Training-Efficient Channel Estimation for the Multiuser Uplink Relying on Generalized Flexible Intelligent Metasurfaces

I. INTRODUCTION

• Training overhead limits how much reconfigurability can be exploited because the large configuration space must be learned within one coherence time.The introduction frames accurate CSI as a prerequisite for RIS, movable-antenna, fluid-antenna, and flexible-metasurface architectures.Source: §I · p. 2

A. State-of-the-Art for RISs and Flexible Intelligent Metasurfaces

• Prior FIM channel-estimation methods use sparse recovery, measurement diversity, sensing-design optimization, or learned shape-to-channel interpolation.This work instead uses a structural parametric tensor model to predict responses at unseen configurations without a shape-space dictionary.Source: A. State-of-the-Art for RISs and… · pp. 3–4

B. State-of-the-Art for Movable Antennas and Fluid Antenna Systems

• Unlike movable-antenna work that indexes diversity by pose catalogues, this paper makes rigid translation a Vandermonde tensor mode in a receive-side aperture.The G-FIM uses translation as controlled diversity and morphing to modulate the normal-axis projection of each wavevector.Source: B. State-of-the-Art for Movable… · p. 4

C. State-of-the-Art for Twin-Time-Scale Training

• The twin-time-scale G-FIM protocol uses slow planar translation and fast morphing as mechanical—not statistical—sources of instantaneous channel information.Their separability induces a tensor model that further compresses the estimation problem.Source: C. State-of-the-Art for… · pp. 4–5

D. State-of-the-Art for Tensor-Based Channel Estimation

• Prior tensor channel-estimation work covers RIS, related apertures, and multiuser uplinks, but the cited flexible-aperture models use one physical mechanism or stop at estimation.The paper targets two mechanical motions at distinct timescales, their fourth-order CP structure and identifiability conditions, and selection of an untrained panel configuration.Source: D. State-of-the-Art for Tensor-Based… · pp. 5–6

E. Novel Contributions

• The contributions are a rigid-panel translation-plus-morphing model, a fourth-order per-user CP model, complementary ALS–ESPRIT and tensor MUSIC estimators, and a shared MDL rank detector.The paper also derives Kruskal conditions, reconstructs channels beyond the training grid, and uses that reconstruction for a multiuser precoder search without pilot cost.Source: E. Novel Contributions · p. 7
• The MDL front end uses the largest-mode unfolding’s eigenvalue spectrum, with the four-way structure supplying a multiplicatively growing effective sample size.The paper argues this supports reliable rank decisions at modest per-mode dimensions.Source: E. Novel Contributions · p. 7
• The paper derives closed-form Kruskal identifiability conditions relating system dimensions to recoverable paths and decouples user separation from spatial estimation.It also positions the work against closest prior art in Table I.Source: E. Novel Contributions · p. 7, Table I · p. 8

System geometry

• The panel translates rigidly over I-by-J planar positions, while each element morphs along the normal across K states on the fast timescale.The paper restricts morphing to a common piston displacement across elements; this keeps the morphing phase separable from the element mode.Source: §II · p. 9

Twin-time-scale training

• The pilots are mutually orthogonal with P ≥ U, enabling matched-filter separation of users before per-user channel estimation.The receiver’s pilot-mode projection reduces the multiuser observation to independent per-user channel tensors.Source: §II · pp. 10, 13

Per-user channel model

• Under a far-field plane-wave model, each path’s steering entry factors into four phases indexed by elements, morphing states, and the two planar-grid axes.Collecting per-user channel vectors across configurations gives a fourth-order tensor that is a sum of at most Lu rank-one tensors.Source: §II · pp. 10–11

Twin-time-scale training

• At each of the KIJ configurations, all users transmit pilots and the receiver collects one N-by-P observation matrix.The received matrices form a five-way tensor, with element, pilot-symbol, morphing, and two planar-grid modes.Source: §II · pp. 12–13

A. Mode-wise PARAFAC Model

• The per-user channel tensor is expressed using four factor matrices and a superdiagonal core whose nonzero entries are the path gains.The core has entries βu,r on its superdiagonal for r = 1, …, Lu.Source: A. Mode-wise PARAFAC Model · pp. 13–14
• The superdiagonal-core form is equivalent to a weighted-factor CP form in which path gains are absorbed into the spatial factor.The paper uses this convention for its mode-wise unfoldings.Source: A. Mode-wise PARAFAC Model · p. 14
• The four mode-wise PARAFAC unfoldings provide the expressions used to estimate factors through alternating updates on each mode.The mode ordering is N, K, I, J.Source: A. Mode-wise PARAFAC Model · p. 14

B. Problem Formulation

• The formulation poses two coupled tasks: acquire per-user channel parameters from the training tensor, then select a panel configuration and transmit precoder.The second task depends on the first task’s solution.Source: B. Problem Formulation · pp. 14–15
• The estimation objective fits each user’s despread observation with a channel tensor generated from path parameters, subject to visible-hemisphere and identifiability constraints.Because the problem is non-convex and the model order is unknown, MDL estimates the order before continuous parameters are recovered.Source: B. Problem Formulation · pp. 14–15
• The configuration-and-precoder design is a mixed discrete-continuous problem; fixing the precoder rule reduces it to exhaustive search over the KIJ grid points.The search evaluates predicted multiuser channels from the estimated user parameters.Source: B. Problem Formulation · p. 15
• The same mechanical motion creates both the training diversity and the feasible configuration set exploited by the precoder.Because the estimates are parametric, the design objective can also be evaluated outside the training grid without pilot cost.Source: B. Problem Formulation · p. 15

III. CHANNEL ESTIMATION PARADIGMS

• Two estimation pipelines share a rank-detection front end: ALS CP decomposition followed by paired ESPRIT, or tensor spatial-smoothing MUSIC.The MUSIC pipeline preserves the multilinear array manifold.Source: §III · pp. 15–16
• The framework outputs per-user parameters from which the channel at any configuration follows; reconstruction and uniqueness conditions complete the estimation framework.The section’s framework first separates users and applies MDL, then uses one of the two estimation pipelines.Source: §III · pp. 15–16

A. Multiuser Separation via Orthogonal Pilots

• The framework separates users by matched-filter projection when pilots satisfy SHS = P IU and P ≥ U, reducing the five-way problem to U independent four-way problems.The projected tensor is an unbiased estimate corrupted by filtered noise.Source: A. Multiuser Separation via Orthogonal… · p. 16

B. Automatic Rank Detection via MDL

• MDL selects the rank by balancing noise-eigenvalue fit against model complexity, with overestimation penalized and underestimation retaining signal eigenvalues in the noise set.Under a rank-r hypothesis, the p − r smallest covariance eigenvalues should be approximately equal, so their geometric-to-arithmetic mean ratio approaches unity.Source: B. Automatic Rank Detection via MDL · p. 17
• The MDL criterion is consistent as the effective sample size grows under mild regularity conditions, but a weak path can still be underestimated when it sinks into the noise floor.The largest-mode unfolding stacks spatial snapshots; its effective sample size is set by KIJ, or by N when N is smaller.Source: B. Automatic Rank Detection via MDL · p. 17

C. Per-User CP Factorization via ALS and Paired ESPRIT

• ALS estimates the per-user channel’s rank-Lu CP factors by alternating least-squares updates over four modes until the reconstruction error converges.Each mode update is an exact linear least-squares minimizer, so the reconstruction cost is non-increasing; convergence of the cost does not guarantee convergence of the iterates.Source: C. Per-User CP Factorization via ALS… · pp. 17–18
• ALS initialization affects which limit point is reached; GEVD initialization and multiple restarts make spurious limit points rare, not absent.Their residual frequency sets the high-SNR error floor reported for the method.Source: C. Per-User CP Factorization via ALS… · p. 19
• Paired ESPRIT recovers spatial frequencies from shift invariance in each Vandermonde-structured factor, then assembles the wavevector components into angles of arrival.Each mode’s spatial step must satisfy |µℓ|δ < π for unambiguous phase recovery.Source: C. Per-User CP Factorization via ALS… · pp. 19–20
• When training uses only one morphing state, the missing wavevector component is closed using the far-field norm constraint, but only for a planar panel and sources in one half-space.For translation modes, unambiguous recovery requires δx, δz < λ/2; for the morphing mode, it requires δy < λ/2.Source: C. Per-User CP Factorization via ALS… · p. 20

D. Tensor Spatial-Smoothing MUSIC

• The 4-D per-user tensor is reshaped into five modes by separating the UPA element dimension into its x- and z-array axes, enabling spatial smoothing along both.Subarray windows slide independently along every mode; each vectorized sub-tensor contributes to the sample covariance.Source: D. Tensor Spatial-Smoothing MUSIC · pp. 21–22
• Multi-mode spatial smoothing creates a large effective snapshot set from overlapping sub-tensors, improving covariance conditioning, but shortened apertures make MUSIC resolution fall behind ESPRIT near the identifiability ceiling.The number of independent sub-tensors grows multiplicatively across modes.Source: D. Tensor Spatial-Smoothing MUSIC · p. 22
• Forward–backward averaging uses the steering manifold’s conjugate symmetry under uniform spacing to improve the covariance rank and effectively double the snapshots.The operation averages the sample covariance with its conjugate-reversed form.Source: D. Tensor Spatial-Smoothing MUSIC · pp. 22–23
• MUSIC separates signal and noise subspaces by eigendecomposing the smoothed covariance, then evaluates a Kronecker steering-vector pseudo-spectrum that preserves the separable G-FIM manifold.The highest local peaks on the two-dimensional angle grid give the AoA estimates.Source: D. Tensor Spatial-Smoothing MUSIC · p. 23
• The angular search grid limits attainable resolution to roughly 180°/G per axis, producing grid-limited behavior.Both pipelines then use the same least-squares gain fit and deliver parameters for channel reconstruction.Source: D. Tensor Spatial-Smoothing MUSIC · p. 23

E. Comparison of the Two Pipelines

• The two estimators coincide below 10 dB and at low path count, separate in favor of ESPRIT near the Kruskal ceiling, and favor MUSIC above 20 dB.At low SNR, both inherit the same MDL model order, which governs error there.Source: E. Comparison of the Two Pipelines · p. 24
• ALS + ESPRIT provides high-resolution estimates when the CP model is well conditioned, but initialization-sensitive spurious limit points bound its mean error at high SNR.GEVD initialization and restarts reduce, but do not eliminate, spurious limit points.Source: E. Comparison of the Two Pipelines · p. 24
• Tensor MUSIC avoids factorization convergence failures, but its resolution depends on subarray dimensions and angular-grid density.Its computation is dominated by G2 steering-vector evaluations and the covariance eigendecomposition.Source: E. Comparison of the Two Pipelines · p. 24
• Tensor MUSIC needs K ≥4 to match ALS on the morphing axis, whereas ESPRIT needs only one shift invariance and can use the norm constraint when appropriate.The smoothed covariance needs a fast mode long enough to smooth over.Source: E. Comparison of the Two Pipelines · p. 24

• Each column of the spatial factor matrix encodes a path’s wavevector.Source: F. Relationship Between Wavevectors… · p. 24
• Under far-field propagation, the wavevector is related to the angle-of-arrival pair.Source: F. Relationship Between Wavevectors… · p. 24
• Elevation is measured from the z-axis, and azimuth from the x-axis in the x-y plane.Source: F. Relationship Between Wavevectors… · p. 25

G. Channel Reconstruction for Arbitrary Configurations

• Estimated factors and path gains reconstruct the channel at configurations beyond the training grid, within the angular support.The steering vectors are evaluated at the new configuration indices using the estimated wavevectors; training overhead covers only the estimation grid.Source: G. Channel Reconstruction for… · p. 25

H. Identifiability of the Twin-Time-Scale CP Model

• With nonzero path gains and steps below λ/2, distinct projections give motion-factor Kruskal ranks min(K, Lu), min(I, Lu), and min(J, Lu).The spatial-factor rank is min(N, Lu) for almost every angle realization under a continuous angular distribution.Source: H. Identifiability of the… · p. 27
• The four-way CP model is essentially unique when its factor Kruskal ranks sum to at least 2Lu + 3.This is a sufficient condition for uniqueness, up to a common column permutation and diagonal scalings whose product is the identity.Source: H. Identifiability of the… · pp. 26–27
• Fast morphing adds a mode to the identifiability budget, so even K = 2 can help recover paths.When Lu exceeds every mode dimension, the condition becomes N + K + I + J ≥ 2Lu + 3; adding a mode of dimension at least two is strictly beneficial.Source: H. Identifiability of the… · p. 28
• Sharing one wavevector projection can reduce a factor rank without destroying identifiability; distinct paths remain identifiable unless two modes collapse.Near-degenerate geometry affects conditioning at finite SNR, rather than exact uniqueness.Source: H. Identifiability of the… · p. 28
• The Kruskal condition is sufficient, not necessary, and its noiseless uniqueness guarantee does not rank finite-SNR estimation accuracy.At the reference geometry, parameter counting permits Lu ≤ 40, compared with the Kruskal value of 9; translation-only can have a higher ceiling yet a larger estimation penalty.Source: H. Identifiability of the… · pp. 28–29
• The constrained parametric model has 4Lu real unknowns per user and can predict channels across configurations from IJKP pilot symbols.An unstructured estimator would need to measure each configuration it intends to use.Source: H. Identifiability of the… · p. 29

I. Training Allocation Under the Identifiability Constraint

• The training budget B = KIJ is allocated across motion modes, while pilot length P is set separately by the orthogonality requirement P ≥ U.Source: §I · p. 29
• For B = 64 at N = 16, allocation (1, 4, 16) nearly doubles the identifiability ceiling of the balanced (4, 4, 4) protocol at the same training-symbol budget.In practical panels, the ceiling grows with the sum of motion dimensions rather than their product; the fully concentrated (64, 1, 1) allocation is excluded.Source: §I · pp. 29–30, Table IV · p. 30
• A higher ceiling need not improve finite-SNR estimation: (2, 8, 4), with ceiling 11, has a smaller estimation penalty than (1, 16, 4), with ceiling 17, above four paths.The paper recommends meeting the required path count with a margin, then balancing modes; every mode should have at least two states.Source: §I · p. 30

IV. MULTIUSER BEAMFORMING WITH CONFIGURATION SELECTION

• The base station can search the K × I × J configuration grid using predicted channels, with no additional pilot overhead.Each candidate configuration requires channel prediction, precoder computation, and SINR evaluation.Source: §IV · p. 31

A. Downlink Signal Model

• The downlink model has U single-antenna users, data-symbol vector d, total power Pt, and unit-norm precoder columns.The sum-rate is the sum over users of log2(1 + SINRu), in bps/Hz.Source: A. Downlink Signal Model · p. 31

B. Linear Precoding Schemes

• The evaluated linear precoders are zero-forcing, MMSE, and maximum-ratio combining.Source: B. Linear Precoding Schemes · pp. 31–32

1) Zero-forcing:

• ZF requires N ≥ U and eliminates inter-user interference, but its channel inversion amplifies noise when the channel is ill-conditioned.Source: 1) Zero-forcing · p. 32

2) Regularized zero-forcing:

• The MMSE TPC regularizes ZF to trade residual interference for reduced noise amplification, and it becomes ZF as SNR increases.The regularization weight is U/SNR; the two TPCs coincide numerically through U = N/2, where the ZF inversion first becomes ill-conditioned.Source: 2) Regularized zero-forcing · p. 32

3) Matched filtering:

• MRC maximizes desired-signal power by matched filtering but does not suppress inter-user interference, so its sum-rate saturates as U or SNR increases.It is optimal in the single-user or noise-dominated regime.Source: 3) Matched filtering · p. 32

C. Channel Prediction and Configuration Selection

• Estimated per-user parameters predict each channel at arbitrary panel configurations, enabling exhaustive configuration search without additional pilot overhead.The search evaluates KIJ configurations while training remains IJKP symbols, regardless of the searched grid size.Source: C. Channel Prediction and… · p. 33

D. Fixed-Panel Baseline

• The fixed-panel baseline holds the array at the reference position and uses the same aperture and training frame as the reconfigurable curves.It spends the training frame on the fixed channel for noise averaging and uses the resulting channel directly for precoding, with no configuration selection.Source: D. Fixed-Panel Baseline · pp. 33–34

E. Complexity Analysis

• Configuration search costs O[KIJ · N(UL + U^2)] per coherence block and is under 1% of the measured cost, so estimation dominates.Search evaluates each grid configuration; estimation costs are reported separately for ALS + ESPRIT and tensor MUSIC.Source: E. Complexity Analysis · p. 34

V. NUMERICAL RESULTS

• The evaluation averages 1000 independent trials, reports training-grid NMSE, extrapolation NMSE, AoA RMSE and achieved sum-rate, and compares estimated CSI with genie and fixed-panel references.Extrapolation NMSE evaluates a position half a step beyond the training grid; achieved sum-rate uses the true channel at the configuration selected from the estimate.Source: §V · p. 35

A. Channel Estimation

• The MDL variants fall 6 to 8 dB short of the genie-rank bound below 0 dB, while ALS flattens near −45 dB above 20 dB because of occasional spurious stationary points.At Lu = 9 and 20 dB, ALS retains −35.6 dB while tensor MUSIC saturates at −27.0 dB; the sufficient Kruskal uniqueness bound is not an accuracy cliff.Source: A. Channel Estimation · pp. 35, 36, 39, Figure 6 · p. 36
• MDL never overestimates the true order Lu = 3, and correct detection rises from 0 at −10 dB to 0.63 at 0 dB and 0.96 at 10 dB.This detection transition explains why MDL and genie curves are indistinguishable above 10 dB.Source: A. Channel Estimation · pp. 36–37, Figure 7 · p. 37
• Increasing morphing states from K = 1 to K = 8 improves both pipelines by roughly 10 dB, while raising the Kruskal bound from Lu ≤ 6 to Lu ≤ 13.The improvement includes noise averaging as the training frame grows, as well as an added tensor mode that improves factor-matrix conditioning; tensor MUSIC catches up at K = 4.Source: A. Channel Estimation · p. 37, Figure 8 · p. 38
• At ∆y = 0.5λ, ALS prediction NMSE degrades from −31.4 to −26.7 dB and AoA RMSE rises from 0.30° to 0.95°, although fitting NMSE is unaffected.The ESPRIT phase limit is δy < λ/2; a step slightly below half a wavelength is recommended for both pipelines.Source: A. Channel Estimation · p. 38, Figure 9 · p. 39

B. TPC Design

• Above −5 dB, estimated-CSI and genie sum-rate curves are indistinguishable to within 0.01 bps/Hz, while MRC saturates at 17.1 bps/Hz as ZF continues rising.The predicted-rate bias never exceeds 0.06 bps/Hz above −5 dB; the predicted rate is not plotted because the same estimate selects and scores the configuration.Source: B. TPC Design · p. 40, Figure 11 · p. 41
• Reconfiguration gain rises from 0.52 bps/Hz at U = 2 to 6.22 at U = 16, with identical training budgets for the compared motion protocols.The fixed panel degrades fastest as load grows; poorly separated user pairs become more frequent, while the G-FIM can select a better-separated configuration.Source: B. TPC Design · pp. 39, 42, 43, Figure 12 · p. 42

1) Contribution of each motion mechanism:

• With KIJ = 64 candidates, the reference protocol uses both translation and morphing, allowing the comparison to isolate the kind of diversity rather than candidate count.The reference setting uses K = I = J = 4.Source: 1) Contribution of each motion mechanism · p. 43
• Under perfect CSI, translation alone was best at every path count, but by less than 0.2 bps/Hz; the half-space geometry makes morphing redundant for beamforming separation.The normal-direction component is determined by the in-plane components.Source: 1) Contribution of each motion mechanism · pp. 43–44
• At Lu = 8, translation alone incurred a 1.95 bps/Hz estimation penalty, versus 0.15 bps/Hz for the combined protocol, despite translation alone having the higher Kruskal ceiling.The combined protocol continues rising to 55.17 bps/Hz as the translation-only curve falls from 54.82 bps/Hz at Lu = 4 to 53.52 at Lu = 8.Source: 1) Contribution of each motion mechanism · p. 44
• Morphing alone is ill-posed for estimation: its rate fell from 47.5 bps/Hz at Lu = 1 to 5.3 bps/Hz at Lu = 8, although its genie stayed above 53.With I = J = 1, the tensor collapses to an N × K matrix that Eq. (46) cannot certify for any Lu.Source: 1) Contribution of each motion mechanism · p. 44

2) Cost of the configuration search:

• The exhaustive configuration search took 0.006 s versus 0.504 s at U = 2 and 0.013 s versus 3.60 s at U = 16, staying below 1% of per-coherence-block cost.The search is dominated by the estimator that precedes it.Source: 2) Cost of the configuration search · p. 45

VI. CONCLUSION

• Estimated CSI matched perfect CSI within 0.01 bps/Hz above −5 dB, while the gain over a fixed panel grew from 0.52 bps/Hz at two users to 6.22 bps/Hz at sixteen.The comparison uses identical aperture and training budget.Source: §VI · p. 45
• Fast morphing conditions the factorization rather than adding beamforming gain, and finite-SNR conditioning can favor a protocol with a lower Kruskal ceiling.Near-field propagation and continuous panel-displacement parametrization are identified as next steps.Source: §VI · pp. 45–46