e H ( Y n 1 | W 1 s ) = n X i =1 H ( Y 1 i | Y i − 1 1 , W 1 s ) ≥ n X i =1 H ( Y 1 i | Y i − 1 1 , T i +1 , T i +2 , · · · , T n , W 1 s ) . (182) Define the following auxiliary random variables, W i = Y i − 1 1 , T i +1 , T i +2 , · · · , T n , W 1 s , i = 1 , 2 , · · · , n. (183) Further define Q as a rando m variable that is un iform on the set { 1 , 2 , · · · , n } and independent of e verything else. Al so, define the following auxili ary random variables: W = ( W Q , Q ) , X 1 = X 1 Q , Y 1 = Y 1 Q and T = T Q . (184) 30 Then, from (181) and (182), we hav e n − 1 ( H ( T n | W 1 s ) − H ( Y n 1 | W 1 s )) = H ( T | W ) − H ( Y 1 | W ) and (185) n − 1 H ( Y n 1 | W s ) ≥ H ( Y 1 | W ) . (186) Due to the memoryless natu re of ¯ V n 1 and ˆ V n 2 , the fact that Q is independent of ev erything else, and the M arkov chain relation ship T i → X 1 i → Y 1 i , for i = 1 , 2 , · · · , n , t he jo int dist ribution of W , X 1 , Y 1 , T satisfies p ( w , x 1 , y 1 , t ) = p ( w ) p ( x 1 | w ) V 1 ( y 1 | x 1 ) V 2 ( t | x 1 , ¯ x 2 ) . (187) From (185) and (186), we may conclude that there exists a nu mber γ ≥ 0 such that 1 n H ( T n | W 1 s ) = H ( T | W ) + γ , 1 n H ( Y n 1 | W 1 s ) = H ( Y 1 | W ) + γ . (188) By com bining (165), (169), (175), (180), (187), and (188), and allowing n → ∞ , we obtain the following con verse result: for any Z-interference channel that sati sfies Condi tion 1 and the case where Recei ver 2 has side information W 1 s , t he achiev able rate t riplets ( R 1 s , R 1 p , R 2 p ) m ust sati sfy R 1 p ≤ H ( Y 1 | W ) + γ − H ( Y 1 | X 1 ) , (189) R 1 s + R 1 p ≤ I ( X 1 ; Y 1 ) and (190) R 2 p ≤ τ − H ( T | W ) − γ , (191) for some number γ ≥ 0 and distribution p ( w ) p ( x 1 | w ) , where the mut ual informatio ns and entropies are ev aluated using p ( w , x 1 , y 1 , t ) = p ( w ) p ( x 1 | w ) V 1 ( y 1 | x 1 ) V 2 ( t | x 1 , ¯ x 2 ) . B. Achievability Result The achie vability result derived in this subsection is valid for any Z-interference channel. W e design a codebook at T ransmit ter 1 such that the inner codebook carries t he side in formation at the Receive r 2, i.e., W 1 s , and part of W 1 p , and the o uter codeboo k carries the remaining part of W 1 p . M ore s pecifically , the inner cod ebook is of rate R 1 s + γ , and the outer codebook is of rate R 1 p − γ . Then, we hav e the achie vable rate region as the uni on over all p ( w ) p ( x 1 | w ) p ( x 2 ) of R 1 p ≤ H ( Y 1 | W ) + γ − H ( Y 1 | X 1 ) (192) R 1 s + R 1 p ≤ I ( X 1 ; Y 1 ) ( 193) R 2 p ≤ I ( X 2 ; Y 2 | W ) and (194) R 2 p ≤ I ( W, X 2 ; Y 2 ) − γ , (195) 31 where the mutual informations are e valuated using p ( w , x 1 , x 2 , y 1 , y 2 ) = p ( w ) p ( x 1 | w ) p ( x 2 ) V 1 ( y 1 | x 1 ) V 2 ( y 2 | x 1 , x 2 ) . C. 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