Multi-Beam Time-Frequency Slot Assignment

• MB-TFSA is a framework that assigns users, flows, or pilots to resource units indexed by beam, time, and frequency under interference, power, and QoS constraints.
• It integrates various formulations—including OFDMA-SDMA and satellite scheduling—using techniques such as zero-forcing, RZF, and QUBO to solve complex coupled optimization problems.
• The approach leverages decomposition and tailored algorithmic strategies to balance efficiency, performance, and computational complexity in multi-beam wireless communications.

Searching arXiv for the cited MB-TFSA-related papers to ground the article in current records. Multi-Beam Time-Frequency Slot Assignment (MB-TFSA) denotes the assignment of communication opportunities across multiple beams, time slots, and, where applicable, frequency channels or other time-frequency structures, under coupling constraints induced by interference, beam exclusivity, power budgets, quality-of-service targets, and physical-layer beamforming or detection. Across the cited literature, the term covers several closely related formulations: downlink OFDMA-SDMA scheduling with zero-forcing precoding, GEO high-throughput satellite user scheduling with full frequency reuse, return-link MF-TDMA scheduling with free-slot assignment, dynamic LEO beam-direction and subchannel-time allocation, time-frequency pilot scheduling in massive MIMO-OFDM, and QUBO-based flow scheduling for digital satellites (Perea-Vega et al., 2012).

1. Core definition and problem abstractions

In its canonical form, MB-TFSA assigns users, flows, or pilot patterns to resource units indexed by beam, time, and frequency. In the digital-satellite flow-scheduling formulation, a resource unit is a tuple $u=(b,f,s)\in U=B\times F\times S$, where $B$ is the set of beams, $F$ the set of frequency channels, and $S$ the set of time slots; the binary decision variable $x_{k,u}\in\{0,1\}$ indicates whether flow $k$ is served on resource unit $u$ (Yan et al., 28 Feb 2026). In dynamic LEO systems, the corresponding time-frequency items are subchannel-time-slot units $(k,t)\in B_{SA}=K\times T$, assigned through $b^t_{k,n}\in\{0,1\}$, while beam pointing is represented separately by coordinate-time-slot units $(c,t)\in C\times T$ via $B$0 (Yuan et al., 2024). In OFDMA-SDMA downlink systems, the assignment variable is $B$1, indicating whether user $B$2 is scheduled on subcarrier $B$3 at time $B$4, with beam occupancy implicit in the precoder columns associated with the scheduled set $B$5 (Perea-Vega et al., 2012).

Several works instantiate only a restricted version of MB-TFSA. In the GEO precoded MB-HTS setting with full frequency reuse, there is effectively a single frequency slot, so MB-TFSA reduces to multi-beam time-slot assignment, or MB-TSA, over an observed window time of $B$6 time slots (Chien et al., 2021). In the multibeam frequency-management framework for mobile users, the baseline problem is frequency assignment over a planning horizon $B$7, and the MB-TFSA interpretation arises by discretizing the time axis into slots $B$8 and allocating discrete frequency channels $B$9 per slot (Casadesus-Vila et al., 2024). In the massive MIMO-OFDM pilot-design setting, MB-TFSA corresponds to assigning time-frequency phase shifts $F$0 and available TF slots or pilot segments so that equivalent triple-beam supports do not overlap (Tang et al., 8 May 2025).

The main structural commonality is that the resource-assignment decision is never isolated. It is always coupled to another layer: beamforming and power loading in OFDMA-SDMA (Perea-Vega et al., 2012), linear precoding and QoS enforcement in GEO forward-link MB-HTS (Chien et al., 2021), MMSE-SIC detection and free-slot assignment in the return link (Boussemart et al., 2012), beam direction control and power allocation in LEO (Yuan et al., 2024), pilot-phase design and tensor estimation in massive MIMO-OFDM (Tang et al., 8 May 2025), or slack-bit-augmented QUBO penalties in digital satellites (Yan et al., 28 Feb 2026). A plausible implication is that MB-TFSA is best viewed as a family of coupled combinatorial resource-allocation problems rather than a single fixed formulation.

2. Physical-layer models and resource semantics

In downlink OFDMA-SDMA with zero-forcing precoding, a single base station equipped with $F$1 transmit antennas serves $F$2 single-antenna users over $F$3 OFDMA subcarriers across time slots $F$4, and up to $F$5 users can be multiplexed simultaneously in each slot $F$6 by linear precoding with zero-forcing constraints (Perea-Vega et al., 2012). For a scheduled set $F$7 with $F$8, the received signal is

$F$9

with $S$0 for all $S$1 under ZF, yielding

$S$2

Once $S$3 is fixed, beam design collapses to power loading through effective gains

$S$4

with $S$5 (Perea-Vega et al., 2012).

In GEO multi-beam high-throughput satellite systems with full frequency reuse, a single gateway precodes the signals to all beams that reuse the same band. The per-slot scheduled channel matrix is $S$6, and the paper uses regularized zero-forcing precoding,

$S$7

with $S$8 (Chien et al., 2021). With symbols $S$9, powers $x_{k,u}\in\{0,1\}$0, and noise $x_{k,u}\in\{0,1\}$1, the received signal is

$x_{k,u}\in\{0,1\}$2

and

$x_{k,u}\in\{0,1\}$3

Because reuse factor is $x_{k,u}\in\{0,1\}$4, interference coupling is across all beams that are simultaneously active in the same slot (Chien et al., 2021).

In the multibeam return link with universal frequency reuse, MB-TFSA is framed as gateway-controlled scheduling of return-link transmissions across multiple spot beams into MF-TDMA slots, with one user per active beam per slot and optional free slots to lower instantaneous co-channel interference (Boussemart et al., 2012). The slot-level received model is

$x_{k,u}\in\{0,1\}$5

and multiuser detection is performed by MMSE filtering with optimal successive interference cancellation ordering (Boussemart et al., 2012).

In dynamic LEO networks, the physical model additionally includes beam-direction control. The transmit antenna gain from beam $x_{k,u}\in\{0,1\}$6 pointing to center $x_{k,u}\in\{0,1\}$7 toward user $x_{k,u}\in\{0,1\}$8 on subchannel $x_{k,u}\in\{0,1\}$9 at time $k$0 is

$k$1

with $k$2 and $k$3 (Yuan et al., 2024). The achievable rate on $k$4 is

$k$5

where $k$6 and $k$7 includes inter-beam interference from simultaneous reuse on the same subchannel (Yuan et al., 2024).

A common source of ambiguity is that “beam” need not always mean a transmit beam serving payload traffic. In the triple-beam pilot-scheduling model, “triple-beam” refers to simultaneous structure across spatial, frequency, and time beam matrices $k$8, $k$9, and $u$0, with the space-frequency-time channel tensor expressed as

$u$1

for user $u$2 (Tang et al., 8 May 2025). This suggests that, in some contexts, MB-TFSA generalizes from data scheduling to structured pilot-domain assignment.

3. Optimization formulations and constraints

The optimization problems associated with MB-TFSA are consistently mixed-integer and non-convex. In OFDMA-SDMA with ZF, the per-slot resource-allocation problem is

$u$3

and is described as nonlinear, non-convex, and mixed-integer due to ZF structure and binary assignments (Perea-Vega et al., 2012).

In GEO MB-HTS with individual QoS constraints, scheduling and power allocation are coupled through per-user aggregate data demands $u$4 over a window of $u$5 slots. The paper enforces a per-slot surrogate QoS condition

$u$6

for each slot in which user $u$7 is scheduled, guaranteeing $u$8 if the user occupies $u$9 slots (Chien et al., 2021). For fixed scheduled sets $(k,t)\in B_{SA}=K\times T$0, the power-allocation subproblem remains nonconvex because SINR constraints and objective terms involve sums of ratios of posynomials (Chien et al., 2021).

In the return-link MF-TDMA case, the core MB-TFSA formulation uses binary assignment variables $(k,t)\in B_{SA}=K\times T$1 and optional free-slot indicators $(k,t)\in B_{SA}=K\times T$2, with objective

$(k,t)\in B_{SA}=K\times T$3

subject to at most one user or a free slot per beam and slot,

$(k,t)\in B_{SA}=K\times T$4

at most one slot per scheduled user,

$(k,t)\in B_{SA}=K\times T$5

and, under free-slot assignment, one free slot per beam across $(k,t)\in B_{SA}=K\times T$6 slots,

$(k,t)\in B_{SA}=K\times T$7

The rates depend jointly on the selected multi-beam path and SIC order, so the problem is combinatorial and non-separable across beams (Boussemart et al., 2012).

In LEO systems, the master problem jointly optimizes subchannel-time allocation $(k,t)\in B_{SA}=K\times T$8, beam-direction variables $(k,t)\in B_{SA}=K\times T$9, and powers $b^t_{k,n}\in\{0,1\}$0 to maximize the long-term fairness-aware objective

$b^t_{k,n}\in\{0,1\}$1

where

$b^t_{k,n}\in\{0,1\}$2

The constraints include within-beam OFDMA orthogonality, per-user subchannel caps, one center per beam and slot, one beam per center and slot, minimum elevation angle, minimum SINR, per-beam power limits, and per-satellite power limits (Yuan et al., 2024).

In digital satellites, the pre-QUBO MB-TFSA formulation is a binary weighted-throughput maximization,

$b^t_{k,n}\in\{0,1\}$3

subject to resource exclusivity,

$b^t_{k,n}\in\{0,1\}$4

per-slot power budgets,

$b^t_{k,n}\in\{0,1\}$5

and per-flow queue-capacity constraints,

$b^t_{k,n}\in\{0,1\}$6

(Yan et al., 28 Feb 2026). Optional interference and guard-band extensions are noted but not instantiated in the core QUBO experiments (Yan et al., 28 Feb 2026).

The following variants summarize the range of formulations.

Setting	Assignment unit	Dominant constraints
OFDMA-SDMA with ZF	$b^t_{k,n}\in\{0,1\}$7 with scheduled set $b^t_{k,n}\in\{0,1\}$8	ZF orthogonality, RT minimum rate, power budget, $b^t_{k,n}\in\{0,1\}$9
GEO MB-HTS forward link	Time slot $(c,t)\in C\times T$0	One user per beam per slot, QoS surrogate $(c,t)\in C\times T$1, sum power
Return-link MF-TDMA	Beam-slot pair $(c,t)\in C\times T$2	Max-min fairness, one user or free slot per beam per slot
Dynamic LEO	Subchannel-time $(c,t)\in C\times T$3 plus coordinate-time $(c,t)\in C\times T$4	Interference-aware matching, beam direction, SINR, power
Digital satellite QUBO	Beam-frequency-time unit $(c,t)\in C\times T$5	Exclusivity, per-slot power, queue capacity

4. Algorithmic approaches

The dual-based method for OFDMA-SDMA with ZF begins by relaxing the total power budget and real-time minimum-rate constraints with dual variables $(c,t)\in C\times T$6 and $(c,t)\in C\times T$7 (Perea-Vega et al., 2012). Using effective gains, the Lagrangian decomposes additively over time-frequency slots $(c,t)\in C\times T$8, and, for a fixed candidate set $(c,t)\in C\times T$9, the power-loading subproblem is

$B$00

with $B$01 for unit weights (Perea-Vega et al., 2012). The optimal power is waterfilling-like,

$B$02

and the scheduled set is chosen by maximizing the dual metric $B$03 over all candidate sets with $B$04 (Perea-Vega et al., 2012). Subgradient updates on $B$05 and $B$06 produce a dual upper bound, while a primal-feasible recovery gives a lower bound (Perea-Vega et al., 2012).

In GEO MB-HTS systems, the paper proposes a successive optimization approach with two stages: heuristic scheduling with fixed powers and GP-based power allocation for the chosen scheduled sets (Chien et al., 2021). The scheduling stage has Strict and Relax variants. It sorts users by channel norm, seeds each slot with the strongest user, tentatively adds candidates by recomputing the RZF precoder, and accepts additions subject to monotonic sum-rate growth and, in Strict mode, immediate per-slot QoS feasibility (Chien et al., 2021). The power-allocation stage reformulates

$B$07

using auxiliary variables $B$08, lower-bounds the signomial term by an AM-GM monomial, and solves the resulting geometric program iteratively, updating the AM-GM weights until convergence to a KKT point (Chien et al., 2021).

The return-link scheduling algorithm uses a bipartite-graph approach. A path is a $B$09-tuple selecting one user from each beam for a slot, and for $B$10 users per beam the number of candidate paths is

$B$11

Each path is weighted by the minimum per-user rate under MMSE-SIC with optimal ordering, and a minimum-deletion algorithm selects $B$12 feasible paths by iteratively deleting the worst path if feasibility is preserved, otherwise marking it as selected (Boussemart et al., 2012). When a target minimum rate cannot be met, the algorithm enables free-slot assignment by spreading $B$13 users over $B$14 slots, with one free slot per beam (Boussemart et al., 2012).

In dynamic LEO networks, the problem is decomposed into beam direction control and time-slot allocation, user subchannel assignment, and beam power allocation (Yuan et al., 2024). The first two are handled by matching with externalities. Beam-direction control uses an initial deferred-acceptance phase followed by swap matching with externalities until an exchange-stable matching is reached (Yuan et al., 2024). Subchannel assignment applies per-beam deferred acceptance and then a cross-beam negotiation step that drops a reused subchannel from the weaker side when doing so increases total utility across beams at that slot (Yuan et al., 2024). Power allocation is then solved by successive convex approximation using the lower bound

$B$15

with $B$16 and $B$17 at the current iterate (Yuan et al., 2024).

In the pilot-scheduling setting, the MB-TFSA algorithm is explicitly two-stage: DSatur-based grouping of users according to a graph whose edge weights depend on the overlap metric

$B$18

followed by greedy assignment of TF phase-shift pairs $B$19 to groups so that equivalent TB-domain supports remain approximately disjoint (Tang et al., 8 May 2025).

In the digital-satellite QUBO framework, the constrained problem is embedded in

$B$20

with objective term

$B$21

resource-conflict penalty

$B$22

and slack-bit-based penalties for power and queue inequalities (Yan et al., 28 Feb 2026). The resulting Ising Hamiltonian is solved by QAOA with layer-wise training, optional warm starts, and SPSA-like stochastic hill climbing (Yan et al., 28 Feb 2026).

5. Performance characterizations and benchmarking results

The dual-bound framework in OFDMA-SDMA is explicitly presented as a benchmark for heuristic MB-TFSA policies (Perea-Vega et al., 2012). In scenarios with $B$23, $B$24, $B$25, and $B$26 dBm, the dual-feasible lower bound was within $B$27–$B$28 of the upper bound for a single RT user with increasing minimum rate $B$29–$B$30 bps/Hz, while the weight-adjustment method had a gap of $B$31–$B$32 (Perea-Vega et al., 2012). With increasing large-scale attenuation $B$33 dB, the dual-feasible gaps were approximately $B$34, $B$35, and $B$36, whereas the weight-adjustment gaps were approximately $B$37, $B$38, and $B$39 (Perea-Vega et al., 2012). As the number of RT users increased from $B$40 to $B$41, the dual-feasible gap grew from approximately $B$42 to approximately $B$43, while the weight-adjustment gap rose from approximately $B$44 to approximately $B$45 and failed to find feasible points for $B$46–$B$47 RT users (Perea-Vega et al., 2012).

In GEO MB-HTS, the numerical evaluation uses $B$48 beams, $B$49 users per beam, $B$50 slots, bandwidth $B$51 MHz, Ka-band carrier $B$52 GHz, and sum power $B$53 dBW (Chien et al., 2021). Scheduling-only results show that Relax mode achieves the highest sum throughput, while Strict mode guarantees QoS for all scheduled users and yields the largest per-user throughput (Chien et al., 2021). Adding GP-based power allocation produces gains of $B$54–$B$55 over fixed power, including example average gains of random $B$56, SUS $B$57 sum, $B$58 per-user), proposed Strict $B$59, and proposed Relax $B$60 (Chien et al., 2021). With power control, Algorithm 2 Relax reaches the highest average sum throughput, approximately $B$61 Mbps, while Algorithm 2 Strict yields the highest per-user rate, approximately $B$62 Mbps (Chien et al., 2021).

The return-link multibeam scheduler quantifies the effect of free-slot assignment on weakest-user performance and efficiency (Boussemart et al., 2012). At SNR $B$63 dB and $B$64, without FSA the slot availability for meeting the target minimum rate is approximately $B$65–$B$66 at $B$67 efficiency (Boussemart et al., 2012). With FSA, the reported examples are: for $B$68 and $B$69 bits/s/Hz, slot availability $B$70, efficiency approximately $B$71, and FSA use approximately $B$72; for $B$73 and $B$74, availability $B$75, efficiency approximately $B$76, and FSA use approximately $B$77; and for $B$78 and $B$79, availability $B$80, efficiency approximately $B$81, and FSA use approximately $B$82 (Boussemart et al., 2012). The paper also reports that $B$83 nearly matches the min-rate CDF of scheduling without FSA at $B$84, while delivering substantial complexity reduction (Boussemart et al., 2012).

In dynamic LEO networks, the integrated framework is evaluated with $B$85 orbits, $B$86 satellites per orbit, altitude $B$87 km, inclination $B$88, $B$89 users, $B$90, $B$91 subchannels per beam, $B$92, $B$93 W per beam, and $B$94 W per satellite (Yuan et al., 2024). The outer loop converges in a few iterations, and with $B$95 beams per satellite it converges in under $B$96 iterations (Yuan et al., 2024). Relative to baselines, the proposal improves the number of served users by up to two times and the sum user data rate by up to $B$97, with beam direction optimization contributing larger gains than power allocation alone (Yuan et al., 2024).

The dynamic frequency-assignment framework for mobile users reports that, in scenarios with more than $B$98 beams, the method is able to serve over $B$99 of the fixed and mobile users (Casadesus-Vila et al., 2024). In the $F$00-user high-uncertainty case, reserving $F$01 spectrum yields $F$02, while a configuration combining $F$03 reserve with proactive protection achieves approximately $F$04 with approximately $F$05 fewer reconfigurations (Casadesus-Vila et al., 2024).

In the pilot-scheduling problem, TFPSP-IGA achieves approximately $F$06 dB NMSE reduction relative to APSP-IGA for $F$07 at SNR $F$08 dB, and more than $F$09 dB reduction for $F$10 at SNR $F$11 dB (Tang et al., 8 May 2025). The IGA method reaches target NMSE in approximately $F$12 iterations at $F$13 dB, whereas GAMP/EPV require more than $F$14 iterations (Tang et al., 8 May 2025). In the quantum MB-TFSA study, shallow QAOA with $F$15 finds the optimal throughput for the $F$16-flow and $F$17-flow instances, while $F$18 underperforms on hardware because increased depth raises noise and exacerbates barren plateaus (Yan et al., 28 Feb 2026).

6. Design implications, ambiguities, and limitations

A recurring design principle is decomposition. In OFDMA-SDMA, per-$F$19 decomposition emerges after dualization and ZF collapse to effective gains (Perea-Vega et al., 2012). In GEO MB-HTS, slotwise scheduling and per-slot GP-based power control are separated within a successive optimization loop (Chien et al., 2021). In LEO, the decomposition into beam direction control, subchannel assignment, and power allocation is explicit (Yuan et al., 2024). In pilot scheduling, grouping and phase assignment are separated (Tang et al., 8 May 2025). This suggests that exact end-to-end MB-TFSA is generally avoided in favor of structured decompositions that preserve tractability.

A second recurring issue is that interference is handled differently across domains. In downlink OFDMA-SDMA, zero-forcing removes intra-slot inter-user interference inside a scheduled set, but poor conditioning of $F$20 inflates the ZF column norms and makes power loading expensive (Perea-Vega et al., 2012). In GEO forward-link MB-HTS, RZF mitigates rather than cancels mutual interference, and scheduling seeks strong channels and semi-orthogonality (Chien et al., 2021). In return-link MF-TDMA, interference is addressed through path selection, MMSE-SIC ordering, and, when necessary, free-slot assignment (Boussemart et al., 2012). In dynamic frequency assignment for mobile users, interference is largely represented through adjacency, reuse-group, and polarization constraints rather than explicit SINR optimization (Casadesus-Vila et al., 2024). A common misconception is that MB-TFSA is merely a slot-labeling problem; in the cited works, the assignment is valuable only because it reshapes interference geometry.

The literature also reveals ambiguity in whether “frequency” must be explicit. One paper states that with reuse factor $F$21, there is effectively a single frequency slot and MB-TFSA reduces to MB-TSA (Chien et al., 2021). Another begins from dynamic frequency assignment and becomes MB-TFSA only after time discretization (Casadesus-Vila et al., 2024). In the pilot domain, the “frequency” dimension is a phase-shift and support-separation mechanism rather than a user data channel (Tang et al., 8 May 2025). This suggests that MB-TFSA is better interpreted as a generalized multi-beam resource-index assignment over a two-dimensional temporal-spectral structure, even when one dimension is degenerate or embedded in pilot design.

Complexity remains a central limitation. In OFDMA-SDMA, per-dual-iteration complexity is dominated by pseudo-inverses for each candidate SDMA set, yielding overall per-iteration complexity $F$22, though pseudo-inverses can be precomputed once per $F$23 (Perea-Vega et al., 2012). In the return-link scheduler, exhaustive search requires $F$24 rate evaluations, while bipartite-path search reduces this to $F$25 (Boussemart et al., 2012). In pilot scheduling, DSatur grouping is $F$26 worst-case, and phase-grid search adds $F$27 (Tang et al., 8 May 2025). In the QUBO formulation, total qubits equal the number of main decision variables plus slack qubits, and the slack-bit count can dominate without aggressive rescaling (Yan et al., 28 Feb 2026).

Finally, the reported limitations are domain-specific but structurally similar. ZF requires full column rank and $F$28 (Perea-Vega et al., 2012). GEO static-operation assumptions rely on quasi-static channels and accurate gateway CSI (Chien et al., 2021). Return-link MMSE-SIC ordering assumes perfect CSI (Boussemart et al., 2012). The TFPSP separability theorem is asymptotic in large $F$29, and finite grids or synchronization impairments may preclude perfect non-overlap (Tang et al., 8 May 2025). The quantum formulation is currently restricted to very small instances because qubit counts and circuit depth remain the bottlenecks (Yan et al., 28 Feb 2026). Across all cases, a plausible implication is that MB-TFSA research is less constrained by modeling expressiveness than by the computational and physical cost of enforcing coupled constraints at operational scale.