SS-ESPRIT: Single-Snapshot DOA & Spectral Estimation

• SS-ESPRIT is a technique that converts single-snapshot measurements into Hankel matrices to enable robust subspace extraction for high-resolution DOA and frequency estimation.
• It leverages rotational invariance and multi-shift ambiguity dealiasing to achieve accurate and unambiguous angular and spectral estimates even under sparse snapshot conditions.
• Designed for rapid, low-latency applications, SS-ESPRIT offers computational efficiency and quantifiable stability, making it ideal for advanced automotive radar and other real-time sensing systems.

Single-Snapshot ESPRIT (SS-ESPRIT) is a direction-of-arrival (DOA) and spectral estimation technique designed to extract high-resolution frequency or angular estimates from a single measurement vector. SS-ESPRIT leverages subspace-based estimation and rotational invariance principles, and is specifically tailored for scenarios in which only a single temporal or spatial snapshot is available, such as fast-moving or low-latency sensing with sparse extremely large aperture arrays (ELAAs) or in ultra-low snapshot regimes (Hu et al., 22 Sep 2025, Fannjiang, 2016, Yang, 2022). Unlike classic multi-snapshot ESPRIT, SS-ESPRIT circumvents the requirement for averaging by embedding single-snapshot data into structured Hankel (or block Hankel) matrices, enabling robust subspace extraction and subsequent frequency or DOA estimation.

1. Signal and Array Model Foundations

In SS-ESPRIT, the observed data is typically a vectorized measurement from an array of $N$ sensors (or $M+1$ equispaced time samples for spectral estimation), contaminated by additive noise:

$$\mathbf{x} = \mathbf{A}(\bm{\theta})\mathbf{s} + \mathbf{n} \in \mathbb{C}^N$$

where $\mathbf{A}(\bm{\theta})$ is the array manifold (steering) matrix, $\bm{\theta} = [\theta_1,\dots,\theta_K]^T$ contains the $K$ unknown DOAs or frequencies, $\mathbf{s}$ are the source amplitudes, and $\mathbf{n}$ is spatial or temporal white Gaussian noise (Hu et al., 22 Sep 2025, Fannjiang, 2016). For ELAA or ULA structures, the spatial locations are leveraged to impose rotational invariance using shift-invariant subarrays.

In the frequency estimation domain, the data $y[n]=\sum_{j=1}^s x_j e^{-2\pi i \omega_j n}$ provides a Vandermonde structure, with measurement length $M+1$, sparsity $s$, and normalized frequencies $\omega_j$ (Fannjiang, 2016).

2. Core Algorithmic Procedure

SS-ESPRIT processes a single-snapshot vector by embedding it into a Hankel matrix to artificially create a multi-snapshot structure. For a vector $\mathbf{y}\in\mathbb{C}^M$ and embedding parameter $L<M$, the Hankel embedding is

$$\mathcal{H}(\mathbf{y}) = \begin{bmatrix} y_1 & y_2 & \cdots & y_{M-L} \ y_2 & y_3 & \cdots & y_{M-L+1} \ \vdots & \vdots & \ddots & \vdots \ y_{L+1} & y_{L+2} & \cdots & y_M \end{bmatrix} \in \mathbb{C}^{(L+1)\times (M-L)}$$

For shifted subarray configurations (as with ELAAs split into two ULAs), separate Hankel matrices are formed for each subarray output. The critical steps then are:

• Compute the rank-$K$ truncated SVD of the embedded data matrices, $U_{\ell,s}$, to estimate the empirical signal subspace.
• Stack subspace estimates from all subarrays to form a composite signal subspace.
• Impose rotational invariance via selection matrices representing the known spatial shift, yielding the linear system $E_{s,2} = E_{s,1}\Psi$ for some nonsingular $\Psi$.
• Estimate $\Psi$ via least squares, and extract its eigenvalues $\widehat{\xi}_k$, which encode the desired frequencies or angles via

$$\widehat{\theta}_k = \arcsin\left(\frac{\lambda}{2\pi \Delta_s} \angle \widehat{\xi}_k \right)$$

for spatial problems, or

$$\hat{\omega}_j = -\frac{1}{2\pi} \arg(\hat\lambda_j)$$

for spectral estimation (Hu et al., 22 Sep 2025, Fannjiang, 2016, Yang, 2022).

3. Ambiguity Dealiasing and High-Resolution Estimation

When the subarray shift $\Delta_s$ exceeds $\lambda/2$ (in spatial settings), aliasing (grating lobe) ambiguities arise in the raw angle estimates. SS-ESPRIT neutralizes these ambiguities by repeating the process with multiple values of $\Delta_s$, corresponding to different subarray shifts. For each shift, all candidate estimates are computed, and a multi-shift intersection enforces physical consistency by retaining only those angles present in all candidate sets. This approach ensures unique, unambiguous DOAs can be extracted even for arbitrarily sparse or widely separated sensor placements (Hu et al., 22 Sep 2025).

For closely spaced sources, the ability of SS-ESPRIT to exploit the full aperture of an ELAA allows angular resolution to scale as $\lambda/D_a$ rather than $\lambda/d$ for local subarray estimation, constricting the resolution cell by up to an order of magnitude in typical automotive radar architectures (Hu et al., 22 Sep 2025). In the frequency domain, stable separation is guaranteed provided the minimum source separation exceeds $2/M$ (twice Rayleigh's Resolution Length) (Fannjiang, 2016).

4. Stability, Resolution, and Nonasymptotic Guarantees

Exact recovery of all $K$ DOAs or frequencies is theoretically achievable in the noiseless case provided the number of measurements meets minimal constraints: $M+1 \geq 2K$ for the spectral setting, or sufficient spatial diversity in the array (Fannjiang, 2016, Yang, 2022). In the presence of noise, SS-ESPRIT exhibits quantifiable robustness and resolution as follows:

• Under a separation condition $\delta > 2/M$ between distinct frequencies, and sufficiently small noise, the matrix perturbation of the estimated $\Psi$ operator admits the explicit bound

$$\|\widehat\Psi - \Psi\|_2 \leq C\,\Bigl( \frac{\|E_1\|_2}{M} + \frac{\|E_2\|_2}{M} \Bigr)$$

where $E_1,E_2$ are structured Hankel perturbations due to noise, and $C$ depends on amplitude dynamic range and singular value conditioning (Fannjiang, 2016).

• The root-mean-square frequency/angle error demonstrates $O(\sqrt{\log M}/\sqrt{M})$ decay with data length and is further scaled by the amplitude dynamic range (Fannjiang, 2016).
• For single-snapshot ESPRIT, nonasymptotic error bounds of the form

$$\max_k|\hat{f}_k - f_k| \lesssim C \cdot \max(\sigma, \sigma^2)$$

hold with overwhelming probability, provided identifiability conditions are satisfied (full-rank source subspace via spatial smoothing if the sources are coherent) (Yang, 2022).

• In the single-snapshot regime, ESPRIT retains phase transition behavior: high SNR yields errors $O(\sigma)$, while low SNR induces a rapid breakdown as the subspace cannot be resolved.

5. Algorithmic and Computational Considerations

SS-ESPRIT is optimized for computational efficiency in high-channel-count applications such as automotive radar. Principal steps and their complexities are:

• Hankel formation: $O((L+1)(M-L+1))$
• Truncated SVD of each Hankel matrix: $O(\min(L+1, M-L+1)\,K^2)$
• Small-size eigenvalue problems for $\Psi$: $O(K^3)$ per subarray shift

The choice of $L \approx (M+1)/2$ yields optimal subspace conditioning. For $K\ll M$, all dominant operations remain practical for real-time hardware implementations (Hu et al., 22 Sep 2025, Fannjiang, 2016). SS-ESPRIT can be an order of magnitude faster than SS-MUSIC for the same data (Fannjiang, 2016).

6. Extensions, Limitations, and Practical Guidelines

Spatial smoothing is essential if the sources are coherent, as plain ESPRIT cannot extract a $K$-dimensional subspace from a rank-deficient measurement. Forward-only or forward-backward smoothing (FOSS/FBSS) creates an effective abundance of subarrays, ensuring subspace identifiability. For smoothing, the number of subarrays $P \geq K$ is required, and the subarray length $M \geq K+1$ (Yang, 2022).

In practical scenarios:

• Use multiple subarray shifts for ambiguity dealiasing, especially with sparse or extremely large arrays (Hu et al., 22 Sep 2025).
• Ensure the separation between sources exceeds the required threshold for stable recovery in noise ($\delta \gtrsim 2/$data length).
• Maintain adequate signal-to-noise ratio ($\sigma\lesssim 0.1$ for $O(\sigma)$ error regime).
• For array design, exploit the maximum aperture possible to achieve $\sim \lambda/D_a$ angular resolution.

SS-ESPRIT demonstrates favorable recovery and stability properties in the single-snapshot regime relative to compressed sensing approaches and outperforms subarray-restricted methods, especially under extreme aperture or snapshot constraints (Hu et al., 22 Sep 2025, Fannjiang, 2016, Yang, 2022). It enables high-resolution, ambiguity-free DOA and spectral estimation under minimal measurement assumptions, provided array shifts, separation, and dynamic range conditions are met.