Channel Estimation for Reconfigurable Intelligent Surface Assisted Upper Mid-Band MIMO Systems

The upper mid-band (UMB) spectrum is a key enabler for 6G systems, yet reconfigurable intelligent surface (RIS)-assisted UMB communications face severe channel estimation challenges due to near-field propagation and transitional scattering, which induce strong spatial correlation and ill-conditioned least-squares (LS) formulations. To overcome this limitation, we propose a conditioning-aware channel estimation framework that transforms the inherently ill-conditioned high-dimensional problem into multiple well-conditioned subproblems via greedy column grouping. By systematically separating highly correlated RIS elements into distinct sub-blocks via piecewise RIS phase design, the proposed method directly improves Gram matrix conditioning and stabilizes piecewise LS reconstruction without relying on sparsity assumptions. Simulation results demonstrate that the proposed method significantly outperforms conventional LS and OMP-based estimators in pilot-limited and transitional UMB regimes, achieving robust performance with low computational complexity.

💡 Research Summary

The paper addresses the channel estimation problem in reconfigurable intelligent surface (RIS)‑assisted multiple‑input multiple‑output (MIMO) systems operating in the upper‑mid‑band (UMB) spectrum (7–24 GHz), a frequency range that is expected to play a pivotal role in 6G deployments. In this band, the physical size of a RIS becomes comparable to the wavelength, causing two critical effects: (i) near‑field propagation with spherical wavefronts, and (ii) a transitional scattering environment where the number of dominant scatterers is moderate rather than extremely sparse or fully rich. Both effects lead to strong spatial correlation among RIS elements, which in turn makes the Gram matrix of the sensing matrix ill‑conditioned. When the conventional two‑timescale channel estimation (2TCE) framework is applied, the least‑squares (LS) inversion amplifies noise dramatically, especially when the number of pilot symbols is limited (the minimal pilot length is T = ⌈M/N⌉). Moreover, the matrix inversion complexity scales as O(M³), which is prohibitive for large‑scale RIS (M in the hundreds or thousands).

To overcome these limitations, the authors propose a “conditioning‑aware” channel estimation framework that (1) partitions the RIS into Q disjoint groups, (2) designs piecewise RIS phase patterns that orthogonalize the contributions of each group across Q time slots within a sub‑frame, and (3) applies a greedy column‑grouping algorithm that deliberately separates highly correlated RIS elements into different groups. The piecewise phase design uses a unitary Hadamard matrix Φ for inter‑group orthogonalization and a per‑sub‑frame Hadamard vector v_b for intra‑group modulation. After stacking the Q observations of a sub‑frame and right‑multiplying by Φᴴ, the received signal decomposes into Q independent sub‑problems:

 z_{b,q} = F_q diag(v_b) h_q + ñ_{b,q},

where F_q contains the columns of the RIS‑BS channel corresponding to group q. For each group, an LS estimator

 ĥ_q = (∑{b=1}^{B} F̂{b,q}ᴴ F̂_{b,q})^{-1} ∑{b=1}^{B} F̂{b,q}ᴴ z_{b,q}

is computed. The stability of this estimator depends on the condition number η(G_q) of the group‑wise Gram matrix G_q = ∑{b} F̂{b,q}ᴴ F̂_{b,q}.

The central optimization problem (P₀) seeks an equal‑cardinality partition that minimizes the worst‑case condition number across all groups. Because P₀ is combinatorial, the authors introduce a surrogate (P₁) that minimizes the maximum intra‑group sum of pairwise column correlations w_{i,j} = |f_iᴴ f_j|/(‖f_i‖‖f_j‖). They then propose a low‑complexity greedy algorithm:

1. Seed initialization – compute all w_{i,j}, select the Q/2 largest non‑overlapping pairs, and assign each element of the selected pairs to a distinct group, guaranteeing that the most strongly correlated columns start in different groups.
2. Sequential assignment – for each remaining column c, compute its average correlation with the members of each group, choose the group that yields the smallest average increase, and append c to that group.

The greedy procedure requires O(N M²) operations for the correlation matrix (N is the number of BS antennas) and is performed only once per RIS‑BS coherence interval, as it depends solely on the quasi‑static channel estimate ˆF. After grouping, the LS inversions are carried out on Q matrices of size M′ = M/Q, reducing the overall computational load from O(M³) to O(M³/Q²).

Simulation results (N = 64, M = 256, Q = 4) demonstrate that the proposed method achieves a substantial NMSE improvement over the conventional LS‑based 2TCE and over OMP‑based compressed‑sensing estimators, especially in pilot‑limited regimes (T = 2–3) and under transitional scattering conditions. The condition numbers of the group‑wise Gram matrices are reduced by an order of magnitude, directly explaining the observed robustness. Complexity analysis confirms that the runtime is dominated by the LS inversions of the smaller sub‑problems, while the offline grouping cost is negligible in practice.

The paper’s contributions are threefold: (i) identification of the conditioning bottleneck specific to RIS‑assisted UMB channels, (ii) a novel piecewise phase design combined with a correlation‑aware greedy grouping that transforms an ill‑conditioned high‑dimensional LS problem into several well‑conditioned low‑dimensional ones, and (iii) a thorough complexity‑performance trade‑off analysis showing scalability to large RIS arrays. Limitations include the assumption of a single‑user scenario, perfect knowledge of the quasi‑static RIS‑BS channel, and static grouping during the RIS‑BS coherence time. Future work could extend the framework to multi‑user settings, adaptive regrouping under fast channel variations, and experimental validation on hardware prototypes. Overall, the study provides a practical and theoretically grounded solution for reliable channel estimation in RIS‑enhanced upper‑mid‑band 6G systems.