i.i.d. Mixed Inputs and Treating Interference as Noise are gdof Optimal for the Symmetric Gaussian Two-user Interference Channel

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1 iid Mixed nputs and Treating nterference as Noise are gdof Optimal for the Symmetric Gaussian Two-user nterference Channel Alex Dytso Daniela Tuninetti and Natasha Devroye University of llinois at Chicago Chicago L USA odytso2 danielat uicedu Abstract While a multi-letter limiting expression of the capacity region of the two-user Gaussian interference channel is known capacity is generally considered to be open as this is not computable Other computable capacity outer bounds are known to be achievable to within /2 bit using Gaussian inputs and joint decoding in the simplified Han and Kobayashi singleletter achievable rate region This work shows that the simple scheme known as treating interference as noise without timesharing attains the capacity region outer bound of the symmetric Gaussian interference channel to within either a constant gap or a gap of order Olog logsnr for all parameter regimes The scheme is therefore optimal in the generalized Degrees of Freedom gdof region sense almost surely The achievability is obtained by using iid mixed inputs ie a superposition of discrete and Gaussian random variables in the multi-letter capacity expression where the optimal number of points in the discrete part of the inputs as well as the optimal power split among the discrete and continuous parts of the inputs are characterized in closed form An important practical implication of this result is that the discrete part of the inputs behaves as a common message whose contribution can be removed from the channel output even though joint decoding is not employed Moreover time-sharing may be mimicked by varying the number of points in the discrete part of the inputs NTRODUCTON The memoryless real-valued additive white Gaussian noise interference channel G-C has input-output relationship Y n = h X n + h 2 X2 n + Z n a Y2 n = h 2 X n + h 22 X2 n + Z2 n b where Xj n := X j X jn and Yj n := Y j Y jn are the length-n vector inputs and outputs respectively for user j [ : 2] the noise vectors Zj n have iid zero-mean unitvariance Gaussian components for n the block length The input Xj n is a function of the independent message W j that is uniformly distributed on [ : 2 nrj ] where R j is the rate for user j [ : 2] and is subject to a per-block power constraint n n i= X2 ji Receiver j [ : 2] wishes to recover W j from the channel output Yj n with arbitrarily small probability of error Achievable rates and capacity region are defined in the usual way [] For sake of simplicity we shall focus from now on on the symmetric G-C only defined as h 2 = h 22 2 = S 0 h 2 2 = h 2 2 = 0 and denote the capacity region as CS The restriction to the symmetric case is just to reduce the number of parameters in our derivations; we believe that the results of this paper hold for the general asymmetric case Past Work The general information stable two-user interference channel was first introduced in [2] and its capacity may be characterized as C = lim n co P X n X n 2 =P X n P X n 2 R n Xn ; Y n R 2 n Xn 2 ; Y2 n where co denotes the convex hull operation Unfortunately the capacity expression in 2 is considered uncomputable in a sense that it is not known explicitly how to characterize the input distributions that attain its convex closure Moreover it is not clear whether there exists an equivalent single-letter form for 2 in general For the G-C an equivalent single-letter form is known in the strong interference region [3] given by S in the symmetric case Because of it uncomputability the capacity expression in 2 has received little attention except for [4] where it was shown that jointly Gaussian inputs may not be optimal nstead the field has focussed on finding alternative ways to characterize single-letter inner and outer bounds The best known inner bound is the Han and Kobayashi HK achievable scheme [5] which is capacity achieving in all cases where capacity is known [] Except for the strong interference regime the sum-capacity of the G-C is known exactly only for the Z-channel and in 2 S + 2 the noisy interference regime defined by in the symmetric case for which iid Gaussian inputs in 2 are optimal [] So instead of pursuing exact results the community has recently focussed on giving performance guarantees on approximations of the capacity region n [6] the authors showed that the HK scheme with Gaussian inputs and without time-sharing is optimal to within /2 bit/sec/hz irrespective of the channel parameters This constant gap result provides an exact characterization of the generalized Degrees of Freedom gdof region defined as R i S = S α Dα := d d 2 : d i := lim S i [ : 2] 2 log + S R R 2 is achievable /5/$ EEE 76 ST 205

2 and where Dα : d d 2 d + d 2 [ α] + + maxα d + d 2 max α α + max α α 2d + d 2 [ α] + + max α + max α α d + 2d 2 [ α] + + max α + max α α P X X 2 =P X P X2 4a 4b 4c 4d 4e Contribution n this work we focus on the following capacity inner bound obtained by using iid inputs in 2 R TN R X in S = ; Y 5 R 2 X 2 ; Y 2 commonly referred to as the treating interference as noise TN inner bound Note that the TN region may not be convex because a time-sharing / convex-hull operation is not considered Our major contribution is to show that iid mixed inputs ie a superposition of discrete and Gaussian random variables in the TN region in 5 achieve the capacity region outer bound to within a gap and thus are optimal for the whole gdof region in 4 This work extends our past work in [7] where we demonstrated that iid mixed inputs achieve the symmetric sum-capacity while here we consider the whole capacity region to within a constant gap or to within Olog log S up to a outage set whose Lebesgue measure is a controllable parameter We note that a similar result ie optimality of TN in all parameter regimes was pointed out in [ Remark 62] However the proof is not constructive and is based on showing that TN is optimal for the linear deterministic model LDA and then using a universal gap between the LDA and G-C to arrive at the result Unfortunately this approach results in a very large gap and characterization of the inputs for the G-C from the LDA is not immediate On the other hand our proof is constructive and provides in closed form the optimal number of points in the discrete part of the inputs as well as the optimal power split among the discrete and continuous parts of the inputs An important practical implication of this result is that the discrete part of the inputs behaves as a common message whose contribution can be removed from the channel output even though joint decoding is not employed Moreover time-sharing may be mimicked by varying the number of points in the discrete part of the inputs Notation convention Throughout the paper we adopt the following notation Lower case variables are instances of upper case random variables rv which take on values in calligraphic alphabets We let N d x := + x 6 g x := log + x 7 2 d X := [ ] + HX gap 8 8 gap 8 := 2πe 2 log log + 2 d 2 9 minx where the subscripts d and g remind the reader that discrete and Gaussian respectively inputs are involved HX is the entropy of the discrete rv X and [x] + := max0 x [n : n 2 ] is the set of integers from n to n 2 n f A is a rv we denote its support by suppa The symbol denotes: A is the cardinality of the set A X is the cardinality of suppx of the random variable X or x is the absolute value of the real-valued x For x R x is the largest integer not greater than x d mins := min i j:sis j S s i s j is the minimum distance among the points in the set S With some abuse of notation we also use d minx to denote d minsuppx for a rv X X PAM N d minx denotes the uniform probability mass function over a zero-mean Pulse Amplitude Modulation PAM constellation with suppx = N points minimum N 2 2 distance d minx and average energy E[X 2 ] = d 2 minx ms denotes Lebesgue measure of the set S For i [ : 2] we let i [ : 2] to be i i MAN TOOLS n the rest of the paper due to space limitations proofs are omitted and may be found in [8] At the core of our proofs is the following lower bound on the rate achieved by a discrete input on a point-to-point additive noise channel Prop Let X D be a discrete random variable whose support has size N minimum distance d minxd and average power E[XD 2 ] Let Z be a zero-mean unit-variance random variable independent of X D not necessarily Gaussian Then d X D X D ; X D + Z HX D 0 where d is defined in 8 Rem The upper bound in 0 for Z = Z G N 0 may be tightened to X D ; X D + Z G min HX D g E[X 2 D] since a Gaussian input is capacity achieving for the powerconstrained point-to-point Gaussian noise channel n the following we shall use a mixed input at each user; this implies that a receiver sees a linear combination of two discrete constellations; in order to apply Prop we need to lower bound the minimum distance of the resulting sumset The following set of sufficient conditions will play an important role in evaluating the minimum distance of mixed input constellations in our TN inner bound Prop 2 Let h x h y R 2 be two constants Let X PAM X d minx and Y PAM Y d miny Then h x X + h y Y = X Y d minhxx+h yy = min h x d minx h y d miny under the following conditions either Y h y d miny h x d minx or X h x d minx h y d miny 2a 2b 77

3 When Prop 2 is not applicable we shall use: Prop 3 Let X PAM X d minx and Y PAM Y d miny Then for h x h y R 2 we have h x X + h y X = X Y almost surely as 3a and for any γ > 0 there exists a set E R such that for all h x h y E we have d minhxx+h yy κ γ X Y min h x d minx hx d minx h y d miny max h y d miny 3b Y X where the complement of the set E referred to as the outage set has Lebesgue measure smaller than γ and where γ κ γ X Y = 2 + lnmax X Y MAN RESULT For the G-C in we now evaluate the TN region in 5 with mixed inputs X i = δ i X id + δ i X ig i [ : 2] : 4a 2 X id PAM N i Ni 2 4b X ig N x ig ; 0 P := [N N 2 δ δ 2 ] N N [0 ] [0 ] 4c 4d where the random variables X ij are independent for i [ : 2] j D G A careful choice of the parameters in P will lead to the desired results in different parameter regimes From the TN region in 5 we have: Prop 4 For the G-C the following region is achievable R in S := R in S ; P 5a P N 2 [0] 2 where R in S ; P is a lower bound on the TN region in 5 evaluated for the mixed input in 4 with fixed parameter vector P := [N N 2 δ δ 2 ] given by Sδi R in S ; P := R R 2 : R i d S i + g min logn i g δi + δ i i [ : 2] + δ i 5b and where the equivalent discrete constellations seen at the receivers are δi SXiD + δ i Xi D S i := 5c + Sδi + δ i for i i [ : 2] and the functionals d and g are defined in 8 and 7 respectively From Prop 4 it follows that the following gdof region is achievable: Prop 5 Parametrize the number of points and the power splits in the mixed inputs in 4 as N i = N d S βi β i 0 δ i = + S p pi i 0 6a for i [ : 2] where N d is defined in 6 For the G-C an achievable gdof region ae is D in α = D in α; p 6b p:=[β β 2p p 2]p R 4 + D in α; p = d d 2 : d i β + β 2 + [ p i [α p i ] + ] + minβ i α [α p i ] + i [ : 2] 6c provided that at least one of the following conditions hold for all i [ : 2] α β i and max p i α p i 0 min β i α β i = 0 2 α β i and max p i α p i 0 min β i α β i = 0 3 max p i α p i 0 min β i α β i max β i β i α β i β i = 0 The main result of this paper is the following theorem: 6d 6e 6f Theorem For the symmetric G-C the TN-based achievable region in 5a is optimal to within a constant gap or a gap of order Olog logs up to an outage set and it is therefore gdof optimal up to a set of measure zero Proof: Depending on the parameters S = S α we identify the following operational regimes: Very Strong nterference: α 2 2 Strong nterference: α < 2 3 Weak nterference Type : 2/3 α < 4 Weak nterference Type : /2 α < 2/3 5 Very Weak nterference: 0 α < /2 We shall consider each regime individually by picking the parameters of the mixed inputs such that inner and outer bounds approximately match Here we view the discrete and the Gaussian part of the mixed input as a common and a private message respectively in the HK scheme Thus a critical aspect of our analysis is to identify the optimal rate split for each point on the convex closure of the capacity region outer bound By using [9 Lem 4] which can be thought of as the gdof region in 4 before Fourier-Motzkin elimination we found that either it is optimal to user p i = α in 6a or equally split the rate between common and private messages a Very Strong nterference Regime: α 2: n this regime it is optimal to send only common messages and to perform successive decoding starting with the interfering message While joint decoding is not allowed in our TN region the discrete part of the inputs behaves as common message ie as if it could be decoded at non intended destination The gdof region in 4 is D Very Strong : d d 2 78

4 which is easily matched by the achievable scheme in Prop 5 by using p = [ ] n order to complete the proof we show that the condition in 6d is satisfied: α β i i [ : 2] α β 2 α where the last implication comes from the definition of regime For the finite S the capacity region in maxr R 2 g S We set δ i = 0 N i = N d S i [ : 2] and thus for sum-sets in 5c we have S = S 2 := S; the gap is thus readily computed as gap = g S d S 6πe 2 log 75 2 b Strong nterference: α < 2: Capacity in this regime is achieved by sending only common messages Thus similar to the very strong interference regime we set δ = δ 2 = 0 However unlike in the very strong interference regime we vary the number of points in the discrete part in order to mimic time sharing for a reason that will become clear soon n this regime the gdof region is that of a Compound MAC and in parametric form is given by D Strong α = t [0] d t + t α := β t d 2 t α + t := β 2 t Note that the parameter t [0 ] is used to time-share between the two corner points of the capacity region in this case a MAC-region This parametric form of the gdof region is matched to the achievable scheme in Prop 5 by using pt = [β t β 2 t ] t [0 ] To complete the proof we choose to verify the condition in 6f for i = and for all t [0 ] by symmetry it also holds for i = 2; we have min β t α β 2 t max β t β 2 t α β t β 2 t = max t t α α t α t 0 = 0 t [0 ] and the last equality follows since in this regime we have α β t α β 2 t t [0 ] For finite S the outer bound for this regime can be written as R t g + t g S R Strong S = t [0] +S := g S t R 2 t g +S + t g S := g S 2t By choosing δ i = 0 N i = N d S it i [ : 2] we have that the gap is gap = max gs it d S it i [:2] 2 log 2πe ln + S 2 3 γ 2 where we used Prop 3 to bound minimum distance of the sum-set constellations at the receivers and where such a bound holds everywhere except a set of measure γ c Weak Type : 2/3 α < : n this regime the best known strategy is to send common and private messages with a power split as in [6] Thus as in the strong interference regime we vary the number of points in the discrete parts to mimic time sharing but unlike the strong interference regime we also use the Gaussian part of the inputs ie δ δ 2 are non-zero to allow for a private message As we shall see for our TN-based scheme a single power split as in [6] does not suffice to achieve all points in the gdof region; we will thus also vary δ δ 2 in addition to β and β 2 n this regime the gdof region in 4 can be written in the parametric form as D Weak α = D d+d 2 α t D 2d+d 2 α t t [0] α t D d+d 2 d t2α + t α + α := β a t + α d 2 t α + t2α + α := β 2a t + α d D 2d+d 2 d 2 t α + t α := β 2b t + t α d β 2b t + t α β 2a t + α d 2 t2α + tα + α := β 2a t + α where again t [0 ] is used to time-share Next D d+d 2 α t is matched to Prop 5 by picking pt =[β a t β b t α α] t [0 ] To complete the achievability of D d+d 2 α t we show that 6f is satisfied Due to the symmetry it is enough to verify condition 6f for i = for all t [0 ] which gives α min β a t α β 2a t max β a t β 2a t α β a t β 2a t = α min t2α t α α t α t2α α t [0 ] = α α = 0 t [0 ] The region D 2d+d 2 α t is matched to Prop 5 with pt =[β 2a t β 2b t α t α] t [0 ] 79

5 To complete achievability of D 2d+d 2 α t we verify condition 6f for S the one for S 2 follows similarly that is max0 p α p 2 min β 2a t α β 2b t max β 2a t β 2b t α β 2a t β 2b t = max0 α α t α min t2α + tα α t α α = 0 t [0 ] n order to achieve D 2d+d 2 and ie points on the closure on the region but not on the sum-rate face we had to vary both the common ie β i s and private ie p i s part of the messages to mimic time sharing and power control Another important observation is that we exactly characterized the set of parameters needed to achieve every rate pair on the closure of D Weak a characterization that is not obvious from the work in [6] which in fact seems to suggest that a single power split suffices to achieve all points in the gdof region The proof of optimality of the proposed scheme to within a gap of order Olog logs is quite tedious and it is not reported here for sake of space; the details can be found in [8] Rem 2 We remark that in order to bound the minimum distance of the received constellations in strong and weak type interference regimes we have used Prop 3 which implies that there exists a set of measure 0 where 5a is strictly below the outer bound 4 This comes as no surprise from [0] we know that in the vicinity of α = for two-user G- C constellation based schemes perform poorly when channel gains are rationally dependent and outage sets have been used to circumvent this Moreover for more than two users this becomes an artifact of the capacity region itself d Weak nterference Type : /2 α < 2/3: n this regime the gdof region in 4 can be written as D Weak : α = Dd H +d 2 α t Dd L +d 2 α t D d+d 2 t [0] D 2d+d 2 α t α t d t2 2α +t4α 2 := β a t + t α + t2α d 2 t4α 2 +t2 2α := β b t + t2α + t α d t2 2α + t := β D 2d+d 2 2a t + α d 2 t4α 2 := β 2b t + t2α d β 2b t + t2α β 2a t + α d 2 Next D d+d 2 α t is achievable for all t [0 ] by picking pt = [β a t β b t β a t β b t] in Prop 5 and verifying the condition in 6e which we do not report here as it is similar to the previous regime For α t we pick pt = [β 2a t β 2b t α β 2b t] in Prop 5 and verify the condition 6d for receiver one and condition 6e for receiver two By symmetry α t is achievable by swapping the users The proof of optimality of the proposed scheme to within a constant gap is quite tedious and it is not reported here for sake of space; the details can be found in [8] e Very Weak nterference: 0 α < /2: Gaussian inputs with power control and treating interference as noise are optimal to within /2 bit from the work of [6] Since this scheme is a special case of the TN region with mixed inputs with N = N 2 = the claimed optimality follows V CONCLUSON n this paper we proved that a very simple generally applicable lower bound that does neither require joint decoding nor time sharing is optimal to within an additive gap either constant uniformly over the channel gains or of order Olog logs up to an outage set of controllable measure and thus achieves the optimal gdof region of the twouser symmetric G-C for all channel parameters Our result demonstrates that properly accounting for the distribution of the interference ie not Gaussian with our mixed inputs when treating interference as noise results in near optimal rates in all channel parameters Moreover an exact characterization of the closure of gdof region together with the fact that we used PAM signals with closed form expressions for the number of points and the power splits makes the scheme practical ACKNOWLEDGMENT The work of the authors was partially funded by NSF under awards and 4225 REFERENCES [] A El Gamal and Y-H Kim Network nformation Theory Camrbidge University Press 202 [2] R Ahlswede Multi-way communication channels in Proc EEE nt Symp nf Theory Mar 973 pp [3] A Carleial nterference channels EEE Trans nf Theory vol T-24 no pp Jan 978 [4] R Cheng and S Verdu On limiting characterizations of memoryless multiuser capacity regions EEE Trans nf Theory vol 39 no 2 pp [5] T Han and K Kobayashi A new achievable rate region for the interference channel EEE Trans nf Theory vol T-27 no pp Jan 98 [6] R Etkin D Tse and H Wang Gaussian interference channel capacity to within one bit EEE Trans nf Theory vol 54 no 2 pp Dec 2008 [7] A Dytso N Devroye and D Tuninetti On Gaussian interference channels with mixed gaussian and discrete inputs in Proc EEE nt Symp nf Theory June 204 pp [8] A Dytso D Tuninetti and N Devroye nterference as noise: Friend or foe? Submitted to EEE Trans nf Theory 205 [9] G Bresler and D Tse The two-user Gaussian interference channel: A deterministic view EuroTrans Telecom vol 9 pp Apr 2008 [0] R Etkin and E Ordentlich The degrees-of-freedom of the K-user Gaussian interference channel is discontinuous at rational channel coefficients EEE Trans nf Theory vol 55 no pp Nov