Rate Splitting is Approximately Optimal for Fading Gaussian Interference Channels

Transcription

1 Rate Splitting is Appoximately Optimal fo Fading Gaussian Intefeence Channels Joyson Sebastian, Can Kaakus, Suhas Diggavi, I-Hsiang Wang Abstact In this pape, we study the -use Gaussian intefeence-channel with feedback and fading links. We show that fo a class of fading models, when no channel state infomation at tansmitte CSIT is available, the ate-splitting schemes fo static intefeence channel, when extended to the fading case, yield an appoximate capacity egion chaacteized to within a constant gap. We also show a constant-gap capacity esult fo the case without feedback. Ou scheme uses ate-splitting based on aveage intefeence-to-noise atio in. This scheme is shown to be optimal to within a constant gap if the fading distibutions have the quantity log E in] E log in unifomly bounded ove the entie opeating egime. We show that this condition holds in paticula fo Rayleigh fading and Nakagami fading models. The capacity egion fo the Rayleigh fading case is obtained within a gap of.83 bits fo the feedback case, and within.83 bits fo the non-feedback case. I. INTRODUCTION The -use Gaussian intefeence channel IC is a simple model that captues the effect of intefeence in wieless netwoks. Significant pogess has been made in the last decade in undestanding the capacity of the static Gaussian IC. But in pactice the links in the channel could be vaying athe than static. In this pape, we study the -use Gaussian IC with fading links. Pevious woks have chaacteized the capacity egion to within a constant gap fo the static Gaussian IC with and without feedback. The capacity of the -use Gaussian IC without feedback was chaacteized to within bit in 3]. In 7], Suh et al. chaacteized the capacity of the Gaussian IC with feedback to within bits. These esults wee based on the Han-Kobayashi scheme, whee the tansmittes split thei messages into common and pivate pats. To the best of ou knowledge no pevious wok has povided a simple scheme to completely chaacteize the capacity egion fo the case of the continuously fading channel with no channel state infomation at tansmitte CSIT. The geneal Han-Kobayashi scheme fo discete memoyless IC ] indeed holds fo the poblem at hand, but it is vey complex due to the time-shaing involved. In ], Wang et al. consideed the busty IC whee the intefeence is eithe pesent o not pesent. In 9], Vahid et al. studied the binay fading model fo the two-use intefeence channel, whee the channel gains, the tansmit signals and the eceived signals ae in the binay field. In 5], Gou et al. poposed an intefeence neutalization scheme and showed a degees of Univesity of Califonia, Los Angeles. {joysonsebastian, kaakus, suhasdiggavi}@ucla.edu National Taiwan Univesity. ihwang@ntu.edu.tw feedom esult fo fading intefeence channel with full channel state infomation at the elays and destinations. In 6], Kang et al. consideed intefeence alignment fo the fading K-use IC with delayed feedback and showed a esult K K+ of degees of feedom. Tuninetti 8] studied powe allocation policies fo fading Gaussian IC with CSIT and numeically showed that fo Rayleigh fading, thei scheme is close to optimal fo some system paametes. In 4] Fasani showed a one bit capacity esult fo fading Gaussian IC with patial CSIT. Thei scheme 4] employs dynamic powe allocation depending on the available CSIT to achieve any given ate point and equies a union of all powe allocation policies fo both tansmittes to achieve the whole inne bound. In this pape, we show that the Han-Kobayashi type atesplitting schemes ], 3], 7] can be extended to a class of fading models that satisfy a condition on the distibution of cosslink stengths. We will show that ate-splitting based on in fo the static case in 3], 7] can natually be extended to schemes with ate-splitting based on Ein] fo the fading case to appoximately obtain the whole capacity egion. Ou schemes use fixed powe allocation to achieve any given ate point, and to achieve the whole inne bound fo the feedback case we need to vay only a single powe allocation paamete othe than choosing the common and pivate message ates, inheiting these popeties fom 3], 7]. The condition on distibution of cosslink stengths is equied to ensue that the splitting is indeed optimal within a constant gap. In paticula, we will show that common fading models, including Rayleigh and Nakagami fading, satisfy the equied condition. The pape is oganized as follows. In section II we set up the poblem and explain the notations used. In section III we discuss the main esults of ou pape, in section IV we go though the details of the scheme fo the feedback case, in section V we go though the esults fo the non feedback case, and in section VI we show that common fading models including Rayleigh and Nakagami fading satisfy the equied conditions. II. NOTATION We conside the -use Gaussian fading IC Y = g X + g X + Z Y = g X + g X + Z whee the links g ij ae fading, the ealizations of g ij fo any fixed i, j ae i.i.d acoss time, and the ealizations fo diffeent i, j ae independent. The instantaneous intefeence-

2 to-noise atios in ae given by g and g. We assume that the tansmittes has no knowledge of the channel states befoe tansmission. Fo the feedback case, afte each eception, each eceive eliably feeds back the eceived symbol and the channel states to its coesponding tansmitte, fo example the eceive feeds back Y, g, g to tansmitte. We nomalize the signal powe and noise powe to, i.e., P k =, Z k CN,. We denote E g ii ] as SNR i and E g ij ] when i j, as INR i. We use the vecto notation g = g, g ], g = g, g ] and g = g, g, g, g ]. Fo a complex numbe z, we use Re z to indicate its eal pat. The natual logaithm is denoted by ln and the logaithm with base is denoted by log. III. SUMMARY OF MAIN RESULTS Fo a given wieless system, the signal stengths would have some pobability distibution; depending on the opeating conditions, the distibutions might vay and hence we get a class of distibutions. We identify W = g ij fo ease of notation. Suppose the class of distibutions satisfy log a + µ W E log a + W c fo some constant c independent of the distibution, fo all a, then we claim that we have the following constant gap capacity chaacteization fo the feedback and non-feedback cases. We assume that we have the same c fo both cosslinks, but ou esults can be easily modified to the case when thee ae two diffeent c and c fo the two cosslinks. The following two theoems summaize ou esults. The achievability schemes fo both cases ae Han-Kobayashi schemes with ate-splitting based on E in]. Fo both cases we obtain capacity gaps in tems of the constant c fom condition, whee a capacity gap of δ means that fo any ate pai R, R in the oute bound egion, the ate pai R δ, R δ is contained in the inne bound egion Theoem. Fo the fading intefeence channel with feedback, the ate egion descibed by 7 is achievable fo ρ, θ < π with λ pk = min R E, ρ : log g + g + ρ Re e iθ g g + R E log + ρ g + λ p g + λ p g 3 R E log g + g + ρ Re gg e iθ + 4 R E log + ρ g + λ p g + λ p g 5 R + R E log g + g + ρ Re gg e iθ + + λ p g + λ p g 6 R + R E log g + g + ρ Re e iθ g g + + λ p g + λ p g 7 An oute bound fo the feedback case is given by 8 3 fo complex numbe ρ with ρ : R E log g + g + Re ρg g + 8 R E log + ρ g ρ g + E log + + ρ 9 g R E log g + g + Re ρgg + R E log + ρ g ρ g + E log + + ρ g R + R E log g + g + Re ρgg + ρ g + E log + + ρ g R + R E log g + g + Re ρg g + ρ g + E log + + ρ, 3 g and if the channel satisfies the condition, the gap between oute bound and inne bound is at most c + bits. Poof. The details ae in Section IV. Theoem. Fo the fading intefeence channel without feedback and the tansmittes having no channel state infomation, the ate egion descibed by 4 is achievable with λ pk = min, : R E log + g + λ p g 4 R E log + g + λ p g 5 R + R E log + g + g R + R E log + g + g + λ p g + λ p g 6 + λ p g + λ p g 7

3 R + R E log + λ p g + g + λ p g + g 8 R + R E log + g + g 3 + λ p g + g + λ p g + λ p g 9 R + R E log + g + g 3 + λ p g + g + λ p g + λ p g An oute bound fo the non-feedback case is given by 7. R E log + g R E log + g R + R E log + g + g + g + g 3 R + R E log + g + g + g + g 4 R + R E log + g + g + g + g + g + g 5 R + R E log + g + g + g + g + g + g + g 6 R + R E log + g + g + g + g + g + g + g, 7 and if the channel satisfies the condition, then the gap between oute bound and inne bound is at most c + bits. Poof. The details ae in Section V. Ou esults can be applied fo most of the well behaved continuous fading models by showing that they satisfy the condition. It tuns out that the busty intefeence model do not satisfy the condition as we discuss in Section VI. IV. THE FADING INTERFERENCE CHANNEL WITH FEEDBACK Fo the inne bound we use the block Makov scheme fom 7] which gives the following achievable egion: R I U, U, X ; Y, g 8 R I U ; Y, g U, X + I X ; Y, g U, U, U 9 R I U, U, X ; Y, g 3 R I U ; Y, g U, X + I X ; Y, g U, U, U 3 R + R I X ; Y, g U, U, U + I U, U, X ; Y, g 3 R + R I X ; Y, g U, U, U + I U, U, X ; Y, g 33 fo all p u p u u p u u p x u, u p x u, u. Note that we have modified the expessions fom 7] by eplacing Y with Y, g and Y with Y, g to take the fading into account, since the eceive knows the eceived symbol itself and the channel stengths of the links. Now simila to 7] we choose the following Gaussian input distibution: U CN, ρ, U k CN, λ ck, X pk CN, λ pk X = e iθ U + U + X p X = U + U + X p with ρ, θ < π,λ ck + λ pk = ρ and λ pk = min, ρ. With this choice of λ pk we pefom the ate splitting accoding to the aveage in in place of ate splitting based on the constant in fo static channels. Note that we have intoduced an exta otation θ fo the fist tansmitte, which will become helpful in poving the capacity gap see Appendix III. On evaluating the tems in 8 33 fo this choice of input distibution, we get the inne bound descibed by 7, the calculations ae defeed to Appendix I. The oute bounds can be easily deived following the poof techniques fom 7] using E X X ] = ρ while making changes elevant to the fading case. The calculations ae defeed to the Appendix II. Claim 3. The gap between the inne bound 7 and the oute bound 8 3 fo the feedback case is atmost c + bits. Poof. The condition we imposed on the fading distibution becomes impotant in poving a constant gap capacity esult. We illustate it by computing the gap δ between the second inequality 9 of the oute bound and the second inequality 3 of the inne bound. ρ g δ = E log + + ρ g

4 E log + λ p g + λ p g + a E log + ρ INR + ρ g E log + ρ INR + c E log + λ p g + λ p g + ρ g E log + + ρ INR E log + λ p g + + c whee a follows fom condition on the distibution of g ij and Jensen s inequality. We have λ p = min INR, ρ ; on consideing the cases λ p = ρ and λ p = INR sepaately, it can be shown that ρ g + + ρ < + λ p g. INR Hence δ c+ follows. Moe details on the computation of capacity gap fo the feedback case ae in Appendix III. V. THE FADING INTERFERENCE CHANNEL WITHOUT FEEDBACK Fom ] we obtain that a Han-Kobayashi scheme fo IC can achieve the following ate egion fo all p u p u p x u p x u. Note that we use Y i, g i instead of Y i in the actual esult fom ] to account fo the fading. R I X ; Y, g U 34 R I X ; Y, g U 35 R + R I X, U ; Y, g + I X ; Y, g U, U 36 R + R I X, U ; Y, g + I X ; Y, g U, U 37 R + R I X, U ; Y, g U + I X, U ; Y, g U 38 R + R I X, U ; Y, g + I X ; Y, g U, U + I X, U ; Y, g U 39 R + R I X, U ; Y, g + I X ; Y, g U, U + I X, U ; Y, g U. 4 Now simila to that in 3], choose the Gaussian input distibution U k CN, λ ck, X pk CN, λ pk, k {, } X = U + X p X = U + X p whee λ ck + λ pk = and λ pk = min,. Hee we intoduced the ate splitting using the aveage in. On evaluating the egion descibed by 34 4 with this choice of input distibution, we get the egion descibed by 4 ; the computations ae simila to that of the feedback case. The oute bounds easily follow fom the esults in 3] by suitably modifying fo the fading case, and by using the oute bounds and 3 fo feedback case afte setting E X X ] = ρ =. Claim 4. The gap between the inne bound 4 and the oute bound 7 fo the feedback case is atmost c + bits. Poof. The poof fo the capacity gap again uses the condition on the fading distibution. Moe details ae given in Appendix IV. VI. FADING MODELS Hee we discuss the fading models that satisfy the equied condition. We fist note that φ a = log a + µ W E log a + W due to Jensen s inequality, so the quantity we ae inteested in is a Jensen s gap. We will simplify the condition futhe. On taking the deivative with espect to a and again using Jensen s inequality we get ln φ a = a + µ W E a + W ]. Hence φ a achieves the maximum value at a = in the ange,. Hence we just have to show log µ W E log W c. Note that if the distibution gets scaled by a facto k, then we have the same gap log kµ W E log kw = log µ W E log W, and hence we can fix the scaling by equiing the mean of W to be. So let µ W =, then the equied condition educes to E log W c. 4 Hence it follows that fo any distibution that has a point mass at, we cannot guaantee a constant capacity gap, since it has E log W =. Now we discuss a few distibutions that can be easily shown to satisfy the equied condition fo the scheme. A. Gamma distibution The pobability density function fo Gamma distibution is given by f w = wk e w θ θ k Γk fo w > and k, θ >. We use the sufficient condition 4 hee. So we need to bound E log W when E W ] =. It is known fo the Gamma distibution that E W ] = kθ

5 and E ln W = ψ k + ln θ, whee ψ is the digamma function. Since E W ] = we get θ = k and hence E ln W = ψ k + ln θ = ψ k ln k. If it is specified that < α k, we fist use the following popety of digamma function ψ k = ψ k + k, and then use the inequality fom ] ln k + < ψ k + < ln k + e γ. Hence ln E log W > ln k + ln k k = ln + k k E log W > log + α α ln. 4 The last steps follows because the function involved is deceasing in k in the ange,. Rayleigh fading: In Rayleigh fading model the g ij is exponentially distibuted with mean INR i. The exponential distibution itself is a special case of Gamma distibution with k =. Substituting α = in 4 we get E log W >.86 = c. Using the above value of c, we get the feedback capacity gap as c + =.86 bits, and fo the non feedback case we get a gap of c + =.86 bits. Also the distibutions fo the Nakagami fading can be obtained as special cases of the Gamma distibution; then the capacity gap will depend upon the paametes used in the model. B. Weibull distibution The pobability density function fo Weibull distibution is given by f w = k w k e w/λ k λ λ fo x > with k, λ >. Hee E W ] = λγ + k and E ln W = ln λ γ k, whee Γ w denotes the gamma function and γ is the Eule s constant. By setting the mean to be unity, we get λ =. Hence if it is specified that < α k, then we get E log W log Γ+ k Γ + γ α α ln. Note that exponential distibution can be obtained fom Weibull distibution also by setting k =. We get E log W = γ ln fo the Rayleigh fading case. This gives the capacity gap computed moe accuately fo the Rayleigh fading case as.83 bits fo the feedback case, and.83 bits fo the non feedback case. C. Othe distibutions Hee we give a lemma that can be used to veify whethe fo a given fading model, ou scheme of ate-splitting at aveage in yield a constant gap capacity esult. Lemma 5. If the cumulative distibution function F w of W is such that F w aw b fo w, ɛ] whee a, < b, < ɛ, then E ln W ln ɛ + aɛ b ln ɛ aɛb b. Poof. The condition in this lemma ensues that the pobability density function fw gows slow enough as w so that fw ln w is integable at. Also the behaviou fo lage values of w is not elevant hee, since we ae looking fo a lowe bound on E ln W. The detailed poof is given in Appendix V. Hence if the fading model guaantees F w aw b fo w, ɛ] with some a, < b, < ɛ fo the scaled vesion of W with unit mean, then we can find a c satisfying the equied condition. VII. CONCLUSION We poved that the ate-splitting schemes fo the static - use Gaussian IC without CSIT 3], ] and that with delayed feedback 7], can be extended to the fading case fo a class of fading distibutions. The poof fo optimality to within a constant gap, elies on the sufficient condition, which the fading distibution is assumed to satisfy. Ou technique does not wok fo the busty intefeence case since it does not satisfy the condition. It would be inteesting to study if busty intefeence scheme ] and ou poposed scheme can be combined to tackle abitay fading distibutions. VIII. ACKNOWLEDGEMENTS The wok was suppoted in pat by NSF gants 34937, 5453 and a gift by Intel Cop. APPENDIX I PROOF OF ACHIEVABILITY FOR FEEDBACK CASE We evaluate the tem in the fist inne bound inequality 8. The othe tems can be similaly evaluated. I U, U, X ; Y, g a = I U, U, X ; Y g vaiance Y g = h Y g h Y g, U, U, X, = vaiance g X + g X + Z g, g = g + g + g g E X X ] + g g E X X ] + = g + g + ρ Re g g e iθ +

6 h Y g, U, U, X = h g X + g X + Z g, U, U, X = h g X p + Z g = E log + λ p g + log πe b E log + g + log πe INR c log + log πe = + log πe, I U, U, X ; Y, g E log g + g + ρ Re g ge iθ + whee a uses independence, b uses the monotonicity of expectation and λ pi INR i, and c follows fom Jensen s inequality. APPENDIX II PROOF OF OUTERBOUNDS FOR FEEDBACK CASE Following the Suh-Tse methods 7], we let E X X ] = ρ. We use the notation g = g, g ], g = g, g ], g = g, g, g, g ], S = g X +Z, and S = g X +Z. We let E X X ] = ρ = ρ e iθ. On choosing a unifom distibution of messages we get nr ɛ n a I W ; Y n g n b h Yi g i h Zi = E gi h Yi g i = g i h Zi ] c = E g h Yi g i = g h Zi R E log g + g + ρ gg + ρg g + whee a follows fom Fano s inequality, b follows fom the fact that conditioning educes entopy, and c follows fom the fact that g i ae i.i.d. Now we bound R in a second way as done in 7]: nr ɛ n I W ; Y n, g n I W ; Y n, g n, Y n, g n, W = I W ; g n, g n, W + I W ; Y n, Y n g n, g n, W = + I W ; Y n, Y n g n, W = h Y n, Y n g n, W h Y n, Y n g n, W, W = h Yi, Y i g n, W, Y i, Y i h Z i + h Z i = h Yi g n, W, Y i, Y i + h Yi g n, W, Y i, Y i h Z i + h Z i a = h Yi g n, W, Y i, Y i, X i + h Yi g n, W, Y i, Y i, S i, X i h Z i + h Z i b h Yi g i, X i h Zi ] + h Yi g i, S i, X i h Zi ] c = E g h Yi X i, g i = g h Z i + E g h Yi S i, X i, g i = g h Z i d R E log + ρ g ρ g + E log + + ρ g whee a follows fom the fact that X i is a function of W, Y i, g i and S i is a function of Y i, X i, b follows fom the fact that conditioning educes entopy, c follows fom the fact that g i ae i.i.d., and d follows fom the Suh-Tse esults 7]. The othe oute bounds can be deived similaly following 7] and making suitable changes to account fo fading as we illustated in the pevious two deivations. APPENDIX III PROOF OF THE CAPACITY GAP FOR FEEDBACK CASE We compae the coesponding equations in oute and inne bounds. Denote the gap between the fist oute bound and inne bound by δ, fo the second pai denote the gap by δ, and so on. Choose θ in the inne bound to match ag ρ in the oute bound. We get + δ = E = E E log g + g + ρ Re e iθ g g log g + g + ρ Re e iθ g g g + g + ρ Re e iθ g g + log = E log +. + g + g + ρ Re e iθ g g Ree + + ρ iθ g g g + g g + g + + ρ Ree iθ g g g + g g + g

7 We have Re e iθ g g g + g = e iθ gg + e iθ g g g + g, hence we call e iθ g g+eiθ g g g + g = sin φ and let g + g =. Theefoe + + ρ sin φ δ = E log + + ρ +. sin φ If sin φ <, then If sin φ >, then since Hence + + ρ sin φ + + ρ sin φ. + + ρ sin φ + + ρ sin φ = + ρ ρ sin φ + + ρ sin φ ρ ρ sin φ and + + ρ sin φ >. δ. We have aleady shown in Section IV that δ + c. By inspection of the othe bounding inequalities, it is clea that the capacity gap is at-most + c bits. APPENDIX IV PROOF OF CAPACITY GAP FOR NON-FEEDBACK CASE Denote the gap between the fist oute bound and fist inne bound 4 by δ, δ fo the second pai and so on. Clealy Now δ δ. δ 3 = + g + g E log + λ p g + λ p g a + c + g + INR E log + λ p g. The step a follows fom Jensen s inequality and condition on g. We have λ p = min INR, INR, + hence E log + g E log + λ p g. + INR Theefoe δ 3 + c. Similaly δ 4 + c δ 5 + c δ c δ c. Fo δ 5, δ 6, and δ 7 we have to use the condition twice and hence c appeas. Hence it easily follows that the capacity gap is at-most c + bits. APPENDIX V PROOF OF LEMMA 5 Poof. We have F w aw b fo w, ɛ] whee a, b >, ɛ >. Now using integation by pats we get E ln W = ˆ ˆ ɛ f w ln w f w ln w + ˆ ɛ f w ln w ˆ ɛ = F w ln w ɛ F w w + ˆ ɛ f w ln w aw b ln w ] ˆ ɛ ɛ aw b w aɛ b ln ɛ aɛb + ln ɛ. b Note that ln w is negative in the ange,, thus we get the desied inequalities in the pevious steps. REFERENCES ] Necdet Bati. Inequalities fo the gamma function. Achiv de Mathematik, 96: , 8. ] Hon-Fah Chong, Mehul Motani, Hai Kishna Gag, and H El Gamal. On the Han-Kobayashi egion fo the intefeence channel. IEEE Tansactions on Infomation Theoy, 547: , 8. 3] Raul H Etkin, David Tse, and Hua Wang. Gaussian intefeence channel capacity to within one bit. Infomation Theoy, IEEE Tansactions on, 54: , 8. 4] R.K. Fasani. The capacity egion of the wieless egodic fading intefeence channel with patial CSIT to within one bit. In Infomation Theoy Poceedings ISIT, 3 IEEE Intenational Symposium on, pages , July 3. 5] Tiangao Gou, Syed Ali Jafa, Chenwei Wang, Sang-Woon Jeon, and Sae-Young Chung. Aligned intefeence neutalization and the degees of feedom of the intefeence channel. Infomation Theoy, IEEE Tansactions on, 587: , July. 6] Myung Gil Kang and Wan Choi. Egodic intefeence alignment with delayed feedback. axiv pepint axiv:33.3, 3. 7] Changho Suh and David Tse. Feedback capacity of the gaussian intefeence channel to within bits. Infomation Theoy, IEEE Tansactions on, 575: , May. 8] D. Tuninetti. Gaussian fading intefeence channels: Powe contol. In Signals, Systems and Computes, 8 4nd Asiloma Confeence on, pages 7 76, Oct 8. 9] A. Vahid, M.A. Maddah-Ali, and A.S. Avestimeh. Capacity esults fo binay fading intefeence channels with delayed csit. Infomation Theoy, IEEE Tansactions on, 6:693 63, Oct 4. ] I-Hsiang Wang, Changho Suh, Suhas Diggavi, and Pamod Viswanath. Busty intefeence channel with feedback. In Infomation Theoy Poceedings ISIT, 3 IEEE Intenational Symposium on, pages 5. IEEE, 3.