# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping
## 3 Signaling Schemes
### Probabilistic Amplitude Shaping  
[PERSON] et al. introduced in [28] the PAS framework which couples an outer shaping code and an inner FEC code to realize _shaped-coded modulation_. Figure 3 (middle) shows the basic PAS architecture where first, an amplitude shaping block maps a \(k\)-bit uniform information sequence \(u^{k}\) to an \(n\)-amplitude sequence \(a^{n}=(a_{1},a_{2},\ldots,a_{n})\) in an invertible manner, where \(a_{j}\in\mathcal{A}\) for \(j=1,2,\ldots,n\). After this mapping block, these amplitudes are transformed into bits using the last \(m-1\) bits of the corresponding BRGC, i.e., the amplitude bits. We note that due to the shaped nature of \(a^{n}\), the bits at the output of the amplitude-to-bit conversion in Figure 3 (middle) are nonuniform. These \(n(m-1)\) nonuniform bits \(c_{2}^{n},c_{3}^{n},\ldots,c_{m}^{n}\) are then used as the input of a systematic, rate \(R_{\rm c}=(m-1)/m\) FEC code which is specified by an \(n\)-by-\(nm\) parity-check matrix \(\mathbf{P}\). The \(n\)-bit parity output of this code is employed as the sign bit-level, i.e., the first bit of the BRGC, to determine the sign sequence \(s^{n}=(s_{1},s_{2},\ldots,s_{n})\). Finally, \(x^{n}=s^{n}a^{n}\in\mathcal{S}^{n}\mathcal{A}^{n}\) is transmitted over the channel. The transmission rate of this scheme is \(R=k/n\) bit/1-D.  
Since symbol-level shaping strategies determine \(m-1\) amplitude bits prior to FEC encoding, they can only be combined with FEC code rates \(R_{\rm c}\geq(m-1)/m\). To employ lower FEC code rates \(R_{\rm c}<(m-1)/m\), bit-level shaping strategies which only determine a subset of \(m-1\) amplitude bits should be employed as in [51; 57; 73]. To use a higher FEC code rate \(R_{\rm c}>(m-1)/m\), a modified PAS architecture is proposed in [28] as shown in Figure 3 (bottom). The code rate in this scheme is \(R_{\rm c}=(m-1+\gamma)/m\) where \(\gamma=R_{\rm c}m-(m-1)\) specifies the number of extra data bits that will be transmitted per symbol. In this modified structure, in addition to the \(n(m-1)\) bit output of the shaper, extra \(\gamma n\) information bits \(\tilde{u}^{\gamma n}\) are fed to the FEC code which is now specified by an \(n(1-\gamma)\)-by-\(nm\) parity-check matrix \(\mathbf{P}\). The \((1-\gamma)n\) bit parity output of the FEC code is then multiplexed with the uniform bits \(\tilde{u}^{\gamma n}\) to form an \(n\)-bit sequence that will select the signs \(s^{n}\). The transmission rate of this scheme is \(R=k/n+\gamma\) bit/1-D.  
**Example 2** (Shaping, FEC and transmission rates in PAS).: _Consider the PAS architecture with 8-ASK, a rate \(R_{c}=5/6\) FEC code, and a target rate \(R=2.25\) bit/1-D. The rate of the extra data that will be carried in the signs of the channel inputs is \(\gamma=R_{c}m-(m-1)=0.5\) bit/1-D. Therefore, the rate of the amplitude shaper should be \(k/n=R-\gamma=1.75\) bit/1-D. If the length of the FEC code is \(n_{c}=648\) bits, the blocklength is \(n=n_{c}/m=216\) real symbols. Then the output set of the amplitude shaper must consist of at least \(2^{k}=2^{216.175}=2^{378}\) sequences._