SeMAC: Sequential Modulation for AirComp

• SeMAC is a multi-symbol digital AirComp framework that maps each input to a sequence of modulated symbols, enlarging the effective modulation space.
• It generalizes the ChannelComp concept by designing slot-dependent constellations and formulating a non-convex QCQP, later relaxed to an SDP for tractable optimization.
• The approach achieves significant performance gains, with simulation results showing up to an 18 dB NMSE improvement in computing functions like product and max under noisy MAC conditions.

Searching arXiv for SeMAC and closely related AirComp papers. arxiv_search.search with query "Sequential Modulation for AirComp SeMAC ChannelComp digital AirComp" Sequential Modulation for AirComp (SeMAC) is a multi-symbol digital over-the-air computation (AirComp) framework in which each input value is mapped to a sequence of modulated symbols, with distinct constellation diagrams across multiple time slots, so that a receiver infers a desired function from the noisy superposition produced by a multiple access channel (MAC) rather than decoding each transmitter separately (Yan et al., 11 Jul 2025). It is positioned as a multi-symbol extension of ChannelComp, a general digital AirComp framework for arbitrary function computation over a MAC, and its central premise is that enlarging the effective modulation space from a single complex symbol to a vector in $\mathbb{C}^L$ can improve computation reliability under channel noise (Razavikia et al., 2023).

1. Conceptual origin and scope

SeMAC emerged from the digital AirComp line initiated by ChannelComp. ChannelComp established that digital superposition over a MAC need not be treated as a meaningless overlap of symbols; instead, if per-user modulations and the receiver-side tabular mapping are jointly designed, the finite constellation induced by superposition can encode arbitrary functions with finite input domains (Razavikia et al., 2023). In that framework, each function input is encoded into a single digital modulation symbol, and the computation point applies a lookup rule to the received superposed constellation point.

SeMAC generalizes this single-symbol paradigm by replacing one complex symbol with a length-$L$ symbol sequence. In the formulation introduced in "Multi-Symbol Digital AirComp via Modulation Design and Power Adaptation" (Yan et al., 11 Jul 2025), each node encodes its input across multiple slots, and different slots may use different constellations. This gives SeMAC more geometric degrees of freedom than a one-shot constellation in $\mathbb{C}$, and the intended consequence is greater separation between received representations of different function outputs.

A common misconception is to identify SeMAC with repetition coding. The literature distinguishes the two. ReMAC repeats the same symbol over multiple slots, whereas SeMAC allows different constellations across time slots; repetition is therefore only a restricted special case of the broader SeMAC design space (Yan et al., 11 Jul 2025). A second misconception is to view SeMAC as a bit-sliced architecture by definition. Subsequent work treats it instead as a multi-symbol ChannelComp-style mapping of whole quantized inputs to sequences, and explicitly contrasts it with bit-partitioned designs that model bit significance (Yan et al., 25 Nov 2025).

2. Multi-symbol signal model and function representation

The SeMAC system model considers $K$ single-antenna nodes and a single computation point. Each node $k$ holds a discrete input $x_k \in \mathbb{F}_Q$ with $Q=2^b$ elements, represented by $b$ bits, and perfect synchronization in carrier frequency and symbol timing is assumed (Yan et al., 11 Jul 2025). At node $k$, the encoder maps the input to a sequence of $L$ complex symbols,

$L$0

At time slot $L$1, the received MAC sample is

$L$2

with complex channel coefficient $L$3, transmit factor $L$4, and complex AWGN $L$5 (Yan et al., 11 Jul 2025).

Under per-slot channel inversion,

$L$6

the effective model becomes

$L$7

so the receiver observes a noisy superposition sequence across $L$8 time slots (Yan et al., 11 Jul 2025). This is the multi-symbol analogue of the ChannelComp sum channel

$L$9

obtained after channel inversion in the single-symbol case (Razavikia et al., 2023).

The framework is presented as general with respect to the target function. In the SeMAC formulation, the function output range has cardinality $\mathbb{C}$0, and the possible input tuples are indexed by $\mathbb{C}$1. For each such tuple, a binary selector vector $\mathbb{C}$2 with $\mathbb{C}$3 chooses one constellation row from each user-specific modulation matrix. If

$\mathbb{C}$4

then the noiseless aggregated sequence associated with tuple $\mathbb{C}$5 is

$\mathbb{C}$6

The receiver’s tabular mapping is designed so that $\mathbb{C}$7 corresponds to the desired function value $\mathbb{C}$8 (Yan et al., 11 Jul 2025).

The simulation study in the SeMAC paper uses the sum, product, and max as representative functions (Yan et al., 11 Jul 2025). Subsequent work also treats SeMAC as a multi-symbol ChannelComp-type design for arbitrary functions represented through a precomputed table, and highlights product computation as a particularly informative stress case (Yan et al., 25 Nov 2025).

3. Modulation-sequence design and optimization

The central design objective in SeMAC is to separate the aggregated symbol sequences associated with different function outputs. The sequence-space overlap-avoidance condition is written as

$\mathbb{C}$9

or, with $K$0,

$K$1

This is the multi-symbol generalization of the ChannelComp distance constraints used to separate function outputs in the complex plane (Yan et al., 11 Jul 2025).

For node $K$2, the modulation matrix $K$3 contains one row per input value and one column per slot. The SeMAC feasibility problem is stated as

$K$4

The formulation is a non-convex QCQP, since the separation constraints are quadratic in $K$5 and the lifted version carries rank constraints (Yan et al., 11 Jul 2025).

Matrix lifting introduces

$K$6

with diagonal blocks $K$7, and defines

$K$8

Then

$K$9

so the relaxed SDP becomes

$k$0

The rank constraints are dropped in the relaxation, yielding a convex semidefinite program (Yan et al., 11 Jul 2025).

Recovering a feasible modulation from the relaxed solution requires low-rank approximation. If

$k$1

is the eigen-decomposition of the optimal diagonal block, Proposition 1 in the SeMAC paper gives the structured recovery

$k$2

with

$k$3

where $k$4 is the $k$5-th eigenvector, $k$6 the corresponding eigenvalue, and $k$7 the $k$8-th standard basis vector (Yan et al., 11 Jul 2025). This construction enforces rank and column-norm constraints while approximating the relaxed block $k$9 in Frobenius norm.

A plausible implication is that SeMAC should be viewed less as a conventional symbol alphabet and more as a function-aware sequence-constellation design problem in $x_k \in \mathbb{F}_Q$0. That interpretation is consistent with later work that reconstructs SeMAC as a multi-symbol digital AirComp baseline whose design objective is to improve the separation between superposed sequences corresponding to different function outputs (Yan et al., 25 Nov 2025).

4. Detection, tabular decoding, and SeMAC with power adaptation

At the computation point, SeMAC performs slotwise maximum-likelihood detection over the set of noiseless constellation points $x_k \in \mathbb{F}_Q$1. For each slot $x_k \in \mathbb{F}_Q$2, the receiver forms Voronoi regions $x_k \in \mathbb{F}_Q$3 and selects the nearest point,

$x_k \in \mathbb{F}_Q$4

After all $x_k \in \mathbb{F}_Q$5 slots are processed, a tabular mapping $x_k \in \mathbb{F}_Q$6 combines the recovered sequence into the final function estimate $x_k \in \mathbb{F}_Q$7 (Yan et al., 11 Jul 2025). The numerical evaluation uses normalized mean squared error,

$x_k \in \mathbb{F}_Q$8

as the primary metric (Yan et al., 11 Jul 2025).

The original SeMAC formulation assumes flexible constellation design and channel inversion. For scenarios in which modulation formats cannot be changed, the same paper introduces a power-adaptation variant. Subsequent literature refers to this variant as SeMAC-PA (Yan et al., 25 Nov 2025). The idea is to keep a fixed per-slot modulation pattern $x_k \in \mathbb{F}_Q$9 and optimize only amplitude and phase through power vectors $Q=2^b$0 under fading channels (Yan et al., 11 Jul 2025).

With

$Q=2^b$1

the power-adaptation problem is

$Q=2^b$2

Lifting $Q=2^b$3 and defining

$Q=2^b$4

yields the convex SDP relaxation

$Q=2^b$5

Feasible vectors $Q=2^b$6 are then recovered by Cholesky factorization when the lifted matrices are rank one, or by Gaussian randomization otherwise (Yan et al., 11 Jul 2025).

This power-adaptation formulation is function-aware: the amplitude-phase control is not intended for user separation but for enlarging distances between effective received sequences associated with different function outputs. That is one of the distinctive structural features of SeMAC-PA relative to conventional link adaptation (Yan et al., 11 Jul 2025).

5. Relation to ChannelComp, repetition-based designs, and bit-aware successors

The most direct antecedent of SeMAC is ChannelComp, which formulates digital AirComp as function-aware constellation design for a single MAC use (Razavikia et al., 2023). ChannelComp can compute arbitrary functions with finite input domains by optimizing the per-user modulation vector and receiver-side mapping. SeMAC inherits that tabular, function-centric viewpoint but replaces one superposed point in $Q=2^b$7 with one superposed sequence in $Q=2^b$8 (Yan et al., 11 Jul 2025).

Scheme	Core mechanism	Relation to SeMAC
ChannelComp	Single-symbol function-aware digital modulation	Precursor framework
ReMAC	Repetition over multiple slots	Restricted multi-slot baseline
Bit-Slicing	Bit blocks sent separately	Tailored to linear functions
SeMAC	Input mapped to an $Q=2^b$9-symbol sequence with slot-dependent constellations	General multi-symbol design
UBP / IABP	Bit-partitioning plus modulation design	Bit-aware refinements beyond SeMAC

ReMAC and Bit-Slicing are the principal multi-symbol comparators in the SeMAC paper. ReMAC improves robustness through repetition but uses identical constellation in all slots, whereas SeMAC permits different constellations across slots and thus a larger geometric design space (Yan et al., 11 Jul 2025). Bit-Slicing splits inputs into bit blocks transmitted separately; it works well for linear functions such as sums, but for nonlinear functions the required pre-processing and post-processing approximations can introduce large approximation error, which is the reason it performs poorly for product computation in the reported simulations (Yan et al., 11 Jul 2025).

A later paper, "Joint Bit-Partitioning and Modulation Design for Digital AirComp" (Yan et al., 25 Nov 2025), places SeMAC in a broader lineage of multi-symbol digital AirComp schemes. It describes SeMAC as a method that encodes each input into a sequence of modulation symbols and employs an iterative algorithm based on successive convex approximations to reduce computational cost while retaining high computation accuracy. The same work identifies a limitation: SeMAC does not consider bit-level significance, even though erroneous reception of more significant bits leads to larger computational error (Yan et al., 25 Nov 2025). That paper proposes uniform bit-partitioning and importance-adaptive bit-partitioning as successors, and reports that IABP achieves up to a 5 dB reduction in computation error compared to SeMAC, especially for product computation (Yan et al., 25 Nov 2025).

These comparisons clarify two points. First, SeMAC is broader than repetition-based multi-slot transmission. Second, the original SeMAC formulation is sequence-aware but not bit-aware. Subsequent designs therefore refine SeMAC by imposing additional discrete structure on the time slots, rather than rejecting the multi-symbol premise itself (Yan et al., 25 Nov 2025).

6. Performance characteristics, limitations, and broader waveform context

The reported numerical results establish SeMAC as a high-reliability digital AirComp method for nonlinear computation tasks. In one study with $b$0 nodes, inputs $b$1, and target functions equal to the sum, product, and max, moving from $b$2 to $b$3 reduces NMSE for all three functions, with the improvement particularly marked for the product and max (Yan et al., 11 Jul 2025). In a second study with $b$4, inputs $b$5, product computation, and $b$6, SeMAC achieves the lowest NMSE among Bit-Slicing, ReMAC, and ChannelComp with repetition, and the numerical results show that SeMAC can achieve a reliable computation by reducing the computation error up to 18 dB compared to other existing methods, particularly for the product function (Yan et al., 11 Jul 2025).

Under Rayleigh fading with fixed QAM constellations, the SeMAC power-adaptation variant improves NMSE relative to naive fixed-power QAM, but the same experiments indicate that SeMAC with full modulation design under channel inversion still performs better than SeMAC-PA because it can freely shape constellations (Yan et al., 11 Jul 2025). This separation between full constellation design and fixed-format power adaptation is important in practice: the former gives the strongest function-specific geometry, whereas the latter is closer to constrained hardware settings.

The main limitations are equally explicit. SeMAC assumes perfect synchronization in carrier frequency and symbol timing, and the modulation-design formulation relies on perfect CSI and channel inversion (Yan et al., 11 Jul 2025). The relaxed optimization variable has dimension determined by $b$7, and the constraints are indexed by all pairs $b$8, so complexity grows with constellation size, number of nodes, and the cardinality of the function range (Yan et al., 11 Jul 2025). The examples in the original paper focus on the sum, product, and max, even though the framework is presented as general (Yan et al., 11 Jul 2025).

Related waveform research suggests several adjacent directions rather than direct replacements. The survey "Waveforms for Computing Over the Air" frames sequential or time-structured AirComp as layered nomographic computation across resources, with slot-dependent pre-processing and post-processing functions (Pérez-Neira et al., 2024). A plausible implication is that SeMAC can be interpreted as one concrete realization of that broader idea in the digital-constellation domain. Other work on frequency-modulation-based AirComp shows that type-based aggregation via MFSK/TBMA can recover histograms rather than direct sums, with constant-envelope signaling and reported $b$9 dB PAPR, while Log-FSK computes in the frequency domain by extracting a dominant DCT component after nonlinear processing (Martinez-Gost et al., 2024, Martinez-Gost et al., 2024). These results do not redefine SeMAC, but they indicate that sequential modulation for AirComp need not be confined to one-shot linear amplitude mappings. This suggests a broader research agenda in which multi-slot sequence design, bit significance, constant-envelope constraints, and frequency-domain computation are treated as complementary design axes rather than mutually exclusive alternatives.