Covariance-Guided Beam Selection

• The paper introduces a framework that reconstructs a virtual fully digital subarray, fits a structured signal-plus-noise covariance model, and selects contiguous DFT beams to optimize DoA estimation.
• It leverages covariance denoising and Toeplitz-PSD projection to concentrate beamforming energy while maintaining a large effective array aperture.
• The approach achieves near-CRB accuracy with reduced complexity and robust performance under dynamic RF constraints and sector-edge scenarios.

The covariance-guided beam selection framework is a principled methodology for direction-of-arrival (DoA) estimation in hybrid analog/digital millimeter-wave (mmWave) MIMO receivers that employ DFT beamspace processing with limited RF chains. This approach reconstructs a virtual fully digital subarray, fits a structured signal-plus-noise covariance model, and utilizes the resulting denoised covariance to select, for each coarse sector, a small contiguous block of DFT beams under explicit beam-budget constraints. The selected beams are then used in a sparse beamspace Unitary ESPRIT stage, resulting in an efficient process in which overall complexity is dominated by a single low-dimensional ESPRIT call, while retaining a large effective aperture and achieving robust estimation performance (Şenyuva, 30 Nov 2025).

1. System Model and Problem Formulation

The foundational signal model leverages a uniform linear array (ULA) comprising $M$ antennas, observing $d$ far-field narrowband sources parameterized by spatial frequencies $\mu_k = -\pi \sin \theta_k$, $k=1, \ldots, d$. The array manifold matrix $$A(\mu) = [a(\mu_1), \ldots, a(\mu_d)] \in \mathbb{C}^{M \times d}$$, where the steering vector $$a(\mu_k) = [1, e^{j\mu_k}, \ldots, e^{j(M-1)\mu_k}]^T$$. The system receives $N_{\mathrm{snap}}$ snapshots $$\mathbf{Y} \in \mathbb{C}^{M \times N_{\mathrm{snap}}}$$ according to

$\mathbf{Y} = A(\mu) \mathbf{S} + \mathbf{N}$

with $$\mathbf{S} \in \mathbb{C}^{d \times N_{\mathrm{snap}}}$$ and additive white Gaussian noise $\mathbf{N}[n] \sim \mathcal{CN}(0, N_0 I_M)$.

Hybrid analog/digital architecture is assumed, with $N_\mathrm{RF} \ll M$ RF chains and the analog combiner $$W_\mathrm{RF} \in \mathbb{C}^{M \times N_\mathrm{RF}}$$ (constant-modulus), implemented using a DFT codebook from which columns are chosen. The digital combiner $$W_\mathrm{BB} \in \mathbb{C}^{N_\mathrm{RF} \times N_\mathrm{RF}}$$ is typically orthonormal. With this setup, beamspace measurements $\mathbf{Y}_b = W^H\mathbf{Y}$ are obtained, and the sample beamspace covariance is

$$\hat{R}_b = \frac{1}{N_{\mathrm{snap}}} \mathbf{Y}_b \mathbf{Y}_b^H.$$

A virtual fully digital subarray of size $N_\mathrm{RF}$ is synthesized around broadside using a centro-symmetric index set $$\mathcal{M} = \{ m : | m - (M+1)/2| \leq (N_\mathrm{RF} - 1)/2 \}$$, enforcing $$J_\mathcal{M}W_\mathrm{RF}W_\mathrm{BB} = I_{N_\mathrm{RF}}$$ to extract the desired subarray data.

2. Covariance Fitting and Denoising

Denoising is performed via structured covariance estimation on the virtual subarray. The sample covariance is computed and averaged using forward–backward averaging:

$$\hat{R}_\mathrm{FBA} = \frac{1}{2}[\hat{R}_\mathcal{M} + \Pi \hat{R}_\mathcal{M}^* \Pi],$$

where $\Pi$ is the reversal matrix. The parametric signal-plus-noise model is

$$R_\mathcal{M}(p, N_0) = A_\mathcal{M}(\hat{\mu}_\mathrm{coarse}) \operatorname{diag}(p) A_\mathcal{M}(\hat{\mu}_\mathrm{coarse})^H + N_0 I_{N_\mathrm{RF}}$$

with non-negative source powers $p$ and noise variance $N_0$ estimated by non-negative least squares:

$$\min_{p \geq 0, N_0 \geq 0} \| \operatorname{vec}_H(\hat{R}_\mathrm{FBA}) - \operatorname{vec}_H(R_\mathcal{M}(p,N_0)) \|_2^2,$$

where $\operatorname{vec}_H(\cdot)$ vectorizes the Hermitian matrix.

The estimated signal covariance $\hat{R}_s$ is projected onto the cone of Hermitian Toeplitz PSD matrices by solving

$$\tilde{R}_s = \arg\min_{R \in \mathcal{T}^+}\|R - \hat{R}_s\|_F^2,$$

with $\mathcal{T}^+$ denoting Hermitian Toeplitz PSD matrices of size $M$. This projection is parameterized through Toeplitz structure by the first column $t = [t_0, t_1, \ldots, t_{M-1}]^T$, and can be formulated as a real-valued quadratic program.

3. Beam Selection Optimization

Coarse DoA estimates are partitioned into $G$ disjoint sectors, and for each sector $g$ a candidate pool of DFT beams $\mathcal{B}_g \subset \{1, \ldots, M\}$ is identified. For every sector, a user-defined beam budget $K_g \geq 2$ constrains the search to contiguous blocks $S_g \subset \mathcal{B}_g$, $|S_g|=K_g$. Candidate blocks are evaluated according to a data-dependent score, computed as:

• Covariance-capture term:

$$\mathrm{cap}(S_g) = \operatorname{tr}[(G_g(S_g) + \gamma I_{K_g})^{-1} C_g(S_g)],$$

where $G_g(S_g) = B_g^H B_g$ is the Gram matrix, $C_g(S_g) = B_g^H\tilde{R}_sB_g$, $\gamma \geq 0$ is the Tikhonov regularizer.

• Numerical robustness: $\kappa_g^2(S_g) = \operatorname{cond}^2(G_g(S_g))$.
• Final score:

$$\mathrm{score}(S_g) = \frac{\mathrm{cap}(S_g)}{1 + \alpha \kappa_g^2(S_g)},$$

with trade-off parameter $\alpha \geq 0$.

The optimal contiguous beam block per sector is $$S_g^\star = \arg\max_{S_g \subset \mathcal{B}_g, |S_g|=K_g} \mathrm{score}(S_g)$$. Optionally, candidate windows can be pruned to retain only those covering high-energy beams as determined by the diagonal of the full DFT-beamspace covariance $R_b$.

4. Algorithmic Procedure

The complete covariance-guided beam selection process operates as follows:

1. Precompute the entire DFT beamforming matrix $B = [b_1, ..., b_M] \in \mathbb{C}^{M \times M}$.
2. Compute the beamspace covariance $R_b = B^H\tilde{R}_sB$ and the per-beam energies $\rho_m = [R_b]_{mm}$.
3. Within each sector $g=1, ..., G$:
   • Sort the candidate beam pool $$\mathcal{B}_g = \{\kappa_1 < \ldots < \kappa_{L_g}\}$$; fix $K_g \leq L_g$.
   • Optionally prune windows to those covering at least one of the top-$q$ beams in $\rho_{\kappa_i}$.
   • For all contiguous starting indices $s=1, ..., L_g-K_g+1$, form $S_g(s) = \{\kappa_s, ..., \kappa_{s+K_g-1}\}$ and evaluate the score.
4. Select the best block $S_g^\star$ for each sector.
5. Aggregate the selected beams into the fine-stage set $K_\mathrm{fine} = \bigcup_g S_g^\star$.

This process yields sectorwise beam blocks that collectively form the input to the downstream sparse beamspace ESPRIT estimator.

5. Sparse Beamspace Unitary ESPRIT Stage

Fine-stage beamspace measurements are obtained by programming the analog combiner to select all indices in $K_\mathrm{fine}$, such that $$|\ K_\mathrm{fine}\ | = N_\mathrm{RF}^\mathrm{fine}$$. The measurements $$\mathbf{Y}_b^\mathrm{fine} \in \mathbb{C}^{N_\mathrm{RF}^\mathrm{fine} \times N_{\mathrm{snap}}}$$ undergo a real-valued transform via forward–backward averaging and the $\Pi$-real operation:

$$\mathbf{Y}_\mathrm{UE} = \sqrt{2} [\mathrm{Re} \{ \mathbf{Y}_b^\mathrm{fine} \}, \mathrm{Im} \{ \mathbf{Y}_b^\mathrm{fine} \}]$$

and SVD $\mathbf{Y}_\mathrm{UE}=U_S\Sigma V^T$ is performed, retaining $U_S$ of size $N_\mathrm{RF}^\mathrm{fine} \times d$. Valid forward shifts are established among contiguous beam indices, yielding selection matrices $J_1$ and $J_2$; the shift-invariance property is enforced by solving

$\hat{\Phi} = (J_1U_S)^\dagger (J_2U_S)$

and the DoA estimates are $\hat{\mu}_{k,\mathrm{fine}} = \arg(\lambda_k)$, where $\lambda_k$ are the eigenvalues of $\hat{\Phi}$.

6. Complexity and Performance Characteristics

The computational complexity is dominated by a single small SVD and low-dimensional optimization steps:

• Coarse subspace extraction: SVD of size $$N_\mathrm{RF}^\mathrm{coarse} \times N_{\mathrm{snap}}$$.
• Covariance fitting: non-negative least squares (dimension $d+1$), Toeplitz-PSD projection (real QP, dimension $2M-1$).
• Beam selection: for each sector, $O(L_g - K_g + 1)$ contiguous windows, with per-window cost $O(MK_g^2 + K_g^3)$, and typically small $K_g$ (2–4).
• Fine stage: one SVD & LS of small matrices.

Monte Carlo simulations illustrate the empirical performance for a $M=32$ element ULA, $d=3$ sources, and $N_{\mathrm{snap}}=100$:

• The covariance-guided approach attains near Cramér–Rao bound (CRB) accuracy (gap $\approx 1$–$2$ dB) for array SNR (ASNR) $\geq 4$ dB; sectorization methods lag by $4$–$6$ dB.
• Failure probability (outlier) falls below $10\%$ near ASNR $\approx 1$ dB for covariance guidance vs. $6$ dB for sectorization.
• Largest principal angles between true and estimated signal subspaces are strongly correlated with angle error (correlation $\rho \approx 0.99$); covariance guidance yields smaller angles.
• Empirical cumulative distribution functions (ECDFs) demonstrate that covariance-guided selection yields fewer large-error trials at any error threshold.
• On the RMSE-runtime Pareto frontier, covariance-guided configurations outperform sectorization for dynamic RF budgets.
• In sector-edge scenarios (multiple sources straddling boundaries), covariance-guided selection maintains RMSE close to the CRB and exhibits robust failure probabilities, while sectorization fails over a broader boundary range.
• In fine-budget ablation, covariance guidance achieves near-CRB performance with $K_f=2$; sectorization requires $K_f \geq 4$ for comparable results.

7. Significance and Methodological Implications

By reconstructing and denoising the full-aperture covariance matrix and leveraging it to score contiguous DFT beam blocks under explicit beam-budget constraints, the covariance-guided beam selection framework enables effective concentration of beamforming energy onto dominant signal paths, preservation of effective array aperture, and substantial improvements in DoA estimation accuracy, robustness to outliers, and computational efficiency relative to standard sectorization-based selections. The framework's use of denoised and Toeplitz-PSD projected covariance effectively exploits array structure and achieves reliable performance under demanding settings, including dynamic RF allocations and sector-edge source placements (Şenyuva, 30 Nov 2025).