Hierarchical Constellation Design

• Hierarchical constellation design is a structured arrangement of signal points that enables simultaneous transmission of high- and low-priority data streams under power and PAPR constraints.
• It employs advanced optimization techniques such as multi-start primal-dual interior-point methods and nested lattice constructions to improve coding gains and overall performance.
• The approach underpins applications in wireless broadcast, visible light communications, and satellite mega-constellations, offering scalability and backward compatibility for legacy systems.

Hierarchical constellation design refers to the structured arrangement of signal points in the complex plane (or high-dimensional Euclidean space) to enable simultaneous transmission of multiple data streams with distinct priorities and protection levels. This methodology generalizes traditional constellations (e.g., QAM, PSK) by exploiting the structure of multi-bit labeling, power and peak constraints, coding gains, and, in space networking, recursive topological symmetry. Hierarchical constellations are central to hierarchical modulation (HM) in wireless broadcast, upgrades of legacy systems, high-dimensional shaping for visible light communications, and recursive layouts of satellite mega-constellations.

1. System Models and Definitions

In hierarchical modulation, the transmitter maps $m = m_H + m_L$ bits per symbol to complex or real-valued vectors $Z$, with high-priority (HP) and low-priority (LP) bit streams, such that $\mathbf B_H = (B_H^1,\ldots,B_H^{m_H})$, $\mathbf B_L = (B_L^1,\ldots,B_L^{m_L})$, and $Z = \mu(\mathbf B_H, \mathbf B_L)$ for a bijection $\mu$ onto the constellation $\mathbb X \subset \mathbb C$ (Xiao et al., 2016). In non-negative intensity modulation for VLC, $\mathbb X \subset \mathbb R_+^n$ with peak $\Vert x \Vert_\infty \leq 1$ and average $$\frac{1}{n|\mathbb X|}\sum_{x \in \mathbb X}\Vert x \Vert_1 \leq \alpha$$ (Guo et al., 2023). Legacy system upgrades adopt a basic/secondary form, e.g., for QPSK/16QAM, $s = s_{\rm basic} + \alpha s_{\rm secondary}$, with $\lambda = d_2/d_1$ controlling performance penalty and payload (Jiang et al., 2013).

Satellite constellations such as the Rosette and Fractal Rosette employ recursive, orbitally symmetric placement of $N^k$ satellites, where the hierarchical structure applies to both physical geometry (multi-level nested orbits) and protocol stack (address/route assignment) (Li et al., 2021).

2. Hierarchical Modulation: Achievable Rates and Optimization

HM enables the simultaneous transmission of HP and LP streams over the same channel. In an AWGN broadcast model, two receivers at SNR$_H$ and SNR$_L$ ($\sigma_H^2 \geq \sigma_L^2$) observe $Y_H = Z + N_H$ and $Y_L = Z + N_L$ (Xiao et al., 2016). The bit-interleaved coded modulation (BICM) with successive interference cancellation (SIC) provides achievable rates:

• HP stream: $r_H = \sum_{i=1}^{m_H} I(B_H^i; Y_H)$
• LP stream: $r_L = \sum_{j=1}^{m_L} I(B_L^j; Y_L | \mathbf B_H)$

Design aims to maximize $r_L$ subject to $r_H \geq r^*$, average power constraint $\mathbb E[|Z|^2] \leq p$, and optional peak-to-average power ratio (PAPR) $$\max_{z\in\mathbb X}|z|^2 /\mathbb E[|Z|^2] \leq \xi$$. The constellation positions $\{z_k\}$ and label assignments are optimized by multi-start primal–dual interior-point methods over the non-convex objective (Xiao et al., 2016). Symmetry (e.g., central symmetry for $m_H=2$) reduces dimensionality.

Empirical studies show hierarchical constellations achieve rate regions strictly containing those of orthogonal transmission (such as time division), with optimized HM outperforming two-scale H-QAM in both rate and PAPR metrics (Xiao et al., 2016). When PAPR restrictions are imposed, HM consistently exceeds the corresponding H-QAM rate region.

3. Geometric Shaping and Hierarchical Coding in Multidimensional Constellations

Optimal shaping for finite blocklength transmissions, particularly in IM/DD VLC systems, involves constructing the constellation $\mathbb X$ as a subset of the truncated cube $T_n(t) = \{x \in [0,1]^n : \sum_i x_i \leq t\}$ maximizing volume (shaping gain) under average intensity constraints (Guo et al., 2023). Second-order asymptotics and large-deviation input-tail bounds are derived, with the optimal truncation parameter satisfying $\tau_n^* = \alpha + (1/\mu^*) \cdot (1/n) + o(1/n)$, where $\mu^*$ solves $\alpha = 1/\mu^* - 1/(e^{\mu^*} - 1)$.

Hierarchical construction proceeds via nested Construction B lattices: a coarse boundary of $D_n$ (even-sum integer lattice) points inside $T_n(t_n^*)$ selects shaping bits, while fine coding is achieved by cosets of binary code $\mathcal C$ and set $\mathcal A$ of coset representatives, generating final constellation points $\lambda = 4d + 2c + a$ (Guo et al., 2023). This "coarse-shaping, fine-coding" architecture is designed for rapid mapping ($O(n)$ via FFT-assisted shell mapping) and low-complexity bounded-distance decoding. An explicit example using the 24-dimensional Leech lattice (Golay code $G_{24}$) yields nominal coding gain of $\approx 3$ dB and operational bit error rates far below benchmarks, with efficient scaling to meet both peak and average constraints (Guo et al., 2023).

4. Hierarchical Constellation Design for Legacy System Upgrades

To upgrade digital broadcast (e.g., satellite TV/radio), hierarchical modulation overlays a secondary constellation atop the existing basic constellation in a backward-compatible arrangement (Jiang et al., 2013). Specifically,

• The primary (basic) layer retains QPSK points at $(\pm d_1, \pm d_1)$.
• Secondary (upgrade) bits modulate smaller offsets $\pm d_2$ ($\lambda=d_2/d_1$) within each cluster, yielding a 16QAM overall structure.

Performance analysis shows that legacy QPSK receivers experience a modulation noise-induced penalty, $$P_{\rm MNR (dB)} = 10\log_{10}(1 + \lambda^2 (1 + \mathrm{CNR}))$$, while upgraded receivers decode both layers with soft iterative decoders. The secondary payload rate scales as $$R_{\rm sec}/R_{\rm basic} \approx \lambda^2/(1-\lambda)^2$$, so a penalty of 0.25–0.5 dB yields $1\%$–$3\%$ additional payload (Jiang et al., 2013). Strong coding (e.g., turbo/LDPC) is necessary for secondary bits; basic layer coding must match the deployed standard.

5. Hierarchical Design in Satellite Mega-Constellations

In large-scale LEO satellite networks, hierarchical constellations are realized in both geometric and protocol domains. The F-Rosette structure recursively nests $N$-orbit Rosette base constellations, creating $N^{k+1}$ satellites each with a time-invariant, multi-level address $s_0.s_1\ldots s_k$, embedding both geographical and topological hierarchy (Li et al., 2021). Earth coverage is partitioned into $(N-m)^2N^{2k}$ stable “cells,” where each user’s fixed IPv6 address is tied to its cell and does not change as satellites move.

Routing embeds hierarchical cell and satellite addresses, guaranteeing minimum-hop and low-delay paths (≤1.4% delay stretch); resource costs per satellite are marginal (<2 MB memory, <1% CPU). The addressing never requires re-convergence under mobility, ensuring network stability and scalability absent in non-hierarchical layouts (Li et al., 2021). The satellite count for coverage matches Starlink with 35–57% fewer satellites at $k=1$, and higher $k$ enhances efficiency.

6. Design Guidelines, Constraints, and Practical Considerations

For constellation optimization in HM (Xiao et al., 2016), the following principles are established:

1. Begin from a well-shaped baseline (QAM, APSK) scaled to full power.
2. Impose natural or quasi-Gray labeling; divide the constellation into $2^{m_H}$ clusters of $2^{m_L}$.
3. Leverage central or rotational symmetries ($m_H > 1$) to reduce optimization variables.
4. Apply multi-start primal-dual interior-point optimization on the negative LP rate, subject to HP rate, average power, and PAPR constraints.
5. Retain solutions with maximal $r_L$, $E[|Z|^2]=p$, and acceptable PAPR.
6. For stricter PAPR, incorporate the constraint and reoptimize; the HM rate region remains strictly above optimized H-QAM.

In legacy upgrades (Jiang et al., 2013), select $\lambda$ to bound legacy receiver penalty (0.08–0.12: <0.3 dB penalty, $~$1% rate; 0.12–0.16: <0.5 dB penalty, $~$2–3% rate). Maintain original coding for basic bits, exploit dense channel coding on secondary bits, and adopt iterative receivers for best performance.

Multidimensional shaping (Guo et al., 2023) employs “truncated cube” constraints, nested-lattice shaping/coding, and mapping/decoding algorithms optimized for computational and BER efficiency. Protocol hierarchies in F-Rosette follow recursive enumeration, stable cell addressing, and prefix/coordinate-based routing (Li et al., 2021).

7. Future Directions and Connections

Hierarchical constellation design is foundational for layered broadcast, 5G-6G superposition coding, future satellite network layout, energy-efficient high-dimensional communication, and iterative upgrades to legacy digital transmission systems. Continued investigation targets non-AWGN channels, more general coding/labeling schemes, and further integration of geometric shaping and network protocol hierarchies to approach theoretical limits on spectral efficiency, latency, and stability.