[Paper] Direction Finding with Sparse Arrays Based on Variable Window Size Spatial Smoothing

Source: arXiv - 2512.22024v1

Overview

The paper presents a Variable‑Window‑Size (VWS) spatial smoothing technique that dramatically improves direction‑of‑arrival (DOA) estimation when using sparse linear antenna arrays. By intelligently shrinking the smoothing aperture, the authors devise new versions of the popular Coarray‑MUSIC and Coarray‑root‑MUSIC algorithms that achieve higher accuracy and lower computational cost, making high‑resolution angle finding feasible for cost‑constrained hardware.

Key Contributions

• Variable‑Window‑Size (VWS) framework for spatial smoothing of co‑array data, replacing a portion of noisy rank‑one terms with clean low‑rank components.
• VWS‑CA‑MUSIC and VWS‑CA‑rMUSIC algorithms that preserve the signal subspace while enlarging the gap between signal and noise subspaces.
• Theoretical identifiability bounds that define admissible compression parameters, guaranteeing that the true source directions remain recoverable.
• Complexity analysis showing that VWS reduces the size of the covariance matrix to be eigendecomposed, cutting runtime and memory usage.
• Extensive simulations on classic sparse geometries (e.g., nested, coprime, and minimum‑redundancy arrays) demonstrating up to several dB improvement in root‑mean‑square error (RMSE) over fixed‑window coarray MUSIC, especially in low‑SNR or limited‑snapshot regimes.

Methodology

1. Coarray concept recap – Sparse physical arrays generate a difference coarray that can be treated as a virtual uniform linear array (ULA). Traditional coarray MUSIC applies spatial smoothing with a fixed window size to obtain a full‑rank covariance matrix.
2. Variable window size – Instead of using the largest possible smoothing window, the VWS approach selects a smaller window (L) (compression parameter) and augments the smoothed covariance with additional low‑rank terms derived from the unperturbed outer products of the original data.
3. Signal‑subspace preservation – The added low‑rank terms are constructed so that they lie entirely within the true signal subspace, ensuring that the eigen‑structure used by MUSIC remains unchanged for the signal part.
4. Algorithmic steps
   • Form the difference coarray and compute its sample covariance.
   • Apply VWS smoothing: partition the coarray into overlapping sub‑arrays of size (L), average their covariance contributions, and inject the low‑rank corrections.
   • Perform eigendecomposition (now on a smaller matrix) to separate signal and noise subspaces.
   • Apply the standard MUSIC (or root‑MUSIC) spectral search on the virtual ULA to estimate DOAs.
5. Identifiability analysis – By bounding (L) relative to the number of sources (K) and the coarray aperture, the authors prove that the true DOAs remain uniquely identifiable.

Results & Findings

• Accuracy boost: VWS consistently halves the RMSE in challenging low‑SNR, few‑snapshot conditions.
• Resolution gain: The separation between signal and noise eigenvalues widens, allowing the algorithm to resolve sources that are closer than the Rayleigh limit of the underlying physical array.
• Complexity reduction: Because the smoothing window is smaller, the eigen‑decomposition operates on a matrix of size (L \times L) instead of the full coarray size, yielding up to 40 % runtime savings.

Practical Implications

• Cost‑effective radar / sonar: Engineers can deploy fewer physical sensors (sparse arrays) while still achieving high‑resolution angle estimation, reducing hardware cost, weight, and power consumption.
• 5G/6G massive MIMO: Base‑station antenna panels can be thinned (e.g., using coprime layouts) and still deliver precise beam‑forming and user‑direction tracking, easing deployment in dense urban environments.
• IoT edge devices: Low‑power embedded platforms (e.g., drones, autonomous vehicles) can run VWS‑CA‑MUSIC with modest CPU resources, thanks to the smaller covariance matrix and fewer required snapshots.
• Software‑defined radio (SDR) toolkits: The algorithm fits naturally into existing MUSIC implementations—only the smoothing step changes—making it straightforward to add as a plug‑in for open‑source DSP libraries.

Limitations & Future Work

• Compression parameter selection: The optimal window size (L) depends on SNR, snapshot count, and array geometry; the paper provides theoretical bounds but not an adaptive rule for real‑time systems.
• Model assumptions: The analysis assumes narrowband, far‑field sources and ideal calibration; performance under wideband or mutual‑coupling effects remains to be explored.
• Scalability to massive arrays: While VWS reduces complexity, extremely large coarrays may still pose memory challenges; hierarchical or block‑wise VWS could be investigated.
• Extension to other subspace methods: Future research could apply the VWS concept to ESPRIT, compressive sensing DOA techniques, or joint angle‑delay estimation for broadband signals.

Bottom line: Variable‑window‑size spatial smoothing offers a practical pathway to unlock the full potential of sparse antenna arrays, delivering sharper DOA estimates with less computation—a win for anyone building next‑generation sensing or communication systems.

Authors

• Wesley S. Leite
• Rodrigo C. de Lamare
• Yuriy Zakharov
• Wei Liu
• Martin Haardt

Paper Information

• arXiv ID: 2512.22024v1
• Categories: cs.LG
• Published: December 26, 2025
• PDF: Download PDF