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Semi-Blind Channel Estimation for Dynamic NTN Systems via Spiked Random Matrix Theory

Xue Zhang, Abla Kammoun, Mohamed-Slim Alouini

arXiv:2608.25694v1eess.SP

TL;DR

Dynamic NTN channels and high-dimensional systems make covariance-based channel estimation unreliable because sampling noise degrades acquisition. The paper combines training and blind subspace information in an RMT-optimized regularized estimator, whose asymptotic MSE characterization supports parameter selection and whose performance is validated in realistic 3GPP NTN simulations.

  • Problem

    Dynamic channels and high-dimensional systems make reliable channel acquisition difficult, while conventional covariance-based estimators suffer degradation from sampling noise.

  • Method

    The paper minimizes a regularized least-squares criterion combining training-based information with blind subspace structure and uses spiked covariance RMT to optimize regularization.

  • Results

    The proposed estimator’s asymptotic expressions are validated in realistic 3GPP NTN simulations, where its minimum MSE closely approaches the deterministic CRB at λ=0.13 versus theoretical λ̂*=0.1350.

  • Takeaways & Limitations

    The resulting estimator is computationally efficient and suited to high-dimensional dynamic NTN channel estimation.

Abstract

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Semi-blind channel estimation offers an attractive tradeoff between pilot overhead and estimation accuracy in large-scale wireless systems. However, reliable channel acquisition becomes particularly challenging in highly dynamic environments such as non-terrestrial networks (NTNs), where rapidly varying channels and high system dimensionality significantly degrade the performance of conventional covariance-based estimators due to sampling noise. In this paper, we propose a robust semi-blind channel estimation framework for multi-user uplink systems operating in NTN systems. The proposed approach introduces an optimally regularized least-squares formulation that balances training-based information and blind subspace structure. By exploiting the spiked covariance model within a random matrix theory (RMT) framework, we derive a closed-form characterization of the resulting channel mean-squared error and obtain an analytically tractable design of the optimal regularization parameter. The resulting estimator is computationally efficient and particularly well suited to high-dimensional regimes. Simulation results under realistic Third Generation Partnership Project (3GPP) NTN channel models demonstrate substantial performance improvements over conventional semi-blind and training-based estimators.

I. INTRODUCTION

Dynamic NTN and large-scale MIMO channels make reliable CSI acquisition difficult, motivating semi-blind estimation that combines limited pilots with blind subspace information. The paper develops an RMT-based regularized estimator and validates it in realistic NTN settings.

  • Rapidly varying channels and high-dimensional signal spaces make reliable channel acquisition challenging in dynamic wireless and large-scale MIMO systems.
  • Semi-blind estimation jointly exploits pilot symbols and unknown data symbols to improve accuracy while reducing pilot overhead.
  • Conventional covariance-based estimators degrade in high-dimensional regimes because of sampling noise and scalability limitations.
  • The proposed estimator minimizes a regularized least-squares cost combining training-sequence information with a blind subspace criterion.
  • Spiked covariance models from RMT analytically optimize the regularization parameter and minimize the resulting channel MSE.
  • Simulations under realistic 3GPP NTN environments validate the asymptotic expressions and demonstrate advantages over classical methods.

IV. OPTIMIZED CHANNEL MATRIX DESIGN

The optimized channel design uses the spiked covariance structure of the received data to select a regularization parameter and construct a channel estimate that minimizes the objective.

  • The method leverages the known spiked covariance structure to obtain an optimized regularization parameter λ̂* for high-mobility and non-stationary environments.
  • The resulting optimized channel matrix Ĝ* is designed to minimize the semi-blind estimation objective.

A. Spiked Covariance Models

The sample covariance follows a spiked random covariance model: a Marchenko–Pastur bulk represents non-spiked eigenvalues, while sufficiently strong population spikes produce outliers. These spectral limits support consistent spike estimation and data-driven regularization.

  • Spiked Covariance Models: With fixed K and jointly growing M, N, and L, the sample covariance has a bulk of M−K eigenvalues and at most K separated spikes.
  • Spiked Covariance Models: The empirical eigenvalue distribution converges almost surely to the Marchenko–Pastur distribution for sufficiently large dimensions.
  • Spectral behavior of the non-spiked eigenvalues: The MP law continues to describe non-spiked eigenvalues under fixed-rank perturbations, while finite-rank covariance models are called spiked models.
  • Localization of the spiked eigenvalues: A population spike above the stated threshold generates a sample outlier that separates from the empirical bulk with probability tending to one.
  • Localization of the spiked eigenvalues: The outlier-to-population relation enables a strongly consistent estimator of population spike strength.
  • The regularization design replaces unknown population spikes and eigenvectors with consistent sample-based estimates, yielding a fully data-driven channel estimate.

B. Deterministic Equivalent MSE(λ) and the Optimal ¯λ∗

The paper derives a deterministic equivalent for the estimator’s MSE and minimizes it to obtain the optimal regularization parameter. This provides an asymptotically grounded basis for selecting λ.

  • Under the asymptotic regime, sample signal-eigenvector projections converge to their population counterparts, enabling deterministic MSE characterization.
  • Theorem 1 establishes the deterministic equivalent governing the asymptotic convergence of the semi-blind estimator’s MSE.
  • Minimizing the deterministic MSE with respect to λ yields the optimal regularization parameter.

C. Estimated Optimal ˆλ∗and Proposed Algorithm

The paper replaces unobservable optimal-parameter quantities with a consistent estimator based on observable sample eigenvalues and eigenvectors, then constructs a closed-form semi-blind channel estimate. For weak spikes, it adapts the analysis to detectable components and uses bulk-edge detection to identify them.

  • Estimated optimal regularization: The practical regularization estimator approximates the unobservable optimal parameter using sample eigenvalues η_i and eigenvectors ˆu_si.The oracle parameter depends on z_k and t_k, which are unavailable for direct implementation.
  • Estimated optimal regularization: Theorem 3 establishes the asymptotic performance of the estimated optimal regularization parameter.
  • Proposed algorithm: The proposed semi-blind estimator combines the training-based channel estimate with the estimated signal-subspace projection in closed form.The method is summarized in Algorithm 2 using covariance construction, eigendecomposition, regularization estimation, and channel reconstruction.
  • Weak-spike regime: For weak spikes satisfying t_i < √c, sample eigenvectors do not asymptotically align with population eigenvectors, and sample eigenvalues do not reliably estimate t_i.The weak-spike regime therefore requires adjusting the analysis to the effective number of strong spikes.
  • Weak-spike regime: Weak-spike sample eigenvalues converge to the right edge of the bulk, independently of the underlying t_i values, making eigenvalue-based estimation unreliable.The training-based estimator remains sufficient for constructing a consistent estimate in this regime.
  • Proposed algorithm: The detectable spike count is inferred by counting sample eigenvalues above the bulk’s right edge, while the remaining eigenvalues represent non-detectable spikes.This criterion can be incorporated into Algorithm 2.

E. Complexity Analysis

The proposed algorithms avoid full eigendecomposition by extracting only the dominant signal subspace, while retaining an overall computational order determined mainly by covariance formation and pilot-based estimation.

  • Evaluation setting: Fig. 2 reports MSE performance versus N for K = 3, α = 1/2, β = 1/8, and SNR = 15 dB.
  • Complexity: Only the K dominant eigenpairs are required, so partial eigendecomposition can replace full eigendecomposition at approximately O(M^2K) complexity.The power method or Lanczos method can extract the dominant signal subspace.
  • Complexity: The regularization parameter computation adds O(K) scalar operations in Algorithm 1 and only O(MK) extra operations for Algorithm 2’s trace term.The additional trace cost does not change the overall computational order because it is dominated by O(MLK) pilot-based estimation.
  • Reconstruction: The channel estimate is formed as a weighted combination of the training estimate and projected training estimate using ˆλ∗.The projection is evaluated without explicitly forming the M × M projection matrix.
  • Complexity: The overall computational complexity of both proposed algorithms is O(M^2(N −L) + M^2K + MLK + MK^2).This includes covariance construction, subspace extraction, pilot-based estimation, regularization computation, and reconstruction.

V. SIMULATION RESULTS

Simulations evaluate the proposed semi-blind channel estimation algorithm under representative 3GPP NTN propagation conditions, using asymptotic and finite-dimensional experiments across system dimensions, SNR, and regularization.

  • Evaluation Scope: The experiments evaluate asymptotic behavior and finite-dimensional performance of the proposed semi-blind channel estimation algorithm.Results are averaged over 1000 independent Monte Carlo realizations.
  • Evaluation Axes: Fig. 3 measures MSE performance versus the regularization parameter λ at K = 3, α = 1/4, β = 1/8, N = 256, and SNR = 15 dB.
  • Simulation Setup: The simulations use Sionna/OpenNTN implementations of 3GPP TR 38.811 NTN channel models in a Dense Urban uplink scenario.The setup uses a 2 GHz carrier, a 600 km satellite altitude, a 10° elevation angle, and a 1×512 satellite antenna array.
  • Evaluation Axes: Fig. 4 measures NMAE versus N under different α and β values with K = 3 and SNR = 15 dB.
  • Evaluation Axes: Fig. 5 measures NMAE versus SNR under different N values with K = 3, α = 1/4, and β = 1/8.

A. Experiments for Validating Accuracy

The accuracy experiments test convergence to asymptotic MSE expressions, identify the optimal regularization parameter, and assess normalized approximation error across dimensions and SNR.

  • Asymptotic Validation: As transmission block length N increases, the proposed algorithm’s MSE converges to its respective asymptotic limit.The comparison uses predictions from Theorems 1 and 2 under random and optimal regularization parameters.
  • Regularization Selection: 0.13 is the minimum-MSE regularization value, close to the theoretical optimum λ̂∗ = 0.1350.The result is obtained for K = 3, N = 256, α = 1/2, β = 1/4, and SNR = 15 dB.
  • Approximation Accuracy: NMAE is defined as the normalized mean-squared error approximation error.
  • Approximation Accuracy: NMAE decreases as system dimensions increase across various α and β values, confirming the asymptotic expression’s accuracy.This evaluation uses K = 3 and SNR = 15 dB.
  • Approximation Accuracy: The asymptotic approximation remains accurate across SNR values from 5 dB to 20 dB under different N values.SNR is varied in 1 dB increments.

B. Comparison with Baseline Methods

The proposed algorithm is compared with EM and subspace-based estimators across SNR and user counts, consistently achieving lower MSE in the tested settings.

  • Baseline Comparisons: The comparison evaluates the proposed algorithm against the EM algorithm from [12] and the subspace-based algorithm from [39].
  • SNR Comparison: The proposed algorithm consistently achieves lower MSE than both baselines across SNR values from 5 dB to 20 dB.The SNR comparison uses N = 512, K = 3, α = 1/4, and β = 1/8.
  • User-Count Comparison: As K increases from 2 to 10, all three algorithms show decreasing MSE, while the proposed algorithm consistently outperforms both baselines.The user-count comparison fixes SNR at 15 dB.

C. Impact of Pilot Overhead and Channel Temporal Variation

The proposed estimator balances pilot overhead against channel temporal variation by combining pilot-based estimates with data-aided covariance information. It provides the greatest benefit under slower channel variations and limited pilots, while additional pilots improve robustness as coherence shortens.

  • Experimental setup: With fixed Nd = 128, pilot ratio β is varied through the training length L, while M = 128, K = 3, and SNR = 10 dB remain fixed.Temporal correlation is modeled using a first-order process with coefficient ρ.
  • Temporal variation: A smaller ρ represents faster temporal variation and a shorter effective channel coherence interval.Lower temporal correlation makes data samples less consistent with the channel observed during training.
  • Temporal variation: As temporal variation accelerates, MSE increases because data-aided covariance information becomes less reliable, especially at small pilot ratios.Increasing β reduces differences among temporal-variation conditions by improving the pilot-based estimate.
  • Estimator comparison: Under moderate temporal variation, the proposed estimator achieves the lowest MSE versus β among the proposed, EM, and subspace-based methods.Its advantage is more pronounced in the low-pilot-overhead regime because data covariance complements limited pilot observations.
  • System-level implications: The tradeoff is that slower variations yield greater data-aided gains, whereas increasing β partially compensates when the coherence interval becomes shorter.The evaluation uses BER and achievable sum rate versus SNR with N = 256, K = 3, α = 1/4, and β = 1/16.
  • Pilot requirements: For a target MSE, the semi-blind method requires fewer than half the pilot symbols of the training-based approach under the reported Fig. 11 settings.The comparison uses N = 512, α = 1/2, K = 3, and SNR = 15 dB.
  • Method: The proposed high-dimensional estimator uses an optimally weighted quadratic criterion combining blind subspace and training-based information.RMT yields an asymptotic MSE characterization and a consistent estimator of the optimal regularization parameter.

APPENDIX A PROOF OF THEOREM 1

The proof establishes uniform convergence of the proposed MSE function over the regularization interval by separately controlling three asymptotic terms. It then constructs consistent estimators of the quantities needed for the asymptotically optimal regularization parameter.

  • Uniform convergence: The proof expands the MSE difference, applies the triangle inequality, and analyzes three terms uniformly for λ ∈ [0, 1].The first term is exactly zero, while the remaining terms are controlled asymptotically.
  • Uniform convergence: Under M/N → α and L/N → β with fixed K, the ratios M/L → α/β and K/L → 0 support the third-term limit.These high-dimensional limits are used in the asymptotic decomposition.
  • Uniform convergence: Analyticity of MSE(λ), combined with convergence of all three terms, establishes uniform convergence over λ ∈ [0, 1].The resulting conclusion is stated explicitly after combining the asymptotic bounds.
  • Consistency: Consistent estimators of t_i and z_i are obtained from their corresponding estimated quantities and the relation 1 + c/ˆt_i.These estimates are then substituted into the expression for the optimal regularization parameter.
  • Consistency: Substituting ˆz_i and ˆt_i for z_i and t_i yields consistent estimators ˆλ* of the asymptotically optimal regularization parameter.This connects the asymptotic design to a data-dependent estimator.

APPENDIX C PROOF OF THEOREM 3

The proof of Theorem 3 bounds the difference between the MSE at the estimated and optimal regularization parameters. It combines convergence results with Lipschitz continuity of MSE(λ) on [0, 1].

  • Proof strategy: The proof begins by applying the triangle inequality and separately analyzing the two resulting terms.The decomposition provides the structure for the subsequent convergence bounds.
  • Proof strategy: A bound from Theorem 1 controls one term, while the derivative expression in (31) provides an upper bound for the other.The argument combines theorem-level convergence with derivative control.
  • Continuity: MSE(λ) is Lipschitz continuous with constant Q on [0, 1], enabling control of MSE(ˆλ*) − MSE(λ*).This continuity condition links parameter convergence to MSE convergence.
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