Per-Codeword Power Shaping: Techniques & Gains

Updated 2 December 2025

Per-codeword power shaping is a method that precisely engineers the energy distribution of each codeword to meet strict peak and average power constraints.
It leverages techniques like probabilistic amplitude shaping and enumerative sphere shaping to achieve significant energy efficiency gains and near-optimal performance at finite blocklengths.
This approach enhances decoding thresholds in spatially coupled codes and enables envelope control in peak-power systems, offering practical benefits for multiuser and resource-allocation scenarios.

Per-codeword power shaping refers to the explicit control or management of the power, energy, or symbol composition of each individual codeword in a communication or coding system, with the objectives of minimizing average/peak power, maximizing rate, optimizing for channel constraints (e.g., peak, average, or per-symbol limitations), or enhancing system-level performance such as decoding threshold, block error rate, or peak-to-average power ratio (PAPR). This concept spans a range of coding, modulation, and resource allocation techniques, extending from variable-length prefix coding to blockwise shaping in probabilistic amplitude shaping (PAS) and energy allocation in multiuser and spatially-coupled systems.

1. Foundations and Core Principles

At its core, per-codeword power shaping imposes explicit constraints or objectives on physical-layer transmitted sequences: instead of relying solely on ensemble-level symbol probabilities, the distribution of energy (or higher-order amplitude statistics) is engineered on a per-codeword or per-block basis. This can be formalized as bounding $\sum_{i=1}^N f(x_i)$ for each codeword $(x_1,\ldots,x_N)$ by some deterministic (e.g., average, peak, or composition) constraint, or prescribing individual codeword-level measures (such as symbol weights in binary codes or amplitude sums in multilevel constellations).

Maxwell–Boltzmann distributions are a central theoretical reference, as they provide the minimum-energy, maximum-entropy distribution for many channels. However, with finite blocklengths or strict per-codeword constraints (e.g., peak or composition shaping), dyadic or distribution-matched codebooks strive to approximate their performance.

2. Probabilistic Amplitude Shaping and Enumerative Sphere Shaping

In PAS, a shaped source node (distribution matcher) generates amplitude sequences with a prescribed empirical distribution. Classical CCDM guarantees only constant composition of amplitudes per block; however, Enumerative Sphere Shaping (ESS) sharpens the per-codeword constraint by producing amplitude vectors within an energy-bounded "sphere" $S =\left\{a^N \in A^N: \sum_{i=1}^N a_i^2 \leq E_{\max}\right\}$ . ESS constructs an indexable mapping between input bits and shaped amplitude sequences, minimizing the finite-length rate loss versus ideal Maxwell–Boltzmann shaping.

The ESS encoding algorithm builds a bounded-energy trellis, where state transitions are determined only by energy feasibility. Complexity is quasi-linear in blocklength, making ESS preferable to shell mapping (cubic) and significantly more efficient than existing alternatives for moderate $N$ . ESS yields negligible shaping loss (e.g., $0.15$ bit/amp at $N=96$ and 8-ASK, compared to $0.75$ bit/amp for CCDM). AWGN and frequency-selective simulations confirm up to $1.6\,\text{dB}$ energy efficiency gain at short blocklengths, with both convolutional and LDPC FEC (Gültekin et al., 2019).

3. Energy Shaping for Threshold Gains in Spatially Coupled Codes

For spatially-coupled LDPC ensembles, per-codeword power shaping (energy shaping) targets iterative decoding thresholds. In the tailbiting, non-terminated regime, the decoder's threshold can be restored to the terminated case without rate loss by boosting the transmitted energy of a carefully-chosen contiguous fraction $\lambda$ of bits per codeword. Under an average-energy constraint, bits are partitioned into two classes: a prefix $\lambda N$ of high-energy $E_1$ and a remainder $(1-\lambda)N$ at $E_2 < E_1$ .

The influence of these classes is analyzed through protograph EXIT (P-EXIT) density evolution, leading to an optimization of $(\lambda,\beta)$ to minimize overall SNR threshold. Empirically, energy shaping allows tailbiting SC-LDPC code ensembles (rate $1/2$) to achieve $0.210\,\text{dB}$ BP threshold—the same as the terminated ensemble but without the associated rate loss. Practical implementations require only two energy levels per codeword and no changes to FEC structure, enabling full threshold gains for large $N$ and over $1.1\,\text{dB}$ gain at moderate blocklengths (Jerkovits et al., 2018).

4. Per-Codeword Envelope Control in Peak-Power Constrained Systems

Envelope control methods (such as Dynamic Selective Mapping, DSLM) address per-codeword PAPR, peak amplitude, and waveform smoothness in peak power-constrained (PPC) intensity modulation/direct detection (IM-DD) fiber systems. Standard probabilistic shaping does not control envelope burstiness, which is critical when transmitter memory effects and nonlinearities are present.

DSLM is a causal, memory-aware mapping operating after PAS: for each symbol, the mapping consults the preceding $L-1$ output symbols and selects among candidate outputs (including the standard Gray-mapped modulation symbol) to avoid forbidden patterns defined by envelope cost functions (e.g., variance times max amplitude). This ensures that each codeword, within a frame, satisfies deterministic or probabilistic envelope/peak-power constraints, with composition derived from forbidden pattern avoidance and hard or soft constraints on the sequence window.

At the receiver, a turbo equalizer with a modified M-BCJR algorithm resolves the symbol sequence, accounting for DSLM ambiguities. Empirical results demonstrate $1\,\text{dB}$ receiver sensitivity gain at $10^{-4}$ BER in 56 GBaud PAM8 fibers, as well as up to $0.7\,$ dB reach extension relative to PAS-only architectures (Zou et al., 24 Jul 2025).

5. Variable-Length and Prefix-Free Codeword Shaping

Variable-length prefix-free or variable-to-fixed coding schemes provide another class of per-codeword power shaping by leveraging codeword assignments that minimize average energy subject to a given entropy rate. V2F, F2V, and V2V prefix-free code constructions assign higher-probability inputs to codewords with lower energy (e.g., shorter codewords mapped to lower-amplitude or zero-energy symbols), often approaching the dyadic approximation of Maxwell–Boltzmann.

To enable framing and synchronization on finite frames, hybrid schemes time-share between efficient variable-length codes and fixed-to-fixed backup codes, with a bounded penalty on average energy per codeword (≤ 0.2 dB for $n\leq10^4$ ). Such methods offer fine-grained shaping, negligible framing penalty, and near-theoretic shaping gain in AWGN channels, and can allow per-codeword PAPR or composition control (Cho, 2018).

6. Power Shaping for Special Channels and Constraints

On-off keying (OOK) and asymmetric channels present further opportunities for per-codeword power shaping. For channels with a non-symmetric capacity-achieving input, per-codeword shaping mandates codewords with non-uniform empirical weight (fraction of '1's), as in the Honda–Yamamoto polar-code approach. This jointly realizes distribution matching and FEC: codeword construction partitions input indices into information, frozen, and shaping classes, with synthetic channel selection ensuring codeword-level match to the optimal input pmf. This yields the full asymptotic SNR gain of shaped OOK (1.8 dB gain observed for $N=65536$ , $R=0.25$ bpcu) (Wiegart et al., 2019).

Alternatively, per-codeword bounds on the number of '1's in binary codes (variable-length prefix or alphabetic codes with at most $D$ ones per codeword) are handled by efficient $O(n^2 D)$ -time dynamic programming. This is critical for applications under strict energy draws, optical symbol intensity, or "costly" event constraints (Bruno et al., 19 Jan 2025). Such codes admit explicit existence criteria (Kraft-like capacity inequalities) and global optimality.

7. Per-Codeword Power Shaping in Multiuser and Resource-Allocation Systems

In multiuser OFDM and spectrum-sharing scenarios, per-codeword—or more accurately, per-block or per-transmit-frame—power shaping arises in "iterative spectrum shaping" (ISS) contexts. Carrier-joint encoding imposes a global power constraint across all subcarriers per codeword. The optimal allocation is generally a water-filling-type solution, potentially with opportunistic multiuser detection (OMD): each subcarrier's power level depends on the joint decodability of interfering user codewords, with per-codeword spectrum realized via dual optimization (bisection in the Lagrange multiplier corresponding to average-power constraint). This achieves maximal rate given codeword-level interference and power constraints, and allocates power "over the codeword's spectrum" rather than symbolwise (0901.2194).

Per-codeword power shaping thus encompasses a large set of techniques for channel input design, code construction, and resource allocation, unified by the explicit enforcement of energy, composition, or envelope constraints on the codeword/frame rather than merely on the ensemble distribution. Strategic exploitation of per-codeword constraints delivers significant gains in average or peak power, decoding threshold, and implementation feasibility across practical communication and coding systems.