# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 1 Introduction  
Coded modulation (CM), which combines multi-level modulation with forward error correction (FEC), is indispensable for digital communication strategies targeting high transmission rates. To realize CM, different techniques have been proposed in the literature, such as multilevel coding (MLC) [1, 2], trellis CM [3], and bit-interleaved CM (BICM) [4, 5, 6, 7, 8]. Among the many proposed CM architectures, the de-facto standard is to combine a high-order modulation format with a binary FEC code using a binary labeling strategy, frequently in the absence of an interleaver, and to use bit-metric decoding (BMD) at the receiver [7], which corresponds to the BICM paradigm. As the modulation order increases, the maximum rate that can be achieved with uniform signaling starts to suffer from a loss with respect to the capacity of the additive white Gaussian noise (AWGN) channel. As an example, the maximum achievable information rate (AIR) for MLC in combination with multi-stage decoding (MSD) [2] is the mutual information (MI) of the channel input and output. If a uniform signaling strategy is employed with MLC-MSD, the MI is bounded away from capacity. This gap is called the _shaping gap_ and is up to 0.255 bits per real channel use (bit/1-D) for the AWGNchannel. When translated into an increase in required signal-to-noise ratio (SNR) to obtain a certain MI, this so-called _ultimate shaping gap_ corresponds to a 1.53 dB loss in power efficiency [9]. There exist numerous techniques in the literature, most of them proposed in the late 1980s and early 1990s, that attempt to close the shaping gap. Motivated by the fact that the capacity-achieving distribution for the AWGN channel is Gaussian, these techniques fundamentally take one of the following approaches. The first is to construct a signal constellation with a Gaussian-like geometry as illustrated in (Figure 4.49 in [10]), which is called geometric shaping (GS) [11; 12; 13; 14; 15; 16; 17; 18]. The other approach is to induce a Gaussian-like distribution over the signal structure, which is called probabilistic shaping (PS) [19; 20; 21; 22; 23]. PS techniques can be further classified into two subgroups using the terminology introduced by [PERSON] and [PERSON] in [19]. The _direct_ approach is to start with a target distribution (which is typically close to the capacity-achieving distribution) on a low-dimensional signal structure and have an algorithm try to obtain it [19; 21]. Following recent literature [24], the direct approach can also be called distribution matching (DM). The _indirect_ approach is to start with a target rate and bound the \(n\)-dimensional signal structure by a sphere, which we call sphere shaping (SpSh) [22; 23]. Here, a (sampled) Gaussian distribution is obtained indirectly (when \(n\rightarrow\infty\)) as a by-product. Finally, there exist some _hybrid shaping_ approaches in which GS and PS are combined [25; 26; 27]. We refer to (Section 4.5 in [10]) for a detailed discussion on GS, and to (Chapter 4 in [10]) and (Section II in [28]) on PS. GS, PS, and hybrid shaping are shown on the top layer of Figure 1 where the taxonomy of constellation shaping (as discussed in the current paper) is illustrated. We call this first layer _shaping approach_. On the second layer which we call _shaping method_, PS is split into two following the Calderbank/Ozarow terminology [19]. In the context of BICM, constellation shaping techniques again attracted a considerable amount of attention in the 2000s. GS was investigated for BICM in [29; 30; 31], and PS was studied in [32; 33; 34; 35]. An iterative demapping and decoding architecture with PS was proposed in [36]. The achievability of the so-called generalized MI (GMI) was shown for independent but arbitrarily distributed bit-levels in [37]. In [38], it was demonstrated that the GMI is a nonconvex function of the input bit distribution, i.e., the problem of computing the input distribution that maximizes GMI is nonconvex.