Hybrid Beam Domain Strategy

• Hybrid Beam Domain Strategy is a framework that splits beamforming into an analog stage for coarse spatial filtering and a digital stage for refined signal processing.
• The analog stage employs techniques like beamsteering, sectorization, and angular-support selection to capture dominant propagation directions and reduce channel complexity.
• The digital stage processes the reduced effective channel using standard methods (SVD, ZF, MMSE) to approach near-digital performance while managing hardware constraints.

In the cited literature, a hybrid beam domain strategy denotes a class of hybrid analog/digital beamforming designs in which the analog stage first selects, steers, partitions, or shapes beams, sectors, or angular supports in a beamspace or angle domain, and the digital stage then operates on the resulting low-dimensional effective channel for multiplexing, multi-user separation, interference suppression, beam management, or robustness enhancement. Across mmWave, THz, cell-free, XL-MIMO, and electromagnetically reconfigurable settings, this structure appears as analog beamsteering with array-response transposes, beamspace processing with discrete lens arrays, virtual sectorization, angular-dictionary selection, multi-level beam codebooks, beam-delay sparse representations, and environment-aware or timing-aware beam selection (Zou et al., 2017, Shuang et al., 2018, Bychkov et al., 2024, Wu et al., 2022).

1. Defining principle and architectural rationale

The central architectural split is consistent across the surveyed designs. The analog stage performs coarse, geometry-aware spatial filtering: it captures dominant propagation directions, forms virtual sectors, selects a sparse beam subset, or aligns radiation patterns and timing in the beam domain. The digital stage then processes only the reduced effective channel, where conventional linear or eigenmode methods become feasible with far fewer RF chains than fully digital architectures require (Zou et al., 2017, Bychkov et al., 2024).

This division is motivated by the directional and sparse nature of mmWave and THz propagation and by the hardware constraint that fully digital precoding would require one RF chain per antenna, which is too costly and power-hungry at mmWave frequencies (Shuang et al., 2018). In multi-user settings, the same logic is expressed differently: a single analog beam cannot simultaneously give high gain to users spread across different directions, so users are grouped into clusters or sectors, and digital processing is confined to the users inside each analog sector (Bychkov et al., 2024).

The same principle persists when the objective is not only spectral efficiency. In beam management, the analog stage defines search sectors and beam hierarchies, while digital combining aggregates panel outputs or evaluates beam candidates (Alexandropoulos et al., 2021). In environment-aware designs, a channel knowledge map reduces the analog search space to location-specific angles or beam indices, after which digital beamforming is computed on the equivalent channel (Wu et al., 2022). In asynchronous cell-free systems, timing is also managed per beam, so that the beam domain becomes both a spatial and a synchronization layer (Xin et al., 2024).

2. Canonical mathematical formulation

A canonical narrowband or frequency-flat starting point is the sparse geometric channel model

$$\mathbf{H}=\mathbf{A}_{UE}\,\mathrm{diag}(\mathbf{g})\,\mathbf{A}_{BS},$$

with array-response matrices built from path AoAs and AoDs. In the analog beamsteering formulation, the analog beamformers are chosen as the transposes of the array-response matrices,

$$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$

so that the effective channel is

$$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$

For large arrays, the off-diagonal terms in the effective beam-domain channel become small, and the effective channel becomes approximately diagonal, which explains why the method can approach SVD-based digital beamforming at low to medium SNR in the single-user case (Zou et al., 2017).

A second canonical formulation is beamspace transformation by a discrete lens array or spatial Fourier matrix. There the physical channel $\mathbf{H}$ is mapped to a beamspace channel

$\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$

whose significant entries correspond to a small number of dominant predefined orthogonal beams. If the selected beam index set is $\mathcal{S}$, the reduced beamspace channel $\overline{\mathbf{H}}_s$ is the object on which digital precoding is performed (Shuang et al., 2018).

In wideband THz MIMO-OFDM, the formulation becomes frequency dependent. The channel on subcarrier $k$ is expressed as a sum over paths with steering vectors that depend explicitly on $f_k/f_c$, and beam split is described through the frequency-scaled angular components

$$\Theta_{k,\ell}=\frac{f_k}{f_c}\sin(\theta_\ell)\cos(\psi_\ell),\qquad \Psi_{k,\ell}=\frac{f_k}{f_c}\sin(\theta_\ell)\sin(\psi_\ell).$$

The resulting normalized gain is approximated by sinc or Dirichlet terms, so a beam aligned at the carrier frequency is misaligned at subcarriers away from $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$0 (Yildirim et al., 24 Mar 2025, Ma et al., 2023).

A further extension replaces the angle-only representation with a beam-delay model. In mmWave XL-MIMO, the spatial-frequency channel is represented in a beam-delay domain with angle, slope, and delay structure,

$$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$1

where $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$2 is sparse under a beam-delay-domain dictionary. This formulation is designed to preserve inter-subcarrier correlation under near-field and beam-squint effects (Hou et al., 2023).

3. Analog-domain mechanisms

The analog stage admits several distinct mechanisms, but all are beam-domain operations that reduce the dimensionality or uncertainty presented to the digital stage.

One mechanism is direct analog beamsteering toward dominant paths. Under perfect AoD/AoA knowledge and infinite-precision codebooks, the analog beamformers are the array-response transposes, and the analog stage steers beams directly along the $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$3 dominant propagation paths. With finite codebooks, the selected steering directions are the nearest codebook points to the true path wavenumbers, and the rate loss depends on the path gain or SNR term and a beam-quantization term that depends on antenna count and codebook size. The derived condition is that the codebook size should be larger than the number of implemented antennas, with about $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$4 the number of antennas described as a good rule of thumb to minimize degradation (Zou et al., 2017).

A second mechanism is quantized angular-support selection. In wideband THz systems, coarse AoD/AoA information and angular spread define an angular support region $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$5, the angular domain is discretized into quantized steering directions, and subcarrier-specific RF beams are selected from the quantized dictionary for each subcarrier. The broader support better covers beam-split-induced offsets across subcarriers, without true-time-delay units (Yildirim et al., 24 Mar 2025).

A third mechanism is hierarchical beam management. In uplink mmWave cellular MIMO with hybrid beamforming, the analog beamformer uses a multi-level beam codebook comprising flat top beams with variable widths. These beams exhibit an almost constant array gain for the whole desired angle width. The search procedure is hierarchical, and dynamic beam ordering starts from the beam used in the previous TTI and then tests adjacent beams first, rather than using a fixed global sweep (Alexandropoulos et al., 2021).

A fourth mechanism is interactive beam search in the beam domain. The group-testing formulations AGTBA and HGTBA reinterpret beam alignment as the identification of defective angular intervals among $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$6 bins. A beam probe is a test, and the BS uses one beam in analog BA or multiple beams simultaneously in hybrid BA, with ACK and NACK outcomes guiding subsequent scans (Yildiz et al., 2021).

A fifth mechanism is sectorization and clustering. In partially connected hybrid beamforming, users are partitioned into clusters, each cluster corresponds to one fixed analog beam or sector, and the phase shifters in each subarray are set from the dominant singular direction of the cluster channel. This virtual sectorization converts an otherwise mismatched multi-user analog beam into a sectorized front end (Bychkov et al., 2024).

A sixth mechanism is radiation-domain pattern selection. In tri-hybrid and electromagnetic hybrid beamforming, the analog layer is extended beyond phase shifters. Pattern-reconfigurable antennas select one pattern from a predefined dictionary on a long timescale, while holographic or superdirective arrays adjust antenna excitation current vectors in real time to shape the radiation pattern throughout the spatial domain (Liu et al., 5 Mar 2025, Ji et al., 2024).

Mechanism	Analog-domain action	Representative source
Array-response beamsteering	Steer along dominant paths	(Zou et al., 2017)
Angular-support dictionary selection	Select subcarrier-specific beams inside $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$7	(Yildirim et al., 24 Mar 2025)
Multi-level flat-top codebook	Broad-to-narrow sector search	(Alexandropoulos et al., 2021)
Group-testing beam probing	Interactive ACK/NACK beam search	(Yildiz et al., 2021)
Virtual sectorization	One sector beam per user cluster	(Bychkov et al., 2024)
EM-domain pattern shaping	Radiation-pattern alignment or excitation control	(Liu et al., 5 Mar 2025, Ji et al., 2024)

4. Digital-domain processing on the reduced effective channel

Once the analog stage has defined the beam-domain support, the digital stage operates on a low-dimensional effective channel. In the simplest single-user setting, this reduced channel is $$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$8, and the paper on analog beamsteering uses the SVD-based digital benchmark

$$\mathbf{F}_A=\mathbf{A}_{BS}^T,\qquad \mathbf{W}_A=\mathbf{A}_{UE}^T,$$9

to explain why analog beamsteering can approach unconstrained digital beamforming in the low-to-medium-SNR regime (Zou et al., 2017).

In wideband THz angular-based hybrid beamforming, the per-subcarrier effective channel is

$$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$0

followed by an SVD and a baseband precoder

$$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$1

The RF stage handles angular coverage and beam split mitigation, whereas the baseband stage handles multiplexing and eigenmode optimization (Yildirim et al., 24 Mar 2025).

In beamspace massive MIMO with a discrete lens array, the digital stage is zero forcing on the selected beamspace channel,

$$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$2

and sum-rate maximization is equivalently rewritten as minimizing

$$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$3

This makes the conditioning of the selected beam submatrix the key digital objective (Shuang et al., 2018).

In multi-cell uplink hybrid beamforming via Kronecker decomposition, the analog beamformer is designed first to null inter-cell interference and coherently combine desired paths, and the digital beamformer then performs MMSE multiuser decoupling on the reduced channel $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$4 (Zhu et al., 2017). In environment-aware hybrid beamforming, once the analog beams have been identified through CAM or BIM, the equivalent channel $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$5 is decomposed by SVD and the digital design follows the standard eigenmode rule with water-filling (Wu et al., 2022). In wideband THz beam-squint mitigation, the digital precoder is water-filling on the dominant singular modes of the effective channel, and the digital combiner is the MMSE solution, both in closed form (Ma et al., 2023).

This repeated pattern shows that hybrid beam-domain designs do not replace conventional baseband linear algebra; they relocate it. The digital stage is typically standard—SVD, ZF, MMSE, water-filling, or SIC—once the analog beam-domain front end has produced a manageable effective channel. This suggests that the main innovation in many such systems lies in how the analog stage defines the beam-domain support rather than in altering the algebra of reduced-channel processing.

5. Major variants across deployment regimes

The beam-domain logic is sufficiently general that it appears in several deployment-specific variants.

In overloaded downlink mmWave systems, multi-beam NOMA uses beam splitting: one RF chain can split the antenna array into multiple subarrays, each subarray is steered toward a different user AOD, and the beam-domain structure becomes an additional resource dimension. The asymptotic analysis shows that the strongest user’s antenna allocation dominates sum-rate, and the paper derives sufficient and necessary conditions under which the proposed scheme outperforms TDMA (Wei et al., 2018).

In sequential multi-link transmission, hybrid beamforming is treated as an add-on feature. A baseline analog-beamformed link is established first, and if an extra RF chain is available at the UE, a parallel beam search probes for a secondary beam pair without interrupting baseline data transmission. If resources and channel conditions are favorable, a second link is activated and digital processing is applied on the resulting $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$6 effective channel (Zou et al., 2017).

In environment-aware designs, the beam domain is indexed by location. CAM maps user location to candidate AoA/AoD sets, and BIM maps user location directly to candidate transmit and receive beam indices. CAM reduces the training problem to path-gain estimation on a predicted angular subspace, while BIM performs light beam sweeping over a reduced beam subset (Wu et al., 2022).

In asynchronous cell-free mmWave OFDM, the beam-domain abstraction incorporates timing. PBTA inserts a timing-advance unit before the beam-selection network so that each RF chain is assigned to one beam direction and the transmission time of that beam is adjusted individually. The residual offset becomes beam dependent, and for the intended UE on that beam the desired-signal phase offset is removed (Xin et al., 2024).

In XL-MIMO channel estimation, the beam-domain representation is extended to a beam-delay domain with a Bernoulli-Gaussian prior on sparse coefficients and constrained Bethe free energy minimization. The proposed HMP algorithms jointly estimate the beam-delay-domain channel and prior hyperparameters, and the MDGPP extension refines multidimensional grid points in angle, slope, and delay (Hou et al., 2023).

In electromagnetically richer architectures, hybrid beamforming becomes tri-hybrid or electromagnetic hybrid. The tri-hybrid design introduces three layers—radiation beamforming in the EM domain, analog beamforming in the RF domain, and digital beamforming in the BB domain—and updates them over long, medium, and short timescales, respectively (Liu et al., 5 Mar 2025). The electromagnetic hybrid beamforming design based on a 3D superdirective holographic antenna array uses antenna excitation current vectors as analog beamforming and digital precoding matrices as the baseband layer, with an electromagnetic channel model that incorporates radiation patterns and mutual coupling (Ji et al., 2024).

Variant	Beam-domain role	Representative source
Beamspace massive MIMO	Select sparse orthogonal beams and apply ZF	(Shuang et al., 2018)
Multi-beam NOMA	Create multiple analog beams from one RF chain	(Wei et al., 2018)
Sequential multi-link HBF	Add secondary beam-domain links after baseline analog BF	(Zou et al., 2017)
Environment-aware HBF	Use location-specific angle or beam subsets	(Wu et al., 2022)
Asynchronous cell-free PBTA	Align transmission timing per beam	(Xin et al., 2024)
Beam-delay XL-MIMO estimation	Exploit sparse beam-delay structure	(Hou et al., 2023)
Tri-hybrid / EHB	Extend beamforming to EM-domain radiation control	(Liu et al., 5 Mar 2025, Ji et al., 2024)

6. Performance characteristics, limitations, and design tradeoffs

Several quantitative results recur across the literature. Analog beamsteering using array-response transposes can achieve performance very close to SVD-based digital beamforming at low to medium SNR in the single-user case, but finite codebooks introduce a rate loss that grows with antenna count, received SNR per path, and the ratio of antennas to codebook size. The closed-form condition for avoiding more than $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$7 dB instantaneous loss is approximately $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$8, with the practical interpretation $$\mathbf{H}_{\mathrm{eff}}=\mathbf{W}_A\mathbf{H}\mathbf{F}_A.$$9 and about $\mathbf{H}$0 the number of antennas described as a good rule of thumb (Zou et al., 2017).

In wideband THz systems, the principal limitation is beam split or beam squint. Angular-based hybrid beamforming broadens effective angular support through angular spread and can keep the normalized array gain close to $\mathbf{H}$1 over different angles and subcarriers, with rates comparable to fully digital precoding and better behavior than narrowband spatially sparse precoding as $\mathbf{H}$2 grows (Yildirim et al., 24 Mar 2025). A complementary analysis for UPAs gives the beam squint ratio

$\mathbf{H}$3

shows that the severity is governed by the more severely affected array dimension, and states that a near-square UPA reduces squint relative to a ULA by a factor of $\mathbf{H}$4. For $\mathbf{H}$5, the reported BSR is about $\mathbf{H}$6, and the considered HBF methods except HBF-DCF can achieve up to $\mathbf{H}$7 of optimal SE (Ma et al., 2023).

In multi-user beamspace selection, explicitly accounting for channel correlation among users materially changes the result. The ACO-based joint beam selection scheme achieves almost the same system sum rate as exhaustive search, about $\mathbf{H}$8 of the system performance is already achieved when $\mathbf{H}$9 increases to $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$0, and one iteration already yields about $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$1 of performance. In one reported case with $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$2 and $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$3, the method achieves $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$4 bits/s/Hz while requiring only $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$5 matrix inversions, compared with at least $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$6 inversions for the IA method under the same conditions (Shuang et al., 2018).

In virtual sectorization, the hierarchical clustering-PC scheme is reported to be better than the multi-center variant by about $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$7 dB, better than generic hierarchical clustering by about $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$8 dB, and within about $\overline{\mathbf{H}}=\mathbf{U}^H\mathbf{H},$9 dB of the fully digital receiver, while reducing the number of required ADC chains from $\mathcal{S}$0 to $\mathcal{S}$1 for $\mathcal{S}$2 clusters (Bychkov et al., 2024). In sequential hybrid beamforming, the two-stream design has about $\mathcal{S}$3 dB performance loss relative to the optimal HBF reference, and the secondary beam pair can also serve as a backup beam pair if the baseline link is blocked (Zou et al., 2017).

For beam management, exhaustive narrow-beam search remains an upper bound on search cost. Group-testing beam alignment reduces expected BA duration by about $\mathcal{S}$4 relative to the analog GT-based method and by more than $\mathcal{S}$5 relative to a hybrid exhaustive search baseline in narrow-beam regimes, while the HGTBA$\mathcal{S}$6 variant is the best-performing hybrid method in the reported studies (Yildiz et al., 2021). The multi-level flat-top-beam codebook with dynamic beam ordering similarly targets latency reduction, and the proposed algorithm consistently achieves the smallest average number of beam searches among the compared schemes while keeping received SNR close to exhaustive search (Alexandropoulos et al., 2021).

Some limitations are equally explicit. Beam split becomes more severe with larger fractional bandwidth and larger effective aperture (Ma et al., 2023). In asynchronous cell-free systems, more RF chains do not necessarily help under asynchrony, because more simultaneously served beams can increase exposure to residual ICI and ISI; PBTA improves the situation substantially but still leaves a small gap to perfect synchronization (Xin et al., 2024). In THz beam-split mitigation, true-time-delay architectures can compensate beam squint more directly, but they require extra analog hardware, cost, power, and design changes; the cited angular-support-based method is lighter because it exploits channel sparsity in the angular domain rather than implementing frequency correction in hardware (Yildirim et al., 24 Mar 2025).

A common misconception is that hybrid beam-domain design is only a low-cost approximation to fully digital precoding. The surveyed results do not support so narrow a view. Some methods indeed pursue near-digital performance with fewer RF chains, but others introduce capabilities that are absent from a purely digital-versus-hybrid comparison: virtual sectorization for clusterized multi-user reception, multi-beam NOMA from beam-split subarrays, beam-management decision trees, environment-aware reduction of training overhead, per-beam timing advance, and EM-domain pattern shaping (Wei et al., 2018, Wu et al., 2022, Xin et al., 2024, Liu et al., 5 Mar 2025). A plausible implication is that the beam domain is best understood not merely as a hardware compromise, but as a design domain in which geometry, sparsity, timing, and radiation-pattern control are made explicit.