A Formulation of the Channel Capacity of Multiple-Access Channel

1 A F ormulation of the Chann el Capacity of Multiple-Ac cess Channel Y oichiro W atanabe , Member , IEEE an d K oichi Kamoi, Memb er , IEEE Abstract — The necessary and suffi cient condition of the ch an- nel capacity is rigorously fo rmulated for the N -user d iscrete memoryless multiple-access chann el (MA C). The essence of the fo rmulation is to in vok e an elementary MA C wher e sizes of input alphabets are not gre ater than the size of ou tput alphabet. The main objectiv e is to demonstrate that the channel c apacity of an M A C is achieved by an elementary MA C included in the original MA C. The proof is quite straightforward by the very definition of the elementary MA C. M oreo ver it is p ro ved th at the Kuhn-T u cker conditions of the elementary MA C are strictly sufficient and obv iously necessary f or the channel capacity . The latter proof requires some step s such that for the elementary MA C ev ery solution of the Kuhn-T ucker conditions rev eals itself as loca l max imum on the domain of all possible input probability distributions an d then it achieves the channel capacity . As a result, in respect of the channel capacity , t he MA C in general can be regarded as an aggr egate of a finite number of elementary MA C’s. Index T erms — multiple-access channel (MA C), elementary MA C, master elementary set, ch annel capacity , Kuhn-T ucker conditions, capacity region, boundary equation I . I N T RO D U C T I O N T HE chann el capacity is without question recognized as an essential sub ject o f the (discrete memory less) multiple- access channe l (MA C) with N input-termin als and one outp ut- terminal. Since it is defined as the maximum of the mutua l informa tion, we ar e familiar with the so-called Kuhn-Tucker condition s as necessary to achieve the chann el ca pacity . Up to now , however , the Kuhn-T ucker condition s are not entirely examined as sufficient for the N -user MA C except fo r the simplest case o f single user d iscrete memoryless ch annel (DMC). T hus it is natural to ask how the su fficienc y cou ld be formulated for the case of MA C in gener al. In this pa per , we demo nstrate that there e xists a non-trivial MA C wh ere the Kuhn-T ucker condition s ar e strictly sufficient (and obviously necessary) for the channel capacity . W e refe r to it as a n elementary MA C who se sizes of inp ut a lphabets are not greater than the size of output alphab et. Evide ntly the DMC is an elementary MA C. The most of this paper is de voted to the proof that the Kuhn-T ucker con ditions ar e suffi cient (the n ecessity is self- evident) for the cha nnel capacity of the elementary MA C. On the o ther h and, for any g iv en N -user MA C w e can uniquely determine a finite set of eleme ntary MAC’ s. It is an aggregate o f the lar gest possible elementary MA C’ s included in the g i ven N -user MAC an d is referr ed to as the master Y oichiro W atanabe is with the Department of Intelligent Information Eng. and Sci., Doshisha Uni versity , Kyota nabe, Ky oto, 619-0321 Japan. Ko ichi Kamoi is with Kno wledge X Inc., W ako-Shi, Saitama, 351-0 114 Japan. elementary set to be den oted by Ω N . W e demon strate that the channe l cap acity of the N -u ser MA C is achieved b y the channel ca pacity of an elemen tary MAC of the set Ω N . The proof here appears quite straightfor ward by merely appealing to the very definition of the e lementary MAC witho ut asking for any othe r featur es such that the Kuhn-T ucker condition s are sufficient. Thus an MAC in general can be regarded as simply an aggregate of elementar y MAC’ s where the Kuhn-T ucker con- ditions are n ecessary and sufficient for the channel cap acity . Roughly sp eaking, an MA C comprises a finite n umber o f elementary MA C’ s. This statement is a basic idea behind our formu lation of this pap er . Here we must emp hasize that se veral steps are requ ired to prove th e sufficiency of the Kuhn -T ucker cond itions of the elementary MA C. In fact, we n eed to prove two d istinctiv e features: The first is that f or the elem entary MA C every solution of the Kuhn-T u cker cond itions is local maximu m on the do main o f all po ssible input pro bability distribution ( IPD) (or the proba bility simp lex, see Cover [1] fo r this terminolo gy and we r efer to the m as IPD vectors for our p urposes). The second is that for the elemen tary MA C a set of I PD vectors for which the value of the mutual in formation is not smaller than th e arb itrary p ositiv e nu mber is c onnected o n the dom ain of all possible IPD vectors. T o prove the secon d pr operty of connected ness we requ ire the first p roperty o f local m aximum. Then it follows after a bit of pro cedures that solutions of the Kuhn-T u cker conditions are un iquely dete rmined, th at is, each solution takes the same value for the mutual infor mation an d therefor e it ach iev es the channel capacity . For th e explicit descrip tion o f our co ncept we take a logical stream as follows : After defining the e lementary MAC an d determinin g the set Ω N , we first prove as the main th eorem that the channel capacity of an N -user MA C is achiev ed by the channel capacity of an elem entary MAC of Ω N and then we prove as the seco nd the orem that the Kuhn- T ucker conditio ns are sufficient f or the chann el capacity of the e lementary MAC . These are the main objective of this paper . After Shan non [2], the study of multiuser channel (m ulti- terminal network ) has long been carried ou t in various fields including MAC, broadcast chan nel, relay chan nel, interf erence channel, two-way chann el and so forth. Th e c hannel coding theorem was proved indepen dently by Liao [3], Ahlswede [4] and M eulen [ 5]. These are f ollowed by many auth ors ([ 6], [7], [8], [9], [1 0], [11], [12]) to p rovide a deeper insight into the capacity regio n. Recently , information -theoretic approach has been adopte d to large scale networks, such that code division multiple-access chann el, contin uous time multiple- access chann el and space-time multiple- access channel (e.g ., 2 [13], [14], [15], [ 16], [17]), as we know . Also a com putation proced ure f or the ch annel capacity o f MAC has been devel- oped (e.g., [18]). The p urpose of the study of MAC is mo stly to in ves- tigate the multiu ser coding that retains both reliability and efficiency . T he in vestigation h as been car ried out mostly o n the computation al calculations for pr actical applications. No t much ha s bee n mad e for the mathematical rig orousness of the formu lation since it app ears rather h ard to solve a non-lin ear optimization problem of th e mutual information with several variables u nder constraints. W e have been highly expecting a theoretical found ation, in particular , for the rigorous ev alu ation of the channel capacity and the exact deter mination of th e capacity region for the M A C in gene ral. These can provide us with the mathematical essence as w ell as the fine stru cture inherent in th e MAC . Also we believe th at these can in part co mplement the comp utational appro aches to various applications as well. In the p ast, for the MA C of two-user and binary output [19], we have shown that the Kuhn-T ucker conditio ns are necessary and suf ficient for the channel capacity . T he basic idea w as to identify the channel matrix of the MA C as a linear m apping from the conv ex-closure o f IPD vectors to the range of outpu t probab ility distributions. Now we expan d the idea and remind a clear concep tion to de scribe the MA C as a pair of chann el matrix P and d omain X where X is a set of IPD vectors and P is interpreted as a mapping ( non-linea r in general) from IPD vectors to o utput prob ability distributions. Any quantity such as m utual infor mation and so for th is consider ed as a function of IPD vector s defined on a restricted do main (a sub-set) of X . These ar e seemingly n on-stand ard in contrast to the or dinary description of inf ormation theor y as in [1]. Howev er , we assure ourselves th at the se con ceptions in cluding the notation adopted in this paper are so successful to overcome some difficulties and cumbe rsome procedure s underly ing in the non- linear optim ization pro blem relatin g to th e mutual inform ation of the MA C. In Section II we descr ibe some expressions an d d efini- tions to be used in this pap er . I n pa rticular we introdu ce an elementary MA C and th e master elemen tary set for the MA C. I n Section III we p rove the main theorem of this paper as Theorem 1 followed by indicating the value of this theorem. In Section IV we inv estigate distinctiv e features of the elementar y MA C that are required to prove the succeed ing Theorem 2. I n Section V we investigate a n special case of binary- inputs MA C. In Section VI we prove Theorem 2. In the last Section VII we summarize the paper with some comments. I I . E L E M E N TA RY M AC In this section we introduc e an elementary MAC with some expressions and definitions to be used in this paper . An N -user MA C is specified by N input a lphabets A k with size of n k , k = 1 , · · · , N , an ou tput alph abet B with size of m , and an m by ( n 1 × · · · × n N ) chan- nel matrix P = [ P ( j | i 1 , · · · , i N )] of tr ansmission p rob- abilities P ( j | i 1 , · · · , i N ) ’ s to be given a priori for the MA C, where j = 0 , · · · , m − 1 , i k = 0 , · · · , n k − 1 , a nd P m − 1 j =0 P ( j | i 1 , · · · , i N ) = 1 . Assume that there is no zero row in P , n k ≥ 2 , m ≥ 2 , and tran smission is synchro nized. T he model thus d efined is called an N -user MA C with a type ( n 1 , · · · , n N ; m ) . The IPD vector p k is assigned to an n k -tuple c olumn set of in put p robability ( p k (0) , · · · , p k ( n k − 1 )) T , k = 1 , · · · , N , where ( · · · ) T implies a tran sposition, defin ed o n A k with the probab ility constraint P n k − 1 i k =0 p k ( i k ) = 1 . Thus each p k is located on an ( n k − 1) -dimen sional simplex X k with n k vertices e kℓ , ℓ = 0 , · · · , n k − 1 . Here e kℓ is a u nit co lumn vector and takes 1 in th e ℓ th column componen t and 0 else where. Obvio usly each X k is convex and is observed as a domain of p k . The face F k of X k is de fined by an ( f k − 1) - dimensiona l simplex whose f k vertices are chosen from ver- tices e k 0 , · · · , e k ( n k − 1) of X k , wh ere f k ≤ n k . A set of f k indices of F k is deno ted by Λ( F k ) = { ℓ | e kℓ ∈ F k } . Ther e are several ch oices fo r f k indices. Obvio usly Λ ( F k ) ⊂ Λ( X k ) and Λ( X k ) = { 0 , · · · , n k − 1 } . Zero-dimension al faces ar e vertices, one-dime nsional faces are lin es, and so for th. If an IPD vector p k is on the bou ndary of X k , then there exists a minimum face F k which c ontains p k exactly inside (and not on the boundary of ) F k . Thus if p k is p k ( i k ) = 0 for i k 6∈ Λ( F k ) an d p k ( i k ) > 0 for i k ∈ Λ( F k ) , th en F k for Λ( F k ) is the m inimum face wh ich con tains p k and is u niquely determined . If p k ( i k ) > 0 for all i k ’ s, th en the minimum face which contain s th e p k is X k itself. Here F k is also ref erred to as a sub-domain of X k . Also p k , with p k ( i k ) > 0 fo r i k ∈ Λ( F k ) and p k ( i k ) = 0 for i k 6∈ Λ( F k ) , is naturally regarded as an ( f k − 1) -dime nsional vector on F k ev en it is still an ( n k − 1) -d imensional vector on the whole domain X k . The Kr onecker pr oduct of p 1 and p 2 is defined here by p 1 × p 2 ≡    p 1 (0) p 2 . . . p 1 ( n 1 − 1) p 2    and th en the Kr onecker pro duct of p 1 , · · · , p k is defined b y induction : p 1 × · · · × p k ≡ ( p 1 × · · · × p k − 1 ) × p k , k = 3 , · · · , N . I n the same way we arrang e the Kron ecker pro duct of X 1 , · · · , X N as X ≡ X 1 × · · · × X N . The set X is a d omain of th e IPD vector p = p 1 × · · · × p N of th e N -user MAC. Remark that X is not conv ex as a whole but each X k is con vex. Also we can set a Kronec ker prod uct of faces F 1 , · · · , F N as F ≡ F 1 × · · · × F N which is a sub- domain o f X . Obviously F is n ot co n vex a s a wh ole even each F k is conve x. A p air ( P, X ) is assign ed to th e N -u ser MAC to specify a chan nel matrix P and a dom ain X . Here P has colum ns ( P (0 | i 1 , · · · , i N ) , · · · , P ( m − 1 | i 1 , · · · , i N )) T arrange d in the order of the co mponen ts of p 1 × · · · × p N . An MA C is denote d in more detail by N - user ( n 1 , · · · , n N ; m ) -MAC ( P , X ) . 3 The mutual in formation of the N -u ser ( n 1 , · · · , n N ; m ) - MA C ( P, X ) is defined by I ( p 1 × · · · × p N ) = X j,i 1 , ··· ,i N p 1 ( i 1 ) · · · p N ( i N ) P ( j | i 1 , · · · , i N ) · log P ( j | i 1 , · · · , i N ) q ( j ) (1) where q ( j ) ≡ P i 1 , ··· ,i N p 1 ( i 1 ) · · · p T ( i N ) P ( j | i 1 , · · · , i N ) is an ou tput probability of the j th symbol of B and log is the natural logarithm. For any p ′ , p ′′ ∈ X , a con vex-linea r co m- bination λ p ′ + (1 − λ ) p ′′ , 0 ≤ λ ≤ 1 , d oes not alw ay s belong to X , since X is not con vex except N = 1 . Therefor e, P is considered in gen eral as a no n-linear m apping from p ∈ X to q ≡ ( q (0) , · · · , q ( m − 1)) T : q = P p = P ( p 1 × · · · × p N ) . Also I ( p 1 × · · · × p N ) is r egarded as a m ulti-variables function defined on the domain X = X 1 × · · · × X N and is concave (conve x -above) on each X k , when p ℓ ’ s, ℓ 6 = k , are fixed, but is not concave on the whole dom ain X . The chann el capa city of the N -user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is defined as usual by the maximum v alue of the mutual informa tion (1): C = max p 1 ×···× p N ∈ X I ( p 1 × · · · × p N ) . (2) An IPD vector which achieves the channel capacity is referr ed to as an optimal IPD vector . The K uhn-T u ck e r condition s are introdu ced a s the condition s to obtain the local extrema of a function o f several variables subject to one or more constra ints. For the mutual inform ation (1) of the N -user ( n 1 , · · · , n N ; m ) -MAC ( P , X ) , the condi- tions to take the maxim um value (chan nel capacity) are stated as follows: If p 1 × · · · × p N is optimal, then it satisfies J ( p 1 × · · · × p N ; i k )  = C, p k ( i k ) > 0 ≤ C, p k ( i k ) = 0 i k = 0 , · · · , n k − 1 , k = 1 , · · · , N C = I ( p 1 × · · · × p N ) (3) where J ( p 1 × · · · × p N ; i k ) ≡ ∂ I ( p 1 × · · · × p N ) ∂ p k ( i k ) + 1 = X j,i 1 , ··· ,i k − 1 ,i k +1 ··· ,i N p 1 ( i 1 ) · · · p k − 1 ( i k − 1 ) p k +1 ( i k +1 ) · · · p N ( i N ) P ( j | i 1 , · · · , i N ) log P ( j | i 1 , · · · , i N ) q ( j ) . These equation s (3) are collectively r eferred to as the K uh n- T u ck er con ditions for the mutu al in formation (1). These ar e quite easy to ob tain, for example, by a m ethod of Lagran ge multipliers to maximize the mutua l inf ormation (1) subject to the co nstraints of p k : P n k − 1 i k =0 p k ( i k ) = 1 , k = 1 , · · · , N . Remark that the Kuhn- T ucker conditions (3) are obviously necessary but no t in ge neral su fficient for the chann el capac ity of the MA C ( P, X ) . In the case of DMC, howe ver, they are necessary and sufficient for the c hannel capacity [20]. A sub-MAC ( P, Y ) , or a sub-c hannel, of an N -u ser ( n 1 , · · · , n N ; m ) -MAC ( P , X ) is rea sonably de fined as an N - user MA C where the channel m atrix is set to the same P and th e d omain is assign ed to a non -empty subset Y of X . In the subsequent d iscussions we focus mostly on th e sub- MA C ( P , Y ) where Y is restricted to a su b-doma in F ≡ F 1 × · · · × F N ⊆ X k . Here if p is an I PD vector o f the sub-MAC ( P , F ) , the n each p k of p acts as an ( f k − 1) - dimensiona l vector on F k (i.e., p k ( i k ) = 0 fo r i k 6∈ Λ( F k ) ) ev en it is still an ( n k − 1) - dimensional vector on the whole domain X k as mention ed b efore. The mutual inform ation o f an N -u ser sub-MA C ( P, F ) is given by I ( p ∈ F ) where the ( i 1 , · · · , i k , · · · , i N ) th co lumns of P for i k 6∈ Λ ( F k ) do not affect the mutu al inform ation (1). The Kuhn-T u cker condition s for a n N -user sub-MA C ( P , F ) are also given by the expression (3) where p ∈ F . The elementary MA C now we define in general as follows: If N -user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) satisfies n k ≤ m for all k = 1 , · · · , N , then it is referred to as an elemen tary MA C. The ele mentary MAC is an MAC wh ose sizes o f inpu t alphabets are not greater than the size of outpu t alphabet. The elemen tary (face) set Φ ( m ) N of X is defined by the set of faces as follows: If n k ≥ m , then F k is pu t to an ( m − 1) - dimensiona l face of X k , and if n k < m , then F k is pu t to the ( n k − 1) -dime nsional X k itself. Thus the dimension of each F k is less than o r eq ual to ( m − 1) . I f X is formed by n k ≤ m for all k = 1 , · · · , N , the n Φ ( m ) N = { X } . A ma ster (elemen tary) MAC ( P, F ) of an N -u ser MA C ( P, X ) is defined as the MA C with a d omain F ∈ Φ ( m ) N . Here each p k ∈ F k acts as a n ( f k − 1) -dimen sional vector as mentio ned above, wh ere p k ( i k ) = 0 fo r i k 6∈ Λ( F k ) , k = 1 , · · · , N . Note tha t the m aster MA C ( P, F ∈ Φ ( m ) N ) is regard ed as the la r gest possible elementary M A C of ( P , X ) in the sense that the re is no elementary MA C ( P , F ′ ) su ch that ( P , F ) is an elementary sub-MAC of ( P, F ′ ) . A set of all master MAC’ s is referr ed to as the master (elementar y) set o f the N - user MA C ( P, X ) and is denoted by Ω N . Obviously Ω N is finite and is uniq uely determined. If an M A C ( P, X ) is itself elementary , then Ω N = { MA C ( P, X ) } . The chann el capacity of an MA C ( P, F ) ∈ Ω N is deno ted b y C ( F ) . In later discu ssions we invest igate the IPD vector p which satisfies the Kuhn-T ucker co nditions (3). If p k ( i k ) > 0 f or all i k = 0 , · · · , n k − 1 , k = 1 , · · · , N , then p is located exactly inside (no t on the boun dary of) X . If p k ( i k ) = 0 f or i k 6∈ Λ( F k ) an d p k ( i k ) > 0 f or i k ∈ Λ( F k ) , k = 1 , · · · , N , then the sub -domain F = F 1 × · · · × F N of X f ormed b y Λ( F k ) is the minimum domain which contain s p exactly inside F . More imp ortantly , the n on-elemen tary MAC h as in essence a d e gen erate property as follows: if n k > m for an N -user ( n 1 , · · · , n N ; m ) -MAC, then for a fixed p k ∈ F k ⊆ X k with f k > m , there exists an IPD vector p ′ k ∈ F ′ k where p ′ 6 = p , F ′ k ⊂ ( 6 =) F k , f ′ k = m , and p = p 1 × · · · × p k × · · · × p N , p ′ = p 1 × · · · × p ′ k × · · · × p N , such tha t q = P p ′ = P p . T he elementary M A C has in general no su ch pr operty . This notio n is crucial to the subsequen t discussion s. Finally fo r this section, we rem ark that we are go ing to in vestigate various types of M A C’ s. For example, we examine an MAC ( P, Y ) w ith a domain Y = Y 1 × · · · × Y N ⊂ X wh ere each Y k , k = 1 , · · · N , is formed by th e line segment o f IPD vectors of X k . E ven then we can examine the Kuhn-Tucker 4 condition s in th e same way as mentioned above. I I I . M A I N R E S U LT The master elementary set Ω N as defined above h as an intrinsic p roperty with r espect to th e N -u ser ( n 1 , · · · , n N ; m ) - MA C ( P, X ) . W e can state it as a main theorem : Theor em 1: The channel capacity C of an N -user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is achieved b y the ch annel capacity C ( F ) of an N -user elementar y MA C ( P, F ∈ Φ ( m ) N ) of Ω N as follows: C = max F ∈ Φ ( m ) N C ( F ) . (4) ✷ Pr oof: It is sufficient to pr ove the case that th e original MA C ( P, X ) is not elementary . Let ¯ p = ¯ p 1 ×· · ·× ¯ p k ×· · ·× ¯ p N be an o ptimal IPD vector that ach iev es the chann el capacity C . Let ¯ F k be the minim um face of X k which contains ¯ p k exactly inside ¯ F k , k = 1 , · · · N . It is suf ficient to assume that ¯ F k is th e m or mor e dim ensional face. Th en b y th e degener ate proper ty there exists an ( m − 1) -dimen sional face ˜ F k ⊂ ( 6 =) ¯ F k such that for an IPD vector ˜ p k ∈ ˜ F k , P ¯ p = P ( ¯ p 1 × · · · × ˜ p k × · · · × ¯ p N ) . (5) Put K ( θ ) ≡ I ( ¯ p 1 × · · · × ( θ ¯ p k + (1 − θ ) ˜ p k ) × · · · × ¯ p N ) for the mutu al inform ation of the o riginal MAC ( P, X ) , where 0 ≤ θ ≤ 1 . The deriv ativ e ∂ K ( θ ) / ∂ θ is con stant by (5) and moreover ∂ K ( θ ) /∂ θ is equ al to zero since ¯ p is op timal. Th en it holds I ( ¯ p ) = I ( ¯ p 1 × · · · × ˜ p k × · · · × ¯ p N ) . This implies th at the optimal IPD vector exists in a do main F = F 1 × · · · × ˜ F k × · · · × F N ∈ Φ ( m ) N . Thus T heorem 1 is proved. Theorem 1 states that the cha nnel capacity C of a ny N -u ser ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is rigo rously determin ed by the channel cap acity C ( F ) of an N - user master elementar y MAC ( P, F ) ∈ Ω N . I n other words, an op timal IPD vector exists at least on a domain F ∈ Φ ( m ) N . Howe ver Theorem 1 does not guaran tee that the o ptimal IPD vector exists only o n a domain F ∈ Φ ( m ) N , that is, there might exist in ge neral an op timal IPD vector that is located exactly inside X and n ot on any F ∈ Φ ( m ) N . Note that if the N -user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is elementary , then Theorem 1 appears self-evident since Ω N contains only an MA C ( P, X ) itself. In the remainin g section of th is p aper we focus on the proof that the Kuhn-T ucker con ditions o f an elementar y MAC ( P, X ) are necessary and sufficient f or the channel capacity . W e will state it in advance as a second theorem. Theor em 2: The Kuhn- T ucker co nditions for the chann el capacity C of an N - user elementar y ( n 1 , · · · , n N ; m ) -MAC ( P, X ) , where n k ≤ m for all k = 1 , · · · , N , are necessary and sufficient. ✷ It is suf ficient to p rove on ly the sufficiency since the necessity is self-evident. From these two theorems th e MA C in gener al can be regard ed as simply an agg regate o f a finite number of elementary MA C’ s wher e th e Kuhn-T u cker condi- tions for the chann el cap acity are necessary and sufficient. I V . F E AT U R E S O F E L E M E N TA RY M A C In this section we prepare b asic p roperties that are requ ired to prove the sufficiency o f Theorem 2. The fir st p roperty A is the chain rules [1]: W e recall that the mutual information of an N -user MA C is in genera l decom - posed in to N compo nents with N ! different deco mpositions by the chain rules. The second property B is the capacity r egion : W e d escribe that the capacity region of the N -user MA C is given by the conv ex-closure of all a chiev ab le rate region s of the N ! decom - positions for the mutua l informatio n [ 4]. It is summar ized as Proposition 1. The third property C is the boundary equations : W e in ves- tigate tha t a b ounda ry of an achievable r ate region satisfies by a method of Lag range multipliers a set of c onditions to be referred to as the bou ndary equations f or the capacity region of the N - user MA C. The f ourth pro perty D is a relation b etween th e K uhn- T u ck er equation s an d the boun dary equ ations : W e prove as Proposition 2 that a so lution of th e Kuhn-T ucker condition s of an N -user M A C with some restrictions satisfies the bound ary equations. The fifth pro perty E is loca l maximum : W e prove a s Pro po- sition 3 that every solution of the Kuhn-T ucker co nditions of an elementary MA C ( P , X ) is local maximum in the domain X . T o prove Prop osition 3 we need Proposition 2. Finally , the sixth property F is co nnectedn ess : W e prove as Proposition 4 that a set of IPD vectors of an elementary MAC ( P, X ) , for which the value of the mu tual infor mation is n ot smaller than the arbitrary p ositiv e nu mber, is connected in the domain X . T o prove Prop osition 4 we use Proposition 3. W e emp hasize h ere that the last two pr operties, i.e. local maximum and conne ctedness , are the most distinctive features exclusi ve to the elementar y MA C. Howev er the first f our proper ties, alth ough they hold for any MAC in gen eral, are required to step by step prove the last two. A. Chain Rules The m utual infor mation of an N -u ser ( n 1 , · · · , n N ; m ) - MA C ( P , X ) is de composed into N compo nents by the ch ain rules [1]. F or the IPD vectors p 1 , · · · , p k − 1 , p k , p k +1 , · · · , p N , let ρ { u, ··· ,w } be a Krone cker produ ct of p k , k 6∈ { u, · · · , w } , and let σ { u, ··· ,w } be a Kr onecker pro duct of p k , k ∈ { u, · · · , w } . Obviously σ { u } = p u . The mutual in formation (1) is decompo sed into two com- ponen ts as I ( p 1 × · · · × p N ) = I ( σ { u } | ρ { u } ) + I ( ρ { u } / σ { u } ) . 5 Here I ( σ { u } | ρ { u } ) = X j,i 1 , ··· ,i N p 1 ( i 1 ) · · · p N ( i N ) P ( j | i 1 , · · · , i N ) · log P ( j | i 1 , · · · , i N ) P h p u ( h ) P ( j | i 1 , · · · , h, · · · , i N ) which is the con ditional mutual informatio n of p u with respect to p 1 , · · · , p u − 1 , p u +1 · · · , p N , and I ( ρ { u } / σ { u } ) = X j,i 1 , ··· ,i N p 1 ( i 1 ) · · · p N ( i N ) P ( j | i 1 , · · · , i N ) · log P h p u ( h ) P ( j | i 1 , · · · , h, · · · , i N ) q ( j ) which is the mutual inf ormation of an ( N − 1) -user MA C with the channel matrix [ P h p u ( h ) P ( j | i 1 , · · · , h , · · · , i N )] . Moreover we de compose the latter into I ( ρ { u } / σ { u } ) = I ( p w | ρ { u,w } / σ { u } ) + I ( ρ { u,w } / σ { u,w } ) . In general, I ( ρ { u, ··· ,w } / σ { u, ··· ,w } ) = I ( p x | ρ { u, ··· ,x, ··· ,w } / σ { u, ··· ,w } ) + I ( ρ { u, ··· ,x, ··· ,w } / σ { u, ··· ,x ·· · ,w } ) . Here I ( p x | ρ { u, ··· ,x, ··· ,w } / σ { u, ··· ,w } ) = X j,i 1 , ··· ,i N p 1 ( i 1 ) · · · p T ( i N ) P ( j | i 1 , · · · , i N ) · log h σ { u, ··· ,w } · P i h σ { u, ··· ,x, ··· ,w } · P i I ( ρ { u, ··· ,x, ··· ,w } / σ { u, ··· ,x ·· · ,w } ) = X j,i 1 , ··· ,i N p 1 ( i 1 ) · · · p N ( i N ) P ( j | i 1 , · · · , i N ) · log h σ { u, ··· ,x, ··· ,w } · P i q ( j ) where h σ { u, ··· ,w } · P i ≡ X h u , ··· ,h w p u ( h u ) · · · p w ( h w ) · P ( j | i 1 , · · · , h u , · · · , h w , · · · , i N ) . If { u, · · · , w } is empty , then h σ { u, ··· ,w } · P i redu ces to P ( j | i 1 , · · · , i N ) . Thus successively redu cing th e suffices { u, · · · , w } o f ρ { u, ··· ,w } up to { 1 , · · · , k − 1 , k + 1 , · · · , N } , I ( p 1 × · · · × p N ) is decompo sed into N com ponen ts. Note that there exist as a whole N ! different decompositions. B. Capa city Re gio n A set of all achiev able rates for an N -user ( n 1 , · · · , n N ; m ) - MA C ( P , X ) is called a capacity region (e .g., [ 1], [3 ], [4]). By a decomp osition we obtain I ( p 1 × · · · × p N ) = I ( p 1 | p 2 × · · · × p N ) + I ( p 2 | p 3 × · · · × p N / p 1 ) + I ( p 3 | p 4 × · · · × p N / p 1 × p 2 ) + · · · · · · + I ( p N / p 1 × · · · × p N − 1 ) . (6) There exist as a wh ole N ! different de composition s as men- tioned above. Define a sub-r egion G 1 as [ p ∈ X ( I ( p 1 | p 2 × · · · × p N ) , · · · , I ( p N / p 1 × · · · × p N − 1 )) . This is identified as a set o f achiev able rates G 1 for the decomp osition (6). Other N ! − 1 sets o f achiev ab le rate s G 2 , · · · , G N ! are also defined in the same way as G 1 . T hen the capacity r egion G is determ ined b y those sub-regions G i ’ s as follows [T heorem 15.3.6 in [1]]: Pr opo sition 1 : The capacity region of an N -u ser ( n 1 , · · · , n N ; m ) -MAC ( P , X ) is gi ven b y G = co N ! [ i =1 G i (7) where “co” implies the conve x -closure. ✷ C. Boun dary Equations A boun dary of each sub-r egion G i , i = 1 , · · · , N ! , for an N -u ser ( n 1 , · · · , n N ; m ) -MAC ( P , X ) , c an be determine d b y a m ethod of Lagrange m ultipliers. Th e bou ndary o f G 1 , f or example, is e valuated by a Lagrang e mu ltiplier function, L ( p 1 , · · · , p N , λ 1 , · · · , λ N − 1 , ζ 1 , · · · , ζ N ) = I ( p 1 | p 2 × · · · × p N ) − λ 1 I ( p 1 × · · · × p N ) − λ 2 I ( p 2 | p 3 × · · · × p N / p 1 ) − · · · − λ N − 1 I ( p N − 1 | p N / p 1 × · · · × p N − 2 ) − N X k =1 ζ k X i k p k ( i k ) where λ 1 , · · · , λ N − 1 and ζ 1 , · · · , ζ N are so- called Lagran ge multipliers. The c onditions that an IPD vector p 1 × · · · × p N takes extremum (maximum or min imum) for G 1 are given by the equations (see Fig. 1 for N = 3 ) det         ˜ ∂ I ( p 1 | p 2 × · · · × p N ) ˜ ∂ p 1 ( i 1 ) ˜ ∂ I ( p 1 × · · · × p N ) ˜ ∂ p 1 ( i 1 ) . . . . . . ˜ ∂ I ( p 1 | p 2 × · · · × p N ) ˜ ∂ p N ( i N ) ˜ ∂ I ( p 1 × · · · × p N ) ˜ ∂ p N ( i N ) ˜ ∂ I ( p 2 | p 3 × · · · × p N / p 1 ) ˜ ∂ p 1 ( i 1 ) · · · . . . . . . ˜ ∂ I ( p 2 | p 3 × · · · × p N / p 1 ) ˜ ∂ p N ( i N ) · · · 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • min max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ( p 1 × p 2 × p 3 ) = α 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ( p 3 / p 1 × p 2 ) cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ( p 1 | p 2 × p 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ( p 2 | p 3 / p 1 ) = α 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 1. Sub-regi on G 1 of three-user MAC . · · · ˜ ∂ I ( p N − 1 | p N / p 1 × · · · × p N − 2 ) ˜ ∂ p 1 ( i 1 ) . . . . . . · · · ˜ ∂ I ( p N − 1 | p N / p 1 × · · · × p N − 2 ) ˜ ∂ p N ( i N )         = 0 , i k = 0 , · · · , n k − 2 , k = 1 , · · · , N . (8) Here, we define partial deriv atives as: ˜ ∂ I ( · · · ) ˜ ∂ p k ( i k ) ≡ ∂ I ( · · · ) ∂ p k ( i k ) − ∂ I ( · · · ) ∂ p k ( n k − 1) , i k 6 = n k − 1 . T otal ( n 1 − 1) × · · · × ( n N − 1 ) equations (8) are co llectiv ely referred to as the b ounda ry e quation s f or G 1 . Solutions o f (8 ) include b oth max imization and m inimization as usual. Succes- si vely we can set up th e b oundar y equation s for G 2 , · · · , G N ! with totally the same form as (8). N ote that the b oundar y equations have the same form as (8) for the different choices of starting Lagrang e mu ltiplier function. D. A r ela tion between th e K uhn- T u ck er equation s an d the bound ary equ ations The b oundar y equatio ns thu s obtained have an impo rtant proper ty which we state as a proposition: Pr opo sition 2 : If a solution ¯ p = ¯ p 1 × · · · × ¯ p N ∈ X of the Kuhn-T ucker condition s for the mutu al in formatio n I ( p ) of an N - user ( n 1 , · · · , n N ; m ) -MAC ( P , X ) satisfies J ( ¯ p 1 × · · · × ¯ p N ; i k ) = C, i k = 0 , · · · , n k − 1 , k = 1 , · · · , N C = I ( ¯ p 1 × · · · × ¯ p N ) (9) then ¯ p is a solution of the bound ary equation s for sub-region s G i , i = 1 , · · · , N ! . ✷ Pr oof: It is sufficient to prove that ¯ p satisfies the bound ary equation (8) for G 1 . By the assumption (9), it ho lds ˜ ∂ I ( p 1 × · · · × p N ) ˜ ∂ p k ( i k )      p k = ¯ p k ,i k =0 , ··· ,n k − 2 = J ( ¯ p 1 × · · · × ¯ p N ; i k ) − J ( ¯ p 1 × · · · × ¯ p N ; n k − 1) = 0 . Then the secon d column of (8) reduce s to zeros. Therefor e ¯ p is a solution of the bound ary eq uation (8). Remark that Propo sition 2 ho lds for a ny MA C includ ing the elementary MA C if it satisfies the condition s (9). E. Local Maximum An IPD vector ¯ p is called a local ma ximum point fo r the mutual info rmation I ( p ) , if ther e exists a neigh borho od U ¯ p of ¯ p such that I ( p ) ≤ I ( ¯ p ) for any p ∈ U ¯ p . W e prove h ere that for th e elementary MA C every solution of the Kuhn-Tucker condition s is local ma ximum. W e state it as a proposition: Pr opo sition 3 : If an N - user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is elemen tary , i.e. n k ≤ m , k = 1 , · · · , N , then e very solutio n p ∗ ≡ p ∗ 1 × · · · × p ∗ N ∈ X o f the Kuhn -T u cker cond itions for the mutual informa tion I ( p ) is local maximum in X . ✷ Before p roceedin g we r emark that Propo sition 3 does not hold in general for the non- elementary MA C by the degenerate proper ty as is stated in the b eginning of the proof of Th eo- rem 1. In fact, we note witho ut proof that a non-e lementary two-user (3 , 3; 2 ) -MA C ( P , X ) , for example, with N = 2 , n 1 = n 2 = 3 , m = 2 , for some ch annel matr ix P , has a solution of th e Kuhn-Tucker conditions whic h is not local maximum in X . Pr oof: Since th e N -user MAC ( P , X ) is elemen tary , it is sufficient to in vestigate two cases for the solution p ∗ of the Kuhn-T u cker conditions: the first is that every p ∗ k ( i k ) is non-ze ro and the second is that at least o ne of p ∗ k ( i k ) ’ s is zer o. In the first case, sinc e p ∗ k ( i k ) > 0 for all compon ents, p ∗ satisfies the Kuhn-T u cker cond itions: J ( p ∗ 1 × · · · × p ∗ N ; i k ) = M T p ∗ k ( i k ) > 0 , i k = 0 , · · · , n k − 1 , k = 1 , · · · , N M T = I ( p ∗ 1 × · · · × p ∗ N ) . (10) and there exist p ′ and p ′′ in X such that p ∗ = ( θ ∗ 1 p ′′ 1 + (1 − θ ∗ 1 ) p ′ 1 ) × · · · × ( θ ∗ N p ′′ N + (1 − θ ∗ N ) p ′ N ) where p ∗ k 6 = p ′ k , p ∗ k 6 = p ′′ k , and 0 < θ ∗ k < 1 . Here we put by using θ k , 0 ≤ θ k ≤ 1 , k = 1 , · · · , N , K ( θ 1 , · · · , θ N ) = I (( θ 1 p ′′ 1 + (1 − θ 1 ) p ′ 1 ) × · · · × ( θ N p ′′ T + (1 − θ N ) p ′ N )) and investigate th e Kuhn -T ucker cond itions and the bou ndary equations with respect to this K ( θ 1 , · · · , θ N ) . The Kuhn- T ucker condition s are simp le to see as K k ( θ 1 , · · · , θ N ) = 0 , k = 1 , · · · , N (11) where K k ( θ 1 , · · · , θ N ) ≡ ∂ K ( θ 1 , · · · , θ N ) /∂ θ k . Also by a decomp osition K ( θ 1 , · · · , θ N ) = K ( θ 1 | θ 2 , · · · , θ N ) + K ( θ 2 | θ 3 , · · · , θ N /θ 1 ) + K ( θ 3 | θ 4 · · · , θ N /θ 1 , θ 2 ) + · · · + K ( θ N /θ 1 , · · · , θ N − 1 ) 7 R N R 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • M T bound ary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R ( θ ∗ 1 , · · · , θ ∗ N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gradient ≤ − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gradient ≥ − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 2. Boundary of cross-section G 1 ( R 1 , R N ) in R 1 - R N plain. Rate s R 2 , · · · , R N − 1 are fixed as specified by (14 ). we obtain a set of achievable rates G 1 = [ θ 1 , ··· ,θ N ( K ( θ 1 | θ 2 , · · · , θ N ) , · · · , K ( θ N /θ 1 , · · · , θ N − 1 )) (12) which leads us to the bou ndary eq uation for G 1 as follows: det    K 1 ( θ 1 | θ 2 , · · · , θ N ) K 1 ( θ 1 , · · · , θ N ) . . . . . . K N ( θ 1 | θ 2 , · · · , θ N ) K N ( θ 1 , · · · , θ N ) K 1 ( θ 2 | θ 3 , · · · , θ N /θ 1 ) · · · . . . . . . K N ( θ 2 | θ 3 , · · · , θ N /θ 1 ) · · · K 1 ( θ N − 1 | θ N /θ 1 , · · · , θ N − 2 ) . . . K N ( θ N − 1 | θ N /θ 1 , · · · , θ N − 2 )    = 0 (13) where K k ( · · · ) ≡ ∂ K ( · · · ) /∂ θ k . Since p ∗ satisfies ( 10), ( θ ∗ 1 , · · · , θ ∗ N ) is a solution o f the Kuhn-T u cker cond itions (11). Then by Proposition 2 it satis fies the bou ndary equ ation (13) and M T = K ( θ ∗ 1 , · · · , θ ∗ N ) . Now we examine a g radient of the bou ndary o f G 1 at θ ∗ ≡ ( θ ∗ 1 , · · · , θ ∗ N ) . Note that the solution θ ≡ ( θ 1 , · · · , θ N ) of the bound ary equatio n ( 13) around θ ∗ defines a set of achiev a ble rates (1 2) as G 1 ( R 1 , · · · , R N ) . Obviously K ( θ ∗ ) = M T . At this step we in vestigate a cross-section of (12) subject to th e restrictions such that K ( θ 2 | θ 3 , · · · , θ N /θ 1 ) = K ( θ ∗ 2 | θ ∗ 3 , · · · , θ ∗ N /θ ∗ 1 ) . . . K ( θ N − 1 | θ N /θ 1 , · · · , θ N − 2 ) = K ( θ ∗ N − 1 | θ ∗ N /θ ∗ 1 , · · · , θ ∗ N − 2 ) . (14) W e denote a cr oss-section (subset) of G 1 subject to (14) as G 1 ( R 1 , R N ) . This is compo sed o f R ( θ 1 , · · · , θ N ) ≡ ( K ( θ 1 | θ 2 , · · · , θ N ) , K ( θ ∗ 2 | θ ∗ 3 , · · · , θ ∗ N /θ ∗ 1 ) , · · · , K ( θ ∗ N − 1 | θ ∗ N − 1 /θ ∗ 1 , · · · , θ ∗ N − 2 ) , K ( θ N /θ 1 , · · · , θ N − 1 )) . The cr oss-section G 1 ( R 1 , R N ) is a region in the two- dimensiona l ( R 1 - R N ) p lain as shown in Fig. 2. Since it holds K 1 ( θ 2 | θ 3 , · · · , θ N /θ 1 ) = 0 . . . K 1 ( θ N − 1 | θ N /θ 1 , · · · , θ N − 2 ) = 0 by the restriction s (1 4), then we h av e K 1 ( θ 1 , · · · , θ N ) = K 1 ( θ 1 | θ 2 , · · · , θ N − 1 ) + K 1 ( θ N /θ 1 , · · · , θ N − 1 ) . Thus the g ra- dient of the boun dary of G 1 ( R 1 , R N ) appears K 1 ( θ N /θ 1 , · · · , θ N − 1 ) K 1 ( θ 1 | θ 2 , · · · , θ N ) = − 1 + K 1 ( θ 1 , · · · , θ N ) K 1 ( θ 1 | θ 2 , · · · , θ N ) . (1 5) The right-ha nd side o f (15) is estimated as − 1 + K 1 ( θ 1 , · · · , θ N ) K 1 ( θ 1 | θ 2 , · · · , θ N ) ≤ ( ≥ )1 (16) accordin g to th e maxim ization (min imization) conditio ns of K ( θ 1 | θ 2 , · · · , θ N ) subject to (14) where it ho lds K 1 ( θ 1 , · · · , θ N ) K 1 ( θ 1 | θ 2 , · · · , θ N ) ≤ ( ≥ ) 0 . Also the gradien t o f the boun daries of a ny cro ss-section G 1 ( R 1 , R k ) ( 2 ≤ k ≤ N − 1 ) is given b y (16). For any region G i of N ! d ecompo sitions, the gradien t of the bound ary of G i at θ ∗ takes the same condition as that of G 1 . Since the inequalities (16) are valid f or any p ′ k , p ′′ k , k = 1 , · · · , N , there exists a n eighbo rhood U p ∗ of p ∗ in X , such that I ( p ∗ ) ≥ I ( p ∈ U p ∗ ) . T his mean s that p ∗ is loc al maximum in X . In the second case, since at least one of p ∗ k ( i k ) ’ s is zero, p ∗ satisfies the Kuhn-T u cker cond itions: J ( p ∗ 1 × · · · × p ∗ N ; i k )  = M T , i k ∈ Λ ( F k ) ≤ M T , i k 6∈ Λ ( F k ) k = 1 , · · · , N , M T = I ( p ∗ 1 × · · · × p ∗ N ) (17) where p ∗ k ( i k ) > 0 for i k ∈ Λ( F k ) an d p ∗ k ( i k ) = 0 for i k 6∈ Λ( F k ) . Thus ther e exists a sub-do main F = F 1 × · · · × F N such th at p ∗ ∈ F . This implies that p ∗ is lo cal maximum in F as described in the fir st case and th ere exists a neighbo rhood U 0 p ∗ ⊂ F such that I ( p ∗ ) ≥ I ( p ∈ U 0 p ∗ ) . For any p ′ = p ′ 1 × · · · × p ′ k × · · · × p ′ N ∈ U 0 p ∗ , consider p ′′ = p ′ 1 × · · · × p ′′ k × · · · × p ′ N , where p ′′ k ∈ X k and p ′′ k 6∈ F k . Put for 0 ≤ θ ≤ 1 K ( θ ) = I ( p ′ 1 × · · · × ( θ p ′′ k + (1 − θ ) p ′ k ) × · · · × p ′ N ) . It hold s dK ( θ ) /dθ | θ =0 ≤ 0 , since K ( θ ) is concave, d iffer - entiable, and p ∗ satisfies (17). Therefo re K ( θ ) is monoto ne non-in creasing for θ . Th us th ere exists θ ′ > 0 such that I ( p ′ 1 × · · · × ( θ ′ p ′′ k + (1 − θ ′ ) p ′ k ) × · · · × p ′ N ) < I ( p ∗ ) . Hence, there exists a ne ighborh ood U p ∗ ⊂ X of p ∗ such that I ( p ∗ ) ≥ I ( p ∈ U p ∗ ) . This means th at p ∗ is local m aximum in X . By these two cases Proposition 3 is proved. 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 O θ 1 θ 2 • p ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 2 ˆ p • . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . . . .. . .. . .. . .. . .. . .. . .. . .. . . . . .. . .. . .. . . . . .. . .. . .. . .. . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 3. Pattern of D ( a 0 ) = D 1 ∪ D 2 for the case of N = 2 , n 1 = n 2 = 2 , p 1 (0) = θ 1 , p 2 (0) = θ 2 . F . Connectedn ess Finally in this sectio n, we prove the p roperty of connecte d- ness for the elementary MA C as a propo sition: Pr opo sition 4 : If an N - user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is elementary , i.e. n k ≤ m , k = 1 , · · · , N , then the set D ( a ) ≡ { p | I ( p ∈ X ) ≥ a } (18) is connected for any a ≥ 0 . ✷ Pr oof: Assume that for a ny ε > 0 , there exists a 0 > 0 such that D ( a 0 ) is c onnected and D ( a 0 + ε ) is discon nected. Since I ( p ) is concave on eac h X k , th en there exist subsets D 1 and D 2 of D ( a 0 ) with prop erties as fo llows: 1) D ( a 0 ) = D 1 ∪ D 2 , an d I ( p ∗ ) = a 0 , f or p ∗ ∈ D 1 ∩ D 2 . 2) For any p ′ 1 × · · · × p ′ k × · · · × p ′ N ∈ D 1 , all IPD vectors p ′ 1 × · · · × p k × · · · × p ′ N , p k ∈ X k , k = 1 , · · · , N , satisfying I ( p ′ 1 × · · · × p k × · · · × p ′ N ) ≥ a 0 belongs to D 1 , and also fo r any p ′′ 1 × · · · × p ′′ k × · · · × p ′′ N ∈ D 2 , all IPD vectors p ′′ 1 × · · · × p k × · · · × p ′′ N , p k ∈ X k , k = 1 , · · · , N , satisfying I ( p ′′ 1 ×· · ·× p k ×· · ·× p ′′ N ) ≥ a 0 belongs to D 2 (cf. Fig. 3). Thus f or any ε > 0 , D ( a 0 + ε ) is separated into su bsets D ′ 1 ⊂ D 1 and D ′ 2 ⊂ D 2 such that D ( a ) = D ′ 1 ∪ D ′ 2 and D ′ 1 ∩ D ′ 2 = φ . It is easy to see that f or any p ∗ = p ∗ 1 × · · · × p ∗ k × · · · × p ∗ N ∈ D 1 ∩ D 2 , k = 1 , · · · N , it holds I ( p ∗ 1 × · · · × p k × · · · × p ∗ N ) ≤ a 0 , p k ∈ X k (19) since for any ˆ p = ˆ p 1 ×· · ·× ˆ p N ∈ D 1 (or ˆ p ∈ D 2 ), ˆ p 6∈ D 1 ∩ D 2 , ev ery ˆ p 1 × · · · × p k × · · · × ˆ p N , k = 1 , · · · , N , satisfying I ( ˆ p 1 × · · · × p k × · · · × ˆ p N ) < a 0 , p k ∈ X k belongs to neither D 1 nor D 2 by the prope rty 2) (see Fig. 3). Consider two cases: Every compon ents of p ∗ is non-zer o and at least a compo nent o f p ∗ is zero. In the first c ase, it hold s b y (1 9) th at p ∗ ∈ D 1 ∩ D 2 satisfies the Kuhn-T ucker cond itions J ( p ∗ 1 × · · · × p ∗ N ; i k ) = a 0 , p ∗ k ( i k ) 6 = 0 i k = 0 , · · · , n k − 1 k = 1 , · · · , N . Therefo re p ∗ is local m aximum f or p ∈ X by Proposition 3 since ( n 1 , · · · , n N ; m ) -MAC ( P, X ) is elementary . Then there exists a neighb orhoo d U p ∗ of p ∗ in X such that I ( p ) ≤ a 0 for any p ∈ U p ∗ . On the other han d, by th e pro perties of D 1 and D 2 there exists p ′ in either U p ∗ ∩ D 1 or U p ∗ ∩ D 2 such that I ( p ′ ) > a 0 . This is inconsistent with that p ∗ is local max imum. Th erefore D ( a ) is conn ected. For the seco nd ca se, consider the minimu m dom ain F ≡ F 1 × · · · × F N ⊂ X which con tains the p ∗ exactly inside (and not on the b ounda ry of) F k , where p k ( i k ) = 0 fo r i k 6∈ Λ ( F k ) and p k ( i k ) > 0 fo r i k ∈ Λ ( F k ) , k = 1 , · · · , N . In the same way as in the first case, it is proved that D ( a ) ∩ F is conn ected. Therefo re D ( a ) is con nected. V . B I N A RY - I N P U T S M AC In this section, we in vestigate an N -user binary-inpu ts MAC ( P, Y ) of the N - user ( n 1 , · · · , n N ; m ) -MAC ( P, X ) wh ere each Y k of Y is formed by a line segmen t. For any given ρ ′ k , ρ ′′ k ∈ X k , k = 1 , · · · , N , d efine a line segment Y k by Y k = { θ k ρ ′ k + (1 − θ k ) ρ ′′ k | 0 ≤ θ k ≤ 1 } , k = 1 , · · · , N and deno te Y = Y 1 × · · · × Y N . Reasonably we set θ ≡ ( θ 1 , · · · , θ N ) and write θ k ∈ Y k , θ ∈ Y . Thus we can build up an N - user b inary-inp uts (2 , · · · , 2; m ) -MAC ( P, Y ) w hose channel matrix is P and do main is a subset Y o f X . Obviou sly it is an elementary MA C since m ≥ 2 . The mutual in formatio n of the N -user (2 , · · · , 2; m ) -M A C ( P, Y ) is given by I ( θ 1 , · · · , θ N ; ρ ′ , ρ ′′ ) ≡ I (( θ 1 ρ ′ 1 + (1 − θ 1 ) ρ ′′ 1 ) × · · · × ( θ N ρ ′ N + (1 − θ N ) ρ ′′ N )) (20) where 0 ≤ θ k ≤ 1 , k = 1 , · · · , N , and ρ ′ = ρ ′ 1 × · · · × ρ ′ N , ρ ′′ = ρ ′′ 1 × · · · × ρ ′′ N . It depends on the choice of ρ ′ , ρ ′′ . The Kuhn-T u cker condition s for (2 0) are giv en by I k ( θ 1 , · · · , θ N ; ρ ′ , ρ ′′ ) = 0 , θ k > 0 ≤ 0 , θ k = 0 k = 1 , · · · , N (21) where I k ( · · · ; ρ ′ , ρ ′′ ) = ∂ I ( · · · ; ρ ′ , ρ ′′ ) /∂ θ k . For simplicity we omit ρ ′ , ρ ′′ from the expression a nd d enote I ( θ ; ρ ′ , ρ ′′ ) ≡ I ( θ ) , in the su bsequent discussions. W e prove th e lem ma to be u sed for the pro of of Theorem 2 as follows: Lemma 1: The Kuhn-Tucker co nditions for the N -user bi- nary (2 , · · · , 2 ; m ) -MAC ( P, Y ) as defined above are neces- sary and sufficient fo r optimality . ✷ Pr oof: I t is sufficient to p rove the sufficiency . Assume that th ere exist two solution s ¯ θ = ( ¯ θ 1 , · · · , ¯ θ N ) and ˆ θ = ( ˆ θ 1 , · · · , ˆ θ N ) of the Kuhn- T ucker conditions (21) suc h that I ( ¯ θ ) 6 = I ( ˆ θ ) . W ithout loss of gen erality , assume that I ( ¯ θ ) > I ( ˆ θ ) . Since the N -user bin ary (2 , · · · , 2; m ) -MAC ( P , Y ) is ele- mentary , by Proposition 3 th e solutio n ˆ θ is local ma ximum in Y and there exists a neig hborho od U ˆ θ of ˆ θ such that I ( ˆ θ ) ≥ I ( θ ∈ U ˆ θ ) . Also by Pro position 4 the set D ( I ( ˆ θ )) ≡ { θ |I ( θ ) ≥I ( ˆ θ ) , θ ∈ Y } is connec ted and inclu des both ¯ θ 9 θ 2 θ 1 • ˆ θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • ¯ θ D ( I ( ˆ θ )) U ˆ θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • θ ′ U θ ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆( ˆ θ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig. 4. T wo solutions of the Kuhn-T ucke r conditions for two-user case. and ˆ θ . Then for any θ ∈ D ( I ( ˆ θ )) ∩ U ˆ θ , it is easy to see I ( θ ) = I ( ˆ θ ) . Let θ ∗ and θ † be any po ints in D ( I ( ˆ θ )) ∩ U ˆ θ , and set I ( θ ∗ 1 , · · · , ( αθ ∗ k + (1 − α ) θ † k ) , · · · , θ ∗ N ) as a fu nction of th e variable α . Since I ( θ ) is con cav e for each variable θ k and I ( θ ∗ ) = I ( θ † ) = I ( ˆ θ ) , we have I ( θ ∗ 1 , · · · , ( αθ ∗ k + (1 − α ) θ † k ) , · · · , θ ∗ N ) = I ( ˆ θ ) for 0 ≤ α ≤ 1 . Therefor e, it holds d I ( θ ∗ 1 , · · · , ( αθ ∗ k + (1 − α ) θ † k ) , · · · , θ ∗ N ) dα = ( θ ∗ k − θ † k ) I k ( θ ∗ 1 , · · · , ( αθ ∗ k + (1 − α ) θ † k ) , · · · , θ ∗ N ) = 0 . This implies that any θ ∈ D ( I ( ˆ θ )) ∩ U ˆ θ satisfies the Kuhn- T ucker cond itions (21): I k ( θ 1 , · · · , θ k , · · · , θ N ) = 0 , k = 1 , · · · , N , even if ˆ θ is located on the bound ary of Y . Let ∆( ˆ θ ) be a set ∆( ˆ θ ) ≡ { θ |I k ( θ ) = 0 , k = 1 , · · · , N , I ( θ ) = I ( ˆ θ ) , θ ∈ D ( I ( ˆ θ )) } . Clearly , this includes D ( I ( ˆ θ )) ∩ U ˆ θ and it holds I ( θ ) = I ( ˆ θ ) for θ ∈ ∆( ˆ θ ) (see Fig 4). Note that e ach poin t in ∆( ˆ θ ) is lo cal maximum . Th en f or any θ ′ ∈ ∆( ˆ θ ) , there exists a neighbo rhood U θ ′ , such that I ( θ ) ≤ I ( θ ′ )(= I ( ˆ θ )) fo r any θ ∈ U θ ′ . Here we define a subset of U θ ′ as V ≡ { θ | θ ∈ U θ ′ , θ 6∈ ∆( ˆ θ ) } ∩ D ( I ( ˆ θ )) . Assum e th at V is non- empty . Then it hold s I ( θ ) < I ( ˆ θ ) for any θ ∈ V , since θ ∈ { θ | θ ∈ U θ ′ , θ 6∈ ∆( ˆ θ ) } . On the other h and, it holds th at I ( θ ) ≥ I ( ˆ θ ) for any θ ∈ V since θ ∈ D ( I ( ˆ θ )) by the definitio n of V . Th is is inconsistent with the assumption that V is non -empty . Thus V is empty and ∆( ˆ θ ) = D ( I ( ˆ θ )) . Since both ˆ θ and ¯ θ belo ng to D ( I ( ˆ θ )) , it hold s I ( ˆ θ ) = I ( ¯ θ ) . Therefore the assumption I ( ¯ θ ) > I ( ˆ θ ) is in valid. This means th at any so lution θ of the Kuhn- T ucker co nditions (2 1) for ( P , Y ) gives the same value for I ( θ ) and then it is optimal. Thus the sufficiency is proved. Note that th e Lemma 1 holds for any domain Y of X fo rmed by ρ ′ and ρ ′′ . V I . P RO O F O F T H E O R E M 2 In this section, we prove T heorem 2 by using Lem ma 1. W e state again Theorem 2: Theor em 2: The Kuhn-T ucker cond itions for the ch annel capacity C of an N - user elementar y ( n 1 , · · · , n N ; m ) -MAC ( P, X ) , where n k ≤ m f or all k = 1 , · · · , N , are necessary and sufficient. ✷ Pr oof: It is sufficient to prove the sufficiency . L et ¯ p = ¯ p 1 × · · · × ¯ p N a so lution of th e Kuhn-Tucker con ditions (3) for the N -u ser elementar y ( n 1 , · · · , n N ; m ) -MAC ( P, X ) , where n k ≤ m , k = 1 , · · · , N . W e pr ove that ¯ p is uniquely determined in the sense that any solution p of the K uhn -T ucker condition s (3) gives th e same for I ( p ) . For an arbitrary p ′ k ∈ X k , k = 1 , · · · , N , there exist p ′′ k ∈ X k and ¯ θ k such that ¯ p k = ¯ θ k p ′ k + (1 − ¯ θ k ) p ′′ k , 0 ≤ ¯ θ k ≤ 1 (22) since X k is simplex. Then ¯ p is represented by ¯ p = ( ¯ θ 1 p ′ 1 + (1 − ¯ θ 1 ) p ′′ 1 ) × · · · × ( ¯ θ N p ′ N + (1 − ¯ θ N ) p ′′ N ) . (23) Here we d efine a function of variables ( θ 1 , · · · , θ N ) ≡ θ ( 0 ≤ θ k ≤ 1 ) by I ( θ ; p ′ , p ′′ ) ≡ I (( θ 1 p ′ 1 + (1 − θ 1 ) p ′′ 1 ) × · · · × ( θ N p ′ N + (1 − θ N ) p ′′ N )) (24 ) where p ′ = p ′ 1 × · · · × p ′ N and p ′′ = p ′′ 1 × · · · × p ′′ N . The function (2 4) ca n be regarded as the mutu al info rmation of an N -u ser (2 , · · · , 2; m ) -MA C ( P , Y ) with th e dom ain Y ≡ Y 1 × · · · × Y N , where Y k ≡ { θ k p ′ k + (1 − θ k ) p ′′ k | 0 ≤ θ k ≤ 1 } , k = 1 , · · · , N . The N -user (2 , · · · , 2; m ) -MA C ( P , Y ) is denoted by ( P, Y ) ( p ′ , p ′′ ) , since it depen ds o n p ′ , p ′′ . Since ¯ p is a solution of the Kuhn -T ucker co nditions (3) for ( P, X ) , then ¯ θ is a solution of th e Kuhn- T ucker cond itions for the mutual informa tion (2 4) of ( P, Y ) ( p ′ , p ′′ ) : I k ( θ ; p ′ , p ′′ ) = C , θ k > 0 ≤ C , θ k = 0 k = 1 , · · · , N , C = I ( θ ; p ′ , p ′′ ) (25) where I k ( θ ; p ′ , p ′′ ) = ∂ I ( θ ; p ′ , p ′′ ) /∂ θ k . Therefo re, it fol- lows from Lemm a 1 tha t ¯ θ is op timal f or ( P , Y ) ( p ′ , p ′′ ) , which means I ( ¯ θ ; p ′ , p ′′ ) ≥ I ( θ ; p ′ , p ′′ ) (26) for any θ ∈ Y . Since ¯ θ is giv en by (23), it holds I ( ¯ θ ; p ′ , p ′′ ) = I ( ¯ p ) (27) for any p ′ ∈ X , wher e p ′′ satisfies (22). Thus since (26) and (27) are valid f or any p ′ ∈ X , it holds I ( ¯ p ) ≥ I ( p ) on the whole domain X . This implies that ¯ p is optim al. Thus we proved the theor em. 10 V I I . C O N C L U S I O N S After Shannon [2] multiuser chan nel has long bee n studied in various fields. Howev er not much works have been made for the fu ndamen tal pro perty o f the ch annel capacity of an N - user ( n 1 , · · · , n N ) -MA C ( P , X ) in gen eral except for some specific cases. W e hav e shown that there exists a non-trivial M A C where the Kuhn -T ucker condition s are necessary and sufficient f or the chan nel cap acity . W e called it as an ele mentary MA C that was d efined by the MAC who se sizes of inpu t alp habets must be not greater than the size of outpu t alp habet. Obvio usly the N -u ser b inary inputs (2 , · · · , 2 ; m ) -MAC ( P, X ) is a typica l example of the elemen tary MA C. Also the DMC is a trivial elementary MA C. W e believe th at there is consid erable merit in a concep t of elementary MA C for which the channel capacity is ev alu ated precisely b y the necessary and sufficient conditio n as in the case of DMC. In fact, we have pr oved as T heorem 1 that the chann el cap acity of any MA C is ach ie ved by the channel capacity of an elementary MA C co ntained in the origina l MA C. Thu s an MAC in general can be r egarded as simply an aggr egate of elemen tary MA C’ s. This statemen t is a basic idea behind our formu lation of this pa per . The most of this paper was dev oted to the proof of The- orem 2 such that the Kuhn-T uc ker co nditions are sufficient (the n ecessity is self- evident) for the cha nnel capacity of the elementary MA C. W e h av e shown as Pro position 2 that a solution o f the Kuhn-Tucker conditions if it satisfies the equal- ity portion of the condition s satisfies the boundary eq uations which defin e the b oundar y of the capacity region. Then we could p rove the pro perty of local maximu m as Proposition 3 followed by the proper ty of conn ectedness as Prop osition 4. By using these two distinctive featu res we cou ld prove that any so lution o f the Kuhn- T ucker co nditions of the elementary MA C was uniqu ely determin ed, that is, e ach solution takes the same value f or the th e mutu al information and th erefore it achieves the channe l capacity . In this respect, we rema rk that the non-elemen tary MA C has a degenerate property as explained in Section II. If it exists, then it is difficult to identify which IPD vectors are exactly contributed to the mutual in formation of the MA C. Howe ver we overcome these difficulties by in troducin g the conc ept of elementary MA C where there exists no such degen erate proper ty . Sin ce the we ll-known DMC is elem entary , then th e elementary MA C is identified as an extension of the DMC. Incidentally , ou r notation introdu ced in this paper seems rather non-stand ard in cluding expressions o f IPD vector p , Kronecker prod ucts p = p 1 × · · · × p N , the chann el matrix P regarded as a non-linear mapping , dom ain X , face F , an d so force . However we emp hasize that the notatio n appears effecti ve to r esolve th e cum bersome p rocedur es relating to the extremum ev aluation of the multi-variable mutu al infor mation with constrain ts for the MAC. Before closing we remark that th e very essence of infor- mation theo ry consists in two majo r subjects such as source coding and channel coding as we know . This paper seems to be quite effective in working out the subject of ch annel codin g since we pr ovide for a formalism to determin e the channel capacity of the MAC . W e ar e con fident that two d istinctiv e features of local maximum (Pro position 3) and conn ectedness (Proposition 4) represen t an intrinsic structure of the MAC. 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