# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 1 Introduction
An efficient numerical algorithm to compute optimal input distributions for BICM was introduced in [39]. The effect of mismatched shaping, i.e., not using the true symbol probabilities or reference constellation at the receiver, was examined in [40]. The achievable rates, error exponents and error probability of BICM with PS were analyzed in [41]. Constellation shaping was investigated for BICM at low SNR in [42], PS in BICM was considered for Rayleigh fading channels in [43; 44]. Recently, probabilistic amplitude shaping (PAS) has been proposed to provide low-complexity integration of shaping into existing binary FEC systems with BMD [28]. PAS uses a reverse concatenation strategy where the shaping operation precedes FEC coding, as shown in Figure 2 (top). This construction has been first examined for constrained coding problems [45]. A corresponding soft-decision decoding approach for this structure was studied in [46]. PAS can be considered as an instance of the Bliss architecture [45] where in the outer layer a shaping code is used, and then in the inner layer parity symbols are added. The main advantage of this structure is that amplitude shaping can be added to existing CM systems as an outer code. (We note here that in optical communication literature, the term \"intensity modulation\" is used to designate amplitude modulation in general. To be consistent with the constellation shaping literature, we prefer to use the term \"amplitude shaping\" in this study.) In addition to closing the shaping gap, PAS also has rate adaptation functionality in the shaping layer. This means that instead of using many FEC codes of different rates to obtain a granular set of transmission rates, the rate can be adjusted by the amplitude shaper with a fixed FEC code. Owing to these advantages, PAS has attracted a lot of attention. PAS has been combined with low-density parity-check (LDPC) codes [28], polar codes [47] and convolutional codes [48]. Its performance has been evaluated over the AWGN channel [28], optical channels [49; 50], wireless channels [48] and parallel channels with channel state information available at the transmitter [51]. The key building blocks of the PAS framework are the amplitude shaper and deshaper, i.e., the green boxes in Figure 2 (top). The function of the amplitude shaper is to map uniform binary sequences to shaped amplitude sequences in an invertible manner. A careful selection of the set of sequences that can be outputted by the shaper with the aim of matching a target distribution (direct approach) or constructing an energy-efficient signal space (indirect approach) results in improvement in overall performance. We call the way this selection is accomplished _shaping architecture_ which affects the performance of PAS. (For channels other than the AWGN channel, metrics other than energy may be used to assess the efficiency of the signal space. In such cases, boundary structures other than spheres may lead to more efficient signal spaces. Equivalently, capacity-achieving distribution is expected to be non-Gaussian for these channels.) On the other hand, the actual implementation of this architecture is called here _slaping algorithm_ and determines the complexity of attaining this performance. The third and fifth layers of Figure 1 illustrate shaping architectures and algorithms, respectively. The difference between the shaping architecture and the underlying algorithm is discussed in detail in Section 2.4. For the initial proposal of PAS [28], constant composition distribution matching (CCDM) was employed as the shaping architecture [52]. The basic principle of CCDM is to utilize amplitude sequences having a fixed empirical distribution that is information-theoretically close to the target distribution. To this end, a constant composition constraint is put on the output sequences such that all have the same amplitude composition. To realize such a mapping, arithmetic coding (AC) is used in a way similar to [53].