# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 3 Signaling Schemes
### Selection of Parameters for PAS  
In this section, we study the optimum shaping and FEC coding rates for PAS using AIRs. Therefore, we consider the case where \(n\to\infty\) which implies that \(k/n=\mathbb{H}(A)\) for asymptotically optimum shaping architectures from (9), and consequently, \(R=\mathbb{H}(A)+\gamma\). In the PAS architecture, to obtain a target rate \(R=\mathbb{H}(A)+\gamma\) using the \(2^{m}\)-ASK constellation, a total of \(n(m-R)\) redundancy bits are added to a channel input sequence by shaping and coding operations combined. Shaping is responsible for \(n(m-1-\mathbb{H}(A))\) redundant bits whereas coding adds \(n(\mathbb{H}(A)+1-R)\) bits. This is illustrated in Figure 4 where the content of a channel input sequence produced by the generalized PAS architecture of Figure 3 (bottom) is shown. The striped areas represent the information carried in signs (red) which is \(\gamma n\) bits, and in amplitudes (green) which is \(k=n\mathbb{H}(A)\) bits. Dotted areas show the redundant bits in a sequence. When \(\gamma=0\), i.e., \(R_{\text{c}}=(m-1)/m\), all signs are selected by redundancy bits and thus, the striped red area in Figure 4 vanishes. When \(\mathbb{H}(A)=m-1\), the amplitudes are uniformly distributed, i.e., there is no shaping, and thus, the dotted green area in Figure 4 disappears. We note that a similar illustration was provided for a single ASK symbol in (Figure 9 in [55]). In Table 1, the content of a sequence at the output of a PAS transmitter (in accordance with Figure 4) is tabulated for Example 2 where \(n=216\). Figure 4: Content of a channel input sequence produced by PAS. When the input is constrained to be MB-distributed, \(\mathbb{H}(X)=\mathbb{H}(A)+1\) can be used as a design parameter which tunes the balance between shaping and coding redundancies at a fixed rate \(R\). More specifically, the entropy \(\mathbb{H}(A)\) of the MB distribution (2) is controlled by \(\lambda\). Thus by changing \(\lambda\), the amount of shaping redundancy in an amplitude can be adjusted. The question is then how to choose the optimum \(\lambda\). Following [PERSON], [PERSON] and [PERSON] [92; 2], we use the gap to capacity (normalized SNR), which is defined as  
\[\Delta\text{SNR}=\frac{\text{required SNR such that }R_{\text{BMD}}=R}{2^{2R}-1} \tag{13}\]  
as the metric to be minimized when searching for the optimum MB distribution for a fixed rate \(R\) and constellation size \(2^{m}\). In general, the gap-to-capacity can be computed for any parametric family of distributions. Here we only consider the MB distributions since they have been shown to perform very close to the capacity of ASK constellations over the AWGN channel and maximize the energy efficiency [21]. The numerator in (13) is the SNR value at which \(R_{\text{BMD}}=R\) for a given \(P_{X}\), and the denominator is the SNR value at which the capacity \(C=R\). We note that instead of the MI in (Equation (55) in [2]), we now use the BMD rate of (12). Observing from Figure 4 that \(1-\gamma=1-(R-\mathbb{H}(A))\), the rate of the FEC code that should be employed in PAS to obtain a transmission rate \(R\) for a given constellation entropy \(\mathbb{H}(X)\) is given by  
\[R_{\text{c}}=\frac{m-1+\gamma}{m}=\frac{m+R-\mathbb{H}(A)-1}{m}=\frac{m+R- \mathbb{H}(X)}{m}. \tag{14}\]  
**Example 3** (Optimal PAS parameters). : _In Figure 5, the entropy \(\mathbb{H}(X)\) of an input \(X\) with \(|\mathcal{X}|/2=4\) MB-distributed amplitude levels (i.e., 8-ASK) vs. \(\Delta\text{SNR}\) is plotted for \(R=2.25\) bit/1-D. On the top horizontal axis, the corresponding FEC code rates (14) are also shown. The rightmost point (indicated by a square) corresponds to uniform signaling where the target rate of 2.25 bit/1-D is obtained by using a FEC code of rate \(R_{\text{c}}=R/m=3/4\). In this trivial case, all 0.75 bits of redundancy are added by the coding operation, and the gap to capacity \(\Delta\text{SNR}\) is 1.04 dB.