On the Capacity of Symmetric Gaussian Interference Channels with Feedback

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1 O the Capacity of Symmetric Gaussia Iterferece Chaels with Feedback La V Truog Iformatio Techology Specializatio Departmet ITS FPT Uiversity, Haoi, Vietam latv@fpteduv Hirosuke Yamamoto Dept of Complexity Sciece ad Egieerig The Uiversity of Tokyo, Japa hirosuke@ieeeorg arxiv: v5 [csit] 21 Apr 2015 Abstract I this paper, we propose a ew codig scheme for symmetric Gaussia iterferece chaels with feedback based o the ideas of time-varyig codig schemes The proposed scheme improves the Suh-Tse ad Kramer ier bouds of the chael capacity for the cases of weak ad ot very strog iterferece This improvemet is more sigificat whe the sigal-to-oise ratio SNR is ot very high It is show theoretically ad umerically that our codig scheme ca outperform the Kramer code I additio, the geeralized degrees-of-freedom of our proposed codig scheme is equal to the Suh-Tse scheme i the strog iterferece case The umerical results show that our codig scheme ca attai better performace tha the Suh-Tse codig scheme for all chael parameters Furthermore, the simplicity of the ecodig/decodig algorithms is aother strog poit of our proposed codig scheme compared with the Suh-Tse codig scheme More importatly, our results show that a optimal codig scheme for the symmetric Gaussia iterferece chaels with feedback ca be achieved by usig oly margial posterior distributios uder a better cooperatio strategy betwee trasmitters Idex Terms Gaussia Iterferece Chael, Feedback, Posterior Matchig, Iterated Fuctio Systems I INTRODUCTION The capacity of the iterferece chael with feedback has bee still ukow for may decades although there were some progresses toward solvig this problem Kramer developed a feedback strategy ad derived a outer boud of the Gaussia chael [6], [9] However, the gap betwee the outer ad the ier bouds becomes large uboudedly as sigal to oise ratio SNR ad iterferece to oise ratio INR icrease Furthermore, Suh ad Tse [1], [2] characterized the capacity regio withi 2 bits/s/hz ad the symmetric capacity withi 1 bit/s/hz for the two-user Gaussia iterferece chael with feedback They also idicated that feedback provides multiplicative gai at high SNR However, their codig scheme does ot work well whe the SNR is close to the INR Its symmetric codig rate becomes less tha the Kramer code whe this coditio happes I additio, it also has lower performace tha the Kramer code whe α = log INR/ log SNR is ot very large ad the SNR is low cf Figs 1-2 of this paper or Fig 14 i [2] Later, the Suh-Tse codig scheme was exteded to M-user Gaussia iterferece chaels with feedback for M 3 [7] I this paper, we propose a codig scheme which achieves better performace tha the Suh-Tse code whe α = log INR/ log SNR is ot very large See Figs1-2 of this paper I additio, our code ca attai better symmetric rate tha the Kramer code for all chael Figure 1 Figure 2 Symmetric Rate Comparisio at High SNR Symmetric Rate Comparisio at Low SNR parameters, ad therefore it overcomes all the weak-poits of the Suh-Tse codig scheme ad improves the Suh- Tse ad Kramer ier bouds For the strog iterferece case, our code ca achieve the same geeralized degreesof-freedom as the Suh-Tse codig scheme Furthermore, umerical results show that our codig scheme ideed has better/equal performace tha/to the Suh-Tse codig scheme for all chael parameters II CHANNEL MODEL We cosider the Gaussia iterferece chael show i Fig 3, which has two seders ad two receivers Seder 1 seds a source of iformatio message poits Θ 1, which

2 Figure 3 Gaussia Iterferece Chael with Feedback is uiformly distributed i 0,1, to receiver 1 Seder 2 seds a source of iformatio poits Θ 2, which is also uiformly distributed i 0,1, to receiver 2 Assumig that Θ 1 is idepedet of Θ 2 ad each chael iterferes with the other Specially, we assume that Y 1 = X 1 + ax 2 + Z 1, Y 2 = X 2 + ax 1 + Z 2, where Z 1 N 0, σ1 2 ad Z 2 N 0, σ2 2 are Gaussia oise radom variables, ad the iput power costraits are P 1 ad P 2, respectively I this paper, we cosider the symmetric iterferece chael such that P 1 = P 2 = P ad σ1 2 = σ2 2 = 1 Hece, the sigal-to-oise ratio ad iterferece-to-oise ratio ca capture chael gais SNR = P ad INR = a 2 P We also assume that output symbols are casually feedbacked to the correspodig seder ad the trasmitted symbol X m at time ca deped o both the message Θ m ad the previous chael output sequece Y 1,m := Y m 1, Y m 2,, Y m 1 for m {1, 2} A trasmissio scheme for the two-user Gaussia iterferece chael with feedback is sequeces of measurable fuctios {g m : 0, 1 R 1 R} =1, m {1, 2} so that the iput to the chael geerated by the trasmitter is give by X m = g m Θ m, Y 1,m A decodig rule for the two-user Gaussia iterferece chael with feedback are sequeces of measurable mappigs { m : R E} =1, m {1, 2} where E is the set of all ope itervals i 0, 1 ad m y,m refers to the decoded iterval at receiver m The error probabilities at time associated with a trasmissio scheme ad a decodig rule, is defied as p m e := PΘ m / m Y,m, m {1, 2}, ad the correspodig rate pair R 1, R 2 at time is defied by R m := 1 log m Y,m We say that a trasmissio scheme together with a decodig rule achieves a rate pair R 1, R 2 over a Gaussia iterferece chael if for m {1, 2} we have lim P R m < R m = 0, lim pm e = 0 1 The rate pair is achieved withi iput power costraits P 1, P 2 if the followig is satisfied: 1 lim sup E[X m k ] 2 P m, m {1, 2} 2 k=1 The symmetric capacity is defied by C sym := sup{r : R, R is achievable} A optimal fixed rate decodig rule for the two-user Gaussia iterferece chael with feedback for rate pair R 1, R 2 is the oe that decodes a pair of fixed legth itervals {J 1, J 2 : J m = 2 Rm for m {1, 2}}, which maximizes posteriori probabilities, ie, m y,m = argmax J m E: J m =2 Rm P Θm Y J m y,m It is easy to see that the optimal fixed rate decodig rule for the Gaussia iterferece chael with feedback is the traditioal MAP, MMSE decodig rule A optimal variable rate decodig rule with target error probabilities p m e = δ m is the oe that decodes a pair of miimal-legth itervals J 1, J 2 such that accumulated margial posteriori probabilities exceeds correspodig targets, ie, m y,m = mi J m E:P Θm Y J m y,m 1 δ m J m Both decodig rules use the margial posterior distributio of the message poit P Θm Y which ca be calculated olie at the trasmitters ad the receivers Refer [6] for more details Lemma 1: The achievability i the defiitio 1 ad 2 implies the achievability i the stadard framework Proof: See the detailed proof i [3], [4], ad [5] Notatios: The cumulative distributio fuctio cdf of a radom variable X is give by F X x = P X, x], ad their iverse cdf is defied as F 1 X t := if{x : F Xx > t} The compositio fuctio is defied by f gx = fgx, ad the sig fuctio is defied as sgx := 1 if x 0 ad sgx := 1 if x < 0 We also use x + := maxx, 0 ad log + x := maxlog x, 0 III A CODING SCHEME FOR GAUSSIAN INTERFERENCE CHANNELS WITH FEEDBACK I this sectio, we propose a time-varyig codig scheme for the symmetric Gaussia iterferece chael with feedback as followig: A Ecodig Step 1: Trasmitter m seds X m 1 = F 1 X Θ m where m {1, 2} ad X N 0, P 1 for some P 1 > 0 We also set ρ 1 := E[F 1 X Step + 1 for 1: Both trasmitters estimate Θ 1F 1 X Θ 2] = 0 P 1 ρ +1 = 1 β 2 {ρ 2b sgρ ρ + a +b 2 sgρ [ ρ 1 + a a ] }

3 Trasmitter 1 seds X 1 +1 sgρ +1 where X 1 +1 := 1 β X 1 b sgρ Y 1 Trasmitter 2 seds X 2 +1sga where X 2 +1 := 1 β X 2 b sgay 2 Receiver 1 receives Y 1 +1 = X1 +1 sgρ +1 + a X Z1 +1 Receiver 2 receives Y 2 +1 = sgax a sgρ +1X Z2 +1 Both receivers feedback their received sigals to the correspodig trasmitters Here, β > 0, b should be chose to satisfy the followig costraits: lim sup N B Decodig 1 N N =1 E[X m ] 2 P, m {1, 2} 3 At each time slot, receiver m {1, 2} selects a fixed iterval J m 1 = s m, t m R as the decoded iterval with respect to X m +1 The, set the decoded iterval J m = s m, T m t m, as the decoded iterval T m with respect to X m 1, where T m s := w m 1 w m 2 w m s, s R ad w 1 := β x + b sgρ Y 1, w 2 := β x + b sgay 2 Receiver m sets the decoded iterval for the message Θ m as follows: m Y,m = F Xm J m We call this codig strategy the Gaussia iterferece time-varyig feedback codig strategy, which is a optimal variable rate decodig rule with doubly expoetial decay of targeted error probabilities see the proof of the Lemma 2 i this paper IV A NEW ACHIEVABLE RATE REGION Lemma 2: Uder the coditio that 0 < lim sup β < 1, the time-varyig codig scheme for the symmetric Gaussia iterferece chael with feedback achieves the followig symmetric rate: R sym = lim sup log β bits/chael use Proof: We provide a sketch of the proof The detailed oe ca be foud i papers [3], [4], ad [5] Defie β := lim sup β It is easy to see that R sym = log β 1 For ay R < R sym, we ca fid a ɛ > 0 such that R < logβ + ɛ 1 Choose a N ɛ N such that sup Nɛ β < β + ɛ Usig the Law of Iterated Expectatios Fubii s Theorem, we ca show that PR m We ca also show that < R A ɛ 2 R β + ɛ Nɛ J m 1 p m e = O exp m J 1 2 8P By choosig J m 1 = o 2 logβ+ɛ 1 R o 2 Rsym R, ad t m = s m = J m 1 /2, the aforemetioed coditio 1 is satisfied Here, symbols Ox ad ox are Ladau symbols Besides, the choice of sequeces b, β followig the rule 3 leads to the fact that the iput power costraits are also satisfied Theorem 1: The o-degraded symmetric Gaussia iterferece chael a 0 ca achieve the followig symmetric rate: [ log where ad Deote R sym bits/chael use = 1 2 max ρ [0,ρ 0], b {b 1,b 2 } P P + b 2 [1 + P + a 2 P + 2 a P ρ] 2bP [1 + a ρ] 0 < ρ 0 := a 2 P 2 + P P [2a 2 P 2 + P ] a 2 P 2 < 1, b 2P ρ + a P + a P ρ 2 1,2 = 2 a P + 2P ρ + 2 a 2 P ρ + ρ + 2 a P ρ 2 P 2 a ± 2 ρ 4 2ρ 2 a 2 P 2 + P + a 2 P 2 2 a P + 2P ρ + 2 a 2 P ρ + ρ + 2 a P ρ 2 Proof: From our trasmissio strategy, we have X 1 +1 = 1 β X 1 b sgρ Y 1, 4 X 2 +1 = 1 β X 2 b sgay 2 5 P = E[X 1 ] 2 = E[X 2 ] 2, ρ = E[X1 X 2 ] P We ca show by iductio that E[X 1 ] 2 = E[X 2 ] 2 for all Therefore, it is easy to see that E[X 1 Y 2 E[X 2 Y 1 ] = P ρ + a, ] = P sgasgρ ρ + a, E[Y 1 Y 2 ] = P sga [ ρ 1 + a a ] Note that E[X 1 +1 X2 +1 ] = 1 { β 2 E[X 1 X 2 ] b sgρ E[X 2 Y 1 ] b sgae[x 1 +b 2 sgρ sgae[y 1 Y 2 ] } Y 2 ] ],

4 Fially, we obtai P +1 ρ +1 = P sgρ 1 β 2 { ρ 2b ρ + a Similarly, observe that +b 2 [ ρ 1 + a a ] } 6 E[X 1 Y 1 ] = P sgρ [1 + a ρ ], E[X 2 Y 2 ] = P sga[1 + a ρ ], E[Y 1 ] 2 = 1 + P + a 2 P + 2 a ρ P, E[Y 2 ] 2 = 1 + P + a 2 P + 2 a ρ P From the relatios 4 ad 5 we have P +1 = 1 +b 2 β 2 {P 2P b [1 + a ρ ] [ 1 + P + a 2 P + 2 a ρ P ]} 7 From 6 ad 7, if we ca force P P, b b, β β ad ρ ρ [0, 1], we obtai the followig equatios with three ukows b, ρ, β: P = 1 β 2 [ P 2bP 1 + a ρ + b P + a 2 P + 2 a ρp ], ρ = 1 β 2 [ ρ 2bρ + a + b 2 ρ1 + a a ] 8a 8b By elimiatig β ad cosiderig ρ as a ruig variable, we have the followig quadratic equatio i b for each fixed choice of ρ: b 2 [2 a P + 2P ρ + 2 a 2 P ρ + ρ + 2 a P ρ 2 ] 2b[2P ρ + a P + P a ρ 2 ] + 2P ρ = 0 9 After some simple derivatios, the discrimiat of this quadratic equatio is give by = P 2 a 2 ρ 4 2ρ 2 a 2 P 2 +P + a 2 P 2 := fρ 10 Sice f0 = a 2 P 2 > 0 ad f1 = 2P < 0, there exists the miimum value ρ 0 0, 1 such that fρ 0 = 0 Specifically, the value of ρ 0 is give by a 0 < ρ 0 = 2 P 2 + P P [2a 2 P 2 + P ] a 2 P 2 < 1 O the other had, sice the derivative of fρ satisfies f ρ = 4P 2 a 2 ρ 3 ρ 4P ρ 0, for all ρ [0, 1, we have = fρ 0 for all ρ [0, ρ 0 ] For all these values of ρ, we ca easily show that 9 ca have two positive solutios b 1, b 2 as the theorem statemet As a result, 8a ad 8b have at least oe solutio b, β for each fixed ρ [0, ρ 0 ] Therefore, if we force all the sequeces b, β, P, ρ to coverges to b, β, P, ρ, the symmetric Gaussia iterferece chael with feedback ca achieve the followig rate by the Lemma 2: + R sym = log mi β = 1 max ρ [0,ρ 0] 2 ρ [0,ρ 0],b {b 1,b 2 } log + P P + b 2 [1 + P + a 2 P + 2 a P ρ] 2bP [1 + a ρ] Note that sice Lemma 2 holds oly for 0 < β < 1, the superscript + is ecessary to deal with geeral cases To complete the proof, we show a procedure to force ρ = ρ ρ = 1 ρ for ay ρ [0, ρ 0 ], b = b, P = P, β = β for ay 2 Ideed, from 6 ad 7, we firstly force P 2 = P, ρ 2 = ρ, ad β 2 = β by settig P ρ = P 1 β1 2 {2 a b 2 1 b 1 }, 11 P = 1 β1 2 {P 1 2b 1 P 1 + b P 1 + a 2 P 1 } 12 This procedure is feasible because 11 ad 12 have at least oe solutio which is a triplet b 1, P 1 > 0, β 1 > 0 for each ρ [0, ρ 0 ] Ideed, For ρ = 0, we ca choose b 1 = 0, P 1 = P, β 1 = 1 For ρ 0, from 11 ad 12 we have the followig quadratic equatio i b 1 b 2 1[1 + P 1 + a 2 P 1 ρ 2 a P 1 ] 2ρ a P 1 b 1 + P 1 ρ = 0 13 The discrimiat of this quadratic equatio ca easily be show to be equal to Eρ = a 2 1 ρ 2 P 2 1 P 1 ρ 2 For the case ρ a, we ca choose P 1 such that this discrimiat is equal to zero by settig P 1 = ρ 2 /a 2 1 ρ 2 By this choice of P 1, we obtai b 1 = ρ a P P 1 + a 2 P 1 ρ 2 a P 1 I order for 11 ad 12 to have solutio β 1, we eed to show that the above choices of P 1, b 1 satisfy b 2 1 b 1 > 0 Clearly for ρ < a, this requiremet is satisfied by otig that b 1 < 0 sice ρ a < 0 ad ρ > 2 a ρ2 a 2 + ρ 2 = 2 a P P 1 + a 2 P 1 For ρ > a a < 1, of course, observe that P 1 = ρ 2 a 2 1 ρ 2 > ρ a a 2 ρ It follows that ρ a P 1 > 1+P 1 +a 2 P 1 ρ 2 a P 1 > 0 or b 1 > 1 This also meas that b 2 1 b 1 > 0 For the case ρ = a ay choice of P 1 > 1/1 a 2 works sice we have E a > 0 Besides, the sum of two solutios of the quadratic equatio 13 i b 1 is equal to zero, ad their product is ot equal to zero by usig Vieta s formula Hece, we must fid at least oe b 1 < 0 or b 2 1 b 1 > 0 Fially, we oly eed to set β = β, P = P, b = b which is a solutio of 8a ad 8b for each choice of ρ [0, ρ 0 ] ad for all 3 Here b should be chose to miimize β for each choice of ρ [0, ρ 0 ] i order to maximize the achievable symmetric rate of our codig scheme Last but ot least, we ca show that our code has better performace tha the Kramer code [6], ad therefore the superscript + ca be got rid of from the achievable symmetric rate formula Remark 1: For the degraded Gaussia iterferece chael with feedback a = 0, 8a ad 8b become ρ = 1 β 2 ρb 12, P = 1 β 2 [P b 12 + b 2 ] 14

5 From 14, we must have ρ = 0 ad the achievable symmetric rate becomes R sym = 1 [ ] 2 max log P b R P b b 2 = 1 log1 + P 2 This result coicides with the well-kow capacity of this chael with o iterferece Corollary 1: The proposed time-varyig code outperforms the Kramer code for all chael parameters Proof: A variat of Kramer code is costructed by choosig triplet b, β, ρ, which is a solutio of 8a ad 8b, as follows: b = P 1 + a ρ P 1 + a a ρ + 1, β = a 2 P 1 ρ P 1 + a a ρ + 1, ad ρ is the uique solutio i 0, 1 of the ext equatio: 2 a 3 P 2 ρ 4 + a 2 P ρ 3 4 a P a 2 P + 1ρ 2 2a 2 P + P + 2ρ + 2 a P a 2 P + 1 = 0 15 See also i [2], [6], [9] Therefore, it is iferior to the proposed code i this paper Remark 2: The choice of b ad β for the Kramer code is to maximize the achievable rate or miimize β by usig oly 8a for each fixed value of ρ I order to satisfy 8b, the choice of b, β i the above proof is applicable oly to the fixed value of ρ which is the uique solutio i 0, 1 of 14 For other values of ρ, from 8a ad 8b, we see that b ad β are give by two differet fuctios of ρ Elargig the set of possible choices of ρ the b, β icreases the achievable symmetric rate i this paper Corrolary 2: For α = log INR/ log SNR > 1, the geeralized degrees of freedom of the proposed codig scheme is give by dα := R sym SNR, INR lim = α SNR,INR log SNR 2 Here, the uit of R sym is bits/s/hz As a cosequece, the proposed codig scheme has the same geeralized degrees of freedom as the Suh-Tse codig scheme [1], [2] ad also provides the multiplicative gai at high SNR Note that the Kramer code achieves oly dα = 1 + α/4 for α 1 cf 48 [2] Proof: Sice α = log INR/ log SNR > 1, we have a 2 P = P α, or a = P α 1/2 For P sufficietly large ad ρ 1, observe that b 1,2 P α+1/2 1 + ρ 2 2P α ρ ± P α+1/2 1 ρ 2 2P α ρ By choosig b = b 1 P 1 α/2 ρ, we obtai R sym ρ log + P P + b 2 P α 2bP 1 + P α 1/2 ρ = log 1 ρ 2 2ρP 1 α/2 bits/s/hz 16 From Theorem 1, we also have ρ 2 0 = 1 + P α 2P α + P 2α > 1 3P α/2, ad the ρ, that maximizes the achievable rate, must be i the iterval [0, ρ 0 ] This coditio is satisfied by settig ρ = 1 P γ + P α 1 P α 1/2 for a arbitrary positive umber γ < α/2 although this ρ may ot be optimal Ideed, for P sufficietly large, we have ρ 1 ad ρ < 1 P γ + P α 1 P α 1/2 = 1 P γ Hece ρ 2 < 1 P γ < 1 3P α/2 < ρ 2 0 O the other had, sice 1 ρ 2 2ρ P 1 α/2 = P γ, we obtai dα R sym ρ lim SNR,INR log SNR = γ Sice γ ca take ay arbitrary value less tha α/2, we have dα α/2 From the result of [2], it is kow that dα α/2 Hece, we must have dα = α/2 V NUMERICAL EVALUATION I order to evaluate R sym i Theorem 1 umerically, we determied the optimal ρ by icreasig it from zero by icremetal step 10 5 The umerical results are show i Figs 1-2 i Sectio I We ote that our proposed code ca achieve better/equal symmetric rate tha/to the Suh- Tse ad Kramer codes for all chael parameter a, P This improvemet is more sigificat whe SNR is ot too high VI CONCLUSION The ier boud of the capacity is improved compared with the Suh-Tse ad Kramer ier bouds by aalyzig the performace of our proposed code as a optimized form of the Kramer code Our result also shows that a optimal codig scheme for the Gaussia iterferece chael with feedback ca be achieved by usig margial posterior distributios ACKNOWLEDGMENT This work was supported i part by JSPS KAKENHI Grat Number REFERENCES [1] Chagho Suh ad David Tse, Symmetric Feedback Capacity of the Gaussia Iterferece Chael to Withi Oe Bit, i Proc It Symp Iformatio Theory, Ju 2009 [2] Chagho Suh ad David Tse, Feedback Capacity of the Gaussia Iterferece Chael to Withi 2 Bits, IEEE Tras If Theory, vol 57, No 5, pp , May 2011 [3] La V Truog, Posterior Matchig Scheme for Gaussia Multiple Access Chael with Feedback, [Olie] Available: [4] La V Truog, Posterior Matchig Scheme for Gaussia Multiple Access Chael with Feedback, i Proc IEEE Iformatio Theory Workshop, Nov 2014 [5] La V Truog, A Novel Time-Varyig Codig Scheme for the Gaussia Broadcast Chael with Feedback, [Olie] Available: [6] Gerhard Kramer, Feedback Strategies for White Gaussia Iterferece Networks, IEEE Tras If Theory, vol 48, pp , Ja 2002 [7] Ravi Tado, Soheil Mohajer, ad H Vicet Poor, O the Symmetric Feedback Capacity of the K-User Cyclic Z- Iterferece Chael, IEEE Tras If Theory, vol 59, o5, pp , May 2013 [8] Ravi Tado, Soheil Mohajer, ad H Vicet Poor, O the Feedback Capacity of the Fully Coected K-User Iterferece Chael, IEEE Tras If Theory, vol 59, o5, pp , May 2013 [9] G Kramer, Correctio to Feedback Strategies for White Gaussia Iterferece Networks, ad a Capacity Theorem for Gaussia Iterferece Chaels with Feedback, IEEE Tras If Theory, vol 50, o 6, pp , Ju 2004 [10] O Shayevitz ad M Feder, Optimal Feedback Commuicatio via Posterior Matchig, IEEE Tras If Theory, vol 57, o3, pp , Mar 2011

Posterior Matching Scheme for Gaussian Multiple Access Channel with Feedback

Posterior Matching Scheme for Gaussian Multiple Access Channel with Feedback 1 Posterior Matchig Scheme for Gaussia Multiple Access Chael with Feedback La V. Truog, Member, IEEE, Iformatio Techology Specializatio Departmet ITS, FPT Uiversity, Haoi, Vietam E-mail: latv@fpt.edu.v