Self-Aligned Beamforming

• Self-aligned beamforming is a method that iteratively refines beam directions using endogenous measurements and reduced reliance on full channel estimation.
• It employs diverse techniques including cross-correlation based self-calibration, adaptive sensing in mmWave systems, and analog phase-conjugation in RIS-assisted setups.
• Practical implementations show rapid convergence with low overhead, enhancing performance in phased-array, MIMO, and full-duplex scenarios.

Searching arXiv for recent and related papers on self-aligned beamforming and adjacent formulations. Self-aligned beamforming denotes a class of beam design and calibration mechanisms in which beam directions, phases, or analog weights are adjusted using measurements generated by the array or propagation loop itself, rather than by full explicit channel estimation followed by conventional digital control. Across the literature, the term covers several distinct but related paradigms: iterative gain calibration in phased arrays via beam-based cross-correlations (Gueuning et al., 2020), adaptive mmWave beam alignment driven by posterior updates or learned sensing policies (Sohrabi et al., 2020), grid-free continuous beam synthesis from learned probing measurements (Heng et al., 2022), analog phase-conjugate self-tracking in RIS-assisted terahertz SWIPT (Wei et al., 15 Jul 2025), resonant retro-directive beam formation without CSI estimation (Guo et al., 9 Dec 2025), and low-overhead beam alignment with online self-calibration under array impairments (Jin et al., 25 Feb 2026). The unifying feature is that alignment emerges from repeated interaction between received field measurements, structural priors, and beam synthesis, often with reduced reliance on full correlation matrices, exhaustive codebooks, or explicit CSI reconstruction.

1. Conceptual scope and defining mechanisms

The most direct formulation of self-aligned beamforming appears in phased-array self-calibration, where each antenna’s voltage is cross-correlated with a beamformed voltage formed from the other antennas, and the gain estimates are updated iteratively until the array “self-aligns” its phase and amplitude errors into a calibrated beam (Gueuning et al., 2020). In that setting, the unknown direction-independent complex gains enter both the beam weights and the cross-correlation statistics, so alignment arises through alternating beam formation and gain re-estimation rather than through full visibility-matrix inversion.

In mmWave initial access and alignment, the term shifts from internal gain calibration to adaptive sensing. The beamformer sequence is selected from the current posterior over the angle of arrival (AoA), or from a learned mapping of posterior statistics to analog sensing vectors, so that the array progressively aligns to the dominant path (Sohrabi et al., 2020). A related but broader interpretation appears in grid-free MIMO beam alignment, where probing measurements from a small number of learned beam pairs are mapped directly to continuous transmit and receive beams, avoiding quantized codebooks and allowing the system to “self-align” to the actual path angles rather than the nearest grid point (Heng et al., 2022).

At terahertz frequencies, self-alignment is also used in a strictly analog sense. In RIS-assisted THz-SWIPT, phase-conjugate circuits at the RIS and user equipment return phase-inverted replicas of the incident field, causing constructive build-up along the reciprocal path; the phase profiles then “lock” through an iterative power cycle without digital beam training (Wei et al., 15 Jul 2025). The RF resonant-beam literature generalizes this idea further: retro-directive antenna arrays and phase-conjugation circuits establish a cyclic electromagnetic loop, and the beam locks onto the unique direction satisfying the loop gain and phase conditions, thereby eliminating digital CSI processing altogether (Guo et al., 9 Dec 2025).

A plausible implication is that “self-aligned beamforming” is not a single algorithmic family but an umbrella term for beamforming schemes in which the steering law is induced by endogenous measurements, reciprocity, or analog positive feedback rather than by exogenous, fully explicit channel reconstruction.

2. Beam-based self-calibration of phased arrays

The phased-array self-calibration formulation begins with an array of $N$ antennas, each having an unknown direction-independent complex gain $g_i$, and seeks real-time calibration without forming the full $N\times N$ visibility matrix (Gueuning et al., 2020). The voltage at antenna $i$ is modeled as

$v_i(t)=g_i\,s_i(t)+n_i(t),$

with $s_i(t)$ the sky signal and $n_i(t)$ additive uncorrelated system noise. For each antenna port $i$, a beamformed voltage is constructed from the other $N-1$ channels: $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$

The core statistic is the time-averaged cross-correlation

$g_i$0

Under ergodicity and uncorrelated noise, the pairwise covariance becomes

$g_i$1

which leads to

$g_i$2

where

$g_i$3

Assuming $g_i$4, the gain estimate is

$g_i$5

In practice, one sets $g_i$6 if $g_i$7 is unknown, thereby eliminating the explicit noise term (Gueuning et al., 2020).

The weight choice is central. When the beamforming weights are selected to maximize the average cross-correlation power, the maximizer is

$g_i$8

This produces a beam that coherently integrates the modeled sky contribution from the other elements while excluding the self-noise term. The resulting procedure is both a calibration method and a beamforming operation, because the beam itself acts as the sufficient high-SNR reference against which each element is re-estimated.

The algorithmic cycle is explicit: initialize $g_i$9, form weights from the current gain estimates and model temperatures, compute the beamformed temperature $N\times N$0, measure the beam-based cross-correlation $N\times N$1, update $N\times N$2, and iterate until $N\times N$3 or $N\times N$4 (Gueuning et al., 2020). The reported computational complexity is $N\times N$5 per time sample in a naïve implementation, reducible to $N\times N$6 using fast beamforming such as NUFFT-based methods, with empirically only a handful of iterations, specifically $N\times N$7–$N\times N$8, typically needed for convergence (Gueuning et al., 2020).

3. Relation to StEFCal and iterative solver structure

A key theoretical result is that the beam-based gain update becomes algebraically identical to one iteration of alternating-direction-implicit StEFCal when the weights are chosen as the normalized average cross-correlated power (Gueuning et al., 2020). Defining the model matrix $N\times N$9 and $i$0, the cited StEFCal update for the $i$1th gain is

$i$2

Choosing

$i$3

and forming $i$4 recovers exactly that ratio (Gueuning et al., 2020).

This equivalence is significant because it links a beamforming interpretation to a well-studied self-calibration solver. In the phased-array formulation, each iteration can be viewed physically as building the best current calibration beam and then projecting each element against that beam; in the solver interpretation, the same operation is a StEFCal pass executed without the full visibility matrix. This suggests that self-alignment here is not merely heuristic. It is an alternative realization of an established gain-estimation recursion, with the beam serving as an implicit compression of the correlation information.

The reported performance reflects that structure. For few point sources, convergence is described as rapid, often $i$5–$i$6 iterations to $i$7 error; for extended diffuse scenes, $i$8–$i$9 iterations may be required, though convergence remains robust because each iteration is essentially a StEFCal pass for non-degenerate sky models (Gueuning et al., 2020). In a simulation with a 256-element SKA-like aperture array and five random point sources, the method reached $v_i(t)=g_i\,s_i(t)+n_i(t),$0 dB relative gain-error in three iterations, each iteration costing $v_i(t)=g_i\,s_i(t)+n_i(t),$1 s of integration, yielding $v_i(t)=g_i\,s_i(t)+n_i(t),$2 s total for better than $v_i(t)=g_i\,s_i(t)+n_i(t),$3 gain accuracy (Gueuning et al., 2020).

A common misconception is that self-aligned beamforming necessarily dispenses with modeling. In this formulation it does not: the method requires model array temperatures $v_i(t)=g_i\,s_i(t)+n_i(t),$4 from an EEP + sky model, and if noise temperatures are known they can also enter the update (Gueuning et al., 2020). What it avoids is the explicit formation of the full correlation matrix.

4. Adaptive and learned self-alignment in mmWave and MIMO systems

In mmWave initial alignment, the relevant latent variable is the dominant AoA rather than per-element chain gain. For a single-user uplink with one RF chain at the base station, the observation under analog sensing vector $v_i(t)=g_i\,s_i(t)+n_i(t),$5 is

$v_i(t)=g_i\,s_i(t)+n_i(t),$6

with a single-path channel $v_i(t)=g_i\,s_i(t)+n_i(t),$7 and the beamforming policy written as

$v_i(t)=g_i\,s_i(t)+n_i(t),$8

where $v_i(t)=g_i\,s_i(t)+n_i(t),$9 is the posterior over AoA (Sohrabi et al., 2020). The deep learning method in "Deep Active Learning Approach to Adaptive Beamforming for mmWave Initial Alignment" learns a nonlinear mapping from the posterior distribution, along with SNR and time index, to the next analog sensing vector, while accommodating both unit-norm and constant-modulus constraints through the output activation layer (Sohrabi et al., 2020).

The training process unrolls the entire $s_i(t)$0-step adaptive procedure as a deep network, using cross-entropy loss for on-grid detection and squared-error loss for gridless estimation (Sohrabi et al., 2020). In the reported $s_i(t)$1, $s_i(t)$2 experiments, the DNN with known $s_i(t)$3 outperforms hierarchical posterior matching by about $s_i(t)$4 dB across a wide SNR range in both on-grid detection and gridless estimation, while MMSE plug-in and Kalman variants for unknown $s_i(t)$5 achieve nearly the same performance as the known-$s_i(t)$6 hierarchical baseline; under constant-modulus constraints, the DNN methods continue to perform at near-optimal accuracy, whereas the hierarchical baselines degrade substantially (Sohrabi et al., 2020).

Grid-free MIMO beam alignment pushes the idea further by removing the codebook grid altogether. In "Grid-Free MIMO Beam Alignment through Site-Specific Deep Learning", a small set of learned probing beam pairs $s_i(t)$7 produce diagonal probing measurements whose squared magnitudes are assembled into the feature vector $s_i(t)$8; two real-valued MLPs then map $s_i(t)$9 to continuous Tx and Rx beam vectors with unit-modulus normalization (Heng et al., 2022). The training objective combines

$n_i(t)$0

for beamforming gain and

$n_i(t)$1

for initial-access coverage, weighted by $n_i(t)$2, so that the learned probing beams are jointly optimized for discovery and refinement (Heng et al., 2022).

The online procedure is “one-shot”: one sweep of the learned $n_i(t)$3 probing beams is performed, $n_i(t)$4 is measured and fed back, and continuous beams are synthesized directly, with no further search or grid (Heng et al., 2022). In the cited O1 urban LOS scenario with $n_i(t)$5, the reported average SNR values are $n_i(t)$6 dB for DL-GF, $n_i(t)$7 dB for exhaustive search, $n_i(t)$8 dB for DFT+EGC, and $n_i(t)$9 dB for MRT+MRC; the method also yields a 5–10 dB SNR advantage at a given beam-alignment speed, or equivalently the same SNR with 100× less latency (Heng et al., 2022).

A related low-overhead formulation appears in "Deep Learning-based Low-Overhead Beam Alignment for mmWave Massive MIMO Systems", where self-alignment is decomposed into QSSR, QSSR-Net, and an online parametric self-calibration stage (Jin et al., 25 Feb 2026). QSSR exploits the monotonic power-ratio property between adjacent DFT beams to recover a continuous angle $i$0 with the same total measurement cost as binary search: $i$1 (Jin et al., 25 Feb 2026). QSSR-Net then uses the full sequence of hierarchical measurements to refine the estimate, and the self-calibration module fits diagonal compensation matrices for position and phase errors directly from the same beam measurements, minimizing a reconstruction loss over the final-layer power patterns (Jin et al., 25 Feb 2026). In simulated LoS channels with $i$2, QSSR alone yields $i$3 dB gain over binary search, QSSR-Net adds another $i$4 dB, and at high SNR the learned method exceeds exhaustive search because exhaustive search remains grid-limited; with impairments $i$5, QSSR-Net-Impair recovers up to $i$6 dB over the uncalibrated case (Jin et al., 25 Feb 2026).

These learned formulations differ from phased-array self-calibration in their latent variables and priors, but they retain the same general architecture: a compact set of beam-domain observations is iteratively or inferentially converted into updated beamformers, with alignment emerging from beam-mediated feedback.

5. Analog reciprocity, phase conjugation, and resonant self-alignment

In RIS-assisted terahertz SWIPT, self-alignment is realized through phase conjugation rather than recursive estimation. The incident field

$i$7

is converted into its complex conjugate

$i$8

so the conjugated wave retraces the original propagation path (Wei et al., 15 Jul 2025). Each RIS element and UE antenna element contains a subharmonically injection-locked mixer that heterodyne-mixes the incoming RF to baseband and remodulates it so that the reflected wave carries the conjugate phase (Wei et al., 15 Jul 2025). If the uplink channel vector is $i$9, the resulting beamforming vector is $N-1$0, yielding maximum coherent gain on the downlink (Wei et al., 15 Jul 2025).

The RIS architecture augments this with active amplification and an iterative power cycle. At iteration $N-1$1,

$N-1$2

and convergence occurs when the per-iteration gain approaches the loss (Wei et al., 15 Jul 2025). The operational interpretation is that the RIS phase profiles $N-1$3 “lock” so that round-trip gain compensates round-trip path loss, and the system settles into a self-reproducing mode after $N-1$4–$N-1$5 microsecond-scale iterations (Wei et al., 15 Jul 2025).

The reported metrics are explicitly tied to this analog self-alignment process. For a 70×70 array, the end-to-end transmission efficiency reaches $N-1$6 at $N-1$7 m and remains $N-1$8 at approximately $N-1$9 m; the same array sustains $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$0 dB SNR and $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$1 bit/s/Hz up to $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$2 m, while maintaining $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$3 efficiency over $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$4 at $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$5 m (Wei et al., 15 Jul 2025).

The RF resonant-beam system in "Self-Alignment Resonant Beam Empowers Beamforming without Estimation and Control for 6G IoT" is closely related but described as a native physical-layer phenomenon rather than a communication-system algorithm (Guo et al., 9 Dec 2025). Retro-directive antenna arrays at both ends establish a cyclic electromagnetic resonance. With loop transfer function

$y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$6

resonance occurs when

$y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$7

(Guo et al., 9 Dec 2025). The system then locks onto the unique $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$8 satisfying these conditions. The beam is self-aligning because off-axis paths fail to close the positive-feedback loop, while the LoS direction with matched conjugation satisfies the Barkhausen criteria (Guo et al., 9 Dec 2025).

The comparison drawn in that review is categorical: RF-RBS requires no CSI estimation, no digital precoding, no beam search, and no training overhead, with beam lock occurring on the order of $y_i(t)=\sum_{j\neq i} w_{ij} v_j(t).$9–$g_i$00s at 30 GHz, contrasted with 10–100 ms for 5G mmWave initial access (Guo et al., 9 Dec 2025). A misconception here would be to equate all self-alignment with machine learning or Bayesian adaptation. In the resonant and phase-conjugate literature, the governing mechanism is analog reciprocity and positive feedback, not statistical inference.

6. Distributed, site-specific, and system-level interpretations

Self-alignment also appears in distributed and system-level beamforming contexts where centralized CSI acquisition is impractical. In self-organizing coherent networks, groups of single-antenna transmitters iteratively coordinate their phases through local message passing so that their rank-one superposition becomes coherent at multiple receivers (Shi et al., 2019). Each node updates its phase to maximize a local coherent-gain objective, and the group converges to a fixed point without a central controller (Shi et al., 2019). For $g_i$01 and a single stream, the Iterative Optimization protocol reportedly achieves $g_i$02 of the theoretical $g_i$03 bound; in the two-stream case, distance-based clustering plus “best-target” beamforming gives $g_i$04, versus $g_i$05 for non-coherent schemes (Shi et al., 2019).

In hybrid wideband multiuser mmWave systems, self-alignment can refer to training-based beam selection that quickly steers analog beams without estimating the full channel matrix. The method in "Beamforming Algorithm for Multiuser Wideband Millimeter-Wave Systems with Hybrid and Subarray Architectures" uses hierarchical orthogonal codebooks and three analog training stages to approximately maximize

$g_i$06

(Viteri-Mera et al., 2019). The algorithm attains more than 70 percent of the spectral efficiency of ideal fully-digital beamforming, corresponding to roughly 1.5–3 dB SNR loss in the analyzed scenarios, while retaining linear dependence on the number of antennas and shorter training overhead (Viteri-Mera et al., 2019). Although the paper does not use the same phase-conjugate or StEFCal language, the operational theme remains the same: beam directions are selected from beam-space measurements rather than full channel reconstruction.

A more recent system-level variant is site-specific full-duplex beamforming via implicit channel estimation. In "Site-Specific Beamforming for Full-Duplex Massive MIMO Systems via Implicit Channel Estimation", a transformer-based model selects a small number of probing beam pairs tailored to the deployment and users, measures only the most relevant portions of the self-interference channel $g_i$07, and then synthesizes transmit and receive beams that jointly deliver high user gain and low self-interference (Li et al., 20 May 2026). The optimization target is the normalized sum-spectral efficiency

$g_i$08

while using only $g_i$09 probes (Li et al., 20 May 2026). At $g_i$10 dB, $g_i$11, and $g_i$12, the reported effective SSE with $g_i$13 is approximately $g_i$14, versus approximately $g_i$15 for LMMSE, approximately $g_i$16 for vector CSI, and below $g_i$17 for matrix CSI; for 16×16 arrays, the proposed method remains above $g_i$18 with $g_i$19, while the baselines fall below $g_i$20 (Li et al., 20 May 2026).

This broader literature suggests that self-aligned beamforming is increasingly associated with implicit channel knowledge: the system is not assumed to reconstruct all channel coefficients, but only to extract the beam-space information needed to align to the relevant propagation mode, calibration state, or self-interference null.

7. Comparative themes, limitations, and open directions

The literature supports several recurring design themes.

Theme	Representative mechanism	Representative source
Internal calibration	Cross-correlate each element with a beamformed sum of its peers	(Gueuning et al., 2020)
Adaptive path alignment	Posterior-driven or learned sensing-vector updates	(Sohrabi et al., 2020)
Continuous beam synthesis	Grid-free mapping from probing powers to analog beams	(Heng et al., 2022)
Analog reciprocal locking	Phase conjugation and iterative power-cycle convergence	(Wei et al., 15 Jul 2025)
Physical-layer resonance	Retro-directive positive-feedback beam locking	(Guo et al., 9 Dec 2025)
Joint alignment and self-calibration	Learned angle estimation plus online impairment compensation	(Jin et al., 25 Feb 2026)

A first distinction is between estimation-centric and physics-centric self-alignment. Estimation-centric methods, such as phased-array gain calibration, posterior-based mmWave alignment, or transformer-based implicit SI probing, still optimize an inferred latent quantity from measurements (Gueuning et al., 2020, Sohrabi et al., 2020, Li et al., 20 May 2026). Physics-centric methods, such as phase-conjugate RIS alignment and RF resonant beams, achieve alignment through reciprocity and positive feedback with no digital beam training (Wei et al., 15 Jul 2025, Guo et al., 9 Dec 2025).

A second distinction is between grid-based refinement and grid-free synthesis. Learned posterior policies and QSSR/QSSR-Net begin with structured beam hierarchies or DFT layers but refine toward continuous angles (Sohrabi et al., 2020, Jin et al., 25 Feb 2026). The site-specific grid-free framework synthesizes beams directly in the continuous manifold from a few probing measurements (Heng et al., 2022).

The principal limitations are also heterogeneous. The phased-array self-calibration method depends on model array temperatures from an EEP + sky model and assumes non-degenerate sky structure for the stated convergence behavior (Gueuning et al., 2020). DNN-based mmWave alignment can require extensive offline training and uses approximations when fading is unknown (Sohrabi et al., 2020). Grid-free site-specific beam alignment presumes access to ray-tracing or deployment-specific training data, and its advantages are therefore site-dependent (Heng et al., 2022). The RIS-assisted THz formulation has a finite FoV, with the link collapsing when the feedback gain drops below path loss outside approximately $g_i$21 for the cited 70×70, 1 m setting (Wei et al., 15 Jul 2025). RF-RBS is explicitly LoS-dependent and subject to multi-user loop competition and range thresholds where $g_i$22 (Guo et al., 9 Dec 2025). QSSR-style super-resolution depends on preserving near-ideal beam-pattern monotonicity, which motivates the added self-calibration stage under array impairments (Jin et al., 25 Feb 2026).

An important interpretive point is that self-alignment does not imply the absence of prior structure. In every major variant, alignment relies on some combination of reciprocity, channel sparsity, site specificity, sky modeling, array manifold structure, or slowly varying hardware error models. The distinguishing feature is not the removal of structure, but the relocation of beam control from explicit full-state estimation to endogenous beam-domain interaction.

A plausible implication for future work is convergence toward hybrid architectures that combine these strands: fast analog or reciprocity-based coarse locking, implicit or learned refinement from a small number of beam measurements, and online self-calibration of array impairments. The recent full-duplex implicit-channel and mmWave self-calibration results indicate that shared probing, environment-specific priors, and parameterized hardware compensation can substantially reduce overhead while maintaining or exceeding the performance of explicit-CSI baselines (Li et al., 20 May 2026, Jin et al., 25 Feb 2026).