Higher-order modulation is indispensable in mobile, satellite,

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1 High Throughput Probabilistic Shapig with Product Distributio Matchig Georg Böcherer, Member, IEEE, Fabia Steier, Studet Member, IEEE, Patric Schulte, Studet Member, IEEE [6] as a shapig device with forward error correctio (FEC), see Fig.. PAS achieves the optimal power efficiecy ad eables flexible SE with oly oe FEC code [4, Sec. VIII]. PAS has bee successfully itegrated with low-desity paritychec (DPC) codes [4], turbo codes [7], SC-DPC codes [8], polar codes [9], ad obiary codes [0]. I compariso [], PAS is over 0.3 db more power efficiet tha No-Uiform Costellatios (NUC) [3], a GS implemetatio advocated by the Advaced Televisio Systems Committee (ATSC) 3.0 stadard. PAS is beig cosidered for iclusio i the 5G stadard [2]. The beefits of PAS for fiber-optic commuicatio were recetly showcased i a field trial [3] ad future optical modems will implemet PS [4, Sec. V-A]. The eablig techology for PAS is the DM, which trasforms a biary data sequece ito a sequece of symbols with a desired distributio. For a overview of existig DM algorithms, see [6, Sec. I] ad refereces therei. For implemetatio, fixed-to-fixed legth DMs are desirable. For highthroughput applicatios, efficiet DM ecodig is required. Furthermore, fixed-to-fixed legth DMs require a large bloc legth to wor well [5]. I may practical settigs, the data li is well modelled by a set of o-iteractig parallel chaels. Examples iarxiv: v [cs.it] 24 Feb 207 Abstract Product distributio matchig (PDM) is proposed to geerate target distributios over large alphabets by combiig the output of several parallel distributio matchers (DMs) with smaller output alphabets. The parallel architecture of PDM eables low-complexity ad high-throughput implemetatio. PDM is used as a shapig device for probabilistic amplitude shapig (PAS). For 64-ASK ad a spectral efficiecy of 4.5 bits per chael use (bpcu), PDM is as power efficiet as a sigle fullfledged DM. It is show how PDM eables PAS for parallel chaels preset i multi-carrier systems lie digital subscriber lie (DS) ad orthogoal frequecy-divisio multiplexig (OFDM). The ey feature is that PDM shares the DMs for lower bitlevels amog differet sub-carriers, which improves the power efficiecy sigificatly. A represetative parallel chael example shows that PAS with PDM is 0.93 db more power efficiet tha covetioal uiform sigalig ad PDM is 0.35 db more power efficiet tha idividual per chael DMs. Idex Terms Probabilistic amplitude shapig, Distributio matcher, Rate adaptatio, Parallel chaels, Bit oadig, DS, OFDM, Coded modulatio I. INTRODUCTION Higher-order modulatio is idispesable i mobile, satellite, cable, ad fiber-optic commuicatio to achieve the high spectral efficiecy (SE) required for data applicatios. Trasceivers must be flexible, i.e., they should support differet SEs so they ca adapt to the li quality at had ad deliver the best possible coectivity. Covetioal coded modulatio uses uiform distributios o the costellatio poits. This has two disadvatages. First, uiform distributios suffer a power iefficiecy of up to.53 db. Secod, flexibility ca be achieved oly by supportig a large umber of modcods, i.e., combiatios of modulatio formats ad chael codes. For example DVB-S2X requires supportig 6 modcods []. Oe approach that has bee proposed is geometric shapig (GS) [2], [3] which uses costellatios with o-equidistat sigal poits. While improved power efficiecy was observed, the problem of flexibility remais. A secod approach is probabilistic shapig (PS) that uses equidistat sigal poits with a o-uiform distributio. For a overview of PS schemes, see [4, Sec. II] ad refereces therei. Recetly, we proposed probabilistic amplitude shapig (PAS) [4], a PS architecture that cocateates a distributio matcher (DM) [5], G. Böcherer, F. Steier, P. Schulte are with the Istitute for Commuicatios Egieerig, Techical Uiversity of Muich (TUM). The wor was partly supported by the Germa Federal Miistry of Educatio ad Research i the framewor of a Alexader vo Humboldt Professorship. The ideas preseted i this wor have bee filed as a patet applicatio with the EPO (applicatio umber: EP ) o October 5, 206. data data DM iv. DM biary labelig iv. biary labelig FEC ENC FEC DEC MOD bit-wise DEMOD commuicatio chael Fig.. System model of PAS. The shapig device DM is cocateated i reverse with the FEC device. data bits DEMUX 2 3 m DM 2 DM 3 DM m Mapper amplitudes Fig. 2. The DM implemetatio proposed i this wor: Product Distributio Matchig (PDM) for 2 m -ASK. biary data bits are demultiplexed ito m parallel blocs of sizes 2 to m. Parallel biary compoet DMs output m shaped sequeces of legth. A bit-mapper recombies the m sequeces ad outputs oe shaped amplitude sequece of legth.

2 2 clude multi-carrier trasmissio such as orthogoal frequecy divisio multiplexig (OFDM), discrete multitoe (DMT), ad multi-atea trasceivers whe the sigular value decompositio (SVD) of the chael matrix is used to orthogoalize the system. Employig curret DM algorithms i such scearios is challegig, as techiques lie bit-loadig partitio the trasmitted sequece i several short segmets, each with a idividual costellatio size ad distributio, which potetially causes a sigificat rate loss. I this wor, we propose a ovel DM architecture called product distributio matchig (PDM), which iterally uses a collectio of parallel DMs with smaller output alphabets to sythesize the desired distributio as a product distributio. A preferable implemetatio uses biary output alphabets for the idividual DMs. This approach both facilitates highthroughput applicatios by parallelizatio ad reduces the rate loss for short output legths, which maes the PDM particularly ameable for large costellatios ad high-throughput. I the fial part of this wor, we propose exteded PDM for parallel chaels, which shares the compoet DMs for lower bit-levels amog differet sub-carriers. Exteded PDM ca be applied, e.g., i OFDM ad DMT. We provide a represetative example where exteded PDM is 0.93 db ad 0.35 db more power efficiet tha uiform sigalig ad idividual per subcarrier DMs, respectively, ad operates close to the waterfillig limit. All simulatio results were obtaied usig the DM implemetatios by [6]. This wor is structured as follows. Sec. II reviews DMs ad PAS ad states achievable rate expressios for system desig. I Sec. III, we itroduce the PDM architecture ad preset fiite legth simulatio results for 64-QAM. Sec. IV shows how exteded PDM ca be used to operate PAS close to the waterfillig limit of parallel chaels. We coclude i Sec. V. II. PREIMINARIES A. Distributio Matchig (DM) DMs [5], [6] trasform a sequece of uiformly distributed iput bits ito a output sequece of symbols from a alphabet A with a desired distributio. A fixed-to-fixed legth DM maps iput bits d to output symbols a = dm(d ). The mappig dm is ivertible, i.e., d ca be recovered from a by applyig the iverse mappig dm. Fixed-to-fixed legth DMs ca be implemeted by the costat compositio distributio matcher (CCDM) [6], for biary output alphabets see also [7]. A DM is characterized by the followig parameters. The rate is R dm = [ ] bits. () output symbol The output distributio is d P A (a) = {0,} P dm(d )(a) 2, a A (2) where P a is the empirical distributio of the sequece a, i.e., P a (a) = {i: a i = a}, a A. (3) TABE I TWO ABES FOR 8-ASK. THE AMPITUDE ABE OF NBBC IS NBC AND THE AMPITUDE ABE OF BRGC IS ASO BRGC BRGC NBBC The rate loss is the differece of the DM rate ad the etropy rate of a discrete memoryless source (DMS) P A, i.e., R loss = H(A). (4) By [6, Sec. III.B], the rate loss of CCDM vaishes for large output legths. I this wor, we are iterested i DMs with relatively short output legths ad we therefore eed to accout for the rate loss i our system desig. B. Amplitude Shift Keyig Modulatio We cosider 2 m -amplitude shift eyig (ASK) costellatios with amplitude alphabet X = {±, ±3,..., ±(2 m )} (5) A = {, 3,..., 2 m }. (6) We use label fuctios β : X {0, } m ad correspodig bit mappers χ: {0, } m X. For all labels B = B... B m i this wor, the first bit B labels the sig S accordig to { 0, S = B = (7), S =. Cosequetly, B 2... B m label the amplitudes ad each label fuctio β implies a amplitude label fuctio β A ad each bit-mapper χ implies a amplitude bit-mapper χ A. Two labels are of special iterest, amely the biary reflected Gray code (BRGC) [8] ad the atural based biary code (NBBC) [4, Sec. VI.C] where the amplitude label is a atural biary code (NBC). The two labels are illustrated for 8-ASK i Table I. C. PAS Trasmitter The PAS architecture implemets probabilistically shaped ASK modulatio. The PAS trasmitter is displayed i Fig. 3 ad wors as follows (for a more detailed descriptio, see [4, Sec. IV.]). A DM maps data bits to amplitudes A, which are represeted by (m ) amplitude bits. The amplitude bits ad γ additioal data bits are multiplied with the parity geeratig part P of a systematic geerator matrix [I P ] to geerate ( γ) redudacy bits. The redudacy bits ad the additioal data bits are mapped to sigs S, which are multiplied symbolwise with the amplitudes A. The FEC code istatiated by P has rate c = (m ) + γ m = m + γ m (8)

3 3 data bits DM A...A X... X FEC 0 Amplitude P abelig γ (m ) amplitude bits ( γ) parity bits Fig. 3. The architecture of the PAS scheme. See Fig. for a system view ad [4, Sec. IV.] for a detailed descriptio. ad the fractio of sigs used for data bits is γ = ( c)m. (9) PAS requires 0 γ. The trasmissio rate of PAS is the umber of data bits per ASK symbol give by R t = + γ. (0) Rbmd db D. Chael Model The geerated sigal poits A i S i are multiplied by the costellatio scalig ad trasmitted over a additive white Gaussia oise (AWGN) chael. We defie X i = A i S i. At a geeric time istace, the chael model is Y = X + Z () where Z is zero mea Gaussia oise with uit variace. The sigal-to-oise ratio (SNR) is E. PAS Achievable Rate SNR = E[X 2 ]. (2) We cosider a PAS receiver with a bit-metric decoder. The PAS trasmitter defies the label B fec = β fec (X) where B fec = B is the sig label ad where B2 fec... Bfec m is the amplitude label. We assume a uiform sig distributio, i.e., P B (0) = P B () = P S ( ) = P S () = 2. (3) We refer to [4, Sec. IV.A] for a ustificatio of this assumptio. A biary demapper calculates the soft-iformatios = log P B fec(0) + log () P B fec p Y B fec (y 0) p Y B fec (y ), =, 2,..., m (4) which are passed to a biary decoder. By [9], a achievable rate for a bit-metric decoder is + R bmd = H(X) H(B fec Y ) (5) = where [ ] + = max(0, ). For PAS, R bmd must be evaluated usig P X = P A P S. The rate R bmd is a achievable rate for the PAS receiver with BMD if H(A) + γ = R bmd. (6) log 2 ( + SNR) R bmd R Π bmd R bicm R bicm (uiform) SNR [db] Fig. 4. Achievable rates for 8-ASK. We will also eed the relatios betwee R bmd, P X, ad SNR, amely R = R bmd (P X, SNR) (7) SNR = R bmd (P X, R). (8) III. PRODUCT DISTRIBUTION MATCHING A. NBC Product Distributios Suppose for some amplitude label B2 dm Bdm m ad the correspodig sigal poit label B dm = B B2 dm Bdm m we have P B dm = m = P B dm (9) where P B dm = P B. I particular, the amplitude distributio P A is such that the bits of the label B dm are statistically idepedet. We ca costruct a distributio (9) by choosig a amplitude label ad biary distributios P B dm, = 2,..., m. Note that the geerated distributio depeds both o the label fuctio ad the biary distributios. A achievable rate is Rbmd Π = = H(B dm ) = H(B fec + Y ). (20)

4 4 Note that the label B dm is ot required to be the same as the label B fec that is used by the FEC ecoder ad decoder. We choose the NBC for the amplitude label B2 dm,..., Bdm m, the BRGC for the FEC label B fec ad we optimize (20) over the biary distributios P B dm, = 2,..., m (recall that the sig distributio P B is uiform) ad the costellatio scalig. I Fig. 4, we display the resultig achievable rate for 8-ASK. We observe that the product costrait (9) leads to virtually o performace loss. Remar. The iformatio-theoretic wor [20] cosidered oly the case whe B dm = B fec, i which case (20) becomes R bicm = = I(B fec ; Y ) (2) which is the so-called BICM capacity. As show i Fig. 4, R bicm is less power efficiet tha Rbmd Π, although the differece is small. B. PDM PDM ca efficietly geerate the product distributios itroduced i the previous subsectio. The PDM is displayed i Fig. 2. biary data bits are demultiplexed ito m parallel blocs of legths 2 to m. The m parallel biary DMs output m shaped biary sequeces of legth. A bit mapper χ dm A recombies the m sequeces ad outputs oe shaped amplitude sequece of legth. C. PDM Rate oss The rate ad the output distributio of the th DM is ad P B dm, respectively. The total rate of the PDM is = m (22) ad the total rate loss of the PDM is the sum of the idividual rate losses, i.e., [ R loss = H(B dm ) ]. (23) =2 D. PDM for the AWGN Chael For the AWGN chael, we use the NBC for the bit-mapper χ dm A ad we choose biary DM distributios that miimize the overall power. Igorig the rate loss for ow, the optimizatio problem is miimize E[X 2 ] P B2,...,P Bm subect to H(B ) = R dm =2 X = χ bbc (B). (24) To accout for the rate loss, we replace the sum-etropy costrait i (24) by a sum-rate costrait, where the th rate is the rate required to implemet the DM output SNRloss ary DM PDM: Bit shaped PDM: 2 Bits shaped PDM: 3 Bits shaped PDM: 4 Bits shaped PDM: 5 Bits shaped Fig. 5. Rate loss compariso for 64-ASK ad SE = 4.5 bpcu. distributio P B. Altogether, we choose the compoet DMs via E. Simulatio Results miimize P B2,...,P Bm E[X 2 ] subect to =2 = R dm X = χ bbc (B). (25) We umerically compare differet DM implemetatios by usig 64-ASK ad a target SE of R t = 4.5 bpcu. We employ a 32-ary DM as a referece as suggested i [4, Sec. V]. The performace of this system is compared to a PDM setup with (B2 dm ), 2 (Bdm 2, Bdm 3 ), 3 (Bdm 2, Bdm 3, Bdm 4 ), 4 (B2 dm, Bdm 3, Bdm 4, Bdm 5 ) ad 5 (Bdm 2, Bdm 3, Bdm 4, Bdm 5, Bdm 6 ) idividually shaped bit-levels ad correspodig biary DMs. The product distributio has bee obtaied by followig the approach of Sec. III-D, while imposig a uiform distributio o the ushaped bit-levels. We first cosider the results of Fig. 5 which illustrates the fiite legth loss of all cosidered cofiguratios. The DM rate loss (4) ad the PDM rate loss (23) is coverted to a SNR loss by ( R bmd SNR loss = 0 log (P ) X, R t + R loss ) 0 R bmd (P. (26) X, R t ) As a rule of thumb, the followig expressio may be useful as a rough estimate: ( 2 2(R t+r ) loss) SNR loss,awg = 0 log 0 2 2Rt R loss 20 log 0 2 R loss 6 db. (27) We observe that the PDMs have a aggregated rate loss that is sigificatly lower tha the rate loss of the 32-ary DM. The resultig performace is comparable oly for output legths of more tha 0 4 symbols. To further illustrate the flexibility of the trasmitter desig, we cosider a coded sceario with a rate 90 DPC bloc

5 5 FER ary DM PDM: Bit shaped PDM: 2 Bits shaped PDM: 3 Bits shaped PDM: 4 Bits shaped PDM: 5 Bits shaped SNR [db] Fig. 6. Performace compariso of the proposed PDM for 64-ASK ad a target SE of 4.5 bpcu ad differet umber of shaped bits. TABE II REQUIRED SNRS FOR DIFFERENT DM CONFIGURATIONS AND A TARGET SE OF 4.5 BPCU. (CAPACITY: DB) DM cofiguratio Required SNR [db] 32-ary DM 27.3 PDM Bit shaped PDM 2 Bits shaped PDM 3 Bits shaped PDM 4 Bits shaped PDM 5 Bits shaped 27.3 code from the DVB-S2 stadard [2] of bloc legth bits ad a correspodig DM output legth of symbols. This choice allows for a fair compariso, as both the parallel biary DMs ad the 32-ary DM have a similar performace. Fifty iteratios are used for the belief propagatio (BP) decodig. As show i Fig. 6, a PDM with 3 shaped bit-levels achieves a similar performace as the 32-ary DM. If oly 2 bit-levels are shaped, the loss i eergy efficiecy is 0.4 db at a target frame error rate (FER) of 0 3. Table II illustrates that these observatios are reflected by the asymptotic achievable rates of Sec. II-E, which were evaluated for the correspodig optimized distributios. While the required SNRs to achieve a SE of 4.5 bpcu are close for 3, 4 ad 5 shaped bit-levels, larger gaps ca be observed for or 2 shaped bit-levels. IV. PROBABIISTIC SHAPING FOR PARAE CHANNES A. System Model We cosider parallel chaels with the IO relatio Y l = h l X l + Z l, l =, 2,...,. (28) The oise terms Z l are zero mea Gaussia with uit variace. The h l model the chael gais ad we assume that both the receiver ad trasmitter have full chael state iformatio, i.e., they both ow the chael gais h l ad the oise variace. We cosider codig over chael uses of each chael, which results i total i chael uses. This choice is for clarity of expositio; the scheme ca easily be geeralized. B. Waterfillig [22, Sec ] The trasmitter has a average power budget P, i.e., the iputs are subect to the sum-power costrait The average SE E[Xl 2 ] P. (29) 2 log 2( + h 2 lp l ) (30) is achievable with the chael iputs X l beig idepedet zero mea Gaussia with variace P l. The average SE is maximized by waterfillig, i.e., [ Pl = λ ] + h 2, λ: Pl = P. (3) l Suppose that Pl the is positive. The SE allocated to chael l is C l = 2 log h 2 l 2 λ. (32) Based o C l, we choose the costellatio size 2 m l so that m l C l + (33) to avoid reduced SE because of too small costellatio sizes. et m = max l m l deote the maximum costellatio size. C. PAS for Parallel Chaels PAS ca easily be combied with parallel chaels. This is illustrated i Fig. 7. A DM device trasforms data bits ito a sequece of amplitudes for each chael, which are the combied with sig bits origiatig from a commo ecodig device. I its simplest form, this DM device cosists of idividual DMs, each with its output alphabet size matched to the correspodig costellatio size, see Fig. 8. D. PDM for Parallel Chaels The PDM suggests a alterative way to geerate amplitude sequeces for distict costellatio sizes. For example, suppose we have = 2 differet chaels ad eed a legth amplitude sequece for 4-ASK ad a legth sequece for 8-ASK. The PDM eeds oe biary DM for 4-ASK ad two biary DMs for 8-ASK. As illustrated i Fig. 0, the idea is ow to use for the first amplitude bit-level B 2 of 4-ASK ad 8-ASK a sigle biary DM with output legth 2 = 2 ad to geerate the secod amplitude bit-level B 3 for 8-ASK by a secod biary DM with output legth 3 =. The potetial beefit of this approach is twofold: first, usig PDM should reduce the rate loss, ad secod, replacig two DMs of legths by oe sigle DM of legth 2 should reduce the rate loss eve further. Fig. 9 shows this exteded PDM scheme. It provides the same iterface to PAS as the aive approach that uses idividual DMs.

6 6 data bits DM A A 2. h h 2 2 m -ASK 2 m 2 -ASK Chael Chael 2 A h 2 m -ASK Chael label label Mux P Demux label γ Fig. 7. Illustratio of PAS for parallel chaels. Each of the chaels has a idividual costellatio size 2 m l, l =, 2,...,. PAS for parallel chaels exteds the PAS for sigle chaels show i Fig. 3. The costellatio scaligs h l are explaied i Sec. IV-F. The DM device ca be implemeted by idividual DMs, as illustrated i Fig. 8, or by exteded PDM illustrated i Fig. 9. Exteded PDM ca be much more power efficiet, see Fig. for a example. idividual DMs DM A DEMUX 2 DM 2 A 2 DM A Fig. 8. Idividual DM implemetatio for PAS for parallel chaels show i Fig. 7. The data bits are demultiplexed ad fed to idividual DMs. exteded PDM 2 DM 2 2 A DEMUX 3 DM 3 3 Mapper Ba A 2 m DM m m A Fig. 9. Exteded PDM implemetig the DM of PAS for parallel chaels show i Fig. 7. The exteded PDM trasforms data bits ito amplitude sequeces of legth. Iterally, the exteded PDM uses m biary compoet DMs, where 2 m -ASK is the largest supported costellatio. The output legths 2,..., m of the compoet DMs are give by (37). The iput legths fulfill m =2 = ad reflect the compoet DM rates. Simultaeously usig oe DM o more tha oe costellatio size imposes restrictios o the distributio families that ca be geerated by exteded PDM. We ext argue how exteded PDM ca be used to geerate families of Gaussialie distributios. The maximum costellatio size is 2 m ad we choose the m DM output distributios so that a NBBC mapper geerates a Gaussia-lie distributio. By groupig 2 eighbourig sigal poits together, the distributio of theses sigal poit groups is still Gaussia-lie, ad it is give by the product distributio geerated by the first m DMs. This suggests that by usig oly m DMs, we ca simultaeously geerate Gaussia-lie distributios o 4, 8,..., 2 m -ASK costellatios. A example is show i Fig. 0. E. Parametrizatio We ext state the parameters of the FEC code ad the PDM so that the parallel PAS operates at a specific SE. For the cosidered case where we use each of the chaels times, the bloc legth of the biary FEC code is code = m l (34)

7 7 DM 2 2 bits DM 3 3 bits Bit-Mapper 8-ASK Bit-Mapper 4-ASK Fig. 0. Simultaeously geeratig two Gaussia-lie amplitude distributios for 4-ASK ad 8-ASK by reusig the DM of bit-level 2. ad formulas (8) ad (9) geeralize to c = (m l + γ) m l (35) 0 0 Idividual DMs Exteded PDM γ = ( c) The DM output legths are give by = m l. (36) (m l ), = 2, 3,..., m (37) ad the correspodig DM iput legths are 2, 3,..., m. The average SE of the overall system is ow m =2 R t = + γ (38) = [ m ] H(B dm ) + γ R loss. (39) F. Waterfillig for PAS =2 For the parallel chaels, suppose we have chose the costellatio sizes 2 m l, l =,..., ad suppose further we have chose the code rate c ad thereby the fractio γ of sigs used for data bits. To achieve the target rate R t, the rate assiged to the amplitudes is thus R dm = R t γ, which results i the followig costrait for the amplitude distributios (igorig the rate loss): H(A l ) = R dm. (40) Recall that the iputs X l are give by l A l S l where A l S l {±, ±3,..., ±(2 m l )}. (4) The average power o the lth chael is E[( l A l ) 2 ] ad depeds o the distributio P Al ad the costellatio scalig l. We use the followig strategy: to esure a similar detectio reliability o each chael, idepedet of the chose amplitude distributios, we choose l = h l. (42) FER Uiform Achiev db P [db] Fig.. Coded performace compariso of idividual DMs ad exteded PDM (DPC code with bloc legth 584 bits) for parallel chaels. I this way, two eighbourig costellatio poits have the distace 2 o all chaels. The average power o each chael is 2 E[A 2 h 2 l ] E[A 2 l h 2 l ]. Next, we calculate the l amplitude distributios by miimize P A,...,P A subect to h 2 l E[A 2 l] (43) H(A l ) = R dm. (44) To accout for rate loss, the sum-etropy costrait is replaced by a DM sum-rate costrait. For exteded PDM, the sumetropy ad sum-rate expressios from (39) ad (38) are used, respectively. G. Simulatio Results To evaluate the performace of parallel PAS ad exteded PDM, we employ the followig example of 3 parallel chaels,

8 8 3 3 R bmd,l R t Waterfillig Idividual DMs Exteded PDM Uiform OP pdm,virt 0.32 db OP pdm 0.4 db OP ref,virt OP ref 0.93 db R loss,pdm R loss, P Fig. 2. Asymptotic aalysis of the icurred rate loss. TABE III PROPERTIES OF THE EXTENDED PDM SETUP Observe i Fig. that the PDM setup improves over the referece strategy at a FER of 0 2 by 0.35 db. This is maily because of the decreased rate loss as show i the asymptotic achievability plot of Fig. 2. We plot the average achievable rate over all parallel chaels vs. the average sum power for both schemes ad their specific iput distributios. The power assigmet is optimized via mercurywaterfillig. We also plot three horizotal lies at 3.09 bpcu, bpcu ad 3.6 bpcu, which deote R t, R t + R loss,pdm ad R t + R loss,ref, respectively. The crossig of the last two horizotal lies with their respective achievability curves are labeled as OP pdm,virt ad OP ref,virt. They idicate virtual operatig poits that would be achievable with the curretly used iput distributios. Because of the rate loss, the actual operatig poits are give by the orthogoal proectios of these poits o the actual SE curve, however. Their differece i SNR of 0.4 db accurately predicts the gap of 0.35 db that we observe i the coded result i Fig.. Compared to uiform distributios, the asymptotic gai (accoutig for the rate loss) is 0.93 db. The gap to the waterfillig solutio is 0.32 db. give as DM P B dm(0) H(B dm ) Y = 2.0 X + Z Y 2 =.3 X 2 + Z 2 Y 3 = 0.6 X 3 + Z 3 V. CONCUSION We proposed product distributio matchig (PDM), a architecture that uses biary DMs i parallel. This parallelizatio eables high-throughput implemetatios of DMs. The biary compoet DMs of PDM reduce complexity. We have show that PDM performs as well as higher-order DMs for log bloc legths ad that PDM ca perform much better tha higherorder DMs for short bloc legths. We have proposed exteded PDM, which eables PAS to operate close to the waterfillig limit of multi-carrier trasmissio schemes such as OFDM. ad a average power costrait of P = 7.23 db. Performig the waterfillig as show i Sec. IV-B, we arrive at the followig rate assigmet: C = 3.87 C 2 = 3.25 C 3 = 2.4. Cosequetly, we target a SE of R t = 3 3 C l = 3.09 bpcu ad select costellatio sizes of 2 m = 32, 2 m2 = 6 ad 2 m3 = 8 poits followig (33). We use each of the three chaels = 432 times. As a referece, we choose a architecture with idividual 6-ary, 8-ary ad 4-ary DMs. For the PDM setup, we employ four parallel biary DMs. Their respective output legths ad distributios are summarized i Table III. We have a maximum costellatio size of 32-ASK, i.e., four bits ca be shaped. Bitlevels 2 ad 3 are shared by all three costellatios, whereas bit-level 4 is used oly by 6-ASK ad 32-ASK. Bit-level 5 appears i 32-ASK oly. The distributios of the idividual DMS ad the biary distributio of the exteded PDM have bee chose followig Sec. IV-F. I the followig, we use a bloc legth 584 DPC code of rate c = 56 from the G.h stadard [23]. As before, 50 BP iteratios are performed. ACKNOWEDGMENT The authors would lie to tha Gerhard Kramer for fruitful discussios ad commets o drafts of this mauscript. REFERENCES [] Digital Video Broadcastig (DVB); Secod geeratio framig structure, chael codig ad modulatio systems for Broadcastig, Iteractive Services, News Gatherig ad other broadbad satellite applicatios; Part 2: DVB-S2 Extesios (DVB-S2X), Europea Telecommu. Stadards Ist. (ETSI) Std. EN , Rev..., 204. [2] M. F. Barsoum, C. Joes, ad M. Fitz, Costellatio Desig via Capacity Maximizatio, i Proc. IEEE It. Symp. If. Theory (ISIT), Ju. 2007, pp [3] N. S. oghi, J. Zöller, B. Mouhouche, D. Asorregui, J. Kim, ad S. I. Par, No-Uiform Costellatios for ATSC 3.0, IEEE Tras. Broadcast., vol. 62, o., pp , Mar [4] G. Böcherer, F. Steier, ad P. Schulte, Badwidth efficiet ad rate-matched low-desity parity-chec coded modulatio, IEEE Tras. Commu., vol. 63, o. 2, pp , 205. [5] G. Böcherer ad R. Mathar, Matchig dyadic distributios to chaels, i Proc. Data Compressio Cof., Sowbird, UT, USA, 20, pp [6] P. Schulte ad G. Böcherer, Costat compositio distributio matchig, IEEE Tras. If. Theory, vol. 62, o., pp , 206. [7] P. Yua, Rate-matched coded modulatio for wireless trasmissio, Master s thesis, Techical Uiversity of Muich, Istitute for Commuicatios Egieerig, 205. [8] F. Buchali, F. Steier, G. Böcherer,. Schmale, P. Schulte, ad W. Idler, Rate adaptatio ad reach icrease by probabilistically shaped 64- QAM: A experimetal demostratio, J. ightw. Techol., vol. 34, o. 8, Apr. 206.

9 [9] T. Priz, Polar codes for higher-order modulatio ad probabilistic amplitude shapig, Master s thesis, Techical Uiversity of Muich, Istitute for Commuicatios Egieerig, 206. [0] J. J. Boutros, F. Jardel, ad C. Méasso, Probabilistic shapig ad obiary codes, arxiv preprit arxiv: , 207. [] F. Steier ad G. Böcherer, Compariso of Geometric ad Probabilistic Shapig with Applicatio to ATSC 3.0, i It. ITG Cof. Source Chael Codig, Hamburg, Germay, Feb [2] Sigal shapig for QAM costellatios, Huawei, Athes, Greece, Tech. Rep., Feb. 208, 3GPP TSG RAN o. 88. [Olie]. Available: rawg RTSGR 88DocsR zip [3] W. Idler, F. Buchali,. Schmale, E. ach, R.-P. Brau, G. Böcherer, P. Schulte, ad F. Steier, Field trial of a Tbits super-chael etwor usig probabilistically shaped costellatios, J. ightw. Techol., vol. 99, o. PP, pp., 207. [4] K. Roberts, Q. Zhuge, I. Moga, S. Gareau, ad C. aperle, Beyod 00 Gbs: Capacity, Flexibility, ad Networ Optimizatio, Joural of Optical Commuicatios ad Networig, vol. 9, o. 4, pp. C2 C24, Apr [5] P. Schulte ad B. C. Geiger, Divergece scalig of fixedlegth, biary-output, oe-to-oe distributio matchig, arxiv preprit arxiv: , 207. [6] [7] T. Ramabadra, A codig scheme for m-out-of- codes, IEEE Tras. Commu., vol. 38, o. 8, pp , Aug [8] F. Gray, Pulse code commuicatio, U. S. Patet , 953. [9] G. Böcherer, Achievable rates for shaped bit-metric decodig, arxiv preprit, 204. [Olie]. Available: [20] A. G. i Fabregas ad A. Martiez, Bit-iterleaved coded modulatio with shapig, i Proc. IEEE If. Theory Worshop (ITW), 200. [2] Digital Video Broadcastig (DVB); 2d Geeratio Framig Structure, Chael Codig ad Modulatio Systems for Broadcastig, Iteractive Services, News Gatherig ad Other Broadbad Satellite Applicatios (DVB-S2), Std. EN , Rev..2., [22] D. Tse ad P. Viswaath, Fudametals of wireless commuicatio. Cambridge uiversity press, [23] Uified high-speed wire-lie based home etworig trasceivers - System architecture ad physical layer specificatio, ITU Std. G.9960, Oct