Hybrid Algorithm Improves 6G Massive MIMO Spectral Efficiency in Simulated IoT Networks

In simulated wireless communications research, an optimization algorithm known as the Differential Evolution-Enhanced Adaptive Black-Winged Kite Algorithm demonstrated substantial performance improvements for joint hybrid beamforming and underlay spectrum sharing across dense sixth-generation Internet-of-Things networks. Conducted by researchers Wenjian Zhang and Ping Li and published in Discover Internet of Things, the numerical study evaluated how coordinated radio-frequency beamforming and dynamic frequency allocation can overcome severe physical-layer bottlenecks in massive multiple-input multiple-output base stations. Under a default simulated configuration comprising 128 transmit antennas, eight radio-frequency chains, 50 single-antenna devices, and 16 shared subchannels, the proposed algorithm attained an aggregate spectral efficiency of 15.87 ± 0.19 bits per second per Hertz over 30 independent Monte Carlo trials. This performance represents an advantage of 27.5 percent over the manifold-optimization alternating minimization baseline and 8.1 percent over the static hybrid BKAPI method. Furthermore, the search procedure achieved its one-percent convergence stopping threshold in an average of 185 ± 12 iterations, substantially reducing the iteration count required by standard metaheuristic benchmarks while enforcing strict hardware and interference boundaries.

The engineering challenge addressed by the study stems from the fundamental architecture of massive multiple-input multiple-output systems envisioned for upcoming 6G applications, including industrial automation, immersive communication, and intelligent transportation infrastructure. To mitigate these hardware burdens, network designers implement hybrid analog and digital beamforming, which cascades a low-dimensional digital baseband precoder with an analog phase-shifting network driven by a drastically smaller pool of radio-frequency chains. While hybrid architectures preserve much of the spatial multiplexing and array gain of fully digital setups, the physical phase shifters impose non-convex constant-modulus constraints where the magnitude of each analog precoding element must remain fixed to a constant unit value while only its phase angle is adjusted. Compounding this challenge, massive Internet-of-Things deployments require multiple devices to concurrently share identical subchannels, demanding meticulous multi-user interference management alongside strict interference-temperature constraints to protect primary licensed incumbents.

Formulated mathematically, the joint beamforming and spectrum-sharing task represents a complex non-convex mixed-integer nonlinear programming problem that combines continuous analog phase variables, complex-valued baseband digital precoding matrices, and discrete binary subchannel link-activation indicators. Conventional mathematical programming techniques encounter acute computational hurdles when applied to such coupled combinatorial spaces. Exact deterministic global optimization remains practically intractable at the evaluated dimensional scales, while classic matrix relaxation and alternating minimization methods typically require successive matrix inversions and subsequent heuristic projections to recover feasibility. Conversely, contemporary deep reinforcement learning and data-driven physical-layer schemes can deliver millisecond-level online inference times once trained, but their operational fidelity depends heavily on extensive offline computational budgets, fixed training distributions, and susceptibility to out-of-distribution wireless channel shifts. Standard population-based metaheuristics avoid offline training requirements entirely, yet conventional evolutionary mechanisms frequently suffer from premature diversity loss or erratic convergence trajectories when navigating high-dimensional landscapes governed by simultaneous power budgets, constant-modulus constraints, and primary-user leakage limits.

To overcome these limitations without relying on prior training data, the authors developed DEBKA by establishing a phased dual-branch evolutionary search framework that assigns distinct, complementary roles to two separate optimization mechanisms. Rather than aggregating diverse algorithmic operators into an unstructured static mixture, the methodology assigns local neighborhood refinement to an enhanced Black-Winged Kite Algorithm attack phase while delegating wide-area exploration to an adaptive Differential Evolution migration phase. In DEBKA, the migration branch completely replaces the leader-centric movement of the original BKA with a DE/current-to-pbest mutation operator and binomial crossover borrowed from the JADE evolutionary algorithm. This design leverages difference vectors constructed between elite solutions, active population members, and discarded candidate states, thereby preventing the population from prematurely collapsing around a single leader coordinate during complex multi-user precoder evaluations.

A critical architectural component of DEBKA is its deterministic problem-specific encoding, decoding, and constraint-handling pipeline, designed to guarantee physical consistency across every candidate evaluation. Each candidate solution in the population is encoded as a single real-valued concatenated vector containing the vectorized analog phase shifter angles, the vectorized real and imaginary components of the subchannel-dependent digital precoders, and continuous spectrum-activation genes. During fitness evaluations, this vector undergoes a rigid, sequential decoding sequence. First, the analog components are projected directly onto the hardware constant-modulus set using phase normalization. Next, the digital precoding matrices are reconstructed across all subchannels and scaled by a power normalization factor if the total radiated power exceeds the maximum budget of 30 dBm (1 Watt). Subsequently, the continuous spectrum genes are mapped through an elementwise sigmoid activation function and evaluated against a fixed binary threshold of 0.5 to determine active device-subchannel links, allowing flexible spatial reuse without enforcing mutually exclusive subchannel locks.

To maintain absolute fidelity to system operational constraints without distorting the primary performance metric, DEBKA implements a feasibility-first selection rule rather than optimizing an arbitrary penalty-weighted objective function. The evaluation pipeline isolates the primary objective—maximizing the weighted aggregate spectral efficiency across active links—from a separate scalar violation function that quantifies excess radiated transmit power and aggregate primary-user interference leakage on each reused subcarrier. In any pairwise selection contest between a parent individual and an offspring trial, a candidate with a lower total constraint violation is unconditionally preferred. When both competing vectors are fully feasible, the candidate delivering the higher weighted spectral efficiency is retained. Furthermore, discarded parent vectors that lose selection tournaments are automatically transferred into an external archive of bounded capacity. This rejected-parent archive preserves exploratory directions that have already departed the active population, supplying diversified difference vectors during subsequent current-to-pbest mutation operations and mitigating evolutionary stagnation.

The search process dynamically balances its dual operators through self-adaptive parameter updating and population-diversity monitoring. Rather than imposing static mutation and crossover hyperparameters throughout an execution, DEBKA tracks the success history of accepted trial vectors following the principles of adaptive JADE. The algorithm samples individual mutation scaling factors from a Cauchy distribution whose location parameter updates dynamically via a Lehmer mean whenever successful trials are accepted. Similarly, individual crossover probabilities are drawn from a normal distribution whose mean adapts iteratively through an arithmetic average of successful values. To orchestrate the choice between local refinement and wide-area migration, the algorithm computes a normalized fitness-diversity statistic across the current population at each iteration. This diversity index dynamically modulates the attack branch probability between bounded extremes, ensuring that the optimizer increases wide-area differential migration when population diversity drops and intensifies Cauchy-Gaussian local exploitation when candidate diversity remains sufficiently healthy.

Within the local refinement branch, DEBKA replaces the original sinusoidal perturbation of BKA with an enhanced Cauchy-Gaussian dual mutation mechanism. At each dimension of a selected individual, the search stochastically chooses between a standard normal Gaussian perturbation and a heavy-tailed Cauchy distribution with equal probability. While the Gaussian operator facilitates fine-grained, smooth exploitation around the current global best candidate, the heavy-tailed Cauchy operator injects occasional long-range jumps that empower the optimizer to escape deceptive local basins in the non-convex precoding landscape. Both perturbations are scaled by a monotonically decaying time-dependent coefficient that progressively contracts the search radius as iteration counts advance toward the global budget limit. The analytical properties established by the authors demonstrate that the combination of bounded search domains, deterministic constant-modulus projection, and elitist feasibility-first selection ensures a monotonically non-decreasing sequence of feasible objective values that converges toward a finite random limit.

The authors validated DEBKA using a simplified geometric millimeter-wave channel simulation inspired by the propagation principles of 3GPP Technical Report 38.901. Operating at a carrier frequency of 28 GHz with a 100 MHz channel bandwidth, the simulation modeled sparse multipath propagation via a geometric Saleh-Valenzuela vector formulation tailored to single-antenna IoT terminals, incorporating path clusters with path counts of 3, 5, and 8 alongside an urban-microcell distance-dependent path-loss exponent of 2.5. Across 30 independent Monte Carlo runs, DEBKA was benchmarked against ten locally implemented baselines spanning alternating matrix methods such as MO-AltMin and orthogonal matching pursuit, standard metaheuristics including genetic algorithms, particle swarm optimization, differential evolution, and the original BKA, as well as modern hybrids like BKAPI and JADE, and deep reinforcement learning represented by proximal policy optimization.

Beyond raw spectral efficiency, the experimental protocol evaluated convergence rates, normalized spectral energy efficiency, and scalability across network dimensions. When evaluating normalized spectral energy efficiency under an aggregate base station power consumption model totaling 3.2 Watts—comprising 1 Watt of transmit power, eight 250-milliwatt radio-frequency chains, and a 200-milliwatt baseband processing overhead—DEBKA yielded 4.96 ± 0.06 bps/Hz/W compared to 4.76 ± 0.08 for JADE and 3.89 ± 0.10 for MO-AltMin. In scalability assessments extending device density up to 100 IoT terminals, aggregate sum rates expanded across all algorithms, but DEBKA widened its margin over MO-AltMin to 29.1 percent and over JADE to 6.8 percent, demonstrating resilient interference management in increasingly congested spatial-reuse settings.

Comprehensive ablation experiments conducted under the default 128-antenna environment confirmed the empirical necessity of each integrated architectural component. Stripping the adaptive Differential Evolution mutation mechanism from DEBKA inflicted the steepest performance degradation, causing spectral efficiency to plummet from 15.87 to 14.92 bps/Hz while expanding the required convergence iterations from 185 to 234. Removing the adaptive parameter update rules in favor of fixed control settings produced the slowest convergence among all variants at 248 ± 21 iterations. Furthermore, beam-pattern radiation visualizations demonstrated that DEBKA synthesized spatial lobes with sidelobe suppression approximately 3.2 dB deeper than standard BKA and 5.8 dB deeper than MO-AltMin, directly explaining its superior multi-user and primary-user interference containment.

The study established rigorous statistical validation through eight prespecified Mann-Whitney U pairwise comparisons between DEBKA and the primary baselines, with Holm adjustments maintaining family-wise significance below an alpha threshold of 0.05 across all tested pairs alongside moderate to large rank-biserial effect sizes. Computational runtime profiles recorded on an Intel Xeon Gold 6248 processor and NVIDIA RTX A6000 GPU indicated that DEBKA required approximately 0.140 seconds per generation, yielding a descriptive mean time-to-convergence of 25.9 seconds compared to 33.4 seconds for JADE. In scaling tests varying antenna counts from 32 to 256 elements, DEBKA exhibited runtime scaling that was substantially more gradual than matrix-intensive alternating solvers, maintaining a favorable computational balance between lightweight vector-only heuristics and computationally burdensome iterative matrix decompositions across diverse propagation environments.

Notwithstanding these positive numerical results, the researchers explicitly delineated several vital theoretical and practical boundaries governing their conclusions. The simulation framework operated under the foundational assumption of perfect, instantaneous channel state information that remained static throughout each optimization epoch, omitting the deleterious impacts of channel estimation noise, feedback delay, user mobility, and physical blockage. Moreover, the mathematical model assumed continuous-resolution analog phase shifters and linear radio-frequency stages, neglecting hardware imperfections such as phase quantization, amplifier nonlinearities, phase noise, and control-plane reconfiguration latency. While DEBKA demonstrated clear computational efficiency within the examined parameter grid, its multi-second iterative execution time exceeds the millisecond-scale coherence intervals of real-world dynamic wireless environments, meaning real-time deployment readiness has not yet been established. The authors emphasized that future investigations must incorporate standardized 3GPP clustered channel generators, hardware-in-the-loop validation on software-defined radio testbeds, and robust uncertainty-aware formulations before the optimizer can transition into field-ready telecommunication infrastructure.

Subject of Research: Optimization of joint hybrid beamforming and underlay spectrum sharing in 6G massive MIMO IoT networks

Article Title: Adaptive DEBKA for joint beamforming and spectrum sharing in 6G massive MIMO IoT networks

Article References: Zhang, W., & Li, P. (2026). Adaptive DEBKA for joint beamforming and spectrum sharing in 6G massive MIMO IoT networks. Discover Internet of Things, 6(1), Article 166. https://doi.org/10.1007/s43926-026-00509-1

Image Credits: AI Generated

DOI: 10.1007/s43926-026-00509-1

Keywords: 6G networks, massive MIMO, hybrid beamforming, spectrum sharing, Internet of Things, DEBKA, Black-Winged Kite Algorithm, differential evolution, spectral efficiency, constant-modulus constraint, millimeter wave, interference temperature, feasibility-first selection, JADE, energy efficiency, radio frequency chains, Saleh-Valenzuela model