Probabilistic Amplitude Shaping (PAS)

• Probabilistic Amplitude Shaping is a modulation technique that shapes signal amplitudes via a Maxwell–Boltzmann distribution to approach capacity on AWGN channels.
• It employs a distribution matcher and systematic FEC encoding with staircase codes to achieve significant gains in spectral efficiency and robust performance.
• PAS offers modular, low-complexity implementations ideal for high-throughput optical communications and other spectrally constrained environments.

Probabilistic amplitude shaping (PAS) is a coded modulation paradigm that achieves near-capacity transmission rates on the additive white Gaussian noise (AWGN) channel using discrete constellations. By shaping the amplitude distribution of real or complex modulation symbols—typically according to a Maxwell–Boltzmann law—PAS enables significant gains in spectral efficiency relative to uniform signaling. The architecture is modular: bits are parsed into shaped amplitude bits using a distribution matcher and parity bits for the sign, which are provided by a systematic forward error correction (FEC) encoder. When implemented with hard-decision decoding (HDD) and staircase codes, PAS delivers robust performance and practical efficiency for high-throughput optical communications and other spectrally constrained channels (Sheikh et al., 2017).

1. System Model and Maxwell–Boltzmann Shaping

PAS is usually formulated on the discrete-time real AWGN channel: $$Y_i = \Delta X_i + Z_i, \quad Z_i \sim \mathcal{N}(0,1)$$ where $\mathbb{E}[(\Delta X)^2]=P$ and SNR $=P$. The signaling alphabet is $M$-ary amplitude-shift keying (M-ASK),

$\mathcal{X} = \{\pm1, \pm3, \ldots, \pm(2^m-1)\}$

PAS shapes the input distribution according to a Maxwell–Boltzmann law: $$P_X^\lambda(x) = \frac{e^{-\lambda x^2}}{\sum_{\tilde{x} \in \mathcal{X}} e^{-\lambda \tilde{x}^2}}, \quad \lambda > 0$$ where the parameter $\lambda$ is optimized to maximize the achievable information rate.

2. Achievable Information Rates and Shaping Gain

For PAS with bit-wise HDD and Gray-labeled mapping, the achievable rate is

$R = \left[ H(X) - m\,H_b(p) \right]^+ \tag{1}$

where $p = \Pr\{\hat{b} \neq b\}$ is the pre-FEC bit error rate, and $H_b(\cdot)$ is the binary entropy function. Compared to uniform inputs, Maxwell–Boltzmann shaping yields up to $\approx 2$ dB gain at the same spectral efficiency.

For reference, the soft-decision mutual information is

$$I(X;Y) = \mathbb{E}\left[ \log_2 \frac{p(y|x)}{\sum_{x'} p(y|x') P_X(x')} \right]$$

The paper reports that, for shaped signaling, the system achieves performance within $0.57$–$1.44$ dB of the achievable rate over a wide SE range.

Shaping gains (uniform vs shaped M-ASK):

Constellation	SE (bpcu)	Gain (dB)
4-ASK	1	0.78
8-ASK	2	1.56
64-ASK	5	2.37

In coded system operation with staircase codes, up to $2.88$ dB gain for 256-QAM and $1.77$ dB for 64-QAM (over uniform signaling) is observed.

3. Distribution Matcher (Shaping Encoder) Design

The shaping encoder is realized by a distribution matcher (DM) that maps uniform bits to amplitude sequences with prescribed empirical distributions. The paper employs constant-composition distribution matching (CCDM), characterized by:

• Fixed-length mapping, avoiding catastrophic error propagation.
• Arithmetic coding implementation, no look-up tables.
• Complexity is $O(n)$, where $n$ is the block length.

The shaping rate for DM is

$R_{\text{shaping}} = \frac{\gamma n}{n} \to H(A)$

as $n \to \infty$.

4. PAS Coding Structure with Staircase Codes and HDD

The PAS coded modulation transmitter executes the following workflow:

1. Input Bit Parsing: Split information bits $u = (u^{(1)}, u^{(2)})$ into $u^{(1)}$ for amplitude shaping and $u^{(2)}$ for sign mapping.
2. Amplitude Labeling: Map each amplitude to its $(m-1)$-bit Gray code.
3. FEC Encoding: Systematic staircase encoder takes $(b^n, u^{(2)})$ as info bits and produces parity bits $p^n$.
4. Sign-bit Mapping: Combine parity bits and info bits row-wise to form $n$ sign bits, mapped to $\{-1, +1\}$.
5. Symbol Generation: Transmit $x_i = \Delta(2s_i-1)a_i$.

Due to the symmetry of $P_X^\lambda$, the channel input decomposes as $X=A \cdot S$ with $P_A(a) = 2 P_X^\lambda(a)$, $P_S(+1)=P_S(-1)=\frac{1}{2}$. PAS ensures the amplitude histogram via CCDM and approximates uniform sign distribution via parity bits.

FEC rate adaptation is governed by BCH code parameters $(v, t, s)$ and overall spectral efficiency is tuned using the shortening parameter $s$: $$R_s = 1 - \frac{2(n_c - k_c)}{n_c} = \frac{m-1+\gamma}{m}$$

Receiver operation uses a symbol-wise MAP hard detector followed by staircase code decoding using a sliding window BDD algorithm, low-latency, and up to 8 iterations.

5. Implementation, Complexity, and Practical Advantages

PAS with staircase codes for HDD offers several practical advantages:

• Single-code operation: A fixed staircase code supports multiple spectral efficiencies by adjusting shortening, simplifying hardware and reducing decoder area.
• Streaming CCDM: Arithmetic-coder-based DM scales with throughput; parallel and product DMs further enhance implementation efficiency.
• Receiver simplicity: Sliding-window HDD yields low decoding latency and power compared to soft-decision LDPC alternatives.

6. Comparison with Uniform Signaling and Overall Impact

Relative to conventional uniform signaling and staircase-coded modulation, PAS consistently increases spectral efficiency and reduces required SNR for equivalent post-FEC error rate. Gains are robust across a broad range of M-ASK constellations and QAM formats. At the core, PAS schemes operate within 0.57–1.44 dB of the corresponding achievable information rate for a wide range of spectral efficiencies.

The flexibility and modularity of PAS have led to its widespread adoption in high-throughput fiber-optic systems and rate-adaptive coded modulation, with foundational theory resting on entropy-optimal amplitude shaping and superior implementation efficiency (Sheikh et al., 2017).