TRACE: A Trilinear Channel Estimation Framework for Tri-Hybrid Architectures with Reconfigurable Antennas

I. INTRODUCTION

• Tri-hybrid beamforming selects radiation patterns in the EM domain, combines signals in RF, and processes them digitally in baseband.Accurate channel acquisition is needed to realize the stated gains of tri-hybrid architectures.Source: §I · p. 1
• Earlier reconfigurable-array estimators treat EM, spatial, and fading components sequentially rather than exploiting their multilinear coupling.Prior approaches include dictionary-based and parametric estimators; tensor decompositions have been used for reconfigurable surfaces in other settings.Source: §I · p. 1
• TRACE models received pilots as a third-order PARAFAC tensor and fits it by ALS before recovering physical parameters.The paper also translates Kruskal uniqueness into pilot, RF, and EM probing design rules and validates TRACE against a greedy dictionary baseline and an oracle-support bound.Source: §I · p. 1

II. SYSTEM AND TENSOR MODEL

• The single-user uplink uses Nr pattern antennas, NRF RF chains, P pilot slots, and T EM probing periods, with RF-chain observations retained before digital combining.The RF weights are commanded phase-shifter weights, whereas FEM characterizes the array offline.Source: §II · pp. 1–2
• The base station cycles the same T reconfiguration states in each slot, so EM training acts in a structured way on the angular mode.All antennas share the same pattern in each probing period.Source: §II · p. 2
• The received pilot tensor has RF-projected steering, EM angular, and pilot-weighted fading factors, with one rank-one term per propagation path.The tensor rank equals the number of resolvable paths, and pilot gains can be recovered by dividing by the known pilot symbols.Source: §II · p. 2
• The independent uniform-phase EM-state model is a statistical surrogate for diverse states, not a physical-realizability claim; neighboring angular samples are correlated.Low mutual coherence among FEM columns motivates the surrogate for support detection.Source: §II · pp. 2–3

A. Tensor factor estimation

• ALS updates each factor by an exact conditional solution because each tensor unfolding is linear in one factor.The update uses the corresponding unfolding of the noise tensor.Source: A. Tensor factor estimation · p. 3
• TRACE initializes factors from dominant left singular vectors, then retains the lowest-residual fit across five starts.The reported settings are tolerance 10^-8 and at most 200 iterations.Source: A. Tensor factor estimation · p. 3
• ALS converges only linearly, while Gauss–Newton is a standard remedy for ill-conditioned fits; both solvers feed the same recovery stage.ALS was adopted because it is exact per block and was indistinguishable from Gauss–Newton in the simulated scenarios.Source: A. Tensor factor estimation · p. 3

B. Recovery of the physical parameters

• EM indices are detected by matching estimated EM-factor columns to the known pattern dictionary, while AoAs are matched to RF-projected steering vectors over an angular grid.Each search is performed per component, using the EM dictionary’s M columns or the angular grid Θ.Source: B. Recovery of the physical parameters · p. 3
• Each PARAFAC component pairs an EM index with the AoA of the same path, and the two recoveries avoid a joint combinatorial search.The common permutation ambiguity only relabels paths; EM support detection and AoA matching are separable across components.Source: B. Recovery of the physical parameters · p. 3
• The detected supports are projected through the known training matrices to form structured factors, which are used to re-estimate the third factor by least squares.This re-referencing to the true training matrices removes the scaling ambiguity.Source: B. Recovery of the physical parameters · p. 3
• When pilot symbols are nonzero, each row of the re-estimated matrix is divided by its corresponding pilot symbol to recover the path-gain matrix without combinatorial assignment.The closed-form recovery is Ĉ = diag(s)^−1 B̂.Source: B. Recovery of the physical parameters · p. 3

IV. IDENTIFIABILITY AND COMPLEXITY

• In the compressive configuration, the greedy dictionary baseline is more accurate below about 7 dB but reaches a grid-induced floor, while TRACE keeps the oracle-support bound’s SNR slope.The comparison is shown for Nr = 8, NRF = 3, M = 24, T = 10, P = 6, and L = 4.Source: Figure 2 · p. 4
• Under well-conditioned codebooks, distinct AoAs, nonrepeated projected EM signatures, and sufficiently diverse fading, Kruskal’s condition is min{T, L} + min{NRF, L} + min{P, L} ≥ 2L + 2.The PARAFAC model is identifiable up to permutation and scaling ambiguities, which do not affect its reconstructed tensor.Source: §IV · pp. 3–4
• Full-column-rank factor updates generically require PT ≥ L, PNRF ≥ L, and TNRF ≥ L, exposing trade-offs among pilot slots, EM probing states, and RF chains.The steering, EM, and fading updates depend on PT, PNRF, and TNRF, respectively.Source: §IV · p. 4
• Although identifiability may hold at P = L and T = L, choosing P > L and T > L improves factor-estimate conditioning and stability.The paper reports these overdetermined dimensions as improving conditioning and stability in its simulations.Source: §IV · p. 4
• When M/T is an integer, subsampling an M-point DFT can make columns m and m + T of F_EM identical, collapsing k_H to 1 and violating the uniqueness condition.Randomized probing avoids this structure; deterministic codebooks are safe when T ≥ M.Source: §IV · p. 4
• Each ALS iteration scales as O(PTNRF L + L^3), while the stated EM-support, beamspace-search, and pilot-division recovery costs are lower order.The recovery costs are O(LMT), O(LGNRF), and O(PL), respectively.Source: §IV · p. 4

Evaluation setup

• Curves report median NMSE over 200 realizations because low-SNR non-convex fits can terminate at stationary points with NMSE > 1; oracle-support LS estimates gains with true supports.Source: §V · pp. 4–5

Fig. 2(a)

• In the non-compressive setting, increasing T from 16 to 24 improves NMSE uniformly by about a factor of 1.6, while P>L improves conditioning and robustness.Source: §V · p. 5

Fig. 2(b)

• At medium-to-high SNR, TRACE tracks the oracle-support LS bound’s SNR slope within a factor of 1.3 to 1.4 and beats SOMP by almost an order of magnitude at 20 dB.The residual offset does not grow with SNR.Source: §V · p. 5
• At 1 dB, SOMP is more accurate by a factor of 2.8, but above the crossover near 7 dB its dictionary constraint produces a high-SNR floor while TRACE has none.SOMP improves by only 1.25 per 3 dB at high SNR, versus the ideal factor of 2 for TRACE and the oracle.Source: §V · p. 5

Fig. 2(c)

• At 10 dB, T=4 meets the sufficiency bound with zero margin, while doubling to T=8 eliminates failures; benefits diminish with further probing.SOMP’s support detection cannot separate the EM columns at T=4, where mutual coherence is 0.985; it falls to 0.52 at T=24.Source: §V · p. 5

• TRACE models received pilots as a PARAFAC tensor that preserves coupling among the EM angular response, RF-projected steering matrix, and pilot-weighted path gains.Source: §VI · p. 5
• The two-stage estimator jointly fits the three factors and maps them back to physical channel parameters in closed form, with identifiability translated into training design rules.Source: §VI · p. 5
• Simulations show TRACE retains the oracle-support bound’s SNR slope, while dictionary-based greedy estimation saturates at a grid-set floor despite better low-SNR accuracy.Source: §VI · p. 5
• Extending TRACE to the multi-user uplink is identified as a future direction.Source: §VI · p. 5