Symmetric Two-User Gaussian Interference Channel with Common Messages

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1 Symmetric Two-User Gaussia Iterferece Chael with Commo Messages Qua Geg CSL ad Dept. of ECE UIUC, IL Tie Liu Dept. of Electrical ad Computer Egieerig Texas A&M Uiversity, TX arxiv:40.509v [cs.it] 0 Ja 04 Abstract We cosider symmetric two-user Gaussia iterferece chael with commo messages. We derive a upper boud o the sum capacity, ad show that the upper boud is tight i the low iterferece regime, where the optimal trasmissio scheme is to sed o commo messages ad each receiver treats iterferece as oise. Our result shows that although the availability of commo messages provides a cooperatio opportuity for trasmitters, i the low iterferece regime the presece of commo messages does ot help icrease the sum capacity. I. INTRODUCTION Iterferece chael is a fudametally importat commuicatio model i iformatio theory []. While the exact capacity regio of iterferece chael i the simplest settig with two trasmitter-receiver pairs is still ukow i geeral, recet research efforts have sigificatly improve our uderstadig of the capacity regio. I particular, [] characterizes the capacity regio of two-user Gaussia iterferece chael withi oe bit. The exact capacity regio has also bee derived i certai regimes, e.g. the strog iterferece regime [3], ad the low iterferece regime [4] [6], which show that i the low iterferece regime treatig iterferece as oise is optimal ad achieves the sum capacity. I this paper, we cosider the symmetric two-user Gaussia iterferece chael with commo messages, where each trasmitter wats to sed a private message to its correspodig receiver ad both trasmitters also ited to sed a commo message to both receivers. We derive a upper boud o the sum capacity usig a geie-aided method [], [4], ad show that the upper boud is tight i the low iterferece regime, where the optimal trasmissio scheme is to sed o commo messages ad each receiver treats iterferece as oise. Our result shows that although the availability of commo messages provides a cooperatio opportuity for trasmitters, i the low iterferece regime the presece of commo messages does ot help icrease the sum capacity. A. Orgaizatio This paper is orgaized as follows. We describe the chael model i Sectio II, ad derive a upper boud o the sum capacity i Sectio III. I Sectio IV, we give a atural lower boud o the sum capacity, ad show that the upper boud matches the lower boud i certai low iterferece regime. I Sectio V, we prove that i the low iterferece regime the availability of the commo messages does ot help icrease the sum capacity ad thus treatig iterferece as oise is optimal. Sectio VI cocludes this paper. II. SYSTEM MODEL We cosider a symmetric two-user Gaussia iterferece chael with commo messages. The chael iput-output relatio is give by Y = X + cx + Z Y = X + cx + Z where X i is the sigal set by the ith trasmitter, ad Y i is the sigal received by the ith receiver, Z k is N 0, for k =,, ad E[Xk ] P for k =,. Without loss of geerality, we assume that c 0. There are a set of three idepedet messages W 0, W, W, where W 0 is available at both trasmitters ad iteded for both receivers, W is available at trasmitter oly ad iteded for receiver oly, ad W is available at trasmitter oly ad iteded for receiver oly. We use R i to deote the trasmissio rate for messages W i, for i = 0,,.

2 Fig.. Chael Model III. UPPER BOUND FOR THE SUM CAPACITY I this sectio, we use the geie-added techique to derive a upper boud o the sum rates. Our mai result is give i Theorem. Theorem. The sum rate R 0 + R + R ca be bouded from above as where gp, P is defied as ad gp, P AP, P := R 0 + R + R max gp, P, 3 0 P =P P mi a,a,var Z,Var Z AP,P f P, P, a, Var Z, a, Var Z, 4 { a, a, Var Z, Var Z 0 Var Z a a Var Z Var Z Var Z c P 0 Var Z a a Var Z Var Z Var Z c P 5 ad f P, P, a, Var Z, a, Var Z := 4 log P + c P + c P P P P + c P + c P + P + c P + c P + Var Z cp + a Var Z log c P + a Var Z + P + c P + c P + Var Z cp + a Var Z log c P + a Var Z + 6 Proof: We first prove that R 0 + R + R max gp, P. 7 0 P,P P

3 From Fao s iequality, we have for k =,, ay ɛ > 0, ad sufficietly large R 0 ɛ/3 IW 0 ; Yk 8 = hyk hyk W 0 9 = hyk hyk W 0 hyk Xk, W 0 hyk Xk, W 0 0 = hyk IXk ; Yk W 0 hyk Xk, W 0 Also have from Fao s iequality, we have for k =,, ay ɛ > 0, ad sufficietly large R k ɛ/3 IW k ; Yk IW k ; Yk, W 0 3 = IW k ; Yk W 0 4 IXk ; Yk W 0 5 where 4 follows from the idepedece betwee W k ad W 0, ad 5 follows from the fact that give W 0, W k X k Y k forms a Markov chai. The sum rate R 0 + R + R ɛ = R 0 ɛ/3 + R 0 ɛ/3 + R ɛ/3 + R ɛ/3 6 hy IX ; Y W 0 hy X, W 0 + hy IX ; Y W 0 hy X, W 0 + IX ; Y W 0 + IX ; Y W 0 7 = hy + IX ; Y W 0 hy X, W 0 + hy + IX ; Y W 0 hy X, W 0 8 hy + IX ; Y, U W 0 hy X, W 0 + hy + IX ; Y, U W 0 hy X, W 0 9 = hy + IX ; U W 0 + IX ; Y U, W 0 hy X, W 0 + hy + IX ; U W 0 + IX ; Y U, W 0 hy X, W 0 0 = hy + hu W 0 hu X, W 0 + hy U, W 0 hy X, U, W 0 hy X, W 0 + hy + hu W 0 hu X, W 0 + hy U, W 0 hy X, U, W 0 hy X, W 0 = hu X, W 0 hu X, W 0 + hy + hy + hy U, W 0 + hy U, W 0 + hu W 0 hy X, U, W 0 hy X, W 0 + hu W 0 hy X, U, W 0 hy X, W 0 for ay geie sigals U, U. Motivated by the problem of two-user Gaussia iterferece chael without commo iformatio, we shall choose ad Z ki are i.i.d. N 0, Var Z k ad are correlated with the oise sigal Z ki as Z ki = U ki = cx ki + Z ki 3 a k Z ki + N ki 4 Var Z k

4 Let Note that P k := VarX ki W 0 5 hu k X k, W 0 = hcx k + Z k X k, W 0 = h Z k = log πevar Z k 6 Next, we shall boud from above the rest of the terms o the RHS of i terms of P, P, a, Var Z, a ad Var Z. First, let us cosider hy ad hy. We have hy hy i 7 where 9 is due to the cocavity of the log fuctio. The variace where the cross term log πevary i 8 [ ] log πe VarY i 9 VarY i = VarX i + cx i + Z i 30 = VarX i + cx i + VarZ i 3 = VarX i + c VarX i + ce[x i E[X i ]X i E[X i ]] + VarZ i 3 E[X i E[X i ]X i E[X i ]] = E [E [X i E[X i ]X i E[X i ] W 0 ]] 33 = E [E [X i E[X i ] W 0 ] E [X i E[X i ] W 0 ]] 34 = E [E[X i W 0 ] E[X i ] E[X i W 0 ] E[X i ]] 35 [ E E[X i W 0 ] E[X i ] ] [ E E[X i W 0 ] E[X i ] ] 36 where 34 follows from the idepedece of X i ad X i give W 0, ad 36 follows from the Cauchy-Schwartz iequality. Furthermore, [ E E[X ki W 0 ] E[X ki ] ] [ = E E[X ki W 0 ] ] E[X ki ] 37 = E[Xki] E[X ki ] [ E[Xki] E E[X ki W 0 ] ] 38 = VarX ki VarX ki W 0 39 Substitutig 36 ad 39 ito 3, we may obtai VarY i VarX i + c VarX i + c VarXi VarX i W 0 VarX i VarX i W 0 + VarZ 40 VarX i + c VarX i + c VarX i VarX i W 0 VarX i VarX i W 0 + VarZ P + c P + c P P P P + 4 4

5 where 4 follows from the fact that gx, x, y, y = y x y x 43 is joitly cocave for x y, x y. Hece, we have hy [πe log P + c P + c ] P P P P + ad similarly hy [πe log P + c P + c ] P P P P + Next, we cosider hy U, W 0 ad hy U, W 0. We have hy U, W 0 hy i U i, W 0 46 = hx i + cx i + Z i cx i + Z i, W 0 47 [ log πevarx i + cx i + Z i cx i + Z ] i, W 0 log [ πe ] VarX i + cx i + Z i cx i + Z i, W 0 [ log πe VarXi W 0 + c VarX i W 0 + cvarx i W 0 + a Var Z c VarX i W 0 + Var Z log [πe c VarX i W 0 + c VarX i W 0 + a Var Z c VarX i W 0 + Var Z = log πe P + c P + VarX i W 0 + cp + a Var Z c P + Var Z 5 5 where 5 follows from the fact that gx, y := x + c y + is joitly cocave for x 0, y 0. Similarly, we may also obtai that hy U, W 0 log πe P + c P + cx + a Var Z c x + Var Z cp + a Var Z c P + Var Z 53 54

6 Fially, let us cosider hu W 0 hy X, U, W 0 hy X, W 0 ad hu W 0 hy X, U, W 0 hy X, W 0. We have Assumig that by the coditioal etropy-power iequality, we have hu W 0 hy X, U, W 0 hy X, W 0 = hcx + Z W 0 hx + cx + Z X, cx + Z, W 0 hx + cx + Z X, W 0 55 = hcx + Z W 0 hcx + N W 0 hcx + Z W 0 56 Var Z VarN = a 57 hcx + N W 0 log e hcx + Z W0 + πe a Var Z 58 ad hcx + Z W 0 e log hcx + Z W0 + πe Var Z 59 Substitutig 58 ad 59 ito 56, we may obtai where hu W 0 hy X, U, W 0 hy X, W 0 gt gt := t e log t + πe a Var Z log e t + πe Var Z 60 ad t := hcx + Z W 0 6 hcx i + i W 0 6 = log [πe [ log πevarcx i + Z ] i W 0 [ log πe c VarX i W 0 + Var Z ] c ] VarX i W 0 + Var Z = [πe log c P + Var Z ] The derivative g t = πe a Var Z Var Z e 4t e t + πe a Var Z e t + πe Var Z so gt is a mootoe icreasig fuctio for t [πe 4 log a Var Z Var Z ] 67 Assumig that we have from 66 c P a Var Z Var Z Var Z 68 hu W 0 hy X, U, W 0 hy X, W 0 [ g log πe c P + Var Z ] 69 [ log πe c P + Var Z log πe c P + a log πe c P + ] 70

7 Similarly, assumig that ad Var Z a 7 c P a Var Z Var Z Var Z 7 we have hu W 0 hy X, U, W 0 hy X, W 0 [ log πe c P + Var Z log πe c P + a log πe c P + ]. 73 Therefore, we have R 0 + R + R max gp, P P,P P Next we argue that we oly eed to cosider the case whe P = P. Recall that P k k =, is defied as P k := VarX ki W We show that i this symmetric model, give ay trasmissio scheme, oe ca easily costruct aother trasmissio scheme achievig the same sum rate with P = P. Ideed, suppose i the give trasmissio scheme P P. We costruct aother trasmissio scheme as follows: I the first time block, we use the same code book of the give trasmissio scheme. I the secod time block, sice the chael is symmetric, we ca switch the roles of user ad user ad use the same trasmissio scheme achievig the same sum rate. Hece the ew trasmissio scheme achieves the same sum rate with P = P = P+P. Therefore, we have This completes the proof of Theorem. R 0 + R + R max gp, P P =P P IV. TIGHTNESS OF UPPER BOUND IN THE LOW INTERFERENCE REGIME I this sectio, we first give a lower boud o the sum capacity, ad the show that the upper boud give i Theorem matches the lower boud i the low iterferece regime. A simple codig scheme is that each trasmitter splits the power P ito two parts, oe for commo message M 0 ad oe for the privacy message, ad does chael codig for each message idepedetly, ad each receiver decodes the iteded messages by usig successive iterferece cacellatio. The trasmissio sum rates of this superpositio codig scheme are a atural lower boud for the sum capacity. Lemma. Give P = P = P P 0, the maximum sum rates achieved by the above superpositio codig scheme is where R 0 + R + R log + P + + c P 0 P c + log + P + c P + =IX G, X 0G ; Y G + IX G, Y G X 0G =IX G, X 0G ; Y G + IX G, Y G X 0G, X G = X G + P P X 0G, X G = X G + P P X 0G, Y G = X G + + c P P X 0G + cx G + Z, Y G = X G + + c P P X 0G + cx G + Z, ad X G, X G, X 0G are zero mea Gaussia radom variables with variaces P, P ad. 77

8 Proof: Gaussia radom variables X G, X G ad X 0G correspod to the codebooks for messages W, W ad W 0 i the superpositio codig scheme. The chael iput-output relatio is Y G = X G + + c P P X 0G + cx G + Z Y G = X G + + c P P X 0G + cx G + Z, Treatig iterferece as oise, we ca write dow the expressios of the MAC capacity regio six iequalities i total ad get the maximum sum rates 77 by usig Fourier-Motzki elimiatio. More specifically, give a, b, c, d, e ad f, from the followig iequalities we ca get a tight upper boud for R 0 + R + R via Fourier-Motzki elimiatio: R 0 a 78 R b 79 R 0 + R c 80 R 0 d 8 R e 8 R 0 + R f 83 R 0 + R + R mi{a + b + e, b + d + e, c + e, b + f}. 84 I our MAC chael, we have the implicit coditios a + b c, d + e f ad by symmetry a = d, b = e, c = f. Thus mi{a + b + e, b + d + e, c + e, b + f} = b + f = c + e. This completes our proof. We show that the lower boud 77 ad the upper boud match for all 0 P = P P if the parameters P ad c satisfy certai coditios. More precisely, Theorem 3. Give P ad c, if there exist oegative parameters a ad b such that the the lower boud 77 ad the outer boud 3 match, i.e., gp, P = for 0 P = P P. c + c P = ab 85 c P a b b b 86 a + b. 87 log + P + + c P 0 c + log + P + P c P + Before provig Theorem 3, ote that by settig a = to be a valid solutio of the above costraits, we ca derive a sufficiet coditio uder which the lower boud 77 ad the outer boud match. Corollary 4. If the parameters P ad c satisfy the followig low iterferece coditios: 88 c 4 P + 4c P + 3c + c P 89 ad c + c P, 90 the the lower boud 77 ad the outer boud match. Proof: The idea is very simple. We fid the regio i which a = is a valid solutio of 85, 86 ad 87. Cosider 85 ad 87 ad use the familiar iequality a + b ab, we have Let m := c + c P, so m. From 85 ad 87 we get c + c P = ab a + b. 9 a + m, 9 a

9 thus I 86, let a =, the we get c equivaletly, 4m P a + 4m. 93 m m m, 94 c 4 P + 4c P + 3c + c P. 95 So if P ad c satisfy 89 ad 90, the a = ad b = m are valid solutios of 85, 86 ad 87. Therefore, the lower boud 77 ad the outer boud match due to Theorem 3. A. Proof of Theorem 3 The followig Lemma 5 is a restatemet of 57, 68, 7, 7 i Theorem, which is the so-called useful geie coditio defied i [4]. Lemma 5 says that if the useful geie coditio is satisfied, the the capacity of geie aided chael is achieved by chael iput with Gaussia distributios. To simplify the otatio, defie b k := Var Z k. Lemma 5. Give the coditioal variaces P ad P, if the b a 96 c P a b b b 97 b a 98 c P a b b b 99 R 0 + R + R ɛ IX G; Y G, U G X 0G + IY G ; X G, X 0G +IX G ; Y G, U G X 0G + IY G ; X G, X 0G, 00 where X kg are the zero mea Gaussia radom variables, ad U kg, Y kg are the correspodig Gaussia geie ad output. More specifically, X G := X G + P P X 0G, 0 X G := X G + P P X 0G, 0 where X G, X G ad X 0G are idepedet zero mea Gaussia radom variables with variace P, P ad, respectively. Ad accordigly, Proof: I the proof of Theorem, 9 is The RHS is exactly Y G = X G + cx G + Z 03 Y G = X G + cx G + Z 04 U G = cx G + Z 05 U G = cx G + Z 06 R 0 + R + R ɛ hy + IX ; Y, U W 0 hy X, W 0 +hy + IX ; Y, U W 0 hy X, W 0. IX ; Y, U W 0 + IY ; X, W 0 + IX ; Y, U W 0 + IY ; X, W 0, 07 ad for each term we have derived a outer boud c.f., 4, 5, 54, 70, 70 i the proof of Theorem. It is easy to verify that the derived boud for each term ca be obtaied by replacig every term i the mutual iformatio ad

10 etropy expressios by the correspodig Gaussia radom variables. I this way, we ca equivaletly write the fuctio fp, P, a, Var Z, a, Var Z as IX G; Y G, U G X 0G + IY G ; X G, X 0G + IX G ; Y G, U G X 0G + IY G ; X G, X 0G. 08 This completes the proof. As stated before, Lemma 5 is just a restatemet of the upper boud i Theorem i terms of mutual iformatio amog Gaussia radom variables. The advatage of doig so is that we ca easily compare the lower boud ad outer boud, ad study uder what coditios they match. The followig lemma deals with the so-called smart geie coditio defied i [4]. Uder this coditio, the geie-aided chael sum capacity is same as the oe achieved by superpositio codig ad successive iterferece cacellatio i the geie-free chael. Lemma 6. Give fixed coditioal variaces P = P, if there exist parameters a ad b satisfyig c + c P = ab 09 ad the useful geie coditios i Lemma 5, the the sum capacity gp, P of the geie aided chael is same as the oe achieved by superpositio codig ad successive iterferece cacellatio i the geie-free chael. Proof: We emphasize that here the ier boud ad outer boud are bouds for specific give P ad P, where 0 P = P P. By Lemma 5 ad Lemma, give P = P, the gap betwee outer boud ad ier boud is IX G; U G X 0G, Y G + IX G ; U G X 0G, Y G. 0 If the gap is zero, i.e., outer boud ad ier boud match, each term must be zero sice mutual iformatio is oegative. Ideed, by symmetry we have IX G ; U G X 0G, Y G = IX G ; U G X 0G, Y G. Recall that X G := X G + P P X 0G, X G := X G + P P X 0G, 3 where X G, X G ad X 0G are idepedet zero mea Gaussia radom variables with variace P, P ad, respectively. By defiig X kg := X kg E[X kg X 0G ] = X kkg, 4 we have IX G ; U G X 0G, Y G = 0 IX G ; cx G + Z X 0G, X G + cx G + Z = 0 I X G ; c X G + Z X 0G, X G + c X G + Z = 0 I X G ; c X G + Z X G + c X G + Z = 0, where i the last step we use the fact that X 0G is idepedet of X kg, i.e., X 0G is idepedet of X kkg. The last coditio is equivalet to the Markov Chai coditio: X G X G + c X G + Z c X G + Z. 5 Sice all the radom variables are Gaussia, by the fact that Gaussia radom variables X Y Z if ad oly if CovX, Z = CovX, Y CovY CovY, Z, 6 we get the smart geie coditio c + c P = ab. 7 So far, we have show that give P = P, uder what coditios ier boud ad outer boud match. The ext step is to show for all P ad P, where 0 P = P P, there exist parameters ap ad bp satisfyig both useful geie ad smart geie coditios, ad this will coclude our proof.

11 More specifically, we wat to show that i some low iterferece regime, for ay P P, there exist oegative parameters a ad b we emphasize here a ad b ca be a fuctio of P satisfyig the followig coditios c + c P = ab 8 c P a b b b 9 a + b. 0 It is easy to see that we oly eed to cosider the case that P = P for the above costraits, sice if there exist a ad b satisfyig the coditios for the case P = P, which are exactly equatios 85, 86 ad 87, the it has solutios for all P P as P decreases, we ca fix a ad decrease the value of b to satisfy all the costraits. This completes the proof of Theorem. V. OPTIMA COMMON MESSAGE RATE IN THE LOW INTERFERENCE REGIME I this sectio, we show that i the low iterferece regime defied i 89 ad 90, the optimal power allocatio is to set P 0 to be zero to achieve sum capacity. Lemma 7. If the the sum rate 77 c 4 + c 3 + c P + c + c 0, RP 0 := log + P P c P 0 P P 0 c + log + P P 0 + c P P 0 + is a decreasig fuctio of P 0 ad thus is maximized by settig P 0 to be zero. So Proof: To simplify the otatio, defie ad the derivative of RP 0 is a := P + c 3 b := P + + P c 4 d := c 5 e = + c. 6 RP 0 = logb + dp 0 a P 0 + log b ep 0 a P 0, 7 dr dp 0 = I total there are three poles ad oe zero, which are It is easy to see bd aed bep 0 + abd + b abe. 8 dp 0 + bp 0 aep 0 b p = b d = cp + + P c p = a = P + c > 0, < 0, p 3 = b e = P + + c < p P + +P c [ + z = c + c P + c + c ] 3 c 4 + c + c + c + 3 c P + 4 c + c + 3 c c. 5 p < 0 < P < p 3 < p. 9 Now we oly eed to cosider the value of z. If c is sufficietly small, the z is egative, ad thus dr dp 0 is egative o [max{p, z }, t 3 ], so RP 0 is a mootoically decreasig fuctio o [0, P ], sice p, z < 0 ad t 3 > P. The coditio of z 0 is exactly the iequality. Lastly, we prove that the coditios 89 ad 90 imply. Deote by Γ A the regio of c, P determied by 89 ad 90, ad deote by Γ B the regio of c, P determied by.

12 Theorem 8. Γ A Γ B. Thus, whe 89 ad 90 hold, the optimal rate for the commo message is zero to achieve the sum capacity. Proof: Note that LHS of 89, 90 ad are icreasig fuctios of P. Fix c, ad let c, P A ad c, P B be the poits o the boudary of Γ A ad Γ B, respectively. To show Γ A Γ B, it is sufficiet to show P A P B. Equivaletly, it is sufficiet to show either c 4 P B + 4c P B + 3c + c P B, 30 or Ideed, we will prove c + c P B. 3 c 4 P B + 4c P B + 3c + c P B, 3 c + c P B. 33 First from we get Therefore, P B = c c c 4 + c 3 + c. 34 c + c P B 35 The last step holds obviously, so c + c P B. Next we prove c 4 P B + 4c P B + 3c + c P B. c c c + c + c + 36 c c + c c c + c c c c. 40 c 4 P B + 4c P B + 3c + c P B 4 c c c c + c + c + 4 c c c + c + + c c + 7 c + c + c + c + + 3c + 4c c + c + c c c + c c c c + c + + 4c c c + 7 c c + c 4 4c + 4c 3 c + c + 4c 4 8c 3 + 8c c c + c 4 + c 6 + 4c 3 + 6c 5 c + 4c 4 8c 3 + 8c c c 6 + 6c 5 + 7c 4 4c 3 + 3c c + 47 c6 + 6c 5 + 5c 4 + 0c 3 + 5c + 6c + 48 c 6 + 6c 5 c 4 8c 3 + 3c 0c c c + 4c Sice the last step holds, we have c 4 PB + 4c P B + 3c + c P B. This completes the proof of Theorem 8.

13 VI. CONCLUSION We cosider symmetric two-user Gaussia iterferece chael with commo messages. We derive a upper boud o the sum capacity, ad show that the upper boud is tight i the low iterferece regime, where the optimal trasmissio scheme is to sed o commo messages ad each receiver treats iterferece as oise. Our result shows that although the availability of commo messages provides a cooperatio opportuity for trasmitters, i the low iterferece regime the presece of commo messages does ot help icrease the sum capacity. ACKNOWLEDGMENT We thak Prof. Pramod Viswaath ad Dr. Sreekath Aapureddy for the helpful discussios. Research of Qua Geg was supported i part by Prof. Pramod Viswaath s Natioal Sciece Foudatio grat No. CCF REFERENCES [] T. M. Cover ad J. A. Thomas, Elemets of iformatio theory. Joh Wiley & Sos, 0. [] R. Etki, D. Tse, ad H. Wag, Gaussia iterferece chael capacity to withi oe bit, IEEE Trasactios o Iformatio Theory, vol. 54, o., pp , 008. [3] H. Sato, The capacity of the gaussia iterferece chael uder strog iterferece corresp., IEEE Trasactios o Iformatio Theory, vol. 7, o. 6, pp , 98. [4] V. Aapureddy ad V. Veeravalli, Gaussia iterferece etworks: Sum capacity i the low-iterferece regime ad ew outer bouds o the capacity regio, IEEE Trasactios o Iformatio Theory, vol. 55, o. 7, pp , 009. [5] X. Shag, G. Kramer, ad B. Che, A ew outer boud ad the oisy-iterferece sum rate capacity for gaussia iterferece chaels, IEEE Trasactios o Iformatio Theory, vol. 55, o., pp , 009. [6] A. Motahari ad A. Khadai, Capacity bouds for the gaussia iterferece chael, IEEE Trasactios o Iformatio Theory, vol. 55, o., pp , 009.