Generalized Constellation-Splitting in Noncoherent MACs

• Generalized Constellation-Splitting is a joint-constellation construction that partitions a single-user Grassmannian constellation into disjoint user subsets for noncoherent multiple-access channels.
• It leverages ML error analysis metrics like pairwise log-likelihood ratios and chordal distances to optimize signal separation and improve performance at high SNR.
• The design framework spans from random bipartitioning to refined manifold optimization, offering a scalable, low-complexity solution for multi-user detection.

Generalized Constellation-Splitting (GCS) is a joint-constellation construction for the noncoherent multiple-access channel (MAC) in which a well-designed single-user Grassmannian constellation is partitioned into disjoint subsets assigned to different users, and the resulting Cartesian product is decoded jointly by the noncoherent maximum-likelihood (ML) rule. In the communication-theoretic literature, the term is associated primarily with the two-user noncoherent MAC construction developed from ML error analysis and with its later extension to $K$-user noncoherent MIMO MACs, where GCS is positioned as a low-complexity alternative to direct joint constellation optimization over all users (Ngo et al., 2020, Ngo et al., 2020).

1. Noncoherent MAC formulation

The canonical setting is a synchronous noncoherent Rayleigh block-fading MAC with coherence block length $T \ge 2$, $K$ users, user-$k$ transmit matrix $X_k \in \mathbb{C}^{T \times M_k}$, receiver with $N$ antennas, and no instantaneous channel state information at any terminal. The per-block received signal is

$$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$

with $X := [X_1 \ \cdots \ X_K]$, i.i.d. Rayleigh block fading, and additive noise $Z$ with i.i.d. $\mathcal{CN}(0,1)$ entries. Each user transmits symbols from a finite constellation $T \ge 2$0, and the joint constellation is the Cartesian product $T \ge 2$1 (Ngo et al., 2020).

Conditioned on a joint symbol $T \ge 2$2, the columns of $T \ge 2$3 are i.i.d. complex Gaussian with covariance $T \ge 2$4. The noncoherent likelihood is

$T \ge 2$5

and the ML detector is

$T \ge 2$6

A fundamental identifiability constraint follows immediately: distinct joint symbols must satisfy

$T \ge 2$7

This condition is necessary because the output distribution depends on $T \ge 2$8 only through $T \ge 2$9. In the MAC papers where GCS is introduced, the construction is therefore not a receiver-side heuristic but a joint codebook design problem constrained by the geometry of the noncoherent likelihood.

2. ML error analysis and design metrics

The design program for GCS is driven by pairwise error analysis. For the joint symbol error rate $K$0, the worst-case pairwise error probability (PEP) controls both upper and lower union-type bounds:

$K$1

Accordingly, the design problem is reduced to making the worst joint-symbol pair as distinguishable as possible under ML (Ngo et al., 2020).

Two closely related families of metrics appear in the literature. In the two-user paper, the pairwise log-likelihood ratio $K$2 is used to motivate the max-min criterion

$K$3

In the large-$K$4 regime, the Chernoff–Stein lemma implies that maximizing $K$5 maximizes the pairwise error exponent with respect to $K$6. In the $K$7-user generalization, nonasymptotic PEP bounds are expressed through the eigenvalues $K$8 of

$K$9

and the metric

$k$0

satisfies

$k$1

This motivates the nonasymptotic criterion $k$2, while the relaxed Chernoff-bound metric $k$3 provides another tractable objective (Ngo et al., 2020).

At high signal-to-noise ratio, both papers isolate the same dominant term,

$k$4

and define the simplified design objective

$k$5

This simplification is structurally important because $k$6 is the only term in the mean PLLR that scales linearly with transmit power $k$7 when $k$8; the determinant and logarithmic terms scale only as $k$9. The $X_k \in \mathbb{C}^{T \times M_k}$0-user paper further gives a geometric interpretation: the PEP exponent grows linearly with a Riemannian distance between the positive definite matrices $X_k \in \mathbb{C}^{T \times M_k}$1 and $X_k \in \mathbb{C}^{T \times M_k}$2, with $X_k \in \mathbb{C}^{T \times M_k}$3 bounded above and below by the affine-invariant Riemannian distance on $X_k \in \mathbb{C}^{T \times M_k}$4.

3. Grassmannian signaling and the GCS construction

GCS is most natural under unitary space-time modulation or, more generally, Grassmannian signaling. In the symmetric case, each user employs symbols satisfying

$X_k \in \mathbb{C}^{T \times M_k}$5

so each codeword corresponds to a point on a Grassmann manifold. In the single-user case, the high-SNR metric reduces to a chordal-distance form, and maximizing $X_k \in \mathbb{C}^{T \times M_k}$6 is equivalent to maximizing the minimum pairwise chordal distance. GCS transfers this single-user Grassmannian packing principle to the MAC setting (Ngo et al., 2020).

For two users, the construction starts from a single-user Grassmannian constellation $X_k \in \mathbb{C}^{T \times M_k}$7 with good minimum chordal distance. One then partitions $X_k \in \mathbb{C}^{T \times M_k}$8 into two disjoint subsets $X_k \in \mathbb{C}^{T \times M_k}$9, assigns them to the two users, and forms the joint constellation $N$0. The key analytical point is that the relevant two-user high-SNR separation can be controlled by user-wise worst-pair metrics $N$1 and $N$2, and the design objective becomes

$N$3

The paper then derives necessary and sufficient conditions linking these MAC metrics to three classes of Grassmannian overlaps: intra-user overlaps in $N$4, intra-user overlaps in $N$5, and cross-user overlaps between $N$6 and $N$7. This leads to the simplified distance-based criterion

$N$8

The sufficient construction is correspondingly simple: choose $N$9 so that

$$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$0

split it into disjoint subsets, and decode jointly with the noncoherent ML rule. A random bipartition already suffices to enforce the overlap conditions, while min-max graph bipartitioning can further improve the criterion. In the two-user analysis, this yields a lower bound on the worst-case separation metric $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$1 that grows linearly in $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$2 when $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$3 is sufficiently small (Ngo et al., 2020).

The $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$4-user extension preserves the same principle: split a sparse single-user Grassmannian constellation into $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$5 disjoint subsets and control both within-subset and cross-subset overlaps. The paper states that at high SNR the single-user USTM criterion carries over to the joint MAC design via splitting, and introduces per-user worst-pair surrogates $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$6 satisfying

$$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$7

4. Optimization, refinements, and related constructions

GCS sits within a broader design toolbox that ranges from direct manifold optimization to structured constructions. The central practical distinction is between optimizing constellation points themselves and optimizing only a partition of an already good single-user constellation.

Method	Mechanism	Role
GCS partitioning	Split $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$8 into disjoint subsets	Low-complexity structured design
Alternating optimization	Optimize one user at fixed others via $$Y = \sum_{k=1}^{K} X_k H_k^{\mathsf H} + Z \;=\; X H^{\mathsf H} + Z,$$9, $X := [X_1 \ \cdots \ X_K]$0, or $X := [X_1 \ \cdots \ X_K]$1	Reduced-complexity refinement
Manifold optimization	Optimize max-min criteria on Grassmann or oblique manifolds	Direct high-performance design
Precoding construction	Map lower-dimensional constellations through user-specific precoders	Increase inter-user separation
Power optimization	Optimize $X := [X_1 \ \cdots \ X_K]$2 or $X := [X_1 \ \cdots \ X_K]$3	Asymmetric-rate refinement

For direct numerical optimization, the $X := [X_1 \ \cdots \ X_K]$4-user paper smooths the min-max objective by replacing $X := [X_1 \ \cdots \ X_K]$5 with $X := [X_1 \ \cdots \ X_K]$6, stacks the representatives on the oblique manifold, and uses the Riemannian gradient

$X := [X_1 \ \cdots \ X_K]$7

A manifold toolbox such as manopt is then used for gradient descent. The same paper also considers the nonasymptotic metric $X := [X_1 \ \cdots \ X_K]$8, the high-SNR metric $X := [X_1 \ \cdots \ X_K]$9, and the KL-based metric $Z$0 as optimization targets (Ngo et al., 2020).

A more geometric generalization is the precoding-based construction for multi-user MIMO. Each user starts from a lower-dimensional Grassmannian constellation $Z$1 and transmits

$Z$2

where $Z$3 is a user-specific full-rank precoder. Type-I and Type-II precoders are designed so that user subspaces share as few dimensions as possible, directly reducing cross-inner-products and improving $Z$4.

For asymmetric two-user SIMO systems, the same framework is extended to per-user power optimization. The paper states that $Z$5 decreases in $Z$6, $Z$7 increases in $Z$8, and that solving $Z$9 provides an approximate maximizer of $\mathcal{CN}(0,1)$0. This suggests that GCS is not restricted to equal-power symmetric designs, although the cleanest analytical guarantees are derived in that regime.

Related structured-splitting ideas exist outside the precise GCS definition. “Cube-Split” partitions $\mathcal{CN}(0,1)$1 into cells and maps Euclidean bent-hypercube grids into each cell for noncoherent SIMO signaling, while translation-pattern GSM splits bits across complex symbols, active antenna subsets, and SPC-coded translation patterns in the spatial domain (Ngo et al., 2019, Singla et al., 2020).

5. Performance evidence and complexity

The two-user paper reports detailed simulations for the SIMO case $\mathcal{CN}(0,1)$2, $\mathcal{CN}(0,1)$3, symmetric rates $\mathcal{CN}(0,1)$4, and Grassmannian signaling on $\mathcal{CN}(0,1)$5. For $\mathcal{CN}(0,1)$6, $\mathcal{CN}(0,1)$7, and SNR from $\mathcal{CN}(0,1)$8 to $\mathcal{CN}(0,1)$9 dB, the constellations optimized with the ML-based criteria $T \ge 2$00 and $T \ge 2$01 achieve the lowest joint SER across the whole range; at $T \ge 2$02 dB the reported SER is approximately $T \ge 2$03, and at $T \ge 2$04 dB approximately $T \ge 2$05. Alternating optimization is slightly worse, with SER approximately $T \ge 2$06 at $T \ge 2$07 dB. GCS partitioning by random split yields SER approximately $T \ge 2$08 at $T \ge 2$09 dB, comparable to the noncoherent precoding baseline at approximately $T \ge 2$10, while the pilot-based ML receiver gives approximately $T \ge 2$11. The paper also states that metric values confirm that $T \ge 2$12 closely tracks $T \ge 2$13 at high SNR, validating the simplified criterion (Ngo et al., 2020).

For larger constellations, where full numerical optimization becomes cumbersome, GCS alone remains effective. In the case $T \ge 2$14, $T \ge 2$15, and $T \ge 2$16, the partitioning construction outperforms the pilot-based scheme, which the paper presents as evidence of its practicality.

The $T \ge 2$17-user generalization gives a similar ranking. In the two-user case $T \ge 2$18, $T \ge 2$19, $T \ge 2$20, $T \ge 2$21, and $T \ge 2$22 bpcu, $T \ge 2$23 gives the best ML SER, $T \ge 2$24 and $T \ge 2$25 perform similarly, and both are significantly better than the pilot-based and adapted point-to-point metrics. In the three-user case $T \ge 2$26, $T \ge 2$27, $T \ge 2$28, $T \ge 2$29, and $T \ge 2$30, the same hierarchy persists, with $T \ge 2$31 remaining best and pilot-based reception inferior to $T \ge 2$32 at moderate and high SNR (Ngo et al., 2020).

Receiver complexity is dominated by noncoherent ML detection over the joint constellation. For each candidate $T \ge 2$33, one computes $T \ge 2$34, its inverse, and its log-determinant, and evaluates the ML metric using $T \ge 2$35. In the two-user paper this is stated as $T \ge 2$36 per candidate, with $T \ge 2$37, although the Woodbury identity can exploit the low rank of $T \ge 2$38. By contrast, GCS design itself requires only splitting a single-user constellation; random bipartition is trivial, graph partitioning is standard, and full manifold optimization can be performed offline and reused across SNRs.

6. Scope, limitations, and later uses of the term

The MAC formulation of GCS is derived under i.i.d. Rayleigh block fading, synchronous users, known noise variance, and coherence block length $T \ge 2$39. The papers emphasize several limitations: ML detection complexity can be high; manifold optimization is nonconvex and returns local minima; high-SNR metrics can be suboptimal at low SNR; and the size of the underlying single-user constellation is constrained by Grassmannian packing bounds. The two-user paper also notes that systematic linear receivers used in the pilot-based baseline do not apply to the noncoherent GCS signals (Ngo et al., 2020, Ngo et al., 2020).

Several extensions are explicit. More than two users can be handled by $T \ge 2$40-way partitioning, though the associated graph partitioning and cross-overlap control become harder. Correlated fading can be incorporated by pre-whitening transformations and modified power constraints. Asymmetric rates and powers require joint optimization of subset sizes and power allocation. These points indicate that GCS is best viewed as a structured design principle rather than a closed-form universal optimum.

After the MAC papers, the label “generalized constellation-splitting” was reused in other settings. In blind SISO-OFDM channel estimation, it denotes a finite-alphabet side-information mechanism that partitions standard constellations into subsets with distinct phase and/or amplitude properties and constrains which subset each subcarrier may use, thereby enabling a modified phase-directed estimator to resolve the residual complex-scalar ambiguity of second-order-statistics channel estimation without pilots (H. et al., 27 Jul 2025). A related paper describes alternate-subcarrier constellation splitting for M-ary PAM systems with non-redundant frequency-domain precoding and frames the method as blind ambiguity resolution in OFDM (H. et al., 27 Jul 2025). In detection under generalized hardware impairments, the phrase is used in a different sense: the PAD-D detector separates amplitude and phase residuals in the polar domain and weights them by their distortion-dependent variances, yielding a symbol-dependent metric $T \ge 2$41 (Oikonomou et al., 12 Nov 2025). The phrase also appears in combinatorics as a “generalized constellation-splitting correspondence” for planar $T \ge 2$42-constellations, where geodesic slicing yields multicontinued fractions and generalized Hankel determinants (Albenque et al., 2011).

This suggests a polysemous terminology. In the noncoherent MAC literature, GCS has a precise meaning: partitioning a sparse single-user Grassmannian constellation into disjoint user codebooks so that intra-user and inter-user subspace overlaps remain controlled under noncoherent ML detection. In later usages, the same phrase denotes more general forms of structured partitioning of symbol sets or combinatorial objects.