The pre-log of Gaussian broadcast with feedback can be two

The pre-log of Gaussian broadcast wi th feedback can be tw o Mich ` ele A. W igger Signal and Infor mation Processing Laboratory ETH Z ¨ urich CH-8092 Z ¨ urich , Switzerland Email: wigger @isi.ee.ethz.ch Michael Gastpar W iFo, Department of EECS University of Californ ia, Berkeley , Berkeley , CA 9472 0-177 0, USA Email: gastpar@berkeley .edu Abstract — A generic intuition says that the pre-log, or multi- plexing gain, cannot be larger than the mini mum of the number of transmit and recei ve dimensions. This suggests that for the scalar broadcast channel, the pre-log cannot exceed one. By contrast, in this n ote, we show that when the n oises are anti- correla ted and feedback is pre sent, then a pre-log of two can be attained. In other words, in thi s special case, in the l imit of h igh SN R, the scalar Gaussian broadca st channel turns into two parallel A WGN channels. Ach iev ability is establish ed via a coding strategy d ue to Sch alkwijk, Kailath, and Ozar ow . I . I N T RO D U C T I O N The significance of fe edback in a cap acity sense has been thorou ghly stu died for point-to -point an d se veral network scenarios. Many results point to the lack of such a signifi- cance, starting with Shannon ’ s pro of that the capacity o f a memory less ch annel is un changed by feedb ack. For n etworks, ev en for memoryless o nes, feed back can increase c apacity , as first shown b y Gaard er and W o lf [1]. Howe ver, in most cases, the in crease in capacity d ue to feedb ack remains m odest, as expressed for example in a general conjecture in [2]. The exact feedb ack capacity remain s unk nown for mo st networks, with the notab le exceptio n of the two-user Gaussian multiple-acce ss channel (MA C), whose capacity was f ound by Ozarow [3]. Some recent prog ress concerns th e M - user Gaussian MAC [4]. Ag ain, these results emp hasize the lack of significance of feedb ack in a capacity sense. By con trast, the resu lt presen ted in this short no te shows that feedback can have a rath er sign ificant impact on capacity in a certain br oadca st setting. Mo re specifically , we consider the p roblem of two-u ser broadcast subject to additive wh ite Gaussian noise. This scenario has b een studied previously by Ozarow [5], Ozarow and Leung [ 6], as we ll as Wil lems and van d er Meulen [7]. The main result of this paper is that fo r th e special case where the two no ises are ( fully) a nti-correlated, in the limit as the av ailable power becomes large, the trade-off between t he two broadca st clien ts vanishes, and each clien t attains a rate as if the oth er did not exist. Forma lly , the result is presented as a “pre-lo g, ” or “multip lexing gain. ” T o o ur knowledge the consider ed setting is the fir st examp le of a channel where th e “pr e-log” is larger than the numb er of transmit antenna s. T his behavior is surprising in v iew of the result by T elatar [8] who showed that f or uncorrelated noise sequences, th e “pre-log” is upper b ounded by the number o f transmit anten nas and b y th e nu mber of rec ei ve antennas, even if th e two recei vers a re allowed coo perate. Ther efore, in th e setting at h and when the noise sequences are uncorrelated th e “pre-log ” is up per bounded by one. An extension of our result c oncerns the two-user Gaussian interferen ce chann el where all channel gains are equal. A gain, we can sho w that when feedback is a vailable and the noises are (fully) anti- correlated, th e ”pre-lo g” is tw o. W ithout f eedback the ”pre-log ” is on e, irrespective of the noise correlatio n. The only situation known to date where a ”pre-log” of two is achiev ab le for a two-user Gau ssian interfer ence chan nel (only for no n-equa l channel gains) is when both transmitters k now the other transmitter’ s message, which co rrespond s to a setting where the two transmitters can fully co operate. If only one of the two transmitters kn ows the o ther transmitter’ s message the ”pre-log” r emains one [9]. He nce, th is specific form of limited transmitter -coope ration does not increase the “pre- log”. Our result h ere sh ows that in gen eral limited transmitter- cooper ation can b e sufficient to incr ease th e ”pre- log” to two, e.g., when the limited tran smitter-cooperation is established throug h full cau sal feedb ack links. For interferen ce networks with m ore th an two users the fact th at limited tran smitter- cooper ation ca n in crease the “pre-log ” h as be en observed in [10] for the case where som e of th e transmitters know some of the other transmitters’ messages. One mo ti vation for the study of anti- correlated noises is that th e signals Z 1 and Z 2 in Figure 1 are d ue to one a nd the same outside interfer er , but appear w ith dif f erent (mo re precisely , opposite) phase shifts at the two receiv ers. I I . T H E M O D E L The com munication system stu died in this n ote is illustrated in Figure 1. For a given time- t channe l inpu t x t the channel outputs observed at receiv ers 1 and 2 are Y k,t = x t + Z k,t , k ∈ { 1 , 2 } , (1) where the seque nce of pairs of rand om v ariables { ( Z 1 ,t , Z 2 ,t ) } is drawn indepen dently an d identically distributed (iid) fo r a ✲ M 1 , M 2 E N C X ✲ ✲ Y 1 ✐ ❄ Z 1 ✛ D ❄ ✲ ✲ Y 2 ✐ ❄ Z 2 ✛ D ✻ D E C 1 ✲ ˆ M 1 D E C 2 ✲ ˆ M 2 Fig. 1. The tw o-user A WGN broadca s t channe l with full cau sal output feedbac k. normal distribution with zero mean and covariance matrix K =  σ 2 1 ρ z σ 1 σ 2 ρ z σ 1 σ 2 σ 2 2  (2) for σ 1 , σ 2 > 0 and − 1 ≤ ρ z ≤ 1 . The goal o f the transmission is to con vey m essage M 1 to Receiv er 1 and an indep endent message M 2 to Re- ceiv er 2, where M 1 is unifo rmly distrib u ted over th e set { 1 , . . . , ⌊ 2 nR 1 ⌋} and M 2 is uniformly distributed over th e set { 1 , . . . , ⌊ 2 nR 2 ⌋} , n being the blo ck-length and R 1 and R 2 the respective rates of transmission. Having acce ss to p erfect feedba ck the encode r can produce its time- t ch annel inputs not only as a function of the messages M 1 and M 2 but also based on th e previous chann el ou tputs. Thus a block-len gth n encodin g scheme consists of n function s f ( n ) t , for t = 1 , . . . , n , suc h that X t = f ( n ) t  M 1 , M 2 , Y t − 1 1 , Y t − 1 2  where Y t − 1 1 , ( Y 1 , 1 , . . . , Y 1 ,t − 1 ) and Y t − 1 2 , ( Y 2 , 1 , . . . , Y 2 ,t − 1 ) . W e impose an a verage block-power constraint P > 0 on the sequen ce of channel inputs: 1 n E " n X t =1 X 2 t # ≤ P . (3) Based on the observed sequ ence o f ch annel o utputs Y n 1 and Y n 2 , respectively , the two receivers per form the fo llowing guess of their corr esponding m essage: ˆ M k = φ ( n ) k ( Y n k ) , k ∈ { 1 , 2 } (4) for some decod ing functions φ ( n ) k for k ∈ { 1 , 2 } . An error occurs in the communication whenev er ( M 1 , M 2 ) 6 = ( ˆ M 1 , ˆ M 2 ) . W e say that a r ate pair ( R 1 , R 2 ) is achiev ab le if for every block-len gth n there exist encoding function s { f ( n ) 1 , . . . , f ( n ) n } satisfyin g (3) an d two d ecoding function s φ ( n ) 1 and φ ( n ) 2 such that lim n →∞ Pr h ( M 1 , M 2 ) 6 = ( ˆ M 1 , ˆ M 2 ) i = 0 . Of par ticular interest to th is note is the sum-rate ca pacity C ( P, σ 2 1 , σ 2 2 , ρ z ) , namely , the supremum of R 1 + R 2 for which reliable commu nication is feasible, i.e., where the pair ( R 1 , R 2 ) is achievable. I I I . T H E M A I N R E S U LT The main result of this note con cerns the so-called “pre- log”, defined as follows. Definition 1: Letting the sum- rate capacity be given b y C ( P, σ 2 1 , σ 2 2 , ρ z ) , its corre sponding p re-log is defined as κ = lim P →∞ C ( P, σ 2 1 , σ 2 2 , ρ z ) 1 2 log 2 (1 + P ) . (5) In the context of fading commun ication chann els, the pr e-log is often referr ed to as the multiplexing gain . W e start by noting that a pre-log of one is tr i vially attainable. Moreover , fr om the fact that for a bro adcast channel witho ut feedback , the capacity region only depends on the conditiona l marginals ( see e.g. [11, p.599 ]), we h av e: Lemma 1: For the two-user A WGN broadcast channel wit h- out feed back, the pr e-log is 1 ir respective of th e no ise corre- lation ρ z . Also, by merely merging the two deco ders into a single decoder, thus turnin g th e problem into a point-to-po int com- munication system, we find: Lemma 2: For the two-u ser A WGN bro adcast ch annel with full (causal) feedback, if th e noise correlation satisfies | ρ z | < 1 , then the pre-lo g is 1 . The main result of this no te is the following: Theor em 1: For the two-u ser A WGN broadcast channe l with full (causal) feedback, if the noise correlation is ρ z = − 1 , then the pre-log is two. The conv erse follows trivially by observ ing that with or without feedbac k, the following simp le “single-user” upper bound s ho ld: R k ≤ 1 2 log 2  1 + P σ 2 1  , k ∈ { 1 , 2 } . (6) Thus, the pre-log canno t exceed two. The some what more interesting part of the theorem concerns the achiev ability . T he proof is g i ven in Appen dix A and is based on a strategy by Ozarow [5] , [6] (see Section V). I V . S O M E E X T E N S I O N S A. Limited F e edback In the broadc ast setting it can be shown that ev en if on ly one of th e two ch annel outpu ts ar e fed back, a pr e-log of two is attainable for the case of fu lly anti-correlate d noises. This follows direc tly by noting that in this case one can compute one of th e chann el outputs based on the cha nnel input and on the other chann el ou tput. B. Mor e than T wo Receivers Consider a real scalar A WGN broadcast chan nel with K > 2 receiv ers. It can be shown th at for more than two receivers Lemma 2 and Theorem 1 do not scale with the number o f receivers, i.e., the maxim um pre-log remains two irrespective of K . Mo re precisely , let { Z k,t } deno te the noise seq uence corrup ting the outpu ts of Receiver k , for k ∈ { 1 , . . . , K } . Then, extend ing Lem ma 2 and Th eorem 1 to K > 2 r eceiv ers, the following two results can be der i ved: If for any k , k ′ ∈ ✲ M 1 ✲ M 2 E N C 1 E N C 2 ✲ ❆ ❆ ❆ ❆ ❯ X 1 ✲ Y 1 ✐ ❄ Z 1 ✛ D ❄ ✲ ✁ ✁ ✁ ✁ ✕ X 2 ✲ Y 2 ✐ ❄ Z 2 ✛ D ✻ D E C 1 ✲ ˆ M 1 D E C 2 ✲ ˆ M 2 Fig. 2. The interfere nce channel with ipsilater al causal output feedba ck. { 1 , . . . , K } with k 6 = k ′ the sequences { Z k,t } and { Z k ′ ,t } are not perfectly po siti vely correlated or an ti-correlated , then the pre-log eq uals o ne; if for any k , k ′ ∈ { 1 , . . . , K } with k 6 = k ′ the sequences { Z k,t } a nd { Z k ′ ,t } are n ot perfectly c orrelated and if additionally there exist k 1 , k 2 ∈ { 1 , . . . , K } such that { Z k 1 ,t } an d { Z k 2 ,t } ar e per fectly anti-co rrelated, then the pre- log equals two. C. Interference Chann el An extension of our result c oncerns the two-user Gaussian interfer enc e channel, see Fig ure 2. Th e main difference to the previously considered br oadcast setting is that here two transmitters wish to com municate. The goal of the tra nsmission is that Transmitter 1 c on veys message M 1 to Receiver 1 and Transmitter 2 con veys Message M 2 to Recei ver 2, where M 1 and M 2 are defined as befo re. W e assume that all links are of unit-gain (ev en though our results require only all equal gain s), and hence the ch annel is described as follows. For given time- t channel inputs x 1 ,t at T r ansmitter 1 and x 2 ,t at Transmitter 2 the ch annel outp uts a t the two receiving termin als are giv e n b y Y k,t = x 1 ,t + x 2 ,t + Z k,t , k ∈ { 1 , 2 } , (7) where the noise sequ ences { ( Z 1 ,t , Z 2 ,t ) } are as in Section II. Having ac cess to f ull causal output f eedback of their re- spectiv e chann el outputs, th e two encoders can pro duce their time- t channel inpu ts as X k,t = f ( n ) k,t  M k , Y t − 1 k  , k ∈ { 1 , 2 } . for some sequences of encoding functio ns f ( n ) 1 ,t and f ( n ) 2 ,t , for t = 1 , . . . , n . As in the br oadcast s etting we impo se an a verage block-p ower constrain t P > 0 on the sequences of channel inputs: 1 n E " n X t =1 X 2 k,t # ≤ P , k ∈ { 1 , 2 } . (8) The notio n of dec oding f unctions, pro bability of erro r , achiev- able rate pairs, and sum-rate capacity are in an alogy to Section II. The main result in this section is that Lemma 2 and Theo- rem 1 can be extend ed also to two-user Ga ussian interfer ence channels with all unit-g ains. Lemma 3: For the two-user A WGN interfe rence chan nel with all u nit chan nel gains and with full (causal) feedb ack to both transmitters, if th e noise c orrelation satisfies | ρ z | < 1 , then the pre-log is 1 . Theor em 2: For the two-user A WGN interf erence chan nel with all unit channel gains and with full (causal) feed back to both tran smitters, if the noise c orrelation is ρ z = − 1 , then the pre-log is two. The con verse fo llows simply by T heorem 1 beca use letting the two tra nsmitters cooperate can only increase capacity . The achiev a bility is b ased o n the following obser vations: In the broadc ast strategy leadin g to Theorem 1 (see Sec tion V) the single transmitter sends a weig hted sum of the current estimation errors at the tw o receivers. In our interf erence setting due to the feedback links Transmitter 1 c an comp ute the estimation error o f Recei ver 1 and T ransmitter 2 can compute th e estimatio n e rror o f Recei ver 2. Therefor e, the two transmitter s can mimic the single- transmitter strategy in Section V by each sending a scaled version of the cor respond - ing estimation error beca use the chan nel implicitly add s up the two inputs. Hence, we can con clude that any rate pair achiev ed by the strategy in Section V for the broadcast ch annel is also achiev ab le fo r the interference c hannel. Note howe ver th at it requires unit gain (or equal gain) on all chann el lin ks. V . E N C O D I N G S C H E M E A N D A NA LY S I S The scheme we propose to prove th e achiev ability of pre-log 2 in The orem 1 follows alo ng the lines of the scheme in [5] , [6]. Ou r main contribution lies in the choice of the parameter γ an d the asymptotic analysis. Just fo r completen ess we give a short descrip tion of the scheme fo llowed by a more detailed analysis of p erforma nce. Prior to tr ansmission, the enco der maps bo th messages with a on e-to-one mappin g into m essage po ints θ 1 and θ 2 in (1 / 2 , 1 / 2] . More precisely , θ ν = 1 / 2 − M ν − 1 ⌊ 2 nR ν ⌋ , ν ∈ { 1 , 2 } . T o start, the first channel use is dedica ted to user 1 on ly , and the en coder transmits q P V ar ( θ 1 ) θ 1 . The second ch annel use is dedicated to user 2 on ly , and the en coder tran smits q P V ar ( θ 2 ) θ 2 . Ther eafter, each u ser forms an estimate o f its message point, namely ˆ θ 1 , 1 = q V ar ( θ 1 ) P Y 1 , 1 and ˆ θ 2 , 2 = q V ar ( θ 2 ) P Z 2 , 2 respectively , incurring errors of ǫ 1 , 2 = r V ar ( θ 1 ) P Z 1 , 1 and ǫ 2 , 2 = r V ar ( θ 2 ) P Z 2 , 2 . ( 9) In subsequen t iterations, the enco der transmits a linear co m- bination of the cur rent receivers’ estimation errors on θ 1 and θ 2 , respectively . Th us, at time k the chann el in put is X k = s P 1 + γ 2 + 2 γ | ρ k − 1 | (10) ·  ǫ 1 ,k − 1 √ α 1 ,k − 1 + γ sign ( ρ k − 1 ) ǫ 2 ,k − 1 √ α 2 ,k − 1  (11) where ǫ 1 ,k − 1 and ǫ 2 ,k − 1 denote the receiver’ s estimation error of θ 1 and θ 2 after the observation o f the ( k − 1 ) -th channel outp ut; where α 1 ,k − 1 and α 2 ,k − 1 denote th e variances of the estimation error s and ρ k − 1 denotes th eir correlatio n coefficient; wh ere sign ( · ) den otes the signum fun ction, i.e., sign ( x ) = 1 if x ≥ 0 and sign ( x ) = − 1 oth erwise; and where we choose (possibly sub-o ptimally) γ = σ 1 σ 2 . (12) After the reception of each k -th channel output each receiv er perfor ms a linear minimum mean squ are estimation (LMMSE ) to estimate the respecti ve error ǫ 1 ,k − 1 and ǫ 2 ,k − 1 , and based on it they update their estimate of the respective message po int. At the end o f each b lock of n chan nel uses, each decoder guesses tha t the m essage has b een transmitted which corre- sponds to the message poin t closest to its final estimate. A. Analysis A detailed analysis of perfo rmance can be found in [5], [ 6]. Here we pr esent the most im portant q uantities o f the ana lysis: the variances o f the estimation error s at time- k α 1 ,k = α 1 ,k − 1 σ 2 1 σ 2 2 P (1 − ρ 2 k − 1 ) + ( σ 2 1 + σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | ) (1 + σ 2 1 σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | )( P + σ 2 1 ) and α 2 ,k = α 2 ,k − 1 P (1 − ρ 2 k − 1 ) + ( σ 2 1 + σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | ) (1 + σ 2 1 σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | )( P + σ 2 2 ) , and th e corre lation coefficient, see (13) on top of th e next page. Note that Recursion (13) ha s at least one “fix poin t” ρ ∗ in the in terval [0 , 1 ] in the sense that if ρ 2 = ρ then the sequence { ρ k } alternates in sign but is constant in mag nitude. This can seen b y noting that fo r ρ k − 1 = 0 it follows th at | ρ k | > 0 an d fo r | ρ k − 1 | = 1 it follows that | ρ k | < 1 , an d thus by the con tinuity of the recursion there must exist a “fix point” ρ ∈ [0 , 1] . By a slight mo dification of the scheme as suggested in [3] one can en sure that ρ 2 equals the “fix point” ρ and one can show that any no n-negative rate pair ( R 1 , R 2 ) is achiev able if it satisfies R 1 < 1 2 log 2 P + σ 2 1 P 2 (1 − ρ ) + σ 2 1 ! (14) R 2 < 1 2 log 2 P + σ 2 2 P 2 (1 − ρ ) + σ 2 2 ! (15) where ρ is a solution in [0 , 1] of ρ 3 + aρ 2 + bρ + c = 0 (16) where a = − 2 σ 1 σ 2 P − P + σ 2 1 + σ 2 2 + ρ z σ 1 σ 2 p P + σ 2 1 p P + σ 2 2 − 2 σ 2 1 σ 2 2 P p P + σ 2 1 p P + σ 2 2 , (17) b = − 1 − σ 2 1 + σ 2 2 P − ρ z ( σ 2 1 + σ 2 2 ) p P + σ 2 1 p P + σ 2 2 − σ 1 σ 2 ( σ 2 1 + σ 2 2 ) P p P + σ 2 1 p P + σ 2 2 , (18) c = P + σ 2 1 + σ 2 2 − ρ z σ 1 σ 2 p P + σ 2 1 p P + σ 2 2 . (19) A P P E N D I X A. Pr oof of The or em 1 In this section we p rove that for ρ z = − 1 the sche me described in Section V achieves a pre- log of 2. The proof of the theorem follows directly by the achiev- ability of rate pairs ( R 1 , R 2 ) satisfying (1 4) and ( 15) an d th e following lem ma. Lemma 4: For ρ z = − 1 the fun ction ρ ( P ) implicitly defined by solutions in [0,1 ] to (16) satisfies lim P →∞ P 1 − δ (1 − ρ ( P )) = 0 , ∀ δ > 0 . (20) Pr oof: Note first that the fu nction ρ ( P ) must satisfy lim P →∞ ρ ( P ) = 1 . (21) This follows by the continu ity o f the coefficients a , b , and c in P , and by observ ing that for large P E quation (16) tends to ρ 3 − ρ 2 − ρ + 1 = 0 fo r wh ich the o nly solu tions ar e − 1 and +1 . Next, define the function g ( P ) ∈ [0 , 1] as g ( P ) , 1 − ρ ( P ) . (22) By (22) and (16) the fun ction g ( P ) mu st satisfy 0 = 1 + a + b + c + g ( P )( − 3 − 2 a − b ) + ( g ( P )) 2 (3 + a ) − ( g ( P )) 3 , (23) or equiv a lently by (1 7)–(19), 0 = − ( g ( P )) 3 + Λ 2 ( P )( g ( P )) 2 + Λ 1 ( P ) g ( P ) + Λ 0 ( P ) (2 4) where Λ 2 ( P ) = 3 − 2 σ 1 σ 2 P − P + σ 2 1 + σ 2 2 + ρ z σ 1 σ 2 p P + σ 2 1 p P + σ 2 2 − 2 σ 2 1 σ 2 2 P p P + σ 2 1 p P + σ 2 2 , Λ 1 ( P ) = − 2 1 − P p P + σ 2 1 p P + σ 2 2 ! + (2 + ρ z ) σ 2 1 + (2 + ρ z ) σ 2 2 + 2 ρ z σ 1 σ 2 p P + σ 2 1 p P + σ 2 2 + σ 2 1 + σ 2 2 + 4 σ 1 σ 2 P + σ 1 σ 2 ( σ 2 1 + 4 σ 1 σ 2 + σ 2 2 ) P p P + σ 2 1 p P + σ 2 2 , Λ 0 ( P ) = − σ 2 1 + 2 σ 1 σ 2 + σ 2 2 P 1 + ρ z P p P + σ 2 1 p P + σ 2 2 ! − σ 1 σ 2 ( σ 2 1 + 2 σ 1 σ 2 + σ 2 2 ) P p P + σ 2 1 p P + σ 2 2 . ρ k = p P + σ 2 1 p P + σ 2 2 P (1 − ρ 2 k − 1 ) + ( σ 2 1 + σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | ) · 1 σ 1 σ 2 ·  ρ k − 1 ( σ 2 1 + σ 2 2 + 2 σ 1 σ 2 | ρ k − 1 | ) − ( σ 1 + σ 2 | ρ k − 1 | )( σ 2 + σ 1 | ρ k − 1 | ) sign ( ρ k − 1 ) P ( P + σ 2 1 + σ 2 2 − ρ z σ 1 σ 2 ) ( P + σ 2 1 )( P + σ 2 2 )  (13) In the remain ing we will prove that lim P →∞ P 1 − δ g ( P ) = 0 , ∀ δ > 0 , (25) which by (22) establishes Lem ma 4 and thu s also concludes the proo f of Theorem 1. Start the proof by noting th at lim P →∞ P 1 − δ g ( P ) ≥ 0 , ∀ δ > 0 , follows trivially since g ( P ) ≥ 0 . Thus, we are left with proving lim P →∞ P 1 − δ g ( P ) ≤ 0 , ∀ δ > 0 , (26) which we shall prove by contrad iction. More precisely , we will show that if there exists a δ > 0 such that g ( P ) satisfies lim P →∞ P 1 − δ g ( P ) > 0 , (27) then Cond ition (24) on the f unction g ( P ) is violated . T o this end, assume tha t there exists a δ > 0 satisfying (27). Then, define δ ∗ , sup n δ : lim P →∞ P 1 − δ g ( P ) > 0 o , (28) and note that by assump tion, δ ∗ > 0 . Also, by (2 1) and (22) lim P →∞ g ( P ) = 0 , (29) and hence δ ∗ ≤ 1 . Next, choo se 0 < ǫ < δ ∗ and consider the asymptotic expression ∆ , lim P →∞ P 2 − δ ∗ − ǫ  − ( g ( P )) 3 + Λ 2 ( P )( g ( P )) 2 +Λ 1 ( P ) g ( P ) + Λ 0 ( P )) . (3 0) W e shall an alyze th e lim iting expr ession (30) and show that it tends to a no n-zero value. Bu t th is violates Condition (2 4) and therefor e leads to the desired contradiction . In the analysis of expression (30) we shall separately co nsider the sum of th e first two summan ds, the third summan d, and forth summ and. W e start with the sum of th e first two summand s. Note first that lim P →∞ Λ 2 ( P ) = 2 . (31) Next, note that since 1 − δ ∗ + ǫ 2 > 1 − δ ∗ by (28) lim P →∞  P 1 − δ ∗ + ǫ 2 g ( P )  2 > 0 , ( 32) Therefo re, also usin g (29) we can conclude that lim P →∞  P 1 − δ ∗ + ǫ 2 g ( P )  2 (Λ 2 ( P ) − g ( P )) > 0 . (33) In order to analyze the third summ and we notice that lim P →∞ P 1 − ǫ/ 2 Λ 1 ( P ) = 0 , (34) where we used that by Bernou lli-de l’H ˆ o pital’ s rule: lim P →∞ P 1 − P p P + σ 2 1 p P + σ 2 2 ! = σ 2 1 + σ 2 2 2 . (3 5) Hence, by (34) and becau se 1 − δ ∗ − ǫ/ 2 < 1 − δ ∗ , lim P →∞ P 2 − δ ∗ − ǫ Λ 1 ( P ) g ( P ) = 0 . (36) Finally , for the last summ and o ne can show that for ρ z = − 1 lim P →∞ P 2 − δ ∗ − ǫ Λ 0 ( P ) = 0 , (37) which is again based on the limiting expression (35). Thus, by (33), ( 36), an d (37) we ob tain that ∆ > 0 wh ich contradicts Condition (2 4). This conclud es the proo f bo th of Lemma 4 and Theor em 1. A C K N O W L E D G M E N T The au thors than k Prof. Frans M. J. W illems, TU Ein d- hoven, for pointing them to [ 7], which insp ired the inves tiga- tion leading to this work. R E F E R E N C E S [1] N. T . Gaarder and J. K. W olf, “The capac ity regi on of a multiple- access discrete memoryless channel can increase with feedback , ” IEEE T ransacti ons on Information Theory , vol. IT –21, pp. 100–102, January 1975. [2] T . M. Co ver and B. Gopina th, Open pr oblems in communicat ion and computat ion , Springer V erlag, New Y ork, 1987. [3] L. H. 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