The Capacity Loss of Dense Constellations

Transcription

1 The Capaity Loss of Dense Constellations Tobias Koh University of Cambridge Alfonso Martinez Universitat Pompeu Fabra Albert Guillén i Fàbregas ICREA & Universitat Pompeu Fabra University of Cambridge guillen@ieee.org Abstrat We determine the loss in apaity inurred by using signal onstellations with a bounded port over general omplex-valued additive-noise hannels for suitably high signalto-noise ratio. Our expression for the apaity loss reovers the power loss of 1.53dB for square signal onstellations. I. INTRODUCTION As it is well known, the hannel apaity of the omplexvalued Gaussian hannel with input power at most P and noise variane isgivenby1] C G (P,= P. (1 Although inputs distributed aording to the Gaussian distribution attain the apaity, they suffer from several drawbaks whih prevent them from being used in pratial systems. Among them, espeially relevant are the unbounded port and the infinite number of bits needed to represent signal points. In pratie, disrete distributions with a bounded port are typially preferred in this ase, the number of points is allowed to grow with the signal-to-noise ratio (NR. Ungerboek omputed the rates that are ahievable over the Gaussian hannel when the hannel input takes value in a finite onstellation ]. He observed that, when transmitting at a rate of R bits per hannel use, there is not muh to be gained from using onstellations with size N larger than R+1. Ozarow and Wyner provided an analyti onfirmation of Ungerboek s observation by deriving a lower bound on the rates ahievable with finite onstellations 3]. In both works, the hannel inputs are assumed to be uniformly distributed on a lattie within some enlosing boundary, where the size of the boundary is saled in order to ensure unit input-power. A related line of work onsidered signal onstellations with favorable geometri properties, e.g., minimum Eulidean distane or minimum average error probability. For signal onstellations with a large number of points, i.e., dense onstellations, Forney et al. 4] estimated the loss in NR with πe respet to the Gaussian input to be dB by omparing the volume of an n-dimensional hyperube with that of an n-dimensional hypersphere of idential average power. Later, Ungerboek s work led to the study of multidimensional onstellations based on latties 5] 8]. The researh leading to these results has reeived funding from the European Community s eventh Framework Programme (FP7/ under grant agreement No and from the European Researh Counil under ERC grant agreement Reently, Wu and Verdú have studied the information rates that are ahievable over the Gaussian hannel when the input takes value in a finite onstellation with N signal points 9]. For every fixed NR, they show that the differene between the apaity and the ahievable rate tends to zero exponentially in N. For the optimal onstellation, the peak-to-average-power ratio grows linearly with N, induing no apaity loss. This is in ontrast to the onstellations onsidered by Ungerboek ] and Ozarow and Wyner 3], whih have a finite peak-toaverage-power ratio. In this work, we adopt an information-theoreti perspetive to study the apaity loss inurred by signal onstellations with a bounded port over the Gaussian hannel for suffiiently small noise variane. In partiular, we use the duality-based upper bound to the mutual information in 10] to provide a lower bound on the apaity loss. The results are valid for both peak- and average-power onstraints and generalize diretly to other additive-noise hannel models. For suffiiently high NR, our results reover the power loss of 1.53dB for square signal onstellations without invoking geometrial arguments. II. CHANNEL MODEL AND CAPACITY We onsider a disrete-time, omplex-valued additive noise hannel, where the hannel output Y k at time k Z (where Z denotes the set of integers orresponding to the time-k hannel input x k is given by Y k = x k + W k, k Z. ( We assume that W k,k Z is a sequene of independent and identially distributed, entered, unit-variane, omplex random variables of finite differential entropy. We further assume that the distribution of W k does neither depend on >0 nor on the sequene of hannel inputs x k,k Z. The hannel inputs take value in the set, whih is assumed to be a bounded Borel subset of the omplex numbers C. We further assume that has positive Lebesgue measure and that 0. The set an be viewed as the region that its the signal points. For example, for a square signal onstellation, itisa square: x C: A Re (x A, A Im (x A (3 for some A > 0. HereRe(x and Im (x denote the real and imaginary part of x, respetively. imilarly, for a irular signal onstellation, x C: x R, for some R > 0. (4

2 We study the apaity of the above hannel under an average-power onstraint P on the inputs. ine the hannel is memoryless, it follows that the apaity C (P, (in nats per hannel use is given by C (P,= I(X; Y (5 where the remum is over all input distributions with essential port in that satisfy E X ] P. We fous on C (P, in the it as the noise variane tends to zero. In partiular, we study the apaity loss, whih we define as L 0 C C (P, C (P,. (6 (Theorem 1 ahead asserts the existene of the it. Here C C (P, denotes the apaity of the above hannel when the port-onstraint is relaxed, i.e., C C (P,= For small, wehave1] I(X; Y. (7 E X ]P C C (P,= P + (πe h(w +o(1 (8 where the o(1-term vanishes as tends to zero. (Here ( denotes the natural arithm and h( denotes differential entropy. The apaity loss (6 an thus be written as L =P + (πe h(w 0 I(X; Y 1. (9 By hoosing an input distribution that does not depend on, we an ahieve 1 Indeed, we have L P + (πe h(x. (10 I(X; Y =h(x + W h(w + 1 (11 whih follows from the behavior of differential entropy under deterministi translation and under saling by a omplex number. Extending 10, Lemma 6.9] (see also 11] to omplex random variables yields then that, for every E X ] < and E W ] <, thefirst differential entropy on the right-hand side (RH of (11 satisfies h(x + W =h(x. (1 0 Consequently, we obtain 0 I(X; Y 1 0 I(X; Y 1 = h(x h(w (13 1 We define h(x = if the distribution of X is not absolutely ontinuous with respet to the Lebesgue measure. whih together with (9 yields (10. Let P U denote the average power of a random variable that is uniformly distributed over, i.e., P U x dx dx. (14 A small modifiation of the proof in 1, Th ] shows that the density that maximizes h(x for X with probability one and E X ] P has the form e λ x f (x = I x, x C (15 e λ x dx where λ =0for P P U,andwhereλsatisfies e λ x x dx = P (16 e λ x dx for P < P U.HereIstatement denotes the indiator funtion: it is equal to one if the statement in the brakets is true and it is otherwise equal to zero. Applying (15 to (10 yields ( L P + (πe e λ x dx λ P. (17 For P = P U (and hene λ =0, this beomes ( ( L (πe+ x dx dx. (18 peializing (18 to a square signal onstellation (3 yields (irrespetive of A L πe (19 6 whih orresponds to a power loss of roughly 1.53dB. Hene, we reover the rule of thumb that square signal onstellations have a 1.53dB power loss at high signal-to-noise ratio. For a irular signal onstellation (4, the upper bound (18 beomes (irrespetive of R L e (0 reovering the power loss of 1.33dB 4]. The inequality in (17 holds with equality if the apaityahieving input-distribution does not depend on, f. (13. However, this is in general not the ase. For example, for irularly-symmetri Gaussian noise and a irular signal onstellation (4, it was shown by hamai and Bar-David 13] that, for every >0, the apaity-ahieving input-distribution is disrete in magnitude, with the number of mass points growing with vanishing. Nevertheless, the following theorem demonstrates that the RH of (17 is indeed the apaity loss. Theorem 1 (Main Result: For the above hannel model, we have ( L =P + (πe e λ x dx λ P (1 where λ = 0 for P P U,andwhereλ satisfies (16 for P < P U. Proof: ee etion III.

3 Note 1: It is not diffiult to adapt the proof of Theorem 1 to other regions and moment onstraints. For example, the same proof tehnique an be used to derive the apaity loss when is a Borel subset of the real numbers and the hannel input s first-moment is ited, i.e., E X ] A. Equations (11 (13 demonstrate that the apaity loss (1 an be ahieved with a ontinuous-valued hannel input having density f (. Using the lower-semiontinuity of relative entropy 14], it an be further shown that (1 an also be ahieved by any sequene of disrete hannel inputs X N for whih the number of mass points N grows with vanishing, provided that L X N X as N ( where X is a ontinuous random variable having density f (. (Here L denotes onvergene in distribution. uh a sequene an, for example, be obtained by approximating the distribution funtion orresponding to f ( by twodimensional step funtions. III. PROOF OF THEOREM 1 In view of (9, in order to prove Theorem 1 it suffies to show that I(X; Y 1 0 ( e λ x dx + λ P h(w. (3 The laim follows then by ombining (3 with (17. To this end, we use the upper bound on the mutual information 10, Th. 5.1] I(X; Y D ( W ( x R( dq(x (4 where Q( denotes the input distribution; W ( x denotes the onditional distribution of the hannel output, onditioned on X = x; andr( denotes some arbitrary distribution on the output alphabet. Every hoie of R( yields an upper bound on I(X; Y, and the inequality in (4 holds with equality if R( is the atual distribution of Y indued by Q( and W (. To derive an upper bound on I(X; Y, we apply (4 with R( having density r(y = where K, e λ y K,, y 1 1 K, e λ y 1 π y 1+ y /, dy + y / (5 1 1 π y 1+ y dy (6 / is a normalizing onstant; where denotes the - neighborhood of y C: y x, for some x ; (7 where denotes the omplement of ;andwhereλis zero for P P U and satisfies (16 for P < P U. ome useful properties of K, are summarized in the following lemma. Lemma : The normalizing onstant K, satisfies inf K, > 0 >0, >0 (8a K, = e λ y dy. (8b 0 0 Proof: Omitted. We return to the analysis of I(X; Y and apply (4 together with the density (5 to express the upper bound as D ( W ( x R( dq(x = h(y X p(y x r(y dy dq(x (9 where p(y x denotes the onditional probability density funtion of Y, onditioned on X = x. Evaluation of the onditional differential entropy gives h(y X =h(w 1 (30 and some algebra applied to the seond summand in (9 allows us to write it as p(y x r(y dy dq(x =K, + λ E I Y ] + ( π Pr ( ( ] + E I Y ] + E I. (31 Combining (30 and (31 with (9 and (4 yields I(X; Y h(w + 1 +K, + λ E I Y ] + ( π Pr ( ( ] + E I Y ] + E I Y. (3 We next show that, for >0, E I Y ] P (33a ( π Pr ( =0 (33b ( ] E I =0 (33 ] E I =0. (33d

4 The first laim (33a follows by upper-bounding E I Y ] E ] = E X ] + E W ] P + (34 where the seond step follows beause X and W are independent, and the third step follows beause E X ] P and E W ] =1. To prove (33b, we first note that Pr ( ( Pr W >. (35 Indeed, if w, thenwehave y x = x+w x for x = x,soy. By Chebyshev s inequality 15, e. 5.4], this an be further upper-bounded by Pr (. (36 It then follows that, for 1 π, 0 ( π Pr ( ( π (37 where the right-most term vanishes as tends to zero. This proves (33b. We next turn to (33. We first note that every y must satisfy y >, sine otherwise y x for x =0,whih by assumption is in. Therefore, ( ] ( E I Y Pr ( Y 0, for. (38 To prove (33, it thus remains to show that ( ] E I Y 0 X,E X ] 0. (39 By Jensen s inequality, we have ( ] E I Y Pr ( (E Y I ] Pr ( ( 1 Pr( P + Pr ( (40 where the last step follows from the Cauhy-hwarz inequality E I ] E ] Pr (. (41 Using (36 together with the fat that ξ ξ ξ is monotonially inreasing for ξ e 1, we obtain for e 1/ ( ] E I Y 1 P (4 from whih (39 and hene (33 follows by noting that the RH of (4 vanishes as tends to zero. To prove (33d, we use Jensen s inequality and (34 to obtain E ] I Y Pr ( ( 1+ E I ] Pr ( Pr ( ( 1+ P + Pr( Pr ( ( Pr Y. (43 Using (36 together with the fat that ξ ξ ξ is monotonially inreasing for ξ e 1, we obtain for e 1/ ] 0 E I P + (44 from whih (33d follows by noting that the RH of (44 vanishes as tends to zero. Combining (33a (33d with (3 yields I(X; Y 1 0 h(w + K, + λ P 0 ( = h(w + K, + λ P (45 0 where the last equation follows from the ontinuity of x (x for x>0. Letting tend to zero, and using (8b in Lemma, we prove (3 and therefore the desired ( L =P +(πe e λ y dy λ P. (46 IV. NONAYMPTOTIC CAPACITY LO A natural approah to prove Theorem 1 would be to generalize (1 to 0 h(x + W = h(x. (47 While this approah may seem simpler, our approah has the advantage that it also allows for a lower bound on the nonasymptoti apaity loss L( C C (P, C (P,, > 0. (48 Indeed, ombining (43, (40, and (34 with (3 yields I(X; Y h(w + 1 +K, + λ ( P + + +( π Pr ( Y + 1 Pr( Y ( 1+ P + Pr ( ( 1+ P + Pr( 3 Pr( ( Pr Y (49

5 L( asymptoti apaity loss L 10 -ary QAM 16 -ary QAM -ary QAM / db] lower bound on L( Fig. 1. The apaity loss L( for irularly-symmetri Gaussian noise and square onstellations with P = P U. where + (ξ max0, ξ, ξ>0. By upper-bounding K, e λ y dy +1 ( π tan 1 (50 (where tan 1 ( denotes the artangent funtion, and by using (35 together with the fat that ξ ξ ξ is monotonially inreasing for ξ e 1 and that ξ ξ 1/e for 0 <ξ<1, we obtain, upon minimizing over >0, C (P, inf h(w + 1 >0 + λ( P + ( + e λ y dy +1 ( π tan 1 + +( π Pr ( W > + 1 Pr( W > P + Pr ( W > P + Pr( W > 3 Pr( W > (Pr ( W > I Pr ( W > 1/e + 3 e I Pr ( W > > 1/e. (51 This together with (48 yields a lower bound on L(. Figure 1 shows the lower bound on L( for irularlysymmetri Gaussian noise and a square signal onstellation (3 with P = P U. It further shows the information-rate losses of m -ary quadrature amplitude modulation (QAM for m =10, 16, and, whih were numerially obtained using Gauss-Hermite quadratures 16], as desribed for example in 17, e. III]. ine for a fixed m the information rate orresponding to m -ary QAM is bounded by m bits, the rate loss of m -ary QAM tends to infinity as tends to zero. We observe that the lower bound on L( onverges to L = (πe/ as tends to zero, but is rather loose for finite. However, in the proof of Theorem 1 we hose the density (5 to deay suffiiently slowly, so as to ensure that the lower bound on L holds for every unit-variane noise of finite differential entropy. For Gaussian noise, a density an be hosen that deays muh faster, giving rise to a tighter bound. ACKNOWLEDGMENT The authors would like to thank Alex Alvarado for helpful disussions and for providing the QAM urves in Figure 1. REFERENCE 1] C. E. hannon, A mathematial theory of ommuniation, Bell ystem Tehn. J., vol. 7, pp and , July and Ot ] G. Ungerboek, Channel oding with multilevel/phase signals, IEEE Trans. Inform. Theory, vol. 8, pp , Jan ] L. H. Ozarow and A. D. Wyner, On the apaity of the Gaussian hannel with a finite number of input levels, IEEE Trans. Inform. Theory, vol. 36, pp , Nov ] G. D. Forney, Jr., R. G. Gallager, G. R. Lang, F. M. Longstaff, and G. R. Qureshi, Effiient modulation for band-ited hannels, IEEE J. elet. Areas Commun., vol. AC-, pp , ept ] G. D. Forney, Jr. and L.-F. Wei, Multidimensional onstellations Part I: Introdution, figures of merit, and generalized ross onstellations, IEEE J. elet. Areas Commun., vol. 7, no. 6, pp , Aug ] G. D. Forney, Jr., Multidimensional onstellations Part II: Voronoi onstellations, IEEE J. elet. Areas Commun., vol. 7, no. 6, pp , Aug ] A. R. Calderbank and L. H. Ozarow, Nonequiprobable signaling on the gaussian hannel, IEEE Trans. Inform. Theory, vol. 36, no. 4, pp , July ] F. R. Kshishang and. Paathy, Optimal nonuniform signaling for gaussian hannels, IEEE Trans. Inform. Theory, vol. 39, no. 3, pp , May ] Y. Wu and. Verdú, The impat of onstellation ardinality on Gaussian hannel apaity, in Pro. 48th Allerton Conf. Comm., Contr. and Comp., Allerton H., Montiello, Il, ept. 9 Ot. 1, ] A. Lapidoth and. M. Moser, Capaity bounds via duality with appliations to multiple-antenna systems on flat fading hannels, IEEE Trans. Inform. Theory, vol. 49, no. 10, pp , Ot ] T. Linder and R. Zamir, On the asymptoti tightness of the hannon lower bound, IEEE Trans. Inform. Theory, vol. 40, no. 6, pp , Nov ] T. M. Cover and J. A. Thomas, Elements of Information Theory, nded. John Wiley & ons, ]. hamai (hitz and I. Bar-David, The apaity of average and peak-power-ited quadrature Gaussian hannels, IEEE Trans. Inform. Theory, vol. 41, no. 4, pp , July ] E. C. Posner, Random oding strategies for minimum entropy, IEEE Trans. Inform. Theory, vol. 1, pp , July ] R. G. Gallager, Information Theory and Reliable Communiation. John Wiley & ons, ] M. Abramowitz and I. A. tegun, Handbook of Mathematial Funtions with Formulas, Graphs, and Mathematial Tables. Dover Publiations, ] A. Alvarado, F. Brännström, and E. Agrell, High NR bounds for the BICM apaity, in Pro. Inform. Theory Workshop (ITW, Paraty, Brazil, Ot. 16 0, 011.