PaperlayerView exact revision on arXiv

Source-linked AI summary

Multiuser Channel Estimation in FIM Systems: A Structured Block Term Decomposition Approach

Xi Han, Junqiao Chen, André L. F. de Almeida, George C. Alexandropoulos

arXiv:2610.30000v1eess.SP

TL;DR

Flexible intelligent metasurfaces provide geometric spatial diversity, but their configuration-dependent array responses complicate multiuser channel estimation. The paper develops a pilot-constrained structured BTD estimator that preserves known pilot directions and diagonal spatial factors. Simulations demonstrate angle and channel recovery in tested settings, with equal total pilot-symbol overhead in rigid-panel comparisons.

  • Problem

    Multiuser channel estimation across common calibrated FIM morphing configurations remains largely unexplored, although each user's distinct spatial direction and the changing geometry complicate acquisition.

    Source: §I · pp. 1–2

  • Method

    The estimator models multi-configuration observations as user-specific structured BTD blocks and uses structured updates that preserve known pilot directions and diagonal spatial factors.

    Source: §I · p. 2

  • Results

    Simulations demonstrate two-dimensional angle estimation and channel reconstruction in tested user settings, with the rigid-panel comparison using the same total pilot-symbol overhead.

    Source: §I · p. 2

  • Takeaways & Limitations

    In the tested settings, estimation accuracy improves with morphing diversity and degrades with increased user loading or highly correlated pilots.

    Source: §I · p. 2

    As pilot correlation approaches unity, least-squares updates become ill-conditioned and amplify noise even while the pilot matrix remains full row rank for correlation below one.

    Source: D. Robustness to Pilot Correlation · p. 11

Abstract

from arXiv · show

Flexible intelligent metasurfaces (FIMs) create geometry-dependent array responses that provide spatial diversity but complicate multiuser channel estimation. This paper studies uplink estimation using pilot observations collected under multiple prescribed and calibrated morphing configurations. Each user contributes a matrix--vector block that couples its diagonal reference-array response with its morphing-dependent phases, while its third-mode vector is the known pilot scaled by the complex channel gain. We develop a pilot-constrained structured block term decomposition (BTD) estimator whose alternating least-squares updates exploit both the known pilot directions and the diagonal spatial factors. The undeformed configuration sets the columnwise scaling convention, and a unit first-entry normalization assigns the remaining block scale to the channel gain. Matching the recovered factors to the calibrated spatial--morphing manifold then yields the physical angles and configuration-dependent channels. We characterize the rank conditions for unique conditional least-squares updates, distinguish these conditions from global factor and physical-angle identifiability, and summarize the training, coherence-time, and computational requirements. Simulations show accurate two-dimensional angle and channel recovery in the tested settings, equal total pilot-symbol overhead in rigid-panel comparisons, improved steering-vector estimates as morphing diversity increases, and degradation as user loading or pilot correlation increases.

I. INTRODUCTION

FIM morphing provides geometric spatial diversity but makes the array response configuration-dependent, complicating multiuser channel estimation. The paper proposes a pilot-constrained structured BTD approach and evaluates recovery under the stated training model.

I. INTRODUCTION

  • FIM element displacements change the spatial manifold across morphing states, adding geometric diversity but complicating channel acquisition.Prior work addressed several FIM channel-estimation approaches, while separating multiple users under common calibrated morphing configurations remained largely unexplored.Source: §I · p. 1
  • The proposed estimator models observations as user-specific structured BTD blocks and preserves known pilot directions and diagonal spatial factors during recovery.It uses the undeformed configuration to fix scaling, then matches recovered factors to the calibrated FIM manifold to estimate angles and reconstruct configuration-dependent channels.Source: §I · p. 2
  • The paper distinguishes conditional least-squares update rank conditions from global factor uniqueness and physical-angle identifiability.It also discusses pilot conditioning, morphing diversity, coherence-time requirements, and computational and memory costs.Source: §I · p. 2
  • Simulations demonstrate two-dimensional angle estimation and channel reconstruction, with the rigid-panel comparison using the same total pilot-symbol overhead.Estimation accuracy varies with morphing diversity and user loading, while highly correlated pilots degrade performance through ill-conditioning.Source: §I · p. 2

A. FIM Geometry and Uplink Training Model

  • Training assumes unchanged physical channel parameters across all M configurations, with the pilot observations acquired within the channel coherence time.Each FIM element has a calibrated sensing and readout chain; analog combining and compressed element-domain readout are outside the stated scope.Source: A. FIM Geometry and Uplink Training… · pp. 3–4

B. Tensor Observation for Angularly Separated Users

The observations from the morphing configurations are stacked into a third-order tensor whose modes represent elements, pilot symbols, and configurations. Angularly separated users have distinct spatial–morphing contributions, which the structured formulation processes jointly rather than estimating each configuration independently.

B. Tensor Observation for Angularly Separated Users

  • The observation tensor stacks the received pilot matrices across morphing configurations, with modes for FIM elements, pilot symbols, and configurations.The pilot and morphing modes are interchanged for the structured formulation without changing the measurements.Source: B. Tensor Observation for Angularly… · p. 4
  • Angularly separated users have distinct reference-array and morphing responses, yielding K user-specific spatial–morphing contributions.The physical parameters to recover are the users’ elevation and azimuth angles, and the objective also includes their gains and configuration-dependent channels.Source: B. Tensor Observation for Angularly… · p. 4
  • When the pilot matrix has full row rank, configuration-wise least squares can estimate each channel separately but does not exploit shared geometric structure.The structured FIM-BTD formulation jointly processes the configurations while preserving known pilot directions and user-specific spatial–morphing structure.Source: B. Tensor Observation for Angularly… · p. 4

III. STRUCTURED FIM-BTD FORMULATION

The structured FIM-BTD represents the observation as a sum of user-specific blocks, each coupling spatial and morphing responses with its pilot. Its constraints encode the known pilots and diagonal spatial factors in the decomposition.

III. STRUCTURED FIM-BTD FORMULATION

  • Each angularly separated user has a distinct reference-array response and morphing-dependent phase profile, so the observation is a sum of K blocks.The FIM-element and morphing modes are coupled through a user-specific matrix.Source: §III · p. 5

A. User-Specific Block-Term Representation

  • The user-specific block uses a diagonal spatial factor constructed from that user’s undeformed reference-array response.The block also includes the user’s morphing responses over the prescribed configurations.Source: A. User-Specific Block-Term… · p. 5
  • During factor recovery, the morphing factor is treated as unknown, and its angle dependence is enforced by matching it to the calibrated FIM manifold.This matching connects the recovered factor to the physical angles.Source: A. User-Specific Block-Term… · p. 5
  • Stacking the user’s contributions over pilot symbols forms its tensor block as an outer product between a matrix and a vector.The outer product assigns each matrix entry to the corresponding vector component in the tensor block.Source: A. User-Specific Block-Term… · p. 5
  • The complete observation tensor is expressed through the collection of user-specific blocks.The representation retains each user’s coupling across the FIM-element, morphing, and pilot dimensions.Source: A. User-Specific Block-Term… · p. 5, §III · p. 5
  • The structured model fixes each block’s third-mode direction with its known pilot and constrains its first-mode factor to be diagonal.The diagonal factor has N free spatial coefficients, embedding user-to-pilot association and the elementwise FIM response in the factorization.Source: A. User-Specific Block-Term… · p. 5

B. Block Multilinear Rank

The section characterizes the tensor blocks’ multilinear ranks and develops complementary matrix representations that expose the model’s structure. The diagonal spatial constraint reduces the number of spatial unknowns.

B. Block Multilinear Rank

  • Under the ideal steering model, nonzero diagonal spatial coefficients preserve the morphing factor’s rank; rank-deficient morphing responses yield block rank (rE,k, rE,k, 1).The same reduced rank applies when M < N.Source: B. Block Multilinear Rank · p. 5
  • The multilinear rank rankML is the tuple of matrix ranks of a tensor block’s three mode unfoldings.The commonly used rank-(N, N, 1) description is not valid in every case.Source: B. Block Multilinear Rank · p. 5

C. Structured Matrix Representations

  • The matrix representations provide linear models for estimating morphing factors, diagonal spatial factors, and user gains, using KN spatial coefficients instead of KN^2 unconstrained entries.The diagonal constraint means only the N diagonal coefficients of each user’s spatial factor need estimation.Source: C. Structured Matrix Representations · pp. 5–6

IV. PROPOSED STRUCTURED FIM-BTD ESTIMATOR

The estimator alternates structured least-squares updates while preserving known pilot directions and diagonal spatial factors. Reference-state normalization fixes factor scaling and assigns the remaining block scale to the gain.

A. Alternating Structured Factor Updates

  • The alternating updates preserve known pilot directions and enforce diagonal spatial factors at every factor-recovery step.The spatial update estimates KN diagonal coefficients rather than the KN^2 entries of an unconstrained block model.Source: §IV · p. 6, A. Alternating Structured Factor Updates · p. 6
  • The morphing and spatial factors are updated in sequence using structured regressions, with the spatial update estimating only diagonal coefficients.The morphing update uses the current estimates; the spatial regression then uses the updated morphing factor while retaining the previous gain estimate.Source: A. Alternating Structured Factor Updates · p. 6

B. Reference-State Normalization

  • The undeformed reference configuration fixes the columnwise scaling between spatial and morphing factors, and unit first-entry normalization assigns the remaining block amplitude and phase to the channel gain.Normalization requires nonzero entries in the estimated reference row.Source: B. Reference-State Normalization · p. 6

C. Direct User-Gain Update

The estimator updates user gains through a linear model that retains the known pilots, keeping estimated blocks associated with their users. It initializes from pilot-domain least-squares estimates and retains the lowest-residual iterate.

C. Direct User-Gain Update

  • The gain update keeps known pilot vectors embedded in its model, so recovered blocks remain associated with their users without a separate block-permutation matching stage.The gain model follows by vectorizing the mode-3 representation.Source: C. Direct User-Gain Update · p. 6

D. Residual Criterion and Initialization

  • Initialization uses pilot-domain least-squares channel estimates, with the undeformed reference configuration initializing user-specific responses and gains and other estimates initializing morphing factors.The undeformed state supplies the reference-array responses; pilot-separated estimates from the remaining configurations initialize the morphing factors.Source: D. Residual Criterion and Initialization · p. 7
  • Because the alternating optimization is nonconvex and global-optimum convergence is not guaranteed, the estimator retains the factor set with the smallest encountered reconstruction residual.Iterations stop at the prescribed maximum or when the stopping criterion is met.Source: D. Residual Criterion and Initialization · p. 7

E. Physical Parameter and Channel Recovery

After factor recovery, the estimator matches each user’s spatial and morphing factors to the calibrated FIM manifold to estimate angles, then reconstructs configuration-dependent channels.

E. Physical Parameter and Channel Recovery

  • Each recovered block is already associated with its user because the known pilot directions are preserved throughout factorization.No additional block-permutation matching is required.Source: E. Physical Parameter and Channel… · p. 7
  • The estimator extracts each user’s angular parameters by jointly matching recovered spatial and morphing factors to the calibrated FIM manifold.The morphing-factor mismatch is weighted relative to the spatial steering-vector mismatch, over the admissible angular region.Source: E. Physical Parameter and Channel… · p. 7
  • It reconstructs each user’s channel separately for each morphing configuration after estimating the angles.The reconstruction is configuration-dependent.Source: E. Physical Parameter and Channel… · p. 7

V. IDENTIFIABILITY AND COMPUTATIONAL ANALYSIS

The section separates uniqueness of conditional least-squares updates from global factorization and physical-angle identifiability. It also specifies how rank, manifold, and user-separation conditions constrain recovery.

A. Algebraic Well-Posedness and Physical Identifiability

  • Conditional least-squares updates are unique when their design matrices have full column rank; positive Tikhonov regularization can make an update unique despite rank deficiency.Source: A. Algebraic Well-Posedness and… · pp. 7–8
  • The dimensional bound MT ≥ K is necessary but not sufficient for the relevant design-matrix rank conditions.Source: A. Algebraic Well-Posedness and… · p. 8
  • Unique conditional updates do not establish global uniqueness of the nonconvex factorization or guarantee convergence to its global optimum.Source: A. Algebraic Well-Posedness and… · p. 8
  • The diagonal spatial factor removes within-block transformation ambiguity, the undeformed reference fixes columnwise scaling, and the known pilots associate blocks with users when design matrices have full column rank.The remaining block amplitude and phase are assigned to each user's gain.Source: A. Algebraic Well-Posedness and… · p. 8
  • A strict rank-(N, N, 1) user block requires M ≥ N and rank(Ek) = N; otherwise the structured representation remains valid with reduced block rank.Source: A. Algebraic Well-Posedness and… · p. 8
  • Physical-angle identification requires an injective spatial–morphing mapping, while stable multiuser recovery also requires sufficiently distinct user-specific responses.Source: A. Algebraic Well-Posedness and… · p. 8

B. Pilot and Morphing Requirements

Pilot and morphing dimensions provide complementary forms of user and geometric diversity, subject to rank and coherence-time requirements. The structured estimator's spatial-factor regression dominates its computation, while configuration-wise least squares is cheaper but ignores cross-configuration coupling.

B. Pilot and Morphing Requirements

  • Pilot-domain separation distinguishes user contributions, while morphing diversity supports recovery of configuration-dependent FIM responses; the undeformed reference fixes scaling and nonzero out-of-plane displacement adds y-axis sensitivity.Source: B. Pilot and Morphing Requirements · p. 8
  • A strict rank-(N, N, 1) characterization requires the additional condition (80), but lower-rank user blocks remain valid structured BTD components.Source: B. Pilot and Morphing Requirements · pp. 8–9
  • Training produces NMT complex observations; increasing T primarily improves pilot-domain separation and noise averaging, increasing M adds morphing diversity, and acquisition must fit within one coherence interval.Source: B. Pilot and Morphing Requirements · p. 9

C. Computational Complexity

  • The structured spatial-factor regression dominates the estimator's arithmetic cost, whereas configuration-wise LS is substantially cheaper but does not exploit geometric coupling across morphing states.The structured method's spatial update is the dominant operation; matrix-free Kronecker and Khatri–Rao products can avoid explicitly constructing its regression matrix.Source: C. Computational Complexity · p. 9

VI. SIMULATION RESULTS AND DISCUSSION

The supplied simulation passages describe the evaluation setup, metrics, and comparison protocol. They identify the quantities plotted and the equal-overhead baseline, but do not state the plotted outcomes.

A. Simulation Setup

  • The default angularly separated-user experiments use DFT pilots with SSH = TIK, K = 3, T = 12, and M = 16, with element displacements restricted to the y-axis.Source: A. Simulation Setup · p. 9
  • Fig. 2 plots final reconstructed-channel NMSE and joint angular RMSE versus SNR for an angularly separated-user case with N = 16, K = 3, T = 12, and M = 16.Source: Figure 2 · p. 10
  • Rigid-panel LS comparisons match total pilot-symbol overhead by using Trigid = MT; other reference estimators ignore morphing-dependent phase variation or impose a common steering response across users.Source: A. Simulation Setup · p. 10

B. Estimation Performance versus SNR

The paper’s simulations demonstrate two-dimensional angle and channel recovery in tested settings, with accuracy varying with SNR, morphing diversity, pilot correlation, and user loading. The structured estimator exploits user-specific spatial–morphing structure at higher computational cost than LS-based baselines.

B. Estimation Performance versus SNR

  • As SNR increases, the proposed estimator’s channel NMSE and angular RMSE decrease, while baselines have larger errors, particularly at moderate and high SNR.The rigid-panel comparison uses the same total pilot-symbol overhead; calibrated out-of-plane morphing adds sensitivity to sin θ sin ϕ.Source: B. Estimation Performance versus SNR · p. 10

C. Impact of Morphing Diversity

  • The proposed estimator achieves the lowest steering-vector NMSE over the considered morphing-configuration range, with accuracy improving as M increases.The improvement reflects both more observations and greater geometric diversity; the equal-overhead rigid-panel baseline gains pilot duration but retains one fixed geometry.Source: Figure 3 · p. 10, C. Impact of Morphing Diversity · pp. 10–11

D. Robustness to Pilot Correlation

  • Both angular RMSE and reconstructed-channel NMSE increase gradually with pilot correlation ρs and deteriorate rapidly when pilots are highly correlated.As ρs approaches unity, the pilot Gram-matrix condition number grows rapidly; full row rank alone does not ensure numerical stability.Source: Figure 4 · p. 11, D. Robustness to Pilot Correlation · p. 11
  • Steering-vector estimation error generally increases with K, but the proposed estimator maintains the lowest NMSE over the considered user-loading range.More user-specific spatial–morphing components must be recovered from the same pilot and morphing dimensions, while total-power normalization reduces average gain power per user.Source: Figure 5 · p. 11, D. Robustness to Pilot Correlation · p. 11
  • The structured FIM-BTD estimator requires more processing than LS-based baselines, whose noniterative matrix operations remain computationally inexpensive.Its runtime decreases as M is reduced because fewer observations enter the factor updates; the tradeoff is processing cost versus accuracy gains from exploiting spatial–morphing structure.Source: D. Robustness to Pilot Correlation · pp. 11–12, Figure 6 · p. 12

VII. CONCLUSION

  • The estimator recovers channels and two-dimensional angles in tested settings, while steering-vector error increases with user loading or pilot correlation.The conclusion also identifies multipath, wideband channels, imperfect morphing calibration, and compressed or partially observed measurements as future work.Source: §VII · p. 12
Loading 2610.30000v1…