Sparse Beamspace Unitary ESPRIT

• Sparse Beamspace Unitary ESPRIT is a hybrid DoA estimation method that combines covariance-guided DFT beam selection with sparse Unitary ESPRIT to achieve near–Cramér–Rao bound accuracy under limited RF chains.
• It reconstructs a denoised, structured signal covariance from partial measurements using nonnegative least squares and PSD projection to optimally select contiguous DFT beam subsets.
• Designed for mmWave MIMO and 6G systems, the approach delivers robust performance, low failure rates, and efficient runtime by concentrating high-resolution estimation in a sparse beamspace.

Sparse Beamspace Unitary ESPRIT refers to a hybrid analog/digital direction-of-arrival (DoA) estimation methodology that couples covariance-guided DFT beam selection with sparse application of Unitary ESPRIT in mmWave MIMO receivers, typically under stringent RF chain constraints. The approach reconstructs a structured, denoised signal covariance from partial array measurements, employs it for optimal contiguous DFT beam subset selection, and applies Unitary ESPRIT in the selected sparse beamspace. This procedure enables near Cramér–Rao bound (CRB) accuracy with minimal loss of aperture, making it compatible with next-generation 6G systems where hardware efficiency is critical (Şenyuva, 30 Nov 2025).

1. Theoretical Foundations and Signal Model

Sparse Beamspace Unitary ESPRIT is grounded in the classical shift-invariant subspace framework of ESPRIT, which is optimized for uniform linear arrays (ULAs) and assumes a superposition signal model:

• The array output is $Y = A(\mu) S + N$, where $A(\mu)\in\mathbb{C}^{M\times d}$ denotes the array manifold for $d$ far-field sources, $S$ the source signals, and $N$ additive noise.
• For hybrid analog/digital (HAD) architectures, the analog combiner $W_\mathrm{RF}$ is chosen from a unitary DFT codebook, and the reduced-dimension measurements are $Y_b = W^H Y$ (Şenyuva, 30 Nov 2025).
• The central problem is to estimate the spatial frequencies $\mu_k$ (or DoAs) from $Y_b$, given the constraint $N_\mathrm{RF}\ll M$.

The ESPRIT estimator exploits the array’s shift-invariance but, in the hybrid setting, can only be applied to effective subarrays determined by the chosen DFT beams.

2. Covariance-Guided Beam Selection Strategy

Effective DoA estimation in hybrid settings is highly sensitive to which beams are selected for digital processing. Covariance-guided beam selection addresses this by:

• Reconstructing a denoised, structured signal covariance $\tilde R_s$ using data from a small virtual subarray. This involves a sequence of operations: coarse ESPRIT, source-power and noise-variance estimation via non-negative least squares, and Hermitian Toeplitz–positive-semidefinite (PSD) projection (Şenyuva, 30 Nov 2025).
• Scanning over beam windows within each angular sector and scoring each contiguous block $S$ of beams using

$$\mathrm{score}(S) = \frac{\operatorname{Tr}\left[(G(S) + \gamma I)^{-1}C(S)\right]}{1+\alpha\cdot\mathrm{cond}(G(S))^2}$$

where $G(S)$ is the Gram matrix of the beams, $C(S)$ the projected covariance, $\gamma$ a Tikhonov parameter, and $\alpha$ a robustness weight.

• Implementing beam-budget constraints for each angular sector, permitting only a fixed number of beams per sector, and selecting those with the highest scores.

This selection maximizes the DoA-informative signal subspace captured under stringent hardware limitations, which is crucial for sparse ESPRIT performance.

3. Sparse Beamspace Unitary ESPRIT Algorithm

Given the covariance-guided DFT beam selection, the algorithm proceeds as follows:

• Measurement Formation: Beamspace measurements are collected on the selected DFT beam indices $\mathcal{K}_\mathrm{fine}$, giving $$Y_b \in \mathbb{C}^{N_\mathrm{RF}^{\mathrm{fine}} \times N_\mathrm{snap}}$$.
• Preprocessing: The real-valued FBA (Forward–Backward Averaging) is applied, with the data matrix

$$Y_\mathrm{UE} = \sqrt{2}[\operatorname{Re}\{Y_b\}\quad \operatorname{Im}\{Y_b\}]$$

for Unitary ESPRIT compatibility.

• Signal Subspace Estimation and Shift-Invariance: The SVD of $Y_\mathrm{UE}$ yields the rank-$d$ signal subspace. Sparse shift-invariance equations are constructed using only adjacent pairs of selected beams, enforcing the ESPRIT structure in the surviving sparse beamspace (Şenyuva, 30 Nov 2025).
• Estimation: Solution of the shift-invariance equation in least squares yields the rotational invariance matrix. Its eigenvalues encode the estimated DoAs.

By concentrating the ESPRIT operations in this covariance-selected beamspace, the approach achieves high-resolution estimation with a small number of RF chains and negligible complexity compared to a full-array digital implementation.

4. Performance Metrics and Comparative Results

Rigorous Monte Carlo simulations on a 32-element ULA with three paths assess:

• Root mean-square error (RMSE) versus the stochastic Cramér–Rao bound (CRB)
• Failure rate ($P_\mathrm{fail}$: fraction with estimation error exceeding $3\sqrt{\mathrm{CRB}}$)
• Largest principal angle (LPA) between true and estimated subspaces
• Computation time per trial

Key findings demonstrate:

• The covariance-guided sparse beamspace Unitary ESPRIT achieves RMSE within 1–2 dB of the CRB for array SNR (ASNR) $\gtrsim 4$ dB, closing a 4–6 dB gap relative to standard sectorization-based selectors (Şenyuva, 30 Nov 2025).
• Failure rates below 10% are attained for ASNR as low as 0–1 dB, while sectorization baselines require significantly higher SNR for equivalent reliability.
• Pareto-optimal trade-off between runtime and estimation accuracy; covariance-guided configurations form the efficiency frontier.
• Robustness to sector-edge aliasing and low failure rates are maintained across operating conditions.

A summary of exemplary configurations appears in the following table:

Method	RMSE (rad) @ ASNR=3dB	Runtime (ms) per trial	Failure Rate (%)
Cov-guided, 12→6	0.0155	3.4	<10
Sector, 12→6	0.455	2.4	>80

5. Stress Testing and Ablation Analyses

Performance under dynamic RF budgets and sector-edge source positioning reveals:

• Covariance-guided selectors sustain near-CRB accuracy and low failure rates even as the number of fine-stage beams is reduced to as few as two per sector.
• Standard sectorization methods exhibit large RMSE and near-unity failure rates when sources approach sector boundaries.
• Increasing the fine-stage beam budget benefits both methods, yet covariance-guidance consistently delivers superior per-beam performance (Şenyuva, 30 Nov 2025).

Analysis of subspace alignment confirms that improved estimation error correlates tightly with improved principal-angle subspace proximity.

6. Significance, Hardware Context, and Extensions

Sparse Beamspace Unitary ESPRIT enables high-accuracy DoA estimation with minimal RF hardware, making it highly relevant for mmWave and 6G hybrid arrays where full digital access is infeasible. The approach leverages prior information (denoised signal covariance) to inform beam selection, achieving a level of performance that previously required dense digital architectures.

Extensions include:

• Adaptation to rapidly time-varying channels by updating covariances
• Application to arbitrary ULA or sectorization geometries, contingent on DFT codebook availability and sector mask design
• Integration with real-time firmware and FPGA/GPU acceleration for low-latency operation
• Potential adaptation to MIMO radar transmit/receive selection by analogy to hybrid covariance-guided frameworks developed for radar applications (Bose et al., 2020)

A plausible implication is that similar covariance-guided selection and sparse subspace techniques could generalize to other shift-invariant estimation frameworks beyond ESPRIT, particularly in hardware-constrained or interference-dominated environments.

7. Relationship to Prior Beam Selection and Covariance-Driven Architectures

The Sparse Beamspace Unitary ESPRIT framework is situated in the broader context of beam and antenna selection for MIMO systems:

• Beam selection via unitary transforms and codebooks has roots in DFT lens array architectures, as illustrated in mmWave hybrid precoding with zero-forcing (Shuang et al., 2018).
• Covariance-guided joint design for beam-pattern and correlation minimization has been established in MIMO radar, incorporating cyclic covariance optimization and greedy antenna selection (Bose et al., 2020).
• Sparse beamspace ESPRIT transposes these developments to the DoA estimation problem, uniting covariance-structure exploitation with shift-invariant subspace methods for efficient, robust hybrid receiver architectures.

This approach defines a principal direction for scalable, high-accuracy spatial parameter estimation in massive MIMO and radar systems under sparse hardware constraints.