Partition-Based Distribution Matching

Transcription

1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES Partition-Based Distribution Matching Fehenberger, T.; Millar, D.S.; Koike-Akino, T.; Kojia, K.; Parsons, K. TR January 2018 Abstract Distribution atching is a fixed-length invertible apping fro a uniforly distributed bit sequence to shaped aplitudes and as such plays an iportant role in the probabilistic aplitude shaping fraework. With conventional constantcoposition distribution atching (CCDM), all output sequences have identical coposition. In this paper, we propose partitionbased distribution atching (PBDM) where the coposition is constant over all output sequences. When considering the desired distribution as a ultiset, PBDM corresponds to partitioning this ultiset into equal-size subsets. We show that PBDM allows to address ore output sequences and thus has lower rate loss than CCDM in all nontrivial cases. By iposing soe constraints on the partitioning, a constructive PBDM algorith is proposed which coprises two parts. A variable-length prefix of the binary data word deterines the coposition to be used, and the reainder of the input word is apped with a conventional CCDM algorith, such as arithetic coding, according to the chosen coposition. For a specific target distribution and a fixed rate loss, we nuerically find that PBDM gives a four-fold reduction in block length in coparison to CCDM. Siulations of 64-ary quadrature aplitude odulation over the additive white Gaussian noise channel deonstrate that the block-length saving of PBDM over CCDM for a fixed gap to capacity varies with the signal-to-noise ratio (SNR) and is approxiately a factor of 2.5 to 5 at ediu and high SNRs, respectively. arxiv This work ay not be copied or reproduced in whole or in part for any coercial purpose. Perission to copy in whole or in part without payent of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by perission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgent of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payent of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Copyright c Mitsubishi Electric Research Laboratories, Inc., Broadway, Cabridge, Massachusetts 02139

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3 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 1 Partition-Based Distribution Matching Tobias Fehenberger, Meber, IEEE, David S. Millar, Meber, IEEE, Toshiaki Koike-Akino, Senior Meber, IEEE, Keisuke Kojia, Senior Meber, IEEE, and Kieran Parsons, Senior Meber, IEEE Abstract Distribution atching is a fixed-length invertible apping fro a uniforly distributed bit sequence to shaped aplitudes and as such plays an iportant role in the probabilistic aplitude shaping fraework. With conventional constantcoposition distribution atching (CCDM), all output sequences have identical coposition. In this paper, we propose partitionbased distribution atching (PBDM) where the coposition is constant over all output sequences. When considering the desired distribution as a ultiset, PBDM corresponds to partitioning this ultiset into equal-size subsets. We show that PBDM allows to address ore output sequences and thus has lower rate loss than CCDM in all nontrivial cases. By iposing soe constraints on the partitioning, a constructive PBDM algorith is proposed which coprises two parts. A variable-length prefix of the binary data word deterines the coposition to be used, and the reainder of the input word is apped with a conventional CCDM algorith, such as arithetic coding, according to the chosen coposition. For a specific target distribution and a fixed rate loss, we nuerically find that PBDM gives a four-fold reduction in block length in coparison to CCDM. Siulations of 64-ary quadrature aplitude odulation over the additive white Gaussian noise channel deonstrate that the block-length saving of PBDM over CCDM for a fixed gap to capacity varies with the signal-to-noise ratio (SNR) and is approxiately a factor of 2.5 to 5 at ediu and high SNRs, respectively. Index Ters Distribution Matching, Probabilistic Aplitude Shaping, Coded Modulation. I. INTRODUCTION The cobination of high-order odulation, such as quadrature aplitude odulation (QAM), and strong binary codes (such as turbo-codes [1] or low-density parity-check codes [2]) that operate within a fraction of a decibel (db) of the additive white Gaussian noise (AWGN) channel capacity [3] have becoe standardized in any digital counication systes. Bit-interleaved coded odulation (BICM) has achieved near universal adoption, due to its low coplexity and close-tooptial perforance [4]. Most coded odulation systes eploy unifor signaling where each constellation point is sent with equal probability. A ethod to further increase the inforation rates is to eploy constellation shaping, which gives signal-to-noise-ratio (SNR) iproveents of up to 1.53 db for the AWGN channel [5, Sec ] [6, Sec. IV- B] [7, Sec. IV-B]. Various techniques have been devised to integrate probabilistic shaping into a coded odulation syste, see [8, Sec. II] for a review. Recently, probabilistic aplitude shaping (PAS) [8] has been proposed in which the shaping blocks T. Fehenberger was with Mitsubishi Electric Research Laboratories. He is now with Technical University of Munich, Gerany. E-ail: tobias.fehenberger@tu.de. D. S. Millar, T. Koike-Akino, K. Kojia and K. Parsons are with Mitsubishi Electric Research Laboratories. E-ails: illar@erl.co; koike@erl.co; kojia@erl.co; parsons@erl.co. are placed outside the forward error correction (FEC) encoder and decoder (see Fig. 1). This reverse-concatenation principle allows a low-coplexity and powerful integration into existing BICM systes. Since its proposal, PAS has attracted a lot of attention, particularly in fiber-optic counications [9], [10], [11]. We focus on PAS as shaping fraework in this paper. An integral subsyste of a PAS syste is the apping function fro the uniforly distributed data bits to shaped aplitudes and its inverse apping. In [8, Sec. V], constantcoposition distribution atching (CCDM) is eployed. 1 In siplified ters, CCDM is a fixed-length invertible operation that aps a block of Bernoulli- 1 2 distributed data bits to a sequence of shaped aplitudes [12]. The constant-coposition principle describes that each output sequence ust have an identical epirical distribution. Designing CCDMs suitable for real-tie processing is challenging largely for two reasons. Any finite-length DM fundaentally suffers fro a rate loss that increases with decreasing length. Hence, it would be beneficial to have CCDM block lengths in the order of 500 shaped output sybols and possibly ore, as can be seen fro, e.g., [12, Fig. 2]. The ost widely used algorith for ipleenting CCDM is arithetic coding [12, Sec. IV], which is an inherently sequential algorith. The cobination of sequential apping and long block lengths currently akes a real-tie ipleentation of CCDM a highly challenging task, particularly in the context of optical fiber counications where sybol rates ay be 30 GBaud or ore. In this work, we exaine distribution atching techniques for which the constant-coposition property of conventional CCDM is lifted. Non-constant-coposition DMs are based on the principle that the enseble average over all output sequences ust have the desired coposition, as opposed to the CCDM principle that every output has identical coposition. While the reoval of this constraint enables large gains over CCDM in the range of block lengths where bruteforce coputation or nuerical optiization are feasible, these techniques are ipossible in the block-length regie where low absolute rate loss is achievable. For exaple, a DM with a sequence length of erely 10 sybols and 1.5 bits of entropy for 4 shaped aplitudes will select 2 15 sequences fro a possible Due to the scaling of this cobinatorial proble, it is clear that we need to have a constructive algorith for generating non-constant-coposition distribution atchers. We propose partition-based distribution atching (PBDM), which fors output sequences that are equal-length parti- 1 Note that PAS is not restricted to the use of algebraic distribution atchers (DMs) such as CCDM. Potential alternatives are, for exaple, lookup tables (whose size could becoe prohibitively large) or shell apping (which has liited granularity) [5, Sec. 4.3].

4 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 2 U Û DM Inv. DM à Binary Labeling Inv. Labeling FEC Encoding FEC Decoding B 1. B. Mod Deod X Y Channel Fig. 1. Diagra of the probabilistic aplitude shaping (PAS) building blocks. This paper studies finite-length distribution atchers (DMs) (gray boxes) that ipleent an invertible apping function fro the unifor data bits U to the shaped aplitudes Â. On the receiver side, an inverse DM undoes this operation such that U = Û in the case of error-free FEC output. The PAS logic of cobining shaped aplitudes with unifor sign bits is explained in detail in [8, Sec. IV]. tionings of a ultiset with the desired distribution. In the binary partitioning case, for exaple, we consider sequences which pairwise follow the target distribution. This set of sequences will have the set of CCDM sequences as a subset, and is therefore a generalization of CCDM. We deonstrate nuerically that PBDM requires significantly shorter block lengths than CCDM at a particular rate loss, which is conjectured to ease realization in hardware. PBDM is fundaentally different fro product distribution atching [13] where the target distribution is factorized such that parallel CCDMs can be used for constituent distributions of saller alphabet size. Our approach, in contrast, is ore general in that the alphabet size reains unchanged. For an ipleentation of PBDM, two constraints on the choice of sequences are iposed. Firstly, pairwise partitioning is deployed where each sequence has a copleent such that their average has the desired coposition. By further requiring the nuber of sequences of a particular coposition to be a power of 2, PBDM with a binary-tree structure is proposed. This enables ipleenting PBDM by splitting the binary data word into a prefix that chooses the coposition and the payload that is apped in the conventional CCDM fashion. In this paper, we focus on ipleentation aspects and perforance coparisons of distribution atchers. A nuerical analysis finds that pairwise tree-based PBDM has significantly lower rate loss and thus better AWGN perforance than conventional CCDM. To the best of our knowledge, the proposed PBDM is the first distribution atcher that lifts the constant-coposition principle for a fixed alphabet size. II. FUNDAMENTALS OF DISTRIBUTION MATCHING A DM is an injective apping fro a sequence of length k of uniforly distributed data bits U to n shaped aplitudes. Its integration in the PAS fraework is shown in Fig. 1. We consider fixed-length block-wise distribution atching only since variable-length DMs have practical disadvantages such as varying buffer sizes. For finite-length DM, the target distribution P A ust be quantized to Pà such that the nuber of occurrences of each aplitude is integer. Following [12], [14], this quantization is carried out to iniize inforational divergence between P A and PÃ. The resulting aplitudes à have the probability ass function (PMF) PÃ, also referred to as type [15, Sec. 11.1], [16, Sec. II], on the alphabet A = {a 1,..., a A }. Consider a DM output sequence x n = {x 1, x 2,..., x n } of length n where each eleent x j with j 1,..., n is chosen fro A according to PÃ. The nuber of occurrences n (a i ) of an aplitude a i in the sequence x n is n (a i ) = { j : xj = a i }, j 1,..., n, i 1,..., A, (1) and we have A i=1 n (a i) = n. In the following, we write n i instead of n (a i ) to indicate the nuber of occurrences of the i th aplitude a i. We call the ordered set of occurrences C = {n 1,..., n A } a coposition, which has the type PÃ. We say that a sequence has coposition C if (1) corresponds to C. The set of unique perutations of x n for a given C is referred to as type class [15, Sec. 11.1], and its size is the ultinoial coefficient [15, Eq. (11.17)] ( ) n M(C) = = n 1, n 2,..., n A n! n 1! n 2!... n A!. (2) In the CCDM approach [12], each of the 2 k output sequences x n is of type PÃ, and we denote the single typical CCDM output coposition as C typ = {npã(a 1 ),..., npã(a A )}. The constant-coposition apping of a unifor input sequence to a sequence that has C typ is denoted as f ccd (C typ ) and can, for exaple, be carried out via arithetic coding [12, Sec. IV]. The nuber of input bits for CCDM of a particular C typ is given by k = log 2 M ( C typ ), (3) where denotes rounding down to the closest integer. Fro (3), we can copute the finite-length rate loss [13, Eq. (1)] R loss = H ( à ) k n, (4) where H ( ) denotes entropy in bits. The rate loss vanishes for large n (see [12, Eq. (23)]), which eans that an infinite-length CCDM can achieve the target rate H ( à ). For distribution atching with fixed n and C, it is desirable to ake k as large as possible to in order to iniize the rate loss. In the following, we introduce a new class of distribution atcher that has a significantly saller rate loss than a conventional CCDM. III. PARTITION-BASED DISTRIBUTION MATCHING A. Principle PBDM is based on the observation that not every possible DM output sequence x n ust be of type Pà (or equivalently have the coposition C) in order to achieve on average the target distribution. As the input bits U are uniforly distributed and a DM establishes an injective apping, it is by the law of large nubers sufficient if the enseble average of all output sequences has the target coposition. Thus, we choose those output sequences whose copositions C l satisfy Ncop l Ncop l c l C l c l! = C typ, (5)

5 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 3 n (ai) C typ a 1 a 2 a 3 a 4 n (ai) C 1 a 1 a 2 a 3 a 4 n (ai) C 2 a 1 a 2 a 3 a 4 Fig. 2. Illustration of non-constant coposition for A = 4. Cobining one sequence that has C 1 and one that has C 2 gives the target coposition C typ. where l indices the N cop possible copositions of the PBDM output sequences and c l is the nuber of occurrences of C l at the PBDM output, with 0 c l M(C l ). Hence, (5) states that the average type of all sequences that are the output of a DM ust be PÃ. The nuber of distinct copositions is given by ( ) n + A 1 N cop =, (6) n which can be proven, for exaple, with the stars-and-bars technique [17, Sec. II-5]. Exaple 1 (Non-Constant Coposition): Figure 2 shows a set of figures deonstrating the concept of non-constant coposition. We have the typical coposition C typ = {4, 3, 2, 1} for CCDM with n = 10, which gives the entropy H ( Ã ) = 1.85 bits. The total nuber of distinct sequences which have this coposition is M ( ) C typ = This deterines the rate of the binary distribution atcher to be log 2 ( )/10 = 1.3 bits per sybol. By (4), we have a rate loss R loss of = 0.55 bits for CCDM. By cobining one occurrence of a sequence that has C 1 = {4, 2, 3, 1} and one of C 2 = {4, 4, 1, 1}, the average behavior is that of C typ. The nuber of distinct sequences with C 1 and C 2 are M(C 1 ) = and M(C 2 ) = 6300, respectively. If, in addition to C typ, these two copositions are used, 6300 additional sequences can each be generated such that (5) is fulfilled. Hence, by considering all three copositions in Fig. 2, we ay now use sequences fro C typ ; 6300 sequences fro C 1 ; and 6300 fro C in total. This increases the rate of the nonconstant coposition distribution atcher by 0.1 bit/sybol and reduces the rate loss fro 0.55 bits to 0.45 bits. Suppose the nuber of input bits k and the DM output length n is fixed. Then, the accuulated coposition of all used output sequences is C acc = 2 k C typ. The non-trivial proble is now to find the partitioning of C acc into 2 k integer subsets while fulfilling (5) and obeying two constraints in the subset selection: the su of the integer eleents in each subset ust be equal to n in order to have a fixed-length DM, and each subset cannot occur ore often than their ultinoial coefficient M(C) (see (2)) such that an injective apping function is established. Exaple 2 (General PBDM): Consider a DM with k = 17, n = 10, and PÃ = [0.4, 0.3, 0.2, 0.1]. We have C typ = {4, 3, 2, 1}, and the accuulated coposition is C acc = {524288, , , }. A general PBDM seeks to find those 2 17 out of the 2 20 possible DM output sequences whose nuber of occurrences of each aplitude gives C acc, i.e., that fulfills (5). By (6), there are N cop = 286 copositions for these sequences. The proble is equivalent to finding the non-unique integer sets (each of which corresponds to a particular C) that su up to C acc, given the constraints that the su over each set ust be n = 10 and that each set occurs at ost M(C) ties. Many partitioning probles are known to be NP-coplete [18, Sec ], yet algoriths giving approxiate solutions with reasonable coplexity are known for special cases [18, Sec. 4.2]. However, even if a general solution to the partitioning proble could be found, it would reain a challenging task to establish an ipleentable apping function between DM input and output sequences, in particular if the DM diensions prohibit the use of a lookup table. By iposing soe structure onto the partitioning, a construction of a PBDM device is ade feasible at the expense of a slightly increased rate loss, as we will show next. B. Pairwise PBDM To facilitate the ipleentation of PBDM, we siplify the general partitioning proble (5) by considering pairwise typical sequences only. Note that other structured partitioning schees, for instance into triples or quadruples, are also possible. 2 In this pairwise case, we require that for every coposition C l, a copleentary coposition C l ust exist such that C l + C l = 2 C typ. (7) The nuber of valid pairs that fulfill this relation can be coputed by odifying (6) with the inclusion-exclusion ethod [19, Theore 4.2], taking into account that certain copositions can never occur in a constrained setting such as the considered pairwise PBDM. The PBDM output sequences that have C l or C l ust have the sae probability of occurrence, which iplies that the total nuber of perutations for a pair is governed by the constituent coposition that has fewer perutations. We denote the perutation count of a pair {C l, C l } as ( ) M {C l, C l } = { ( ) )) 2 in M (C l, M (C l, C l C l, M ( C typ ), Cl = C l, where the first case corresponds to non-degenerate pairs and the latter case is the degenerate CCDM pair. For a pairwise PBDM with N pairs distinguishable pairs {C l, C l } that satisfy (7), the total nuber of perutations is N pers = N pairs l=1 (8) ( ) M {C l, C l }. (9) Note that {C l, C l } is invariant to perutations of the copositions and switching the two copositions (i.e., {C l, C l } instead of {C l, C l }) does not give a new unique pair. The rate loss iproveent of PBDM over CCDM (see Sec. IV) is the result of including non-degenerate pairs in (9) in addition to 2 With PBDM, PÃ can also be requantized such that the divergence between the target PMF and PÃ is reduced. Potential benefits of this approach reain for future work.

6 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 4 k bits 0 1 { }} { data word: }{{}} {{ } 0: f ccd (C 1 ) 0 1 prefix k l bits 1: f ccd (C 1 ) 0: f ccd (C 2 ) 1: f ccd (C 2 ) 0: f ccd (C 3 ) 1: f ccd (C 3 ) : f ccd (C 4 ) 1: f ccd (C 4 ) 0: f ccd (C 5 ) 1: f ccd (C 5 ) 0: f ccd (C 6 ) 1: f ccd (C 6 ) Fig. 3. Illustration of a tree-structured pairwise PBDM with six pairs. The apping operation for a 17-bit input sequence (top right) to the coposition C 4 is exeplified. A 4-bit prefix (blue) chooses the pair. The next bit (red) selects the coposition within the pair. Mapping the 12-bit payload (black) onto a sequence of shaped aplitudes that has C 4 can be carried out with a conventional CCDM algorith, such as arithetic coding. the typical CCDM coposition. 3 For any DM with binary input, (9) is rounded down to the nearest power of 2, i.e., we have 2 k = N pers 2. (10) The sae requireent is ade for binary CCDM in (3). Exaple 3 (Pairwise PBDM): Consider P A = [0.4415, , , ] (taken fro [12, Exaple A]) and n = 10. We use [14, Algorith 2] to quantize P A to PÃ = [0.4, 0.3, 0.2, 0.1], which has H ( Ã ) = 1.85 bits and C typ = {4, 3, 2, 1}, see Exaple 1. For pairwise PBDM, there are N pairs = 49 pairs (including the degenerate one) that fulfill (7). The new total perutation count is , which increases the nuber of input bits to k = 17 and thus reduces the rate loss to 0.15 bits. Although considering pairwise-typical copositions greatly siplifies the search for valid partitionings, the ipleentation of such a pairwise PBDM is not straightforward if a lookup table is not to be used. In the following, we ipose another constraint, again at the expense of transission rate, that enables ipleentation of PBDM with reasonable coplexity. C. Ipleentation with a Binary Tree Structure Any DM with binary input uses N pers 2 out of N pers output sequences. In order to design an ipleentable pairwise PBDM, we require, ( in) addition to (7), that the nuber of perutations M {C l, C l } of each pair {C l, C l } ust be a power of 2. A pair {C l, C l } thus represents an integer k l bits, and (8) becoes { ( ) ) 2 k 2 in M (C l = l ) 2, M (C l, C l C l, ( ) 2 M Ctyp 2, C (11) l = C l. 3 It is only in the trivial case where C typ has only one non-zero eleent that the nuber of perutations of CCDM and PBDM is identical. The total nuber of perutations with this power-of-2 constraint is thus 2 k = N pairs l=1 2 k l, (12) where N out of the initially available N pairs pairs pairs are selected as to axiize N pers (while keeping it a power of 2 to have a binary DM) and thus, to axiize k. The selection of pairs can be done by sorting {C l, C l } according to k l in descending order and including only the first N pairs in pairs that ranked list until k is integer. We note that the constraint (11) can lead to fewer perutations than for unconstrained pairwise PBDM (see (9)) and thus an increased rate loss. Exaple 4 (Pairwise PBDM with Binary Tree Structure): Consider the case of Exaple 3. With (11) and (12), the total nuber of perutations is coputed to be , which gives k = 16 input bits and R loss = 0.25 bits, see the arker in Fig. 4. The nuber of pairs that is necessary to address the 16 bits is N pairs = 9 out of the initial N pairs = 49. Note that C typ is not included in these 9 pairs as they already axiize the integer-valued k. The power-of-2 constraint of (11) allows to ipleent PBDM in a binary-tree structure as follows. The N pairs different pairs are sorted by their k l in ascending order, and the copositions within a pair are labeled 0 and 1, respectively. Note that this single-bit label is oitted in the special case of the CCDM coposition C typ. The two pairs with the sallest k l (i.e., the least perutations) for a branch, with one eleent labeled 1 and the other 0. If ore than one branch reains, i.e., if there are pairs that have not been used in the tree yet, the erging and labeling process is repeated. When only a single branch reains, the prefix tree is copleted. This standard sourcecoding technique gives an optial labeling for the prefix. Once this binary tree is generated, the apping fro k-bit unifor data word to shaped aplitude sequence is done by splitting the PBDM input sequence into three parts. The first k k l bits are the prefix that identifies the pair. The next bit chooses the coposition within that pair. For the apping of the final k l 1 bits onto the shaped sybols according to the selected coposition, conventional CCDM based on arithetic coding can be eployed [12, Sec. IV]. This tree-structured design is illustrated in Fig. 3. At the receiver, inverse PBDM of a shaped sequence x n ust be perfored in order to recover the initially transitted data word. Note that PBDM is designed as an invertible function and hence will not introduce any errors if the input, i.e., the FEC decoder output, is error-free. First, the coposition of x n is deterined by, e.g., a siple histogra operation, fro which the binary prefix of length k k l + 1 can be looked up. In order to obtain the reaining k l 1 payload bits, an inverse CCDM algorith based on arithetic coding can be eployed. This recovers the transitted bit sequence. In the next section, we copare pairwise PBDM with a binary-tree-structured ipleentation (which we siply refer to as PBDM) to conventional CCDM.

7 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 5 Rate loss [bit/aplitude] DM block length n [aplitude sybols] CCDM PBDM AIRDM [bit/2d-sy] Capacity DM n PBDM n = 30 PBDM n = 100 PBDM n = 250 Unifor SNR [db] 0.75 db Fig. 4. Rate loss over block length for conventional CCDM and pairwise PBDM ipleented with a tree structure. The target PMF is P A = [0.0722, , , ] fro [12, Exaple A]. The arker for PBDM at n = 10 refers to Exaple 4. Fig. 5. AIR in bit/2d-sy over SNR in db for bit-etric decoding 64QAM. The AWGN capacity log 2 (1 + SNR) is shown as reference. The inset zoos into the region around AIR DM = 4 bit/2d-sy where PBDM of length n = 250 is 0.75 db ore power-efficient than unifor 64QAM. IV. NUMERICAL RESULTS This section nuerically studies the rate loss and AWGN perforance of pairwise PBDM with the tree-structure design outlined in Sec. III. Rate loss is coputed with (4), where the input length k is coputed fro (3) for CCDM and fro (12) for PBDM. For the AWGN channel results, we consider quadrature aplitude odulation (QAM) channel input as the concatenation of two one-diensional aplitude-shift keying (ASK) sybols. The figure of erit is the achievable inforation rate (AIR) for bit-etric decoding reduced by the DM rate loss, [ ] AIR DM = H (B) H (B i Y) R loss. (13) i=1 The derivation of (13) is given in the Appendix. A quantized version of the optial Maxwell-Boltzann distribution [20, Sec. IV] for each SNR is used as PÃ. We focus on 64QAM for the AWGN rate analysis, ephasizing that PBDM is feasible with any odulation forat that is copatible with PAS. Figure 4 shows rate loss over block length for the target PMF of [12, Exaple A]. We observe that the pairwise PBDM achieves a saller rate loss copared to CCDM for all block lengths. For a rate loss of bits per aplitude sybol, PBDM can operate with approxiately n = 140 sybols, whereas a conventional CCDM requires a fourfold increase in length. Note that the jagged shape of CCDM and PBDM rate loss is due to flooring operations in (3) and (10), respectively. In Fig. 5, AIR DM in bits per 2D-sybol (bit/2d-sy) is shown over SNR in db for 64QAM. In addition to the AIRs for PBDM of short (n = 30), ediu (n = 100), and large size (n = 250), the AWGN capacity (solid black), an infinite-length DM without rate loss (dotted) and unifor 64QAM (dashed) are included as references. We observe that PBDM with length as sall as n = 30 has larger AIR than unifor 64QAM over the relevant SNR range. For significantly shorter PBDMs AIRDM [bit/2d-sy] bit/ 2D-sy 3 90% 4.4 Capacity DM n 4.3 PBDM CCDM Unifor DM block length n [aplitude sybols] Fig. 6. AIR in bit/2d-sy over the block length n for 64QAM at 14 db SNR. PBDM with n = 60 (arker) operates within 0.2 bit/2d-sy of capacity, whereas a conventional CCDM requires three ties the length. At n = 250, the PBDM achieves 90% of the axiu available shaping gain at this SNR, which is given by an infinite-length DM. (not shown), the rate loss is larger than the shaping gain. By increasing n to 250 sybols, the PBDM closely approaches the asyptotic liit. In Fig. 6, a coparison of PBDM and CCDM as a function of the block length n in 1D aplitude sybols is shown for a fixed SNR of 14 db. At this SNR level and for 64QAM, an infinite-length CCDM without any rate loss operates within approxiately 0.1 bit/2d-sy of the AWGN capacity. PBDM with n = 60 is able to operate within 0.1 bit/2d-sy of the infinite-length liit and thus within 0.2 bit/2d-sy of capacity. We further note that for n = 60, PBDM achieves half of the available shaping gain of 0.24 bit/2d-sy. By increasing the PBDM length to n = 250, 90% of the shaping gain are attainable.

8 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 6 SNR gap to AWGN capacity [db] DM n PBDM n = 50 PBDM n = 100 PBDM n = 250 CCDM n = 250 Unifor 0.1 db AIR DM [bit/2d-sy] Aplitude bits: 1 R sh R FEC 1 R sh Sign bit: 1 1 R FEC Fig. 8. Coposition of an ASK sybol with shaping according to the PAS schee. The fixed PAS boundary (thick vertical line) between 1 shaped aplitude and 1 unifor sign bit is shown for 8ASK ( = 3). The striped areas show the inforation in the aplitude bits (blue) and in the sign bit (red). The gray areas represent the redundancy of the shaping code and the FEC code, respectively. Fig. 7. SNR gap to AWGN capacity in db over AIR. The gray arkers show that the block length reduction of PBDM copared to CCDM is between a factor of 2.5 and 5. Figure 7 shows the SNR gap to capacity in db over AIR DM. For all considered rates, a length-250 PBDM operates within approxiately 0.1 db of its asyptotic liit. When coparing CCDM with n = 250 to PBDM of various lengths, we observe that the PBDM length reduction is approxiately a factor of 2.5 at low AIRs (left gray arker). At AIR DM = 3 bit/2d-sy, PBDM of length 100 has an SNR gap to capacity of 0.3 db. When increasing AIR DM, CCDM approaches (and eventually crosses) the PBDM curve of n = 50, which corresponds to a fivefold length reduction (right gray arker). For large AIRs beyond 5.5 bit/2d-sy (not shown) where the QAM PMF is close to unifor, the length reduction is up to a factor of 10, indicating that the PBDM benefit depends on how strongly the quantized PMF is shaped. For a heavily shaped distribution, pairwise PBDM only gives few additional perutations over a conventional CCDM. As we have seen fro the results in this section, this nuber and thus the potential input sequence k increases drastically when the PMF is closer to a unifor one, leading to superior perforance of PBDM over CCDM. V. CONCLUSION We have proposed partition-based distribution atching (PBDM) which generalizes conventional CCDM by lifting the constant-coposition property of the distribution atcher output sequences. As a result, PBDM has saller rate loss for a fixed block length in all relevant cases of distribution atching. By iposing constraints on the choice of partitionings and their nuber of occurrences, a constructive algorith for distribution atching and deatching is devised. PBDM is nuerically found to allow the block length to be reduced by a factor of 4 for the sae rate loss as CCDM. AWGN siulations with 64QAM deonstrate that this reduction depends on the SNR (i.e., the target distribution) and aounts to a factor of 2.5 to 5 for a fixed gap to AWGN capacity. APPENDIX AIR EVALUATION FOR FINITE-LENGTH DM The following derivation shows how an achievable inforation rate (AIR) for a finite-length DM can be coputed fro the conventionally estiated AIR (assuing infinitelength DM) and the DM rate loss. Both DM and FEC are considered as codes with rates R sh and R FEC, respectively. We then evaluate the inforation content per ASK sybol for an infinite-length and finite-length DM. In the PAS schee, 1 shaped aplitude bits of each ASK sybol are cobined with 1 unifor sign bit. The source of these sign bits can be the unifor data that is to be transitted or the parity bits of the FEC code. A scheatic illustration of such an ASK-sybol coposition is given in Fig. (8). The cobined striped areas represent the overall aount of inforation contained in each sybol. This inforation content I PAS in bits per ASK sybol is I PAS = [ 1 R sh + ( R FEC 1 )], (14) where the first ter inside the brackets corresponds to the inforation contained in the shaped aplitudes (blue striped area in Fig. (8)) and the second ter is the inforation within the sign bits (red striped area), soeties denoted γ [8, Sec. IV-D]. For PAS, R FEC ust be at least 1 [8, Sec. IV-B]. The shaping rate R sh in (14) is defined as R sh = k n 1 H(Ã) 1 for finite-length DM, for infinite-length DM, (15) respectively, and thus describes the ratio of inforation contained in the shaped aplitude bits. With the definitions (14) and (15), we can state a perforance easure of a finite-length DM. We define the efficiency η of a finite-length DM as the ratio of I PAS for finite-length and infinite-length DM, i.e., η = 1 1 k ( n 1 + H(Ã) 1 + ( R FEC 1 R FEC 1 ) ) (16)

9 FEHENBERGER et al.: PARTITION-BASED DISTRIBUTION MATCHING 7 = k n (R FEC 1) H ( Ã ) (R FEC 1). (17) In the following, we use this shaping efficiency to copute AIRs for finite-length DM. Consider an AIR in bits per sybol that is coputed without including any DM rate loss. A highly relevant AIR for PAS and binary FEC is the bit-etric decoding (BMD) rate R BMD defined as [8, Eq. (63)] [ ] R BMD = H (B) H (B i Y), (18) where B = [B 1, B 2,..., B ] is the binary input vector and Y the channel output (see Fig. 1). The coputation of R BMD can, for exaple, be carried out via nuerical integration if the channel law is known, or in Monte Carlo siulations and by isatched decoding for an unknown channel [21, Sec. VI]. We use nuerical integration for the AWGN results in Sec. IV. Note that R BMD according to (18) is achievable only for a DM without rate loss. The BMD rate for PAS and a finite-length DM, referred to as AIR DM, is given by i=1 AIR DM = η R BMD. (19) This eans that by using a finite-length DM, the inforation content of each sybol is reduced by η. Note that (19) also holds for AIRs other than R BMD, such as utual inforation. To siplify (19), we consider PAS with capacity-achieving codes that operate at their thresholds, in which case we have Solving (14) for the FEC rate then gives R FEC = R BMD (1 + H ( Ã ) ) 1 By inserting (21) into (17) we get R BMD I PAS. (20) η = (21) k n H ( Ã ), (22) R BMD which, inserted into (19), finally gives a siplified expression for the BMD rate of a finite-length DM, AIR DM = R BMD + k n H ( Ã ) (23) = R BMD R loss, (24) where R loss was introduced in (4). We use (24) in Sec. IV to copare the AIRs of DMs that have different rate losses, in particular of a conventional CCDM and the proposed PBDM. [4] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded odulation, IEEE Transactions on Inforation Theory, vol. 44, no. 3, pp , May [5] R. F. H. 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