Coded Modulation for Gaussian Channels: Dispersion- and Entropy-Limited Regimes

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1 Coded Modulatio for Gaussia Chaels: Dispersio- ad Etropy-Liited Regies Ebrahi MolaviaJazi Departet of Electrical Egieerig Pesylvaia State Uiversity Uiversity Park, PA 6 Eail: eu@psu.edu J. Nicholas Laea Departet of Electrical Egieerig Uiversity of Notre Dae Notre Dae, IN 6556 Eail: jl@d.edu Abstract We study coded odulatio over poit-to-poit Gaussia chaels with discrete iputs ad fiite blocklegth usig recet tools fro iforatio theory. We show that fiitelegth chael codig has quite a sigificat effect at low SNR, ad we idetify SNR values below which the coded odulatio is dispersio liited. Ulike the ifiite-legth case, coded PSK outperfors coded QAM i this dispersio-liited regie, based upo aalysis of i.i.d. rado codes. At high SNR, we show that fiite blocklegth codig barely affects the perforace so that ucoded odulatio is essetially optial, ad we quatify the SNR values beyod which the odulatio is etropy-liited. I. INTRODUCTION Power efficiecy ad ipleetatio coplexity are two critical requireets of ay oder couicatio systes, both i stadard cellular etworks ad eergig wireless applicatios such as Machie-to-Machie MM) couicatio ad the Iteret of Thigs IoT). The ubiquitous applicatio of battery-powered devices liits the axiu available power, ad ipleetatio issues call for low-coplexity processig. As a iitial ivestigatio ito soe of these aspects, i this paper, we cosider a coded odulatio schee over a poit-to-poit coplex Gaussia chael which is costraied i ters of the allowable odulatio size ad chael codig blocklegth, capturig the coplexity liitatios, as well as the operatig sigal-to-oise ratio SNR), takig ito accout the power liitatio. I particular, we focus o two coo digital odulatio schees: quadrature aplitude odulatio QAM) ad phase shift keyig PSK). Our aalysis builds upo the recet thee of fiite blocklegth iforatio theory to characterize the correspodig chael dispersio [] ad secod-order codig rate []. The ai idea i both [] ad [] is that fiite-legth chael codig itroduces a back-off fro chael capacity, which depeds upo the codig blocklegth ad the target block error probability. For exaple, give a poit-to-poit cotiuous iput coplex Gaussia chael with SNR P, fiite blocklegth, ad block error probability ɛ, the axiu achievable rate ca be expressed as [], [] ) V P ) log R, ɛ) = CP ) Q ɛ) + O, ) P P +) P +) where CP ) := log + P ) ad V P ) := log e) are the capacity ad dispersio of the Gaussia chael, respectively, ad Q ) deotes the Gaussia copleetary CDF. The ai techical cotributio of this paper is providig a explicit forula for chael dispersio ad secod-order codig rate of discrete iput Gaussia chaels with coded PSK ad coded QAM ad coo i.i.d. codes; Sectio V poits to the case of o-i.i.d. codes. To the best of our kowledge, this work is a first step towards the dispersio aalysis of coded odulatio schees for the Gaussia chael. Our results suggest iterestig desig isights about coded odulatio schees at fiite blocklegth. I particular, we show that, depedig upo the operatig SNR, the codig rate perforace of fiite blocklegth codes over coplex Gaussia chaels with PSK ad QAM iputs ca be essetially categorized ito three regies: a) a low-snr dispersio-liited regie i which the perforace is alost idepedet of the odulatio size ad is aily dictated by the fiite-legth chael codig ad the correspodig chael dispersio; b) a high-snr etropy-liited regie i which the perforace is alost uaffected by the chael codig legth ad is aily deteried by the odulatio costellatio size, ad c) a oderate-snr balacig regie i betwee the previous two, i which the perforace is the resultig balace of the costellatio size ad the chael dispersio of the associated fiite-legth chael code). I particular, roughly below -5) db, these coded odulatio schees are dispersio-liited. O the other had, coded QPSK becoes etropy-liited roughly above db, 6-QAM above db, ad 6-PSK above db. Quite iterestigly, the sallest gap to the Shao liit occurs i the oderate-snr balacig regie ad is achieved about -5 db SNR, for ost blocklegth ad block error probability values of iterest. Aother key cotributio of this paper is that coo idepedet ad idetically distributed i.i.d.) codes do ot achieve the axiu possible codig rate over discrete-iput Gaussia chaels. I particular, we will show that, at low SNR, coded-psk surprisigly outperfors coded-qam whe used with i.i.d. codes. The key reaso for this pheoeo is that i.i.d. codes are ot power-efficiet at fiite blocklegth. Figure illustrates the typical fiite-blocklegth perforace for the coplex Gaussia chael with PSK ad QAM iputs. We will precisely defie the otio of SNR or used i this figure later i Sectio IV, but it basically represets

2 SNR or 6 Optial ifiite Optial =5 6 PSK ifiite 6 QAM ifiite 6 PSK =5 6 QAM = although ot optial i geeral, are ore typical i practice due to ipleetatio costraits, so we will refer to C uif P ) defied i 3) also as the practical capacity of the Gaussia chael with discrete QAM iput. It is well-kow that [6], for all differet cases of -PSK ad -QAM, C uif P ) coicides with CP ) at low SNR, so there is o gap to the Shao capacity i this regie if we costrai the iput to take oly a few discrete values. For other values of SNR, however, C uif P ) stays below CP ), as expected, ad saturates to log at high SNR. I the followig, we explore the fiite-blocklegth perforace of the coplex Gaussia chaels with the aforeetioed discrete odulatio schees. We state secod-order approxiatios to the chael codig rate for these schees ad use the to copare differet odulatio schees i ters of chael dispersio ad gap to the Shao liit. Fig. : SNR or copariso, as a fuctio of the operatig SNR, for the optial cotiuous odulatio, the 6-PSK ad the 6-QAM odulatios. Paraeters for the secod-order approxiatios are blocklegth = 5 ad block error probability ɛ = 3. how ay db a schee is operatig away for the Shao liit, so that a SNR or of zero eas operatig exactly at the Shao liit ad the ore SNR or gets suggests larger gap to the Shao liit. Oe ca observe all of the aforeetioed effects i Figure : the doiat effect of fiite blocklegth at low SNR ad its egligible effect at high SNR, the SNR thresholds for each regie, ad the superiority of PSK over QAM at low SNR. II. BACKGROUND For the discrete iput coplex Gaussia chael with PSK or QAM costellatios of size ad average power xk) = P, ) k= the achievable rate ca ever exceed log bits per chael use. I the liit of asyptotically large chael code blocklegth, the capacity of -PSK is give by [3] C uif P ) := log )] E Z [log e Z xk)+z xl), k= l= 3) where Z deotes a zero-ea uit-variace coplex Gaussia distributio. This capacity is achieved by the i.i.d. discrete uifor distributio over the sybols of the costellatio, due to the syetry properties of the chael ad PSK iput, so why the otatio C uif P ). For -QAM, however, a i.i.d. uifor distributio is capacity-achievig oly for very high or very low SNR. For iterediate SNR values, a coplicated o-uifor distributio achieves the capacity [], whose output iics the Gaussia distributio as uch as possible [5]. Note that uifor iputs, III. FINITE-BLOCKLENGTH CODING RATES I this sectio, we characterize the secod-order codig rate over discrete-iput coplex Gaussia chaels. It turs out that our results fudaetally deped upo the dispersio iduced by the i.i.d. discrete uifor distributio defied as U uif P ) := )] Var Z [log e Z xk)+z xl) k= l= [ ) ]] + Var X E Z [log e Z X+Z xl) X l= where X deotes the i.i.d. discrete uifor distributio over the sybols. Now, we suarize our ai results i the followig theore. Theore. The axiu achievable codig rate for the Gaussia chael with power costrait P ad -PSK iput is expressed as ) RPSK, Uuif P ) log ɛ) = C uif P ) Q ɛ) + O, 5) ad that with i.i.d. discrete uifor -QAM iput is ) RQAM, Uuif P δ) ɛ) = C uif P δ) Q ɛ)+o, 6) where δ > is a arbitrarily sall costat, ad C uif P ) ad U uif P ) are defied i 3) ad ), respectively. The coplete proof of Theore is give elsewhere [7, Sec ], whose coverse side for 5) follows the lies of reasoig as i []. We suarize the ore iportat secod-order achievability proof i the Appedix. We ote that, the results i Theore are basically aalogous to the correspodig result ) for the cotiuousiput Gaussia chael, with capacity ad dispersio replaced by their odulatio-specific versios. So, fiite blocklegth agai itroduces a back-off fro capacity C uif P ) for the iterestig regie of average block error probability ɛ /. )

3 Codig Rate b/cu) Optial ifiite Optial =5 6 PSK ifiite 6 QAM ifiite 6 PSK =5 6 QAM = Fig. : Secod-order approxiatio of the codig rate as a fuctio of SNR for the coplex Gaussia chael with the optial cotiuous iput, 6-PSK iput, ad 6-QAM iput. Paraeters are blocklegth = 5 ad block error probability ɛ = 3. Figure illustrates a typical plot of the chael codig rate at fiite blocklegth for the coplex Gaussia chael with PSK ad QAM iputs. As ca be see i Figure, at low SNR, the secod-order codig rates for all differet costellatios essetially achieve the optial cotiuous-iput secod-order perforace. At high SNR, the codig rate over the discrete-iput Gaussia chael is barely affected by the fiite blocklegth code. However, oe observes a key differece betwee the result 5) for PSK ad the result 6) for QAM, which is the power argi δ itroduced for the QAM result 6). The reaso is that a sigificat fractio of the QAM sybols have idividual powers above the available power budget P, so although a i.i.d. uifor sequece cosues power P o average, the probability of geeratig codewords above the power budget is cosiderable at fiite blocklegth. To esure that the costrait-violatio probability is ot the doiat factor i the block error probability of QAM, we back off fro the available power budget P by a positive costat argi δ ad geerate codewords accordig to a i.i.d. discrete uifor distributio with power P δ; check the Appedix for ore details. I priciple, oe ca optiize the choice of δ for ay give, P, ad ɛ, but we have ot perfored such a optiizatio for Figure ad have siply chose it to be δ = i.5p,.5). Note that, with PSK iputs, o such power argi is eeded, sice all of its costellatio sybols cosue exactly power P. This is the crucial reaso why the fiite-blocklegth codig rate of -PSK beats that of -QAM at low SNR, if coo i.i.d.-like codes are eployed. Figure 3 copares, as a fuctio of SNR P, the optial dispersio V P ) of the cotiuous-iput coplex Gaussia chael with the practical dispersio U uif P ) correspodig to the discrete-iput coplex Gaussia chael for several TABLE I SNR THRESHOLD IN db) TO ACHIEVE NEGLIGIBLE DISPERSION OVER DISCRETE-INPUT GAUSSIAN CHANNEL Dispersio bpcu) Modulatio i SNR for Correspodig Schee U uifp ) <. Capacity QPSK PSK QAM PSK QAM Dispersio Copariso 6 QAM 6 QAM i.i.d. Gaussia 6 PSK Optial Cotiuous 6 PSK Fig. 3: Dispersio copariso as a fuctio of SNR for the coplex Gaussia chael with the optial cotiuous iput, i.i.d. Gaussia iput, PSK iput, ad QAM iput. cases of -PSK ad -QAM, ad also the dispersio iduced by the i.i.d.-gaussia iput CN, P ). Figure 3 provides iterestig coparisos regardig the fiite blocklegth perforace. First, siilar to capacities, odulatio-iduced dispersio U uif P ) coicides with the optial dispersio V P ) at low SNR, for all differet cases of -PSK ad -QAM. So agai, costraiig the Gaussia chael iput to oly a few discrete values does ot affect the perforace at low SNR. Secod, very iterestigly, the odulatio-iduced dispersio U uif P ) coverges to zero at high SNR, cosistet with the well-kow fact that ucoded odulatio is essetially optial at high SNR. Oe ca therefore cosider Figure 3 as a desig tool that quatifies the SNR values for which this qualitative stateet holds. Table I suarizes the uerical SNR threshold values beyod which the dispersios of -PSK ad -QAM costellatios are less tha. bits-squared per chael use bpcu); the correspodig capacity values i bits per chael use bpcu) at those threshold SNR values are also provided for copariso. It is clear fro both Figure 3 ad Table I that, copared with PSK costellatios, QAM costellatios require a saller SNR threshold to achieve egligible dispersio values.

4 IV. GAP TO THE SHANNON LIMIT Although the plots of secod-order approxiatio to the codig rate, such as those i Figure, provide a basis for copariso of fiite blocklegth perforace of differet odulatio schees, it ay be difficult to clearly recogize ad evaluate their differeces. I particular, fro a couicatios egieerig stad poit, it is iterestig to ivestigate the horizotal gaps betwee differet schees either as a fuctio of SNR or as a fuctio of target rate. I other words, we are iterested to kow, at a give blocklegth ad block error probability ɛ, how ay dbs a specific odulatio is away fro aother schee or fudaetal liit), whe operatig with a certai SNR or a certai spectral efficiecy. Note that, we ca select ay of the perforace liits as the baselie ad copare all other perforace results agaist it, sice the gaps betwee ay other two schees ca be evaluated fro such a plot. A atural baselie to choose is the Shao capacity CP ), which has bee widely used as a referece i the couicatio ad codig literature. Adoptig this baselie, we ca ow characterize the horizotal gap o the plots of codig rate versus SNR. Cosider a arbitrary schee operatig at a certai rate R ad SNR P. The questio of iterest is to fid the SNR P of a coplex Gaussia chael whose capacity is equal to R. Fidig P is straightforward, sice CP ) = R log + P ) = R P = R. 7) Oe otes that the gap to the Shao liit of our schee of iterest is the differece P P i a db scale, that is log P/P ). This is exactly Forey s SNR or [6], ) P SNR or := log R. ) As was oted i [6], for R = C we obtai SNR or =, which is cooly referred to as the Shao liit, but for ay rate R < C soe gap exists which is cosistet with SNR or >. Before ivestigatig SNR or for discrete odulatios, let us explore it for the optial cotiuous iput. Substitutig the optial fiite blocklegth codig rate ) ito the SNR or defiitio ) yields the gap to capacity for the optial fiite blocklegth code as ) P SNR or, opt = log. 9) CP ) V P ) Q ɛ) We would also like to give low SNR ad high SNR asyptotes of this expressio. I the liit of high SNR, the capacity ca be approxiated by CP ) log P, ad the dispersio V P ) log e. Moreover, for sufficietly large ad P, the first ter i the deoiator of 9) doiates the secod ter, so we obtai SNR or, high Q ɛ). ) l I the liit of low SNR, however, the capacity ca be approxiated by CP ) P log e, ad the optial dispersio by V P ) P log e. Accordigly, 9) siplifies to ) SNR or, low log P Q ɛ), ) SNR or Optial Low SNR asyptote High SNR asyptote Fig. : SNR or of the optial cotiuous iput with blocklegth = 5 ad block error probability ɛ = 3 alog with its low-snr ad high-snr asyptotes. where ) follows fro the Taylor expasio for the expoetial fuctio e x + x aroud x =. Figure illustrates the uerical evaluatio of SNR or for the optial secod-order code as well as its low ad high SNR asyptotes. The low-snr ad high-snr asyptotes are iterestigly observed to well atch the actual curve roughly below -5)dB ad roughly above +)db, respectively. We ow tur to the SNR or perforace aalysis for -PSK ad -QAM odulatios. Recallig the practical capacity C uif P ) ad the fiite-blocklegth chael codig rate RPSK, ɛ) or R QAM, ɛ) of Theore for these odulatios, we ca uerically copute the SNR or fro defiitio ). We already observed a copariso of SNR or for PSK ad QAM i Figure, where agai we have ot optiized the power argi δ required to eet the power costrait for QAM, but siply have chose this to be δ = i.5p,.5). For further copariso, we show i Figure 5 the SNR or copariso aog oly PSK ad oly QAM odulatios for several values of the odulatio size. Figures ad 5 show that, at low SNR, all PSK ad QAM odulatios, regardless of the costellatio size, have alost the sae gap to the Shao liit, which is equal to the optial gap ). However, the gap is quite sigificat ad icreases for lower SNRs. For -QAM odulatios with i.i.d. codes, we observe a eve larger gap to the Shao liit at low SNR copared to the -PSK odulatio of the sae size, as a cosequece of the ecessary power back-off that deteriorates the perforace. O the other had, at high SNR, the gap to the Shao liit of all -PSK ad -QAM odulatios coverges to that based oly o the capacity C uit P ) log. Therefore, ) P SNR or, high, od log, ) that is a costat back-off log ) i db fro the operatig SNR. However, the thresholds at which these

5 SNR or 6 QPSK 6 PSK 6 PSK SNR or 6 6 QAM QAM 6 QAM Fig. 5: SNR or copariso, as a fuctio of the operatig SNR, for the optial cotiuous odulatio ad several PSK/QAM odulatios solid lies deote ifiite blocklegth, arkers deote = 5). Block error probability is ɛ = 3. asyptotic values are relevat deped upo the odulatio type, as was idicated i Table I, explaiig why -QAM approaches this asyptote faster tha -PSK. V. DISCUSSION I the aalysis above, we have observed that -QAM with i.i.d. discrete uifor distributio ca be iferior to -PSK i the low SNR regie, due to a) the i.i.d. codig requirig a power argi δ to avoid a power costrait violatio ad b) the i.i.d. discrete uifor distributio ot beig optial for -QAM, thus iducig a larger chael dispersio. We have show elsewhere [7] that, resolvig this issue requires slightly ore structured codes such as the costat copositio codes. The certai structure of these codes guaratees power efficiecy, i that o power argi δ is ecessary, ad it also exhibits a saller chael dispersio. A fial coet is regardig the oe-diesioal pulse aplitude odulatio PAM) over the real Gaussia chael. It is straightforward to see that the perforace of -PAM over the real Gaussia chael is aalogous to that of the - QAM over the coplex Gaussia chael, after proper scaligs i variaces, etc.) i the expressios. ACKNOWLEDGMENT This work has bee supported i part by the NSF grats CCF5-66 ad CPS-39. APPENDIX ACHIEVABILITY PROOF FOR THEOREM The secod-order achievability proof uses rado codig accordig to the i.i.d. uifor distributio over the -PSK or -QAM sybols alog with typicality threshold) decodig. Usig stadard arguets [], [7], [], the error probability ɛ of this code is bouded as [ ] ɛ Pr ix ; Y ) R [ + R Pr + Pr ] i X ; Y ) > R ], 3) [ X > P where X ad Y are the true iput-output pair of the chael; X deotes the wrog codewords which are idepedet of the output but copete to look typical; ad ix ; y ) is the realizatio of the utual iforatio rado variable defied as the log-likelihood ratio of the chael law P Y X y x ) ad the iduced output distributio P Y y ), ix ; y ) = log t= e yt xt l= e yt xl). ) The costrait violatio probability P cv ) := Pr[ X / > P ] is exactly zero for coded PSK, sice all PSK sybols cosue power equal to P. For code QAM, this probability P cv ) is expoetially sall i via Cheroff type bouds, provided that P is above the ea of X /, so we eed to geerate X with average power P δ < P. O the other had, usig strog large deviatio aalysis [, Lea 7], oe fids that the secod ter i 3) does ot exceed B / for soe positive costat B. The fial ad ost iportat step is to evaluate the first ter i 3) usig the Cetral Liit Theore. Notig that Y t = X t +Z t with the i.i.d. uit-variace Gaussia oise Z t of the chael, oe ca readily verify that the utual iforatio rado variable ix ; Y ) give i ) for the coded PSK has a ea equal to C uif P ) give i 3) ad a variace equal to U uif P ) itroduced i ). Therefore, [ ] Pr ix ; Y ) R Q C uif P ) R Uuif P )/ ) + B, 5)

6 where B is aother positive costat. A aalogous boud holds for coded QAM, except that P is replaced by P δ due to the codebook geeratio for X. Upper boudig the RHS of 3) as above ad settig the su to be o greater tha ɛ proves the secod-order achievability of Theore. REFERENCES [] Y. Polyaskiy, H. V. Poor ad S. Verdú, Chael codig rate i the fiite blocklegth regie, IEEE Tras. If. Theory, vol. 56, o. 5, pp ,. [] M. Hayashi, Iforatio spectru approach to secod-order codig rate i chael codig, IEEE Tras. If. Theory, vol. 55, o., pp , 9. [3] G. Ugerboeck, Chael codig with ultilevel/phase sigals, IEEE Tras. If. Theory, vol., o., pp , 9. [] A. R. Calderbak ad L. H. Ozarow, Noequiprobable sigalig o the Gaussia chael, IEEE Tras. If. Theory, vol. 36, o., pp. 76 7, 99. [5] Y. Wu ad S. Verdú, The ipact of costellatio cardiality o Gaussia chael capacity, i Proc. Allerto, pp. 6 6,. [6] G. D. Forey Jr ad G. Ugerboeck, Modulatio ad codig for liear Gaussia chaels, IEEE Tras. If. Theory, vol., o. 6, pp. 3 5, 99. [7] E. MolaviaJazi, A Uified Approach to Gaussia Chaels with Fiite Blocklegth, Ph.D. Dissertatio, Uiversity of Notre Dae, IN,. [] E. MolaviaJazi ad J. N. Laea, A Fiite-Blocklegth Perspective o Gaussia Multi-Access Chaels, IEEE Tras. If. Theory, subitted for publicatio,, arxiv preprit, arxiv:39.33.