Capacity Bounds for the Gaussian Interference Channel

1 Capacity Bounds for the Gaussian Inte rferenc e Ch annel Abolfazl S. Mot ahari, Student Member , IEEE, and Amir K. Khandani, Member , IEEE Coding & Signal Trans mission Laboratory (www .cst.uw aterloo.ca) { abolfa zl,khandani } @cst.uw aterloo.ca Abstract The capac ity region of the two-user Gaussian Interference Channel (I C) i s studied. Three classes of channels are considered: weak, one-sided, an d mixed Gaussian I C. For the weak Gaus sian IC, a new outer bound on the capacity r egion is obtained that outperforms prev iously known outer bounds. The sum capacity for a certain range of channel parameters is deriv ed. For this range, it is proved that using Gaussian co debook s and t reating interference as noise is optimal. It is shown that when Gaussian codeboo ks are used, the f ull Han-K obayash i achie v able rate region can be obtained by using the naiv e Han-K obaya shi achie v able scheme ov er three frequ ency ban ds (equi v alently , three subspac es). For the on e-sided Gaussian IC, an alternati v e proof for the Sato’ s outer bound is presented. W e deri v e the full Han-K obaya shi achiev able ra te region when Gaussian codebook s are utilized. For the mixed Gaussian IC, a new outer bound is obtained that ou tperforms pre viously known o uter bo unds. Fo r th is case, the sum capacity for the entire range of chan nel parameters is deriv ed. It is prov ed that the full Han-K obayashi achiev able rate region using Gaussian codebook s is equiv alent to that of the one-sided Gaussian IC for a particular ran ge of channel parameters. Index T erms Gaussian interference channels, capacity region, sum capacity , con vex regions. I . I N T R O D U C T I O N O NE of the fund amental prob lems in In formation Theor y , origin ating from [1], is the full characterization of the capacity region of the interferenc e chan nel (IC). The simplest form of IC is the two-user case in which two tran smitters aim to conv ey ind ependen t m essages to th eir cor respond ing re ceiv ers through a common chan nel. Despite some special cases, such as very stro ng and stro ng interfer ence, where th e exact capacity r egion has bee n derived [2], [3], the characte rization of the capacity region for the general case is still an op en p roblem. A limiting expression f or the capacity region is ob tained in [4] (see also [5]). Unfo rtunately , due to excessi ve comp utational complexity , this type of expression d oes not r esult in a tractable a pproach to fu lly char acterize th e capacity region. T o show the weakness of the limiting expression, Cheng and V erd ´ u have shown tha t fo r the Gau ssian Multiple A ccess Channel (MA C), which c an be conside red as a special case of the Gaussian I C, the limiting expression fails to f ully ch aracterize the cap acity region by rely ing only on Gau ssian distributions [6]. Howe ver , there is a po int o n the bounda ry of the capacity region of the MA C that can be obtain ed directly from the limiting expression. This poin t is achievable by using simp le scheme of Frequency/Time Di vision (FD/TD). The co mputationa l complexity inh erent to the limiting expression is due to the fact that the corre sponding encodin g and decodin g strate gies are of the simplest possible form. The en coding strategy is based on mapp ing data to a code book constructed from a unique prob ability d ensity and the decoding strategy is to treat the interferen ce as noise. In contrast, using mo re sophisticated encod ers and decoder s may result in collapsing the limiting expression into a single letter fo rmula for the capacity region . As an evidence, it is kn own that the joint typical dec oder for th e MAC ac hiev es the capacity region [ 7]. Moreover , th ere are some special cases, su ch as strong I C, wh ere the exact char acterization of the capa city region h as been derived [2], [3] wh ere decod ing the interferen ce is th e key idea behin d this result. In their pioneerin g work, Han and K obayashi (HK ) p roposed a coding strategy in which the receivers are allowed to deco de part o f the in terference as well as their own d ata [8]. The HK ach iev able region is still th e best inner bo und for th e capacity region. Speciﬁcally , in their scheme, the me ssage o f each user is split into two indep endent par ts: the co mmon part and the priv ate par t. Th e commo n part is en coded such that both user s can decod e it. The p riv ate part, on the oth er h and, can b e decoded only by the intended receiver a nd th e other receiver treats it as n oise. In summary , the HK achievable regio n is the intersection of the capacity regions o f two three-u ser MACs , pro jected o n a two-dim ensional subspa ce. The HK scheme can be directly ap plied to the Gaussian IC. Non etheless, there are two sources of difﬁculties in characterizin g the full HK achiev able rate region. First, the optimal d istributions are un known. Seco nd, even if we con ﬁne the distributions to be Gau ssian, comp utation of the full HK region und er Gaussian d istribution is still difﬁcult due to num erous degrees of freed om in volv ed in the pro blem. The main reason b ehind this com plexity is the comp utation of the cardinality of the time-sharin g parameter . 1 An earlie r version of this work containing all the results is repor ted in Library and Archive s Canada T echnical Report UW-E CE 2007-26 , Aug. 2007 (see http://www .cst.uwaterloo.ca/pub tech rep.html for details). 2 Recently , refer ence [ 9], Chong et al. has presen ted a simpler expression with less inequalities for the HK ach ie vable region. Since the cardinality of the time-sharin g pa rameter is d irectly related to the number of inequalities appear ing in the achiev able rate region, the com putational co mplexity is d ecreased. Howe ver , ﬁn ding the full HK achiev able r egion is still proh ibitiv ely complex. Regarding outer boun ds on the capacity region, there are three main resu lts known. T he ﬁrst one ob tained b y Sato [1 0] is originally derived for the degraded Gaussian IC. Sato has shown that the cap acity region of the degraded Gaussian IC is outer boun ded by a certain degrad ed broa dcast c hannel whose capacity region is fully character ized. In [11], Costa has p roved that the ca pacity region of the degrade d Gaussian bro adcast chann el is eq uiv alent to th at of th e o ne-sided weak Gaussian IC. Hence, Sato outer bou nd ca n be u sed for the o ne-sided Gaussian IC as we ll. The secon d outer bo und o btained for the weak Gaussian IC is d ue to Kramer [12]. Kram er outer bound is based on the fact that removing o ne of the interfer ing link s enlarges the cap acity region. Theref ore, th e cap acity r egion of the two-user Gaussian IC is inside th e intersectio n of the cap acity region s of th e und erlying one- sided Gau ssian ICs. For the case of weak Gaussian IC, th e und erlying one- sided I C is we ak, for wh ich the capacity region is unkn own. Howe v er , Kramer has used the outer bo und ob tained by Sato to d eriv e an o uter boun d f or the weak Gau ssian IC. The third o uter b ound due to Etkin, Tse, and W ang (ETW) is based o n the Genie aided tech nique [13]. A genie that provide s some extra inf ormation to the receivers can on ly enlarge th e cap acity region. At ﬁrst glanc e, it seem s a clever gen ie must provide some info rmation ab out the interf erence to the receiver to h elp in d ecoding the sign al b y removing the interfe rence. In contrast, the genie in the ETW scheme pr ovides info rmation abou t th e inten ded sign al to the r eceiv er . Remar kably , ref erence [13] shows that their prop osed outer bo und outp erforms Kram er b ound fo r c ertain range of parameters. Moreover , using a similar meth od, [13] presents an outer bo und for th e m ixed Gaussian IC. In this paper, by in troducin g the notion of admissible I Cs, we prop ose a new o uter boun ding techniqu e for the two-user Gaussian IC. Th e pro posed techniqu e r elies on an extremal inequality recen tly proved b y Liu and V isw anath [14]. W e show that by using this schem e, on e can o btain tighter outer bo unds for both weak and mixed Gaussian ICs. More impor tantly , the sum capacity o f the Gau ssian weak I C for a cer tain range o f the chan nel paramete rs is der iv ed. The r est of this pap er is o rganized as follows. In Section II , we p resent some basic d eﬁnitions and revie w the HK ach iev able region when Gaussian co debook s are used. W e study th e tim e-sharing an d the conv exiﬁcation method s as means to enlarge the basic HK achievable region. W e investigate condition s for which the two regions obtained from time-sharing and con caviﬁcation coincide. Finally , we consider an op timization problem based on extremal inequ ality and com pute its optimal solu tion. In Section II I, th e notion of an ad missible IC is intro duced. Some classes of ad missible ICs for the two-user Gau ssian c ase is studied and ou ter boun ds on the capacity region s of these classes are comp uted. W e also obtain the sum cap acity of a spe ciﬁc class of admissible IC wh ere it is shown that using Gau ssian c odeboo ks and treatin g interfere nce as noise is op timal. In Sectio n IV , we study the cap acity region of th e weak Gau ssian IC. W e ﬁrst derive th e sum cap acity of this chan nel for a certain rang e of p arameters where it is proved that user s should treat the interfer ence a s no ise and tr ansmit at their h ighest possible r ates. W e then derive an ou ter bo und on the capacity region wh ich ou tperform s the known results. W e ﬁnally prove that the b asic HK ach iev able region results in the sam e e nlarged r egion by using either time-shar ing or concaviﬁcation. This reduces the co mplexity o f the char acterization of the fu ll HK a chiev able region wh en Gaussian codebo oks are used. In Sectio n V , we study the capacity region of th e on e-sided Gaussian IC. W e pr esent a new pro of for the Sato outer bo und using the extrem al in equality . T hen, we present m ethods to simplify th e HK achievable region such that the f ull region can be characterized . In Section VI, we study the capacity region o f th e mixed Gaussian IC. W e ﬁrst ob tain the sum capacity o f this channel and then der i ve an outer bo und which outp erforms other known r esults. Finally , by investigating the HK achiev able region fo r different cases, we prove that f or a certain range of c hannel parameters, the f ull HK ach ie vable r ate region using Ga ussian codebo oks is equiv alent to th at of the one-sided IC. Finally , in Section VII, we con clude the p aper . A. Notatio ns Throu ghout this pap er , we use the following notations. V ectors are rep resented b y bold faced letters. Rand om variables, matrices, an d sets a re den oted by cap ital letters where the difference is clear from the co ntext. | A | , tr { A } , an d A t represent the determin ant, trace, and transpo se of the squa re matrix A , resp ecti vely . I denotes the identity matrix . N and ℜ are th e sets of n onnegative in tegers and real nu mbers, resp ecti vely . The unio n, intersection , and Minkowski sum o f two sets U and V are represented by U ∪ V , U ∩ V , and U + V , respectively . W e u se γ ( x ) as an ab breviation for th e f unction 0 . 5 log 2 (1 + x ) . I I . P R E L I M I NA R I E S A. The T wo-user Interference Chan nel Deﬁnition 1 (two-user IC): A two-user discrete mem oryless IC consists of two ﬁnite sets X 1 and X 2 as input alphabe ts and two ﬁnite sets Y 1 and Y 2 as th e correspon ding o utput alphabets. Th e channel is governed by cond itional pro bability distributions ω ( y 1 , y 2 | x 1 , x 2 ) where ( x 1 , x 2 ) ∈ X 1 × X 2 and ( y 1 , y 2 ) ∈ Y 1 × Y 2 . 3 00000000 00000000 00000000 00000000 00000000 00000000 00000000 11111111 11111111 11111111 11111111 11111111 11111111 11111111 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 One−sided Mixed Mixed One−sided Degraded Strong Weak 1 1 b a ab = 1 P 1 = P 2 Symmetric Fig. 1. Classe s of the two-user ICs. Deﬁnition 2 (capa c ity r e gion of the two-user IC): A code ( 2 nR 1 , 2 nR 2 , n, λ n 1 , λ n 2 ) f or the two-user I C c onsists of the fol- lowing com ponen ts for User i ∈ { 1 , 2 } : 1) A u niform distributed m essage set M i ∈ [1 , 2 , ..., 2 nR i ] . 2) A co deboo k X i = { x i (1) , x i (2) , ..., x i (2 nR i ) } wher e x i ( · ) ∈ X n i . 3) An encodin g functio n F i : [1 , 2 , ..., 2 nR i ] → X i . 4) A d ecoding fun ction G i : y i → [1 , 2 , ..., 2 nR i ] . 5) The average prob ability of er ror λ n i = P ( G i ( y i ) 6 = M i ) . A rate pair ( R 1 , R 2 ) is ach iev able if there is a sequ ence of codes ( 2 nR 1 , 2 nR 2 , n, λ n 1 , λ n 2 ) with vanishing average error probab ilities. The cap acity region of the I C is d eﬁned to be the sup remum of the set of achiev able rates. Let C I C denote the capacity region of th e two-user IC. Th e limiting expression for C I C can be stated as [5 ] C I C = lim n →∞ cl osure   [ P ( X n 1 ) P ( X n 2 )  ( R 1 , R 2 ) | R 1 ≤ 1 n I ( X n 1 , Y n 1 ) R 2 ≤ 1 n I ( X n 2 , Y n 2 )    . (1) In this p aper, we focus o n the two-user Gaussian IC wh ich can be re presented in stan dard form as [15], [1 6] y 1 = x 1 + √ ax 2 + z 1 , y 2 = √ bx 1 + x 2 + z 2 , (2) where x i and y i denote the inpu t an d ou tput alp habets of User i ∈ { 1 , 2 } , re spectiv ely , and z 1 ∼ N (0 , 1 ) , z 2 ∼ N (0 , 1 ) are standard Gaussian ran dom variables. Constan ts a ≥ 0 and b ≥ 0 represent the gain s of th e interfer ence lin ks. Fu rthermo re, T ransmitter i , i ∈ { 1 , 2 } , is subject to the power con straint P i . Achievable rates and the c apacity r egion o f the Gaussian IC can be deﬁned in a similar fashion as th at o f th e general IC with the condition that the codew ords mu st satisfy th eir correspond ing power co nstraints. The cap acity region of the two-user Gaussian IC is denoted b y C . Clearly , C is a function of th e param eters P 1 , P 2 , a , an d b . T o emp hasize this re lationship, we ma y write C as C ( P 1 , P 2 , a, b ) a s needed . Remark 1: Since th e capacity r egion o f the general IC depends only on the m arginal d istributions [ 16], the ICs can be classiﬁed into equiv alent classes in which ch annels within a class hav e th e same capacity region. In particular, for the Gaussian IC g i ven in (2), any choice of jo int distributions for the pair ( z 1 , z 2 ) do es not affect the capac ity region as long as the marginal distributions r emain Ga ussian with zer o mean and unit variance. Dependin g on the values of a a nd b , the two-user Gau ssian IC is classiﬁed into weak, strong , m ixed, one-sided , and degraded Gau ssian IC. In Fig ure 1, regions in ab -plan e to gether with their associated nam es ar e shown. Brieﬂy , if 0 < a < 1 and 0 < b < 1 , then the chann el is called weak Gaussian IC . If 1 ≤ a and 1 ≤ b , then the chann el is called str on g Gau ssian IC . I f either a = 0 or b = 0 , th e chann el is called one- sided Gaussian IC . If ab = 1 , th en the channel is called de graded Gaussian IC . If either 0 < a < 1 and 1 ≤ b , or 0 < b < 1 and 1 ≤ a , then the chann el is called mixed Gaussian I C . Finally , the symmetric Gaussian IC (used throu ghout th e p aper for illustration pu rposes) cor responds to a = b and P 1 = P 2 . Among all classes shown in Figur e 1, the capacity region of the stron g Gau ssian I C is fully ch aracterized [ 3], [2]. In th is case, the ca pacity region can be stated as th e collectio n of all r ate pairs ( R 1 , R 2 ) satisfyin g R 1 ≤ γ ( P 1 ) , R 2 ≤ γ ( P 2 ) , R 1 + R 2 ≤ min { γ ( P 1 + aP 2 ) , γ ( bP 1 + P 2 ) } . 4 B. Su pport Function s Throu ghout this paper, we use the fo llowing facts fro m con vex analysis. There is a o ne to one correspo ndence b etween any closed co n vex set and its suppor t function [ 17]. Th e suppor t function of a ny set D ∈ ℜ m is a f unction σ D : ℜ m → ℜ d eﬁned as σ D ( c ) = sup { c t R | R ∈ D } . (3) Clearly , if the set D is compac t, then the sup is attained and can be replac ed b y max. In this case, the solutio ns of (3) correspo nd to the bound ary p oints of D [ 17]. The fo llowing relation is th e dual o f (3) and h olds when D is closed and conve x D = { R | c t R ≤ σ D ( c ) , ∀ c } . (4) For any two closed conve x sets D an d D ′ , D ⊆ D ′ , if and only if σ D ≤ σ D ′ . C. Ha n-K obayashi Achievable R e gion The best inner bou nd for the two-user Gau ssian I C is th e full HK ach iev able region deno ted by C H K [8]. Despite having a single letter f ormula, C H K is not fully characterized yet. In fact, ﬁnding the optimum d istributions achieving bo undary points of C H K is still an o pen problem . W e deﬁne G as a subset of C H K where Gaussian d istributions are u sed for codeb ook generation . Using a shor ter descrip tion o f C H K obtained in [9], G can be d escribed as follows. Let us ﬁrst deﬁne G 0 as the collectio n of all rate p airs ( R 1 , R 2 ) ∈ ℜ 2 + satisfying R 1 ≤ ψ 1 = γ  P 1 1 + aβ P 2  , (5) R 2 ≤ ψ 2 = γ  P 2 1 + bαP 1  , (6) R 1 + R 2 ≤ ψ 3 = min { ψ 31 , ψ 32 , ψ 33 } , (7) 2 R 1 + R 2 ≤ ψ 4 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ  αP 1 1 + aβ P 2  + γ  β P 2 + b (1 − α ) P 1 1 + bαP 1  , (8) R 1 + 2 R 2 ≤ ψ 5 = γ  β P 2 1 + bαP 1  + γ  P 2 + b (1 − α ) P 1 1 + bαP 1  + γ  αP 1 + a (1 − β ) P 2 1 + aβ P 2  , (9) for ﬁxed α ∈ [0 , 1] and β ∈ [0 , 1 ] . 1 ψ 3 is the minimum of ψ 31 , ψ 32 , and ψ 33 deﬁned as ψ 31 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ  β P 2 1 + bαP 1  , (10) ψ 32 = γ  αP 1 1 + aβ P 2  + γ  P 2 + b (1 − α ) P 1 1 + bαP 1  , (11) ψ 33 = γ  αP 1 + a (1 − β ) P 2 1 + aβ P 2  + γ  β P 2 + b (1 − α ) P 1 1 + bαP 1  . (12) G 0 is a p olytope and a fu nction of fou r variables P 1 , P 2 , α , and β . T o em phasize this r elation, we may write G 0 ( P 1 , P 2 , α, β ) as n eeded. It is co n venient to represent G 0 in a matrix form as G 0 = { R | A R ≤ Ψ( P 1 , P 2 , α, β ) } where R = ( R 1 , R 2 ) t , Ψ = ( ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 ) t , and A =  1 0 1 2 1 0 1 1 1 2  t . Equiv alently , G 0 can be r epresented as th e c on vex hull o f its extreme points, i.e., G 0 ( P 1 , P 2 , α, β ) = conv { r 1 , r 2 , . . . , r K } , where it is assumed that G 0 has K extreme points. I t is easy to sh ow that K ≤ 7 . Now , G c an be deﬁne d as a region obtain ed from enlargin g G 0 by makin g use o f the time-shar ing p arameter, i.e. , G is the collection of all rate p airs R = ( R 1 , R 2 ) t satisfying A R ≤ q X i =1 λ i Ψ( P 1 i , P 2 i , α i , β i ) , (13) 1 In the HK scheme, two independent messages are encode d at each transmitter , namely the common message and the private message . α and β are the paramete rs that determine the amount of power allocated to the common and priv at e m essages for the two users, i.e., αP 1 , β P 2 and (1 − α ) P 1 , (1 − β ) P 2 of the total po wer is used for the transmission of the pri v at e/common messages to the ﬁrst/sec ond users, respecti ve ly . 5 where q ∈ N an d q X i =1 λ i P 1 i ≤ P 1 , (14) q X i =1 λ i P 2 i ≤ P 2 , (15) q X i =1 λ i = 1 , (16) λ i ≥ 0 , ( α i , β i ) ∈ [0 , 1] 2 ; ∀ i ∈ { 1 , . . . , q } . (17) It is easy to show that G is a closed , bound ed an d conve x r egion. In fact, the capa city region C which contains G is inside the rectan gle d eﬁned by ine qualities R 1 ≤ γ ( P 1 ) an d R 2 ≤ γ ( P 2 ) . Mo reover , (0 , 0) , ( γ ( P 1 ) , 0) , and (0 , γ ( P 2 )) are extreme points of both C and G . Hence, to ch aracterize G , we n eed to o btain all extreme p oints of G that are in th e interior of the ﬁrst q uadrant (th e same argument hold s for C ). In other words, we nee d to obtain σ G ( c 1 , c 2 ) , th e sup port fu nction of G , either when 1 ≤ c 1 and c 2 = 1 o r when c 1 = 1 and 1 ≤ c 2 . W e also deﬁne G 1 and G 2 obtained by e nlarging G 0 in two d ifferent mann ers. G 1 is deﬁned as G 1 ( P 1 , P 2 ) = [ ( α,β ) ∈ [0 , 1] 2 G 0 ( P 1 , P 2 , α, β ) . (18) G 1 is no t ne cessarily a co n vex region . Hence , it can be fur ther enlarged by th e conve x h ull oper ation. G 2 is deﬁn ed as the collection of all rate p airs R = ( R 1 , R 2 ) t satisfying R = q ′ X i =1 λ i R i (19) where q ′ ∈ N an d A R i ≤ Ψ( P 1 i , P 2 i , α i , β i ) , (20) q ′ X i =1 λ i P 1 i ≤ P 1 , (21) q ′ X i =1 λ i P 2 i ≤ P 2 , (22) q ′ X i =1 λ i = 1 , (23) λ i ≥ 0 , ( α i , β i ) ∈ [0 , 1] 2 ; ∀ i ∈ { 1 , . . . , q ′ } . (24) It is e asy to show that G 2 is a clo sed, b ounde d and co n vex re gion. In fact, G 2 is o btained b y u sing the simple metho d of TD/FD. T o see this, let us divide the av ailable fre quency band into q ′ sub-ban ds w here λ i represents the len gth o f the i ’th ba nd an d P q ′ i =1 λ i = 1 . User 1 an d 2 allocate P 1 i and P 2 i in the i ’th sub-b and, respecti vely . Therefore, all rate pairs in G 0 ( P 1 i , P 2 i , α i , β i ) are achievable in the i ’th sub-b and for ﬁxed ( α i , β i ) ∈ [0 , 1] 2 . Hence, all rate pairs in P q ′ i =1 λ i G 0 ( P 1 i , P 2 i , α i , β i ) ar e achiev able provided th at P q ′ i =1 λ i P 1 i ≤ P 1 and P q ′ i =1 λ i P 2 i ≤ P 2 . Clearly , th e c hain of inc lusions G 0 ⊆ G 1 ⊆ G 2 ⊆ G ⊆ C H K ⊆ C always h olds. D. Conca viﬁcation V ersus T ime-Sharing In this subsection, we follow two objectiv es. First, we aim at providing some n ecessary condition s such that G 2 = G . Second, we b ound q an d q ′ which are p arameters in volved in th e description s o f G and G 2 , respec ti vely . Howe ver , w e derive the requir ed conditio ns for the more gener al c ase wh ere there are M user s in th e system . T o this end , assume an achievable scheme for an M -user chann el with the power con straint P = [ P 1 , P 2 , . . . , P M ] is giv en. Th e co rrespond ing achiev able region can be represented as D 0 ( P , Θ) = { R | A R ≤ Ψ ( P , Θ) } , (25) where A is a K × M matrix an d Θ ∈ [0 , 1] M . D 0 is a p olyhedr on in general, but f or the purpo se of th is paper, it sufﬁces to assume tha t it is a poly tope. Sin ce D 0 is a co n vex region, th e co n vex h ull operation do es not lead to a new e nlarged r egion. Howe ver , if the extreme points of the region are not a con cave func tion of P , it is possible to enlarge D 0 by using two 6 different method s which are explained next. Th e ﬁrst metho d is based on using the time sharing pa rameter . Let u s d enote the correspo nding region as D which can be written as D = ( R | A R ≤ q X i =1 λ i Ψ( P i , Θ i ) , q X i =1 λ i P i ≤ P , q X i =1 λ i = 1 , λ i ≥ 0 , Θ i ∈ [0 , 1] M ∀ i ) , (26) where q ∈ N . In the second method, we use TD/FD to enlarge the achievable rate region. This results in an achiev able region D 2 represented as D 2 =    R = q ′ X i =1 λ i R i | A R i ≤ Ψ( P i , Θ i ) , q ′ X i =1 λ i P i ≤ P , q ′ X i =1 λ i = 1 , λ i ≥ 0 , Θ i ∈ [0 , 1] M ∀ i    , (27) where q ′ ∈ N . W e refer to this meth od as co ncaviﬁcation. It can be r eadily shown that D and D 2 are clo sed and conv ex, an d D 2 ⊆ D . W e are interested in situa tions where the inv erse inclu sion holds. The suppo rt fu nction of D 0 is a functio n of P , Θ , an d c . Henc e, we have σ D 0 ( c , P , Θ) = max { c t R | A R ≤ Ψ( P , Θ) } . (28) For ﬁxed P and Θ , (28) is a linear progr am. Using strong du ality o f linear p rogram ming, we obtain σ D 0 ( c , P , Θ) = min { y t Ψ( P , Θ) | A t y = c , y ≥ 0 } . (29) In g eneral, ˆ y , the minim izer of (29), is a fun ction of P , Θ , and c . W e say D 0 possesses the unique minimizer pr operty if ˆ y merely dep ends o n c , for all c . In th is case, we have σ D 0 ( c , P , Θ) = ˆ y t ( c )Ψ( P , Θ) , (30) where A t ˆ y = c . This cond ition means that for any c th e extrem e p oint o f D 0 maximizing the ob jectiv e c t R is a n extreme point obtain ed by intersecting a set o f sp eciﬁc hype rplanes. A necessary con dition for D 0 to possess the uniqu e m inimizer proper ty is th at each in equality in describin g D 0 is eithe r redun dant or active fo r all P and Θ . Theor em 1 : If D 0 possesses the unique minim izer pr operty , th en D = D 2 . Pr o of: Since D 2 ⊆ D always ho lds, we need to show D ⊆ D 2 which can be e quiv alently veriﬁed by showing σ D ≤ σ D 2 . The suppo rt fu nction of D can b e w ritten as σ D ( c , P ) = max  c t R | R ∈ D  . (31) By ﬁx ing P , P i ’ s, Θ i ’ s, and λ i ’ s, the above maximizatio n become s a linear program . Hence, relying o n weak du ality of linear progr amming, we obtain σ D ( c , P ) ≤ min A t y = c , y ≥ 0 y t q X i =1 λ i Ψ( P i , Θ i ) . (32 ) Clearly , ˆ y ( c ) , th e solution of (29), is a fea sible p oint for ( 32) and we have σ D ( c , P ) ≤ ˆ y t ( c ) q X i =1 λ i Ψ( P i , Θ i ) . (33) Using (30), we obtain σ D ( c , P ) ≤ q X i =1 λ i σ D 0 ( c , P i , Θ i ) . (34) Let us assume ˆ R i is the maximizer of (28). In this case, we have σ D ( c , P ) ≤ q X i =1 λ i c t ˆ R i . (35) Hence, we have σ D ( c , P ) ≤ c t q X i =1 λ i ˆ R i . (36) By deﬁnitio n, P q i =1 λ i ˆ R i is a p oint in D 2 . The refore, we conc lude σ D ( c , P ) ≤ σ D 2 ( c , P ) . (37) This com pletes the pro of. 7 Cor o llary 1 (Ha n [18]): If D 0 is a p olymatroid , then D = D 2 . Pr o of: It is easy to show th at D 0 possesses th e uniq ue minimizer proper ty . In fact, f or given c , ˆ y can be obtained in a greedy fashion ind ependen t o f P and Θ . In what follows, we u pper boun d q and q ′ . Theor em 2 : Th e cardinality of the time sh aring parameter q in (26) is less th an M + K + 1 , wh ere M and K are the dimensions of P and Ψ( P ) , respectively . Moreover, if Ψ ( P ) is a continu ous f unction o f P , then q ≤ M + K . Pr o of: Let us deﬁn e E as E = ( q X i =1 λ i Ψ( P i , Θ i ) | q X i =1 λ i P i ≤ P , q X i =1 λ i = 1 , λ i ≥ 0 , Θ i ∈ [0 , 1] M ∀ i ) . (38) In fact, E is the collection of all possible boun ds for D . T o prove q ≤ M + K + 1 , we deﬁne ano ther r egion E 1 as E 1 = { ( P ′ , S ′ ) | 0 ≤ P ′ , S ′ = Ψ( P ′ , Θ ′ ) , Θ ′ ∈ [0 , 1] M } . (39) From the direct consequ ence of the Carathe odory ’ s theor em [19], the conve x h ull of E 1 denoted by conv E 1 can b e obtained by conve x com binations of no more th an M + K + 1 po ints in E 1 . Mor eover , if Ψ( P ′ , Θ ′ ) is continu ous, th en M + K po ints are sufﬁcient d ue to th e extension of the Cara theodor y’ s theorem [19]. N ow , we d eﬁne the region ˆ E as ˆ E = { S ′ | ( P ′ , S ′ ) ∈ conv E 1 , P ′ ≤ P } . (40) Clearly , ˆ E ⊆ E . T o show th e other inclusion, let us consider a point in E , say S = P q i =1 λ i Ψ( P i , Θ i ) . Since ( P i , Ψ( P i , Θ i )) is a poin t in E 1 , P q i =1 λ i ( P i , Ψ( P i , Θ i )) be longs to conv E 1 . Having P q i =1 λ i P i ≤ P , we conclu de P q i =1 λ i Ψ( P i , Θ) ∈ ˆ E . Hence, E ⊆ ˆ E . This completes the p roof. Cor o llary 2 (E tkin , P a rakh, an d Tse [20]): For the M -user G aussian IC where u sers u se Gaussian codebo oks f or d ata transmission and treat the interferen ce as n oise, the card inality o f the time sharing pa rameter is less than 2 M . Pr o of: In th is case, D 0 = { R | R ≤ Ψ( P ) } wher e bo th P and Ψ ( P ) have dimen sion M and Ψ( P ) is a contin uous function of P . Apply ing T heorem 2 yield s the desired result. In the following theo rem, we obtain an upper bou nd on q ′ . Theor em 3 : T o char acterize bou ndary poin ts of D 2 , it su fﬁces to set q ′ ≤ M + 1 . Pr o of: Let us assume ˆ R is a bou ndary p oint of D 2 . Hence , there exists c such that σ D 2 ( c , P ) = max R ∈ D 2 c t R = c t ˆ R , (41) where ˆ R = P q ′ i =1 ˆ λ i ˆ R i and th e optimum is achieved for the set of para meters ˆ Θ i , ˆ λ i , and ˆ P i . The optimiz ation prob lem in (41) can be written as σ D 2 ( c , P ) =max q ′ X i =1 λ i g ( c , P i ) (42) subject to: q ′ X i =1 λ i = 1 , q ′ X i =1 λ i P i ≤ P , 0 ≤ λ i , 0 ≤ P i , ∀ i ∈ { 1 , 2 , . . . , q ′ } , where g ( c , P ) is deﬁn ed as g ( c , P ) =max c t R (43) subject to: A R ≤ Ψ( P , Θ) , 0 ≤ Θ ≤ 1 , In fact, σ D 2 ( c , P ) in (42) can be v iewed as the r esult of the co ncaviﬁcation o f g ( c , P ) [19]. Hence, usin g Theore m 2.16 in [19], we c onclude that q ′ ≤ M + 1 . Remarkable p oint abou t T heorem 3 is that th e up per bou nd o n q ′ is ind ependen t of th e num ber of in equalities inv olved in the de scription of the ach iev able rate r egion. Cor o llary 3: For th e M -user Ga ussian IC wher e u sers use Gau ssian codeboo ks and treat the inter ference as noise, we h av e D 2 = D and q = q ′ = M + 1 . 8 E. Extremal Ine q uality In [14], th e following o ptimization pro blem is studied: W = ma x Q X ≤ S h ( X + Z 1 ) − µh ( X + Z 2 ) , (44) where Z 1 and Z 2 are n -dimension al Gau ssian random vectors with the strictly positive deﬁnite covariance matr ices Q Z 1 and Q Z 2 , r espectiv ely . The o ptimization is over all ran dom vectors X indep endent of Z 1 and Z 2 . X is also sub ject to the covariance matrix constraint Q X ≤ S , where S is a p ositiv e deﬁnite matrix. In [14], it is shown that for all µ ≥ 1 , this optimization prob lem has a Gaussian optimal solution for all positive d eﬁnite matrices Q Z 1 and Q Z 2 . Howe ver , for 0 ≤ µ < 1 this optim ization prob lem has a Gau ssian o ptimal solu tion provide d Q Z 1 ≤ Q Z 2 , i.e., Q Z 2 − Q Z 1 is a positive semi-d eﬁnite matrix. It is worth notin g that for µ = 1 this pro blem wh en Q Z 1 ≤ Q Z 2 is studied und er the n ame of the worse add iti ve no ise [21], [22]. In this pap er , we co nsider a special case of (44) whe re Z 1 and Z 2 have the covariance matrices N 1 I and N 2 I , re spectiv ely , and the trac e constra int is consider ed, i.e., W = max tr { Q X }≤ nP h ( X + Z 1 ) − µh ( X + Z 2 ) . (45) In the following lemm a, we provide the op timal solution fo r the above op timization prob lem when N 1 ≤ N 2 . Lemma 1: If N 1 ≤ N 2 , the o ptimal solution o f (45) is iid Gau ssian for all 0 ≤ µ and we have 1) F or 0 ≤ µ ≤ N 2 + P N 1 + P , the o ptimum covariance matrix is P I and th e o ptimum solution is W = n 2 log [(2 πe )( P + N 1 )] − µn 2 log [(2 πe )( P + N 2 )] . (46) 2) F or N 2 + P N 1 + P < µ ≤ N 2 N 1 , the o ptimum covariance matrix is N 2 − µN 1 µ − 1 I and th e o ptimum solution is W = n 2 log  (2 π e ) N 2 − N 1 µ − 1  − µn 2 log  µ (2 π e )( N 2 − N 1 ) µ − 1  . (47) 3) F or N 2 N 1 < µ , the o ptimum covariance matrix is 0 and the o ptimum solution is W = n 2 log(2 π eN 1 ) − µn 2 log(2 π eN 2 ) . (48) Pr o of: From the gener al r esult f or (44), we know that the op timum inpu t distribution is Gau ssian. Hence , we n eed to solve th e following max imization problem : W =max 1 2 log ((2 π e ) n | Q X + N 1 I | ) − µ 2 log ((2 π e ) n | Q X + N 2 I | ) (49) subject to: 0 ≤ Q X , tr { Q X } ≤ nP . Since Q X is a positi ve semi- deﬁnite matrix, it c an be decompo sed as Q X = U Λ U t , where Λ is a diagon al matrix with nonnegative e ntries an d U is a u nitary ma trix, i. e., U U t = I . Substituting Q X = U Λ U t in (49) an d u sing the identities tr { AB } = tr { B A } and | AB + I | = | B A + I | , we o btain W =max 1 2 log ((2 πe ) n | Λ + N 1 I | ) − µ 2 log ((2 π e ) n | Λ + N 2 I | ) (50) subject to: 0 ≤ Λ , tr { Λ } ≤ nP. This op timization pr oblem can b e simpliﬁed as W =max n 2 n X i =1 [log(2 π e )( λ i + N 1 ) − µ lo g(2 π e )( λ i + N 2 )] (51) subject to: 0 ≤ λ i ∀ i, n X i =1 λ i ≤ nP. By intro ducing L agrange multipliers ψ and Φ = { φ 1 , φ 2 , . . . , φ n } , we o btain L (Λ , ψ, Φ) = max n 2 n X i =1 [log(2 π e )( λ i + N 1 ) − µ log(2 π e )( λ i + N 2 )] + ψ nP − n X i =1 λ i ! + n X i =1 φ i λ i . (52) 9 N 2 − µN 1 µ − 1 P + N 2 P + N 1 N 2 N 1 1 P V ariance µ Fig. 2. Optimum v aria nce versus µ . The ﬁrst o rder KKT nec essary conditio ns for the optimu m solutio n of (5 2) can be wr itten as 1 λ i + N 1 − µ λ i + N 2 − ψ + φ i =0 , ∀ i ∈ { 1 , 2 , . . . , n } , (53 ) ψ nP − n X i =1 λ i ! =0 , (54) φ i λ i =0 , ∀ i ∈ { 1 , 2 , . . . , n } . (55 ) It is ea sy to show that whe n N 1 ≤ N 2 , λ = λ 1 = . . . = λ n and the only solution for λ is λ =      P, if 0 ≤ µ ≤ N 2 + P N 1 + P N 2 − µN 1 µ − 1 , if N 2 + P N 1 + P < µ ≤ N 2 N 1 0 , if N 2 N 1 < µ (56) Substituting λ into the objective function gives the desire d result. In Figure 2, the optimum variance as a fu nction of µ is plotted. This ﬁgur e shows that for any value of µ ≤ P + N 2 P + N 1 , we need to use the maxim um p ower to op timize the o bjective function , whereas fo r µ > P + N 2 P + N 1 , we use less power than what is permissible. Lemma 2: If N 1 > N 2 , the optimal solution of (45) is iid Gaussian for all 1 ≤ µ . In this case, the op timum variance is 0 and the op timum W is W = n 2 log(2 π eN 1 ) − µn 2 log(2 π eN 2 ) . (57) Pr o of: The p roof is similar to th at of Le mma 1 and we om it it here. Cor o llary 4: For µ = 1 , the op timal solution o f (45) is iid Gaussian and the op timum W is W =    n 2 log  P + N 1 P + N 2  , if N 1 ≤ N 2 n 2 log  N 1 N 2  , if N 1 > N 2 . (58) W e freq uently app ly the following optimization pr oblem in the r est of the p aper: f h ( P, N 1 , N 2 , a, µ ) = max tr { Q X }≤ nP h ( X + Z 1 ) − µh ( √ a X + Z 2 ) , (59) where N 1 ≤ N 2 /a . Using the id entity h ( A X ) = log( | A | ) + h ( X ) , (59) can be written as f h ( P, N 1 , N 2 , a, µ ) = n 2 log a + max tr { Q X }≤ nP h ( X + Z 1 ) − µh ( X + Z 2 √ a ) . (60) Now , Lemma 1 can be ap plied to ob tain f h ( P, N 1 , N 2 , a, µ ) =      1 2 log [(2 πe )( P + N 1 )] − µ 2 log [(2 πe )( aP + N 2 )] if 0 ≤ µ ≤ P + N 2 /a P + N 1 1 2 log h (2 π e ) N 2 /a − N 1 µ − 1 i − µ 2 log h aµ (2 πe )( N 2 /a − N 1 ) µ − 1 i if P + N 2 /a P + N 1 < µ ≤ N 2 aN 1 1 2 log(2 π eN 1 ) − µ 2 log(2 π eN 2 ) if N 2 aN 1 < µ (61) 10 ˆ y 1 ˆ y 2 ˜ y 1 ˜ y 2 f 1 f 2 ω ( ˜ y 1 , ˜ y 2 | x 1 , x 2 ) x 1 x 2 Fig. 3. An admissible channe l. f 1 and f 2 are determi nistic functions. I I I . A D M I S S I B L E C H A N N E L S In this section, we aim at building I Cs whose capa city regions contain the c apacity region o f the two-user Gaussian IC, i.e., C . Since we ultima tely use th ese to ou ter boun d C , these ICs need to h av e a tractab le expression (or a tractable ou ter boun d) for their ca pacity regions. Let us con sider an IC with th e same inp ut letters as that of C and the outp ut letters ˜ y 1 and ˜ y 2 for Users 1 and 2, respectively . The capacity region o f this cha nnel, say C ′ , con tains C if I ( x n 1 ; y n 1 ) ≤ I ( x n 1 ; ˜ y n 1 ) , (62) I ( x n 2 ; y n 2 ) ≤ I ( x n 2 ; ˜ y n 2 ) , (63) for all p ( x n 1 ) p ( x n 2 ) and f or all n ∈ N . One way to satisfy (62) and (63) is to provide some extra infor mation to either on e o r to b oth receivers. T his techniq ue is kn own as Genie a ided oute r boun ding . In [ 12], Kram er has used such a genie to provid e some extra info rmation to b oth receivers such that they can decode both users’ messages. Since the capacity region of this new interferen ce chann el is equiv alent to th at of the Compoun d Multiple Access Chan nel who se capac ity region is k nown, referen ce [12] obtain s an ou ter bou nd on th e capac ity region. T o o btain a tighter outer bo und, refer ence [1 2] fur ther uses the fact that if a genie provid es the exact informa tion about the in terfering sign al to on e o f the receivers, th en the n ew channel becomes th e o ne-sided Gaussian IC. Although the cap acity r egion o f the one-side d Gau ssian I C is unk nown for all r anges o f parameter s, ther e exists an outer bound for it d ue to Sato and Costa [23], [11] that can be app lied to the orig inal chan nel. I n [13], Etkin et al. use a d ifferent genie that pr ovides some extra info rmation ab out the inten ded sign al. Even thou gh at ﬁr st glance the ir propo sed meth od a ppears to be far fr om ach ieving a tig ht b ound , remarkab ly they show th at th e co rrespon ding bound is tig hter th an the o ne due to Krame r for certain ranges of parameter s. Next, we intro duce the notion of a dmissible cha nnels to satisfy ( 62) and (6 3). Deﬁnition 3 (Admissible Chan nel): An IC C ′ with inp ut letter x i and outp ut letter ˜ y i for User i ∈ { 1 , 2 } is an admissible channel if th ere exist two deterministic fu nctions ˆ y n 1 = f 1 ( ˜ y n 1 ) and ˆ y n 2 = f 2 ( ˜ y n 2 ) such that I ( x n 1 ; y n 1 ) ≤ I ( x n 1 ; ˆ y n 1 ) , (64) I ( x n 2 ; y n 2 ) ≤ I ( x n 2 ; ˆ y n 2 ) (65) hold fo r all p ( x n 1 ) p ( x n 2 ) an d for all n ∈ N . E denotes the collection of all adm issible ch annels ( see Figure 3). Remark 2: Genie aid ed chann els are among adm issible chann els. T o see this, let u s assume a genie provides s 1 and s 2 as side in formation f or User 1 an d 2 , r espectiv ely . In this case, ˜ y i = ( y i , s i ) for i ∈ { 1 , 2 } . By choo sing f i ( y i , s i ) = y i , we observe th at ˆ y i = y i , and hence, (64) and (65) tr i vially hold . T o ob tain the tightest outer boun d, we need to ﬁnd the intersection of the cap acity r egions of all ad missible channels. Nonetheless, it may hap pen that ﬁnding the ca pacity r egion of an a dmissible ch annel is as hard a s that o f the orig inal one (in fact, based on the deﬁnition , the chann el itself is on e of its admissible chann els). Henc e, we need to ﬁnd classes of adm issible channels, say F , wh ich possess two impo rtant prop erties. First, their capacity regions ar e clo se to C . Second, either their exact ca pacity regions are comp utable or there exist good outer bo unds for them . Since F ⊆ E , we have C ⊆ \ F C ′ . (66) Recall that ther e is a o ne to o ne corresp ondenc e between a closed c on vex set an d its suppo rt functio n. Since C is closed an d conv ex, the re is a on e to one corr esponden ce be tween C and σ C . In fact, bo undary po ints o f C correspo nd to th e so lutions of the following op timization pr oblem σ C ( c 1 , c 2 ) = max ( R 1 ,R 2 ) ∈ C c 1 R 1 + c 2 R 2 . (67) Since we ar e interested in th e bou ndary p oints excluding the R 1 and R 2 axes, it sufﬁces to con sider 0 ≤ c 1 and 0 ≤ c 2 where c 1 + c 2 = 1 . 11 Admissible Channel f 2 ( ˜ y 22 , ˜ y 21 ) = (1 − √ g 2 ) ˜ y 22 + √ g 2 ˜ y 21 ˆ y 1 ˆ y 2 f 1 ( ˜ y 1 ) = ˜ y 1 ˜ y 1 x 2 z 21 z 22 √ a x 1 z 1 √ g 2 ˜ y 21 ˜ y 22 1 − √ g 2 √ b ′ Fig. 4. Class A1 admissible channe ls. Since C ⊆ C ′ , we have σ C ( c 1 , c 2 ) ≤ σ C ′ ( c 1 , c 2 ) . (68) T ak ing the minim um o f the r ight hand side, we o btain σ C ( c 1 , c 2 ) ≤ min C ′ ∈ F σ C ′ ( c 1 , c 2 ) , (69) which can b e written as σ C ( c 1 , c 2 ) ≤ min C ′ ∈ F max ( R 1 ,R 2 ) ∈ C ′ c 1 R 1 + c 2 R 2 . (70) For co n venience, we use the following two optimization pr oblems σ C ( µ, 1) = max ( R 1 ,R 2 ) ∈ C µR 1 + R 2 , ( 71) σ C (1 , µ ) = max ( R 1 ,R 2 ) ∈ C R 1 + µR 2 , ( 72) where 1 ≤ µ . I t is easy to show that the solutions o f (71) and ( 72) cor respond to the b oundar y points of the capacity r egion. In the rest of this section, we introduce c lasses of ad missible channels an d ob tain upper bo unds o n σ C ′ ( µ, 1) an d σ C ′ (1 , µ ) . A. Classes of Admissible Channe ls 1) Class A1: This class is de signed to obtain an up per bo und on σ C ( µ, 1) . Th erefore, we need to ﬁnd a tight upp er bound on σ C ′ ( µ, 1) . A membe r of this class is a channel in which User 1 has on e tr ansmit an d on e receive anten na wh ereas User 2 has on e transmit antenn a and two receive antennas (see Figure 4). T he chann el m odel can b e w ritten as ˜ y 1 = x 1 + √ ax 2 + z 1 , ˜ y 21 = x 2 + √ b ′ x 1 + z 21 , ˜ y 22 = x 2 + z 22 , (73) where ˜ y 1 is the signal at the ﬁrst r eceiv er , ˜ y 21 and ˜ y 22 are the signals at th e seco nd r eceiv er , z 1 is additive Gau ssian n oise with unit variance, z 21 and z 22 are ad ditiv e Gaussian noise with variances N 21 and N 22 , r espectively . Transmitters 1 and 2 are subject to the power constrain ts of P 1 and P 2 , respec ti vely . T o inves tigate admissibility conditio ns in (6 4) and (65), we intro duce two d eterministic fun ctions f 1 and f 2 as f ollows (see Figure 4) f 1 ( ˜ y n 1 )= ˜ y n 1 , (74) f 2 ( ˜ y n 22 , ˜ y n 21 )= (1 − √ g 2 ) ˜ y n 22 + √ g 2 ˜ y n 21 , (75) where 0 ≤ g 2 . For g 2 = 0 , the chann el can be co n verted to the one -sided Gau ssian I C by letting N 21 → ∞ an d N 22 = 1 . Hence, Class A1 contains the on e-sided Gaussian IC o btained by r emoving the link b etween Transmitter 1 and Receiver 2. Using f 1 and f 2 , we obtain ˆ y n 1 = x n 1 + √ ax n 2 + z n 1 , (76) ˆ y n 2 = p b ′ g 2 x n 1 + x n 2 + (1 − √ g 2 ) z n 22 + √ g 2 z n 21 . (77) 12 Hence, this channel is admissible if the corre sponding par ameters satisfy b ′ g 2 = b, (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 . (78) W e furth er ad d the following constraints to the con ditions o f the ch annels in Class A1: b ′ ≤ N 21 , aN 22 ≤ 1 . (79) Although the se additio nal c onditions r educe the number of a dmissible ch annels within the class, they ar e need ed to g et a closed form formu la for an upp er bo und on σ C ′ ( µ, 1) . In the f ollowing lemma, we obtain the requ ired upp er boun d. Lemma 3: For th e chan nels mod eled b y (73) a nd satisfying ( 79), we have σ C ′ ( µ, 1) ≤ min µ 1 2 log [2 π e ( P 1 + aP 2 + 1)] − µ 2 2 log(2 π e ) + 1 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  (80) + µ 2 f h  P 1 , 1 , N 21 , b ′ , 1 µ 2  + f h ( P 2 , N 22 , 1 , a, µ 1 ) subject to: µ 1 + µ 2 = µ, µ 1 , µ 2 ≥ 0 . Pr o of: Let us assum e R 1 and R 2 are achiev able r ates f or User 1 and 2, respectively . Furtherm ore, we split µ in to µ 1 ≥ 0 and µ 2 ≥ 0 such that µ = µ 1 + µ 2 . Using Fano’ s inequ ality , we o btain n ( µR 1 + R 2 ) ≤ µI ( x n 1 ; ˜ y n 1 ) + I ( x n 2 ; ˜ y n 22 , ˜ y n 21 ) + nǫ n = µ 1 I ( x n 1 ; ˜ y n 1 ) + µ 2 I ( x n 1 ; ˜ y n 1 ) + I ( x n 2 ; ˜ y n 22 , ˜ y n 21 ) + nǫ n ( a ) ≤ µ 1 I ( x n 1 ; ˜ y n 1 ) + µ 2 I ( x n 1 ; ˜ y n 1 | x n 2 ) + I ( x n 2 ; ˜ y n 22 , ˜ y n 21 ) + nǫ n = µ 1 I ( x n 1 ; ˜ y n 1 ) + µ 2 I ( x n 1 ; ˜ y n 1 | x n 2 ) + I ( x n 2 ; ˜ y n 21 | ˜ y n 22 ) + I ( x n 2 ; ˜ y n 22 ) + nǫ n = µ 1 h ( ˜ y n 1 ) − µ 1 h ( ˜ y n 1 | x n 1 ) + µ 2 h ( ˜ y n 1 | x n 2 ) − µ 2 h ( ˜ y n 1 | x n 1 , x n 2 ) + h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 ) + h ( ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 ) + nǫ n =  µ 1 h ( ˜ y n 1 ) − µ 2 h ( ˜ y n 1 | x n 1 , x n 2 )  +  µ 2 h ( ˜ y n 1 | x n 2 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 )  +  h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 )  +  h ( ˜ y n 22 ) − µ 1 h ( ˜ y n 1 | x n 1 )  + nǫ n , (81) where (a) f ollows from the fact that x n 1 and x n 2 are indepe ndent. Now , we separ ately u pper boun d the terms within each bracket in (8 1). T o maximize the terms within the ﬁrst bra cket, we u se the fact that Gaussian distribution m aximizes the differential entropy subject to a con straint on the covariance matrix . Hence, we have µ 1 h ( ˜ y n 1 ) − µ 2 h ( ˜ y n 1 | x n 1 , x n 2 )= µ 1 h ( x n 1 + √ ax n 2 + z n 1 ) − µ 2 h ( z n 1 ) ≤ µ 1 n 2 log [2 π e ( P 1 + aP 2 + 1)] − µ 2 n 2 log(2 π e ) . (82) Since b ′ ≤ N 21 , we can make use of Lemm a 1 to u pper boun d the seco nd bra cket. In this case, we have µ 2 h ( ˜ y n 1 | x n 2 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 )= µ 2  h ( x n 1 + z n 1 ) − 1 µ 2 h ( √ b ′ x n 1 + z n 21 )  ≤ µ 2 nf h  P 1 , 1 , N 21 , b ′ , 1 µ 2  , (83) where f h is deﬁned in (61). W e upp er bo und the terms within the th ird bracket as f ollows [13]: h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 ) ( a ) ≤ n X i =1 h ( ˜ y 21 [ i ] | ˜ y 22 [ i ]) − h ( z n 22 ) ( b ) ≤ n X i =1 1 2 log  2 π e  N 21 + b ′ P 1 [ i ] + P 2 [ i ] N 22 P 2 [ i ] + N 22  − n 2 log (2 π eN 22 ) ( c ) ≤ n 2 log " 2 π e N 21 + 1 n n X i =1 b ′ P 1 [ i ] + 1 n P n i =1 P 2 [ i ] N 22 1 n P n i =1 P 2 [ i ] + N 22 !# − n 2 log (2 πe N 22 ) ≤ n 2 log  2 π e  N 21 + b ′ P 1 + P 2 N 22 P 2 + N 22  − n 2 log (2 πe N 22 ) ≤ n 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  , (84) 13 Admissible Channel f 1 ( ˜ y 11 , ˜ y 12 ) = (1 − √ g 1 ) ˜ y 11 + √ g 1 ˜ y 12 √ a ′ x 2 x 1 ˜ y 2 ˜ y 12 ˜ y 11 f 2 ( ˜ y 2 ) = ˜ y 2 ˆ y 1 ˆ y 2 √ g 1 1 − √ g 1 z 11 z 12 z 2 √ b Fig. 5. Class A2 admissible channe ls. where ( a) fo llows fro m the chain rule an d the fact tha t rem oving independ ent co nditions do es n ot decr ease differential entro py , (b) follows fro m the fact that Gaussian d istribution m aximizes the con ditional entropy fo r a given covariance matrix , and (c) follows for m Jenson ’ s in equality . For th e last br acket, we again make use o f the d eﬁnition of f h . In fact, since a N 22 ≤ 1 , w e have h ( ˜ y n 22 ) − µ 1 h ( ˜ y n 1 | x n 1 )= h ( x n 2 + z n 22 ) − µ 1 h ( √ ax n 2 + z n 1 ) ≤ nf h ( P 2 , N 22 , 1 , a, µ 1 ) . (85) Adding all in equalities, we o btain µR 1 + R 2 ≤ µ 1 2 log [2 π e ( P 1 + aP 2 + 1)] − µ 2 2 log(2 π e ) + 1 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  + µ 2 f h  P 1 , 1 , N 21 , b ′ , 1 µ 2  + f h ( P 2 , N 22 , 1 , a, µ 1 ) , (86) where the fact th at ǫ n → 0 as n → ∞ is used to eliminate ǫ n form the r ight h and side of th e in equality . Now , b y tak ing the minimum of th e righ t h and side of ( 86) over all µ 1 and µ 2 , we obtain the desired result. T his comp letes the proo f. 2) Class A2: This class is the comp lement of Class A1 in th e sense that we use it to u pper bo und σ C (1 , µ ) . A memb er of this c lass is a chann el in which U ser 1 is equipp ed with o ne tra nsmit and two rece i ve an tennas, wh ereas User 2 is equipp ed with on e antenna at b oth transmitter and receiver sides (see Fig ure 5). T he chann el mod el c an be wr itten as ˜ y 11 = x 1 + z 11 , ˜ y 12 = x 1 + √ a ′ x 2 + z 12 , ˜ y 2 = x 2 + √ bx 1 + z 2 , (87) where ˜ y 11 and ˜ y 12 are the signals at the ﬁrst rec ei ver , ˜ y 2 is the signal at the secon d receiver , z 2 is additive Gau ssian n oise with un it variance, z 11 and z 12 are ad ditiv e Gau ssian no ise with variances N 11 and N 12 , respe cti vely . Transmitter 1 and 2 ar e subject to the power constraints P 1 and P 2 , respec ti vely . For th is class, we consider two linear function s f 1 and f 2 as f ollows (see Figure 5 ): f 1 ( ˜ y n 11 , ˜ y n 12 )= (1 − √ g 1 ) ˜ y n 11 + √ g 1 ˜ y n 12 , (88) f 2 ( ˜ y n 2 )= ˜ y n 2 . (89) Similar to Class A1 , wh en g 1 = 0 , the ad missible cha nnels in Class A2 becom e the one-sid ed Gaussian IC by letting N 12 → ∞ and N 11 = 1 . Ther efore, we have ˆ y n 1 = x n 1 + p a ′ g 1 x n 2 + (1 − √ g 1 ) z n 11 + √ g 1 z n 12 , (90) ˆ y n 2 = √ bx n 1 + x n 2 + z n 2 . (91) W e conc lude that the ch annel modeled by ( 87) is ad missible if the correspo nding param eters satisfy a ′ g 1 = a, (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 . (92) 14 Admissible Channel ˜ y 12 x 2 x 1 z 11 z 12 z 21 z 22 ˜ y 22 ˜ y 21 ˜ y 11 √ g 1 1 − √ g 1 ˆ y 1 ˆ y 2 1 − √ g 2 √ g 2 √ b ′ √ a ′ f 1 ( ˜ y 11 , ˜ y 12 ) = (1 − √ g 1 ) ˜ y 11 + √ g 1 ˜ y 12 f 2 ( ˜ y 22 , ˜ y 21 ) = (1 − √ g 2 ) ˜ y 22 + √ g 2 ˜ y 21 Fig. 6. Class B admissible chann els. Similar to Class A1, we furth er add the f ollowing constrain ts to th e condition s o f Class A2 ch annels: a ′ ≤ N 12 , bN 11 ≤ 1 . (93) In the following lemm a, we obtain th e r equired upp er b ound. Lemma 4: For th e chan nels mod eled b y (87) a nd satisfying ( 93), we have σ C ′ (1 , µ ) ≤ min µ 1 2 log [2 π e ( bP 1 + P 2 + 1)] − µ 2 2 log(2 π e ) + 1 2 log  N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11  (94) + µ 2 f h  P 2 , 1 , N 12 , a ′ , 1 µ 2  + f h ( P 1 , N 11 , 1 , b, µ 1 ) subject to: µ 1 + µ 2 = µ, µ 1 , µ 2 ≥ 0 . Pr o of: The p roof is similar to th at of Le mma 3 and we om it it here. 3) Class B: A member of this class is a chan nel with one transm it an tenna an d two receive antennas for eac h user m odeled by (see Figure 6) ˜ y 11 = x 1 + z 11 , ˜ y 12 = x 1 + √ a ′ x 2 + z 12 , ˜ y 21 = x 2 + √ b ′ x 1 + z 21 , ˜ y 22 = x 2 + z 22 , (95) where ˜ y 11 and ˜ y 12 are the sign als at the ﬁr st receiver , ˜ y 21 and ˜ y 22 are the sign als at the second receiver , a nd z ij is add itiv e Gaussian noise with variance N ij for i, j ∈ { 1 , 2 } . Transmitter 1 and 2 are sub ject to th e power constraints P 1 and P 2 , respectively . In fact, this chann el is designed to u pper bo und bo th σ C ( µ, 1) and σ C (1 , µ ) . Next, we in vestigate admissibility of th is channel an d the con ditions th at must be impo sed on the und erlying param eters. Let us con sider two linear deter ministic fu nctions f 1 and f 2 with parameter s 0 ≤ g 1 and 0 ≤ g 2 , respectively , as follows (see Figure 6) f 1 ( ˜ y n 11 , ˜ y n 12 )= (1 − √ g 1 ) ˜ y n 11 + √ g 1 ˜ y n 12 , (96) f 2 ( ˜ y n 22 , ˜ y n 21 )= (1 − √ g 2 ) ˜ y n 22 + √ g 2 ˜ y n 21 . (97) Therefo re, we h av e ˆ y n 1 = x n 1 + p a ′ g 1 x n 2 + (1 − √ g 1 ) z n 11 + √ g 1 z n 12 , (98) ˆ y n 2 = p b ′ g 2 x n 1 + x n 2 + (1 − √ g 2 ) z n 22 + √ g 2 z n 21 . (99) T o satisfy (64) and (6 5), it sufﬁces to have a ′ g 1 = a, b ′ g 2 = b, (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 , (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 . (100) 15 Hence, a chan nel modele d by (95) is adm issible if there exist two nonn egati ve n umbers g 1 and g 2 such that the equalities in (100) are satisﬁed. W e fu rther add the f ollowing two con straints to the eq uality c onditions in ( 100): b ′ N 11 ≤ N 21 , a ′ N 22 ≤ N 12 . (101) Although adding mo re constrain ts red uces the num ber o f the adm issible channels, it enab les us to co mpute an ou ter boun d on σ C ′ ( µ, 1) and σ C ′ (1 , µ ) . Lemma 5: For th e chan nels mod eled b y (95) a nd satisfying ( 101), we have σ C ′ ( µ, 1) ≤ µγ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  + f h ( P 2 , N 22 , N 12 , a ′ , µ ) + µ 2 log((2 π e )( a ′ P 2 + N 12 )) − 1 2 log((2 π e )( P 2 + N 22 )) , (102) σ C ′ (1 , µ ) ≤ γ  P 1 N 11 + P 1 a ′ P 2 + N 12  + µγ  P 2 N 22 + P 2 b ′ P 1 + N 21  + f h ( P 1 , N 11 , N 21 , b ′ , µ ) + µ 2 log((2 π e )( b ′ P 1 + N 21 )) − 1 2 log((2 π e )( P 1 + N 11 )) . (103) Pr o of: W e only upp er bo und σ C ′ ( µ, 1) and an u pper bo und on σ C ′ (1 , µ ) can b e similarly o btained. Let u s assume R 1 and R 2 are ach iev able rates for User 1 and User 2, respectively . Using Fano’ s in equality , we obtain n ( µR 1 + R 2 ) ≤ µI ( x n 1 ; ˜ y n 11 , ˜ y n 12 ) + I ( x n 2 ; ˜ y n 22 , ˜ y n 21 ) + nǫ n = µI ( x n 1 ; ˜ y n 12 | ˜ y n 11 ) + µI ( x n 1 ; ˜ y n 11 ) + I ( x n 2 ; ˜ y n 21 | ˜ y n 22 , ) + I ( x n 2 ; ˜ y n 22 ) + nǫ n = µh ( ˜ y n 12 | ˜ y n 11 ) − µh ( ˜ y n 12 | x n 1 , ˜ y n 11 ) + µh ( ˜ y n 11 ) − µh ( ˜ y n 11 | x n 1 ) + h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 ) + h ( ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 ) + nǫ n =  µh ( ˜ y n 12 | ˜ y n 11 ) − µh ( ˜ y n 11 | x n 1 )  +  h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 )  +  µh ( ˜ y n 11 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 )  +  h ( ˜ y n 22 ) − µh ( ˜ y n 12 | x n 1 , ˜ y n 11 )  + nǫ n . (104) Next, we upp er bo und the terms within each bracket in (10 4) sep arately . For the ﬁrst bracket, we h av e µh ( ˜ y n 12 | ˜ y n 11 ) − µh ( ˜ y n 11 | x n 1 ) ( a ) ≤ µ n X i =1 h ( ˜ y 12 [ i ] | ˜ y 11 [ i ]) − µn 2 log (2 πe N 11 ) ( b ) ≤ µ n X i =1 1 2 log  2 π e  N 12 + a ′ P 2 [ i ] + P 1 [ i ] N 11 P 1 [ i ] + N 11  − µn 2 log (2 π eN 11 ) ( c ) ≤ µn 2 log " 2 π e N 12 + 1 n n X i =1 a ′ P 2 [ i ] + 1 n P n i =1 P 1 [ i ] N 11 1 n P n i =1 P 1 [ i ] + N 11 !# − µn 2 log (2 πe N 11 ) ≤ µn 2 log  2 π e  N 12 + a ′ P 2 + P 1 N 11 P 1 + N 11  − µn 2 log (2 πe N 11 ) = µn 2 log  N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11  , (105) where (a) follows fro m th e ch ain r ule and the fact th at removing in depende nt c onditions increases differential entropy , (b) follows fro m the fact th at Ga ussian distribution op timizes co nditional en tropy f or a given covariance matr ix, and (c) follows form Jenson’ s in equality . Similarly , th e terms within the second bracket ca n be u pper boun ded as h ( ˜ y n 21 | ˜ y n 22 ) − h ( ˜ y n 22 | x n 2 ) ≤ n 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  . (106) Using Lemma 1 and the fact that N 11 ≤ N 21 /b ′ , the ter ms with in the th ird bracket can b e upper bo unded as µh ( ˜ y n 11 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 )= µ  h ( x n 1 + z n 11 ) − 1 µ h ( √ b ′ x n 1 + z n 21 )  ≤ µnf h  P 1 , N 11 , N 21 , b ′ , 1 µ  . (107) Since 1 ≤ µ , from (61) we obtain µh ( ˜ y n 11 ) − h ( ˜ y n 21 | x n 2 , ˜ y n 22 ) ≤ µn 2 log((2 π e )( P 1 + N 11 )) − n 2 log((2 π e )( b ′ P 1 + N 21 )) . (108) 16 For th e last br acket, again we use Lem ma 1 to obtain h ( ˜ y n 22 ) − µh ( ˜ y n 12 | x n 1 , ˜ y n 11 )= h ( x n 2 + z n 22 ) − µh ( √ a ′ x n 2 + z n 12 ) ≤ nf h ( P 2 , N 22 , N 12 , a ′ , µ ) . (109) Adding all in equalities, we h av e µR 1 + R 2 ≤ µ 2 log  N 12 N 11 + a ′ P 2 N 11 + P 1 P 1 + N 11  + 1 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  + µ 2 log((2 π e )( P 1 + N 11 )) − 1 2 log((2 π e )( b ′ P 1 + N 21 )) + f h ( P 2 , N 22 , N 12 , a ′ , µ ) , (110) where the fact th at ǫ n → 0 as n → ∞ is used to e liminate ǫ n from the r ight h and side o f the in equality . By rearra nging the terms, we o btain µR 1 + R 2 ≤ µγ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  + f h ( P 2 , N 22 , N 12 , a ′ , µ ) + µ 2 log((2 π e )( a ′ P 2 + N 12 )) − 1 2 log((2 π e )( P 2 + N 22 )) . This com pletes the pro of. A uniq ue feature of the channels within Class B is that for 1 ≤ µ ≤ P 2 + N 12 /a ′ P 2 + N 22 and 1 ≤ µ ≤ P 1 + N 21 /b ′ P 1 + N 11 , the up per boun ds in (1 02) and (103) bec ome, respectively , µR 1 + R 2 ≤ µγ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  (111) and R 1 + µR 2 ≤ γ  P 1 N 11 + P 1 a ′ P 2 + N 12  + µγ  P 2 N 22 + P 2 b ′ P 1 + N 21  . (112) On the o ther han d, if the receivers trea t th e interfe rence as n oise, it can be shown that R 1 = γ  P 1 N 11 + P 1 a ′ P 2 + N 12  (113) and R 2 = γ  P 2 N 22 + P 2 b ′ P 1 + N 21  (114) are achievable. Comparin g upper b ounds and achievable r ates, we con clude th at the u pper bound s are in deed tight. In fact, this pro perty is ﬁrst o bserved b y Etkin et al. in [13]. W e summar ize this result in the fo llowing theorem : Theor em 4 : Th e sum capac ity in Class B is attained when transmitters use Gaussian codebo oks and receivers treat the interferen ce as n oise. In this case, the sum cap acity is C ′ sum = γ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  . (115) Pr o of: By substituting µ = 1 in (1 11), we obtain th e d esired result. 4) Class C: Class C is design ed to up per bound σ C ( µ, 1) for th e mixed Gaussian IC where 1 ≤ b . Class C is similar to Class A1 (see Figu re 4) , h owe ver we imp ose different constra ints on the parameter s of the chan nels with in Class C. These constraints assist us in p roviding up per bound s by u sing the fact tha t at o ne of the re ceiv ers b oth sign als are decod able. For channels in Class C, we use th e same mod el that is given in ( 73). T herefor e, similar to channels in Class A1 , this channel is admissible if the corre sponding par ameters satisfy b ′ g 2 = b, (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 . (116) Next, we chan ge the constra ints in (79) as b ′ ≥ N 21 , aN 22 ≤ 1 . (117) Throu gh this chang e of constraints, th e seco nd receiver a fter deco ding its own sign al will have a less noisy version of the ﬁrst u ser’ s sig nal, and co nsequently , it is able to decod e the sign al o f the ﬁrst u ser as well as its own sign al. Relying on this observation, we h av e the following lem ma. 17 Lemma 6: For a chann el in Class C, we have σ C ′ ( µ, 1) ≤ µ − 1 2 log (2 πe ( P 1 + aP 2 + 1)) + 1 2 log  2 π e  P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21  − 1 2 log(2 π eN 21 ) − 1 2 log(2 π eN 22 ) + f h ( P 2 , N 22 , 1 , a, µ − 1) . (118) Pr o of: Since the second user is able to decode bo th users’ messages, we have R 1 ≤ 1 n I ( x n 1 ; ˜ y n 1 ) , (119) R 1 ≤ 1 n I ( x n 1 ; ˜ y n 21 , ˜ y n 22 | x n 2 ) , (120) R 2 ≤ 1 n I ( x n 2 ; ˜ y n 21 , ˜ y n 22 | x n 1 ) , (121) R 1 + R 2 ≤ 1 n I ( x n 1 , x n 2 ; ˜ y n 21 , ˜ y n 22 ) . (122) From aN 22 ≤ 1 , we have I ( x n 1 ; ˜ y n 1 ) ≤ I ( x n 1 ; ˜ y n 21 | x n 2 ) = I ( x n 1 ; ˜ y n 21 , ˜ y n 22 | x n 2 ) . He nce, ( 120) is redund ant. It can be shown that µR 1 + R 2 ≤ µ − 1 n I ( x n 1 ; ˜ y n 1 ) + 1 n I ( x n 1 , x n 2 ; ˜ y n 21 , ˜ y n 22 ) . (123) Hence, we have µR 1 + R 2 ≤ µ − 1 n h ( ˜ y n 1 ) − µ − 1 n h ( ˜ y n 1 | x n 1 ) + 1 n h ( ˜ y n 21 , ˜ y n 22 ) − 1 n h ( ˜ y n 21 , ˜ y n 22 | x n 1 , x n 2 ) = µ − 1 n h ( ˜ y n 1 ) + 1 n h ( ˜ y n 21 | ˜ y n 22 ) − 1 n h ( ˜ y n 21 , ˜ y n 22 | x n 1 , x n 2 ) +  1 n h ( ˜ y n 22 ) − µ − 1 n h ( ˜ y n 1 | x n 1 )  (124) Next, we bou nd the different term s in (12 4). For th e ﬁrst term, we have µ − 1 n h ( ˜ y n 1 ) ≤ µ − 1 2 log (2 π e ( P 1 + aP 2 + 1)) . (125) The second term can b e bou nded as 1 n h ( ˜ y n 21 | ˜ y n 22 ) ≤ 1 2 log  2 π e  P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21  . (126) The third ter m can be boun ded as 1 n h ( ˜ y n 21 , ˜ y n 22 | x n 1 , x n 2 ) = 1 2 log(2 π eN 21 ) + 1 2 log(2 π eN 22 ) . (127) The last terms can b e bou nded as 1 n h ( ˜ y n 22 ) − µ − 1 n h ( ˜ y n 1 | x n 1 )= 1 n h ( x n 2 + z n 22 ) − µ − 1 n h ( √ ax n 2 + z 1 ) (128) ≤ f h ( P 2 , N 22 , 1 , a, µ − 1) . (129) Adding all in equalities, we o btain the d esired result. I V . W E A K G AU S S I A N I N T E R F E R E N C E C H A N N E L In th is section , we foc us on the weak Gau ssian IC. W e ﬁrst obtain th e su m capac ity of this chann el for a certain rang e of parameters. Then , we obtain an outer bo und o n the cap acity region which is tighter than the previously known ou ter bo unds. Finally , we show tha t time -sharing and co ncaviﬁcation result in the same achievable region fo r Gaussian cod ebooks. 18 A. Su m Capacity In this sub section, we use the Class B channels to ob tain the sum cap acity o f th e we ak IC for a ce rtain ran ge of par ameters. T o this en d, let us co nsider the f ollowing minimizatio n pr oblem: W =min γ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  (130) subject to: a ′ g 1 = a b ′ g 2 = b b ′ N 11 ≤ N 21 a ′ N 22 ≤ N 12 (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [ a ′ , b ′ , g 1 , g 2 , N 11 , N 12 , N 22 , N 21 ] . The objective function in (1 30) is the sum capacity of Class B channels obtained in Theo rem 4. The constraints are the combinatio n of (10 0) and (101) where app lied to conﬁrm the ad missibility of the chan nel and to validate th e sum capacity result. Since every chan nel in the class is adm issible, we h av e C sum ≤ W . Substituting S 1 = g 1 N 12 and S 2 = g 2 N 21 , we have W =min γ  (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1  + γ  (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2  (131) subject to: b (1 − S 1 ) (1 − √ g 1 ) 2 ≤ S 2 < 1 a (1 − S 2 ) (1 − √ g 2 ) 2 ≤ S 1 < 1 0 < [ g 1 , g 2 ] . By ﬁrst m inimizing with r espect to g 1 and g 2 , the optimizatio n prob lem (1 31) can be decom posed as W =min W 1 + W 2 (132) subject to: 0 < S 1 < 1 , 0 < S 2 < 1 . where W 1 is deﬁned as W 1 =min g 1 γ  (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1  (133) subject to: b (1 − S 1 ) S 2 ≤ (1 − √ g 1 ) 2 , 0 < g 1 . Similarly , W 2 is de ﬁned as W 2 =min g 2 γ  (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2  (134) subject to: a (1 − S 2 ) S 1 ≤ (1 − √ g 2 ) 2 , 0 < g 2 . The optim ization p roblems (133) an d (134) are e asy to solve. In fact, we h av e W 1 =      γ  P 1 1+ aP 2  if √ b (1 + aP 2 ) ≤ p S 2 (1 − S 1 ) γ  bP 1 S 2 + (1 − √ b (1 − S 1 ) /S 2 ) 2 P 1 aP 2 + S 1  Otherwise (135) W 2 =      γ  P 2 1+ bP 1  if √ a (1 + bP 1 ) ≤ p S 1 (1 − S 2 ) γ  aP 2 S 1 + (1 − √ a (1 − S 2 ) /S 1 ) 2 P 2 bP 1 + S 2  Otherwise (136) 19 From (135) an d (1 36), we ob serve that fo r S 1 and S 2 satisfying √ b (1 + aP 2 ) ≤ p S 2 (1 − S 1 ) a nd √ a (1 + bP 1 ) ≤ p S 1 (1 − S 2 ) , the ob jectiv e fun ction b ecomes indep endent of S 1 and S 2 . In th is case, we have W = γ  P 1 1 + aP 2  + γ  P 2 1 + bP 1  , (137) which is achiev able by tr eating interfe rence as n oise. In th e following theor em, we prove that it is possible to ﬁnd a cer tain range of parameter s such that there exist S 1 and S 2 yielding (13 7). Theor em 5 : Th e su m capacity of the two-user Gaussian IC is C sum = γ  P 1 1 + aP 2  + γ  P 2 1 + bP 1  , (138) for the ra nge of p arameters satisfying √ bP 1 + √ aP 2 ≤ 1 − √ a − √ b √ ab . ( 139) Pr o of: Let us ﬁx a and b , and d eﬁne D as D = ( ( P 1 , P 2 ) | P 1 ≤ p S 1 (1 − S 2 ) b √ a − 1 b , P 2 ≤ p S 2 (1 − S 1 ) a √ b − 1 a , 0 < S 1 < 1 , 0 < S 2 < 1 ) . (140) In fact, if D is feasible th en there exist 0 < S 1 < 1 and 0 < S 2 < 1 satisfying √ b (1 + aP 2 ) ≤ p S 2 (1 − S 1 ) and √ a (1 + bP 1 ) ≤ p S 1 (1 − S 2 ) . Ther efore, the sum cap acity of the chann el for all feasible p oints is attained due to (137). W e claim that D = D ′ , wher e D ′ is deﬁned as D ′ = ( ( P 1 , P 2 ) | √ bP 1 + √ aP 2 ≤ 1 − √ a − √ b √ ab ) . (141) T o show D ′ ⊆ D , we set S 1 = 1 − S 2 in (1 40) to get  ( P 1 , P 2 ) | P 1 ≤ S 1 b √ a − 1 b , P 2 ≤ 1 − S 1 a √ b − 1 a , 0 < S 1 < 1  ⊆ D . (142) It is easy to show tha t the left hand side of th e ab ove equation is another represen tation of the region D ′ . Hence, we have D ′ ⊆ D . T o sh ow D ⊆ D ′ , it sufﬁces to prove that for any ( P 1 , P 2 ) ∈ D , √ bP 1 + √ aP 2 ≤ 1 − √ a − √ b √ ab holds. T o this end, we introd uce the fo llowing maximization prob lem: J = max ( P 1 ,P 2 ) ∈ D √ bP 1 + √ aP 2 , (143) which can b e written as J = max ( S 1 ,S 2 ) ∈ (0 , 1) 2 p S 1 (1 − S 2 ) + p S 2 (1 − S 1 ) √ ab − 1 √ a − 1 √ b . (144) It is ea sy to show that the solution to the ab ove optimization prob lem is J = 1 √ ab − 1 √ a − 1 √ b . (145) Hence, we deduce that D ⊆ D ′ . This co mpletes the pro of. Remark 3: The a bove sum ca pacity r esult for the weak Gaussian IC (see also [2 4]) has bee n established inde pendently in [25] and [2 6]. As an example, let us consider the symm etric Gaussian IC. In this ca se, the constraint in ( 139) beco mes P ≤ 1 − 2 √ a 2 a √ a . (146) In Figu re 7, the admissible region for P , where tr eating interf erence as noise is optimal, versus √ a is plotted. For a ﬁxed P and all 0 ≤ a ≤ 1 , the upper b ound in (130) and the lower boun d when receivers tre at th e interferen ce as noise ar e plotted in Figure 8. W e observe that up to a ce rtain value o f a , the u pper bou nd co incides with the lower bo und. 20 a Fig. 7. The shaded area is the regi on where treating interference as noise is optimal for obtaining the sum capa city of the symmetric Gaussian IC. a 2 1 R R  7 2 1 P P Fig. 8. The upper bound obta ined by solving (130). The lo wer bound is obtained by treati ng the interfe rence as noise. 21 B. New Outer Bound For the we ak Gaussian IC, there ar e two ou ter bound s th at a re tighter than th e other known bo unds. The ﬁr st on e, due to Kramer [1 2], is obtain ed b y relying o n the fact th at the capa city region o f the Gau ssian IC is inside the capacity region s o f the two und erlying one- sided Gau ssian ICs. E ven though the capacity region of th e one-sided Gaussian I C is unk nown, the re exists an outer bo und for th is ch annel that can be u sed instead. Kr amers’ o uter bou nd is th e intersectio n of two r egions E 1 and E 2 . E 1 is the collection of all rate p airs ( R 1 , R 2 ) satisfyin g R 1 ≤ γ  (1 − β ) P ′ β P ′ + 1 /a  , (147) R 2 ≤ γ ( β P ′ ) , (148) for all β ∈ [0 , β max ] , where P ′ = P 1 /a + P 2 and β max = P 2 P ′ (1+ P 1 ) . Similarly , E 2 is the collection of all rate pairs ( R 1 , R 2 ) satisfying R 1 ≤ γ ( αP ′′ ) , (149) R 2 ≤ γ  (1 − α ) P ′′ αP ′′ + 1 /b  , (150) for all α ∈ [0 , α max ] , wher e P ′′ = P 1 + P 2 /b and α max = P 1 P ′′ (1+ P 2 ) . The secon d outer b ound , due to Etkin e t al. [13], is obtained by u sing Genie aid ed techniq ue to u pper boun d different linear comb inations of rates that a ppear in th e HK achievable region. Their outer b ound is the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ ( P 1 ) , (151) R 2 ≤ γ ( P 2 ) , (152) R 1 + R 2 ≤ γ ( P 1 ) + γ  P 2 1 + bP 1  , (153) R 1 + R 2 ≤ γ ( P 2 ) + γ  P 1 1 + aP 2  , (154) R 1 + R 2 ≤ γ  aP 2 + P 1 1 + bP 1  + γ  bP 1 + P 2 1 + aP 2  , (155) 2 R 1 + R 2 ≤ γ ( P 1 + aP 2 ) + γ  bP 1 + P 2 1 + aP 2  + 0 . 5 log  1 + P 1 1 + bP 1  , (156) R 1 + 2 R 2 ≤ γ ( bP 1 + P 2 ) + γ  aP 2 + P 1 1 + bP 1  + 0 . 5 log  1 + P 2 1 + aP 2  . (157) In th e outer boun d pr oposed h ere, we derive an u pper bo und on all linear combin ations o f the rates. Recall that to o btain the bound ary po ints o f the capacity r egion C , it sufﬁces to calculate σ C ( µ, 1) and σ C (1 , µ ) for all 1 ≤ µ . T o this end , we ma ke use of channels in A 1 and B classes and channels in A2 and B classes to obtain upp er bound s on σ C ( µ, 1) a nd σ C (1 , µ ) , respectively . In o rder to o btain an upper bou nd on σ C ( µ, 1) , we in troduce two op timization problem s as follows. Th e ﬁrst o ptimization problem is wr itten as W 1 ( µ ) =min µ 1 2 log [2 π e ( P 1 + aP 2 + 1)] − µ 2 2 log(2 π e ) + 1 2 log  N 21 N 22 + b ′ P 1 N 22 + P 2 P 2 + N 22  (158) + µ 2 f h  P 1 , 1 , N 21 , b ′ , 1 µ 2  + f h ( P 2 , N 22 , 1 , a, µ 1 ) subject to: µ 1 + µ 2 = µ b ′ g 2 = b b ′ ≤ N 21 aN 22 ≤ 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [ µ 1 , µ 2 , b ′ , g 2 , N 22 , N 21 ] . In fact, the ob jecti ve of the ab ove minim ization pro blem is an u pper b ound on the supp ort fun ction of a ch annel within Class A1 which is o btained in Lem ma 3. T he constraints are th e combinatio n of (78) an d (79) which ar e a pplied to gu arantee the 22 admissibility of the c hannel and to validate th e uppe r bou nd obtain ed in Lemma 3. Henc e, σ C ( µ, 1) ≤ W 1 ( µ ) . By using a new variable S = (1 − √ g 2 ) 2 N 22 , we obtain W 1 ( µ ) =min µ 1 2 log [2 π e ( P 1 + aP 2 + 1)] + 1 2 log  (1 − √ g 2 ) 2 ( 1 − S + bP 1 g 2 S + P 2 (1 − √ g 2 ) 2 P 2 + S )  (159) + µ 2 f h  P 1 , 1 , 1 − S g 2 , b g 2 , 1 µ 2  + f h ( P 2 , S (1 − √ g 2 ) 2 , 1 , a, µ 1 ) − µ 2 2 log(2 π e ) subject to: µ 1 + µ 2 = µ S ≤ 1 − b S ≤ (1 − √ g 2 ) 2 a 0 ≤ [ µ 1 , µ 2 , S, g 2 ] . The second optimization pr oblem is written as W 2 ( µ ) =min µγ  P 1 N 11 + P 1 a ′ P 2 + N 12  + γ  P 2 N 22 + P 2 b ′ P 1 + N 21  + f h ( P 2 , N 22 , N 12 , a ′ , µ ) (160 ) + µ 2 log((2 π e )( a ′ P 2 + N 12 )) − 1 2 log((2 π e )( P 2 + N 22 )) subject to : a ′ g 1 = a b ′ g 2 = b b ′ N 11 ≤ N 21 a ′ N 22 ≤ N 12 (1 − √ g 1 ) 2 N 11 + g 1 N 12 = 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [ a ′ , b ′ , g 1 , g 2 , N 11 , N 12 , N 22 , N 21 ] . For this p roblem, Class B channels are used. In fact, th e objective is the u pper bo und on the supp ort functio n of channe ls within th e cla ss ob tained in Lemm a 5 and the constrain ts are deﬁned to o btain th e closed form for mula for the upper bound and to conﬁrm that the chan nels are ad missible. Hen ce, we deduce σ C ( µ, 1) ≤ W 2 ( µ ) . By using new variables S 1 = g 1 N 12 and S 2 = g 2 N 21 , we ob tain W 2 ( µ ) =min µγ  (1 − √ g 1 ) 2 P 1 1 − S 1 + g 1 P 1 aP 2 + S 1  + γ  (1 − √ g 2 ) 2 P 2 1 − S 2 + g 2 P 2 bP 1 + S 2  (161) + f h  P 2 , 1 − S 2 (1 − √ g 2 ) 2 , S 1 g 1 , a g 1 , µ  + µ 2 log  (2 π e )( aP 2 + S 1 g 1 )  − 1 2 log  (2 π e )( P 2 + 1 − S 2 (1 − √ g 2 ) 2 )  subject to : b (1 − S 1 ) (1 − √ g 1 ) 2 ≤ S 2 < 1 a (1 − S 2 ) (1 − √ g 2 ) 2 ≤ S 1 < 1 0 < [ g 1 , g 2 ] . In a sim ilar fashion, one can introd uce two other optimiza tion problems, say ˜ W 1 ( µ ) and ˜ W 2 ( µ ) , to obtain uppe r b ounds on σ C (1 , µ ) by using the upp er bo unds on the suppor t fun ctions o f channels in Class A2 and Class B. Theor em 6 (New Outer Bound ): For any rate pair ( R 1 , R 2 ) achievable fo r the two-user weak Gaussian IC, th e in equalities µ 1 R 1 + R 2 ≤ W ( µ 1 ) = min { W 1 ( µ 1 ) , W 2 ( µ 1 ) } , (162) R 1 + µ 2 R 2 ≤ ˜ W ( µ 2 ) = min { ˜ W 1 ( µ 2 ) , ˜ W 2 ( µ 2 ) } , (163) hold fo r all 1 ≤ µ 1 , µ 2 . T o obtain an upp er bo und on the sum rate, w e can ap ply th e following inequality: C sum ≤ min 1 ≤ µ 1 ,µ 2 ( µ 2 − 1) W ( µ 1 ) + ( µ 1 − 1) ˜ W ( µ 2 ) µ 1 µ 2 − 1 . (164) 23 r 3 r 2 r 1 r 4 R 2 R 1 r ′ 1 R 1 + 2 R 2 = ψ 5 R 2 = ψ 2 R 1 + R 2 = ψ 3 2 R 1 + R 2 = ψ 4 R 1 = ψ 1 r ′ 4 r ′ 5 r ′ 2 r ′ 3 r ′ 6 Fig. 9. G 0 for the weak Gaussian IC. r 1 , r 2 , r 3 , and r 4 are ext reme points of G 0 in the interi or of the ﬁrst quadra nt. C. Ha n-K obayashi Achievable re g ion In th is sub-section , we aim at ch aracterizing G fo r the wea k Gaussian IC. T o this end, we ﬁrst inv estigate some p roperties of G 0 ( P 1 , P 2 , α, β ) . First of all, we show that none of the inequa lities in describing G 0 is redu ndant. I n Figure 9, all possible extreme points ar e shown. It is easy to prove that r ′ i / ∈ G 0 for i ∈ { 1 , 2 , . . . , 6 } . For in stance, we consider r ′ 6 =  2 ψ 4 − ψ 5 3 , 2 ψ 5 − ψ 4 3  . Since ψ 31 + ψ 32 + ψ 33 = ψ 4 + ψ 5 (see Section II. C), w e have ψ 3 = min { ψ 31 , ψ 32 , ψ 33 } ≤ 1 3 ( ψ 31 + ψ 32 + ψ 33 ) = 1 3 ( ψ 4 + ψ 5 ) . Howe ver , 1 3 ( ψ 4 + ψ 5 ) is th e sum of the compon ents of r ′ 6 . Ther efore, r ′ 6 violates (7) in the deﬁnition of the HK achievable region. Hence, r ′ 6 / ∈ G 0 . As an other example, let us con sider r ′ 1 = ( ψ 1 , ψ 3 − ψ 1 ) . W e claim th at r ′ 1 violates (8). T o this end , we n eed to show that ψ 4 ≤ ψ 3 + ψ 1 . H owe ver , it is easy to see th at ψ 4 ≤ ψ 31 + ψ 1 , ψ 4 ≤ ψ 32 + ψ 1 , a nd ψ 4 ≤ ψ 33 + ψ 1 reduce to 0 ≤ (1 − α )(1 − b + β (1 − ab ) P 2 ) , 0 ≤ (1 − β )(1 − a + (1 − ab ) P 1 ) , an d 0 ≤ (1 − α )(1 − β ) aP 2 , r espectively . Therefo re, r ′ 1 / ∈ G 0 . W e conc lude that G has four extreme points in the interior of the ﬁrst qu adrant, n amely r 1 = ( ψ 1 , ψ 4 − 2 ψ 1 ) , (165) r 2 = ( ψ 4 − ψ 3 , 2 ψ 3 − ψ 4 ) , (166) r 3 = (2 ψ 3 − ψ 5 , ψ 5 − ψ 3 ) , (167) r 4 = ( ψ 5 − 2 ψ 2 , ψ 2 ) . (168) Most imp ortantly , G 0 possesses the un ique minimizer p roperty . T o prove this, we need to show that ˆ y , th e minim izer of the optimization pro blem σ D 0 ( c 1 , c 2 , P 1 , P 2 , α, β )= max { c 1 R 1 + c 2 R 2 | A R ≤ Ψ( P 1 , P 2 , α, β ) } = min { y t Ψ( P 1 , P 2 , α, β ) | A t y = ( c 1 , c 2 ) t , y ≥ 0 } , (169) is indep endent of the p arameters P 1 , P 2 , α , and β and only depen ds on c 1 and c 2 . W e ﬁrst conside r th e case ( c 1 , c 2 ) = ( µ, 1) for all 1 ≤ µ . It can be shown that for 2 < µ , the maximum of (1 69) is attained at r 1 regardless o f P 1 , P 2 , α , and β . Therefo re, the d ual prog ram has the minimizer ˆ y = ( µ − 2 , 0 , 0 , 1 , 0) t which is clearly indep endent of P 1 , P 2 , α , and β . In this case, we have σ D 0 ( µ, 1 , P 1 , P 2 , α, β ) = ( µ − 2) ψ 1 + ψ 4 , 2 < µ. (170) For 1 ≤ µ ≤ 2 , on e can show that r 2 and ˆ y = (0 , 0 , 2 − µ, µ − 1 , 0) t are the maximizer and the minimizer of (169), respectively . In this case, we have σ D 0 ( µ, 1 , P 1 , P 2 , α, β ) = (2 − µ ) ψ 3 + ( µ − 1) ψ 4 , 1 ≤ µ ≤ 2 . (171) Next, w e consider the case ( c 1 , c 2 ) = (1 , µ ) for all 1 ≤ µ . Again, it can be shown that for 2 < µ and 1 ≤ µ ≤ 2 , ˆ y = (0 , µ − 2 , 0 , 0 , 1) t and ˆ y = (0 , 0 , 2 − µ, 0 , µ − 1 ) t minimizes (16 9), respectively . Hence, we have σ D 0 (1 , µ, P 1 , P 2 , α, β )= ( µ − 2) ψ 2 + ψ 5 , if 2 < µ, (172) σ D 0 (1 , µ, P 1 , P 2 , α, β )= (2 − µ ) ψ 3 + ( µ − 1) ψ 5 , if 1 ≤ µ ≤ 2 . (173) 24 Fig. 10. Compari son between differe nt bounds for the symmetric weak Gaussian IC when P = 7 and a = 0 . 2 . W e con clude that the solutions o f the dual progr am are always independ ent of P 1 , P 2 , α , and β . Hence, G 0 possesses the unique minim izer pr operty . Theor em 7 : For the two-user weak Gaussian IC, time-sh aring a nd concaviﬁcation result in the sam e region. In o ther words, G can b e fully c haracterized by using TD/FD and allocating power over three d ifferent dimension s. Pr o of: Since G 0 possesses the u nique minimizer pr operty , from Theor em 1, we deduce th at G = G 2 . M oreover , using Theorem 3, th e num ber of f requency b ands is at most thr ee. T o obtain the support f unction of G 2 , we need to obtain g ( c 1 , c 2 , P 1 , P 2 , α, β ) deﬁne d in (4 3). Since G 0 possesses the uniqu e minimizer prop erty , (43) can be simpliﬁed. Let us con sider th e ca se where ( c 1 , c 2 ) = ( µ, 1) for µ > 2 . I t can be shown that for this case g = max ( α,β ) ∈ [0 , 1] 2 ( µ − 2) ψ 1 ( P 1 , P 2 , α, β ) + ψ 4 ( P 1 , P 2 , α, β ) . (174) Substituting into (42), we obtain σ G 2 ( µ, 1 , P 1 , P 2 ) =max 3 X i =1 λ i [( µ − 2) ψ 1 ( P 1 i , P 2 i , α i , β i ) + ψ 4 ( P 1 i , P 2 i , α i , β i )] (175) subject to: 3 X i =1 λ i = 1 3 X i =1 λ i P 1 i ≤ P 1 3 X i =1 λ i P 2 i ≤ P 2 0 ≤ λ i , 0 ≤ P 1 i , 0 ≤ P 2 i , ∀ i ∈ { 1 , 2 , 3 } 0 ≤ α i ≤ 1 , 0 ≤ β i ≤ 1 , ∀ i ∈ { 1 , 2 , 3 } . For o ther r anges o f ( c 1 , c 2 ) , a similar op timization pro blem can b e formed . It is worth noting that even though the num ber of param eters in ch aracterizing G is reduc ed, it is still pr ohibitively difﬁcult to cha racterize b ound ary points of G . In Figur es (10) and (11), different b ounds for the symmetric weak Gau ssian IC ar e p lotted. As sh own in these ﬁgure s, the new outer bound is tig hter than th e p reviously kn own bound s. V . O N E - S I D E D G AU S S I A N I N T E R F E R E N C E C H A N N E L S Throu ghout this section, we co nsider th e o ne-sided Gaussian IC ob tained b y setting b = 0 , i.e, the second receiver incur s no interferen ce fr om the ﬁrst tr ansmitter . One can furthe r split th e class of on e-sided ICs into two subclasses: the str o n g on e-sided 25 Fig. 11. Compari son between differe nt bounds for the symmetric weak Gaussian IC when P = 100 and a = 0 . 1 . IC and th e weak one-sided I C . For the former, a ≥ 1 and th e cap acity region is fully ch aracterized [16]. I n th is c ase, the capacity region is th e unio n of all rate pairs ( R 1 , R 2 ) satisfying R 1 ≤ γ ( P 1 ) , R 2 ≤ γ ( P 2 ) , R 1 + R 2 ≤ γ ( P 1 + aP 2 ) . For the latter , a < 1 and the full chara cterization of the cap acity region is still an open pro blem. Ther efore, we always assume a < 1 . Thre e impor tant r esults are proved fo r this channe l. T he ﬁrst one, p roved by Costa in [11], states that the capacity region of the w eak on e-sided IC is eq uiv alent to that of the degrad ed IC with an ap propria te chan ge of parameters. The second one, p roved b y Sato in [1 0], states that th e cap acity region of the degraded Gau ssian IC is outer bou nded by the capacity region of a certain degraded b roadcast chan nel. The th ird on e, proved b y Sason in [ 16], characterizes the sum capacity b y combinin g Costa’ s and Sato’ s results. In this section, we provid e an alter nativ e proo f for the ou ter b ound ob tained b y Sato . W e th en c haracterize the f ull HK achiev able region where Gaussian codeb ooks are used, i.e. , G . A. Su m Capacity For th e sake of completen ess, we ﬁrst state the sum capac ity result obtaine d by Sason. Theor em 8 (S a son): The rate pair  γ  P 1 1+ aP 2  , γ ( P 2 )  is an extreme p oint o f th e capacity region of th e on e-sided Gaussian IC. Mor eover , the sum capacity o f the ch annel is attained a t this p oint. B. Outer Boun d In [10], Sato derived an outer b ound on the capacity of the degraded IC. This o uter bo und can be u sed for the weak on e-sided IC as well. This is du e to Costa’ s result which states that the cap acity region of the d egraded Gaussian I C is equiv alent to that of the weak one -sided I C with an appro priate chan ge of par ameters. Theor em 9 (S a to): If the rate pair ( R 1 , R 2 ) belon gs to the ca pacity region of the we ak one-side d IC, then it satisﬁes R 1 ≤ γ  (1 − β ) P 1 /a + β P  , R 2 ≤ γ ( β P ) , (176) for all β ∈ [0 , 1] where P = P 1 /a + P 2 . Pr o of: Since the sum capacity is attain ed at the point where User 2 transmits at its maximum rate R 2 = γ ( P 2 ) , other boun d- ary po ints of the capacity region can be ob tained by cha racterizing the solution s of σ C ( µ, 1) = max { µR 1 + R 2 | ( R 1 , R 2 ) ∈ C } 26 for all 1 ≤ µ . Using Fano’ s ineq uality , we have n ( µR 1 + R 2 ) ≤ µI ( x n 1 ; y n 1 ) + I ( x n 2 ; y n 2 ) + nǫ n = µh ( y n 1 ) − µh ( y n 1 | x n 1 ) + h ( y n 2 ) − h ( y n 2 | x n 2 ) + nǫ n =[ µh ( x n 1 + √ ax n 2 + z n 1 ) − h ( z n 2 )] + [ h ( x n 2 + z n 2 ) − µh ( √ ax n 2 + z n 1 )] + nǫ n ( a ) ≤ µn 2 log [2 π e ( P 1 + aP 2 + 1)] − n 2 log(2 π e ) + [ h ( x n 2 + z n 2 ) − µh ( √ ax n 2 + z n 1 )] + nǫ n ( b ) ≤ µn 2 log [2 π e ( P 1 + aP 2 + 1)] − n 2 log(2 π e ) + nf h ( P 2 , 1 , 1 , a, µ ) + nǫ n , where (a) fo llows f rom the fact that Gau ssian distribution maxim izes the differential entr opy for a giv en con straint o n the covariance matrix and ( b) follows fr om th e deﬁnitio n of f h in (5 9). Dependin g on the value o f µ , we consider the following two cases: 1- For 1 ≤ µ ≤ P 2 +1 /a P 2 +1 , we have µR 1 + R 2 ≤ µγ  P 1 1 + aP 2  + γ ( P 2 ) . (177) In fact, the po int  γ  P 1 1+ aP 2  , γ ( P 2 )  which is achievable by treating interferen ce as n oise at Receiver 1, satisﬁes (177) with equality . There fore, it belon gs to th e c apacity r egion. Mo reover , by setting µ = 1 , we ded uce that this point correspo nds to the sum c apacity of th e one-sided Gaussian IC. This is in fact an alternative proo f fo r Sason’ s result. 2- For P 2 +1 /a P 2 +1 < µ ≤ 1 a , we have µR 1 + R 2 ≤ µ 2 log ( P 1 + aP 2 + 1) + 1 2 log  1 /a − 1 µ − 1  − µ 2 log  aµ (1 /a − 1) µ − 1  . (178) Equiv alently , we have µR 1 + R 2 ≤ µ 2 log  ( aP + 1)( µ − 1 ) µ (1 − a )  + 1 2 log  1 /a − 1 µ − 1  , (179) where P = P 1 /a + P 2 . Let us de ﬁne E 1 as th e set of all rate p airs ( R 1 , R 2 ) satisfyin g (179), i.e. E 1 =  ( R 1 , R 2 ) | µR 1 + R 2 ≤ µ 2 log  ( aP + 1 )( µ − 1) µ (1 − a )  + 1 2 log  1 /a − 1 µ − 1  , ∀ P 2 + 1 /a P 2 + 1 < µ ≤ 1 a  . (180) W e claim that E 1 is the du al repr esentation o f the region d eﬁned in the statement of the theorem , see (4). T o th is end, we deﬁne E 2 as E 2 =  ( R 1 , R 2 ) | R 1 ≤ γ  (1 − β ) P 1 /a + β P  , R 2 ≤ γ ( β P ) , ∀ β ∈ [0 , 1]  (181) W e ev aluate the suppo rt fu nction of E 2 as σ E 2 ( µ, 1) = max { µR 1 + R 2 | ( R 1 , R 2 ) ∈ E 2 } . (182) It is ea sy to show that β = 1 /a − 1 P ( µ − 1) maximizes the ab ove optimization pr oblem. Th erefore, we have σ E 2 ( µ, 1) = µ 2 log  ( aP + 1)( µ − 1) µ (1 − a )  + 1 2 log  1 /a − 1 µ − 1  . (183) Since E 2 is a closed conv ex set, we can u se (4) to o btain its dual representatio n which is indeed equ i valent to ( 180). This completes the p roof. C. Ha n-K obayashi Achievable R e gion In th is su bsection, we characterize G 0 , G 1 , G 2 , and G for the weak on e-sided Gaussian IC. G 0 can b e characterized as follows. Since there is no link betwee n T ransmitter 1 and Rece i ver 2, User 1’ s message in the HK achievable region is only 27 the pr i vate message, i.e., α = 1 . In th is case, we have ψ 1 = γ  P 1 1 + aβ P 2  , (184) ψ 2 = γ ( P 2 ) , (185) ψ 31 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) , (1 86) ψ 32 = γ  P 1 1 + aβ P 2  + γ ( P 2 ) , (187) ψ 33 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) , (1 88) ψ 4 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ  P 1 1 + aβ P 2  + γ ( β P 2 ) , (189) ψ 5 = γ ( β P 2 ) + γ ( P 2 ) + γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  , (190) It is ea sy to show that ψ 3 = min { ψ 31 , ψ 32 , ψ 33 } = ψ 31 , ψ 31 + ψ 1 = ψ 4 , ψ 31 + ψ 2 = ψ 5 . Hence, G 0 can be represen ted as all rate pa irs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aβ P 2  , (191) R 2 ≤ γ ( P 2 ) , (192) R 1 + R 2 ≤ γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) . (193) W e claim that G 2 = G . T o pr ove this, we need to show that G 0 possesses th e u nique minimizer prope rty . G 0 is a pentag on with two extreme po ints in the in terior o f the ﬁr st quadr ant, n amely r 1 and r 2 where r 1 =  γ  P 1 1 + aβ P 2  , γ  (1 − β ) aP 2 1 + P 1 + β aP 2  + γ ( β P 2 )  , (194) r 2 =  γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) − γ ( P 2 ) , γ ( P 2 )  . (1 95) Using above, it can be veriﬁed th at G 0 possesses the unique minim izer pr operty . Next, we can u se the optimizatio n problem in (42) to obtain the supp ort function o f G . Howev er , we only need to co nsider ( c 1 , c 2 ) = ( µ, 1) for µ > 1 . Ther efore, we have g ( µ, 1 , P 1 , P 2 , β ) = max 0 ≤ β ≤ 1 µγ  P 1 1 + β aP 2  + γ ( β P 2 ) + γ  (1 − β ) aP 2 1 + P 1 + β aP 2  . (196) Substituting in to (42), we con clude that bou ndary points of G can b e characterize d b y solv ing the f ollowing op timization problem : W =max 3 X i =1 λ i  µγ  P 1 i 1 + β i aP 2 i  + γ ( β i P 2 i ) + γ  (1 − β i ) aP 2 i 1 + P 1 i + β i aP 2 i  (197) subject to: 3 X i =1 λ i = 1 3 X i =1 λ i P 1 i ≤ P 1 3 X i =1 λ i P 2 i ≤ P 2 0 ≤ β i ≤ 1 , ∀ i ∈ { 1 , 2 , 3 } 0 ≤ [ P 1 i , P 2 i , λ i ] , ∀ i ∈ { 1 , 2 , 3 } . For th e sake of completen ess, we p rovide a simp le description for G 1 in the n ext lemm a. 28 1 Fig. 12. Compari son between differe nt bounds for the one-sid ed Gaussian IC when P 1 = 1 , P 2 = 7 , and a = 0 . 4 . Lemma 7: The region G 1 can be represented as the collection of all r ate pairs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aβ ′ P 2  , (198) R 2 ≤ γ ( β ′ P 2 ) + γ  a (1 − β ′ ) P 2 1 + P 1 + aβ ′ P 2  , (199) for all β ′ ∈ [0 , 1] . Moreover , G 1 is convex and any po int that lies o n its bou ndary ca n b e achieved by using super position coding and su ccessi ve dec oding. Pr o of: Let E deno te the set deﬁned in the above lemma. It is easy to sh ow that E is conv ex and E ⊆ G 1 . T o prove the inverse inclusion , it sufﬁces to show that the extrem e poin ts of G 0 , r 1 and r 2 (see (19 4) a nd (1 95)) are in side E for all β ∈ [0 , 1] . By setting β ′ = β , we see that r 1 ∈ E . T o p rove r 2 ∈ E , w e set β ′ = 1 . W e conc lude that r 2 ∈ E if the f ollowing inequality hold s γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) − γ ( P 2 ) ≤ γ  P 1 1 + aP 2  , (200) for all β ∈ [0 , 1] . However , (200) redu ces to 0 ≤ (1 − a )(1 − β ) P 2 which holds fo r a ll β ∈ [0 , 1] . Hence , G 1 ⊆ E . U sing these facts, it is straigh tforward to show that the b ounda ry points G 1 are ach iev able by using sup erposition co ding and successi ve decodin g. Figure 12 compare s different b ounds for the one-sided Gaussian I C. V I . M I X E D G AU S S I A N I N T E R F E R E N C E C H A N N E L S In th is section, we focus on the mixed Gaussian Interferen ce cha nnel. W e ﬁr st characterize the sum capacity o f this chan nel. Then, we provid e a n o uter b ound o n the capacity region. Fin ally , we in vestigate the HK achievable region. Without loss o f generality , we assume a < 1 and b ≥ 1 . 29 A. Su m Capacity Theor em 1 0: The sum capa city of the mixed Ga ussian IC with a < 1 and b ≥ 1 can b e stated as C sum = γ ( P 2 ) + min  γ  P 1 1 + aP 2  , γ  bP 1 1 + P 2  . (201) Pr o of: W e n eed to prove the achievability and converse for the the orem. Achievability part : Transmitter 1 send s a commo n message to b oth r eceiv ers, while the ﬁrst user’ s signal is co nsidered as noise at b oth receivers. In this case, the rate R 1 = min  γ  P 1 1 + aP 2  , γ  bP 1 1 + P 2  (202) is achiev able. At Receiver 2, the signal fro m Transmitter 1 ca n b e decod ed and rem oved. Th erefore, User 2 is left with a channel withou t in terference and it can com municate at its maximum rate wh ich is R 2 = γ ( P 2 ) . (203) By add ing ( 202) and (2 03), we obtain the desired r esult. Con verse pa rt : The sum capacity of the Gaussian IC is upper boun ded by that of the two under lying one-sided Gaussian ICs. He nce, we c an ob tain two upper b ound s on the sum rate. W e ﬁrst r emove the interf ering link betwee n T ransmitter 1 an d Receiv er 2. In th is case, we have a one-sid ed Gaussian IC with weak inter ference. The sum capacity o f this cha nnel is k nown [16]. Hence, w e have C sum ≤ γ ( P 2 ) + γ  P 1 1 + aP 2  . (204) By rem oving th e inter fering link between Transmitter 2 and Receiv er 1, we o btain a one-side d Gaussian IC with strong interferen ce. The sum capacity of this ch annel is known. Hence, we have C sum ≤ γ ( bP 1 + P 2 ) , (2 05) which equ i valently can be wr itten as C sum ≤ γ ( P 2 ) + γ  bP 1 1 + P 2  . (2 06) By taking the minimu m o f the rig ht hand sides o f (204) and (206), we obtain C sum ≤ γ ( P 2 ) + min  γ  P 1 1 + aP 2  , γ  bP 1 1 + P 2  . (207) This com pletes the pro of. Remark 4: In an indepen dent work [2 5], the sum capacity of the mixed Gaussian IC is o btained for a certain range of parameters, whereas in the above theorem, we characterize the sum cap acity of this chan nel for the entire rang e of its parame ters (see also [24]). By compar ing γ  P 1 1+ aP 2  with γ  bP 1 1+ P 2  , we ob serve that if 1 + P 2 ≤ b + abP 2 , then the sum cap acity cor respond s to the sum capacity of th e o ne-sided weak Gaussian IC, where as if 1 + P 2 > b + a bP 2 , th en the sum ca pacity co rrespond s to the sum cap acity o f the one- sided strong IC. Similar to the o ne-sided Gau ssian IC, since the su m capacity is attained at the poin t where User 2 transmits at its maximum rate R 2 = γ ( P 2 ) , other bo undary po ints o f the capacity region can b e obtained by characterizin g the so lutions of σ C ( µ, 1) = ma x { µR 1 + R 2 | ( R 1 , R 2 ) ∈ C } for all 1 ≤ µ . B. New Outer Bound The b est outer b ound to date, due to Etkin et al. [13], is obtain ed by using the Genie aide d techn ique. This bo und is the union of all rate p airs ( R 1 , R 2 ) satisfyin g R 1 ≤ γ ( P 1 ) , (208) R 2 ≤ γ ( P 2 ) , (209) R 1 + R 2 ≤ γ ( P 2 ) + γ  P 1 1 + aP 2  , (210) R 1 + R 2 ≤ γ ( P 2 + bP 1 ) , (211) 2 R 1 + R 2 ≤ γ ( P 1 + aP 2 ) + γ  bP 1 + P 2 1 + aP 2  + γ  P 1 1 + bP 1  . (212) 30 The capacity region o f the mixed Gaussian I C is in side the in tersection of th e capacity regions o f the two underly ing one - sided Gau ssian ICs. Removing the link be tween T ransmitter 1 and Receiver 2 resu lts in a we ak one-side d Gaussian IC whose outer bo und E 1 is the collection of all rate p airs ( R 1 , R 2 ) satisfyin g R 1 ≤ γ  (1 − β ) P ′ β P ′ + 1 /a  , (213) R 2 ≤ γ ( β P ′ ) , (214) for all β ∈ [0 , β max ] , where P ′ = P 1 /a + P 2 and β max = P 2 P ′ (1+ P 1 ) . On th e other han d, removin g the link between T ransmitter 2 and Receiv er 1 results in a strong one-sided Gaussian IC whose capacity region E 2 is fu lly cha racterized as the collection of all r ate pairs ( R 1 , R 2 ) satisfying R 1 ≤ γ ( bP 1 ) , (215) R 2 ≤ γ ( P 2 ) , (216) R 1 + R 2 ≤ γ ( bP 1 + P 2 ) . (217) Using the channels in Class C, we upp er bo und σ C ( µ, 1) based on the fo llowing optimizatio n pr oblem: W ( µ ) =min µ − 1 2 log (2 πe ( P 1 + aP 2 + 1)) + 1 2 log  2 π e  P 2 N 22 P 2 + N 22 + b ′ P 1 + N 21  (218) − 1 2 log(2 π eN 21 ) − 1 2 log(2 π eN 22 ) + f h ( P 2 , N 22 , 1 , a, µ − 1) subject to: b ′ g 2 = b b ′ ≥ N 21 aN 22 ≤ 1 (1 − √ g 2 ) 2 N 22 + g 2 N 21 = 1 0 ≤ [ b ′ , g 2 , N 22 , N 21 ] . By substituting S = g 2 N 21 , we ob tain W ( µ ) =min µ − 1 2 log (2 πe ( P 1 + aP 2 + 1)) + 1 2 log  2 π e  P 2 (1 − S ) (1 − √ g 2 ) 2 P 2 + 1 − S + bP 1 + S g 2  (219) − 1 2 log  2 π eS g 2  − 1 2 log  2 π e (1 − S ) (1 − √ g 2 ) 2  + f h  P 2 , 1 − S (1 − √ g 2 ) 2 , 1 , a, µ − 1  subject to: S < 1 a (1 − S ) ≤ (1 − √ g 2 ) 2 0 ≤ [ S, g 2 ] . Hence, we have the following theorem that provides an outer bound on the capacity region of the mixed Gaussian IC. Theor em 1 1: For any rate pa ir ( R 1 , R 2 ) ach iev able for the two-user mixed Gau ssian IC, ( R 1 , R 2 ) ∈ E 1 T E 2 . M oreover , the ineq uality µR 1 + R 2 ≤ W ( µ ) (220) holds for all 1 ≤ µ . C. Ha n-K obayashi Achievable R e gion In this subsection , we study the HK achievable regio n fo r the m ixed Gaussian IC. Since Receiv er 2 can always decod e the message of the ﬁrst user, User 1 associates all its power to the co mmon message . User 2 , on th e other hand , allocates β P 2 31 R 1 R 2 γ ( P 2 ) r 4 r 3 r 2 r 1 G ′ 0 Alternating Regions Fig. 13. The ne w regi on G ′ 0 which is obtaine d by enlar ging G 0 . and (1 − β ) P 2 of its total power to its p riv ate and common messages, respectively , where β ∈ [0 , 1] . T herefor e, we have ψ 1 = γ  P 1 1 + aβ P 2  , (221) ψ 2 = γ ( P 2 ) , (222) ψ 31 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) , (223) ψ 32 = γ ( P 2 + bP 1 ) , (224) ψ 33 = γ  a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 + bP 1 ) , (225) ψ 4 = γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 + bP 1 ) , (226) ψ 5 = γ ( β P 2 ) + γ ( P 2 + bP 1 ) + γ  a (1 − β ) P 2 1 + aβ P 2  . (227) Due to the fact tha t the sum cap acity is attained a t the point where th e second user tr ansmits at its ma ximum rate, the last ine quality in the d escription of the HK ach iev able region can b e rem oved. Althou gh th e poin t r ′ 5 = ( ψ 3 − γ ( P 2 ) , γ ( P 1 )) in Figure 9 may not be in G 0 , th is po int is always achievable due to the sum c apacity r esult. Hence , we can enlarge G 0 by removing r 3 and r 4 . Let us d enote the resulting region as G ′ 0 . Moreover , one can show that r ′ 2 , r ′ 3 , r ′ 4 , and r ′ 6 are still o utside G ′ 0 . Howe ver , for the m ixed Ga ussian IC, it is p ossible that r ′ 1 belongs to G ′ 0 . In Figure 13, two a lternative cases fo r the regio n G ′ 0 along with the new lab eling of its extreme poin ts are plotted. Th e new extreme points can be written as r 1 = ( ψ 1 , ψ 4 − 2 ψ 1 ) , r 2 = ( ψ 1 , ψ 3 − ψ 1 ) , r 3 = ( ψ 4 − ψ 3 , 2 ψ 3 − ψ 4 ) , r 4 = ( ψ 3 − ψ 2 , ψ 2 ) . In fact, we have either G ′ 0 = con v { r 1 , r 3 , r 4 } or G ′ 0 = conv { r 2 , r 4 } . T o simplify th e characteriza tion of G 1 , we consider three ca ses: Case I: 1 + P 2 ≤ b + abP 2 . Case II: 1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 . Case III: 1 + P 2 > b + abP 2 and 1 − a > abP 1 . Case I ( 1 + P 2 ≤ b + abP 2 ) : I n this case, ψ 3 = ψ 31 . Mor eover , it is easy to verif y that ψ 31 + ψ 1 ≤ ψ 4 which means ( 8) is redund ant for th e entire r ange of param eters. Hen ce, G ′ 0 = conv { r 2 , r 4 } consists of all rate pairs ( R 1 , R 2 ) satisfyin g R 1 ≤ γ  P 1 1 + aβ P 2  , (228) R 2 ≤ γ ( P 2 ) , (229) R 1 + R 2 ≤ γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) , (230) 32 where β ∈ [0 , 1] . Using a reason ing similar to the o ne used to express boun dary points of G 1 for the on e-sided Gaussian IC, we can expr ess b oundar y points of G 1 as R 1 ≤ γ  P 1 1 + aβ P 2  , (231) R 2 ≤ γ ( β P 2 ) + γ  a (1 − β ) P 2 1 + P 1 + aβ P 2  , (232) for all β ∈ [0 , 1] . Theor em 1 2: For the mixed Gaussian IC satisfying 1 ≤ a b , region G is eq uiv alent to that of the one sided Gaussian I C obtained fro m rem oving the interferin g link between T ransmitter 1 an d Receiv er 2. Pr o of: If 1 ≤ ab , then 1 + P 2 ≤ b + a bP 2 holds for all P 1 and P 2 . Henc e, G ′ 0 ( P 1 , P 2 , β ) is a pentago n deﬁn ed by (2 28), (229), and (2 29). Com paring with the co rrespond ing r egion for the one-sided Gaussian I C, we see th at G ′ 0 is equivalent to G 0 obtained for the one-sid ed Gau ssian IC. This directly imp lies tha t G is the same for both channe ls. Case II ( 1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 ): In this case, ψ 3 = min { ψ 31 , ψ 32 } . It can be shown that G 1 is the un ion of three r egions E 1 , E 2 , and E 3 , i.e, G 0 = E 1 S E 2 S E 3 . Region E 1 is the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aβ P 2  , (233) R 2 ≤ γ ( β P 2 ) + γ  a (1 − β ) P 2 1 + P 1 + aβ P 2  . (234) for all β ∈ [0 , b − 1 (1 − ab ) P 2 ] . Region E 2 is the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ  bP 1 1 + β P 2  , (235) R 2 ≤ γ  P 1 + a (1 − β ) P 2 1 + aβ P 2  + γ ( β P 2 ) − γ  bP 1 1 + β P 2  . (236) for all β ∈ [ b − 1 (1 − ab ) P 2 , ( b − 1) P 1 +(1 − a ) P 2 (1 − ab ) P 1 P 2 +(1 − a ) P 2 ] . Region E 3 is the union o f all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ bP 1 (1 + (1 − ab ) P 1 1 − a ) 1 + bP 1 + P 2 ! , (237) R 2 ≤ γ ( P 2 ) , (238) R 1 + R 2 ≤ γ ( bP 1 + P 2 ) . (239) Case III ( 1 + P 2 > b + abP 2 and 1 − a > abP 1 ) : In this case, ψ 3 = min { ψ 31 , ψ 32 } . Sim ilar to Case I I, we ha ve G 1 = E 1 S E 2 S E 3 , where regions E 1 , E 2 , and E 3 are deﬁned as f ollows. Region E 1 is the unio n of all rate pairs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aβ P 2  , (240) R 2 ≤ γ ( β P 2 ) + γ  a (1 − β ) P 2 1 + P 1 + aβ P 2  . (241) for all β ∈ [0 , b − 1 (1 − ab ) P 2 ] . Region E 2 is the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aβ P 2  , (242) R 2 ≤ γ  a (1 − β ) P 2 1 + P 1 + aβ P 2  + γ ( β P 2 + bP 1 ) − γ  P 1 1 + aβ P 2  . (243) for all β ∈ [ b − 1 (1 − ab ) P 2 , 1 ] . Region E 3 is the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ γ  P 1 1 + aP 2  , (244) R 2 ≤ γ ( P 2 ) , (245) R 1 + R 2 ≤ γ ( bP 1 + P 2 ) . (24 6) Remark 5: Region E 3 in Case II an d Case I II r epresents a facet tha t b elongs to the capacity region of the mixed Gaussian IC. It is impor tant to n ote th at, surprisingly , this facet is ob tainable when the second transmitter uses b oth the common message and the pr iv ate message. Different bou nds ar e comp ared for the mixed Gaussian I C for Cases I, II, and III in Figur es 1 4, 15, a nd 16, r espectiv ely . 33 Fig. 14. Compari son between differe nt bounds for the mixe d Gaussian IC when 1 + P 2 ≤ b + abP 2 (Case I) for P 1 = 7 , P 2 = 7 , a = 0 . 6 , and b = 2 . Fig. 15. Comparison between dif feren t bounds for the mixed Gaussian IC when 1 + P 2 > b + abP 2 and 1 − a ≤ abP 1 (Case II) for P 1 = 7 , P 2 = 7 , a = 0 . 4 , and b = 1 . 5 . V I I . C O N C L U S I O N W e have studied the capacity region o f the two-u ser G aussian IC. Th e sum capacities, in ner bou nds, an d outer bou nds have been con sidered for thr ee classes of chann els: weak , one-sided , and mixed Gaussian I C. W e h av e used admissible ch annels as the main tool for deriving outer b ounds on the capacity region s. For the weak Gau ssian IC, we h av e derived th e sum cap acity fo r a certain range of channe l param eters. In this rang e, the sum cap acity is attained whe n Gaussian c odeboo ks are used an d interfere nce is treated as noise. Mor eover , we have deriv ed a new o uter bou nd on the c apacity region. This outer boun d is tighter th an the Kramer’ s bou nd a nd the E TW’ s bound . Regardin g inner boun ds, we h av e redu ced the com putational comp lexity o f the HK ac hiev able region. In fact, we have shown th at when Gaussian co debook s are used, the fu ll HK achievable region ca n be obtain ed by using the naive HK achievable scheme over three freq uency band s. For the one-sided Gau ssian IC, we have presented an altern ati ve pr oof for the Sato’ s outer bou nd. W e have also d eriv ed the full HK ach iev able region when Gau ssian co deboo ks are u sed. For the mixed Gau ssian IC, we have derived the sum capacity f or the en tire r ange o f its par ameters. Mo reover , we have presented a new outer b ound on the cap acity region that outper forms ETW’ s boun d. W e h av e proved that th e full H K achievable region using Gaussian co deboo ks is equiv alent to that of the on e-sided Gaussian IC f or a par ticular range o f chan nel gains. 34 Fig. 16. Comparison between dif ferent bounds for the mixed Gaussian IC when 1 + P 2 > b + abP 2 and 1 − a > abP 1 (Case III) for P 1 = 7 , P 2 = 700 , a = 0 . 01 , and b = 1 . 5 . W e h av e also derived a facet that belo ngs to the capacity region for a certain range o f parameter s. Surp risingly , this facet is obtainable when one of th e transmitters u ses both the commo n message and the p riv ate message. R E F E R E N C E S [1] C. E. Shannon , “T wo-w ay communicat ion channe ls, ” in Pro c. 4th Berke le y Symp. on Mathematical Stati stics and P r obabi lity , vol. 1, 1961, pp. 611–644 . [2] A. B. Carleial, “ A case where interf erence does not reduce capacity , ” IE EE T rans. Inform. Theory , vol. IT -21, pp. 569–570, Sept. 1975. [3] H. Sato, “The capaci ty of the gaussian interfer ence channel under strong interference , ” IEEE Tr ans. Inform. Theory , vol. IT -27, pp. 786–788, Nov . 1981. [4] R. Ahlswede, “Multi-w ay communnicati on channel s, ” in Pr oc. 2nd Internationa l Symp. on Information theory , U. Tsahkadsor , Armenia, Ed., Sep 2-8 1971, pp. 23–52. [5] I. Csisz ´ ar and J. 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Capacity Bounds for the Gaussian Interference Channel

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