Capacity-Approaching Signal Constellations for the Additive Exponential Noise Channel

Transcription

1 Capacity-Approachig Sigal Cotellatio for the Additive Expoetial Noie Chael Stéphae Y. Le Goff School of Electrical, Electroic ad Computer Egieerig Newcatle Uiverity, Uited Kigdom Abtract We preet a ew family of igal cotellatio, called log cotellatio, that ca be ued to deig ear-capacity coded modulatio cheme over additive expoetial oie (AEN) chael. Log cotellatio are deiged by geometrically approximatig the iput ditributio that maximize the AEN chael capacity. The mutual iformatio achievable over AEN chael with both coded modulatio (CM) ad bit-iterleaved coded modulatio (BICM) approache i evaluated for variou igal et. I the cae of CM, the propoed log cotellatio outperform, ometime by over half a decibel, the bet exitig igal et available from the literature, ad ca diplay error performace withi oly 0.12 db of the AEN chael capacity. I the cotext of BICM, log cotellatio do ot offer igificat performace advatage over the bet exitig cotellatio. A the potetial performace degradatio reultig from the ue of BICM itead of CM i larger tha 1 db, BICM may however ot be a uitable deig approach over AEN chael. I. INTRODUCTION The additive expoetial oie (AEN) chael cotitute the atural dicrete-time model for the cotiuou-time Gauia chael whe the tramitted igal i affected by a fat-varyig phae oie [1]. The reultig lack of coherece betwee igal ad oie compoet at the receiver iput make it impoible to detect the complex-valued quadrature amplitude. I uch cae, iformatio ca oly be tramitted by modulatig the igal eergy ad uig a direct (o-coheret) receiver architecture to detect it. A a example, the AEN chael ca be coidered a a good model for 1

2 digital commuicatio ytem uig electromagetic radiatio a a photo ga with o quatum iterferece effect [2]. I 1996, Verdú ivetigated variou iformatio-theoretical apect of the expoetial ditributio ad howed that the AEN chael hare a umber of imilaritie with it additive white Gauia oie (AWGN) couterpart [3]. I particular, he determied the expreio of the AEN chael capacity ad foud it to be idetical to that of a equivalet AWGN chael. More recetly, Martiez coidered the iue of deigig coded modulatio (CM) cheme for the AEN chael [1], [4]. Oe of hi mot iteretig reult wa the itroductio of a family of igal cotellatio that ca perform approximately 0.76 db away from capacity at high igal-to-oie ratio (SNR). I thi paper, we propoe aother family of igal et, hereafter called log cotellatio, that ca igificatly outperform thoe itroduced by Martiez whe ued for CM deig. I particular, it i how that, whe the umber of cotellatio ymbol i greater tha or equal to 1024, a error performace withi oly 0.12 db of the capacity i achievable with a CM ytem uig log cotellatio. We alo coider the deig of bit-iterleaved coded modulatio (BICM) ytem baed o the propoed log cotellatio. Recall that BICM i a low-complexity, ub-optimal alterative to coded modulatio itroduced by Zehavi i 1992 [5]. Due to it implicity ad earoptimal error performace over AWGN ad fadig chael, thi approach ha actually become the de facto tadard for wirele commuicatio ytem (ee, e.g., [6] ad [7]). It i how that, over AEN chael, the error performace i igificatly degraded whe employig BICM rather tha CM. The obtaied reult alo idicate that log cotellatio are ot particularly attractive for BICM. The remaider of the paper i orgaized a follow. I Sectio II, we provide a brief review of the AEN chael ad decribe the techique ued to obtai the propoed log cotellatio. The mutual iformatio achievable with the CM approach uig log cotellatio i the evaluated i Sectio III for differet umber of cotellatio ymbol ad compared to the reult obtaied with Martiez igal et. I Sectio IV, imilar evaluatio ad compario of mutual iformatio are performed i the cotext of BICM. Fially, cocluio are draw i Sectio V. 2

3 II. A NEW FAMILY OF SIGNAL CONSTELLATIONS FOR AEN CHANNELS A. AEN Chael Model Coider the tramiio of a ymbol x {x 1,, x M } draw from a oe-dimeioal M-ary igal cotellatio compoed of o-egative real ymbol x 1, x 2, ad x M. At the output of a AEN chael, the correpodig ample y i give by y x, (1) where deigate a o-egative real oie ample with a probability deity fuctio (PDF) I thi expreio, fuctio defied a follow: x 1 ratio (SNR) i defied a where P x 1 exp E x E u x E deote the average value of the oie ample ad u for 0 x, ad x 0 E. (2) u x i the uit tep u for x 0. The chael igal-to-oie, (3) E E i the average value of the tramitted M-ary ymbol x. To implify the otatio ued throughout thi paper, we will aume from ow o that the igal cotellatio ha uit average value, i.e. 1 E ad thu E 1. I hi 1996 paper, Verdú howed that the capacity of the AEN chael (expreed i bit/chael ue) i give by C log 2 1, (4) ad i therefore idetical to that of a equivalet AWGN chael [3]. I order to achieve thi capacity, the chael iput x mut have the followig ditributio: P x x E E E 2 exp E x E E E E x, x 0, (5) 3

4 which i, with the otatio ued i thi paper, equivalet to P x x where x deote the Dirac delta fuctio. 2 1 exp x x, x 0, (6) B. Sigal cotellatio for the AEN Chael Eq. (6) clearly implie that, i AEN chael, igal cotellatio that are compoed of equiprobable uiformly paced ymbol caot achieve the capacity give by (4). Actually, it wa how i [1] ad [4] that, at high SNR, the gap betwee chael capacity ad uiformly paced equiprobable igalig i approximately equal to 1.33 db. It i worth metioig that thi SNR gap i lightly maller tha that obtaied at high SNR over AWGN chael, the latter beig approximately equal to 1.53 db [8]. Cotellatio hapig mut be ued i order to reduce the SNR gap. Perhap the implet method to implemet cotellatio hapig i called geometrical hapig ad coit of uig a equiprobable igal et for which the ymbol are ot uiformly paced o the oe-dimeioal axi. I particular, it ha bee how that geometrical Gauia-like M-ary igal cotellatio ca achieve the chael capacity of the AWGN chael a M [9]. I [1] ad [4], Martiez propoed a method to implemet geometrical hapig for the AEN chael. Baically, he itroduced a family of M-ary igal et compoed of equiprobable o-egative real ymbol x i, i {1, 2,, M}, defied a follow: x i i 1, (7) where i a deig parameter that ca be et to ay value ad i imply a ormalizatio parameter computed o that the average value of the ymbol x i i equal to the uit, i.e. E 1. Martiez wa able to how that operatio at high SNR at approximately 0.76 db away from the AEN chael capacity ca be achieved whe 1 5/ approximately 0.57 db over uiformly paced equiprobable igalig., which correpod to a hapig gai of 4

5 I thi paper, we propoe a ew family of igal et that outperform the cotellatio propoed i [1] ad [4] over the AEN chael. I order to deig thee et, the idea i to geometrically implemet the iput ditributio give i (6), i.e. place the real ymbol x i, i {1, 2,, M}, over the iterval 0, accordig to (6). Ufortuately, it appear that the dicrete compoet repreeted by the Dirac delta fuctio i (6) actually make thi goal impoible to achieve i practice. Thi i the reao why we have to replace (6) with the followig cotiuou PDF: which i very imilar to (6). Note that the etire iterval,. x exp x 1 P x 2 1, (8) P x x i ow a eve fuctio which take o-zero value over The iitial tep coit of dividig the oe-dimeioal axi ito 2 1 o that the area uder the fuctio P x x are equal for all thee iterval: k 1 Px k x 1 dx 2M 1 M iterval, 1,..., 2M 1 choe k, k 1 k. (9) By combiig (8) ad (9), ad uig 1 ad 2M, we how that the parameter k ca be expreed a ad where l deigate the atural logarithm. 1 2 k 1 l, for k 1,..., M k 2M 1, (10) 1 2 2M k l, for M 1,..., 2M 1 k 2M 1 k, (11) The, we defie 2M 1 igal poit x k a the cetroid of the iterval k, k 1 with repect to the ditributio P x x give i (8): 5

6 For k M 1,..., 2M 1 x k k 1 2M 1 x P x, the evaluatio of (12) yield k x dx, 1,..., 2M 1 k. (12) where f i a fuctio defied a x k f 1 x M x f k f k 1, (13) e 2M 1 2 l 2 2M x. (14) Fially, a M-ary igal et uitable for the AEN chael i geerated by imply keepig the M ymbol x, M,..., 2M 1 k how that M k, ad dicardig the ymbol k x = 0 wherea the value of the 1 computed uig (13) ad (14). M other ymbol k x, k 1,..., M 1. We ca eaily x, M 1,..., 2M 1 k are To implify the otatio, we ca at thi tage replace the idex k with a idex i defied a i k M 1. The et of M ymbol k i 1,..., M, defied a follow: 1 0 x, k M,..., 2M 1 x ad x i gi gi 1, i 2,..., M, thu become a et of M ymbol x i,, (15) where i a ormalizatio parameter computed o that the average value of the ymbol x i i equal to the uit, i.e. 1 E, ad g i a fuctio defied a g x M x e 2M 1 1 l 2 M 1 x. (16) A M, the cotellatio defied by (15) ad (16) actually implemet a geometrical verio of (8) over the iterval 0, ad ca thu be ued over AEN chael to deig capacity-approachig CM ytem. Note that, hereafter, cotellatio obtaied uig (15) ad (16) will be referred to a log cotellatio a the igal poit are defied uig the logarithm fuctio. 6

7 III. MUTUAL INFORMATION ACHIEVABLE WITH CM SYSTEMS USING LOG CONSTELLATIONS I thi Sectio, we coider the deig of coded modulatio (CM) ytem uig the log cotellatio itroduced i Sectio II. Let x ad y deote repectively the tramitted ymbol ad the correpodig received ample after tramiio over the AEN chael. Iformatio theory tell u that the highet rate, expreed i bit/chael ue, at which iformatio ca be tramitted reliably uig a CM cheme i give by the mutual iformatio where the operator I E x y x y xi p xi p, y log, (17) 2 M i p 1 E, deigate the expectatio of with repect to x ad y, x y probability of tramittig a particular ymbol i, wherea p y x ad y x, i 1,..., M p i the x i p are the probability deity fuctio of the ample at the chael output give the tramiio of ymbol x ad x i, repectively. It ca eaily be how that, for equiprobable igalig over the AEN chael, (17) ca be writte a I E x, y log 2 M i 1 M exp exp y x y xi u y xi x i. (18) We have evaluated (18) for everal M-ary log cotellatio, with M ragig from 4 to 256, uig umerical itegratio via the Mote Carlo method. Fig. 1 how the variatio of the mutual iformatio I a a fuctio of the SNR E / E for ome of thee cotellatio. For compario ake, we have alo plotted i Fig. 1 the AEN chael capacity computed uig (4) a well a the reult obtaied with the cotellatio propoed by Martiez ad defied uig (7) with = [1], [4]. Note that, to be able to diplay reult clearly uig a reaoable amout of pace, we decided ot to how i Fig. 1.a the plot obtaied with M = 8, 32, ad

8 It i ee from Fig. 1 that, for all mutual iformatio value, the log cotellatio achieve a better performace tha thoe propoed by Martiez i the ee that they are able to perform cloer to the AEN chael capacity. For itace, for I = 1 bit/chael ue, we oberve that the 16-ary log cotellatio i able to outperform the bet Martiez igal et by approximately 0.29 db. If the deired mutual iformatio i greater tha 1.5 bit/chael ue, the bet log igal et i alway the oe with M = 256 ymbol. For I = 2 ad 3 bit/chael ue, the SNR gap betwee thi cotellatio ad the bet Martiez igal et are about equal to 0.34 db ad 0.44 db, repectively. Whe mutual iformatio value of 4 ad 5 bit/chael ue are deired, the 256-ary log cotellatio outperform the bet Martiez igal et by approximately 0.40 db ad 0.37 db, repectively. It i worthwhile metioig that, for I = 3 ad 4 bit/chael ue, Fig. 1 idicate that a CM cheme deiged uig a 256-ary log cotellatio i actually capable of operatig withi oly 0.23 db of the capacity limit. We remark that icreaig the umber M of cotellatio ymbol clearly ha a beeficial effect o the performace achieved by log cotellatio, which i ot the cae with Martiez igal et. Thi ca merely be explaied a follow: A M, the geometrical implemetatio of (8) become more ad more accurate, thu progreively brigig the mutual iformatio achievable with a log cotellatio cloer to the capacity limit. I order to further clarify thi poit, we have plotted i Fig. 2 the variatio of the mutual iformatio I a a fuctio of the SNR E / E, for a umber M of cotellatio ymbol ragig from 256 to 2048 ad a deired value of I aroud 4 bit/chael ue. Fig. 2 clearly idicate that icreaig M from 256 to 2048 doe ot reult i ay performace improvemet whe the cotellatio propoed by Martiez are employed. A a example, for I 4 bit/chael ue, we ca ee that a error performace withi about 0.64 db of the capacity i achievable whe uig ay of the four Martiez cotellatio coidered here. If the log igal et are employed, we achieve a SNR gai of about 0.1 db at I 4 bit/chael ue a M i icreaed from 256 to However, it appear that o further improvemet ca be achieved by icreaig the umber of cotellatio ymbol beyod I ay cae, thee reult how that a CM cheme deiged uig log cotellatio with M 1024 i able to perform approximately 0.12 db away from 8

9 the capacity limit of the AEN chael. Thi correpod to a igificat SNR gai of about 0.52 db over a equivalet CM ytem employig a Martiez igal et. IV. MUTUAL INFORMATION ACHIEVABLE WITH BICM SYSTEMS USING LOG CONSTELLATIONS We ow coider the deig of bit-iterleaved coded modulatio (BICM) ytem uig the log cotellatio itroduced i Sectio II. BICM i a ub-optimal alterative to CM that ha become very popular over the lat decade or o owig to it low complexity ad ear-optimal error performace over AWGN ad fadig chael [5] [7]. The idea behid BICM i to map the ecoded bit, after iterleavig, to a certai cotellatio uig Gray mappig. The decodig i performed by firt computig the log-likelihood ratio of the coded bit, ad the, after de-iterleavig, uig a biary decoder a if thee log-likelihood ratio were the obervatio at a biary phae-hift keyig/quaterary phae-hift keyig (BPSK/QPSK) chael output. Due to the importace of BICM for practical applicatio, it i iteretig to ivetigate the potetial of log cotellatio for BICM deig over AEN chael. Coider a M-ary modulatio modelled by a igal et S compoed of M ymbol. Let m c c 0, 1 deote a vector of 1, c2,..., c m m log 2 M coded bit at the modulator iput, ad y the correpodig received ample at the chael output. It wa how i [6] that the geeric expreio of the BICM mutual iformatio I, valid uder the cotrait of uiform iput ditributio, i give by where c y I m i1 p y x xs i,ci E c, y 1 log 2, (19) p y x xs E, deigate the expectatio of with repect to c ad y, ad of all cotellatio ymbol S i,c i deote the ubet x S whoe label have the value c 0,1 i poitio i. Aumig tramiio over a AEN chael, we ca eaily how that (19) i equivalet to i 9

10 I m E c, y log 2 i1 xs i,ci m xs exp exp y x uy x y x uy x m. (20) By performig a umerical itegratio via the Mote Carlo method, we have evaluated (20) i order to determie the highet rate at which iformatio ca be tramitted reliably with the BICM approach. We have coidered the ue of variou M-ary log cotellatio, with M ragig from 4 to 256. Fig. 3 how the variatio of the BICM mutual iformatio I a a fuctio of the SNR E / E for ome of thee cotellatio. A we did for the CM cae, we have alo plotted i Fig. 3 the curve obtaied with Martiez cotellatio, defied uig (7) with = 1.618, a well a the AEN chael capacity give by (4). Oce agai, to be able to diplay reult clearly uig a reaoable amout of pace, we have choe to diplay i Fig. 3 oly the mot iteretig reult amog all thoe which have bee obtaied. It i ee from Fig. 3.a that, for low mutual iformatio value, the bet performace i achieved uig Martiez cotellatio with a mall umber of ymbol. For example, for I = 1 ad 2 bit/chael ue, the bet igal et are thoe defied uig (7) with M = 4 ad 8 ymbol, repectively, ad they outperform the bet log cotellatio by approximately 0.15 db ad 0.20 db, repectively. However, a the deired mutual iformatio value i icreaed beyod a value of 3 bit/chael ue, the log cotellatio ted to perform margially better tha Martiez igal et. A a example, for I = 4, 5, ad 6 bit/chael ue, we oberve that the bet cotellatio i alway the 256-ary log igal et, the latter beig, i all three cae, able to outperform the bet Martiez igal et by approximately db. Thee reult idicate that, i the particular cotext of BICM deig, the log cotellatio itroduced i thi paper do ot offer ay igificat performace advatage over the igal et propoed by Martiez. It i worthwhile otig that, for all cotellatio coidered here, the SNR gap betwee the 10

11 AEN chael capacity ad the achievable mutual iformatio i alway greater tha about 1.40 db, which i much larger tha the typical gap value obtaied with the CM approach. We ca thu coclude that, over AEN chael, BICM may ot really be coidered a a ear-optimal techique to combie modulatio ad codig. V. CONCLUSIONS We have itroduced a ew family of igal cotellatio, referred to a log cotellatio, that ca be employed for deigig ear-capacity CM cheme over AEN chael. For thee igal et, the ditributio of the ymbol o the oe-dimeioal axi i a geometrical approximatio of the optimal iput ditributio determied by Verdú i 1996 [3]. We have evaluated the mutual iformatio achievable over AEN chael with both CM ad BICM approache for variou igal et. The reult have how that, i the cae of CM, the propoed log cotellatio igificatly outperform, by over half a decibel i ome cae, the bet exitig igal et available from the literature. We have alo how that operatio withi oly 0.12 db of the AEN chael capacity i poible whe uig log cotellatio with a umber of ymbol greater tha or equal to I the cotext of BICM, it ha bee oberved that log cotellatio do ot offer igificat performace advatage over the bet exitig cotellatio. I ay cae, we believe that, depite it implicity, BICM may ot be a uitable approach to modulatio ad codig over AEN chael. I fact, our reult have idicated that the potetial performace degradatio reultig from the ue of BICM itead of CM i larger tha 1 db, which implie that BICM may ot really be coidered a a ear-optimal approach over AEN chael. REFERENCES [1] A. Martiez, Commuicatio by eergy modulatio: The additive expoetial oie chael, IEEE Tra. Iform. Theory, vol. 57, pp , Jue

12 [2] A. Martiez, Iformatio rate of radiatio a a photo ga, Phy. Rev., vol. 77, pp /7, Mar [3] S. Verdú, The expoetial ditributio i iformatio theory, Prob. Ifo. Tram., vol. 32, pp , Ja.-Mar [4] A. Martiez, Codig ad modulatio for the additive expoetial oie chael, i Proc. IEEE It. Symp. Iform. Theory (ISIT 08), Toroto, Caada, July 2008, pp [5] E. Zehavi, 8-PSK trelli code for a Rayleigh chael, IEEE Tra. Commu., vol. 40, pp , May [6] G. Caire, G. Taricco, ad E. Biglieri, Bit-iterleaved coded modulatio, IEEE Tra. Iform. Theory, vol. 44, pp , May [7] S. Le Goff, A. Glavieux, ad C. Berrou, Turbo code ad high pectral efficiecy modulatio, i Proc. IEEE ICC 94, New Orlea, Louiiaa, May 1994, pp [8] G. D. Forey, Jr., R. G. Gallager, G. R. Lag, F. M. Logtaff, ad S. U. Qurehi, Efficiet modulatio for bad-limited chael, IEEE J. Sel. Area Commu., vol. 2, pp , Sep [9] F. W. Su ad H. C. A. va Tilborg, Approachig capacity by equiprobable igalig, IEEE Tra. Iform. Theory, vol. 39, pp , Sep

13 Mutual Iformatio (bit/chael ue) Mutual Iformatio (bit/chael ue) Capacity M = 4, Mart. M = 4, log M = 16, Mart. M = 16, log M = 64, Mart. M = 64, log M = 256, Mart. M = 256, log E/E (db) (a) Capacity M = 16, Mart. M = 16, log M = 32, Mart. M = 32, log M = 64, Mart. M = 64, log M = 128, Mart. M = 128, log M = 256, Mart. M = 256, log E/E (db) (b) Fig. 1 Mutual iformatio achievable with a CM cheme a a fuctio of the SNR, for variou log cotellatio ( log ), over the AEN chael. For compario purpoe, we have alo plotted the curve correpodig to the chael capacity a well a the mutual iformatio curve obtaied with the igal et propoed by Martiez ( Mart. ). The umber M of cotellatio ymbol rage from 4 to 256. For clarity ake, we decided ot to how the plot obtaied for M = 8, 32, ad 128 i Fig. 1.a. 13

14 Mutual Iformatio (bit/chael ue) 4.2 Capacity M = 2048, log M = 1024, log M = 512, log M = 256, log M = 2048, Mart. M = 1024, Mart. M = 512, Mart. M = 256, Mart E/E (db) Fig. 2 Mutual iformatio achievable over the AEN chael with a CM cheme a a fuctio of the SNR, for variou log cotellatio ( log ) ad deired mutual iformatio value aroud 4 bit/chael ue. For compario purpoe, we have alo plotted the curve correpodig to the AEN chael capacity a well a the mutual iformatio curve obtaied with the igal et propoed by Martiez ( Mart. ). The umber M of cotellatio ymbol rage from 256 to

15 Mutual Iformatio (bit/chael ue) Mutual Iformatio (bit/chael ue) Capacity M = 4, Mart. M = 4, log M = 8, Mart. M = 8, log M = 16, Mart. M = 128, log M = 256, log E/E (db) (a) Capacity M = 32, Mart. M = 32, log M = 64, Mart. M = 64, log M = 128, Mart. M = 128, log M = 256, Mart. M = 256, log E/E (db) (b) Fig. 3 Mutual iformatio achievable with a BICM cheme a a fuctio of the SNR, for variou log cotellatio ( log ), over the AEN chael. For compario purpoe, we have alo plotted the curve correpodig to the chael capacity a well a the mutual iformatio curve obtaied with the igal et propoed by Martiez ( Mart. ). The umber M of cotellatio ymbol rage from 4 to 256. For clarity ake, we decided ot to how ome curve that were of o particular iteret. 15