On Gaussian Interference Channels with Mixed Gaussian and Discrete Inputs

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1 On Gaussian nterference Channels with Mixed Gaussian and Discrete nputs Alex Dytso Natasha Devroye and Daniela Tuninetti University of llinois at Chicago Chicago L USA odytso devroye uic.edu Abstract This paper studies the sum-rate of a class of memoryless real-valued additive white Gaussian noise interference channels C achievable by treating interference as noise TN. We develop and analytically characterize the rates achievable by a new strategy that uses superpositions of Gaussian and discrete random variables as channel inputs. Surprisingly we demonstrate that TN is sum-generalized degrees of freedom optimal and can achieve to within an additive gap of O or Olog logsnr to the symmetric sum-capacity of the classical C. We also demonstrate connections to other channels such as the C with partial codebook knowledge and the block asynchronous C.. NTRODUCTON Consider a memoryless real-valued additive white Gaussian noise interference channel with input-output relationship Y n = h X n + h X n + Z n a Y n = h X n + h X n + Z 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 : ] the noise vectors Zj n are independent and have i.i.d. zero-mean unit-variance Gaussian components and the inputs Xj n is subject to a per-block power constraint n n i= X ji for j : ]. nput Xj n j : ] is a function of the independent message W j that is uniformly distributed on : nrj ] where R j is the rate and n the block length. Receiver j : ] wishes to recover W j from the channel output Yj n with arbitrary small probability of error. Achievable rates and capacity region are defined in the usual way ]. n this work we shall focus on the sum-capacity defined as the largest achievable R + R. t was shown by ] that the sum-capacity of any information stable 3] C is given by max X n NP X n X n =P X n P n ; Y n + X n ; Y n. X n n For the Gaussian noise channel in the maximization in is further restricted to inputs satisfying the power constraint. For sake of simplicity we shall focus from now on on the symmetric Gaussian C only defined as h = h = S 0 h = h = 0 and denote its sumcapacity in as CS. n general little is known about the optimizing distribution in and only some special cases have been solved. n 6] it was showed that i.i.d. Gaussian inputs maximize for S. n contrast the authors of 7] show that in general Gaussian inputs do not maximize expressions of the form of. The difficulty of the problem in arises from the competitive nature of the problem 8]: for example say X is i.i.d. Gaussian taking X to be Gaussian increases X n ; Y n but simultaneously decreases X n ; Y n as Gaussians are known to be the best inputs for Gaussian point-to-point channels but are also the worst type of noise for a Gaussian input. A lower bound to the sum-capacity can be obtained by considering i.i.d inputs in thus giving R L S = max X ; Y + X ; Y 3 P X X =P X P X where the maximization in 3 is further restricted to inputs satisfying the power constraint. n 8] 5] the authors demonstrated the existence of input distributions that outperform i.i.d. Gaussian inputs in R L S for certain asynchronous C. Both works use local perturbations of an i.i.d. Gaussian input: 5 Lemma 3] considers a fourth order approximation of mutual information while 8 Theorem ] uses perturbation in the direction of Hermite polynomials of order larger than three. n both cases the input distribution is assumed to have a density. For the cases reported in 5] 8] the improvement over i.i.d. Gaussian inputs shows in the decimal digits of the achievable rates; it is hence not clear that these inputs can actually provide substantial rate gains compared to Gaussian inputs. Recently in 9] for the C with one oblivious receiver we showed that a properly chosen discrete input has a somewhat different behavior than continuous inputs: it may yield a good X ; Y while keeping X ; Y relatively unchanged thus substantially improving the rates compared to using Gaussian inputs in the same achievable region expression. n this work we seek to analytically evaluate the lower bound in 3 for a special class of mixed Gaussian and discrete inputs by generalizing the approach of 9]. Our contributions and paper organization are as follows. n Section we present the main tools used in our analysis: a lower bound on the the mutual information attained by a discrete input on a point-to-point Gaussian noise channel and tools to compute the cardinality and minimum distances of sum-sets. n Section we present an achievable sum-rate valid for any memoryless C obtained by evaluating R L S with an input that consists of the superposition of discrete and Gaussian components which we term mixed input from now on. n Section V we show that in terms of generalized degrees of freedom gdof mixed inputs in R L S achieve

2 the optimal CS. n Section V we show that at any finite S R L S with mixed inputs lies to within a constant gap of O or of loglogs from CS. n Section V we apply our results to the block asynchronous C and the oblivious C. Section V concludes the paper. We use the following notation convention: if A is a random variable r.v. we denote its support by suppa; A is the cardinality of a set A or cardinality of suppa if A is r.v.; the symbol d mins = min i j:sis j S s i s j denotes the minimum distance among the points in the set S. With some abuse of notation we also use d mina to denote d minsuppa for a r.v. A; X N µ σ denotes a real-valued Gaussian r.v. X with mean µ and variance σ ; log denotes logarithms in base and ln in base e; we let x] + := maxx 0 and log + x := logx] + ; x refers to largest integer less than or equal to x; PAMN d min denotes the uniform distribution over a zero-mean real-valued Pulse Amplitude Modulation PAM constellation with N points and minimum distance d min and average energy E = d min N. n the following we will compare R L S to an upper bound on CS denoted by R U S from the classical C with full codebook knowledge at all nodes and with block synchronous communication. R U S is therefore an upper bound to the capacity of block asynchronous or C with partial codebook knowledge. As mentioned earlier from 6] we have R L S = CS = log + S if S. Moreover CS R U S := min O O O 3 with O := log + S cut-set bound a O := log + S from 0] b + S O 3 := log + + log + S from ]. c. MAN TOOLS n this Section we present the main tools to evaluate the lower bound in 3 under mixed inputs. At the core of our proof is the following new lower bound on the rate achieved by a discrete input on a point-to-point Gaussian noise channel. Theorem 9 Theorem ]. Let X D be a discrete r.v. with N distinct masses and minimum distance d min Z G N 0 and S be a non-negative constant. Then X D ; SX D + Z G d N Sd min 5 where for N N and x R we define d N x := logn e log log + N e x] +. The inequality in 5 nicely captures the effect of the cardinality and minimum distance of the discrete constellation on the achievable mutual information. We are not the first to analytically consider discrete inputs. For point-to-point Gaussian noise channel: ] characterized the optimal discrete input distribution at high and low SNR; 3] found tight high-snr asymptotic of mutual information for any discrete constellation whose size N is independent of SNR; in ] the authors considered the point-to-point power-constrained Gaussian noise channel and derived lower bounds on the achievable rate when the input is constrained to be PAM. However in our work we need a firm lower bound that holds for all SNR s and all input distributions. The reason is that we want to carefully select N as a function of SNR a problem that was left open in ]. Note that we can not directly use ] as the sum of two PAMs is not necessarily another PAM. n multi-user settings we may wish to select one user s input as Gaussian another as discrete or both mixtures of discrete and Gaussian. To handle such scenarios based on 5 we need bounds on the cardinality and minimum distance of sums of discrete constellations. f X and Y are two sets denote the sum-set as X + Y := {x + y x X y Y }. Tight bounds on the cardinality and the minimum distance of X + Y for general X and Y are an open problem in number theory. The following set of sufficient conditions will play an important role in evaluating 3 under mixed inputs. Proposition. Let h x h y R be two constants X PAM X d minx and Y PAM Y d miny. Then d minhxx+h yy = min h x d minx h y d miny and h x X + h y Y = X Y if Y h y d miny h x d minx 6 or if X h x d minx h y d miny 7 Proof: The conditions in 6-7 are such that one PAM constellation is completely contained within two points of the other PAM constellation. When Proposition is not applicable we will use the following proposition inspired by 5]: Proposition. Let X and Y be two PAMN d min. Then h x X + h y X = X almost everywhere and d minhxx+h yy min h x h y min γ d min 8 for all h x h y E R where the complement of E has Lesgebue measure smaller than γ for any γ > 0. Proof: The proof can be found in Appendix A.. ACHEVABLE SUM-RATE WTH MXED NPUTS We now evaluate the lower bound in 3 with inputs X i = δ X id + δ X ig δ 0 ] 9a X id PAM N N X ig N 0 9b where X ij are independent for i : ] j {D G}. The input in 9 has two parameters: the number of points N since here we consider unitary power constraint and the power split

3 TABLE : Parameters used in the proof of Theorems and 3. α N δ d d mins L α Additive Gap 0 /] Not Applicable Not Applicable α bit 0] for S / /3] + ɛ S < +S+ + +S+ N ɛα eq.5 for 3 S + S + /3 + + S ɛ + mins +S+ 0 S N N min γ maxα α except on an outage set of measure γ eq.6 for 3 > S + S + < S + S ɛ eq. for S + S δ. Careful choices of these parameters will lead to the desired results in different regimes. The inputs in 9 are symmetric i.e. same parameters for both users since we restrict attention to a symmetric C. Extensions to a non-symmetric Cs are straightforward but more computationally involved. Proposition 3. For the input in 9 R L S may be further lower bounded as R L S d + log + Sδ min logn log S d mins + δ δ where the sum-set S is defined as { δ S := Sx D + x D : + Sδ + δ δ } x D X D x D X D Proof: Due to the symmetry of the problem X ; Y = X ; Y = R L S /; hence for a Z N 0 we have X ; Y = h X + SX + Z h X + Z δ = h Sx D + x D + Z hz + Sδ + δ = d S d mins by Theorem δ h XD + Z hz δ δ XD +δ +Z;X D minlogn δ log+ +δ + logδ + Sδ logδ where the mutual information upper bound follows since Gaussian maximizes the differential entropy for a given second moment constraint and a uniform input maximizes the entropy of a discrete random variable ]. V. HGH SNR PERFORMANCE n this Section we show that the lower bound in Proposition 3 attains the same generalized degrees of freedom gdof as the upper bound in where gdof is defined as R L S = S α d L α := lim S. 0 log + S Similarly define d U α by replacing R L by R U in 0. With the upper bound in we have 0] α d U α = min max α max α α. The main result of this Section is: Theorem. A mixed input as in Proposition 3 achieves the gdof summarized in Table. Proof: The parameters of the input in 9 are chosen as in Table. When α Gaussian inputs are optimal to within bit 0] hence we set δ =. For α / we choose δ such that the interference from the Gaussian portion of the input which is treated as noise is below the level of the noise 0] by setting δ = +. For α we choose to send only discrete inputs by setting δ = 0. n very strong interference S + S α we set δ = 0 and N = + S ɛ for some ɛ 0 where ɛ < insures a non-vanishing rate as S increases and ɛ > 0 insures that the minimum distance increases as S increases. With this δ and N the condition in 6 is N S which is readily verified in this regime. Therefore S = N and d mins = 3S N Sɛ from Proposition. By plugging these values in Proposition 3 an achievable sum-rate is R L d N 3S N min logn log e logn log log + N e 3S N. Finally in the high-snr limit we obtain eq. lim S 0.5 log+s = ɛ. The moderately weak interference regime α / /3] follows similarly: now the condition in Proposition is N S and the sum-rate is given by 3 see next because δ = +. For the remaining regime α /3 we set N = + S α/ and δ = + and therefore from Proposition S = N and d mins as in Table almost surely for all channel parameters. Here d mins increases in S i.e. no need for the ɛ parameter as in the previous regimes but the result holds for all channel parameters up to an outage set of measure less than γ where γ > 0 also affects the minimum distance. By plugging these values in Proposition 3

4 we can further lower bound the achievable sum-rate as e R L logn log log + N e d mins + log + S. 3 Finally in the high-snr limit we obtain eq.3 lim S 0.5 log+s = α + α]+ = max α α. Theorem shows that a mixed input can achieve the optimal same as the classical C gdof: exactly in very weak interference α 0 /] arbitrarily close in moderately weak interference α / /3] and very strong interference α > and up to an outage set similar to 5] in α /3 ]. This result especially for the strong and very strong interference regimes is unexpected. n the classical C which we use as an upper bound we know that i.i.d. Gaussian inputs with joint decoding of the intended and interfering message is optimal in strong and very strong interference. The lower bound R L S however seems to imply TN; if this interpretation of the mutual information expression of R L S were correct then we would conclude that 0 gdof are achievable if both users are active because the strong interference swamps the useful signal at each receiver and since it is Gaussian it is the worst find of noise; indeed i.i.d. Gaussian inputs give R L S = log + S + that corresponds to 0 gdof for S. Our work shows that by selecting non-gaussian inputs with more structure and by exploiting knowledge of this structure leads to unbounded gains gdof gains even when interference is treated as noise. V. FNTE SNR PERFORMANCE n this Section we refine the gdof result of Theorem by proving an additive gap between a sum-capacity upper bound in and the sum-rate achievable with a mixed input in Proposition 3. This improves the result of Section V and demonstrates that the classical C gdof are exactly achievable. Theorem 3. A mixed input as in Proposition 3 achieves the gap summarized in Table. Proof: We analyze the different regimes separately. We only consider the case S otherwise a trivial gap of bit can be achieved by silencing both users and otherwise a trivial gap of bit can be achieved by using Gaussian inputs and treating interference as noise i.e. δ = in 9. Very strong interference α : in in addition to setting log the parameters as in Table we further set ɛ = so that log + N e 3S N bit. The gap is the difference between R U in a and R L and is bounded as R U R L + S log + S ɛ e + log + a + S log + S ɛ e + log + 3 lns logs ] + b e S ɛ + log + ] c + e 3 lns + log + where: a x x for x b +x +x x ɛ for x ɛ and c definition of ɛ. Moderately weak interference regime / < α /3: the gap analysis is similar to the very strong interference regime but we now use the upper bound in b: we first notice that the difference between the upper bound in b and the last term in 3 can be upper bounded as log + S + log + S + + S + log therefore the gap analysis is the same as that leading to if we replace S with +S+ and we add an extra bit; we therefore conclude that in this regime the gap is R U R L 3 ln ] + e + log S + ] + S e 3 ln + log where the last inequality follows since S in this regime. Very weak interference α 0 /]: Gaussian inputs and treat interference as noise are optimal to within bit 0]. Remaining regime α /3 : the difference between the upper bound in c and the last term in 3 satisfies + S log+ + log + S log + S log + log thus the gap with N = + is R U R L e log logn + + log for 0 + log + N e d mins where the last term is not bounded for α = /3 or α = because at these points d min does not increase with S. We remedy this by slightly changing the number of points of the discrete part of the input to N = + ɛ and pick ɛ = log log ] + ln 3 minγ to ensures that log + N exp d mins /. With this we can show + ɛ R U R L log log log + log e ln ] + e 3 min γ log. 6

5 We note however in strong interference for ln 3 minγ S S ln and in moderate interference for 3 minγ 3 we have that log + N exp d mins / ; therefore ɛ is not needed and the gap is R U R L 5 + log e. V. APPLCATONS OF OUR RESULTS We have evaluated a very simple generally applicable lower bound to the capacity of any C with a mixture of discrete and Gaussian inputs and shown that through careful choice of the discrete input where the number of points may depend on the SNR that such an input may achieve to within an approximately constant gap of the capacity of a classical C. This result is of interest in several channels as outlined next. n 5] the authors studied the block asynchronous C whose sum-capacity is denoted by C AC-C. t was shown in 5] that R L S C AC-C R U S. Hence our results apply directly and imply that using mixed inputs of the type proposed here we may approximately in the sense of Theorem 3 achieve the capacity of the classical synchronous C. Another application is to the C with oblivious receivers introduced in ] where both receivers lack knowledge of the codebook of the interfering transmitter. Lack of codebook knowledge of the interfering signal prevents the decoders from using joint decoding of messages and successive interference cancellation which are known to be capacity achieving in a classical C where all codebooks are available to all nodes. Denote the sum-capacity of the oblivious C by C OB-C. ] demonstrated that R L S = C OB-C is indeed the capacity of the oblivious C however the input distribution that maximizes R L S was not found. Our results again directly apply and demonstrate the surprising fact that even if receivers do not know the interfering codebooks and cannot perform joint decoding of messages using a mixture of discrete and Gaussian inputs one can overcome this lack of codebook knowledge and approximately achieve the capacity of the C where all codebooks are known. V. CONCLUSON We studied the performance of mixed inputs on the Gaussian C. ts application to oblivious and asynchronous Cs somewhat surprisingly implies that much less global coordination between nodes is needed than one might expect: synchronism and codebook knowledge might not be critical if one is happy with approximate capacity results. We showed that TN is gdof optimal and within Olog logs of capacity. APPENDX Let S = h x X + h y Y and z Z. We want to find d mins := min i j { s i s j : s i s j S} with s i s j = h x x i + h y y i h x x j h y y j. Case x i = x j and y i y j or x i x j and y i = y j : s i s j = h y d min Y z i z j h y d min Y or s i s j h x d min X. Case x i x j and y i y j : s i s j = h y d min Y h x d min X h y d min Y z xi z xj z yj z yi. Next we lower bound hz z where h := hxdminx h yd miny. Let E = {h : hz z γ} for some γ > 0. The Lebesque measure of E c this will be our outage set is me c = m{h : γ + z < h < γ + z } = γ z z < γ. Hence on a set E we have the following lower bound s i s j h y d min Y γ. By symmetry s i s j h x d min X γ. Putting all the cases together we have: d mins min h y d min Y h x d min X min γ. Now assume X = Y = N and d miny = d minx. f S = N then there exists s i = h x x i + h y y i and s j = h x x j + h y y j such that s i = s j for some i j which in turn implies that x i y i x j y j such that h x h y = y j y i x i x j = z yi z yj d min Y z xi z xj d min X = z yi z yj z xi z xj 7 h x h y i.e. Q in order for S N. Since rationals have measure zero this implies that S = N a.e.. REFERENCES ] A. El Gamal and Y.-H. Kim Network nformation Theory. Camrbidge University Press 0. ] R. Ahlswede Multi-way communication channels in Proc. EEE nt. Symp. nf. Theory Tsahkadsor Mar. 973 pp ] R. Dobrushin General formulation of Shannon s main theorem in information theory American mathematical society translations: 9 papers on differential equations on information theory p ] O. Simeone E. Erkip and S. 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