Compressive Toeplitz Covariance Estimation From Few-Bit Quantized Measurements With Applications to DOA Estimation

This paper addresses the problem of estimating the Hermitian Toeplitz covariance matrix under practical hardware constraints of sparse observations and coarse quantization. Within the triangular-dithered quantization framework, we propose an estimator called Toeplitz-projected sample covariance matrix (Q-TSCM) to compensate for the quantization-induced bias, together with its finite-bit counterpart termed the $2k$-bit Toeplitz-projected sample covariance matrix ($2k$-TSCM), obtained by truncating the pre-quantization observations. Under the complex Gaussian assumption, we derive non-asymptotic error bounds of the estimators that reveal a quadratic dependence on the quantization level and capture the effect of sparse sampling patterns through the so-called coverage coefficient. To further improve performance, we propose the quantized sparse and parametric approach (Q-SPA) based on a covariance-fitting criterion, which enforces additionally positive semidefiniteness at the cost of solving a semidefinite program. Numerical experiments are presented that corroborate our theoretical findings and demonstrate the effectiveness of the proposed estimators in the application to direction-of-arrival estimation.