EDN Beamforming Framework Overview

• EDN beamforming is a design class that uses structured mappings (e.g., edge-centric, end-to-end, encoder–decoder) to compute beamformers instead of traditional iterative methods.
• It integrates approaches like graph neural networks, raw observation-to-filter mappings, and CSI compression to optimize beamforming for various wireless scenarios.
• EDN frameworks are hardware-aware and scalable, demonstrating improved performance and reduced overhead in cooperative, mmWave, and distributed systems.

Searching arXiv for the cited beamforming frameworks and related EDN formulations. arXiv search query: "(Wang et al., 2022) Learning Cooperative Beamforming with Edge-Update Empowered Graph Neural Networks" EDN beamforming framework denotes a family of beamforming formulations in which the beam design is represented, learned, compressed, predicted, or coordinated through an explicit network structure rather than obtained only by per-instance iterative optimization. In current arXiv literature, the expression appears in several closely related senses: edge-centric graph neural beamforming with beamformers on graph edges, end-to-end deep neural beamforming from multichannel or perceptual inputs, encoder–decoder architectures that compress CSI or target patterns into latent codes, and edge/distributed or end-to-end digital system frameworks that couple feedback, sensing, or hardware constraints to beamforming decisions (Wang et al., 2022, Koyama et al., 2020, Zhang et al., 27 Sep 2025, Zhang et al., 14 Jul 2025, 0806.3023, Guajardo et al., 2024). Taken together, these works suggest that EDN beamforming is not a single standardized architecture but a design class organized around structured representations, low-overhead inference, and task-aligned optimization.

1. Terminological scope and core abstractions

The literature uses the label in more than one technical sense. In one line of work, beamformers are edge variables on a bipartite wireless graph, so the framework must update edge embeddings explicitly rather than only node states (Wang et al., 2022). In another, EDN denotes an end-to-end deep neural mapping from raw observations, such as multichannel STFTs, images, or power profiles, directly to beamforming filters or beam indices (Koyama et al., 2020, Karunasena et al., 2024). A third usage treats EDN as an encoder–decoder network that compresses high-dimensional CSI or target beam patterns into low-dimensional latent vectors and reconstructs beamformers under architectural constraints (Zhang et al., 27 Sep 2025, Zhang et al., 14 Jul 2025). A fourth usage emphasizes edge/distributed or end-to-end system design, including feedback-limited distributed beamforming, split computing, or hardware-level digital beamforming models (0806.3023, Bahadori et al., 2023, Guajardo et al., 2024).

Usage in the literature	Core object	Representative sources
Edge-centric neural beamforming	Beamformers on graph edges	(Wang et al., 2022)
End-to-end deep neural beamforming	Raw observations to beam decisions	(Koyama et al., 2020, Karunasena et al., 2024, Li et al., 2021)
Encoder–decoder beamforming	Latent compression and beam reconstruction	(Zhang et al., 27 Sep 2025, Zhang et al., 14 Jul 2025)
Edge/distributed or system-level beamforming	Feedback, split computing, or hardware pipeline	(0806.3023, Bahadori et al., 2023, Guajardo et al., 2024)

Across these variants, a recurring abstraction is a learned or structured mapping of the form

$$f_\theta:\text{observations or latent state}\longrightarrow \text{beamformer, beam index, or candidate set}.$$

What changes from one framework to another is the choice of state: graph edge features, time–frequency embeddings, compressed CSI, image-derived geometry, predicted mobility parameters, or hardware-constrained digital signals.

2. Edge-centric graph formulations

In cooperative multi-cell beamforming, the natural optimization variable is the link-specific beamformer $\mathbf{v}_{m,k}$, not a node label. This motivates the heterogeneous bipartite graph

$$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$

where $\mathcal{M}$ are BS nodes, $\mathcal{K}$ are UE nodes, and $(m,k)\in\mathcal{E}$ denotes a BS–UE edge carrying both the channel $\mathbf{h}_{m,k}\in\mathbb{C}^N$ and the beamforming vector $\mathbf{v}_{m,k}\in\mathbb{C}^N$ (Wang et al., 2022). The underlying downlink problem is to maximize the sum rate

$$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$

with

$$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$

Conventional GP and WMMSE solve this iteratively on an instance-by-instance basis; the edge-update empowered graph neural network replaces that runtime iteration with a learned mapping from node and edge features to edge beamformers.

The key architectural point is the explicit edge-update mechanism. BS nodes aggregate messages from neighboring UEs, UE nodes aggregate messages from neighboring BSs, and each edge $\mathbf{v}_{m,k}$0 aggregates from the two incident nodes plus neighboring edges that share the same BS or the same UE. In the reported configuration, all MLPs use 3 fully-connected layers with ReLU, the hidden dimension is 64 for both nodes and edges, the number of updating layers is 2, and the aggregators are element-wise max. The network is trained unsupervised by maximizing the differentiable sum rate with RMSProp at learning rate $\mathbf{v}_{m,k}$1, followed by per-BS power normalization to enforce $\mathbf{v}_{m,k}$2 (Wang et al., 2022).

This framework is notable because the edge variables encode exactly the coupling imposed by interference and per-BS power budgets. The paper proves permutation equivariance with respect to BS and UE permutations and reports that the trainable parameter count does not depend on the numbers of BSs and UEs. Empirically, the model is trained at $\mathbf{v}_{m,k}$3 BSs and $\mathbf{v}_{m,k}$4 UEs, then generalizes to UE counts from $\mathbf{v}_{m,k}$5 to $\mathbf{v}_{m,k}$6 and BS counts from $\mathbf{v}_{m,k}$7 to $\mathbf{v}_{m,k}$8. It achieves higher sum-rate than GP and WMMSE in the reported experiments, is $\mathbf{v}_{m,k}$9 faster at inference, and maintains good performance on unseen problem sizes (Wang et al., 2022).

3. End-to-end neural beamforming from raw observations

A second EDN lineage removes explicit beamforming formulas and lets the network learn the entire mapping from multichannel observations to beamforming filters. In the W-Net beamformer, the input is the multichannel noisy STFT represented as

$$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$0

where the last dimension stacks magnitudes and phases for all $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$1 microphones. A first U-Net, $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$2, produces a time–frequency reference $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$3, and a second U-Net maps either $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$4 or $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$5 to a time-varying complex beamforming tensor $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$6. The enhanced signal is

$$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$7

and training minimizes complex-domain MSE to a reference-channel target (Koyama et al., 2020).

The reported results show why this two-stage EDN is technically interesting. On Anechoic static mixtures, SI-SNR improves from $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$8 dB for BLSTM-GEV and $$\mathcal{G} = (\mathcal{M}, \mathcal{K}, \mathcal{E}),$$9 dB for BLSTM-MVDR to $\mathcal{M}$0 dB for a one-stage U-Net beamformer and $\mathcal{M}$1 dB for W-Net with concatenation. On Anechoic moving mixtures, BLSTM-GEV drops to $\mathcal{M}$2 dB, whereas W-Net with concatenation reaches $\mathcal{M}$3 dB. On the ACE corpus, models trained only on simulated reverberant data still achieve $\mathcal{M}$4 dB SI-SNR and $\mathcal{M}$5 STOI for W-Net with concatenation (Koyama et al., 2020). The implication is that the intermediate time–frequency reference preserves some of the generalizability of mask-based approaches while retaining the flexibility of time-varying filter estimation.

The same end-to-end logic appears in the causal multichannel speech-enhancement beamformer EaBNet, which defines two explicit neural modules, EM and BM, and avoids explicit spatial covariance matrix estimation. EM produces a 3-D embedding tensor intended to carry both spectral and spatial discriminative information, BM directly derives beamforming weights for filter-and-sum operation, and a post-processing module further suppresses residual noise. The system is causal because its networks are uni-directional in time and operate frame-by-frame with no noncausal context (Li et al., 2021). This suggests a broader EDN principle: rather than learning SCMs and then imposing a fixed MVDR or GEV map, the framework can learn a task-specific spatial latent directly.

4. Encoder–decoder, feedback compression, and split inference

A third major interpretation of EDN is an encoder–decoder network for CSI compression and beamformer reconstruction. In large-scale sparse MIMO, one proposed EDN consists of a user-side encoder $\mathcal{M}$6, a BS-side beamformer decoder $\mathcal{M}$7, and a BS-side channel decoder $\mathcal{M}$8 (Zhang et al., 27 Sep 2025). Each user compresses its estimated normalized channel into

$\mathcal{M}$9

the BS reconstructs an initial beamformer $\mathcal{K}$0, reconstructs channels $\mathcal{K}$1, and then refines $\mathcal{K}$2 by analytical gradient ascent,

$\mathcal{K}$3

Training combines semi-amortized learning with knowledge distillation through

$\mathcal{K}$4

where the supervised term imitates MMSE beamformers early in training and the unsupervised term gradually dominates. The framework is extended to both far-field and near-field hybrid beamforming (Zhang et al., 27 Sep 2025).

Another encoder–decoder formulation replaces CSI with target beam patterns. A MobileNetV2-based encoder compresses a 2D beam pattern $\mathcal{K}$5 into a latent code $\mathcal{K}$6, and a 4-layer MLP decoder reconstructs digital, analog, or fully connected hybrid beamformers subject to architecture-specific constraints (Zhang et al., 14 Jul 2025). The method uses two-stage training: offline pre-training on $\mathcal{K}$7 randomly generated beam patterns with an auxiliary reconstruction module, followed by online decoder training with only $\mathcal{K}$8 target patterns and a composite loss over a main-lobe region $\mathcal{K}$9, a sidelobe region $(m,k)\in\mathcal{E}$0, and an intermediate region. In the reported single-user mmWave setup, the EDN-based digital beamforming achieves about $(m,k)\in\mathcal{E}$1 of the optimal fully digital spectral efficiency and outperforms representative existing methods; example online training times are about $(m,k)\in\mathcal{E}$2 s for DBF and $(m,k)\in\mathcal{E}$3 s for HBF (Zhang et al., 14 Jul 2025).

SplitBeam pushes the encoder–decoder idea into split computing. In IEEE 802.11ac/ax MU-MIMO, the standard STA-side pipeline performs CSI estimation, per-subcarrier SVD, Givens decomposition, angle quantization, and compressed beamforming feedback. SplitBeam instead trains a split DNN to map CSI directly to the beamforming matrix, with a head at the STA, a bottleneck representation $(m,k)\in\mathcal{E}$4, and a tail at the AP (Bahadori et al., 2023). The bottleneck optimization problem explicitly constrains BER and end-to-end reporting delay while minimizing STA computation and airtime overhead. Experimental results show beamforming feedback size and computational complexity reductions of up to $(m,k)\in\mathcal{E}$5 and $(m,k)\in\mathcal{E}$6, respectively, while maintaining BER within about $(m,k)\in\mathcal{E}$7 of existing approaches; FPGA implementation yields an end-to-end BM reporting delay below $(m,k)\in\mathcal{E}$8 milliseconds in the most complex scenario (Bahadori et al., 2023). A plausible implication is that EDN beamforming can be interpreted not only as a neural mapping, but as a placement problem over distributed compute resources.

5. Sensing-, geometry-, and prediction-driven beamforming

A different branch of EDN beamforming uses side information that is not conventional CSI. In vision-aided mmWave beam selection, the full pipeline is

$(m,k)\in\mathcal{E}$9

where $\mathbf{h}_{m,k}\in\mathbb{C}^N$0 is a sequence of RGB images from a BS-mounted camera, $\mathbf{h}_{m,k}\in\mathbb{C}^N$1 is the mmWave received power profile over a beam codebook, and $\mathbf{h}_{m,k}\in\mathbb{C}^N$2 is the predicted top-$\mathbf{h}_{m,k}\in\mathbb{C}^N$3 beam set (Karunasena et al., 2024). The architecture combines a modified YOLOv8 transmitter-identification network with a custom 3-channel input, a DeepOCSort tracking network, and a custom CNN+MLP beam-prediction network operating on an isolated transmitter image and a geometry-derived reduced beam search space vector. In the reported DeepSense 6G experiments, the beam predictor alone reaches Top-1/Top-3/Top-5 accuracies of $\mathbf{h}_{m,k}\in\mathbb{C}^N$4, $\mathbf{h}_{m,k}\in\mathbb{C}^N$5, and $\mathbf{h}_{m,k}\in\mathbb{C}^N$6 on Scenario 3, and $\mathbf{h}_{m,k}\in\mathbb{C}^N$7, $\mathbf{h}_{m,k}\in\mathbb{C}^N$8, and $\mathbf{h}_{m,k}\in\mathbb{C}^N$9 on Scenario 4. The whole pipeline obtains Top-5 accuracies of $\mathbf{v}_{m,k}\in\mathbb{C}^N$0 and $\mathbf{v}_{m,k}\in\mathbb{C}^N$1 on Scenarios 3 and 4, respectively, and achieves over $\mathbf{v}_{m,k}\in\mathbb{C}^N$2 overhead reduction by restricting beam training to a small candidate subset (Karunasena et al., 2024).

Near-field predictive beamforming replaces explicit channel estimation with state prediction. In the near field, spherical waves and non-uniform Doppler frequencies permit joint sensing of $\mathbf{v}_{m,k}\in\mathbb{C}^N$3, and the beamformer is updated using

$\mathbf{v}_{m,k}\in\mathbb{C}^N$4

with $\mathbf{v}_{m,k}\in\mathbb{C}^N$5 and $\mathbf{v}_{m,k}\in\mathbb{C}^N$6 over a short coherent processing interval (Jiang et al., 10 Jun 2025). The matched near-field beamformer

$\mathbf{v}_{m,k}\in\mathbb{C}^N$7

then focuses in both range and angle and can incorporate Doppler compensation. The paper emphasizes two properties relative to far-field predictive beamforming: Prior-Knowledge-Free Prediction and Low-Complexity and Generalizable System Design. In the case study, a Kalman-filter-based implementation with $\mathbf{v}_{m,k}\in\mathbb{C}^N$8 antennas, $\mathbf{v}_{m,k}\in\mathbb{C}^N$9 GHz, transmit power $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$0 dBm, noise power $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$1 dBm, and $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$2 s tracks both position and 2D velocity accurately (Jiang et al., 10 Jun 2025).

Symbiotic sensing and communication extends sensing-assisted beamforming to a bistatic architecture with a transmitting BS and a passive sensing node (Xia et al., 2024). The sensing node estimates 2D AoAs of VUEs and of a weak target, shares the results with the BS over optical fiber, and the BS reconstructs LoS channels and solves a sum-rate maximization problem under a CRLB constraint on target-angle estimation. Fully digital and hybrid analog–digital beamformers are obtained by PDD-based alternating algorithms. The reported simulations show effective localization capability together with high data rate, and the HAD design yields remarkable performance gain compared with conventional schemes, especially with fewer RF chains (Xia et al., 2024). This usage illustrates an EDN interpretation in which sensing nodes, communication nodes, and beamforming solvers form a coordinated inference network.

6. Distributed, generative, and analytical formulations

EDN beamforming also includes frameworks that are not primarily neural but remain network-structured and low-overhead. In distributed sensor or relay networks with low-rate feedback, adaptive phase alignment can be written as a random search over

$$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$3

The analysis shows that all local maxima are global maxima and that the objective is invariant under a common phase shift. Under one-bit keep/discard feedback, the adaptive scheme converges in probability and in mean, and the time required for convergence in mean scales linearly with respect to the number of sensor or relay nodes (0806.3023). This is a distributed EDN-style beamforming framework in the sense that beam adaptation is driven by a global scalar feedback process rather than full CSI exchange.

Generative site-specific beamforming introduces another route to low-overhead design. For UPA systems, decoupled channel sensing probes azimuth and elevation independently, reducing online sweeping overhead from multiplicative complexity $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$4 to linear complexity $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$5, but the resulting RSRP observations lose explicit azimuth–elevation coupling (Tang et al., 21 Jun 2026). The proposed framework addresses this ambiguity with a bidirectional cross-attention encoder and a conditional normalizing flow that generates a candidate set of beam-space hypotheses. Training is task-oriented:

$$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$6

so the generator is rewarded when at least one candidate beam is high gain. In DeepMIMO scenarios, the reported normalized beamforming gain improvements over full 1024-beam two-dimensional DFT search are $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$7, $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$8, and $$\max_{\{\mathbf{v}_{m,k}\}} \sum_{k=1}^K \log_2(1+\mathrm{SINR}_k) \quad \text{s.t. } \sum_{k=1}^K \|\mathbf{v}_{m,k}\|^2 \le P_m,\ \forall m,$$9 in I2_28, O1B_28, and Boston5G_28, respectively, while the sweeping overhead is reduced by $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$0 (Tang et al., 21 Jun 2026).

Analytical resolution theory supplies a complementary foundation for near-field EDN design. For a UPA in the Fresnel region, the pairwise resolution metric is

$$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$1

which measures how distinguishable two users are in the distance–angle domain (Rao et al., 2023). The framework shows that angular separation can yield $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$2 as array size grows, but distance separation alone does not guarantee perfect resolution across the entire near-field region; if the normalized distance $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$3 is sufficiently large, $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$4 can remain close to $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$5 even as $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$6 (Rao et al., 2023). This analytical result matters because many EDN beamforming frameworks, especially predictive and site-specific ones, rely on the assumption that range–angle structure is exploitable; the letter formalizes when that assumption is and is not justified.

7. Hardware-aware optimization and recurring research directions

Some EDN beamforming frameworks are defined at the system and hardware level rather than at the representation level. An open-source end-to-end digital beamforming model describes a complete uplink chain from multiple UEs to a large BS array, including a free-space LOS narrowband MIMO channel, per-antenna LNA, VGA normalization, $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$7-bit ADC quantization, and digital combining with conjugate beamforming or zero forcing (Guajardo et al., 2024). The paper derives

$$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$8

and shows how a strong interferer degrades effective SQNR because the ADC full-scale must accommodate both desired signal and interferer. The model is explicitly aimed at implementations with low-resolution ADCs and emphasizes ENOB, array gain, and minimum tolerable SIR as design metrics (Guajardo et al., 2024). In this usage, EDN beamforming includes the quantization, interference, and linear-combining pipeline itself.

A distinct hardware-constrained formulation is the hybrid quantum optimization framework for a 32-element ULA with jointly optimized discrete phases and continuous amplitudes (Zeng, 20 Mar 2026). The method combines Gray-code and odd-combination phase encoding, geometric amplitude encoding, a rainbow quantum-inspired algorithm with multiple optimizers, hierarchical-clustering-based candidate refinement, and double outer-product constructions of the coupling matrix and bias vector. Under the 7th National Quantum Computing Hackathon scoring rules, the reported score is $$\mathrm{SINR}_k = \frac{\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,k}\right|^2} {\sum_{l\ne k}\left|\sum_{m=1}^M \mathbf{h}_{m,k}^H \mathbf{v}_{m,l}\right|^2+\sigma_k^2}.$$9, nearly doubling the baseline score, while respecting constraints on near-main-lobe sidelobes, wide-angle sidelobes, beamwidth, and optimization time (Zeng, 20 Mar 2026). Although this work is not an EDN in the encoder–decoder sense, it is an EDN-style framework in that beamforming is posed as a structured optimization stack with modular encoding, candidate generation, and refinement.

Across the literature, several constraints recur: perfect CSI or LoS-dominant reconstruction, narrowband or single-stream assumptions, site-specific training, and strong dependence on the chosen sensing or feedback modality (Jiang et al., 10 Jun 2025, Tang et al., 21 Jun 2026, Zhang et al., 27 Sep 2025). This suggests that the central open problem is not whether EDN beamforming should be graph-based, encoder–decoder, predictive, or generative, but how to combine these representations under realistic mobility, hardware, and overhead constraints. A plausible implication is that future EDN beamforming frameworks will increasingly be hybrid systems in which compressed CSI, sensing outputs, graph structure, and hardware-aware optimization coexist within a single inference-and-control loop.